R E C H i V b: D MAR 2 6 19/n DATA LIBRARY INSTITUTION ARCHIVES W.H.O.I. DATA LIBRARY WOODS HOLE. MA. 02543 THE SEA Ideas and Observations on Progress in the Study of the Seas Volume 1 PHYSICAL OCEANOGRAPHY Volume 2 composition of sea-water COMPARATIVE AND DESCRIPTIVE OCEANOGRAPHY Volume 3 the earth beneath the sea HISTORY Editorial Board: M. N. Hill Department of Geodesy and Geophysics Madingley Rise, Cambridge, England Edward D. Goldberg University of California La Jolla, California C. O'D. Iselin Woods Hole Oceanographie Institution Woods Hole, Massachusetts W. H. MUNK Institute of Geophysics and Planetary Physics University of California La Jolla, California 1 u^ I // MAR ^ t) I9>u /€£^ DATA LIBRARY THE SEA Ideas and Observations on Progress in the Study of the Seas General Editor M. N. HILL ^^nrurjoN archives W.H.O.;. DATA L/SRARY ■ m : to ; rn ; m ■ u~ I o : a czi m a □ VOLUME 1 Physical Oceanography INTERSCIENCE PUBLISHERS a division of John Wiley & Sons • New York • London • Sydney First published 1962 by John Wiley & Sons, Inc. ALL RIGHTS RESERVED Library of Congress Catalog Card Number 62-18366 PRINTED IN THE UNITED STATES OF AMERICA 10 9 8 7 6 ISBN 470 39615 CONTRIBUTORS TO VOLUME I R. H. Backus, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts N. F. Barber, Dominion Physical Laboratory, Department of Scientific and Industrial Research, Wellington, New Zealand K. F. BowDEN, Department of Oceanography, The University, Liverpool, England D. E. Cartwright, National Institute of Oceanography, Wormley, Surrey, England G. L. Clarke, Biological Laboratories, Harvard University, Cambridge, Massachusetts and Woods Hole Oceanographic Institution, Woods Hole, Massachusetts C. S. Cox, Scripps Institution of Oceanography, University of California, La JoUa, California J. Darbyshire, National Institute of Oceanography, Wormley, Surrey, England E. L. Deacon, C.S.I.R.O., Division of Meteorological Physics, Aspendale, Victoria, Australia E. J. Denton, Marine Biological Association, The Laboratory, Citadel Hill, Plymouth, England Seibert Q. Duntley, Visibility Laboratory, Scripps Institution of Oceanography, University of California, San Diego, California C. H. EcKART, Scripps Institution of Oceanography, University of Cahfornia, La Jolla, California N. P. FoFONOFF, Fisheries Research Board of Canada, Pacific Oceanographic Group, Nanaimo, British Columbia Kdward D. CJoldberc, University of California, La Jolla, California P. Groen, Kon. Ned. Meteorologisch Instituut, De Bilt, The Netherlands Gordon W. Groves, Institute of Geophysics and Planetary Physics, University of California, La Jolla, California (contribution written at In ituto de Geofisica, Universidad Nacional de Mexico, Mexico City) Walter Hansen, Institut fiir Meereskunde, Universitat Hamburg, Hamburg Vi CONTRIBUTORS J. B. Hersey, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts E. C. LaFond, Environmental Studies Branch, U.S. Navy Electronics Laboratory, San Diego, California L. N. LiEBERMANN, University of California, San Diego, La Jolla, California Joanne S. Malkus, Woods Hole Oceanographic Institution, Woods Hole, Massa- chusetts and The University of California, Los Angeles W. H. MuNK, Institute of Geophysics and Planetary Physics, LTniversity of California, La Jolla, California E. R. Pounder, Department of Physics, McGiU University, Montreal R. W. Preisendorfer, Visibility Laboratory, Scripps Institution of Oceanography, University of California, San Diego, California R. Revelle, Scripps Institution of Oceanography, University of California, La Jolla, California R. W. Rex, California Research Corporation, La Habra, California J. R. RossiTER, University of Liverpool Tidal Institute and Observatory, Bidston, Birkenhead, England W. E. ScHEViLL, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts Hans E. Suess, Scripps Institution of Oceanography, University of California, La Jolla, California M. J. Tucker, National Institute of Oceanography, Wormley, Surrey, England J. E. Tyler, Visibility Laboratory, Scripps Institution of Oceanography, University of California, San Diego, California P. Vigoureux, National Physical Laboratory, Teddington, Middlesex, England E. K. Webb, C.S.I.R.O., Division of Meteorological Physics, Aspendale, Victoria, Australia A. H. Woodcock, Woods Hole Oceanographic Institution, Woods Hole, Massachu- setts PREFACE Oceanography has surged forward as a subject for research during the past twenty years, and great progress has been made in our understanding of the structure of the water-masses, of the crust of the earth beneath the oceans, and of the processes which are involved in creating these structures. This progress has been particularly rapid in the recent past for two prime reasons. Firstly, techniques of investigation have become available which have made many hitherto intractable problems capable of solution. For example, it has only recently become possible to measure the value of gravity in a surface ship, to determine the deep currents of the oceans by direct methods, or to analyze wave spectra with the detail which modern electronic computers allow. The second reason for the present rapidity of progress lies in the increasing availability of the resources for both theoretical and practical marine investigations; the international character of oceanography and the growing world-wide interest in fundamental research has resulted in the provision of opportunities hitherto impossible. Some years ago Dr. Roger Revelle of the Sjcripps Institution of Oceanography suggested to us that perhaps the production of treatises on the new develop- ments in oceanography were not keeping pace with the progress in the subject. It was true that papers were appearing in healthy numbers throughout the journals of the world, but there were no recent comprehensive works such as The Oceans, written by Sverdrup, Johnson and Fleming and published in 1942. This work had been, and still is, of value to many, if not all, oceanographers ; Revelle suggested that we produce another such volume containing ideas and observations concerning the work accomplished during the twenty years since this masterpiece. It was suggested that this new work should not attempt to be a textbook but a balanced account of how oceanography, and the thoughts of oceanographers, were moving. It was early apparent that a work of this nature would have to be from the pens of many authors ; the subject had become too broad, the oceanographers perhaps too specialized, to allow a small number of contributors to cover a vast field of study. It was also apparent that we could do no more than include biology in so far as it was directly related to physical, chemical and geological processes in the ocean and on the ocean floor. Marine biology could alone fill that space which we thought was the maximum we should allow for these volumes. Biology is, therefore, somewhat scattered through this work; so it has to be, since the contributions of biologists extend throughout the disciplines embraced by the contents of these volumes. It has been said in the past that oceanography as a subject did not, or even should not, include the study of the mode of formation and the structure of the muds and hard rocks forming the floor of the oceans and the seas. This exclusion is no longer possible, and we have included in these volumes geological and geophysical ideas and observations concerning the earth beneath the seas. In the great oceanographical (or oceanological) establishments of the world the earth sciences relating to the oceanic environment are pursued alongside one another without discrimination as to whether the air over the sea, the water, or the solids beneath the sea could claim prior imjjortance. We have adopted the same breadth of outlook in these volumes. With a composite work such as we have tried to produce, it has proved diffi- cult to ensure a precise balance in the emphasis given to our wide variety of topics. Sometimes, maybe, there will be too much written about a narrow field ; sometimes we shall perhaps be criticized for the converse, too little written about a broad field. At least, however, we hope we have covered most of the topics suitable for a work on the new developments in our subject. When we drew up the first list of contents of this work we had supposed that we should be producing one volume. We soon found, however, that we had omitted a number of topics, and the contents list consequently increased. We also found that our contributors could not possibly have provided, within the limits we originally proposed, adequate discourses on their various topics. It was apparent that we should have to subdivide The Sea into three volumes. The first consists of new thoughts and ideas on Physical Oceanography ; the second on the Composition of Sea Water and Comparative and Descriptive Oceano- graphy ; and the third on the Earth Beneath the Sea and History. The three volumes, therefore, are divided from one another by major divisions of oceanography, and we believe for this reason that our readers will not be inconvenienced by constant cross-referencing from one volume to another. This clear division has also allowed us to index the volumes separately ; it should be possible to select the volume containing information on any particular topic without difficulty. We have no overall index, since for a work of this magnitude it would be cumbersome. We considered that for ease of production we should start a new numbering sequence at the beginning of each chapter for figures, equations and tables. In order to avoid confusion in referring back, we have printed the chapter number and section number on alternate pages. In collecting the material for these volumes we have had most helpful cooperation from our contributors. In some topics, however, we have failed to find authors, or potential authors have withdrawn. We must admit, therefore, to omissions, some of which are conspicuous and important. By further effort we could doubtlessly have repaired these gaps in subject matter, but this would have delayed publication, and we believe the contributions we already have make a valuable collection. It is possible that in the future a fourth volume might be produced to deal with topics omitted or new since we first started collecting material for The Sea. We are greatly indebted to the secretarial assistance provided by Mrs. Joyce PREFACE IX Nightingale and Mrs. Joyce Day. Mr. Cameron D. Ovey has provided editorial assistance of outstanding value, and Dr. J. E. Holmstrom has produced an index of great merit. To them and to the publishers and printers we extend our thanks. May, 1962 M.N.H. CONTENTS PHYSICAL OCEANOGRAPHY Section I. Fundamentals ........ CH. 1. Physical Properties of Sea-Water by N. P. Fojonoff The equilibrium thermodynamic state Equation of state for sea- water Entropy ..... Chemical potential difference The non-equilibrium state . 6. Other physical properties of sea-water CH. 2. The Equations of Motion of Sea-Water by Carl Eckart 1. Introduction 2. Thermodynamics 3. Hydrodynamics . 4. The irreversible processes 5. Transformation of the equation 6. The zeroth approximation . 7. The first approximation 8. Definitions related to convective motion 9. The field equations .... Section II. Interchange of Properties between Sea and Air CH. 3. Small-Scale Interactions by E. L. Deacon and E. K. Webb 1 . General considerations of transfer .... 2. Momentum transfer and the wind-profile 3. Drag coefficients of the sea surface .... 4. Transfer of heat and water vapour .... CH. 4. Large-Scale Interactions by Joanne 8. Malkus Introduction How the whole system works Determination of air-sea fluxes Climatology of energy exchange and the global heat and water budgets .......■• 5. Heat and water exchange and its role in tropical circulations 6. Large-scale momentum relations ..... 7. Exchange mechanisms and fluctuations 8. Exchange fluctuations in mid-latitudes and long-period interaction anomaUes 9. Concluding remarks 3 4 8 12 17 22 28 31 31 32 33 33 35 36 37 39 40 43 43 43 49 57 66 88 88 92 100 114 144 178 202 253 285 CONTENTS CH. 5. Insolubles by R. W. Rex and E. D. Goldberg 1 . Meteorology of transport 2. Eolian materials in marine sediments CH. 6. Solubles by A. H. Woodcock . CH. 7. Gases by R. Revelle and H. E. Suess 1. Introduction 2. Oxygen and nitrogen . 3. Rare gases. 4. Carbon dioxide . Section III. Dynamics of Ocean Currents by N. P. Fofonojf . 1. Conservation equations for momentum and mass . 2. Separation into steady and time -dependent motion 3. Magnitude of forces 4. Steady-state circulation 5. Steady inertial circulation 6. Convective circulation 7. Time-dependent motion Section IV. Transmission of Energy within the Sea .... CH. 8. Light by J. E. Tyler and R. W. Preisendorfer . 1. Physical constructs ....... 2. Instrumentation for the measurement of the underwater li field and the determination of the optical properties of the 3. Radiance distribution ...... 4. Attenuation coefficient ...... 5. Volume scattering functions and total scattering coefficient 6. Irradiance ........ 7. Diffuse attenuation function and reflectance function 8. Scalar irradiance (spherical irradiance) .... 9. Absorption coefficient ...... 10. Path function ........ 11. Data 12. Applications ........ 13. List of symbols ....... CH. 9. Underwater Visibility 6?/ *9. (?. i)w?i^?f 2/ • 1. Image transmission ....... 2. Inherent contrast ....... 3. Sighting range ........ CH. 10. Light and Animal Life by G. L. Clarke and E. J. Denton . CH. 11. Other Electromagnetic Radiation by L. N. Liebermann Introduction ........ Electromagnetic properties of sea-water Propagation through sea-water ..... The effect of the sea-surface on electromagnetic propagation Natural electromagnetic radiation or "noise" in the sea . ght sea CONTENTS CH. 12. Sound in the Sea by P. Vigoureux and J. B. Hersey 1 . The nature of sound ...... 2. Propagation of sound in water .... 3. Noise 4. Instruments and applications of sound to oceanography CH. 13. Sound Scattering by Marine Organisms by J. B. Hersey and R. H. Backus ...... 1. Introduction ...... 2. Occurrence and description of scattering layers 3. Identification of sound scatterers . 4. Sound-scattering theory .... 5. Sound-scattering observations 6. What is "the deep scattering layer" ? 7. Ideas and miscellaneous observations . CH. 14. Sound Production by Marine Animals by W. E. Schevill, R. H. Backus, and J. B. Hersey 1. Introduction 2. History 3. Instrumentation. 4. Identification of source 5. Purposeful and adventitious sounds 6. Sound-producing mechanisms 7. Spectra of sounds 8. Functions of sound 9. Hearing as related to sound production 10. Eliciting and suppressing marine animal sounds 11. Exploitation of marine animal sounds by the oceanographer Section V. Waves ....... CH. 15. Analysis and Statistics by D. E. Cartwright 1. Introduction ..... 2. Fundamental equations of wave motion 3. Statistical formulation 4. Properties of a wave system in terms of its directional spectrum ...... 5. Estimating the directional energy spectrum 6. Waves recorded by a single detector 7. Spectral measm'ement . 8. Second-order approximations to energy spectra CH. 16. Long-Term Variations in Sea-Level by J. R. Rossiter 1. Introduction ....... 2. The determination of mean sea-level 3. Causes of variations in sea-level .... 4. The analysis of observations .... 5. Some geophysical implications of long-term variations level ........ 6. Conclusion ....... energy in sea- 476 476 476 489 491 498 498 499 507 512 524 533 534 540 540 540 542 549 550 552 554 558 561 561 562 567 567 567 568 570 573 576 580 584 586 590 590 590 592 601 605 608 CONTENTS CH. 17. Surges hy P. Groen and G. W. Groves 1. Introduction .... 2. Description .... 3. Dynamics and forecasting . CH. 18. Long Ocean Waves by W. H. Munk 1. Introduction 2. The instruments 3. The spectrum 4. Surf beat . 5. Shelf waves 6. Tsunamis . CH. 19. Wind Waves hy N. F. Barber and M. J. Tucker 1 . Kinematics of waves 2. The description of a comphcated wave pattern: the spectrum ..... 3. Theories of wave generation by wind 4. Wave prediction 5. Waves from distant storms . 6. Waves approaching the shore 7. The surf zone .... 8. Ships and waves 9. Methods of observation and analysis — methods taking no account of direction of travel 10. Methods of observation and analysis — the directional power spectrum ...... CH. 20. MiCROSEiSMS by J. Darhyshire . 1 . Relation between sea waves and microseisms 2. The nature of microseisms . 3. Refraction of microseisms . 4. Storm tracking and estimation of the direction of approach of microseisms ..... 5. Estimation of direction from the nature of microseisms 6. Instruments ..... 7. Other work ..... CH. 21. Ripples 6t/ C. >S. Coa: .... 1 . Spectrum and mean square slope . 2. Effect of slicks . ... 3. Shape of a rippled water surface . 4. Growth of ripples .... CH. 22. Internal Waves. Part I by E. C. LaFond 1. Introduction 2. Measurements 3. Observed relations Part II by C. S. Cox 4. Differential equations 5. Spectrum . 6. Internal waves and turbulence CONTENTS XV cm. 23. Tides by W. Hansen 764 1. Introduction ......... 764 2. The hydrodjmamic equations and their apphcation to tidal problems .......... 765 3. Tidal observation . . . . . . . .768 4. Tidal charts 773 5. Classical theory ......... 779 6. Numerical methods for ascertaining the tides and tidal currents. Boundary-value problems ....... 781 7. The application of difference methods to initial -boundary problems .......... 786 8. Numerical solutions of initial-boundary-value problems of tides in one and two dimensions . . . . . . .788 9. Internal tides 799 Section VI. Turbulence by K. F. Bowden .... 1 . General properties of turbulence . 2. Turbulence in the sea ..... 3. Turbulent fluctuations and turbulent transports 4. Vertical turbulence ..... 5. Horizontal turbulence .... 802 802 808 810 814 817 Section VII. The Physics of Sea-Ice by E. B. Pounder 1. Introduction ..... 2. Mechanical properties 3. Thermal properties .... 4. Electrical properties .... 5. Growth and disintegration of an ice cover 6. Theory of sea-ice structure and properties 826 826 829 832 834 834 835 Author Index Subject Index 839 850 Physical Oceanography I. FUNDAMENTALS 1. PHYSICAL PROPERTIES OF SEA- WATER N. P. FOFONOFF Much of our knowledge and understanding of the ocean's behaviour is obtained through the apphcation of the conservation laws of mass, momentum and energy to processes we observe to occur in the ocean. In order to formulate these laws, we have to know certain properties of the fluid medium, such as density, specific heat, viscosity and other physical and chemical characteristics, as functions of temperature, pressure and salt content. The present discussion is confined primarily to properties entering into the conservation equations so that these properties may be described within the framework of a thermo- dynamical system. Such a framework enables us to understand more clearly the relationships among the various properties and will aid in the comprehen- sive development of our knowledge of them. The incentive for much of the early studies of sea-water stemmed from the need to be able to apply the conservation laws with precision in terms of observations taken in the field. This was particularly true in the case of the momentum equations with the development of methods of computing velocities from the density distribution in the ocean. As a result, physical properties entering into these equations have received more attention than others. There are exceptions, of course. A number of properties have been studied because of their importance to life in the sea. Others have been studied for technological reasons and for the development of special techniques for the analysis of sea- water. A complete discussion of all of these properties is beyond the scope of the present undertaking. Our knowledge of sea-water properties has not developed systematically. Early in the present century, a co-ordinated effort was made to determine the equation of state and related physical and chemical properties. Since then the efforts have been sporadic. Nevertheless important gaps in our knowledge are being filled and efforts are being made to encourage and co-ordinate work in this basic field. ^ There are a number of descriptions, reviews and tables of sea-water properties available in the literature. Among these are Kriimmel (1907), Matthews (1923), Thompson (1932), Bein et al. (1935), Subov and Chigrin (1940), and Sverdrup et al. (1942). More recent compilations and descriptions are those of Rouch (1946), Lafond (1951), Dietrich and Joseph (1952), Dietrich (1957), Richards (1957) and Montgomery (1957). 1 See, for example, the report on the conference held at Easton, Maryland, September, 1958 — titled "Physical and Chemical Properties of Sea Water." National Academy of Sciences-National Research Council, Publ. 600. [MS received March, 1960] 3 4 FOFONOFF [chap. 1 There is still a need to determine or re-examine many of the basic properties of sea-water. For this reason, more effort is taken here to develop) the thermo- dynamical basis for these properties in order that the relationships among them may be brought out more clearly and systematically. The properties are discussed, therefore, in three parts. Properties necessary to describe thermo- dynamic equilibrium are considered first, then the additional properties required to characterize the non-equilibrium state, and, lastly, other properties that are of interest. Because of the raj)idly increasing use of electronic digital computers for processing oceanographic data, empirical formulae in current use for some of the more frequently computed properties are included. Unfortunately, recent studies indicate that most of these formulae are of inadequate accuracy and, therefore, cannot be recommended for general future usage. We are sorely in need of a complete overhaul of existing tables and formulae for the routine calculation of sea-water properties. However, it would not be advisable to propose changes that are not supported by more complete and accurate determinations of the properties than we have at present. 1. The Equilibrium Thermodynamic State The continual exchange of energy and matter across the boundaries of the ocean prevents the establishment of a complete thermodynamical equilibrium. In general, the transport of water, heat and salt by currents greatly exceeds that of molecular processes associated with the tendency for the ocean to approach thermodynamical equilibrium. However, at sufficiently small scales of motion, molecular processes become significant and even dominant. Hence, it is necessary to formulate and examine the conditions under which equilibrium is attained and to determine the physical properties required to describe the equilibrium state. We shall examine the equilibrium state from the physical point of view, treating the dissolved salts as a single component and ignoring chemical reactions and equilibrium conditions that alter the ionic structure in sea-water. Because of gravity, the pressure in the ocean increases continuously with depth. This means that we cannot have a uniform phase of finite vertical extent. We must consider the ocean as being made up of an infinite number of phases in its vertical structure. Each of these phases forms an open system free to exchange heat and matter with adjacent phases. It is, therefore, more con- venient to formulate the thermodynamical relationships in terms of a system of unit mass and to consider finite systems in the ocean in terms of integrals with respect to mass over all of the phases within the system. The thermodynamical relationships for a sea-water system can be obtained from a general statement of the first law of thermodynamics for a multi- constituent system within a gravitational field (e.g., Guggenheim, 1950, p. 356). Changes of the total energy, E, of a system of total mass, M, at a gravitational potential (geopotential), 0, are given by SECT. 1] PHYSICAL PROPERTIES OF SEA-WATER 5 dE = TdN-pdV + y{ixi + 0)dmi + Md0, (1) i where T is absolute temperature, N entropy, p pressure and fXi the specific chemical potential of the i-th constituent of mass mtA As defined by (1), the total energy includes both the internal and potential energies. If appreciable motion of the phase occurs, the kinetic energy must also be taken into account. We can convert (1) to apply to a system of unit mass by substituting the intensive quantities : e = E/M = € + 0 specific total energy, where e is the specific internal energy, and 7] = NjM specific entropy V = V jM specific volume Xi = mijM mass fraction of the i-th constituent of sea- water. 2 The substitution yields the three relations among the intensive quantities de = T dr]-pdv + yixidxi + d0 (2) i e = Tr]-pv + 2H^i + ^ (3) i and the Gibbs-Duhem equation -qdT-v dp + ^ Xi diJLi = 0. (4) i Although the development of the thermodynamic relationships can be carried out in terms of the general multi-constituent system, a great simplifica- tion is achieved by introducing the concept oi salinity. Dittmar (1884) showed that the majority of ions in sea- water are present in remarkably constant ratios to one another. To a first approximation, we can consider changes of salt content of sea-water to be brought about by the addition or removal of water. In principle, we could define salt content as the ratio of the total mass of dis- solved salts to the mass of sea-water. In practice, however, this ratio is almost impossible to determine accurately, and, historically, a somewhat different concept of salinity has evolved. 1 The term M d0 represents changes of the total energy due to displacements of the phase relative to the geopotential field. As pointed out by Craig (1960), this term was not taken into account by Fofonoff (1959) in his application of thermodynamical considera- tions to a sea-water system. His derived equations are, therefore, valid only if the phase is fixed in space, i.e. if d0 is set equal to zero. 2 Mass fractions rather than mole fractions are used throughout this section in order to retain the same units as the equations of motion (cf. Eckart, Chapter 2). Although mole fractions are more suitable for expressing many of the thermodynamical relationships, they cannot be introduced conveniently into the dynamical equations. The mole fraction for the i-th constituent in terms of the mass fraction is given by (Mxi/Mi)/'^ mi/Mi, where Mi is the molecular weight of the i-th constituent. * 6 FOFONOFF [chap. 1 Salinity has been defined by Forsch et al. (1902) as the total solid material in grams contained in one kilogram of sea-water when all carbonate has been converted to oxide, the bromine and iodine replaced by chlorine, all organic matter completely oxidized, and the residue heated to constant weight at 480°C. Salinity, under this definition, is not equivalent to the total salt content, but for our purposes it is sufficient to assume that a linear relationship exists between them. Estimates of total salt content, made by Lyman and Fleming (1940), are related to the salinity, S, by the formula ,9totai = 0.043+1.0044*9. Salt content can be specified in terms of the ratio of mass of a single ion or group of ions to that of sea-water. Titration of sea-water with silver nitrate precipitates chlorides, bromides and iodides, so that an accurate determination of the quantities of these ions can be made. The chlorine-equivalent of the pre- cipitated halides in grams per kilogram of sea-water is called the chlorinity of sea-water.^ It has been related to the salinity S by the empirical formula S = 0.03+1.8050(7^, (5) where both salinity S and chlorinity CI are expressed in parts per thousand (%o). If we assume that the ratio of each constituent of sea-water to the salinity is constant, we can introduce a set of multipliers, Aj, giving the mass fraction of the i-th constituent in the form Xi = XiS, (6) where s is the salinity expressed in grams per gram of sea-water. - If we restrict our considerations to processes that do not affect the mass ratios of the constituents, we can introduce a combined chemical potential, /xs, for the dissolved salts. From (2) and (6), we obtain i Introducing /xw for the specific chemical potential of water in sea-water and fjL for the difference of specific chemical potentials ^lis — /xw, we can express (2), (3) and (4) in the form de ^ T d-q-pdv + fjids + dO (7) e = Tt] - pv + iJiS + ixv, + 0 (8) dfjiv, = —rjdT + vdp — sdiJL. (9) By introducing these transformations, we are, in effect, treating the dissolved salts as a single solute. 1 To avoid variation in the chlorine-equivalent due to refinements in the determination of the atomic weights of the halogens, chlorinity has also been defined in terms of the weight of purified silver (prepared by specified techniques) required to precipitate the halogens. Salinity is then considered to be defined by the empirical formula obtained by Forsch et al. (1902) relating salinity and chlorinity (see equation (5)). 2 If salinity expressed in parts per thousand is used, (6) becomes Xi = IQ-^XiS. SECT. 1] PHYSICAL PBOPERTIES OF SEA-WATER 7 The Gibbs-Duhem equation (9) can. also be expressed in the form dfx^ = -L + s^jdT+lv-s^\dp-s^ds, (10) which is equivalent to = —rjvfdT + Vyidp — s-^f-ds. 8s By analogy with a pure-water phase, we can interpret rjv/ as the partial specific entropy and Vw as the partial specific volume of water in the sea-water phase. We can define other thermodynamic potentials in the usual way, i.e. h = €+pv specific enthalpy / = e—Trj specific free energy g = e — T7)+pv specific thermodynamic potential (Gibbs function). Temperature, pressure and salinity are measured directly in the ocean. These variables can then be used to calculate other properties, including the depth of the observation; i.e. its location relative to the geopotential field. Therefore, these variables are considered to be the independent variables in the subsequent development. The appropriate function to describe the thermodynamical properties of a phase in terms of temperature, pressure and salinity is the Gibbs function g. From the first derivatives with respect to the independent variables, we obtain 81 eg dp -k = -" ('^) ■,p = " <•=" a. = " <'*) and from the second derivatives d~g dr) Cp where Cp is the specific heat at constant pressure, and 8^g dfx dtj dT 8s ^ JT ^ ~8s 8^g 8v 8r) 8T 8p ^ ~8T^ "Jp 8^g 8v 8fjL 8s 8p 8s 8p (15) (16) (17) (18) 8 FOFONOFF [chap. 1 The third derivatives give us the additional useful relationships d^g 1 8cp d'^v dp dT^ T dp dT^ d^g 1 dcp d^fjL ds 8T^ ^ ~T~ds ^ W^' (19) (20) To determine the Gibbs function, g, completely — except for an arbitrary constant — we would have to know absolute values of the three first derivatives ; i.e. entropy, specific volume and chemical potential difference as functions of temperature, pressure and salinity. However, this cannot be done as neither entropy nor chemical potential difference is completely specified in terms of equilibrium-state properties. As will be seen later, entropy, rj, can be determined except for a linear function of salinity, i or, from (16), /x except for a linear function of temperature. This implies that the Gibbs function is arbitrary to the extent of a function of the form aT + bTs + cs + d, where a, b, c and d are constants. Conversely, it is clear that we do not require a knowledge of this arbitrary function to describe the equilibrium state. Our knowledge of the three first derivatives — specific volume, entropy and chemical potential difference — is now considered in more detail. 2. Equation of State for Sea-Water Our present knowledge of the specific volume of sea-water as a function of temperature, pressure and salinity is based on three sets of measurements. The first of these was the determination of specific gravity as a function of chlorinity at 0°C and atmospheric pressure. These determinations were made with pycnometers by Forsch et al. (1902) and the results were expressed by the empirical formula cTo = - 0.069 +1.4708(7/- 1.570 X 10-3(7/2 + 3.98 X 10-6OP, (21) where o-q, the specific gravity anomaly, is defined in terms of the specific gravity So by CTo=103(so- 1). A total of 24 samples of sea-water, all from surface waters, were used in the determinations. Most of the samples were collected in the Baltic, North Sea and the North Atlantic Ocean, and are now recognized not to be adequately representative of all ocean waters. Thompson and Wirth (1931) carried out determinations of ao on 36 samples, including sea-water from the Pacific and Indian Oceans, and from depths to 1000 m. A comparison with ctq, computed from (21), showed that their measured values were higher than those given 1 An alternative interpretation of the arbitrary linear function of salinity in the definition of -q is obtained by introducing a combined partial entropy of the salts, r]s= —8ixs/8t- Then, the specific entropy, rj, is given by stjs + ( 1 + 'S)>7w The partial entropies, rjs, tjw, contain arbitrary constants giving rise to an arbitrary linear function of salinity in rj. SECT. 1] PHYSICAL PROPERTIES OF SEA-WATER 9 by the formula by an average of + 0.02 units of ctq- No changes in the formula for CTo were made on the basis of these measurements. Carritt and Carpenter (1958) discussed both series of measurements used to derive the relationships among chlorinity, salinity and density. They conclude from the observations that salinity may vary by ± 0.01 7 %o and ctq by at least ± 0.04 for a given chlorinity. They suggest that these variations in the measure- ments cannot be attributed entirely to experimental error and are probably due to regional variations in the ratios of dissolved ionic constituents in sea- water. Consequently, they interpret these variations as rejaresenting a limit of accuracy to which we may define S or ctq in terms of chlorinity. The second set of measurements by Forsch et al. (1902) was used to establish the thermal expansion of sea-water at atmospheric pressure. These determina- tions were reduced to an empirical formula for the specific gravity anomaly at, expressed in terms of temperature and ctq in the form at = A + Bao + Cao^ (22) where A, B and C are functions of temperature. For pure water, the formula becomes Et = A^BEq + CEq'^, (23) where Et, Eq denote the specific gravity anomaly of pure water. By subtracting (23) from (22), the formula can be put in the form used by Knudsen (1901) ; i.e. at = Et + {ao-Eo)[l-At + Bt{ao^EQ)l (24) where i;«= -(;^-3.98)2(^ + 283)/503.570(7^ + 67.26), Eq= -0.1324, B=\-At = 1_ 4.7867 X 10-3,^ + 9.8185 X 10-57^2 _ 1.0843 X lO-^j^^ C'= ^<= 1.8030 x 10-5,^ - 8.164 X 10-'^j^2_|_ 1.667 X 10-8^3^ and d' is the temperature in degrees Celsius. Formula (22) is more convenient in some applications. The coefficient A is given by A = (4.53168j^-0.545939j?2_ 1.98248 X 10-3^3_ 1.438 X 10-7,^4)/(^+ 67.26) For pure water (cro = i7o= —0.1324), formula (24) reduces to Et. This formula for the specific gravity anomaly of pure water was developed by Thiesen (1897). Tilton and Taylor (1937) constructed a more accurate formula of the same form using observations of density made by Chappius (1907). The revised formula differs from Et by less than 0.005 (Dorsey, 1940, table 100). The revision has not been incorporated into the sea-water formula. Formulas (21) and (24) do not reduce to the pure water formula for either S = Q or Cl = 0 because of the constant terms in the relationships between S and CI, and between o-q and CI. The most recent determinations of density of sea-water are those of Bein et al. (1935). The densities agreed within ±0.02 units of at with values com- puted from Knudsen's (1901) tables. Ekman (1908) carried out the third set of measurements to determine the effect of pressure on specific volume. He used a single sample of sea- water 10 FOFONOFF [chap. 1 collected from a depth of 3000 m off the coast of Portugal. Part of the sample was diluted with distilled water to a salinity of 3 1.13 %o, and the rest evaporated to a salinity of 38.83 %o. Although he carried out some measurements of com- pression by lowering sea-water samples into the ocean to subject them to pressure, his final results were based entirely on subsequent determinations in the laboratory. Pressures in his laboratory apparatus were determined by measuring the compression of distilled water at 0°C, and then calculating pressure using Agamat's (1893) determinations of the compression of pure water. Ekman measured sea-water compression at several temperatures from 0-20°C for pressures of 200, 400 and 600 bars. He used his results to develop an empirical formula for the mean compression of sea-water, fx, in the form ju, = {ao-a)lpao, where a is the specific volume ^ at pressure p and ao the specific volume at atmospheric pressure. Specific volume at pressures above atmospheric pressure is computed from a = ao{\-fxp), (25) where lOV = [4886/(1 + 1.83 X 10-5j9)]- (227 + 28.33r?-0.551j^2 + o.004j^3) + 10-4p(105.5-f 9.50?^- 0.158t?2)_ 1,5 x IQ-^^p^- - 10-i(cto - 28)[(147.3 - 2.T2& + 0M&^) - 10-4p(32.4 - 0.87i^ + 0.02j^2)] + 10-2(cto - 28)2[4.5 - O.lj? - 10-4^9(1.8 - 0.0Q&)] (26) and^ is pressure 2 in decibars (10^ dynes/cm2).3 Eckart (1958), in a careful study of the density and compression of pure- and sea-water, has concluded that the equation of state for both can be represented with sufficient accuracy by the Tumlirz equation of the form {p+po){a-ao) = A, (27) where A= 1779.5+ 11. 25z^-0.0745i^2_(3,804-0.01i^)*S, ao = 0.6980, ;po = 5890 + SSd' — O.SlB&^ + SS, p is total pressure in atmospheres (1 atm= 10.1325 db), and a is specific volume in millilitres per gram. 1 The specific volume, a, is given in millilitres per gram and is equal numerically to the reciprocal of specific gravity. To obtain specific volume v in cubic centimetres per gram, a is multiplied by 1.000027 cm^/ml. 2 The pressure, p, is total pressure less one standard atmosphere. This convention is widely used by oceanographers. 3 Bjerknes and Sandstrom (1910) gave the coefficient 0.002 instead of 0.02 for the (ao — 28)p&^ term in (26). The discrepancy is probably due to a misprint. However, it is not clear in which paper the misprint occurred. The difference between the two formulae appears to be completely negligible for ranges of variables encountered in the ocean. SECT. 1] PHYSICAL PROPERTIES OF SEA-WATER 11 Eckart estimated random errors in measurements of specific volume to be about + 2 X 10"'* nil/g and systematic errors probably greater than + 2 x 10~'^^ ml/g for sea-water. Both formulas (25) and (27) give the same value for specific volume within the limits of error given by Eckart. Over the major part of the range of temperature, pressure and salinity encountered in the ocean, the difference in specific volume from the two formulas is less than 5 x 10"^ ml/g. ^_;^ "^-^ ^ 3Q~~ — :_ -^^-^ ~^~~~^ "^-20 — , 2 _l;^ ' 10 _ '^~^ — ~~~~~~~0.0--__^ ^_ — - (o) p = Odb ~~^-I.O ^ ) "^ <1 _^^ ::::;^ 1 - — :;:: -^ Fig. 1. A comparison of equations of state of sea-water, ax (Kjiudsen, 1901) and as (Eckart, 1958). Values of the differences 104(o:/f — «£:) are shown as functions of temperature (°C), salinity (%o) and pressure (db). However, the difference increases rapidly at the limits of the range. For ex- ample, at atmospheric pressure the difference increases from 5 x 10^^ nil/g at 18°C to 30 x 10-5 ml/g at 30°C for sea-water of 35%o salinity. The difference between the two formulae for various temperatures, pressures and salinities is shown in Fig. 1. Although the specific volumes do not differ greatly, the two formulae pro- posed for the equation of state of sea-water indicate that some of the derived properties, such as the coefficient of thermal expansion, are seriously in doubt. Even for pure water, which has been studied more intensively, the thermal 12 FOFONOFF [chap. 1 expansion coefficient is poorly defined by present measurements. Eckart (1958) found the values 90 x 10-6 deg-i and 190 x 10-6 deg-i at OT and 1000 atm, depending on whether a linear or quadratic expression is assumed for^o in (27). A comparison of the coefficient of thermal expansion, (1/a) dajd^, computed from (24) and (27) is given in Table I. The coefficients of compressibility, — (1/a) da/Bp, and saline contraction, —(1/a) da/ds, as computed from the two formulae, differ by about 1% or less. Table I A Comparison of the Coefficient of Thermal Expansion of Sea- Water at 35 %„ Salinity and Atmospheric Pressure as Computed from Equations of State, a/f (Knudsen, 1901) and as (Eckart, 1958) Temp., °C 106 daK 106 duE 0 52 80 5 114 121 10 167 161 15 214 201 20 256 237 25 297 274 30 335 311 3. Entropy We can determine changes of entropy through the equation = '^dT-^dp-^ds. (28) if we know the specific heat, the thermal expansion and the thermal rate of change of chemical jDotential. If these coefficients are known at atmospheric pressure, their values at elevated pressures can be computed from (18) and (19) using the equation of state. As indicated earlier, djjLJdT cannot be completely specified in terms of equilibrium-state properties, and, hence, entropy is also incompletely specified. Until recently, oceanographers have used specific heat values obtained by Thoulet and Chevallier (1889) for a range of sea-water densities (salinities) at a temperature of 17.5°C and at atmospheric pressure. In the absence of measure- ments at other temperatures, the specific heat was assumed to decrease with temperature in the same way as pure water. Cox and Smith (1959) carried out a complete determination of the specific heat over the range - 2°C to 30°C in temperature, 0%o to 40 %o salinity, at atmospheric pressure. The variation of SECT. IJ PHYSICAL PKOPERTIES OF SEA-WATER 13 specific heat with temperature and sahnity is shown in Fig. 2. Not only is the variation of specific heat of sea-water different from that of pure water, but absohite vahies differ appreciably from those given by Thoulet and Chevallier (Fig. 3). Fig. 2. Specific heat of sea-water, Cp, in absolute Joules per gram per degree Celsius as a function of temperature (°C) and salinity (%o) at atmospheric pressure. ■Cox ond Smith (1959) Thoulet and Chevallier (1889) Fig. 3. A comparison of values of specific heat of sea-water, Cp, at 17.5°C and atmospheric pressure as determined by Thoulet and Chevallier (1889) and Cox and Smith (1959). From an analysis of their determinations. Cox and Smith proposed the formula Cp = r/(i?') - 5.075 X 10-3^ -1.4 X 10-5,S2 (29) for the specific heat at constant pressure, where Cp is the specific heat in abso- lute Joules per gram per degree Celsius and Cp^{d'') is the specific heat of pure water at temperature &' . The value of c^o appropriate to (29) is given by the pure -water formula at an elevated temperature &' , where ^' = ^ + 0.7/S + 0.0175*^2 (30) Cox and Smith did not give a formula for Cp^{&). A simple and accurate 14 FOFONOFF [chap. 1 formula for the specific heat of pure water can be constructed in two parts, using the measurements of Osborne et al. (1937). For temperatures above 33.67°C c/(,^) = 4.1784+8.46x10-6(^-33.67)2 (31) and below 33.67°C CpQ{d') = 4.1784+(8.46 + 26.19e-o-048*)x 10-6(,?-33.67)2, (32) where e is the natural logarithm base. These two formulae reproduce tabulated values of CpO to within 2 parts in 10^ over the range 0°C to 100°C. The use of the two-part formula gives a discontinuity in B^Cp^jdd'^ at 33.67°C of approxi- mately 1% of the magnitude of the second derivative. The primary reason for using the two-part formula is that &' is greater than 33.67°C for salinities in excess of 30 %o for all temperatures above freezing. Thus, for most oceano- graphic work, only (31) is required. The specific heat of sea-water at elevated pressures has not been measured, but is calculated from (19). Coefficients for an empirical formula, based on (24) and (25), for the effect of pressure on specific heat are given in Table II. The specific heat at constant volume, Cy, of sea- water has not been measured, but can be computed by using the auxiliary equation 8v ,^ dv , to (28). We can then eliminate d'p d-q = to obtain ""' dT ^^ ds T dT ' where Cjj = Cj (33) Table II Formula" for the Change of Specific Heat of Sea-Water, Cp, in Absolute Joules per gram per degree Celsius, with Pressure as a Function of Temperature °C, Salinity %o, and Pressure db Cp{0)-Cp{p) T = f„w''p = ^^^"^p''"^' Aioo = +1.8185x10-'? ^120= +1.1649x10-11 ^101 = -5.574x10-9 A200 = -7.3867x10-13 ^102= +1.7417x10-10 ^201= +1.3574x10-13 ^103= -2.599x10-12 ^210= +4.824x10-14 ^110= -1.8117x10-9 ^300= +1.991x10-15 Am = +1.5158x10-11 ^310= -2.5446x10-17 « This formula has been constructed for electronic computer applications by the Pacific Oceanographic Group (Fofonoff and Froese, 1958). Its precision is 0.2% of Cp for the range -2° to 30°C, 20 to 40%o, and 0 to 10,000 db. SECT. 1] PHYSICAL PROPERTIES OF SEA-WATER 15 The ratio of specific heats is Cp y = — Cv 1 + -1 Cp{8vl8p) J ^^*^ and is greater than unity because dv/Bp is negative. The ratio varies from 1.0004 at 0°C to 1.0207 at 30°C for a sahnity of 34.85%o (Matthews, 1923). Two concepts which find particular use in the analysis of deep-water struc- ture in the ocean have been developed from the equation for entropy change (28). These are the adiabatic lapse rate of temperature and potential tempera- ture. The equation for the adiabatic lapse rate of temperature was first derived by Sir William Thomson (1857). Potential temperature was first used in oceanography by Helland-Hansen (1912). If a layer of sea-water is completely mixed so that two elements of water, anywhere within the layer, are indistinguishable when brought together to the same pressure, the layer must have constant salinity and entropy. This may be seen from the fact that the comparison of the elements within the layer can be made by a reversible and, hence, isentropic, process. If either salinity or entropy are different, the mixing cannot be complete. Consequently, for a completely mixed layer of sea-water, we may write (28) as dr] = {cplT)dT-{8vldT)dp = 0 or \8pJ^ Cp where F is called the adiabatic lapse rate of temperature or adiabatic tempera- ture gradient. Values of F for various temperatures, pressures and salinities can be found from the graphs shown in Fig. 4. Coefficients for an empirical formula for F are given in Table III. Both the graphs and formula are based on Table III Formula" for the Adiabatic Lapse Rate of Temperature, F, in Terms of Temperature (°C), Salinity (%o) and Pressure (db) „ . lO^T dv „ „ „ . a V v^/j Cp 8& — Z. Z. Z. -^ijlcj^ f^ '^ t J k ^000 = -4.21 X 10-2 Aq2Q = -6.5x10-6 ^001 = + 1.022 X 10-2 ^100 = 4- 2.291 X 10-5 ^002 = - 1.478 X 10-4 ^101 = -7.62x10-7 ^003 == + 3.45X 10-6 ^102 = -h 7.5x10-9 ^004 = -3.3x 10-8 -4iio = -1.06x10-7 ^010 == -1- 2.427 X 10-3 ^111 = +1.97x10-9 ^011 = -5.0x 10-7 ^200 = -4.53x10-10 ^012 = + 9.0x 10-8 ^201 = +1.56X 10-11 « Fofonoff and Froese, 1958. The precision of the formula is about 0.1% of F for the range 0° to 30X\ 20 to 40%o and 0 to 10,000 db. 16 FOFONOFF [chap. 1 (24) and (25) and should be used with reservations in any apphcations. Absolute accuracy of F is difficult to establish in view of our inadequate knowledge of the coefficient of thermal expansion (see Table I). Values of F differing by 50% can be obtained from the two equations of state, (24) and (27). db 10000 i::^ ^^:- ;::::; ^■" (b) ^- ~\ r = i; t a^ t «/; N^\ >. °C/IOOO db \^ V. (a) r, p = Odb nV" V ■'° (b) SFg S -- 35%. _ \\^ \ (c) sr^ 8 = o°c \\^ \ M .002 \ (c) 0 /-.002 1 Fig. 4. The adiabatic lapse rate of temperature, F, as a function of temperature (°C), salinity (%o), and pressure (db). The units of 7"" are °C/1000 db. The temperature lapse rate, F, enters into studies of temperature inversions in the deep trenches in the ocean. It enters also into the formula used to compute sound speed, V, (Kuwahara, 1938) given by V = [-v^'l{dvidp + Fdvld&)]''-^ (36) and into the calculation of static stability. E, from the equation (Hesselberg and Sverdrup, 1914) " dp (d& _\ dp ds' E = -gjp d&\ ds dz (37) where g is gravity, p density and ;:; the vertical co-ordinate. ^ 1 Ivanov-Frantzkevich (1953) has pointed out that the tables for the density derivatives, dp/8& and dpjds, included in the paper by Hesselberg and Sverdrup contain systematic errors amounting to a maximum of 10% of the values of the derivatives. These errors do not seriously affect stability values for depths under 5000 m. SECT. 1] PHYSICAL PROPERTIES OF SEA -WATER 17 If an element of sea-water, at an initial temperature &i and pressure pi, is raised to the surface of the ocean with no exchange of heat or salt with its surroundings, its final temperature, d'f, at atmospheric pressure will be ^f = & =-&i+r r{&', p, s) dp, (38) where &' = d'i+ r r{&, p, s) dp. Jpi The final temperature & is called the potential temperature. Table IV Formula" for the Cooling A@ of Sea-Water of Temperature d'°C and Salinity S%o for an Adiabatic Change of Pressure from a Pressure p db to Atmospheric Pressure A0CC) = 111 Aiikp^Sm i j k ^100 = -1.60x 10-5 ^120 = + 4.1 X 10-9 ^101 = + 1.014 X 10-5 ^200 = + 9.14X 10-9 ^102 = -1.27X 10-7 ^201 = -2.77X 10-10 ^103 = + 2.7x10-9 ^202 = + 9.5X 10-13 ^110 = + 1.322 X 10-6 ^300 = - 1.557 X 10-13 ^111 = -2.62x 10-8 « Fofonoff and Froese, 1958. The minimum precision of the formula is ± 0.004°C. The potential temperature is useful in discussing the movement of deep water in the ocean, because the effects of pressure on temperature are removed. Thus, temperatures of the water at different depths may be compared. Graphs giving values of the difference &i -■&■/, denoted hy A0, are shown in Fig. 5. Coefficients for an empirical formula for A© are given in Table IV. Absolute accuracy of both the graphs and formula is uncertain because of our inadequate knowledge of the thermal expansion coefficient of sea- water (see Table I). 4. Chemical Potential Difference The chemical potential difference, fx, and the partial chemical potential, jUw, of water in sea-water can be determined in terms of the colligative properties of sea-water, i.e. vapour-pressure lowering, freezing-point depression, boiling-point elevation and osmotic pressure. 18 FOFONOFF [chap. 1 If sea-water is in equilibrium with water vapour at a given temperature, pressure and salinity, the partial chemical potential, /xw, of the water in sea- water must equal the chemical potential of the vapour, [x^. If we compare two ' — 1 r- ' -0.10 .' l-"^ '^^ -0.20 — . • ^^"-"'''^ ^^^^^^^ ^0.30 y y^ ^.^^^^^ ^0.40 - //X (a) - '// Aid = AB^ t °C se. - / / - / / /°-^° i.0) ^e s 35 %<, '/// fs; 8«, e = CC - / / /°-8o - // / /I.OO - '/ / / ,\.zo - / / / / , 1 1 , 30 35 S %.40 Fig. 5. The difference A& between temperature in situ and potential temperature as a function of temperature (°C), salinity (%o) and pressure (db). The difference is given in degrees Celsius. such equilibrium systems of slightly different salinities, but at the same tem- perature and geopotentialj we must have for the differences of chemical potentials J/XW = /1/Xv ^ -|^ ^Pv = Vy Apv, (39) where p^ is the vapour pressure of sea-water and Vy is the specific volume of the vapour. From equation (11), we have J|Hw ^ Vw Apv -s— As, (40) where Vw is the partial specific volume of water. Equation (39) can then be written dfji Apv S^ ^ {Vy- Vw) — 7- ds As = -iVy-V.)—. (41) SECT. 1] PHYSICAL PROPERTIES OF SEA -WATER 19 If we define r to be the fractional lowering of vapour pressure, {pv^ — pv)lpv^, where ^v" is the vapour pressure of pure water at the same temperature, we can write (41) as py^VyO dr ET dr 1 — r ds l—rds dfiM 8s (42) neglecting Vw in comparison with Vy. The constant R has the value 46.15 db cm/g "'C for pressure in decibars (db). From (42), we see that jjlv, can be expressed in the form /xw = [x^^ + RT\n{l-r), (43) where />tw° is the chemical potential of pure water at the same temperature. A similar comparison of two sea-water systems of slightly differing salinities in equilibrium with ice yields 'Ys = -(^--^i-)-aJ' (4*) where T{7]v/ — r)ice) is the heat of fusion of ice. Again, at the boiling point of sea- water, we have S-=irj^-rj^)—, (45) where T{rjy — rjw) is the heat of vaporization of sea-water. We can find the osmotic pressure, tt, by considering it to be the excess of pressure required to bring sea-water into equilibrium with pure water across a semi-permeable membrane. In the equilibrium, we have fl^^ip) = fl^{p + 7T). (46) Comparing two such systems containing sea-water of slightly different salinities, we obtain 0 = /\^.^^A.-s^As, dp 8s so that da 8av! 8iT Stt RT 8r ,^„. ds dp ds ds 1—r 8s The change of osmotic pressure with temperature is given by dn dlT /AO\ Vy,^^ = T7w-i7w" (48) 20 FOFONOFF [chat. 1 where rjw° is the entropy of pure water. Using the definition for the partial entropy of water in sea-water djx drj we obtain ' cT c^s Substitution from (47) yields (49) ^w -V-' = 4rJ^T f -L ^ ds, (50) ' ' c'T Jo I -r c's so that in terms of vapour-pressure lowering ,,._^ _i?ln(l-r)+3--^^- (51) Values of -n have been tabulated by Robinson (1954) and by Wilson and Arons (1955) using equation (47) and measured values of vapour-pressure lowering. Given the vapour-pressure lowering and specific heat at atmospheric pressure, we can make use of (42) and (20) to calculate ju. From (42), we have du. RT dr , ^ cs 5(1 —r) OS and from (20) By integration, the chemical potential difference /x, relative to an arbitrary reference state Tq, sq, is /x = ^o + {T-To)i^^^+\^[\AT^T, + {T-To) ( ldA\ ds j: T' + B{T,s)dT'dT. (52) T„ We cannot evaluate the linear function ijlo + {T - To){diJLldT)o in (52) as both jLto and {d^ldT)o are arbitrary for any reference state To, so- Although present measurements of the lowering of vapour pressure are sufficiently accurate for most purposes where total vapour pressure is required, they are not consistent enough for use in (52). The fractional lowering of vapour pressure, r, is approximately a linear function of salinity. Thus, the ratio rjS SECT. 1] PHYSICAL PROPERTIES OF SEA-WATER 21 is nearly constant and provides an easy method for comparing various results reported in the literature. In the formula given by Sverdrup (1942) and the table by Miyake (1952), the ratio is constant. More recent measurements by Robinson (1954) and Arons and Kientzler (1954) indicate that the ratio in- creases slightly with salinity. A comparison of the ratios obtained by these 6.0 xlO r/S 5.0 4.0 Miyake (1952) Sverdrup (1942) Robinson (1954) Arons and Klenfzler (1954) 10 20 30 S %. 40 Fig. 6. Variation of the ratio r/S with salinity, S, where r is the fractional lowering of vapour pressure of sea-water as compared with pure water. r- .^-4 XlO 0 30 %o 35 7oo 5.0 ~ O □ 40 %<, r/S □ D D n 3 6 0 a o D A O A o D A O 4.8 1 1 1 1 1 1 1 1 10 20 30 40 Fig. 7. Variation of the ratio 7-/S with temperature, where r is the fractional lowering of vapour pressure of sea-water as compared with pure water, and S is salinity. The points are based on the determinations of Arons and Kientzler (1954). authors is given in Fig. 6. Miyake's (1952) table gives the greatest lowering of vapour pressure; about 10% greater than obtained by Arons and Kientzler (1954). The temperature dependence of rjS is small, as may be seen from Fig. 7. Measurements of freezing-point depression (Knudsen, 1903; Miyake, 1939) and boiling-point elevation (Miyake, 1939) are too restricted in temperature range to provide an adequate definition oi s dfi/ds. Further determinations of 22 FOFONOFF [chap. 1 colligative properties of sea- water, particularly vapour-pressure lowering, would be desirable to improve our knowledge of the chemical potential difference of sea-water. 5. The Non-equilibrium State If two adjacent phases of a thermodynamical system are not in equilibrium, they will tend towards an equilibrium by the exchange of heat and matter. Thus, the thermodynamics of non-equilibrium states deals primarily with the transport processes between adjacent phases. In the ocean, the transport pro- cesses include conduction of heat, diffusion of dissolved salts and gases, and rates of exchange of water with the vapour and solid phases. The related trans- port process of momentum diffusion due to viscosity of sea-water can also be included. The properties discussed in the previous section enable us to describe equili- bria between adjacent phases of a sea- water system. In addition, we can use these properties, together with the conservation laws of mass and energy, to determine the equilibrium state towards which adjacent phases, initially not in equilibrium, will proceed by the exchange of heat and matter. The final state can be found by applying the second law of thermodynamics, which requires the entropy of the final equilibrium state to be at a maximum. Thus, by examining the entropy of possible final states having the same total energy and mass, we can determine the distribution of temperature and salinity of the equilibrium. However, we cannot determine the rate at which the system approaches equilibrium. Therefore, the additional properties necessary to characterize non-equilibrium states are the rates of the transport processes. It is evident that entropy is produced by the system in reaching equilibrium. Hence, the concept of entropy production and particularly the rate of entropy production forms the central core of non-equilibrium theory. The theory of non-equilibrium states (more often referred to as the thermodynamics of irreversible processes) can be applied to heat conduction and salt diffusion in the ocean to develop the general equations describing these phenomena. The treatment is simplified by assuming the dissolved salts to behave as a single solute. 1 The basic concepts of the non-equilibrium theory are introduced by con- sidering an elementary sea-water system consisting of two homogeneous elements of sea-water in contact with each other along a portion of their boundary. We assume that these two adjacent phases are at a constant pressure jp and are initially characterized by temperatures d'l and ^2, salinities S\ and Sz, and masses mi and mz. We assume that the system is closed so that no salt or water is exchanged across the outer boundary and, also, that no heat is added 1 This assumption is an over-simplification. Some fractionization of sea-water con- stituents will occur because of differing diffusion rates for various ions. However, because of the lack of experimental studies of diffusion in sea-water and the fact that the treatment presented here can be readily extended to a system of several diffusing components, only the single-component system is discussed. SECT. 1] PHYSICAI. PROPERTIES OF SEA-WATER 23 to the system. Under these conditions, it can be shown that maximum total entropy is attained when the temperature and sahnity of the system become uniform. We characterize the final phase by the temperature d-, salinity s, and total mass m. As the two initial phases proceed towards equilibrium, both total mass and energy must be conserved. We can assume that there is no net exchange of mass between the two phases during diffusion in order to avoid convective terms in the conservation equations. ^ The conservation of total mass and of salt is expressed by Am = m — {mi + m2) = 0 (53) and Ams = ms — {miSi + m2S2) = 0. (54) Therefore, the change of salinity in each phase on reaching equilibrium is Asi = s — si = {m2lm){s2-si) As2 = S — S2 = —{'milm){s2 — si). As no heat, salt or water is exchanged across the outer boundary, the total internal energy me is changed only by the work done on the system by the pressure acting on the outer boundary. Thus, the change of internal energy is Am€ = me— {mi€i + 'm2€2) = —pAmv = —p[mv — {miVi + m2V2)], where Amv is the total change of volume of the system. Rearranging terms, remembering that pressure is assumed to be constant, we obtain A'm{€+pv) = Amh = 0, (55) which indicates that the total enthalpy of the system remains constant. Changes of enthalpy in each phase on reaching equilibrium are Ahi — h — hi = {7n2J'm){h2 — lii) Ah2 = h — h2 — — {milm){h2 — hi). We can compute the change of any other property, cp, from the general equation Anup = m A(p = m(p — {'mi(pi + m2(p2), (56) provided we know 99 as a function of salinity, pressure and enthalpy. If the initial phases differ only slightly from each other, we can express (56) in terms of the initial differences of enthalpy and salinity by expanding cpi and 992 in 1 A more general development of the theory, including convective terms, is given by De Groot (1952). The results are the same for the particular system considered here. 24 FOFONOFF [chap. 1 Taylor series about the final equilibrium values. By substituting the expansions into (56) and neglecting terms of third order and higher, we obtain Aw = 1m? l^r(A.-Ao^.2 Z|_„, _,,)(..-.,) + 1^(^.-0^' (67) = —ra\m%\1m^ h^ Lhs + L sh The flux of enthalpy can be written Fh = Fq — = Fg + hsFs, 8T \T ^ 0. F.. (65) (66) 26 FOFONOFF [chap. 1 where Fq is the flux of heat and hg the heat of transfer resulting from the flux of salt. By subtracting the heat of transfer from (63), we obtain the heat flux equation Fq = {Lhh — hsLsh)Xh+{Lhs — hsLss)Xs. (67) We can express the heat and salt fluxes in more familiar form by substituting ^ T2 dz ^ ~ ~d'z [tJ ~ fd^~T7hdz into (64) and (67). Collecting terms, we obtain (68) (69) az az Avhere T, ,d&^ds ^'= -^''Tz'^dz k = {llT^)[Lhh-hs{Lhs + Lsh) + hs Jqs = -^ -^ (Lhs-hsLss) Ds = {llT^){Lsh-JhLss) D = 1 8^1 From (65), we may conclude that the coefficients of heat conduction, k, and salt diffusion, D, are positive. If a pressure gradient exists within the system, the conditions of final equilibrium, and, hence, the transport processes, are altered slightly. The analysis can be carried out in the same way with the result that the effect of a pressure gradient can be taken into account by replacing dsjdz in (68) and (69) by Avhere ^'~\8pj,- 8f.l8s ^''^ The salinity gradient does not vanish in the equilibrium state in the presence of a pressure gradient. The equilibrium salinity gradient for zero salt flux under the hydrostatic pressure gradient is about 3%o to 4%o per 1000 decibars. Vertical gradients of salinity in the deep water of the ocean are usually less than the equilibrium gradient. Consequently, the effect of the pressure gradient in the ocean is to produce diffusion of salt in the direction of increasing salinity. None of the coefficients, k, D, Dgs or Ds^, has been measured for sea- water. Estimates of thermal conductivity, k, have been made by Kriimmel (1907) by SECT. 1] PHYSICAL PROPERTIES OF SEA-WATER 27 assuming that the thermal diifusivity , kIpCp, of sea-water is equal to that of pure water. The diffusivity D is usually estimated by assuming it to be equal to the diifusivity of sodium chloride in a solution of similar concentration to that of sea-water. Table V Transport Phenomena in Water at a Pressure of 1 atm (Montgomery, 1957) Name, symbol, units Pure Water O^C 20°C Sea-Water, (salinity 35 per mille) 0°C 20°C Dynamic viscosity, r), g cm~l sec~l = poise Thermal conductivity, k, watt cm-1 °C-l Kinematic viscosity, v — T^lp, cm2 sec~l Thermal diffusivity,*^ K = K/cpp, cm2 sec~i Diffusivity, D, cm^ sec~l : NaCl N2 O2 Prandtl number, Np = v/k 0.01787« 0.01002^' 0.00566C 0.00599c 0.01787 0.01004 0.00134 0.00143 0.0000074« 0.0000141/ O.OOOOlOes' 0.0000169? 0.000021'^ 13.3 7.0 0.01877« 0.01075a 0.00563^ 0.00596C 0.01826 0.01049 0.00139 0.00149 0.0000068« 0.0000129/ 13.1 7.0 a Yasuo Miyake and Masami Koizumi. The Measurement of the Viscosity Coefficient of Sea-Water. J. Mar. Res., 7, 63-66 (1948). Values taken from their Table I and reduced by 0.00007 poise to agree with Swindells et al. (Table III presents smoothed values for 0, 1, . . ., 30°C, chlorinity 0, 1, . . ., 20 per mille.) b J. F. Swindells, J. R. Coe, Jr., and T. B. Godfrey. Absolute Viscosity of Water at 20°C. J. Res. Nat. Bur. Standards, 48, 1-31 (1952). <^ L. Riedel. Die Warmeleitfahigkeit von wassrigen Losungen starker Elektrolyte. Chem.-Inst.-Technik, 33, 59-64 (1951). '^ Thermal diffusivity is also called thermometric conductivity. « Values for 0°C calculated from those at 20°C by use of temperature coefficient of L. W. Oholm. tJber die Hydrodiffusion der Elektrolyte. Z. physik. Chem., 50, 309-349 (1904). / A. R. Gordon. The Diffusion Constant of an Electrolyte, and Its Relation to Concentra- tion. J. Chem. Phys., 5, 522-526 (1937). Gordon used measurements by B. W. Clack. On the Study of Diffusion in Liquids by an Optical Method. Proc. Phys. Soc. London, 36, 313-335 (1924). R. H. Stokes. The Diffusion Coefficients of Eight Uni-univalent Electro- lytes in Aqueous Solution at 25°C. J. Amer. Chem. Soc, 72, 2243-2247 (1950). ? Gustav Tammann and Vitus Jessen. Uber die Diffusionskoeffizienten von Gasen in Wasser und ihre Temperaturabhangigkeit. Z. anorg. Chem., 179, 125-144 (1929). * Tor Carlson. The Diffusion of Oxygen in Water. J. Amer. Chem. Soc, 33, 1027-1032 (1911) ; I. M. Kolthoff and C. S. Miller. The Reduction of Oxygen at the Dropping Mercury Electrode. J. Amer. Chem. Soc, 63, 1013-1017 (1941) ; H. A. Laitenen and I. M. Kolthoff. Voltammetry with Stationary Microelectrodes of Platinum Wire. J. Phys. Chem., 45, 1061-1079 (1941). 28 FOFONOFF [chap. 1 The phenomenon of salt flux due to a temperature gradient is known as the Soret effect. It has not been studied for sea- water but has been observed in other solutions. Heat flow due to a concentration gradient (Dufour effect) has not been observed in solutions (De Groot, 1952). Other transport phenomena can be analysed by methods similar to those given here. Present knowledge of transport processes has been recently sum- marized by Montgomery (1957). His compilation of transport coefficients is reproduced in Table V. He points out that diflfusivities of nitrogen and oxygen in sea- water may be uncertain by as much as 15%. The effect of pressure on the transport coefficients is not available for sea-water. For pure water, thermal conductivity increases slightly with pressure ; viscosity decreases with pressure for temperatures under about 25°C and increases at higher temperatures (Dorsey, 1940, table 86). 6. Other Physical Properties of Sea-Water In addition to the properties which enter into the thermodynamical descrip- tion of a sea-water system, there are a number of other properties of sea-water that have been studied. Some of these, such as the optical and acoustical properties of sea-water and properties of sea ice, are discussed in subsequent sections and are, therefore, omitted here. Of the remaining properties, electrical conductivity is important because of the increasing use of conductivity bridges (salinometers) for determining the salinity of sea-water. The electrical conductivity in ohms per cubic centimetre has been measured by Thomas et al. (1934) and Bein et al. (1935) for a range of temperatures and salinities at atmospheric pressure. Their measurements show that conductivity increases with both temperature and salinity. Recently, Hamon (1958) carried out measurements on the effect of pressure on electrical conductivity. He used a single sample of sea- water of chlorinity 19.7 %„ (salinity 35.5 %o) and made measurements of the change of resistance of a cell filled with the sample and sealed in a pressure bomb. Measurements were made over the temperature range 0°C to 20°C and pressures up to 100 atm. Hamon found that conductivity decreased with pressure, the decrease being more rapid at lower temperatures. He gave the fractional decrease of conductivity of 1.50 x 10"^ db"i at 0.5°C and 0.82 x 10"^ db~i at 19.5°C, and estimated the accuracy of his results to be about ±5%. The pressure dependence of electrical conductivity is of particular import- ance in the development of conductivity bridges capable of measuring salinity in situ. References Agamat, E.-H., 1893. Memoires sur I'elasticite et la dilabilite des fluides jusqu'aux tres hautes pressions. Ann. Chi?n. Phys., 29, 505-574. Arons, A. B. and C. F. Kientzler, 1954. Vapor pressure of sea-salt solutions. Trans. Amer. Geophys. Un., 36, 722-728. SECT. 1] PHYSICAL PROPERTIES OF SEA-WATER 29 Bein, W., H.-G. Hirsekoni and L. MoUer, 1935. Konstantenbestimmungen des Meerwassers und Ergebnisse iiber W^asserkorper. Veroffentl. Inst. Meeresk., Univ. Berlin, N.F.S.A., Geog.-Naturwiss., H. 28, 1-240. Bjerknes, V. and J. W. Sandstrom, 1910. Dynamical meteorology and hydrography. Part I. Statics. Publ. Carnegie Inst. Wash., 88, 1-146. Carritt, D. E. and J. H. Carpenter, 1958. The composition of sea water and the salinity- chlorinity-density problems. Physical and chemical properties of sea water. Nat. Acad. Sci.-Nat. Hes. Council, Publ. 600, 67-86. Chappius, P., 1907. Dilatation de I'eau. Trav. Mem. Bur. Intern. Poids Mes., 13, D1-D40. Cox, R. A. and N. D. Smith, 1959. The specific heat of sea water. Proc. Roy. Soc. London, A252, 51-62. Craig, H., 1960. The thermodynamics of sea water. Proc. Nat. Acad. Sci., 46, 1221-1225. De Groot, S. R., 1952. Thermodynamics of Irreversible Processes. North Holland Publishing Co., Amsterdam ; Interscience Publishers, Inc., New York. Dietrich, G., 1957. Allgemeine Meereskunde. Gebriider Borntraeger, Berlin-Nikolassee. Dietrich, G. and J. Joseph, 1952. 3261. Physikalische Eigenschaften des Meerwasser. Landolt- Bornstein Zahlenwerte und Futiktionen. 6 Auflage, Bd. Ill Astronomic und Geophysik, 426—476. Springer-Verlag, Berlin. Dittmar, W., 1884. Report on researches into the composition of ocean water collected by H.M.S. Challenger 1873-76. Sci. Res. Voyage 'Challenger', 1873-76; Chem. Phys., 1, 1-211. Dorsey, N. E., 1940. Properties of ordinary water substance. Amer. Chem. Soc. Mono- graph, Ser. No. 81. Reinhold Publishing Corp., N.Y. Eckart, C, 1958. Properties of water. Part III. The equation of state of water and sea water at low temperatures and pressures. Amer. J. Sci., 256, 225-240. Ekman, V. W., 1908. Die Zusammendriickbarkeit des Meerwassers. Publ. Circ. Cons. Explor. Mer, No. 43, 1-47. Fofonoff, N. P., 1958. Interpretation of oceanographic measurements — thermodynamics. Physical and chemical properties of sea water. Nat. Acad. Sci.-Nat. Res. Council, Publ., 600, 38-66. Fofonoff, N. P. and C. Froese, 1958. Program for oceanographic computations and data processing on the electronic digital computer ALWAC III-E. PSW-I Programs for properties of sea water. Manuscript Rep. Ser. No. 27, Fish. Res. Bd. Can., Pac. Oceanogr. Group, Nanaimo, B.C. (Unpublished Manuscript.) Forsch, C, M. Knudsen and S. P. L. Sorensen, 1902. Berichte iiber die Konstantenbestim- mungen zur Aufstellung der Hydrographischen Tabellen. Kgl. Danske Videnskab. Selskabs, Skifter, Naturvidenskab math., Afdel. XII 1, 1-151. Guggenheim, E. A., 1950. Thermodynamics. An advanced treatment for chemists and physicists. North Holland Publishing Co., Amsterdam ; Interscience Publishers, Inc., New York. Hamon, B. V., 1958. Effect of pressure on conductivity of sea-water. J, Mar. Res., 16, 83-89. Helland-Hansen, B., 1912. The ocean waters. Intern. Rev. ges. Hydrobiol. Hydrogr., Suppl. Bd III, H. 2, 1-84. Hesselberg, Th. and H. U. Sverdrup, 1914. Die Stabilitatsverhaltnisse des Seewassers bei vertikalen Vershiebungen. Bergens Mus. Aarb. 1914-1915, No. 15, 1-17. Ivanov-Frantzkevich, G. N., 1953. Vertical stability of water layers as an important oceanological characteristic. (In Russian.) Trudy Inst. Okeanol., Akad. NaukS.S.S.R., 7, 91-110. Kjiudsen, M., 1901. Hydrographical Tables. G.E.C. Gad, Copenhagen; Williams and Norgate, London. ICnudsen, M., 1903. Gefrierpunkttabelle fiir Meerwasser. Publ. Circ. Cons. Explor. Mer, No. 5, 11-13. Kriimmel, O., 1907. Handbuch der Ozeanographie. J. Engelhorn, Stuttgart. 30 FOFONOFF [chap. 1 Kuwahara, S., 1939. Velocity of sound in sea water and calculation of the velocity for use in sonic sounding. Hydrog. Rev., 16, 123-140. Lafond, E. C, 1951. Processing oceanographic data. U.S. Navy Hydrog. Office, Publ. 614. Lyman, J. and R. H. Fleming, 1940. Composition of sea water. J. Mar. Res., 3, 134r-146. Matthews, D. J., 1923. Oceanography, Physical. A Dictionary of Applied Physics. Vol. Ill, 665-692. MacMillan and Co., London. Miyake, Y., 1939. Chemical studies of the western Pacific Ocean. III.' Freezing point, osmotic pressure, boiling point, and vapor pressure of sea water. J. Chem. Sac. Japan, 14, 58-62. Miyake, Y., 1939. Chemical studies of the western Pacific Ocean. IV. The refractive index of sea water. J. Chem. Soc. Japayi, 14, 239-242. Miyake, Y., 1952. A table of saturated vapour pressure of sea water. Oceanog. Mag., 4, 95-118. Montgomery, R. B., 1957. Oceanographic data. American Institute of Physics Handbook. Sec. 2, Mechanics, 115-124. McGraw-Hill Book Co., New York. Osborne, N. S., H. F. Stimson and D. C. Ginnings, 1939. Measurement of heat capacity and heat of vaporization of water in the range 0° to 100°C. J. Res. Nat. Bur. Standards, 23, 197-260. Richards, F. A., 1957. Some current aspects of chemical oceanography. Progress in Physics and Chemistry of the Earth, 2, 77-128. Pergamon Press, London, New York, Paris. Robinson, R. A., 1954. The vapour pressure and osmotic equivalence of sea water. J. Mar. Biol. Assoc. U.K., 33, 449-455. Rouch, J., 1946. L'eau de mer. Traite d' oceanographic physique, 2, Payot, Paris. Subov, N. N. and N. J. Chigrin, 1940. Oceanological tables. Hidro-Met-Izdat., Moscow. Sverdrup, H. U., M. W. Johnson and R. H. Fleming, 1942. The Oceans. Prentice-Hall, Inc., N.Y. Thiesen, M., 1897. Z. Instrumentenk., 17, 140. Thomas, B. D., T. G. Thompson and C. L. Utterback, 1934. The electrical conductivity of sea water. J. Cons. Explor. Mer, 9, 28-35. Thompson, T. G., 1932. The physical properties of sea water. Nat. Res. Council Bull., No. 85, 63-94. Thompson, T. G. and H. E. Wirth, 1931. The specific gravity of sea water at zero degrees in relation to the chlorinity. J. Cons. Explor. Mer, 6, 232-240. Thomson, W., 1857. On the alterations of temperature accompanying changes of pressure in fluids. Proc. Roy. Soc. London, 8, 564-569. Thoulet, J. and A. Chevallier, 1889. Sur la chaleur specifique de l'eau de mer a divers degres de dilution et de concentration. C. R. Acad. Sci. Paris, 108, 794-796. Tilton, L. W. and J. K. Taylor, 1937. Accurate representation of the refractivity and density of distilled water as a function of temperature. J. Res. Nat. Bur. Standards, 18, 205-214. Wilson, K. G. and A. B. Arons, 1955. Osmotic pressures of sea water solutions computed from experimental vapor pressure lowering. J. Mar. Res., 14, 195—198. 2. THE EQUATIONS OF MOTION OF SEA-WATER Carl Eckart 1. Introduction Prior to the twentieth century, hydrodynamics concerned itself almost exclusively with the motion of idealized fluids. The most common is the in- compressible, homogeneous fluid. The autobarotropic fluid (also called the polytropic fluid by some), whose density is a function only of its pressure, also received attention, notably in the theory of acoustics. More recently it has been studied in connection with steUar structure. These two classic lines of develop- ment were brought to a definitive stage by Rayleigh (1894) and Lamb (1932), In the twentieth century, it became clear that the hydrodynamics of real fluids cannot be developed independently of thermodynamics. The first to attempt a fusion of these two disciphnes was G. Jaumann (1911 ; 1918). In a series of elaborate studies, he and E. Lohr (1916; 1924; 1924a) made very important contributions to the field. Two other independent attempts were made by V. Bjerknes and his collaborators (1933) and by C. Eckart (1940). Since 1947, an extensive literature on the subject has appeared, consisting largely of applications to acoustics and physical chemistry. The most recent review of this work, by Meixner and Reik (1959), contains a complete biblio- graphy. A review of earlier work in geophysics was published in 1956 (Eckart and Ferris, 1956); this also contained a much simplified account of the results of the recent research in thermodynamics. The air and sea-water were treated as chemically pure substances, and the irreversible phenomena of viscosity and heat conduction were not treated. There are other more extensive and systematic accounts of the hydro- dynamics of chemically pure fluids, but these are usually written from the aerodynamical rather than the geophysical point of view. One such account is that of Oswatitsch (1956). These treatments devote special attention to the theory of shock waves and similar discontinuities. The present account will not consider such discontinuities. Since the velocities of sea-water under natural conditions are sub -sonic, shock waves in the common sense of the term are not to be expected. Other forms of discontinuous flow, especially those appro- priately described as highly oblique, very weak shocks, may be of importance, especially in turbulent motion. In the following pages, sea-water will be treated as a solution of a single compound ("salinity") in water, and the equations of motion will include viscosity, heat conduction and diffusion. The equations are a special case of those derived by Jaumann (1911 ; 1916) and Eckart (1940) and are the same (except for notation) as those given in Section 11 y of the article by Meixner and Reik (1959). The derivation of the equations will only be outlined here; more complete derivations will be found in the references just cited. The present objectives will be to introduce certain approximations and to transform the resulting equations into forms that exhibit their essential mathematical structure. [MS received December, 1959] 31 32 ECKART [chap. 2 2. Thermodynamics Sea-water will here be treated as a two-component solution, consisting of a solvent (water) and a single solute (salinity). The basic thermodynamic function, which specifies many of the properties of such a solution, is its internal energy (e erg/g) expressed as a function of its specific volume {v cm^/g), entropy {rj erg/g deg) and salinity (*S' g/g) : €{V, -q). Its pressure, p, absolute temperature, 6, and chemical potential, /x, are then calculable from the equations p = -deldv, e = deldri, /x = dejdS. (1) Other thermodynamic properties are defined in terms of the second deriva- tives of e ; with a view to later application, they will be written in terms of the small increments oi p . . . S: 8p = -X8v+Y87^ + K8S 86 = -YSv + Z8r] + L8S (2) S/x = -K8v + L8r] + M 8S where X, — Y, Z, —K, L and M are the second derivatives of e with respect to V, 7) and S. These derivatives can often be related to more commonplace thermodynamic coefficients. Thus X = p^c^ Y = p{y-l)la (3) z = eiCrs where p= l/v = density, a = coefficient of thermal expansion, c = velocity of sound, and Cvs, Cps = specific heats at constant salinity and constant volume or pressure, while y = CpslCrs = 1+a^cWICps. (4) Similarly commonplace expressions for K, L and M have not yet arisen in the literature. The definitions Hpr = KdjY (5) Hev = LdlZ (6) H,r = MOIL (7) give the three //'s the dimensions erg/g. For reasons that will become clear in Section 5 of this chapter, they will be called heats of diffusion, Hpv being the heat of diffusion at constant pressure and volume, etc. Methods for the measurement of the heats of diffusion or, alternatively, for their calculation from the measurement of related quantities have not yet been devised. Of the three, Hpv is the most urgently needed quantity since it enters into the final equations of motion (cf. equations (37), (43), and (47)). The third and higher derivatives of e may also be of importance, but will not be investigated here. SECT. 1] THE EQUATIONS OF MOTION OF SEA- WATER 33 3. Hydrodynamics The Elder equation may be written P^ + V;) + /5SrVx + pX2xu = V-/ (8) where u = velocity of the water, {DIDt) = {dldt)+u-V, gr = acceleration of gravity, gr^ = gravitational potential, -1^ = angular velocity of the earth's rotation, /^ = stress tensor of molecular viscosity, and the other symbols have the same meaning as above. The conservation of matter is expressed by ^ + V.(^u) = 0, (9) Avhich may be transformed into Dv „ -^^ = vV-u. (10) Let h erg/cm 2 sec be the heat flow and G erg/cm 3 sec the heat generated by irreversible processes such as friction, etc. ; then the rate of change of entropy is given by ^^= {vie){G-V.h); (11) G — Vh is the net accession of heat. Finally, let s g/cm^ sec be the flux of salinity ; then the conservation of salinity is given by DS ^=-.V.s. (12) — V • s is the net accession of salinity. 4. The Irreversible Processes It remains to give expressions for G, h and s. It can be shown (Jaumann, 1911, 1918; Eckart, 1940; Meixner and'Reik, 1959) that G = (/•V)-u-(l/^)h-V^-s-V/x. (13) The second law of thermodynamics states that the laws of viscosity, heat flow and diffusion are such that G is never negative. It is also necessary to specify the viscous stresses, /t, the heat flow, h, and the diffusion flux of salinity, s. The Stokes-Navier laws of viscosity are expressed by the equations /ixx = 2v'{du^l dz) + {v" - 2v' /3)V -u (14) /^xy = v'[{^f^xlSy) + {dUyl8x)] etc., 34 EOKABT [chap. 2 v' and v" being the two coefficients of viscosity. The heat generated by viscosity is then + K-§v')(V.u)2 (15) and will be positive provided that 3v" > 2v' > 0. Fourier's law of heat conduction is h = -kVB, (16) and hence the heat generated by the irreversible conduction of heat is -h-V0 = /c(V^)2 (17) and will be positive if /c> 0. Accepting the Stokes-Navier and Fourier laws, the simplest law of diffusion consistent with the uniform positiveness of 0 is s = -DVjM (18) where D>0. Fick's law of diffusion (see (20), below) is not consistent with this requirement, but is an approximation to (18) under certain conditions. This will now be proven. Equation (1) expresses the potential /a, as a function of v, 17 and S; it can also be expressed in terms of ^, ^ and S. Then (18) may be written Now, in the laboratory experiments that confirm Fick's law, conditions are controlled so that the pressure and temperature gradients are of no importance, while the effects of salinity gradients are maximized. Under these conditions s = -DsVS (20) where Ds = D{diildS) is Fick's coefficient of diffusion. In the laboratory it is easy to modify the experimental conditions so that thermal diffusion occurs ; in extreme cases, the effect of the temperature gradient dominates those of pressure and salinity, and then s = -DrV0 (21) where Dt = D{dfxldd) is the coefficient of thermal diffusion. Finally, in centri- fuge experiments (usually conducted only with solutes of high molecular weight but in principle applicable to any solute) the effect of the pressure gradient dominates, and s = -DpVp (22) where Dp= D{dfjLl8p). SECT. 1] THE EQUATIONS OF MOTION OF SEA-WATER 35 In the ocean, the effect of all three gradients is undoubtedly small, but it is not certain that the effect of any one is smaller than that of the others. This is one of many unsolved problems. It should also be noted that (18) is a consequence of the assumption that the Stokes-Navier and Fourier laws are correct. However, Onsager (1931; 1931a) has shown that concentration gradients may cause heat flow; consequently (16), and hence also (18), probably require modification. For present purposes, the precise forms of (14), (16) and (18) are not essential, but may become important in theories of oceanic turbulence, etc. One further consequence of the second law of thermodynamics is that, at thermodynamic equilibrium, where Vu, s and h vanish, both Vd and Vyu, must also vanish. If the equilibrium occurs in a gravitational field, the pressure gradient will not vanish. Since /x is constant, but a function of p, 6 and S, there must be salinity gradients to balance the pressure gradient. Actually, the ocean is far from thermodynamic equilibrium. This conclusion is, therefore, largely of theortical importance ; it raises the question of the quantitative difference between the actual state of the ocean and the nearest equilibrium state. 5. Transformation of the Equations Returning to (10), (11) and (12); these may be transformed into others that are sometimes more useful ; if they are multiplied by ~X, Y and K and then added term by term, (2), (6) and (7) result in ^ + pc2V.u = '^[G-V-h-Hj,,V.s]. (23) Hence, insofar as pressure changes are concerned, diffusion and heat conduc- tion have similar effects, their equivalence ratio being the heat of diffusion, Hpv, defined by (5). Using —Y,Z and L in the same way, it follows that ^ + ^^Vu = -^[G-V-h-HevV-sl (24) Lft a ply vS so that, in respect to temperature changes, the equivalence ratio of diffusion and heat conduction is the heat of diffusion at constant temperature and volume, (6). Finally, a similar procedure yields ^ + ?^//,„V.u = |5[6'-V.h-fl,„V.s] (25) for the rate of change of the potential of the fluid. If (8) is multiplied by u, and (10), (11), (12) by pp, p6 and p/x respectively, and the results are added, the energy equation is obtained in the form p^^[iu2 + e + Vx] + V-(^u) = u-(V./)-V-h-/x V-s + <5, (26) 3G EOKART fCHAP. 2 use having been made of the equation De Dv ^ Dri DS which is a consequence of (1). Inspection of (26) shows that, insofar as energy is concerned, the equivalence ratio of heat flow and diffusion is /z, the chemical potential of the solution. It is most unfortunate that no numerical values are known for any of these equiva- lence ratios. Until such values become available, it will be almost impossible to give any account of the convection and mixing processes in the ocean. Of the seven equations (10), (11), (12), (23), (24), (25) and (26), only three are independent; for the following purposes, (11), (12) and (23) are the most convenient independent set. 6. The Zeroth Approximation The hydrodynamical equations are too elaborate to be solved except by methods of successive approximation. One such systematic method will be outlined here. It has been developed elsewhere in more detail (Eckart, 1960). It begins with a rough approximation, variously known as the static or zeroth approximation. This assumes that V -/i, Vh — G^ and Vs are all zero, and that all time derivatives vanish. In that case, (9) reduces to Vp + pgVx = 0, (28) and (11), (12) and (13) are identically satisfied. The equations (16) and (18) would be additional restrictions on d and /x, but these will be ignored ; the equation (1) will be retained, however. The equation (28) has the solution P = Po{x)> P = Po'ix)l9 = Po(x)> (29) po being an arbitrary positive, monotonically decreasing function of the gravitational altitude, x, and the accent indicating differentiation with respect tox- If the entropy is eliminated between the equations (cf. (1)) = -— 6 = — dv bi] the result is the equation of state : p = iHv) = F{p,d,S). (30) When (29) is substituted into this, it becomes Po'{x) + 9nPo'{xhdo,So] = 0, (31) the subscript on ^o and So indicating that these quantities also refer only to the zeroth approximation. This approximation, therefore, involves only two SECT. 1] THE EQUATIONS OF MOTION OF SEA-WATER 37 arbitrary functions, which will be taken to be ^o(x) ^^^d Soix, ^' ^)^ 0 ^^^^ ^ being the horizontal co-ordinates, latitude and longitude. The equation (31) can then be solved for ^o : ^o(x, 0, A) = T[x, Soix, > A)]. (32) Having thus obtained ^o from po (or ^o) and >S^o, the spacial distribution of any other thermodynamic variable, such as 17, /x, a, Cvs, Hpv, s, h, can be obtained by using the equations of Section 2 ; these zeroth approximation values will be indicated by a subscript zero — e.g. 170, Hpvo, So — and are to be treated as empirically known functions of x, ^, A in all that follows. Relations between the gradients of the zero-order quantities can be calcu- lated from (2) ; thus V^o = -Xo Vvo + Yo Vrjo + Ko VSq. (33) From (29), it follows that V^o = -pog^x^ Vvo = - (po'/po^) Vx ; since Xq = poCo-, this becomes - ipog + po'c2) Vx = Fo Vr^o + A^o V*So. (34) Two quantities that will be useful in the work below are the Brunt-Vaisala frequency, N, and the adiabatic temperature gradient, Oa' ; these are defined by N'lg = -{po'lpo)-{gM (35) and dA' = Sr(yo-l)/«oCo-. (36) With these definitions, (34) becomes m Vx = ^^'[V^o + Hpvo VSoldol (37) a result that will be needed in the following sections. The Brunt-Vaisala frequency is a measure of the stability of the oceanic stratification. The accepted treatment of this problem is that of Hesselberg and Sverdrup (1914). They give numerical tables for the calculation of Oa' and E= — poN^lg, but these tables should be used with caution, since they are reported to contain numerical errors. 7. The First Approximation To obtain the next approximation, one sets p=po+pi, p = po + pi, u = ui, etc., and substitutes into (8), (11), (12) and (23). Every term is then expanded by Taylor's Theorem, and squares and products of quantities carrying the subscript 1 are neglected. An exception to this rule are the quantities y^, h and s, 38 ECKART [chap. 2 which were neglected for the zeroth approximation and are now to be calcu- lated using Uo = 0, p=po, P = po, etc. In this way, the equations -— i + vo Wpi + vi Vpo + Q X ui = 0, (38) ot ^ + ui.Vr,o = ^(G^o-V.ho), (39) ^ + ui-V^o= -voV-so, (40) ot %i + ui-V^o + /)oCo2V.Ui = [(yo-l)Mo][G^o-V.ho-i^proV-So], (41) ot are obtained. In these equations, ui, 171, pi and Si will be considered as the unknowns, while the quantity vi will be eliminated by using the first of equations (2), which is essentially pi = -XoVi+Yorji + KoSi; it then follows from (6) and (7) that vi Vpo = - pogvi Vx = [piiglpoco^) - ^^'(r?i + HpvoSildo)] Vx- (42) If qi is defined by the equation (analogous to (37)) N^qi = -eA'im + HpvoSildo), (43) it will have the dimension of length ; its interpretation will appear in Section 8. Equation (42) may then be written vi Vpo = [piiglpoco^) + N^i] Vx, (44) while (38) becomes ^ + voVpi + [?)i(g'/poCo2) + iV2gi]Vx + ftxui = 0. (45) ot It is convenient, now, to replace tji as an unknown by qi ; differentiating (43) with respect to t, and using (39) and (40), it becomes m^ = dA'ui-{V-qo + Hj,ro'^Soldo)-{voldo){Go-W-ho-Hp,oV-So), ot which (37) converts into %_ui.Vx = -{dA'lpoeoN^){Go-V-ho-Hp,oV-so). (46) ot The equations (45) and (46), together with (40) and (41), are the most con- venient independent set with which to work. They can be simplified by two additional definitions. SECT. 1] THE EQUATIONS OF MOTION OF SEA- WATER 39 8. Definitions Related to Convective Motion One first remarks that the quantity C = Go-V-ho- Hp,o V • Soldo (47) is known from the zeroth approximation ; it will be called the convectomotive force. The quantity Be = podoN^ie/ (48) is also known, and will be called the convective resistance. Then (41) and (46) become ^ - pogui • Vx + poco^ V • ui = poCoHN^Ig)CIRc, (49) and ^-ui-Vx = -01 Re. (60) It remains to justify the terminology used in these definitions. The quantity tv — Ui- V;\; is the vertical component of velocity. If steady solutions only are considered, so that 8ldt = 0, (50) becomes wRc = C, (51) which is reminiscent of Ohm's law. The convectomotive force is essentially the net accession of heat due to all causes, and w is the vertical component of the convective motion resulting from the heating of the fluid. Equation (49) then becomes V,.„..^(^VA)._^^. (52) lie \ g CoV po -fic Equations similar to (51) and (52) are discussed in Chapter 3 of Eckart (1960). On the other hand, if (7 = 0, (50) becomes |i = »; (53) in the absence of a convectomotive force, gi is thus the vertical displacement of the fluid to the accuracy of the present approximation. Finally, it is to be remarked that the unknown Si does not appear in (45), (49) and (50). These may, therefore, be solved without regard to (40), the solution of which is trivial once Ui has been determined from the other three equations. 40 ECKART [CHAP. 2 9. The Field Equations The equations of the last section can be brought into a form that makes it apparent that they are self-adjoint, and is otherwise more convenient to work with. For this purpose, the change of variables U = Ul(/^oCo)'/^ P = piipoco)'"''', Q = (Zi(poCo)'/s J = {CIEc){poc^y^, (54) is one possibility. Then (45), (49) and (50) become ^+co(VP+rp)+iy^2gvx + ^xU = o, (55) ot ^ + co(V . U - r . U) = coiN^^JIg), (56) ^ + VVx=J, (57) where r = (l/2poCo) V(poco) + iglco^) Vx (58) is a known vector. If (55) is multiplied by U • , (56) by P and (57) by Q, the result is the quadratic integral l_l(U2 + p2 + iV2g2) + v.(PU) = N^J{Plg + Qlco). (59) The quantity ^ = -L (U2 + P2 + A^2g2) erg/cm3 (60) 2co may be called the external energy density (Eckart and Ferris, 1956, and Eckart, 1960). However, it may not be considered to be an approximation for the true energy density, which is (cf. (26)) W = p{hu^+e + gx); the difference between W and E is discussed more fully in Eckart and Ferris (1956) and Eckart (1960). The equations (55), (56) and (57) are essentially identical with those derived in Eckart and Ferris (1956), and Onsager (1931) for a chemically pure fluid. There is only one difference : in the case of a pure fluid, the vector T is always vertical. The horizontal component of F is a consequence of possible horizontal gradients of the salinity. If the salinity is, to the zeroth approxima- tion, independent of latitude and longitude (cf. (32)) then the same will be true of all zero-order quantities, including poCo ; equation (58) shows that T will then SECT. 1] THE EQUATIONS OF MOTION OF SEA-WATER 41 have no horizontal component. In any case, the magnitude of T will be small, and it is probable that the approximation r = 0 is justifiable. If so, this would result in a marked simplification of the mathematics of the field equations. References Bjerknes, V. et al., 1933. Physikalische Hydrodynamik. Berlin. Eckart, C, 1940. The thermodynamics of irreversible processes. I. The simple fluid. Phys. Rev., 58, 267-275. Eckart, C, 1960. H ydrodynaniics of ocean and atmospheres. Eckart, C. and H. G. Ferris, 1956. Equations of motion of the ocean and atmosphere. Rev. Mod. Phys., 28, 48. Hesselberg, T. and H. V. Sverdrup, 1914. Der Stabilitatsverhaltnisse des Seewassers bei vertikalen Vershiebungen. Bergens Mus. Aarh. 1914—15, No. 15. Jaumann, G., 1911. Geschlossenes System physikalischer und chemischer Differential- gesetze. Sitz-her. Akad. Wiss. Wien., Math-yiat. KL, 120, Abt. 2A, 385. Jaumann, G., 1918. Physik der kontinuierlichen Medien. Denkschr. Akad. Wiss. Wien, Math.-nat. KL, 95, 461. Lamb, H., 1932. Hydrodynamics. Cambridge Univ. Press. Lohr, E., 1917. Entropieprinzip und geschlossenes Gleichungssystem. Denkschr. Akad. Wiss. Wien, Math.-nat. KL, 93, 339. Lohr, E., 1924. Zur Differentialform des Entropieprinzips. Denkschr. Akad. Wiss. Wien, Math.-naL KL, 99, 59-91. Lohr, E., 1924a. Des Entropieprinzip der Kontinuitatstheorie. Festschr. Deut. Tech. Hochschule Briinn, 1924, 176-187. Meixner, J. and H. G. Reik, 1959. Thermodynamik der irreversiblen Prozesse. Handb. Phys., 3, (2), 413. Onsager, L., 1931. Reciprocal relations in irreversible processes. I. Phys. Rev., 37, 405. Onsager, L., 1931a. Reciprocal relations in irreversible processes. II. Phys. Rev., 38, 2265. Oswa kitsch, K., 1956. Oas Dynamics (Translated by G. Kuerti). Academic Press, New York. Rayleigh, Lord, 1894. Theory of Sound. Macmillan, London. II. INTERCHANGE OF PROPERTIES BETWEEN SEA AND AIR 3. SMALL-SCALE INTERACTIONS E. L. Deacon and E. K. Webb A knowledge of the rates of exchange of energy between sea and atmosphere is a fundamental requirement for a proper understanding of the general circula- tion of atmosphere and ocean, and of the way in which air- and water-masses undergo modification of their characteristics in moving about the globe. One of the main objectives is to arrive at relationships which would permit computa- tion of the transfer rates of sensible heat, latent heat and momentum from the meteorological elements routinely recorded on board ship, that is, from air temperature and humidity, wind velocity and sea-surface temperature. Fortunately this is a practicable objective, whereas over land the complexities of surface conditions and topography are such that the simple meteorological elements are much less informative. The fact that the observations at sea have this great utility additional to their normal meteorological significance makes it necessary to strive for continued improvement in the standard of accuracy of observations at sea. The methods by which the transfers may be measured and the main results so far obtained are outlined in the following pages. The accent is placed here on small-scale interactions, i.e. events taking place in the layers from sea surface to some 10 or 15 m height. Although, as will be seen, useful progress has been made in recent years, much remains to be done to establish relationships of satisfactory precision. Considerations of the heat income available for evaporation and sensible- heat transfer applied to suitable large oceanic areas have given a foundation on which it has been possible to build useful first approximations to the geo- graphical distributions of these transfers and their seasonal variations (see e.g. Jacobs, 1951, 1951a). Further progress in this direction awaits more detailed knowledge of transfer processes as well as more accurate measurement of the radiant energy income of the sea. The work considered in the following pages has all been directed to the problem of steady-state wind and sea conditions ; unsteady conditions are not dealt with, not because they are unimportant, but because they have not as yet been studied. 1. General Considerations of Transfer The transfer of such an entity as heat from the sea surface to the atmosphere is effected in the first place by purely molecular diffusion through a film of air [MS received Juhj, 1960] 43 44 DEACON AND WEBB [CHAP. 3 of the order of a millimetre in thickness. Above this laminar layer there is a transitional layer in which turbulent exchange increases rapidly and becomes the over-riding process at the base of the region of fully developed turbulent flow. Much of the difficulty in attempting theoretical treatment of the problem of heat transfer between sea and air in relation to the temperature difference between the two media (and the corresponding problems for water vapour, momentum, etc.) resides in this complexity of regimes close to the surface, a complexity aggravated by the wave motion at the interface. That much of the resistance to transfer exists in the air layers close to the sea surface is exhibited by the fact that typically more than half of the difference in temperature between the sea and the air at, say, 5 m height takes place over the lowest few millimetres. For diffusion through the laminar layer the following molecular transfer coefficients of dimension L^T"! are operative for heat, momentum and water vapour respectively : Thermometric conductivity k = thermal conductivity -=-/)Cp Kinematic viscosity v = viscosity/p Diflfusivity of water vapour in air, D, where p is density and Cp is the specific heat at constant pressure. For air the values of these coefficients are given in Table I, reproduced from Montgomery's (1947) critical review of published values. Table I Density, Kinematic Viscosity and Thermometric Conductivity of Air and Diflfusivity of Water Vapour in Air at 1000 mb Temperature, Density, Kinematic Thermometric Diffusivity, °C g cm~3 X 10^ viscosity, cm2 sec-l conductivity, cm^ sec""l cm2 sec-1 -20 1.377 0.117 0.165 0.197 -10 1.325 0.126 0.177 0.211 0 1.276 0.135 0.189 0.226 10 1.231 0.144 0.202 0.241 20 1.189 0.153 0.215 0.257 30 1.150 0.162 0.228 0.273 40 1.113 0.172 0.242 0.289 The molecular transfer coefficients are inversely proportional to pressure at any given temperature. Ratios of quantities in Table I are very nearly iil- dependent of both temperature and, pressure and are as follows : v/k (Prandtl number) = 0.711 + 0.003 v/D = 0.596+0.008 k/D = 0.84+0.01 The specific heat, Cp, for dry air at 0°C has the value 1.004 x 10'^ erg g-i °C-i. SECT. 2] SMALL-SCALE INTERACTIONS 45 To measure the rate of transfer of some property between sea and atmosphere we are fortunately not under the necessity of making observations at the surface of separation itself. Except when conditions are changing rather rapidly, the rate of accumulation of the property in the lowest few metres of atmosphere is negligible in comparison with the flux traversing the layer to affect the atmos- phere above. The existence of this layer of very nearly constant flux makes it possible to obtain the requisite transfer rates by suitable measurements at convenient heights above the sea. A vertical turbulent flux of the entity *S^, whose measure per unit mass of air is s, is accompanied by two manifestations : (^ ) Turbulent fluctuations of S from its mean value s at the level of measure- ment, these fluctuations being correlated with the vertical component of eddy motion. {ii ) A vertical gradient of s. These phenomena provide the two main ways in which knowledge of the vertical fluxes may be derived, the flrst giving rise to the eddy- correlation technique and the second to the "aerodynamic" or "profile" methods. ^ A. The Eddy -Correlation Technique Considering unit area in a horizontal plane, the eddy flux {Fs) of S is given by the covariance of the fluctuations of s and of the product of vertical velocity component, w, and density, p. The relation is Fs = ipwYs' ~ pw's', (1) where the bar denotes the average over a period of time at a fixed point and primes denote instantaneous departures from w and s appropriate to the same period of time. In the particular case of momentum transfer the above expres- sion becomes the familiar Reynolds' formulation of the turbulent stress. Direct measurements of eddy flux can, therefore, be made using instruments of sufficiently rapid response. Fortunately it is not necessary to take full account of all the higher frequencies present in fluctuations of w and s. This is a consequence of the smaller scales of eddy motion being generated by the larger scales : in the progressive handing down of energy from large eddies to smaller ones, the anisotropy of the large eddies, which enables them to effect momentum or heat transfer, etc., is rather rapidly lost. This is well illustrated by the spectral distribution curves given by Panofsky and Deland (1959) of which Fig. 1 is an example. It is apparent that a nearly ten-fold greater responsiveness is needed to take account of all the vertical eddy energy (full line spectrum of Fig. 1) than is requisite for the flux measurement (broken 1 There is, however, another method, outhned by Deacon (1959), which relates the intensity of the higher frequencies in the turbulent fluctuations of velocity and tempera- ture to the respective fluxes. This approach, which has given promising results in pre- liminary work over land, may have considerable advantages for use at sea. 46 DEACON AND WEBB [chap. 3 curve). These and other studies at different heights over land surfaces show that, for near adiabatic or unstable lapse rates of temperature, it can be taken as an approximate rule that the highest frequency which need be catered for is given 4 6 10 20 40 60 100 200 400 cycles /hour Fig. 1. Atmospheric eddy spectra. (After Panofsky and Deland, 1959, Fig. 15.) O ■ Spectrum of vertical velocity (w) X Cospectrum of w and horizontal wind speed. The ordinates are spectral intensities in m^ sec~2 per unit range of In (frequency). by u/z, i.e. the mean wind speed divided by the observation height. The height dependence is a result of the observed fact that the scale of the eddy motion (vertical component) increases in proportion to height above the boundary. The eddy-correlation method of flux measurement has been treated at greater length than is appropriate here by Swinbank (1951) and also in Priest- ley's (1959) book on transfer. Types of instrumentation have been described by Swinbank {loc. cit.), Cramer and Record (1953), Krechmer (1954) and Mcllroy (1955). Mostly hot-wire anemometers have been used to sense the eddy velocity components and fine wire resistance elements the temperature fluctuations : with these, photographically recording mirror galvanometers are commonly employed. Over land the application of the eddy-correlation technique has already led to considerable advances in knowledge on the transfers of heat, water vapour and momentum and to the discovery of relationships of wide application. As yet only a few measurements have been made over the sea but, as will be seen later, even these few are helpful in resolving discrepancies arising from the other methods. More extensive application of the eddy-correlation method should be rewarding. B. Profile Methods To be able to deduce the vertical flux of S from the vertical gradient requires a knowledge of the eddy transfer coefficient, Ks, defined by Fs = pKs dsjdz. (2) SECT. 2] SMALL-SCALE INTERACTIONS 47 Here s, the amount oiS per unit mass of air, becomes u, the mean wind speed, ^ when the vertical transfer of horizontal momentum is under consideration and q, the specific humidity, for water-vapour transfer. For heat transfer the corresponding quantity is the product of specific heat, Cp, and temperature, but the appropriate vertical gradient must take account of the effect of the variation of pressure with height and the adiabatic heating or cooling ex- perienced by a volume of air on changing its level. So the heat flux, H, is related to vertical gradient of temperature, T, as follows : H = pCpKh{dTI8z + r), (3) where F is the dry adiabatic lapse rate of 0.98°C per 100 m. In the atmosphere near sea -level, it is a close approximation to take the term in brackets to be the vertical gradient of potential temperature, 6. The vertical gradient of potential temperature has a marked influence on the intensity of turbulence and three main states may be distinguished. (i) dd/dz positive; stable conditions of stratification: a volume of air dis- placed upwards becomes colder than its environment and tends to sink back to its original level ; turbulence is diminished. (m) dd/dz zero; neutral or adiabatic conditions.^ {Hi) dd/dz negative; unstable stratification with consequent increased turbulence. The coefficient Kg in equation (2) is not a constant as in molecular transfer ; it depends on the turbulent motions in the air and these vary markedly with height because, at the surface itself, the eddy motion must be greatly reduced. It also varies with the stabihty of the stratification and with the roughness of the surface, both factors influencing the intensity of the turbulence. As will be seen in due course, knowledge of Km, the transfer coefficient for momentum, can be obtained from certain aerodynamic relationships, but there is no corresponding way in which Kh and Ke, the transfer coefficients for heat and water vapour, can be derived. It has therefore been customary to assume, as the simplest hypothesis, that the K values for the various entities are equal. However, momentum, unlike other properties of the air flow, is affected by pressure forces so there is no a priori reason why Km should exactly equal the other transfer coefficients. Also heat transfer is affected by buoyancy forces because the turbulent temperature fluctuations are associated with density fluctuations in an expansible fluid such as air. This leads to the expectation, as shown by Priestley and Swinbank (1947), that Kh should differ from the transfer coefficient of a passive entity, i.e. one having no effect on the structure of turbulence. 1 From this point onward the bar denoting a time mean will be omitted except in those cases where confusion might result. 2 Taking account also of the buoyancy effects associated with water vapour stratifica- tion, the condition for neutrality becomes that the lapse-rate of virtual temperature shall be equal to F. This makes a significant difference when the Bowen ratio (see p. 83) is small. 48 DEACON AND WEBB [CHAP. 3 Measurement of the various fluxes by direct methods such as the eddy- correlation technique together with vertical gradient observations enables the equality or otherwise of the transfer coefficients to be put to the test. The trans- fer coefficients for momentum, Km, and for water vapour, Ke, have been measured simultaneously in two investigations using very different techniques by Rider (1954) and by Deacon and Swinbank (1958). These experiments over level grass surfaces gave the following ratios of KeJKm '■ Deacon and Swinbank 1.04+ 0.09 Rider 1.12+0.04. From this it appears that any difference there may be between the two co- efficients is quite small and, in practice, the assumption of equality oi Ke with Km is not likely to lead to serious error. The comparison of Kh and Km has also been made in a similar manner by Swinbank (1955), who found Kh to exceed Km under convectively unstable conditions while the reverse is true for stable conditions. It appears that the magnitude of the difference depends not only on the stability conditions but increases with height above the surface. Under strongly unstable conditions Kh exceeds Km by some 30% at the relatively small height of 1.5 m above the surface, so it may often not be satisfactory to neglect the difference between these two transfer coefficients. In those cases for which Ks = Km, the result of dividing equation (2) by the corresponding flux equation for momentum is Fs^ _ dsjdz T du/dz where t is the shearing stress. Furthermore the profiles of S and of wind would be similar allowing differences to replace differentials giving Fs = -r{S2-Si)l{U2-Ui), (4) in which the subscripts denote values at heights zi and 22- If, therefore, t can be derived from the wind-profile, as dealt with later, then the flux oi S can be obtained from measurements of s at two or more heights. The aim of being able to relate the fluxes to the more easily measured differences between sea surface and air requires knowledge of the appropriate values of the coefficient fa in equations of the form : Flux Oi S = fap{Ss - Sa)Ua. (5) Here subscripts s and a denote values at the sea surface and at some convenient observing point at height a respectively. The approach via this relationship is commonly called the hulk aerodynamic method to distinguish it from that based on equation (2). For the various transfers equation (5) becomes: Momentum r = CapUa} (6) Heat H = hapCp{ds-6a)ua (7) Water vapour E — dap{qs — qa)ua- (8) SECT. 2] SMALL-SCALE INTERACTIONS 49 where 6 denotes potential temperature and q specific humidity. In (6), Ca is known as either the friction or drag coefficient. For heat transfer ha in (7) is the Stanton number, while, for evaporation, it would be appropriate to call da the Dalton number as Dalton first proposed a relationship of the form of equation (8). In dealing with heat transfer the well-known Reynolds analogy treatment is to take the Stanton number equal to the drag coefficient, a procedure which has been found to give approximately correct results in some problems but to be quite considerably in error in others. The corresponding treatment for evaporation is to take the Dalton number equal to the drag coefficient. i It is readily seen from equations (4) and (5) that these treatments would only be accurate if equation (4) applied not only to the fully turbulent layers, but right down to the sea surface itself. Such extension ignores the fact that in the transitional and laminar layers the various transfer coefficients are not equal. Furthermore, in the layer between the troughs and crests of the waves, the momentum-transfer mechanism must differ from that of the other entities owing to the operation of the pressure forces which give rise to the form drag of the waves. The various theoretical attempts that have been made to take account of these effects are discussed later. 2. Momentum Transfer and the Wind-Profile A. The Neutral Wind-Profile Studies both in the laboratory and in the field have established for steady flow over rigid surfaces and neutral conditions of stability a simple relation- ship between the vertical gradient of fluid speed and the shearing stress in the fully turbulent layers. This is 0'll> ^d; / f\\ dz kz in which u = mean speed at height z, u^ = {rlpy''', t = shearing stress, and /) = fluid density. As the shearing stress is the vertical flux of momentum in the direction of the mean flow : TJp = —u'w' and so u^^ has the dimensions of a velocity and is called the friction velocity. The Karman constant, k, has a value of around 0.4; 0.41 according to Clauser (1956) is the best value fitting the various laboratory investigations and, as the work over natural surfaces (Rider, 1954; Deacon, 1955) gives a closely similar value, this is used hereafter. The eddy viscosity, Km, is U:^^l{duldz) and, for neutral conditions, it follows from (9) that Km = ku^z in the fully turbulent layers. 1 In obtaining the evaporation from measurements of the heat available for the two processes, evaporation and sensible-heat transfer together, the apportionment is custo- marily made by assuming equality of Dalton and Stanton numbers. This is the Bowen ratio method applied in such heat-balance analyses as those of Jacobs (1951, 1951a) and Budyko et al. (1954). 3- s. I 50 DEACON AND WEBB [CHAP. 3 Equation (9) applies to fiow over both rough and polished surfaces provided z is sufficiently large for the flow at that level to be fully turbulent and un- affected by the more complex conditions existing very close to the surface. Integration of (9) gives for the layer of constant stress the logarithmic law u = (w^/A;) ln(2/c) (10) in which c is the integration constant. An aerodynamically smooth surface is one for which the roughness elements are small compared with the thickness of the laminar layer so the only drag is that due to skin friction. In the laminar layer of molecular transfer the velocity profile is linear as far as about z = 5vju^, while between 5 and 30 multiples of vju^ conditions are transitional between laminar and fully turbulent flow. In the fully turbulent regime the value of c is such that (10) becomes u — {u^jk) \n{Au^zlv) (11) and with A; = 0.41 the appropriate value of -4 is 7.5 according to Clauser (1956). For rough surfaces the viscous drag at the surface is negligible in comparison with the form drag of the roughness elements and (10) is then found to become u = {u^lk) ln(2/2o), (12) in which zq, the roughness parameter, is a constant for a given uniformly roughened surface under neutral stability conditions. Typical values of zq for land surfaces range mainly between 0.1 cm for fairly smooth snow to 100 cm for scrub -covered country. B. The Non-Neutral Profile When there exists a temperature contrast between surface and air, the effects of thermal stratification and buoyancy have also to be considered, so that in relating the shear stress to the wind shear the counterpart of (9) must contain at least one more basic parameter. This involves consideration of the effects of a heat flux, H, on the production of turbulent kinetic energy. Produc- tion by the shear stress is at the rate t dujdz per unit volume. The working of the buoyancy forces associated with a heat flux introduces an additional term {gH)ICpT, where g is the acceleration of gravity and T air temperature : this augments the energy supply to the turbulence under unstable conditions but decreases it with stable stratification. The dimensionless ratio of these terms is the flux form of the Richardson number, viz. Bf = -gHlicpTrduldz) (13) in which the minus sign is a matter of convenience to make the signs of Bf and of the vertical gradient of potential temperature the same under given conditions. Introducing proportionality of fluxes to gradients in Bf leads to the more familiar gradient form of the Richardson number _ g dd/dz ^' ~ Tiduldz)^ ^ ^ in which 6 is potential temperature. SECT. 2] SMALL-SCALE INTERACTIONS 51 Both Rf and Ri increase numerically with height and roughly in direct proportion. The effect of a heat flux on the flow is, therefore, small near the surface but increases rapidly with height, and this has been confirmed observa- tionally. This has the consequence that, to specify the overall stability condi- tions on any given occasion, it is necessary to give the value of Rf or Ri at some definite height. Difficulties can arise in comparison of results when different reference heights are used and specification in terms of only one quantity, independent of height, would be advantageous. With known values of friction velocity and heat flux this can be done in terms of the stability length, L, introduced by Obukhov (1946). Its definition is L = -pCpTu^^lkgH (15) and zjL is another form of flux Richardson number. For near neutral conditions, for which Kh = Km is a good approximation, the three parameters Rf, Ri and zjL are very nearly equal. i The choice of stability parameter in any given problem is one of convenience; there is no real physical difference between them. The generalization of equation (9) in terms of Ri is du U4. „ „ and/(i?^) must tend to unity as Ri—^0. In Fig. 2 observational evidence as to the dependence on Ri of u^l{z dujdz), denoted by Km, is given for those researches where the shearing stresses were measured by direct methods. The eddy-correlation technique was used in one study (Swinbank, 1955; Deacon, 1959)2 and the measurement of the drag on suitably mounted samples of the surface in the case of Rider's (1954) work. This shows that as yet the functional form of the stability dependence in (16) is but poorly known and further work of this sort is to be desired. Rossby and Montgomery (1935), by a generalization of Prandtl's mixing length approach, arrived at I = 11 (!+.«)■/■ (17) for stable stratification, a being a constant to be determined by observation. Various wind-profile studies have given values of o- ranging between 6 and 12 with an average of around 9. The dotted Curve in Fig. 2 shows that the Rossby- Montgomery formula with cr = 9 has about the right slope to fit the near-neutral 1 The relationship between them in the general case is Ri(KH/KM) = Rf ^ {KMlk){z/L), where Km = u^I{z dujdz). 2 The values of m* have been increased by 10% to correct for under -measurement caused by high-frequency cut-off of the eddy spectrum. 52 DEACON AND WEBB [chap. 3 observations. However, for these conditions, (17) approximates to the more convenient expression (18) which is equivalent to the relationship proposed by Monin and Obukhov (1954).! The full line in Fig. 2 is for a = 4.5 but this appears to apply over the range of Bi from —0.05 to +0.15 at the most. Fig. 2. Observations of Km = u*I{z Sujdz) related to the Richardson number Ri; over grassland. O Results from Swinbank's eddy correlation apparatus. A Rider's (1954) results by the isolated test surface method. The vertical lines indicate the standard errors of the means. Equation (18) with a = 4.5 Rossby-Montgomery formula (17) with ct = 9 Ellison's equation (20) with y= 18. A convenient way of applying (18) in the evaluation of u^ from wind- profiles obtained under stability conditions falling within the prescribed limits is as follows. Assuming that Kh = Km under such conditions, the Richardson 1 Monin and Obukhov used, however, the stabihty parameter zjL. For the near-neutral conditions to which (18) refers, zjL closely approximates to Ri. From wind- and tempera- ture-profiles over the steppes, Monin and Obukhov found a value of a = 0.6, which is very much smaller than the value of ~4.5 corresponding to ct = 9. This has been shown by Taylor (1960) to be largely a result of Monin and Obukhov having fitted their relationship over a much wider stability range than appears to be justifiable. SECT. 2] SMALL-SCALE INTERACTIONS 53 number can be written in terms of wind and potential temperature differences between two levels, a and b, i.e. T du/dz Ua — Ub As a sufficiently good approximation for use in the correction term of (18) we may write duldz = u^j'kz so that it becomes du u^ 8z k 1 ^ akg{da-db) Z U^T{Ua — Ub) which yields on integration between two levels zi and Z2, and division through- out by ln(z2/zi) : U2-U1 _ u^ ag{0a-6b){z2-zi) ln(z2/2i) k T{ua-Ub)\n{z2lzi) If now the values of the L.H.S. of equation (19) are plotted against (22 — 2:1)/ ln(22/2i) and a straight line of slope a{glT){6a — 6b)l{ua — Ub) drawn to fit the points as well as possible, then the intercept, w^^/A:, gives the required value of m^. If air temperatures at the requisite two heights are not available, then approxi- mate allowance for stability may be made by taking level b to be the sea surface. This approximation also leads to the following air-sea temperature differences between which (18) and (19) may be applied when dealing with profile data extending up to 10 m. Wind speed, m/sec 5 10 15 Air-sea temperature differences, °C -0.4to+1.2 -1.5to+4.o - 3 to + 9 If the greater height is 5 m instead of 10 m then the temperature difference ranges are doubled. At large instability, the convective action sets the pattern to the air motion and Ellison (1957) suggests that it might then be expected that Kh/Km should approach a constant value. Should this be so, the form of the wind-profile would be similar to that of potential temperature. Now dimensional analysis (Priestley, 1954, 1959) shows that under these conditions of free convection ddldz oc 2-4/3 and so this line of reasoning would indicate a similar inverse four-thirds power law for the vertical wind gradient. To provide a formulation which would give this result at large instability and approximate to (18) under near neutral conditions, Ellison (loc. cit.) proposed a relationship in terms of the flux Richardson number dujdz = {uJkz){\-yRf)-y* (20) 54 DEACON AND WEBB [CHAP. 3 and this for y= 18 is shown in Fig. 2. Panofsky, Blackadar and McVehil (1960) show that a large number of wind-profile data for unstable conditions are well represented by a form of this equation in which y' Ri replaces yRf — a relation- ship first proposed on theoretical grounds by Olukhov (1946). The value of y' = 18 gave the best fit and is consistent with a = 4.5 in equation (18) for near- neutral conditions (i.e. a = y'l4). This formula of Panofsky et al. gives a curve similar to that of Ellison in Fig. 2 but some 5% lower at Ri= —0.4. C. Observational Considerations To obtain satisfactory exposures for accurate profile observations over the sea is considerably more difficult than over the land, and the smallness of the vertical gradients makes considerable demands on technique. In much of the earlier work with rafts and small boats, it has been a matter of doubt as to the extent to which the profiles have been influenced by such factors as distortion of the wind-field by the boat, etc., motion of the anemometers and some un- certainty as to the datum plane for height measurement — a factor of some importance with wind speeds measured at relatively small heights above the water. Work carried out at coastal sites using rafts or masts in shallow water near sand spits, etc. is also somewhat suspect as, with such sites, the character of the water surface is not representative of open sea conditions nor is it uniform but varies from point to point owing to variation in depth of water and to refraction and reflection of waves. In such circumstances there may be ap- preciable departures from the desired equilibrium wind-profile from which the shearing stress may be determined. As a result of observations over shallow water, Bruch (1940) obtained wind- profiles over the height interval 1 8-200 cm but the parts of the profiles below the 64-cm level were difficult to reconcile with the upper portions. This may have been an exposure effect of the kind mentioned above but there is, of course, the possibility that flow over a water surface disturbed by waves is sufficiently different from that over a rigid surface for the logarithmic law not to hold at all heights over water; see e.g. Stewart (1981). However, Roll (1948), working with what would appear to have been a more uniform expanse of shallow water over the Neuwerk shoals, found profiles following rather closely the logarithmic form from close above the wave tops up to 2 m, his highest level, as may be seen from Fig. 3, but the waves were only around 10 cm high. In another paper, Roll (1949) has shown how a number of earlier observa- tions may have been vitiated by the disturbance to the wind-field caused by a ship's hull, even with anemometers mounted on the bowsprit. To secure profiles characteristic of the Open sea, Deacon, Sheppard and Webb (1956) fitted an anemometer mast to the outboard end of the jib-boom of a small diesel-schooner and were able to investigate, and to correct for, the extent to which the ship, head to wind, influenced the wind-profile. This was done by taking profiles at the lowest possible speed consistent with maintaining SECT. 2] SMALL-SCALE INTERACTIONS 55 steerage way and at a liigher s])eed alternately. In the absence of a "hull effect" the increased ship speed should produce equal increments to the anemometer speeds at all heights ; insofar as this was not the case, hull effect corrections could be calculated. This showed that, even at 5 m forward of the stem of this small vessel, the wind speed was decreased by some 2 or 3% at the lower heights. In view of the small magnitude of the wind gradient, corrections of 1000 4 6 Wind speed , m/sec Fig. 3. Wind-profiles over shallow water. (After Roll, 1948, Fig. 5.) several per cent are undesirably large and undoubtedly detract from the accuracy obtainable with this method. The work supported Roll's (1949) suspicions of earlier profiles involving measurements on ships. A special buoy with a 9-m light-alloy mast has been devised by Brocks (1959) and this appears to be a considerable advance on anything used previously. The general arrangement is shown in Fig. 4. It has guiding fins to maintain orientation with respect to the wind and damping plates to reduce vertical motion. The buoy is towed to the observation area where, by means of cables or radio signals, the measurements are transmitted to recording instruments on board ship. With the ship drifting, the buoy automatically takes station to windward so that undisturbed observations are secured. In waves 3 m high the maximum inclination of the mast was only 10° from the vertical. Some speci- men profiles obtained over the North Sea by Brocks are shown in Fig. 5. Where rolling cannot be eliminated entirely, measurement of the root-mean- square rolling amplitude, a, and period, T, permits correction to be made to the recorded wind speed, Ur, given by a cup anemometer using the following formula (Deacon et at., 1956) UrjUc = l+[iThal{TUr)]^, 56 DEACON AND WEBB [chap. 3 fHond winch Weight , 500 Kg Fig. 4. Buoy-mast for profile observations by Brocks. 20.0 - 4 5 6 7 8 9 10 m/sec Fig. 5. Specimen wind-profiles obtained by Brocks over the Noith Sea. a. 13.8.59 11.30-11.45; Zl^ = 0.0°C b. „ 15.00-15.15; „ 0.7°C c. 11.8.59 15.45-16.00; „ 0.2°C A6 — air-sea potential temperature difference. Observations at heights above 9 m taken on ship's mast. SECT. 21 SMALL-SCALE INTERACTIONS 57 where Uc is the corrected wind speed and h the height of the anemometer above the axis of roll. 3. Drag Coefficients of the Sea Surface A. From Wind-Profiles Values of the drag coefficient Cio = rl{puio^) (the subscript 10 denoting that the wind speed is for 10 m height) are collected together in Fig. 6 from the results under neutral or near-neutral conditions obtained in those studies made with the more satisfactory exposures. The broken line for an aerodynamically smooth surface is that indicated by equation (11) from laboratory results for 0.003 0.002 - a> 0.001 - I I I I 1 1 1 rF $ T S_,,,J 25 m/sec, and any variation there may be with wind speed in this range is obscured by experimental scatter. 0.003 0.002 0.001 20 Wind speed , m/sec 30 Fig. 7. Drag coefficient, ciq, from surface-tilt observation related to wind speed. O Gulf of Bothnia • Ringkobing Fiord A Lake Erie. D. The Dependence of the Drag Coefficient on Wind Speed That the extent of the variation of the drag coefficient with wind speed is still rather uncertain is evident from Figs. 6 and 7. Earlier suggestions that the drag coefficient increases rather abruptly by 100% or more at about 7 m/sec — the wind speed at which white caps start to become numerous — is not supported by the data shown in Fig. 6. A comparatively slow increase of drag coefficient with wind speed is now seen to be much more probable and the simple linear relationship for near-neutral conditions, Cio = (1.00 + 0.07i^io)x 10-3, (22) in which uiq is in m/sec, is shown in Fig. 6. At 20 m/sec this would give Cio = 0.0024, which agrees reasonably well with values from sea-slope data. Sheppard (1958) has also proposed a linear relationship but with a rather greater slope. His proposal for the range 1-20 m/sec is cio = (0.80-f 0.114wio)x 10-3. At wind speeds from 2-9 m/sec this differs from (22) by no more than 10% at any point. At 20 m/sec Sheppard's formula gives Cio = 0.0031, which appears to be somewhat large. 62 DEACON AND WEBB [chap. 3 It is surprising at first sight that the very marked increase in height of the waves with wind si)eed is not accompanied by a more marked rise of drag co- efficient than shown by the observations. This is largely attributable to the fact that the components of large amplitude, in a fully developed sea, travel down- wind with a speed not much less than the wind speed. Francis (1951) has studied the drag coefficient of a water surface in wind- tunnel experiments and found it to increase nearly linearly with wind speed. When he extrapolated his air speeds at 10 cm height to the 10 m anemometer height used for the sea, his drag coefficients were of very similar magnitude to those shown in Fig. 6. Francis considered that this might well show that the mechanism for drag is not controlled by the larger waves but largely by the tiny wind ripples. The form drag of a spectrum of waves travelling on a water surface, each component at its appropriate speed, has been treated theoretically by Munk 0.5 - X\' 1 /'IX 1 1 1 - \ Y \ - form /\. / _ drag \J \. elevation''^ \ v: 1 1 1 1 1 1 1 0.5 1.0 c/u 1.5 2.0 Fig. 8. Contributions to the mean-square elevation, mean-square slope and form drag by waves of different wave age, c/m. c = phase velocity u = wind speed (1955). His analysis is based on a model first considered by H. Jeffreys in which the air flow over a wave separates near the crest with the formation of a lee eddy. As a result the main air current, instead of flowing steadily down into the troughs and over the crests, merely slides over each crest and impinges on the next wave at some point intermediate between the trough and the crest. By analogy with the thrust of a current against a plane lamina inclined to the direction of flow, the reaction^ is assumed to be related to wind speed and water slope by p = spu^ dhjdx, where s, the "sheltering coefficient" of Jeffreys, is of the order of 0.05 to judge from wind-tunnel experiments on solid corrugated surfaces. Munk's analysis brings out clearly the importance of the high frequency components of the wave motion which contribute largely to the mean-square slope and even more largely to the form drag. Fig. 8 gives the contributions to mean-square elevation of the SECT. 2] SMALL-SCALE INTERACTIONS 63 water surface, mean-square slope and form drag by waves of different "wave age", i.e. ratio of phase velocity, c, to wind velocity, u. This shows that mean- square elevation, to which the low-frequency waves make by far the largest contribution, is of very little significance in considering the aerodynamic roughness of the sea surface. On Munk's theory the drag of the sea surface should not vary greatly with fetch of the wind over the water, as the high-frequency wave motion reaches the value for a fully developed sea much more rapidly than the low-frequency components dominating the mean-square elevation. Cox and Munk (1954, 1954a) developed an interesting method for obtaining the slope statistics of the sea surface from aerial photographs of the sun's glitter on the water. Visual observation of the glitter had long been known as a way of determining the maximum water slopes, but Cox and Munk, by means of 0.05 0.04 Q- 0.03 - 0.02 0.01 4 6 8 10 Wind speed , m/sec Fig. 9. Mean-square slope of the sea surface related to the wind speed at 12 m height. (After Cox and Munk, 1954a, Fig. 5.) O Natural sea surface Oil slick on sea densitometric measurements of suitable out-of-focus negatives, were able to derive the frequency distribution of slope and the mean-square slope. The fre- quency distribution they found to depart only slightly from the Gaussian form and this indicates that a whole spectrum of wave motions is present. The variation of mean-square water slope (up-and-down wind component) with wind speed resulting from their trials near Hawaii is shown in Fig. 9. For the cross-wind direction the mean -square slope exhibited a similar variation and averaged about 70% of the up-and-down wind value. Conditions of slight to moderate stability prevailed, the sea being from 0° to 2.5°C colder than the air (mean 1.2°C). Munk (1955) incorporated the linear variation of mean-square slope with wind speed shown in Fig. 9 in his form-drag analysis and found that this com- ponent of the total drag of the sea surface should vary as u^. As the skin- friction component should vary approximately as u'^, the final prediction is a 64 DEACON AND WEBB [CHAP. 3 drag coefficient increasing linearly with wind speed and starting from the smooth-surface value at very low wind speeds. As far as may be judged from Fig. 6. this prediction accords with observation and rough quantitative agree- ment would be secured with a sheltering coefficient, s. of rather less than 0.01. A value ,5 = 0.01 was estimated by Sverdrup and Munk (1947) from observed rates of wave growth. Neumann (1956) suggests that the sheltering effect of the large waves may be rather different from that visualized by Cox and Munk and puts forward some evidence for a drag coefficient decreasing with wind speed (see also Neumann. 1951). This mainly arises from giving undue weight to sea-slope data for light winds (Deacon, 1957a). He argues further that wind-profile results are un- reliable because cup anemometers at low heights over the sea would be sub- jected to considerable fluctuations in wind speed with the passing of the waves and, because of their inertia, record too high a mean wind speed : this would give spuriously low wind gradients and correspondingly low drag coefficients, particularly in light winds for which the over-estimation effect is most marked. Much of the data given in Fig. 6 is, however, almost certainly free from ap- preciable error of this sort. In the studies by Johnson (1927) and Deacon, Sheppard and Webb (1956) and also by Brocks (I960), the heights involved were too great and/or the inertia of the anemometers too small for appreciable over-estimation error. Takahashi (1958) used thermocouple anemometers not subject to the inertia effect, and the eddy-correlation values of Mcllroy (1955) and Vinogradova (1959) give support to the wind-profile results by a com- pletely diiferent technique. E. Dejjejidence of Drag on Atmospheric Stability The state of the sea surface and the drag depend on the wind speed close above the surface. If it were practicable always to measure the wind speed at some small height, such as 1 m, then the drag coefficients would be found very nearly independent of the air-sea temperature difference. With winds measured at greater heights, such as 10 m, the effect of atmospheric stability is no longer negligible. For a given wind speed at 1 m. the 10 m speed wall be greater under stable than under unstable conditions with the consequence that the drag co- efficient, cio- should be smaller for stable conditions and larger for unstable. As yet there is only qualitative evidence on the magnitude of the stability effect over the sea. Darbyshire and Darbyshire (1955) in their analysis of Lough Neagh tilt observations found evidence for a quite marked effect of stability but, as their wind speeds were measured over the land and not over the lake, the interpretation of their result is obscured. That the stability effect is of significant magnitude is shown by the results of studies of the state of the sea in relation to wind speed and air-sea temperature difference. The basic data for these studies were wave-height observations at measured wind speeds from weather ships. Roll (1952) found that, for a given wind speed, the mean wave height when the sea temperature is 6.7^C above air SECT. 2] s:mall-scale interactions 65 temperature is 22% greater than when air and sea temperatures are equal, and Brown (1953) and Fleagle (1956) have reported effects of similar magnitude. When applying drag coefficients to the calculation of surface stresses, the distinction between measured and estimated wind strengths should be re- membered. The values estimated from the appearance of the sea surface are virtually estimates of surface stress or of wind speed close to the surface, and the conversion from Beaufort scale to wind speed at, say, 10 m height should properly depend on the stability of the air (Roll, 1953/54). But, using the customary conversion which neglects any stability effect, the resulting wind speeds are most appropriately used with neutral drag coefficient values. F. Effect of Slicks The sun glitter measurements of mean-square slope shown in Fig. 9 include a number taken after the application of oil to the sea surface to produce artificial slicks some 700 m square. The oil film suppresses the higher-frequency wave motions, which make a large contribution to the mean-square slope. That the supf)ression of the riijples and wavelets has a marked effect on the stress is shown by Van Dorn's (1953) results using the surface-tilt method applied to an artificial pond 250 m long. Stresses measured in this way, with 1 I 1 I 1 - = j^^ >-i^ ^ .^^' ' s^° 1 1 1 1 - 4 6 8 10 12 Wind speed , m/sec Fig. 10. Van Dorn's measurements of stress on a model-yacht pond. O Natural water surface • Detergent applied a During heavy rain and without a surface film, are shown ip Fig. 10. In this case the surface film was produced by the application of detergent dispensed as a powder at the up- wind end of the pond. It seems probable, as suggested by Munk (1955), that we may take Van Dorn's results as indicating the relative magnitude of the skin-friction and form-drag contributions to the total surface stress. The damping of capillary ripples by surface films has been studied by Vines (1961). The natural films present on the surface of the sea as a result of biological activity may be a factor of some importance influencing the state of the sea. 66 DEACON AND WEBB [CHAP. 3 G. Effect of Rainfall Five of Van Dorn's measurements shown in Fig. 10 were taken during a period of moderately heavy rainfall (~0.5 cm/h). They show that rainfall can appreciably augment the transfer of momentum between atmosphere and sea, and analysis of the problem shows that, for large drops, the increment of stress so produced will approximate to the product of the rate of rainfall (g cm~2 sec~i) and the wind speed at some such height as 5 or 10 m. 4. Transfer of Heat and Water Vapour A. Introductory One approach to evaluation of the coefficient da in the evaporation bulk formula (8) has been from annual heat budget considerations over the oceans [see the accounts by Jacobs (1951, 1951a) and Sverdrup (1951)]. This line of attack, valuable as it has proved to be in giving the broader features of the interchanges between ocean and atmosphere, is not capable of much refinement since it is difficult to evaluate all the forms of heat transport with sufficient accuracy. However, for lakes the heat budget over periods of sufficient length — a fortnight and upward — is more reliable [see applications by Anderson (1954) and Webb (I960)]. Other approaches are by eddy fluctuation measurements and by profile observations. Since these two methods are applicable over periods as short as 15 min or so, it is reasonable to expect that they will eventually lead to fully adequate knowledge of the dependence of the bulk coefficients on wind speed and thermal stratification. There is no doubt that, in the near future, the approach by profile measure- ments will prove fruitful, and it is with this method that most of the following discussion is concerned. The measurements demand accurate instrumentation, since, over the open sea, the vertical gradients are small. The small wind gradient is a consequence of the smaller roughness parameter than over land ; while, in the case of temperature and humidity, the small gradients are charac- teristic of an air-mass which has become well modified over a sea surface of almost constant temperature — the diurnal variation of sea-surface tempera- ture is only slight owing to the large thermal capacity of the upper layers of the sea. It is to be remembered that, over the ocean, small gradients of potential temperature do not imply negligible thermal stratification since the wind gradient, which appears in the denominator of the Richardson number, is also small. To gain some idea as to the importance of stratification, it may be noted that the air minus sea temperature difference observed by German weather ships in the Atlantic is — 3°C nearly half as frequently as it is zero (Brocks, 1956). Even with a wind as strong as 6 m/sec such a value corresponds to a Richardson number at 6 m height of about —0.17 (estimated from (34), p. 68) and reference to Fig. 2 shows that under these conditions the thermal stratifica- tion has a marked effect. SECT. 2] SMALL-SCALE INTERACTIONS 67 B. Profile Coefficients Montgomery (1940) introduced a coefficient Fe in order to express the humidity gradient conveniently in terms of the sea to air difference. We may define analogous coefficients Fm and Fh for wind and temperature respectively, the three then being Ua d in z r. = -!-^. (25) qa-qs c In z The F's will have a slight dependence on the height of measurement, but the subscript a is omitted for convenience. They can be related to the coefficients c, h and d in the bulk formulae (6), (7) and (8) by noting that in neutral condi- tions K M = K H = Ke = ku^z or, in non-neutral conditions, Km = KmU^z, Kh = * * Khu^z, and Ke = Keu^z (these being the chosen definitions of the generalized * Karman coefficients, the iC's). The relationships are c = Km^Fm^ (general) = k^Fn^ (neutral case) (26) h = KmKhFmFh (general) = k'^FrnFu (neutral case) (27) * * d = KmKeFmFe (general) = k'^FuFs (neutral case). (28) For the general case, (6), (7) and (8) may be rewritten * U^ = KMFuUa (29) H = CppRffFnu^ids-da) (30) E = pKEFEU^iqs-qa). (31) C. Bulk Stability Parameter Using the above definitions of the profile coefficients, the Richardson number from (14) becomes Now FhIFm^ may be expected to be a function of Ri ; it does in fact vary moderately over a wide range of stability, though the relationship is perhaps not perfect since it may also have some variation with wind strength. At heights of about 4 to 8 m and conditions from neutral to moderate thermal 68 DEACON AND WEBB [CHAP. 3 stratification, Fh and Fm have values around 0.1 (see pp. 70-71). Therefore we may define as a convenient bulk stability parameter which is roughly equal to the Richardson number, Eb^ 10 iz ^-^. (33) When we take a reference height, 2 = 6 m, and a temperature, T = 290°K, in the vicinity of those commonly encountered in practice, and express d in °C and u in m/sec, then we arrive at a particularly convenient expression for the bulk Richardson number i?&6 = 2.o4^^^|f%/ (34) Ua^ (m2/sec2) D. Vapour Pressure at the Sea Surface The air in direct contact with the sea surface has, at least approximately, a water-vapour pressure equal to that of sea-water at the sea-surface temperature. Normally this is approximately 98% of the vapour pressure of pure water at the same temperature. According to Witting {vide Sverdrup, Johnson and Fleming, 1942) the vapour-pressure ratio, sea-water/pure water, is related to chlorinity (CI) as follows : v.p. ratio sea-water /pure water = 1 — 0.00097CL More recently, Arons and Kientzler (1954) have determined the vapour pressures of sea-salt solutions over a very wide range of temperatures and chlorinities and found reasonably good agreement with Witting's equation for chlorinities up to 20 parts per thousand. E. Observational Investigations Only recently have refined measuring techniques been applied to the observa- tion of temperature and humidity profiles in the atmospheric boundary layer over the sea ; ^ and such investigations should provide a wealth of data of high quality during the next few years. However, the pioneering investigations of earlier workers using ordinary meteorological instruments have already indicated approximately the magnitude of the humidity-profile coefficient. There is an important point of experimental technique which is worth mentioning. By employing a pair of electrical thermometers such as thermistors (the application of which is discussed by Haeggblom, 1959), it is possible to measure the temperature difference between two heights rather than the individual temperatures. This is a highly desirable arrangement for profile studies from the standpoint of instrumental accuracy. Furthermore, in cases where measurements are sampled at intervals during the observation period 1 An outstanding exception is the much earlier equipment of Johnson (1927). SECT. 2] SMALL-SCALE INTERACTIONS 69 rather than recorded continuously for the whole period, it is virtually essential to employ the difference method in order to avoid excessive sampling errors. Measurements of temperature and/or humidity profiles, employing sampling at individual heights, have been published by Wiist (1937), Montgomery (1940), Bruch (1940), Sverdrup (1946) and Takahashi (1958). Some of these results have been collated and discussed by Brocks (1955). Measurements using a pair of thermistors to record differences are given by Deacon et al. (1956), and further measurements in Port Phillip Bay and Bass Strait in November, 1956, and May, 1958, are as yet unpublished. In the last-mentioned paper are repro- duced also some earlier results of Johnson and Meredith, who measured dif- ferences with the aspirated platinum resistance thermometers described by Johnson (1927). In all the work referred to above, the observations were made from shipboard. Other recent investigations are those of Fleagle, Deardorff and Badgle}" (1958) using a raft-mounted mast, and of K. Brooks (results not yet published) using the buoy-mast illustrated in Fig. 4. F. Nature of Observed Results Tjrpical forms of temperature profile in lapse and inversion conditions are illustrated in Fig. 11. In (a) the plot against a linear height scale shows the marked height-dependence of the gradient. In (b) the plot against a logarithmic height scale is more nearly linear, but is concave towards the height axis in lapse conditions and convex in inversion conditions. These characteristics are, of course, broadly the same as observed over land surfaces. The potential temperature difference between the surface and a height of a few metres comprises, roughly speaking, about one-fifth across the layer of molecular transfer about a millimetre thick adjacent to the water surface, and the other four-fifths across the overlying region of turbulent transfer ; this is estimated from the theoretical approach to be dealt with later. Profiles of specific humidity (or vapour pressure) are qualitatively similar in form, having the same rule of dependence on thermal stratification as stated above. However, the sign of the humidity gradient is, of course, normally negative, the greatest humidity being right at the water surface and corre- sponding at least approximately to saturation at the temperature there. One might expect as a first approximation that the potential temperature gradient at a given height, or difference between two given heights, would be proportional to the difference of temperature between sea and air ; that is, that Fh would be approximately constant. Broadly speaking, this is found to be true at the lower levels, as illustrated by the measurements plotted in Fig. 12. At higher levels, say above 4 m, the effect of thermal stratification illustrated by the curvature in Fig. lib becomes important. This can be seen in Fig. 13, where measurements of ^12.6 — ^4 are plotted. It is apparent that the greater the sea-air temperature difference and the lighter the wind, the greater is the deviation from the linear relationship (though these measurements unfortu- nately do not include light winds on the inversion side, Ta—Ts> 0). The 70 DEACON AND WEBB [chap. 3 deviation is particularly marked on the lapse side, Ta—Ts< 0. Referring back to the lower-level observations in Fig. 12, the corresponding deviation on the lapse side is in fact discernable in the lightest winds. The slope of the straight line fitted in Fig. 12 indicates a value of Fh at a height of 4 m as 0.10 with no evident dependence on wind strength; and the same relationship converted to the co-ordinate scales in Fig, 13 is seen to be 15 1 1 1 1 E ■ "f 10 I - - 5 \ \ 0 1 1 V , -4 -3 -2 Potential temperature, deg F (q) 0 .1 .2 .3 .4 Potential temperature, deg F -2.6 -2,5 -2.4 -2.3 (b) Potential temperature, deg F LAPSE .2.5 »3 .3.5 Potential temperature, deg F INVERSION Fig. 11. Specimen profiles of potential temperature plotted against : (a) linear height scale, and (b) logarithmic height scale. Lapse: observation No. 134 (10 min), November, 1956, in Bass Strait, wi3 =- 5.8 m/sec, Rh^ = - 0.08. Inversion: observation No. 48 (15 min), May, 1958, in Port Phillip Bay, wi3 = 7.3 m/sec, Rb^ = -t-0.13. also consistent with the measurements plotted there, apart from the deviations produced by thermal stratification. Values of Fh quoted by Sheppard (1958) from earlier published observations are in broad agreement with the value indicated above, while Brocks (1956) has measured temperature gradients by an optical refractive index method, and these indicate a value of Fh virtually identical with it. SECT. 2] SMALL-SCALE INTERACTIONS 71 Turning now to humidity profiles, some values of Fe obtained by Mont- gomery (1940) from his and from Wiist's (1937) observations, and by the present authors from measurements made in May, 1958, are plotted against the bulk Richardson number in Fig. 14. The value of Fe is seen to be about 0.1 in neutral conditions, with a dependence on the stability somewhat as shown by the curve drawn in by eye. There does not appear to be any marked variation of Fe with wind speed, but in view of the paucity of data and the experimental scatter, no definite conclusion on this point is possible as yet. d^-0^. deg F Fig. 12. Potential temperature difference between 2.8 and 4 m plotted against difference between sea and air — observations in Port Phillip Bay and Bass Strait, November, 1956. The slope of the straight line, which is fitted by eye, indicates 7/^ = 0.10. Symbols denote wind speed at 13m: X > 9 m/sec A 3 to 4.5 m/sec + 6 to 9 m/sec O 4.5 to 6 m/sec 2 to 3 m/sec ^ 2 m/sec There are three complicating factors which may influence the behaviour of Fh and Fe, though to what extent is largely unknown at present. These are : i. Spray. As has been remarked by Montgomery (1940), the evaporation of spray, by providing a source of water vapour between the sea surface and the measurement level, will operate to increase Fe. While negligible in light winds this effect may become of some importance at higher wind speeds. ii. Cool surface skin. Montgomery (1947a), Ball (1954) and others have pointed to the existence, in at least some conditions, of a large temperature gradient through a water-skin layer of a millimetre or so depth immediately below the interface ; this will normally mean that, on account of evaporation, 72 DEACON AND WEBB [chap. 3 the true surface temperature is lower than that measured just below the surface, and estimated values of Fe will appear low as a result. iii. Radiative heat exchange. In addition to the transfer of heat between sea and air by the processes already considered, there is some radiative transfer giving rise to a source or sink of heat between surface and measurement level. This effect, pointed out by Sverdrup (1943), is only likely to be of significant magnitude when the other transfers are small, i.e. under stable conditions with light wind. Either of the first two effects would cause a displacement of measured temperatures in the sense actually found in Fig. 12, with the line of best fit zo 1 1 1 1 1 1 1.5 - o X ,y 1.0 X " ^^ y^ - 0.5 - ,t. XX - 0 ° r-,«<^ x^ •"" ^/X XX -0.5 /^ 1 1 1 1 7"2 - Ts , deg. F Fig. 13. Potential temperatvire difference between 4 and 12.6 m, plotted against tempera- ture difference between sea and air — observations in Port Phillip Bay and Bass Strait, October, 1955 (reproduced from Deacon, Sheppard and Webb, 1956, Fig. 10). The straight line is that fitted in Fig. 12 carried over with the appropriate scale con- version. Symbols indicate wind speed at 10 m: X > 5.5 m/sec o 4.5 to 5.5 m/sec ▲ 2.5 to 4.5 m/sec passing to the left of the origin (in the case of spray, this would be the effect of a flux of heat towards the region where evaporation of spray is consuming latent heat). However, it is not yet certain, though it appears probable, that the displacement from the origin is real : in the investigation plotted in Fig. 12, the corresponding displacement is in fact exhibited by the simultaneous measurements at greater heights up to 1 3 m ; and a similar displacement is clearly evident in the measurements of ^22 — ^5 by Johnson and Meredith reproduced by Deacon, Sheppard and Webb (1956). On the other hand, the displacement is not clearly apparent in all investigations, for example that reproduced in Fig. 13. It is clear that examination of extensive temperature SECT. 2] SMALL-SCALE INTERACTIONS 73 measurements in the style of Figs. 12 and 13, with the corresponding treatment of humidity measurements, will throw much light on the effects of the compli- cating factors mentioned above. A tool which may prove to be of considerable value in investigating these eff'ects is the use of the Taylor characteristic diagram, which is a plot of (poten- tial) vapour pressure against (potential) temperature. Montgomery (1950) has 0.2 & • o -0.1 -0 05 0 f 0.05 (a) Wind speed >5m/sec -0.8 -0.6 -0.4 -0.2 0 ^0.2 *0A ^0.6 +0.8 (b) Winds <5m/sec Fig. 14. Observations of Fe (4 m) plotted against the bulk stability parameter, Bh^. Black symbols : grouped observations of Wiist and Montgomery. Open symbols : individual observations, Port Phillip Bay, May, 1958. (a) Strong winds : •, O uq 5 to 7.5 m/sec A, -^ uq above 7.5 m/sec (b) Light winds: •, o uq below 3 m/sec A., ^ iiQ 3 to 5 m/sec given an elegant discussion of this form of plotting, which was introduced by G. I. Taylor in 1917 for the study of water vapour in the atmosphere. Some measurements plotted on the characteristic diagram are shown in Fig. 15. The following is the principle on which the diagram is interpreted. When air, whose characteristics are represented by some point on the diagram, flows over the sea surface, represented by the point on the saturation curve at the appropriate 74 DEACON AND WEBB [chap. 3 temperature, then various samples of air that result from mixing will be represented by various positions along tbe straight line joining these two points. On account of the meagre quantity of data re])resented by the plots in Fig. 15, they can do little more than merely illustrate the principle involved; and, clearly, it would have been advantageous to make observations over an 61 1 1 1 I 1 1 1 - 60 - ^ ?s AIR ~ 59 - ■^ - 58 - \ ^ /' - 57 ~ ^^ ^^ V 56 - ^ -y — 55 1 1 1 1 1 / 1 1 - 10 II 12 13 14 15 16 Potential vapour pressure, mb (a) Strong wind U|3 greater than 7 m/sec 9 10 II 12 13 14 15 Potential vopour pressure, mb (b) Moderate wind U|3 between 4-5 m/sec 9 10 II 12 13 14 Potential vapour pressure, mb (c) Ligtit wind: u^-^ less than 3 m/sec Fig. 15. Temperature and humidity at 12.75 and 4 m and temperature at the sea surface, plotted on the characteristic diagram — observations (15-min averages) in Port Phillip Bay, May, 1958. The saturation curve for air in contact with sea- water is shown. Symbols identify different runs. extended range of heights. However, we can at least note the following conclu- sion. In strong and moderate winds, the extrapolations of the atmospheric observations meet the saturation curve close to the measured sea temperature — - within 0.5°F in every case. In light winds, the extrapolations point generally below the measured sea temperature, the disparity being greater than 0.5°F in nearly every case : this is suggestive of the existence of a cool-water skin layer under the favourable circumstance of a smooth surface. SECT. 2] SMALL-SCALE INTERACTIONS 75 From the drag-coefficient formula (22) which was adopted to fit the observa- tions in Fig. 6, we can estimate the wind-profile coefficient Fm via equation (29). The resulting Fm values for heights 4 m and 8 m are as follows : Wind uq. A/(4 m) /m(8 m) m/sec 2 0.090 0.085 5 0.099 0.092 8 0.107 0.100 14 0.123 0.112 We can now combine the estimates of profile coefficients given above to derive the bulk heat flux and evaporation coefficients, h and d, using (27) and (28) for the neutral case. Let us restrict attention to evaporation, since the measurements are not yet sufficiently refined to detect any difference between values of Fh and Fe and, therefore, between h and d. Humidity measurements have commonly been taken at a height of 4 m and winds at around 6 to 10 m. Taking Fe (4 m) = 0.1 and (for a wind speed of 5 m/sec) Fm (8 m) = 0.092, then (28) gives the bulk evaporation coefficient, (i = 0.00155. This value lies between two earlier estimates of evaporation from the oceans (see Sverdrup, 1951): the average result from the Meteor Expedition is 20% lower, while the result obtained by Jacobs from oceanic heat balances is some 37% higher. Of particular interest is a comparison with the result from an intensive investigation at Lake Hefner, where Marciano and Harbeck (1954) evaluated the bulk aerodynamic coefficient by matching against accurate water-budget data. The Lake Hefner value is in fact 5% higher than that quoted above, that is to say, the two are virtually equal. At two other lakes (Harbeck et al., 1958; Webb, 1960), the Lake Hefner value has also been confirmed ; and it seems to be indicated that this value of the bulk coefficient is applicable to water expanses of a wide range of sizes from the open sea down to lakes a mile or so across. Russian investigations reported by Budyko, Berliand and Zubenok (1954) indicate again similar values: an approximate value quoted for lakes is 3% higher than that derived here, and a value they derive from heat-budget calculations over the oceans is 24% higher. Some discrepancy between the ocean heat-budget values and that derived from the neutral profile is to be expected on account of the stability effect, the former values having been deduced for ocean regions over which the average condition is one of slightly unstable stratification. Furthermore, it is to be remembered that the heat-budget values are obtained on the assumption, needed to be able to utilize climatic averages, that {qs-qa)Ua = {qs-qa)Ua, where the bars denote mean values taken over an extended period of time. 76 DEACON AND WEBB [CHAP. 3 For convenient conversion of F values at different reference heights, the following ratios to 7^(8 m) are given. They are calculated from the logarithmic profile form for the case 7^(4 m) = 0.1, but for other 7^(4 m) values, differing by up to 30%, they remain correct to within a few per cent. They are, of course, equally applicable to Fm, Fh or Fe. Fi/Fs 1.15 A/A 1.07 Ae/A 0.94 A2/A 0.88 If further values are required they may be calculated from F{b)IF{a) - [ln(6/a)r(a)+l]-i. G. Further DisciLssion of the Effects of Thermal Stratification The effects of thermal stratification on the profiles, as illustrated by the curvature in Fig. lib, are qualitatively the same as are observed over land. It is indeed most likely that the effects are virtually identical over land and sea, though this awaits confirmation by a sufficient number of high-quality measure- ments over the sea. We now proceed to outline the information available from observations over land. Even here the subject is still under investigation, and, although a fairly detailed picture of the behaviour of the profiles has been formed, there are still gaps to be filled in and further effort is needed in the improvement of observa- tional methods and site selection. Accounts of the subject up to recent years are to be found in Sutton (1953), Sheppard (1958), Charnock (1958) and Priestley (1959). H. Lajpse Conditions In unstable (lapse) conditions, the transfer coefficients are greater than in inversions, and the unstable case is, therefore, the more important from the point of view of vertical transfer. The case of free convection, with wind-shear effects negligible in comparison with thermal effects, has been considered in the similarity theory of Priestley (1954), which indicates the form of the temperature profile as dejdz oc s-4/3. (35) In a further study Priestley (1955) has examined experimental data in terms » of the corresponding dimensionless heat flux, H, defined by H = HlpCp{glTy/2z^ddiez\^'^, (36) SECT. 2] SMALL-SCALE INTERACTIONS 77 * pointing out that H should be proportional to |i?^|-l/2 j^ a region of forced convection and, according to the theory, a constant independent of Richardson number in a region of free convection. His examination affirmed these relation- ships and gave the constant free convection value of H as about 0.9. The height, Zjn say, at which the region of essentially forced convection merges with the overlying region of essentially free convection, he found to be at the level where \Ri\ reaches a value of approximately 0.03, i.e. 2/«;:i;0.03|L| in terms of Obukhov's scale length, L, defined in formula (15) : the relationship between the two values is * {Zmi\L\YI'^ = k^ I H (tree convection). In a detailed study of observed temperature profiles, Webb (1958) found close agreement wdth the 4/3-power law over a roughly 30-fold height range from 2/|L|=0.03 up to about zl\L\ = l, and found that the gradient ddjdz vanishes at the latter height and remains small or zero in the region above. When the wind is light with a marked lapse, the height at which the 6 gradient vanishes over land or sea is only a few metres; and, correspondingly, Fig. 13 (T2— Ts negative, lightest winds) shows that the observed values of ^12.6 — ^4 are then close to zero. The wind-profile does not exhibit any related singularity — the trend of du/dz continues smoothly through the transition height where dd/dz vanishes. It seems certain that this transition represents the boundary between the superadiabatic convection layer, of depth a few metres or tens of metres, and the neutral or slightly subadiabatic "homogeneous" region that often extends to a considerable height in the troposphere. The homogeneous region has been illustrated by the temperature measure- ments made by Craig (1946, 1947) — see also Craig and Montgomery (1951) — and by Bunker et al. (1949) up to heights of 1000 ft or so over the sea. We must, however, note the difficulty of detecting the superadiabatic layer at these heights, since its potential-temperature gradient would be even smaller there than the 0.2 or 0.3 °F per three-fold height interval which is indicated in Fig. 13. There is a continual buoyant transfer of heat upwards through the neutral or subadiabatic temperature gradient of the homogeneous region, a condition envisaged by Priestley and Swinbank (1947) and treated theoretically by Priestley (1954a). Counter-gradient heat flow in this region has been observed by Bunker (1956). The concepts of transfer coefficient for heat and of Richard- son number cease, of course, to be meaningful in this region. As a guide to the height, 2; = | L| , which is approximately where the transition between superadiabatic and homogeneous regions occurs. Fig. 16 shows \L\ in terms of the bulk conditions represented by Bbe. The relationship is somewhat tentative, but should be approximately correct ; it is based on the forms of temperature and wind profiles described below. Let us now consider the merging across the boundary between the layers of forced and free convection. As the boundary is approached from beneath, the deviation from the neutral profile may be represented by a function/ as in (16). 78 DEACON AND WEBB [chap. 3 Monin and Obukhov (1954) have suggested that it may be convenient to express /as a power series in z/L. Priestley (1960) has shown that, if the power series coefficients are adjusted to provide the smoothest crossover to an over- lying region with gradient ccz~^/^, then the second and higher powers of zjL 1000 100 - 10 Fig. 16. Scale height \L\ plotted against the bulk stability parameter |i?66| defined by (33) or (34), for unstable conditions. \L\ represents approximately the height of transition from superadiabatic to homogeneous structure, and 0.03|iv| the height be- low which the stratification may be regarded as near-nevitral. (Curve derived on the assumption that in neutral conditions /\f(4 ni) = 7^//(4 m) = 0.1.) have rapidly diminishing effect and may as well be ignored for practical purposes. Perhaps the most appropriate scheme, used by Webb (1960), is to take a two-sided linear smoothing factor, i.e. in the case of the temperature gradient to take dd Tz 86 74/3 ■J '- •"in 7 z Z \ 1 Zm for Z ^ Zm, for z ^ Zr (37a) (37b) where A— —Hl{kcppu^). The coefficient 1/7 in the linear factors is the value which provides a smooth crossover at Zm by making dW/dz^ and dWjdz^, as well as ddjdz, continuous there. Since the term —zjlzm in (37b) is simply a convenient alternative form of aizjL, the first-order term in the Monin-Obukhov series, we must have Zml\L\ = l/7ai. This gives, assuming a trial value ai = 4.5, the same as observation in- SECT. 2] SMALL-SCALE INTERACTIONS 79 dicates for the wind-profile, Zml\L\ =0.032 and correspondingly H (free convec- tion) = 0.94. As these are, to within experimental error, eqnal to the values of « Zmj\L\ and H obtained directly from observed temperature-profiles and heat fluxes as mentioned above, this seems to justify the trial assumption of ai = 4.5. This assumption implies, of course, the virtually exact similarity of wind- and potential temperature-profiles over a range extending from the forced convec- tion region into at least part of the free convection region, and, throughout this range, we would, therefore, have KhIKm= 1. This disagrees with the in- dication from eddy-correlation results that KhIKm is greater than unity over the whole unstable range ; and further work is needed to clarify the differences between wind- and temperature-profiles. Assumed similarity between the two profiles leads to a curve of Km against Ri rather similar to that shown in Fig. 2 from Ellison's formula but running about 15% higher in the Ri range — 0.2 to —0.4; it is clear that measurements of shearing stress are not yet sufficiently numerous or refined to arbitrate between the two. One of the possible alternative approaches to this problem is through the examination of humidity -profiles, and of particular value in this direction will be measurements over the sea, where the surface condition is highly uniform and it is possible to secure a large fetch. The point here is that the humidity- profile is in some respects more amenable to accurate and representative measurement than is the wind-profile; and all the evidence is concordant, as mentioned earlier, in indicating that the two are effectively identical in form. Just one further remark may be made regarding the form of the humidity- profile. In the upper parts of the free convection region, the supposed mechan- ism of the vanishing potential-temperature gradient (Webb, 1958) implies that the humidity-profile probably follows fairly closely the form of the poten- tial temperature-profile, including a similar abrupt fall to negligible gradient at about z= \L\. No observations relating to this are yet available, however. /. Inversion Conditions As inversion conditions over land are accompanied by low wind velocities and considerable unsteadiness associated with katabatic drifts, etc., observa- tional difficulties have hampered progress and little is known with certainty. Over the sea, steady conditions of inversion are more frequently encountered particularly in coastal waters, with a warm wind from the land blowing over cooler sea. The observations of Pasquill (1949) and Rider (1954) have indicated that the equality of the transfer coefficients for momentum and water vapour. Km and Ke, extends to inversion conditions. It is therefore to be expected that the Rossby-Montgomery stability relationship (17) should apply to the humidity profile if u and u^ are replaced by q and Ejpu^ respectively and a value cr = 9 used as for the wind-profile. The relationship would thus be Ke = kj{l + aRiy/^. (38) 80 DEACON AND WEBB [CHAP. 3 Information on the behaviour of the temperature-profile is even more limited. The eddy-correlation measurements have given values of KhJKm around 0.7 over the range from near-neutral lapse to i?t = 0.08, the strongest stability investigated. Further work on the effects of stable stratification is clearly needed. J . Theoretical Approaches to Bulk Relationships Several theoretical approaches have been made to the relating of evapora- tion to measured bulk quantities, as in formula (8). No rigorous approach is possible owing to lack of complete knowledge of the transfer characteristics in the lowest layer of air near the water surface ; and the respective theories are based on differing assumptions as to the nature of these characteristics. The subject, as it stood in 1950, has been reviewed by Anderson et al. (1950) and by Sverdrup (1951). Of the various theories which have been proposed, we mention below those which appear to be acceptable by comparison with the observational data available at present. Details of the other theories may be found in the reviews quoted above. In the discussion below, the cases of flow over aerodynamically smooth and aerodynamically rough surfaces are considered separately. K. Aerodynamically Smooth Flow In the case of aerodynamically smooth flow, the treatment proposed by Montgomery (1940) has been adopted by later writers and no alternative has been proposed. His approach proceeds as follows. Over a smooth surface, the shearing stress is transmitted through a thin layer of laminar flow (of the order of a millimetre thick) in the air immediately adjacent to the water surface. On dimensional grounds, the thickness, 8. of the laminar layer is related to the friction velocity, u^, and the kinematic viscosity, ^^ by u^hjv = A, (39) where A is a constant. In the laminar layer, the transfer coefficient for water vapour is the molecular diffusivity, D, so that the evaporation is given by E = Dp{qs-q,)jh, i.e. qs-q, = ESjDp, (40) where subscripts s and 8 refer to the water surface and height respectively. In the turbulent region, reckoning height, z, to be measured from the top of the laminar layer, the transfer coefficient is D + ku^^z, and thus the evaporation is given by E = - p{D + ku^z) dqjdz, SECT. 2] SMALL-SCALE INTERACTIONS from which, on transposing and integrating, we find q,-q^ = {Elpku^)\n[{D + ku^z)ID]. 81 (41) From (40), (41) and (39), neglecting D in comparison with ku^z, the evapora tion is given by pku^{qs-qz) ^ {kXvlD) + ln{ku^zlD) and comparison with (31) shows that this is equivalent to rsiz) = [{kXvlD)+\n{ku^zlD)]-K (42) (43) Curves of the relationship (43) are given in Fig. 17 for two values of the parameter A that have been proposed : 11.5 by von Karman and 7.8 by Mont- gomery (see Montgomery, 1940, for discussion). The value of D, here and later, is taken as 0.24 cm^ sec-i. 0.1 r£(8m) 0.05 c _ 20 30 40 u,, cm/sec I I I 50 60 14 Ug, m/sec Fig. 17. Theoretical values of the humidity coefficient, Fg (8 m), plotted against friction velocity, u^, and [via (22)] wind speed, wg- (a) A = 7.8, (b) A =11.5, (c) A = 27.5 in Sverdrup's (1937) theory; (d) Sheppard's theory; (e) A = 7.8, (f) A= 11.5 in Montgomery's theory for a smooth surface. L. Aerodynamically Rough Flow Several different theories have been proposed for the case of aerodynamically rough flow. Two of these, due to Sverdrup and Sheppard respectively, are now to be discussed. In the theory of Sverdrup (1937), it is assumed that there is a layer of molecular transfer of vapour with diffusivity, D, through a surface air film of thickness, 8, given by (39). In contradistinction, the shearing stress is, of course, transmitted by pressure forces against the surface irregularities, rather than by molecular viscosity. In the region above this layer, turbulent transfer is assumed with the same transfer coefficient as for momentum, i.e. Ke = Km = ku^{z + zo). 4 — s. I. 82 ^ DEACON AND WEBB [CHAP. 3 Thus, as before, we find that for the layer of molecular transfer (40) is applicable, while for the overlying region of turbulent transfer. qs-qz = {Elpku^)\ny{z + Zo)j{h + Zo)l (44) Finally, (40), (44) and (39) lead to rE{z) = {(A:Ai./7)) + ln[(3 + 2o)/(S + 2o)]}-i. (45) Fig. 17 shows curves plotted from (45) for three values of A, namely 7.8 and 11.5, as already quoted, and 27.5 as proposed by Sverdrup (1937). In deriving these curves, the values of 2o inserted in (45) are those implied by the observed drag relationship (22). In Sheppard's formulation, the transfer coefficient for water vapour is taken to be Ke = D + ku^z throughout; thus molecular transfer becomes dominant right near the surface, though no distinct layer of exclusively molecular transfer is assumed. The turbulent transfer coefficient is taken as kU:^z rather than kU:^{z + zo) on the reasonable view that the appearance of zq is a result of pressure forces against the rough surface, which act to transfer momentum but not vapour. Similarly, as in the preceding cases, though more simply, we find rsiz) = [ln{ku^zlD)]-K (46) The curve representing (46) is shown on Fig. 17. It will be seen from Fig. 17 that the theories for aerodynamically rough flow of Sheppard, and of Sverdrup (1937) with A= 7.8, yield values of Fe that are close to that indicated by observation. Sheppard's approach is perhaps to be preferred since it has the merits of simplicity and of freedom from an adjustable constant. The bulk evaporation coefficient, ds, derived from (28), using Fe from the rough-surface theories discussed above and Fm implied by the relationship (22), is shown in Fig. 18. From the theories mentioned above, we may calculate Fh, the profile co- efficient for potential temperature, by replacing D with k. Over the whole range of wind speeds from 2 to 14 m/sec, its ratio to Fe is found to be close to FhIFe = 0.98, within 0.5% and 2% from the formulae of Sheppard and Sverdrup (A = 7.8) respectively. Thus the observational indication of closely similar values of Fh and Fe is backed by the theoretical estimate. In concluding this discussion of theoretical approaches, we may remark that an assumption throughout is that effects of thermal stratification are absent. The only attempt to introduce these effects appears to be an approximate approach by Anderson et al. (1950). Their result should presumably be valid provided the deviation from neutral stratification is small. SECT. 2] SMALL-SCALE INTERACTIONS 83 0.0015 0.001 0.0005 - 0.1 0.05 -I 0 Fig. 18. Coefficient (for height 8 m) in the bulk evaporation formula (8), derived from the theoretical values of Fig. 17 and the drag coefficient relationship (22) shown in Fig. 6. The right-hand scale gives the coefficient for p—1.2x 10~3, pressure 1000 mb, in the equivalent formula E = coeff. x wsl^s "" ^s)- (a) A=7.8; (b) A=11.5; (c) A = 27.5, in Sverdrup's (1937) theory; (d) Sheppard's theory. M, The Bowen Ratio In computing evaporation on the basis of the heat balance (for accounts of which see Sverdrup et at., 1942, pp. 100-125, or Sverdrup, 1951), it is necessary to be able to derive the Bowen ratio, jS, which is the ratio of vertical flux of heat to that of latent heat, i.e. ^ = HjLE, where L is the latent heat of vaporization of water. As first deduced by Bowen in 1926, ^ may be estimated from the sea-air temperature and humidity differences using the relationship ^ = B{ds-da)l{qs-qa). In a turbulent region with equal transfer coefficients for heat and water vapour, the value of B = CplL would apply. In actual circumstances, where there is also a layer of predominantly molecular transfer to be included, the relationship is B = {r„irE){c^iL). As seen above, the indication from the theories, at least in near-neutral condi- tions, is that FhITe is close to 0.98 ; so that taking account of the surface film of molecular transfer the value of the Bowen ratio calculated on the usual assumption of B = CplL is only slightly changed. 84 DEACON AND WEBB [CHAP. 3 Fluctuations of the Bowen ratio can be taken into account using the method given by Webb (1960a). References Anderson, E. R., 1954. 'Energy budget studies' in 'Water-loss investigations: Vol. 1^ Lake Heffner Studies Tech. Rep.' U.S. Geol. Surv. Prof. Paper, No. 269. Anderson, E. R., L. J. Anderson and J. J. Marciano, 1950. A review of evaporation theory and development of instrumentation. U.S. Navy Electronics Lab., Rep. No. 159. Arons, A. B. and C. F. Kientzler, 1954. Vapour pressure of sea-salt solutions. Trans. Amer. Qeophys. Un., 35, 722-728. Ball, F. K., 1954. Sea surface temperatures. Austral. J . Phys., 7, 649-651. Batchelor, G. K., 1950. The application of the similarity theory of turbulence to atmos- pheric diffusion. Q. J. Roy. Met. Soc, 76, 133-146. Brocks, K., 1955. Wasserdampfschichtung iiber dem Meer und "Rauhigkeit" der Meere- soberflache. Arch. Met. Geophys. Bioklim., A, 8, 354-383. Brocks, K., 1956. Atmospharische Temperaturschichtung und Austauschprobleme iiber dem Meer. Ber. deut. Wetterdienstes, No. 22, 4, 10-15. Brocks, K., 1959. Ein neues Gerat fiir storungsfreie meteorologische Messungen auf dem Meer. Arch. Met. Geophys. Bioklim., A, 11, 227-239. Brocks, K., 1961. Ergebnisse von Windprofilmessungen auf See. Proc. Meeting of Advanced Study Institute, Sept. 18-27, Hamburg (to be published). Brown, P. R., 1953. Wave data for the eastern North Atlantic. Mar. Observer, 23, 94—98. Bruch, H., 1940. Die vertikale Verteilung von Windgeschwindigkeit und Temperatur in den untersten metern iiber der Wasseroberflache. Ve7'6ffentl. Inst. Meeresk. Univ. Berlin, n.s.. A, 38, 5-51. Budyko, M. I., T. G. Berliand and L. I. Zubenok, 1954. Heat balance of the earth's surface. Bull. Acad. Sci. U.R.S.S., Sir. Geograph., No. 3. Bunker, A. F. et al., 1949. Vertical distribution of temperature and humidity over the Caribbean Sea. Papers Phys. Oceanog. Met., Mass. Inst. Tech. and Woods Hole Oceanog. Inst., 11, No. 1, 88 pp. Bunker, A. F., 1956. Measurement of counter-gradient heat flows in the atmosphere. Austral. J. Phys., 9, 133-143. Chamock, H., 1958. Wind, temperature and humidity gradients near the ground. Sci. Progr., 46, 470-487. Charnock, H., J. R. D. Francis and P. A. Sheppard, 1956. An investigation of wind structure in the Trades: Anegada, 1953. Phil. Trans. Roy. Soc. London, A249, 179-234. Clauser, F. 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Symposium on Fundamental Problems in Turbulence, 4—9 Sept. 1961, Marseille (to be published). Deacon, E. L., P. A. Sheppard and E. K. Webb, 1956. Wind profiles over the sea and the drag at the sea surface. Austral. J. Phys. 9, 511-541. Deacon, E. L. and W. C. Swinbank, 1958. 'Comparison between momentum and water vapour transfer' in 'Arid Zone Research. XI Climatology and Microclimatology', Proc. Canberra Symp., U.N.E.S.C.O., Paris, 38-41. Ellison, T. H., 1957. Turbulent transport of heat and momentum from an infinite rough plane. J. Fluid Mech., 2, 456-466. Fleagle, R. G., 1956. Note on the effect of air-sea temperature difference on wave genera- tion. Trans. Amer. Geophys. Un., 37, 275-277. Fleagle, R. G., J. W. Deardorff and F. I. Badgley, 1958. Vertical distribution of wind speed, temperature and humidity above a water surface. J. Mar. Res., 17, 141-157. Francis, J. R. D., 1951. The aerodynamic drag of a free water surface. Proc. Roy. Soc. London, A206, 387-406. Francis, J. R. 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Observations of vertical humidity distribution above the ocean surface and their relation to evaporation. Papers Phys. Oceanog. and Met., Mass. Inst. Tech. and Woods Hole Oceanog. lyist., 7, No. 4, 30 pp. Montgomery, R. B., 1947. Viscosity and thermal conductivity of air and diffusivity of water vapour in air. J . Met., 4, 193-196. Montgomery, R. B., 1947a. Problems concerning convective layers. Ann. N.Y. Acad. Sci., 48, 707-714. Montgomery, R. B., 1950. The Taylor diagram (temperature against vapour pressure) for air mixtures. Arch. Met. Geophys. Bioklim., A2, 163-183. Munk, W., 1955. Wind stress on water: an hypothesis. Q. J. Roy. Met. Soc, 81, 320-332. Neumann, G., 1951. Gibt es eine "kritische Windgeschwindigkeit" fiir die Grenzflache Wasser-Luft? Deut. Hydrog. Z., 4, 6-13. Neumann, G., 1956. Wind stress on water surfaces. Bidl. Amer. Met. Sac, 37, 211-217. Obukhov, A. M., 1946. Turbulence in an atmosphere of non-homogeneous temperature. Trans. Inst. Theoret. Geophys. 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U., 1952. tjber Grossenunterschiede der Meereswellen bei Warm- und Kaltluft. Deut. Hydrog. Z., 5, 111-114. Roll, H. U., 1953/54. Beaufortaquivalente auf See bei verschiedenen thermischen Schich- tungen. Ann. Met., 6, 193-201. Rossby, C. G. and R. B. Montgomery, 1935. The layer of frictional influence in wind and ocean currents. Papers Phys. Oceartog. Met., Mass. Inst. Tech. and Woods Hole Oceanog. Inst., 3, No. 3. SECT. 2] SMALL-SCALE INTERACTIONS 87 Sheppard, P. A., 1958. Transfer across the earth's surface and tywough the air above. Q. J. Roy. Met. Soc, 84, 205-224. Stewart, R. W., 1961. The wave drag of wind over water. J. Fluid Mec/i., 10, 189-194. Stewart, R. W., 1961a. Wind stress on water. Pruc. Meeting of Advanced Study Institute, Sept. 18-27, Hamburg (to be published) Sutton, O. G., 1953. Micrometeorology. McGraw-Hill Book Co., New York. Sverdrup, H. U., 1937. On the evaporation from the oceans. J. Mar. Res., 1, 3-14. Sverdrup, H. U., 1943. 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Taylor, R. J., 1960. Similarity theory in the relation between fluxes and gradients in the lower atmosphere. Q. J. Roy. Met. Soc, 86, 67-78. Van Dorn, W., 1953. Wind stress on an artificial pond. J. Mar. Res., 12, 249-276. Vines, R. G., 1959. Wind stress on a water surface: measurements at low wind speeds with the aid of surface films. Q. J. Roy. Met. Soc, 85, 159-162. Vines, R. G., 1960. The damping of water waves by surface films. Austral. J. Phys., 13, 43-51. Vinogradova, O. N., 1959. The tangential stress of the wind on a disturbed sea surface. Bull. Acad. Sci. U.S.S.R., Oeophys. Ser., No. 11, 1646-1655. Webb, E. K., 1958. Vanishing potential temperature gradients in strong convection. Q. J. Roy. Met. Soc, 84, 118-125. Webb, E. K., 1960. An investigation of the evaporation from Lake Eucumbene. C.S.I. R.O., Div. Met. Phys., Technical Paper No. 10, Melbourne. Webb, E. K., 1960a. On estimating evaporation with fluctuating Bowen ratio. J. Geophys. Res., 65, 3415-3417. Wiist, G., 1937. Temperatur- und Dampfdruckgefalle in den untersten Metern iiber der Meeresoberflache. Met. Z., 54, 4-9. 4. LARGE-SCALE INTERACTIONS JOANNK S. Ma I. K US Dedication and Acknowledgements This effort is dedicated to Columbus O'D. Iselin to whom the writer owes her interest in the sea, and her opportunity to pursue it without restriction. His efforts have provided a stimulating atmosphere of broad inquiry into the earth sciences and a group of lively and productive interacting individual colleagues without whose experience and participation such an ambitious endeavor could not have been undertaken. His glorious confidence in us all has given each of us the necessary self-confidence to tackle difficult problems while at the same time his own humility in the face of the sea's complexity has imbued us with the necessity of continuous self-criticism in our attempts to understand its behavior. Among those colleagues who have been particularly helpful in the prepara- tion of this review are Henry Stommel, Andrew Bunker, Kirk Bryan, Jr., Herbert Riehl and Jose Colon. Miss Margaret Chaffee made a large number of the calculations and ably undertook the drafting, while the manuscript was typed by Mrs. Mary C. Thayer, who went far beyond the call of duty in its execution. 1. Introduction The links connecting the atmosphere and oceans are intimate, multifarious and vital to the operation and the understanding of each. Most obvious is the physical proximity, contact and mutual exchange of motion, heat, water, salt, gases, pollutants and biological organisms. The wind systems of the atmosphere are solar-powered mainly through the intermediary of the sea ; the water-vapor fuel, picked up by the trade winds over the sun-warmed tropical oceans, is itself converted from latent to usable form by another intervention of marine in- fluence— giant salt particles which collect the tiny cloud drops in the towering equatorial clouds and allow them to fall out as rain, leaving the released heat air-borne and available for transport. The rough waves of a choppy sea act to brake the winds which build them and remove their momentum, but this same drag is recognized by the oceanographer as the main transporter of water masses, driving the huge ocean-current systems of the globe. The oceans thus regain through wind stress a small fraction of the energy they supplied in evaporated water to the air, and one large circuit in the complex coupling of the two systems is complete. The tiny ratio of energy fed back from air to sea might suggest a stable and uninteresting interaction, except for the enormous heat storage capacity of hydrosphere compared to land or air. Small changes in ocean circulation may significantly alter the atmospheric energy sources, permitting internal instabihties and possibly amplifying fluctuations, as will be suggested later. Another link is that both media are turbulent, differentially heated fluids in [MS received September, I960] 88 SEOT. 2] LARGK-SCALE INTERACTIONS 89 restless motion upon a rotating spheroidal planet. That the basic problems and questions in the sciences of sea and air are so similar is thus, superficially, not surprising. These are epitomized in two closely related basic puzzles : first, the coexistence in each of many interacting scales of motion, from tiny eddy to planetary gyre, supplying and removing energy from one another, coupled in loops within loops of stable and unstable interaction, inseparable and non- linear, where the whole is frequently spectacularly different from the sum of its separate parts. Second, the streakiness of motion in each : restricted regions of concentrated fast streams imbedded in surrounding stagnation, irrespective of driving force, are characteristic of both sea and air, in systems ranging in size from seaweed streaks in harbor waters, through long lines of cumulus clouds and temperature striations in the Antarctic ocean "convergence", up to the famous planetary jets of Gulf Stream and circumpolar westerlies, themselves striated when "fine structure" is examined. These basic problems could be lumped into the broad category called "turbu- lence", which, however, smears over the importance and fascination of mechan- ism, and to some would imply a randomicity, where perhaps most significant is the very opposite : an organized departure from purely random motion, an organization with transport and release of energy as its guiding principle. That many interacting scales of motion and streakiness on all these scales are basic features of rotating f)lanetary fluids has been beautifully demonstrated by the recent model experiments [popularly referred to as "dishpan" studies (see, for example, Fultz, 1951 ; Stommel, Arons and Faller, 1958)], used equally by meteorologists and oceanographers to illuminate their problems and to select, from a vast complexity, the important features of the fluid motion. These rotating dishpan studies are only one example of the third major linkage be- tween atmospheric and oceanic science, namely the common structure and foundation of the tools of investigation, and the community and exchange among the investigators. With the inception of nonlinear hydrodynamics and the introduction simultaneously of the high-speed computers, both meteorology and oceanograf)hy are graduating from the "linearized" age of science, whose limitations almost always precluded fruitful discussion of energy transforma- tions and thus of the fundamentals of heat-driven fluid-engines. These develop- ments, plus some treatments of the fully turbulent hydrodynamic equations (still restricted to simplified prototype models), offer hope that the emphasis may now begin to shift from steady-state analyses of purely dynamic, linear systems in which energy sources, sinks and transports are perforce ignored to at least idealized cases of growing and decaying thermal circulations in which conversions and transports are primary, giving hope at last of modelling the most important interaction between sea and air, namely energy exchange. But theoretical models and analytical tools require critically made measure- ments to test their predictions, to build and improve their foundations by comparison with nature. Meteorology and oceanography are fundamentally still observational sciences, with primary phenomena often difficult or im- possible to scale in laboratory experiment or to analj'ze by differential 9<^ MALKUS [chap. 4 equations. Undoubtedly the most exciting new observational tool contributing advances to marine sciences in the past two decades is the instrumented aircraft, which has provided quantitative data on phenomena ranging from sea-surface waves, through cumulus clouds, hurricane, Gulf and jet stream, and whose results will be drawn upon heavily in the sections to come. As in astrophysics, the complexity of the problems in geophysics has resulted in piecemeal and fragmentary attacks ; within this complexity, specific prob- lems have been isolated for their tractability to the tools and interests pre- vailing. Unlike astrophysics, the practical importance of the weather and the sea in man's daily life have often channelled geophysical effort and observations to those areas of immediate human urgency, but equally have provided motiva- tion and means of attacking problems which pure curiosity-driven science might have avoided as too difficult or costly. In general, the more comprehen- sive the problem, the more parts of the interacting sea-air system considered, the less formal, the more descriptive, and even speculative, the treatment has had to be, and the heavier demands placed upon the breadth and courage of the investigator. Conversely, the more formal and rigorous the approach, the less useful are the results in the practical sense of forecasting or controlling a planetary phenomenon ; at present, even the convective motions in a coffee-cup- sized laboratory cell are only on the threshold of tractability. The challenge is to isolate simplified, prototype problems of this sort, whose rigorous treatment provides key insights into the complexity of natural geophysical systems. Thus quantitative attacks on the joint air-sea interaction problem, with both media simultaneously affecting and altering each other, are still few ; energetic treatments in either meteorology or oceanography alone are just emerging. The usual procedure is to isolate a single scale of phenomenon in the single system, for example, the wind-driven ocean current or the growth of a cumulus cloud, and to parameterize or assume the effects of the other medium (and the other scales of motion in the treated medium) in some empirical or semi- intuitive manner, such as in the first example through an observationally computed "curl of the wind stress" or a known latent heat supply in the second. The problems of air-sea interaction treated to date, then, may be crudely divided into four categories : (1) Isolated, formally tractable problems, in which the hydrodynamic equations are solved for a chosen scale of motion in either ocean/air, with the input or removal of energy from the air/ocean parameterized, assumed from observation or ignored. Among problems treated in this way are the large-scale circulation of the atmosphere (Phillips, 1956), the wind-driven ocean currents (Stommel, 1948; Munk, 1950), the maintenance of the trade-winds (Malkus, 1956), hurricane maintenance (Malkus and Riehl, 1960), the flow of stable air over a heated small island (Malkus and Stern, 1953), and, to some extent, the growth of cumulus clouds (Levine, 1959; Malkus and Witt, 1959). The results of these treatments are all ])redictive, and the limitations upon the degree of solution of the posed ])roblem lie in the importance of the pliysical effects ignored or inadequately parameterized. SECT. 2] LARGE-SCALE INTERACTIONS 91 (2) Budget studies, in which some of the hydrodynamic equations are used as conservation laws, particularly those stating continuity of mass, heat, kinetic energy, momentum and often vorticity. The various terms in the equa- tions are generally evahiated from observations and physical hypotheses ; the results of large-scale field expeditions in meteorology and oceanography com- monly come into their first valuable fruition when utilized in this manner. These investigations have the great value of shedding light upon the energy sources and transports of a geophysical fluid system, and of demonstrating Avhat processes and scales of motion are important in their operation. In meteorology, such studies have surged ahead since World War II due to the efforts of E. Palmen, H. Riehl and their collaborators, and have been largely enabled by extended observational networks and the use of instrumented air- craft. Examples involving oceanic influence include the trade-wind system (Riehl et al., 1951), the equatorial trough (Riehl and Malkus, 1958), several hurricanes (Palmen, 1958; Gangopadhyaya and Riehl, 1959), and, for the first time, a joint air-sea budget study of the Caribbean region (Colon, 1960), most of which will be discussed in some detail later. Oceanographic studies of this sort have been undertaken for whole ocean basins, and often include budgets of various solutes such as salt, carbon dioxide and other gases (Wiist, 1957). All these treatments have the basic limitation of not being predictive dynamic or thermal-dynamic models, thus precluding real understanding of time changes, stabilities and fluctuations, and thereby of causality: but they are usually required before it is known what are the important variables to incorporate in the more formal models, and how to parameterize meaningfully, which is in fact the key to successful treatment of a planetary geophysical problem. They suffer three further operational limitations, namely, first the specification of boundaries, since most geophysical systems are open ; secondly, that radiation energy sources and sinks are at best known on a long-term (monthly or seasonal) basis (daily fluctuations therein are beyond conjecture) ; and, thirdly, the effects of the turbulent (size, meters or less), convective (meters to several kilometers) and even meso-scale (kilometers to about 100 kilometers) motions are difficult to assess directly with existing observational networks and coverage. (3) Quantitative descriptive and statistical studies, directed toward finding "What is there?" and "What happens when?" in terms of numerical values, such as where is the Gulf Stream, how warm is its core, and how do these measurables fluctuate? Or what are the rainfall patterns in Hawaii, are they composed of a uniform monthly and daily precipitation regime, or is the average figure made up of a cruel see-saw between occasional flood and usual starvation? Do the statistical moments of the distribution suggest primarily orographic rainfall, or are storms required to enable the mountains to build up rain-clouds? As these questions are phrased, it is implied that such studies are generally most valuable if they either suggest or test a physical hypothesis. (4) Purely descriptive treatments, in which one attempts to say what happens in words, generally drawing on physical laws, results of fragmentary and purely visual observations, and analogies with other known phenomena. 92 MALKUS [chap. 4 The more complex, extensive or difficult the geophysical problem, the more the temptation to resort to pure description and inference, particularly where data are scarce. Climatic fluctuations in the past, which have been deduced from information ranging from lake levels, through tree rings, to human habits, have largely had to be treated in this way. The danger is that two entirely opposite effects may often be equally deduced from the same cause : for ex- ample, reduction of the transport of the Gulf Stream may be argued to cool the European climate, but it may equally soundly be argued at this level that reduction of the Stream's flow would lead to a warmer Europe (see Stommel, 1958, pp. 175-177)! The danger of misleading the layman, who is unaware of the level of inquiry, is vast and concomitant with that of discrediting sounder geo- physical studies. On the other hand, when recognized for what they are, these descriptions may pave the ground for more quantitative formulations and can help the mariner or the pilot, for example, to interpret coherently what he sees, as did the so-called "bubble model" of atmospheric cumulus, with its attractive and visualizable concepts of cloud-tower "erosion" and dilution of the warm, wet cores by "entrainment" (see, for example. Scorer and Ludlam, 1953). This chapter on air-sea interaction will draw heavily on the first three types of study, occasionally joining together their results by the fourth method. We shall begin in this way by describing how the whole system operates, trying to indicate the level of inquiry and the present state of the knowledge upon which the component parts of the description are made. For the most part, we shall proceed using the following assumptions : (a) The solar radiational input to the planet Earth does not vary from one year to the next, so that the ocean-atmosphere fluctuations are due to seasonal, diurnal and spatial variations in input, and internal instabilities. (6) The mean condition of the air-sea system does not vary from year to year, so that in its main climatological features one July, for example, is just like any other in the major global regions. These assumptions are obviously not strictly true, particularly the second which ignores internal instabilities with periods of several months or more (possibilities of which are briefly suggested at the end of the chaiDter). They place outside our scope the important topics of forced variations from solar changes, secular storage in the ocean, and the entire fascinating labyrinth of climatic fluctuation and control which are still beyond the bounds of being tractable to our science. While the explorers' day in geophysics is ending, and that of the model maker, the theoretician, and the controlled experimenter is just dawning, that of the climatic engineer probably awaits another generation. 2. How the Whole System Works The physical state of a fluid is intimately tied to how it moves or its "general circulation". Planetary fluid circulations are governed by their energy sources (which may in turn be regulated by their motions, as for example the cloudiness SECT. 2] LARGE-SCALE INTERACTIONS 93 affects solar radiation input) and the degrees of freedom or constraints limiting these. For our planet, the ultimate energy source is the sun, and the major constraints are the geometry and rotation of the earth, with an important degree of freedom provided by the change in phase of water. The high ratio of the rotation rate relative to differential heat input [formu- lated in terms of a "Rossby number" by those studying hierarchies of planetary flow regimes in the dishpan (cf. Fultz, 1956; Riehl and Fultz, 1957)] constrains /\ Both Oceons /.A, / i-A''' f'^\ \ ■"\ ^"^^/^ "e }. V o Uj H t\^ At' \ \\ \ ■' \\ \ \ \ \ \\ K"- \ c IP IC , 1 North Lotitude ~-' )0 20° 30° 40° 5 .. L 1 1 0° 6 0° Fig. 1. Seasonal values of total evaporation, "LE in 10^ m^/day, as a function of latitude for the North Atlantic and North Pacific Oceans combined. (After Jacobs, 1951a, Fig. 10.) Computed by 10° -latitude belts from charts of evaporation per unit area of sea surface for each ocean. Value for each 10°-belt entered at mid-point. Total oceanic area included: 1.27 x 10^ km^. Mean annual evaporative loss to both oceans: 112.5 cm/year. the large-scale motions to be almost "geostrophic" or nearly at right angles to the pressure gradient. Since fluid kinetic energy can only be thermally produced by "ageostrophic" flows down the pressure gradient, this strait jacket of rotation markedly complicates planetary circulations in comparison to those of simpler non-rotating thermal fluid engines, and results in the storage of large amounts of energy in "potential" form of sharp air- and water-mass density contrasts, which can be released far in space and time from the input point. Solar energy, in short wave form, is received by our planet almost entirely at the earth's surface, with less than 20% absorbed directly by the air and its 94 [chap. 4 clouds. The atmosphere is thus fuelled mainly from below, with 80% (global average) of its fuel initially latent in the form of water vapor. The latter figure is the heart of air-sea interaction, and the key to the importance of the oceans to meteorology. Of the water- vapor fuel, considerably more than half (see Fig. 1) is supplied to the lowest air by the tropical oceans between 30° North and 30° South latitude. Intervening between the input, by evaporation, of this fuel and the eventual conversion of a tiny fraction into winds and ocean currents are multifold and complex transportations and transformations. The fluid engine is highly in- efficient, converting one or two per cent of its thermal energy input into Fig. 2. Typical photograph of oceanic trade cumulus clouds. Row is parallel to easterly wind (blowing from left to right). Tallest clouds reach about 6000 ft. motion ; the vast majority of the calories received by the atmosphere are con- sumed in balancing its radiation losses to space. The free air is everywhere a radiation sink and would begin to cool off by 1-2°C per day if its heat input from the earth below were suddenly to terminate. The overall radiation balance (energy received minus that lost) of the earth-atmosphere system is positive between about 38° North and South latitudes, and negative poleward. The circulations of sea and air must, therefore, operate in such a fashion as to convey heat energy from regions of positive balance to those of deficit. Heat energy must be carried poleward, and yet, we know that in the tropical input regions the prevailing low-level winds all blow toward the equator. How is this apparent paradox resolved? To find out, we must briefly explore tropical meteorology. SECT. 2] LARGE-SCALE INTERACTIONS 95 Study of the energetics of planetary circulations thus logically begins with the exchanges at the tropical ocean surface and from there ascends into the tropical atmosphere. The fluxes and conditions near the interface have received intensive study, using the data provided by the post-war expeditions of the Woods Hole Oceanographic Institution (Wyman et al., 1946; Bunker et al., 1949; Bunker, 1955; Malkus, 1958). From the sea surface, the water vapor is stirred by turbulent convection, through a well-mixed layer several hundred meters thick, up to the level of water-vapor condensation. From there, it is pumped aloft by the so-called "trade-wind cumulus" clouds, a picturesque feature of tropical skies at all hours of the day and night (Fig. 2). These pro- cesses create a moist convective layer, gradually deepening along the airflow, from 6-10,000 ft in vertical depth. In the trade-wind zones of high evaporation, most of the moisture is retained in vapor form rather than being rained back into the oceans. In both hemispheres the trade-winds ship their accumulated water vapor equatorward, at a rate of energy export easily two orders of magnitude greater than the rate of kinetic energy consumption by all the global winds and sea currents combined. It is in the so-called "equatorial trough zone", a belt about 10°-latitude wide on either side of the thermal equator, that the combustion or condensation of the water-vapor fuel takes place. The "cylinders" in which the vapor is con- verted into liquid water droplets, releasing their 600 calories per gram of water condensed, are a relatively few giant cumulonimbus cloud towers (Fig. 3), over- grown brothers of the trade- wind cumulus mentioned earlier. These "hot towers", though averaging several thousand active at a given instant around the globe, are not uniformly or randomly scattered over the equatorial seas but are highly concentrated into a very few (about 20-50) vortical or wave-like storms, several hundred kilometers in horizontal dimensions, which go through a life cycle of a few days each (Malkus and Ronne, 1960). It is small wonder that the global circulation system operates in fits and starts, with its evanescent cylinders, of transient numbers, whose very existence depends upon the vagaries of the flow itself! The giant cloud towers play the role of fuel pump as well as combustion cylinder in the atmospheric heat engine ; their towers commonly reach 35,000 ft and frequently penetrate the tropical tropopause at about 50,000 ft elevation. The water-vapor fuel is thus converted into sensible heat and potential energy, the latter due to the work done against gravity by the cloud buoyancy. At high levels in the tropics, the average air flow is, as it must be, poleward, but it is much less uniform both spatially and temporally than the almost mono- tonous equatorward stream of the steady trade winds. Thus a direct meridional circulation cell exists in the tropics (Palmen, Riehl and Vuorela, 1958 ; Tucker, 1959). In the low levels, the equatorward component is a rather uniform 2-3 m/ sec superposed on a steady easterly trade regime; aloft, the jDoleward flow component averages out the same in the long run, although it occurs in bursts and preferred longitudes, due to the irregular and restless character of upper level tropical circulations. 96 [chap. 4 Fig. 3. Giant cumulonimbus clouds in the North Pacific equatorial trough zone. Rounded tower in top photograph reaches at least 35,000 ft. SECT. 2] LARGE-SCALE INTERACTIONS 97 In middle latitudes, atmospheric circulations are entirely different, being irregular and eddy-like near the ground, and showing an unsteady wave pattern aloft in the three or four undulations of the subtropical jet stream. No significant global-scale meridional cell circulations appear to exist. The flow is characterized by transience and strong air-mass contrasts. Along these, frontal cyclones develop at the surface, beneath shorter-lived jet-stream branches which often move southward and merge with the more persistent subtropical jet. The eddies or cyclones themselves play a major role in maintaining the westerly jet streams, by release of the potential energy stored in their air-mass contrasts or fronts. The important interaction between latitudes takes place in the energy and momentum export, from the tropics poleward across the subtropical ridge lines at about latitudes 35° North and South. While the seasonal magnitudes of the various transports are now quantitatively estimated, the relative roles of the several scales of motion in carrying them out is not yet settled beyond con- troversy ; it is still not clear whether the tropical cell may be said to maintain the subtropical jet by a rather direct kinetic energy and momentum supply, or whether most of the energy goes around the much more involved circuit of being stored in air-mass contrasts, released in synoptic (cyclonic storm) sized eddies which in turn feed it back up the spectrum to drive the jet. The end fate of the energy and momentum is, however, clear. The kinetic energy is dis- sipated into heat, largely by ground friction and stresses of the rough oceans. This tiny amount of heat suffers the same fate as the huge remainder received from the tropics, most of it being used to balance the radiation losses of the long polar nights. The westerly momentum shipped poleward across the sub- tropical ridge has also been gained from the tropical oceans, where the trade winds have given off their easterly momentum to drag the equatorial currents and pile up the waters for creating Gulf Stream and Kuroshio. This westerly momentum is given back irregularly to the earth's surface in middle latitudes, completing the northern portions of the ocean gyres and the remaining part of the stress curl that maintains the wind-driven ocean-current systems. The average annual heat-energy transports by both ocean and atmosphere together are computable as a function of latitude from the distribution of net radiative sources and sinks for the air-sea system as a whole. The magnitude of the energy transactions required of the earth's fluids in order to maintain our planet's heat balance is suggested in Table I. The transport unit is lO^^ cal/ sec, which is two orders of magnitude greater than the rate of kinetic energy dissipation by all winds and nearly five orders of magnitude greater than the rate of power consumption by human civilization. Total fluxes deduced from the radiation results reported by three recent authors (Houghton, 1954; London, 1957; Budyko, 1956) are shown in the first three columns. On the right, Budyko's figures are used to show separately the contributions from sea and atmosphere, derived in a manner to be described in Section 4 of this chapter. Particularly noteworthy is the significant role attributed to the ocean ; its fluxes are a non-negligible fraction of the total poleward transiDort. While it 98 MALKUS [chap. 4 may be pointed out that the sea's contribution hes within the disagreement of the net radiation figures (discrepancy between columns two and three, for example) we shall see later that this objection is not quite relevant ; independent oceanic heat budgets of several authors converge on this general magnitude, rendering it at least a serious postulate for further examination. Table I Mean Annual Energy Transport across Latitude Circles Unit: 1015 cal/sec Air plus sea total Breakdown (based on fig ures of Bu dyko, 1956) Lat. Houghton London (1954) (1957) Budyko (1956) r Sea Air „ _,. sea Katio, ^— ' air /o Air : sen- sible heat plus potential ^ Air : latent heat in water vapor energy 60°N 0.77 0.53 0.76 0.07 0.69 10 0.68 0.01 50°N 1.11 0.71 1.05 0.13 0.92 14 0.76 0.16 40°N 1.29 0.81 1.20 0.18 1.02 18 0.77 0.25 30°N 1.21 0.77 1.10 0.21 0.89 24 0.68 0.21 20°N 0.89 0.60 0.84 0.13 0.71 18 0.71 0.00 10°N 0.47 0.33 0.46 0.03 0.49 6 0.63 0.14 0° 0 0 0 0.30 0.30 100 0.17 0.47 10°S 0.46 0.48 0.02 — 0.49 0.51 20°S 0.91 0.51 0.40 127 0.58 0.18 30°S 1.20 0.46 0.74 62 0.58 0.16 40°S 1.24 0.32 0.92 34 0.56 0.36 50°S 1.05 0.22 0.83 26 0.54 0.29 60°S 0.71 0.10 0.61 16 0.46 0.15 Underlined fluxes directed toward South Pole, all others toward North Pole While an annual global average picture of the earth's ocean and air circula- tions and their transports is valuable in depicting the overall operation of the media controlling man's environment, one of the most exciting and important paradoxes of earth sciences lies in the changes and fluctuations therein. Except for man himself, the weather is probably the most variable, unreliable, and fiuctuatory phenomenon of which human intelligence has dared to attempt a science. Since the input of short-wave solar radiation is relatively regular, stable and independent to first order of the circulation, ^ we must probably look 1 Except possibly for its dependence upon the average cloud distributions over wide regions, fluctuations of which are not well-known and may be quite small. SECT. 2] LARGE -SCALK INTERACTIONS 99 within the air-sea system itself for the source of the major instabihties we experience. In doing so, a finger is pointed to air-sea exchange. The main roles of the oceans in the global budgets and circulations are, we have seen, as suppliers of the atmosphere's heat (primarily latent) and removers of its momentum. The dishpan experiments (Riehl and Fultz, 1958) suggest that the circulation of a rotating planetary fluid is more critically governed by its energy sources and sinks than by its momentum sinks. The energy sinks of the atmosphere are radiational and to a first order relatively independent of latitude, time and small-scale fluctuations (Rossby, 1959; Budyko, 1956). The direct energy sources for the atmosphere are primarily precipitation, and secondly transport convergence in its own circulation and input of sensible heat from the earth's surface (see Fig. 15, page 135). The indirect source of the first two of these is, as we saw, mainly evaporation from the oceans. The most important demonstration in marine meteorology of the past twenty years (Jacobs, 1951; Montgomery, 1940; Bunker, 1960; Riehl et al., 1951) is that transfer of latent and sensible heat (and momentum) from sea to air is governed largely by just two parameters, the air-sea property difference and the pre- vailing wind speed averaged over about one hour. The clue to the paradox above is that these two parameters at a given place and time are a direct function of the so-called synojjtic pattern, or synoptic scale of motion — the individual circulation features seen on daily weather maps, of sizes 100-1000 km and life-cycle of only several days duration. In middle and high latitudes these are the travelling frontal cyclones and anticyclones, and in the tropics the easterly waves, equatorial vortices, shear lines, and polar troughs whose budgets and dynamics are just beginning to be investigated. Only in the low- level trade-wind region does the average climatological picture bear a close resemblance to the flow one finds on a given weather chart, and even here the synoptic-scale fluctuations are proving more significant than hitherto suspected, particularly with respect to rainfall, which is highly concentrated in limited disturbances. Therefore, while the average air-sea fluxes over long periods and large regions are computable from and constrained by planetary budget requirements, the fluctuations which build this picture and give rise to the enormous departures from it, so important to human life, are directly governed by transient atmospheric phenomena, themselves the product of circulation instabilities. We shall pursue our discussion of air-sea interaction first by an analysis of exchange calculation methods and their range of validity, leading to a presenta- tion of the overall global climatology of exchange, to the extent this is known today. Using this as a framework, we shall then examine exchange and its fluctuations in some more restricted portions of the system, in which the more detailed operation of the parts have been analyzed, working down finally to the dynamics of exchange in individual synoptic-scale systems which give rise to the variations about the climatic picture they build. In the latter, we shall concentrate primarily upon air-sea interaction in the fuelling and firebox regions of the tropical and equatorial zones. 100 MALKUS 3. Determination of Air-Sea Fluxes [chap. 4 The classical works on global air-sea fluxes and the overall climatology of exchange have been a series of heroic researches by W. C. Jacobs beginning in the early 1940's (Jacobs, 1942, 1943, 1949, 1950, 1951, 1951a). Recently, a series of Russian studies, carrying out and extending the same type of computa- tion with a more modern and complete set of data, has become available ; these data have been summarized in a monumental work by Budyko (1956) which will be drawn upon heavily in this chapter. All in all it may be said that both the methods and results of Jacobs and his predecessors (Wiist, 1936; Mosby, 0.5 - 1 I 1 1 1 1 1 -' ^MOSBY 04 - Cwusr r^^Cl^^?^^^'^''^ ^\. 03 - / y '-JACOBS 0.2 - ^y- - 0.1 " 1 1 40° North 30° Lafi tude Fig. 4. Mean annual evaporation in North Atlantic and North Pacific together as function of latitude. Comparison of determinations of several workers: Mosby (1936), Wiist (1936), Jacobs (1951a) and Budyko (1956). 1936) have borne the test of time remarkably well. First the computational procedures and assumptions have been subject to considerable test by use in a variety of situations (cf. Sections 5-7 in this chapter) and some (still limited) direct verification in actual aircraft flux measurements (Bunker, 1960). Secondly, the actual figures arrived at by the Russians and other recent workers differ very little indeed from those presented earlier (see Fig. 4). In short, air-sea flux computations appear to be stable and reproducible, and to fit in consistently with mechanistic, budget and dynamic requirements on numerous scales. Direct measurements of evaporation and sensible heat flow from sea to air are difficult, doubtful and scanty. What is wanted is a method of obtaining reliable exchange values at numerous points over the oceans as a function of time, from measurements cheaply and routinely obtainable. At present, this ideal is realized only approximately. Sensible heat and water-vapor fluxes between ocean and atmospliere may be computed indirectly in two independent SECT. 2] LARGE-SCALE INTERACTIONS 101 ways, the one using overall energy and mass budget requirements, the other based upon the so-called "exchange formulas" which, via the laws of small- scale molecular and turbulent transfer, give the exchange as a function of the air-sea property difference and the wind speed. The budget method, to be described first, is only suitable for arriving at averages over rather large regions ( ~ 5° latitude) and long periods (monthly or longer) since it depends on radiation figures which are only computable for these intervals. The scale of applicability of the transfer formulas, discussed secondly, is dependent on the observations used ; they appear to give satisfactory results from the climato- logical scale down to periods of one hour and to resolve meso-scale variations if desired. A. Energy Budget Method of Air-Sea Flux Computation Let us consider the heat-energy budget of a deep ocean column of unit area in order to arrive at the transfers across its surface. Of the total impinging solar radiation, some is reflected and some is reradiated back to space in long wave lengths. That which is radiated to the atmosphere and reradiated back to the sea, due to the "greenhouse eflFect", does not directly enter the budget equations as a net gain or loss for the ocean column, although it is obviously of major indirect importance in governing the temperatures, radiation balances and other features of the air-sea system and its parts. The net amount absorbed, called the "radiation balance" of the surface, is used in three ways : it may raise the temperature of the ocean column, be carried away by sea transports, or be supplied to the atmosphere in latent and/or sensible form. The law of energy conservation is used to express this heat balance quantita- tively, as follows : R = Qs + Qe+S + Qvo, (1) where the units of each term are heat energy per unit area per unit time, commonly cal cm~2 sec~i. Qs plus Qe is the heat energy supplied by the sea to the atmosphere in sensible, Qs, plus latent, Qe, form. The latter may be written as LE, where L is the latent heat of vaporization in cal g~i and E is evaporation in g cm~2 or in cm of water per cm^ ; the ocean recognizes the sum of these as a sensible heat loss. S is the ocean storage, jDositive for increased heat content of the column ; it is assumed zero in the climatological calculations of annual budgets (assumption (6), p. 92), although its maximum seasonal values may exceed one-third Qe. Shorter term storage fluctuations are generally unknown and pose great difficulty in smaller scale budget analyses. Qvo is the flux di- vergence of oceanic heat transports, assumed for large areas to be due to advective currents although for restricted regions lateral eddy fluxes might require inclusion. Omitted from equation (1) are heat sources due to dissipation of kinetic energy of air and ocean, and that due to radioactive decay in the earth's interior transmitted through the ocean bottom. The former may, in the extreme, amount to 1% of the dominant terms or 3 cal/cm^ per day, while 102 MALKUS [chap. 4 the latter is estimated as not above 0.1% or 0.3 cal/cm^ per day. These are well within the accuracy of presently possible evaluations of the larger terms and surely negligible for the annual and seasonal budget studies to be presented here; for long ])eriods and the consideration of climatic change their omission may not be so readily justified. R is the radiation balance of the surface, and, under certain restrictions, may be independently calculated using the following form of the energy conservation law, namely : where Q is the sum of direct short-wave radiation, q is the sum of diffuse short- wave radiation, a is the albedo of the surface, and Qb the "back" or "effective outgoing" radiation, which is determined by the difference between "green- house" radiation of the atmosphere and outgoing radiation from the surface. Thus the total exchange, Qs plus Qe, may be computed as a function of space and time by combining (1) and (2) to give Q^ + Qe = {Q + q){l-a)-Qo-S-Qro. (3) Equation (3) and its solution are the essence of exchange determination using the energy budget method. Aside from problems raised by ocean storage S and heat flux divergence Qvo, the range of validity and resolution (space- and time-wise) of the method depend upon the adequacy and resolution of deter- minations of JR. The latter has been discussed extensively in the radiation literature (London, 1957; Houghton, 1954; Budyko, 1956). The critical variable entering E, from the meteorological-oceanographic standpoint, is atmospheric cloudiness, which affects both (Q + q) and Qf,. Under cloudless conditions, Budyko among others has presented tables for both {Q + q)Q and Qbo (subscript 0 denotes "cloudless"). The constants in his formulas have been evaluated from extensive actinometric measurements at varying latitudes, mainly in the U.S.S.R. Similar tabulations have been evolved recently by London (1957), who compares his results with those of the earlier workers used by Sverdrup (1942) in his pioneer endeavors to construct heat budgets of the oceans. To obtain {Q + q)o we need to know the solar radiation impinging at the top of the atmosphere and subtract that depleted by the air column due to absorption, scattering, and reflection by the atmospheric constituents. The fact that Budyko's (and other workers') tables give {Q + q)Q as a function of latitude and month alone implies an assumed longitudinal constancy and temporal reproducibility in the air's transmissivity. The albedo of a water surface to direct solar radiation has been generally found less than 10% except for extremely low solar angles ; the reflectivity for diffuse sky radiation is somewhat larger. Equation (1) ignores this diff"erence, as do most radiation tabulations, which commonly also neglect effects of surface roughness and present a as a function of latitude and month, thus in- cluding variations in integrated sun angle only. Under these restrictions, all agree that oceanic albedo ranges from about 6-11% between 40°N and S. SECT. 2] LARGE-SCALE INTERACTIONS 103 The net back radiation under cloudless conditions is the difference between long-wave radiation emitted by the surface, which radiates very nearly like a black body, and that received from the atmosphere, due mainly to its water- vapor content, ^^o thus increases with temperature and decreases with vapor pressure ; within the common range of these parameters the variation is, in fact, a rather slow decrease with rising temperature, due to the warmer air's higher capacity to hold water vapor. Sverdrup (1942, p. Ill) has published a con- venient graph from which Q^q may be read as a function of sea-surface tempera- ture and the relative humidity at ship's deck level. Budyko's table {loc. cit., Table 7), worked out in terms of low-level air temperature and vapor pressure over soil, gives values about one-third lower in the region of overlap. The important departures from these tabulations, causing the actual surface radiation balance to be altered by effects of the circulation lie first in the rather strong reduction in Q + q by clouds and secondly in the much weaker reduction of Qb thereby. Until recently the radiation hterature has asserted that, to the first order, satisfactory corrections for cloudiness to the cloudless tabulations are obtainable from knowledge of the average fractional cloudiness alone, ignoring types, heights, and the exact nature and distribution of the clouds. These correction formulas are based on the assumption that cloud albedo is constant and independent of thickness ; uncomfortable doubts have been creeping into the literature as a result of more measurements (see Budyko, loc. cit., and review by Charnock, 1951). It is at this point that determinations of energetic relations depending on radiation balance, as do all terrestrial energy budgets, lose their resolution. Worse yet, by being forced to assume and introduce mean cloudiness as given, the feedback between source and circulation is removed, and such studies thereby lose their ability to treat dynamics and causality with predictive ability or rigor. Furthermore, even mean cloudiness over wide portions of the oceans can hardly be said to be well known observationally, while temporal and regional variations therein are presently j)ure conjecture. However, satellite measurements may shortly relieve the latter situation and radiation work in progress offers hope that the radiative properties of different cloud forms may be treated. Meanwhile, we must proceed to study ocean-atmosphere energy relations using the best methods available. An important clue to the interaction between heat sources and motions may be provided by the demonstration that the average monthly radiation balance over wide portions of the oceans decreases approximately linearly with mean cloudiness. Budyko shows that the incoming short-wave radiation obeys a relationship of the form (Q + q) = {Q + q)o[l-{l-k)n]. (4) which he calls the Saviiio-Angstrom formula. Here n is the mean fractional cloudiness and k is an empirically determined constant ranging from 0.35 to 0.50 in its weak increase with latitude. This empirical variation of k is an attempt to include the effects of different altitudes and thicknesses of the 104 MALKUS [chap. 4 climatologically prevalent cloud forms. A similar formulation by Mosby (1936) was used by Sverdrup ; detailed discussion and criticism of the physics under- lying these approaches is found in the work of London (1957). For our purposes, it is important to note that incoming radiation has decreased to about half its cloudless value as the mean cloudiness approaches eight-tenths. For the reduction in back radiation due to clouds, Budyko's evidence suggests a faster-than-linear decrease, namely, Qb - ^^o(l-Cw'«)> (-5) with more adjustable constants than the earlier linear formulation used by Mosby and Sverdrup which was Qb = Qboi^-O.SSn). (5a) In (5), m= 1.5-2.0, and C, tabulated taking into account mean frequency of clouds at various heights by latitude, ranges between 0,5 at the equator and 0.8 at high latitudes. Since (5a) gives a sharper reduction in Qi, with n than does (5), we find for average sky covers of about five-tenths that the differences between Sverdruj)'s larger and Budyko's smaller Q^q are considerably com- pensated in the final estimate of Qb. Despite the faster-than-linear decrease in (5), even here Qb is only very weakly dependent on cloudiness. For a given latitude and season, therefore, the mean radiation balance may be well approximated by a relation of the form R ^ A-Bn (6) where the constants A and B are obtainable from the radiation tables. Such a relationship might provide a first start at modelling the interaction between energy input and circulation dynamics, particularly in the tropics where mean cloudiness appears to be, to a first order, directly dependent upon the low-level convergence in the flow (Malkus and Ronne, 1960). For demonstration, a sample calculation of i? as a function of cloudiness in tenths is presented in Fig. 5 for latitude 20°, in February, using the radiation figures of Budyko's Tables 1, 2, 6, 7 and 8. It is also now possible to assess the degree of feasibility and adequacy of quantitative air-sea exchange determinations using the energy balance method set forth in equation (3). For annual averages, the storage S, though probably important in long-period climatic changes, is surely negligible in comparison to the other terms and their accuracy. In the months of February and August, when the ocean heat content reaches its minimum and maximum, respectively, it is zero. The net flux divergence, Qvo, is similarly negligible for some regions, and for whole oceans, and may be calculated and included for those basins, such as the Caribbean-Atlantic (Colon, 1960), where measurements of current transports are available. While the foregoing limitations, especially in the radiation evaluations, greatly restrict the scope of the energy equations, particularly in time- dependent, dynamic, and short-period studies, there are significant and geo- SECT. 2] LARflK-SOALE INTERACTIONS 105 physically valuable occasions when the energy balance method may be used in average ocean-atmosphere budget studies over a month or more. Selected examples will be described in the following sections of this chapter. The test of the usefulness of the approach is whether a consistent quantitative picture can be constructed, supported by a sufficient number of independent determina- tions, whose major features and conclusions do not lie within the margin of error of the computations. In practice, the equations may often be used to obtain the sum of heat-energy exchange, Qs plus Qe, which then may be sepa- rated if either one alone or the ratio between them is determinable. Jacobs, in 1 1 1 Lot 20° - February ■- ^^^ R -- A- Bn ^\^ R = 028-OZn coi/crr 2 . 0,30 ^^\<^» 0.20 ^^^~~~^ - O.iO 1 1 1 1 1 I 1 1 ;;;- Mean fractional cloudiness n — »- Fig. 5. Graph illustrating approximately linear decrease of radiation balance, R, with mean fractional cloudiness, n. R is the difference between Q + q (equation 4) and Qij (equation 5). Constants A and B found graphically by fitting closest straight line to R curve. Constants in equations (4) and (5) taken from tables in Budyko (1956) for latitude 20\ February. Compviting Qg from R and equation (1), ignoring S and Qio and assuming Qs = ~ioQej we find (in kg cal/cm^ month) Qe is 11.1, 7.5 and 3.3 for cloudiness of 0, 40 and 100%, respectively. Compare with Table XI, line 11 and Fig. 29. his classical studies, used the work of Bowen (1926) to assess their ratio r, which may be shown to depend linearly upon the ratio of the respective air-sea property differences as follows "" Qe L Tc qo-qa where Cp is the specific heat of air at constant pressure, L is the latent heat of condensation. To is the sea-surface temperature and Ta is the air temperature at about 6 m height, both in degrees Centigrade, go is the saturation specific humidity at the sea-surface temperature, and qa is the actual atmospheric specific humidity at 6 m, both in grams per gram. The origin and range of validity of (7) will be discussed below in connection with the transfer formulas. 106 MALKUS [chap. 4 Therefore we have for an annual average Q, = LE = {R-Qro)j{\+r). (8) It is worth noting that, since climatologically r is generally small compared to one, particularly in the tropics where it averages less than 0.1, uncertainties in its determinations are not serious in computing evaporation. Jacobs used equation (8) to obtain evaporation over portions of the oceans where Qvo is known to be small. He then used the resulting Qe to obtain the proportionality constant in the transfer formula ; the latter was applied, using extensively available meteorological data, to assess the distribution of evaporation over all the Northern Hemisphere oceans (summarized by the solid curve in Fig. 4). As a retrospective check, he found that in those regions where Qvo should be significant, the departures of the results of (8) from those of the transfer formula were logically explainable in terms of major current transports. Ideally, air-sea budget studies could be completed by independent determina- tion of all terms in (8) ; however, due primarily to inadequate data and still, to some extent, to imperfect or quasi-empirical physical laws of restricted applica- bility, this is rarely possible. Most existing studies of air-sea exchange and budget energetics are, therefore, forced to restrict their range of validity by neglecting terms, by approximating, and often by computing one term as a residual, to be checked by fragmentary observations and consistency. Despite these obstacles, studies of this sort stand today as one of the largest single contributions to meteorology and oceanography, and have been paramount in the enormous increase in our knowledge of the tropical fuelling region, as will be illustrated in the remainder of this chapter. The major step in enabling these studies, and giving confidence in the general soundness of their results, is the development of the exchange formulas and their placement upon a sound foundation, which has progressed greatly since the days of their first courageous use by Jacobs twenty years ago. B. Exchange Computations by the Transfer Formulas The classical works in this area were carried out in the decade 1930-1940 by Rossby and Montgomery (1935; 1936) and by Montgomery (1940). Their aims were a quantitative formulation of the turbulent structure at the air-sea inter- face and a method of deducing thereby the fluxes of heat, moisture and momen- tum from the ocean surface through the lowest decameters of the air above. In the intervening years, this approach has been refined, examined critically, extended and tested, particularly by British and Australian workers (Sheppard, 1958; Deacon, Sheppard and Webb, 1956; Priestley, 1959), some of whom report on it in detail elsewhere in this volume. For present purposes, we shall be interested mainly in the application of the results to the larger-scale air-sea interaction problem ; it is, however, necessary in doing this briefly to re- capitulate the basic foundations and assumptions of the formulations in order to delimit their range of applicability to our problem. Fortunately, as we shall SECT. 2] LARGE-SCALE INTERACTIONS 107 see, the premises used in deriving the transfer formulas are best appHcable to the very regions in which we are most interested, namely, the tropics, sub- tropics and, secondarily, the western portions of middle-latitude ocean basins, where air-sea transfer is largest and dynamically most important. The basic premise underlying the development is that turbulent exchange is the dominant mechanism affecting the vertical distribution of property from the interface to several tens of meters above it, so that the fluxes obey equations analogous to the heat-transfer equation of classical physics, namely, F,= -k/£. (9) where Fp is the flux of the property p involved (unit property per cm^ per sec), dpidz is the vertical gradient of the property, and Kp is the so-called "eddy transfer coefficient" (units g cm~i sec^i), many orders of magnitude larger than the corresponding molecular transfer coefficient. Starting with equations of the form (9) for momentum, water-vapor and sensible heat flux, and introducing several assumptions as to the nature of the turbulence, we can derive equations relating each flux at the boundary to a wind speed u at "anemometer level" (usually 5-10 m) and the difference between the property in question at that level and at the sea surface. The momentum flux formulation is basic to the other two ; in order for an equation of the form (9) to govern vertical transport of horizontal momentum, or shearing stress r, the vertical variations of other forces acting on the air parcels must be negligible in comparison to that of the turbulent frictional forces. In some regions, it has been demonstrated observationally (Sheppard, Charnock and Francis, 1952) that the vertical variation of horizontal pressure forces (thermal wind) is significant even in the lowest few meters ; however, the lower trade-wind air is nearly "barotropic", that is to say the pressure forces are nearly constant in the vertical, so that this difficulty is minimal. Equation (9) predicts the distribution of a property above the sea, if enough can be said about Fp and Kp in order to integrate it in the vertical. In the case of momentum, Fp is constant with height in a steady state if the barotropic condition is met ; for heat and water vapor, Fp is constant in the vertical through a thin boundary layer in which heat and vapor are not accumulating. Furthermore, the classical turbulence work of Prandtl (1932) and von Karman (1935) may be drawn on to argue that, in cases of neutral static stability, Kp should increase linearly upward. Upon integrating (9), we then derive p{z)(x: In z, (10) the famous logarithmic profiles upon which the transfer formulas are based. In recent years, over-extension of the "eddy exchange coefficient" concept and attempts to apply equations of the form (9) to situations where the basic premises are far from realized have led to an unfortunate discrediting in meteorology of the entire approach. Actually the low-level profiles of the 108 MALKUS [chap. 4 several properties in the atmosphere have been extensively measured (Mont- gomery, 1940; Deacon, Sheppard and Webb, 1956) and are indeed logarithmic when the stability is neutral, a wind is blowing, and the other physical assump- tions are fulfilled. Furthermore, recent and more fundamental work on turbu- lent shear flow (W. V. R. Malkus, 1956) has deduced theoretically the logarithmic law, relationship (9), the constancy of t, and the linear ratio {duldz)JT near the boundary in a simplified situation where the assumptions outlined are clearly fulfilled. It should be cautioned, however, that other theoretical work by the same author (W. V. R. Malkus, 1954), equally sup- ported by controlled laboratory experiment, predicts that when heat flow alone occurs in the absence of a shearing current, neither the logarithmic law, nor linear increase of Kp is realized. Therefore, the range of validity of the transfer formulas to be deduced should be restricted to conditions of a significant wind blowing over the ocean, where turbulent shear flow is the major process govern- ing the transports of all properties. Happily this condition is almost always realized over the oceans where large sea-air fluxes are occurring, particularly in the trades. Obviously, since the exchange formulas give fluxes linearly proportional to wind speed u, they will give incorrect (zero) results as the wind speed vanishes. Therefore, it may be said that the use of the exchange formulas to obtain fluxes from the tropical oceans to the overlying air is probably as valid a use of simple physically deduced laws as occurs anywhere in large-scale meteorology or oceanography. In the lowest few tens of meters there the air is well-stirred, shear-turbulence dominated (relatively free of pronounced "selective buoyancy" forces), neutrally stable and barotropic. The useful forms of the exchange formulas are readily derived using the above assumptions from equations of the type (9) for each property of interest, namely, jr du az ^ J. Tde dT ..., Q, ^ -CpKs-^-^x -CpRs-^ (12) and Qe ^ LE = -LKe^^ (13) dz where equation (12) for the dependence of the sensible heat flux, Qs (which strictly speaking should be written proportional to {TI6)d9ldz), has been approximated as proportional to the actual temperature gradient. This is clearly valid in surface layers only a few meters in depth (see Chapter 3, p. 66). It will now be assumed, partly for simplicity in derivation, that, firstly, the height above the sea at which all three atmospheric properties are measured is identical, and, secondly, that Km = Ks = Ke = K (14) SECT. 2] LARGE-SCALE INTERACTIONS 109 SO that we may substitute from A = r/^^ (lla) into (12) and (13) in order to express the fluxes of heat and vapor in terms of \^'hat we may deduce about stresses and wind profiles. Furthermore, Tq = pCDUa^, (15) where the subscript 0 refers to the surface and a to "anemometer height" or the level of measurement. Equation (15) may either be deduced from observa- tions or from the classical turbulence work (see Rossby and Montgomery, 1935), in which case the drag coefficient Cd is expressed in terms of surface roughness, anemometer height and a universal constant (von Karman's). Finally we exjjress the derivatives in (11-13) in terms of finite differences for the vertical interval Az = Za — 0, so that Au = Ua — 0 = Ua AT = Ta-To = -{To-Ta) (16) Aq = qa-qo == -{qo-qa) and substitute relations (16), (14), (lla) and (15) into (11-13) to obtain the three transfer equations, all similar in form, namely. To = pCDUa^, (17) Qs = pCpCD{To-Ta)Ua, (18) Qe = LF = pLcD{qo-qa)ua. (19) The foundations, shortcomings and range of validity of this simplified derivation are examined with more care in the previous chapter, in relation to the internal structure of the interface layer. Operationally, equations (17-19) are the forms in which the exchange formulas are used, expressing the transfer in terms of the low-level wind speed, Ua, the difference in property between sea- level and the level a, a few meters above, and a parameter, usually taken as a constant. Montgomery (1940) has shown that equations of this form are derivable even if the measurement levels are not the same for the three proper- ties (the leading parameter is very insensitive to small variations in measure- ment height) and that the form of the equations is not dependent upon the equality of coefficients as assumed in (14) provided their height dependence is linear. He made detailed models of the lowest few centimeters, in which he believed the transition from molecular to turbulent regime to occur, with quite different modelling assumptions for the so-called "hydrodynamically smooth" and "hydrodynamically rough" sea. He thus deduced a sharp increase, of a factor of 2-3, in Cd as the sea went from smooth to rough. The details of his transition-layer modelling are uncertain and extremely difficult to test. We prefer the above formulation in view of usefulness, simplicity and in the light 110 [chap. 4 of more recent evidence. The latter suggests that the whole approach may- break down when winds are very weak. On the other hand, when winds of several meters per second or more prevail over the open sea its surface is rarely if ever smooth, and the roughness appears to increase more or less continuously. When turbulent shear flow is the dominant process, the assumption that Ks = Ke = Km appears to be satisfactory from observation (Budyko, 1948; Pasquill, 1949; Timofeev, 1951; Rider, 1954), although no sound theoretical foundation for it is as yet available. Recent observational work (see Fig. 6) on the dependence of cd on wind speed (roughness) over the ocean suggests a more gradual increase with increasing wind, the total range being about a factor of two, between a cd of about 1.0 x lO^^ for winds near 10 knots and 2.0 x 10"^ for winds near 20 knots. 0004 0003 0002 K no ts 12 20 24 28 slightly stoble neulrol slightly unstoble unstoble 0 2 4 6 8 10 12 14 Wind speed (m/sec) Fig. 6. Relation between drag coefficient cj) and wind speed at 10 m elevation. (After Deacon, Sheppard and Webb, 1956, Fig. 5. By courtesy of the Journal.) Top abscissa in knots, bottom in m/sec. Squares are unstable observations ; tri- angles slightly unstable ; crosses neutral ; and circles slightly stable. Solid curve drawn for neutral conditions. Dashed curve indicates earlier determination of cj) by Montgomery (1940) with sudden transition between hydrodynamically smooth (wind speed < 6.7 m/sec) and rough (wind speed > 6.7 m/sec) conditions. Therefore, ideally speaking, equations (18) and (19) provide totally in- dependent checks on the calculation of the sum of Qs plus Qe by the energy- budget method using (3). The derivation of the so-called Bowen ratio, r (7), is now also clear; it is merely the result of dividing (18) by (19). Unfortunately, available data are rarely adequate for this ideal to be achieved ; the common situation, as we shall see, is that both methods must be used together to study the energy budgets and operation of a portion of the ocean-atmosphere system. Furthermore, actual usage of equations (17) to (19) in budgetary and dynamic analyses requires further consideration, particularly when average exchanges over long periods are desired. Strictly speaking, hourly measurements of wind SECT. 2] LAKGE-SOALE INTERACTIONS 111 speed and property difference should be obtained, the product made of their product with the cd read from Fig. 6 and the average of this sum taken as the average transfer at a given locahty. Numerous local measurement points should have observations available to be dealt with in this manner in order to draw isopleths of exchange to compare with radiation balances, etc. Clearly this situation is not and never will be realized, and we must usually calculate fluxes from equations (17) to (19) using climatological (monthly or longer) averages of property differences and wind speed, and a mean value of Cj) appropriate to the latter. We may analyze the errors in these determinations by breaking down the factors entering an equation of the type (18) into long- period averages, designated by a bar, and hourly departures therefrom, designated by primes, namely, ^ 2 pCyjCp + c' d){AT + AT'){Ua^ N [18b) pCpCD A T Ua + pCp[CD{A T'u'a) + Ua(c'j) A T'] + AT{c'jju'a) + c'i,AT'u'al (18c) where To—Ta has been written AT, variations in p are assumed negligible, and N is the number of hourly observations, which are indicated in (18a) by unprimed unbarred quantities. The first term on the right in (18c) is the heat flux calculated from the long- period mean values of the observables and we wish to examine the closeness of its approximation to Qs : namely, to find out how good is the relation Q's = il pCpCD ATua)IN ;^ pCpCo ATua. (18d) This depends upon the size of all the neglected terms in the brackets in (18c). If the relationship between c/> and Ua is satisfactorily given by Fig. 6, then the adequacy of using climatological average data in determinations of air-sea ex- change depends entirely upon the correlation between air-sea property dif- ference and wind speed, and the standard deviations of each. A sample examination of these correlations and comparisons of the type indicated in (18d) has been made using the ship observations of the Wyman- Woodcook expedition to the Caribbean in April, 1946 (Wyman et al., 1946). The results are presented in Table II. In considering sensible heat flux Qs, a small but statistically significant negative correlation ( — 0.22) was found between air- sea temperature difference and wind speed, suggesting that the ocean surface layers are allowed to warm up in the absence of stirring by strong winds. The larger negative correlation between the wind speed and surface temperature confirms this. However, the standard deviations in AT and Ua were such that Qs computed in the two ways of (18d) differed by less than 7%. A similar formulation and analysis for latent heat flux Qe was performed with nearly 112 [chap. 4 identical results. In this case, use of climatological mean values for the period would have led to an underestimate of about 7 % in both fluxes and no error in their ratio. Similar analyses have been carried out for two other locations in the tropics (Atlantic and Pacific equatorial trough zones) where synoptic-scale disturbances are more frequent, but no significant differences from Table II were found. Table II Comparisons of Air-Sea Fluxes using Means and Fifty-Nine Three-Hourly Ship Observations (from Wyman- Woodcock (1946) Caribbean Data) Sensible Heat Flow n 1 p(^P^D A Tva cal/cm^/day Qs ^pCpCD ATua, cal/cm^/day Correlation coefficients, R 11.2 10.5 RAT,Ua= -0.220 ■RTo,Ma= -0-248 Rrp^.^^= -0.183 Ma = 6.56 ni/sec Standard deviation Aq= 5.89 X 10 3 Standard deviation Ci,= 1.23x 10-3 Difference of two «a= 1-5 m/sec ZlT = 0.30°C e.'s = 7% Moistuke Flux, Qg = ■ LE n 2 pLcD AqUa Qe ^ cal/cm^/day Qe = pLcj) Aqua cal/cm2/day Correlation coefficients, R 360 r = Qs/ge = 0.031 337 r = QJQe = 0.031 RAq,Ua— —0.46 ^9o,Ma=-0-68 RQa,Ua= +0-23 Standard deviation Zlg = 0.74x 10-3 Difference of two r's = 0% Difference of two It may be concluded tentatively from these small samples that use of mean values for the constant and of climatological data in the transfer formulas for heat and moisture are satisfactory within the purposes for which such long- period calculations are generally made. Many recent workers in meteorology (Riehl et al., 1951 ; Riehl and Malkus, 1958; Colon, 1960) have used equations (18) and (19) in the form Qs = 4.16xlO-7(To-T«)Ma, Qe = LE = 1.71 X 10-^ L{qo-qa)ua, (20) (21) SECT. 2] LARGE-SCALE INTERACTIONS 113 where the constants were obtained from (18) and (19) taking p = l,2x 10^3 g cm""^ Cj, = 0.24 cal g~i deg~i and Cd= 1-4 x 10^3 (mean wind speed ~ 14 knots from Fig. 6), where all units are c.g.s. and the measurables may be mean values for the period considered. The problem of arriving at momentum flux using a formula of the type (17) and climatological mean data is much more difficult and uncertain. Since each component of shearing stress is a directed quantity, the formulation analogous to (18 a-d) is more complex and since the wind speed squared appears, the errors in the climatological approach are almost certain to be larger than those for heat and moisture. Such calculations, their limitations and results are described in more detail in Section 6 on Large-Scale Momentum Relations (p. 180). A method of testing results of computations from these exchange equations by direct flux measurements has recently come into existence with the develop- ment of the instrumented and calibrated research aircraft (Bunker, 1955). While direct comparisons of the two methods are still very limited, the results to date are encouraging. The aircraft method is based on measuring the actual vertical transports of the property involved, using the definitions T = -puW, (22) Fh = pCpT'w', (23) and Fe = pq'W, (24) where r is the vertical momentum flux, Fh the vertical heat flux, and Fe the vertical water- vapor flux through the level in question. The aircraft is flown horizontally at 50-100 ft above the sea and a continuous record is made of wind speed, u, temperature, T, specific humidity, q, and vertical velocity, w, throughout the run. The primes denote the departure from the mean values of these quantities. The method is presently able to include fiuxes accomplished by elements in the size range from about 10 m to 10 km, cutting off at the small end due to limitations in speed of response of instruments and aircraft, and at the large end due to limitations in run length and aircraft calibration to deter- mine w'. Recent calculations by Bunker (1960) of heat fiuxes from (23) and (20) in comparison show that the direct method gives values proportional to ship- board determinations of the product of property difference and wind speed, but about 25% lower numerically. It appears likely, however, that the discrepancy is due to height difference and to the limitations of size spectrum coverage by the aircraft method. Although many more such careful comparisons are required, and will be forth- coming, it appears that the exchange formulas are on a sufficiently sound footing at present to enable useful fiux computations to be made from simply obtained and relatively plentiful shipboard observations, particularly the accurate ones provided by weather ships and research vessels. That we are able in this way 5 — s. I 114 MALKUS [chap. 4 to determine heat, water and momentnm exchanges between sea and air for jDeriods from hours up to yearly averages is, in fact, a major cornerstone upon which the present-day foundations of marine meteorology are built. Further- more, the functional relation that these formulas prescribe between air-sea property difference and wind speed provides a key to the role played by the air circulation features in regulating their own energy sources, and thus lies very close to the heart of planetary circulation dynamics on a wide range of scales. 4. Climatology of Energy Exchange and the Global Heat and Water Budgets In this section, we shall first present mean annual distributions ofQe and Qs, the latent and sensible heat fluxes from sea to air, and then the other heat- balance components (equation (1)) of the ocean surface. The resulting annual heat budget of the ocean forms a foundation for discussing the global heat and water budgets, the climatological picture of exchange and its average seasonal variations in selected regions. We thus develop a quantitative description of the operation of the whole system and the function played by the various parts of the sea and air in its machinery ; the remaining sections will consider the mechanisms of the described energy transactions and their implications to circulation energetics and dynamics. A . Mean Annual Distribution of Latent and Sensible Heat Exchange The basic materials for this discussion are the recent Russian calculations of Qe and Qs and the other heat-balance components over the oceans (Budyko, 1955, 1956; Drozdov, 1953) and their comparison with determinations by Jacobs (1951a), Sverdrup (1957), London (1957), Houghton (1954) and others. Annual mean maps of the geographic distribution of Qe and Qs after Budyko are shown in Figs. 7 and 8. Similar maps for each month are available in the Atlas of the Heat Balance (Budyko, 1955) but are not reproduced here. Depart- ures of these mean annual values from those of Jacobs are indicated in Tables III and IV. Table V gives the corresponding values of the Bowen ratio {r = QslQe)- Particularly in the case ofQe {LE where E is evaporation), the departures are gratifyingly small. Fig. 4 compared the globally integrated evaporation figures by all authors available to date. The closeness of all four curves suggests there is little disagreement in determinations of mean annual evaporation and its variation with latitude. Since Jacobs and Budyko both used the transfer formula method for their determinations of Qe, with a leading coefficient differing by only a few per cent, the discrepancies in Tables III-V are probably attributable to differences in the climatological mean values of air-sea property difference and wind speed used. Since many years have elapsed between the two sets of evaluations, it is possible that the discrepancies at least partially represent real time changes in the parameters. It is interesting to note that among the greatest departures are the much higher turbulent heat transfers reported by Budyko in the North SECT. 2] LARGE-SCALE INTERACTIONS 115 Atlantic, where evidence of recent oceanic warming exists (Bjerknes, 1959). However, the accessible Russian publications do not disclose enough informa- tion about their data sources and processing to assess the similarity of coverage, the method of analysis or averaging, particularly in respect to wind distribution within speed intervals or Beaufort category (see also Section 6, page 181) to resolve this point. 100 80 60 40 20 0 20 40 60 80 100 120 WO 160 180 160 140 120 100 100 80 60 40 20 0 20 40 60 80 100 120 140 160 180 160 140 120 100 Fig. 7. Mean annual distribution of evaporative heat transfer Qg from sea to air in kg cal cm-2 per year. (After Budyko, 1956, Fig. 24.) 100 80 60 40 20 0 20 40 60 80 100 120 140 160 ISO 160 140 120 100 100 80 60 40 20 0 20 40 60 80 100 120 140 160 180 160 140 120 100 Fig. 8. Mean annual distribution of sensible heat transfer Qg from sea to air in kg cal cm~2 per year. (After Budyko, 1956, Fig. 25.) 116 MALKUS [chap. 4 O O O I— I -"^ O © CD C o a o Q w w H o I-:; lO C^l 'M fO >ra 00 CO f- o m lo o ^ o o »-o ^. o ^- M ^ — ' hJ J hJ hJ J J J o o o ^ X l> :o c-i I— I < o '^ O CO O o o o O H w o ZLJ^ I I •^ -H O lO J J ^ J k:] J J h4 J i5' 'M '^ ^ _^ t1< ^ O t^ LO L't 0*0 0 O lO o r— lO J J -- ^^ i-i ^ o o '^'^^ be bt) SECT, 2] LARGE-SOALE INTERACTIONS 117 o o o o d odd ^ I "^ £ "=5 o d d d d •o ^ Tt< Tf< o 5r ■» o ^ fo -:; o -, '-^ o d d d o le O o o c ce o o ce s o O ce o o O f-I O 00 o O ^ ^^ J o§o2^f ^ S «' d ;^ o CO o S^ ;:^ -t 9 o o o o o o o d o -h' d CO t^ M t^ lO t- S~ . (M O o o O ■^ £. ^ ^" o d d d d o o d t-1 o o ^ c^ ST ^ O O I— I o o o d o o d fJ 00 00 c^, ST O O -H o o o d d o 2 ^ ^ '1 o o o o d 3 O fcJO o 118 MALKUS [chap. 4 a. Distribution of evaporative heat flux Fig. 7 was obtained using 650 oceanic sites, in 5° latitude and 10° longitude intervals. Its main features are : (i) Very high evaporation in the subtropics and trade-wind regions, ranging between 80-120 kg cal/cm^ year or 250-375 cal/cm^ day. The isopleth 120 kg cal/ cm 2 year means 2 m of ocean water per cm 2 of surface are lost annually. (ii) Slight diminution in evaporation toward the equator. (iii) Rapid diminution in evaporation poleward, except for very high values near the western ocean boundaries. (iv) Pronounced non-zonal changes in evaporative heat loss such that local Qe values may depart from the latitudinal average by a factor of two or three. The main reason for non-zonal changes in heat losses for evaporation is the presence of warm and cold sea currents. All principal warm currents, such as Gulf Stream, Kuroshio, Brazilian Current, etc., are associated with local maxima in Qe, while cold currents, like the Canary, Benguela, California, Peruvian and Labrador Currents, are associated with low Qe or equatorward dips in the isopleths. The main difference, however, between the Qe pattern of Fig. 7 and that of Jacobs is the relative diminution here of these longitudinal anomalies due to ocean currents. In Jacobs' figure (which covers only the Northern Hemisphere) the Gulf Stream peak, for example, exceeds the tropical North Atlantic values by nearly a factor of two rather than 20% and the Kuroshio peak is far more pronounced than that shown in Fig. 7. Besides the sea currents, atmospheric circulations also contribute to geo- graphic variations in evaporative heat loss. This effect is felt primarily through changes in the radiation balance of the ocean surface (see equation (1)). For example, due to increased cloudiness, the radiation balance of the ocean surface diminishes slightly from subtropical to equatorial regions. In the exchange formulas, this effect shows up in Qe and Qs by way of the diminished sea-air temperature excess in equatorial regions as compared to the trades. b. Distribution of sensible heat flux Fig. 8 (obtained from the same computation points as Fig. 7) shows the amount of sensible heat Qs that is emitted from the underlying sea surface to the air (positive values) or is received by the surface from the air (negative values). The most outstanding feature of Fig. 8 is that the major portion of the ocean surface, on the average for the year, emits heat into the atmosphere. Over most of the ocean surface Qs, the turbulent heat flux, is small in com- parison with the principal components of heat balance such as R and Qe, and, except for parts of the North Atlantic, comprises not more than 10-20% of the latter (Table V). High absolute values of turbulent exchange are reached in regions where the water is, on the average, much warmer than the air, i.e., in regions affected by powerful warm sea currents (such as the Gulf Stream) and SECT. 2] laiu:e-scale interactions 119 in some areas of higher latitudes where tlie sea is still free of ice. Under these conditions, turbulent heat Hux can exceed 50 kg cal/cni^ year (or 137 cal/cm^ per day). Fig. 8 suggests that the other warm currents, except the Gulf Stream, exert a relatively minor influence on Qs. The cold currents, which lower the sea-water temperature, diminish the turbulent streams of heat from the ocean surface into the atmosphere and reinforce streams of opposite flow. As a result, in some regions affected by cold currents, Qs though small is negative (Canary, California, Benguela Currents). More complicated are the causes for the appearance of areas of negative Qs in the southern portions of the Atlantic and Indian Ocean (at 50°S). In this case, the turbulent flux is apparently affected by the advection of warm air masses over a cold ocean surface. Again the main difference between the pattern in Fig. 8 and the similar figure by Jacobs is the relative reduction here in the longitudinal anomalies due to ocean currents, particularly in the Kuroshio, which scarcely shows up in Fig. 8 (see Table IV). The Gulf Stream peak is also reduced, as is that of the Canary current off Africa, where Jacobs shows an isopleth of — 30 cal/cm^ day ( — 10 kg cal/cm2 year) which is absent on Fig. 8. B. Mean Annual Distribution of the Remaining Components of Ocean Surface Heat Balance The mean annual distribution of total incoming short-wave radiation im- pinging upon the ocean surface, {Q + q) in equation (2), is reproduced from the Atlas of the Heat Balance in Fig. 9. The radiation balance, R, of the ocean surface, arrived at by correcting for albedo and by subtracting back radiation, is shown in Fig. 10 (for details of the formulas and methods, see Budyko, 1956). 40 20 0 20 40 100 80 60 40 20 0 20 40 60 80 100 120 140 160 ISO 160 140 120 100 40 20 0 20 40 -T / SH y^^^-lCy^ 4~^m. i-4 ^■N ^ ""JVJ \ "— — \ / /■ ^?v U \ CJ - 160 H)1 ^ 1 -^ - r^ :^' KJl^ -■^^ p^^T?EZi--v-\t /^ j.-^ "^^ '^"'\' y/"^--/^'"^"; ■ "\ y^^^^^ 'f^^4^Jnyl 1 '"i /_ / %^^ \- ^i^irFH \V^ lA \ 00 =^ itn^ T I "M VlV -V^x 1 ^ T 100 80 60 40 20 0 20 40 60 80 100 120 140 160 180 160 140 120 100 Fig. 9. Mean annual distribution of total radiation {Q + q) impinging on the ocean surface, in kg cal cm~2 per year. (After Budyko, 1956, Fig. 16.) 120 MALKUS [chap. 4 00 80 60 40 20 0 20 40 60 80 100 120 140 160 180 160 140 120 100 40- 00 80 60 40 20 0 20 40 60 80 100 120 140 160 160 160 140 120 100 Fig. 10. Mean annual distribution of the radiation balance, B, of the sea surface. (After Budyko, 1956, Fig. 20.) We see from Fig. 10 that the radiation balance of the ocean surface is every- where positive, that is to say on a yearly average the oceans at all latitudes gain considerably more heat by radiation than they lose. The surplus goes into the atmosphere by evaporation and turbulent-heat transfer and into ocean transport (storage in the oceans is negligible on an annual basis). The radiation balance is of the same order of magnitude as Qe (compare Figs. 7 and 10) but shows a much more regular zonal pattern. It is now possible to use Figs. 7, 8 and 10 to complete the annual heat balance of the ocean surface from ( 1 ) by computing the yearly average distribution of Qvo (oceanic heat-flux divergence) as residual at each grid point. The result Fig. 11. Mean annual distribution of the oceanic heat -flux divergence, (?,.o, computed residually from equation (1). (After Budyko, 1956, Fig. 26.) SECT. 2] LARGE-SCALE INTERACTIONS 121 (after Budyko) is presented in Fig. 11. Before deducing any consequences to geophysics, it is well to make careful tests of any residually deduced quantity such as Qio, particularly to determine whether its important features could be due to uncertainties in evaluation. Table VI shows the oceanic heat -balance components integrated longitudinally and averaged over 10°-latitude belts. On the right we compare the earth's surface radiation balance computed from the tables of London (1957) which, unfortunately, could not be separated into land and ocean regions. The agreement with Budyko 's surface radiation balance, B, is excellent equatorward of 30° and fair jJoleward. Table VI Mean Annual Distribution with Latitude of the Heat-Balance Components of the Ocean Surface Units : kg cal/cm^ year Oceans only, after Budyko (1956) Whole earth. Lat. A flffpT TiOTiHon (1957) Q + q R Qe Qs Qs/Qe Qvo R 60°-50°N 88 34 34 18 53% -18 46 50°-40°N 109 54 51 15 29% -12 63 40°-30°N 136 78 73 12 17% -7 82 30°-20°N 151 100 85 7 8°^ 8 96 20°-10°N 156 110 89 5 o /o 16 106 10°N-0° 149 107 76 5 no/ ' /o 26 105 0°-10°S 152 107 81 7 9% 19 10°-20"S 155 107 97 9 9% 1 20°-30°S 147 94 87 10 11% -3 30"-40°S 128 73 77 12 16% -16 40°-50°S 104 53 57 5 9% -9 50°-60°S 84 31 37 12 SS /o -18 whole earth 128 77 68 9 13% 0 Since the mean latitudinal dependence of Qe (Fig. 4) is agreed on by all authors to a much smaller margin than radiation, or the size of the residual it- self, we may substitute London's R for Budyko's in (1) to test Qvo- While the negative values poleward of 30° are much reduced thereby, its main features from 0-30°N are reproduced, as is the change in sign to negative north of 30°N. The next test is a comparison of Fig. 11 with a similar map constructed by Sverdrup (1957) shown in Table VII. In making the balance in equation (1), Sverdrup used the Qs and Qe of Jacobs together with radiation balance maps of Kimball (1928). The agreement between Budyko and Sverdrup is good in the Atlantic, poor in the Pacific and fair when averaged longitudinally. Briefly, Budyko's chart (relative to Sverdrup's) shows much diminished effects of the 122 [chap. 4 0) Oh 02 Ph ^3 (— 1 02 w t3 h^ c m c3 -< H O <-^ o c o o I 12 c §1 h:! k:1 h^ — J - h-1 _ ^ ^ y, 'A J I :ii O' o — y. SECT. 2] I.AROE-SCALE INTERA(!TIONS 123 Kuroshio and much enhanced effects of the cold currents in the eastern Pacific. Nevertheless, the averaged flux divergences have the same sign at each latitude belt, the same general magnitude and mostly differ by less than 50%. All Northern Hemisphere evidence thus converges to support oceanic flux diver- gence south of 30°, convergence north of that, of a magnitude comparable to one-third Qe and thus not negligible in the planet's heat budget. In other words, the oceans are taking on heat equatorward of the subtropical ridges, and emitting it at higher latitudes, thus introducing an important factor in effecting a milder climate of temperate regions during the cold season. We may now make the additional consistency check of the isopleth patterns of Fig. 11 with the known distribution of ocean currents. In examining the longitudinal anomalies, we see that, in general, there is good agreement between areas of high positive values (positive ocean -flux divergence, or ocean currents taking on heat in an annual balanced situation) and regions of cold sea currents. Conversely, there is good association between high negative Qvo values (flux convergence, currents giving up heat) and the warm currents. This agreement is particularly observed in the case of warm currents — the Gulf Stream, Kuroshio, Agulhas, Southwest Pacific Currents — and in the case of cold currents — Canary, Benguela, California Currents and the northern portion of the Peruvian Current. At the same time, in some regions of the ocean, the distribution of isolines in 'Fig. 11 does not coincide with the main locations of warm and cold streams. This is explained partly by the fact that the map shows oceanic heat-transport divergence and not directly the heat transport, which can be obtained by spatial integration oiQvo- For this reason, the greatest absolute values oiQvo in the Gulf Stream region of moderate and high latitudes are located more to the west, away from the main core of the stream. It is quite probable that the greatest loss of heat by the stream takes place in its western portions which are closest to the cold region of the northwestern shores of the Atlantic coast. Regardless of details, which should not be pursued too far, the degree of quahtative agreement encourages belief that the residual quantity Qvo has some real physical meaning and exists outside the uncertainties of the calculation. This point is of the utmost concern to both meteorologists and oceanographers, as we shall continue to see throughout the chapter. To oceanographers, the implications of Fig. 1 1 and Tables VI and VII are crucial since the Qvo distribu- tion provides to date the only known way of arriving at a global picture of heat transports by ocean currents. In order to compute the current fluxes from their divergence, Qvo may be integrated by multiplying it by the ocean area between the given latitude belts and accumulating from some boundary latitude where the flux can be specified. This may be done separately for individual oceans or for the world as a whole. Before describing the transports obtained in this manner, a word of caution is in order. A sample calculation shows that with identical boundary conditions a uniform 20% alteration in the magnitude oiQvo leads easily to factors of two discrepancies in the deduced heat transports. With this in mind, we have 124 MALKUS [chap. 4 integrated Budyko's figures in Table VI and Fig. 11 (all world oceans) to obtain the fluxes in the first column on the right in Table I (entitled "Sea"), under the boundary condition that the fluxes vanish in the polar regions of the Northern Hemisphere. A check is provided in that the necessity for their vanishing also near the South Pole is automatically met. Rather remarkable, and significant if true, is the large resulting southward heat flux carried by the oceans across the equator. Bryan and Webster (1960) have performed the integration from Fig. 11 separately for each ocean basin in the Northern Hemisphere, using a similar boundary condition. Their results are shown by the solid curves in Fig. 12. Since the Indian Ocean was found to contribute negligibly, the bottom solid curve may be regarded equally well as the deduction for all Northern Hemi- sphere oceans (cf. Table I) or the contribution of Atlantic and Pacific together. The dashed curves in Fig. 12 show a similar integration performed by Sverdrup (1957) of his own Qvo figures. Without Southern Hemisphere data, he was beset by less certainty in boundary conditions. He assumed a small northward heat flux across the equator in the Atlantic from direct oceanographic measure- ments. The agreement here (top curves. Fig. 12) with the deduction from Budyko's figures is excellent. In the Pacific, Sverdrup assumed the cross- equator heat fiow to be zero for want of better information. The enormous departure (middle curves in Fig. 12) from Budyko's Pacific curve is by no means attributable solely to the boundary assumption, but depends also on the significant discrepancies in Qio (Table VII). Thus, while the crux of these deductions rests upon the reliability of the radiation evaluations, whose difliculties (particularly in the relatively un- explored Southern Hemisphere) presently preclude any firm conclusions. Fig. 12 stands as a framework to suggest critical measurements to be made. Both authors are in excellent agreement in the Atlantic where data is be- coming plentiful, while the predicted large southward cross-equator flow is confined entirely to the Pacific, whose equatorial regions are experiencing oceanographic exploration. Bryan and Webster (1960) have offered independent dynamic and chemical evidence to support the reality of this feature ; its further confirmation or rejection would form not only an important corner- stone in oceanography but a significant test of the overall usefulness and reliability of both Budyko's figures and this type of budget approach to plane- tary energetics. In any case, the possibility that ocean currents play so large a role in poleward heat transport is of sufficient consequence to meteorology to warrant its vigorous investigation. We conclude our discussion of the oceans' heat budget with an analysis of the results in Table VI. Important deductions concerning atmosphere-ocean energetics and dynamics may be drawn from this Table. The total radiation, increasing regularly from high to low latitudes, has its maximum not on the equator but in the belts of high pressure near the 20° latitudes. The equatorial minimum is apparently explained by the considerable increase in cloudiness at the equator, so that we see here an example of circulation features regulating SECT. 2] LARGE-SC!ALE INTERACTIONS 125 their own energy input. Tlie apparent paradox raised concerning how equatorial regions may still serve as the atmosphere's "firebox" will be resolved at the end of this section and in Section 5 (page 164). The radiation balance of the sea surface grows rapidly with decreasing latitude only in temperate zones, while in tropical regions it is only slightly dependent on the latitude. ^ -^\ Budyko - Atlantic 1 I 1 Sverdrup 1 1 80 f-O ■10 M 0 O -I - Svefdrup - Botti Oceans 1 1 \ Sverdfup 1 80 1 1 60 40 1 \ 20 1 \ ° \ Budyko _ \ Fig. 12. Net ocean-current heat transports in Northern Hemisphere obtained by in- tegrating Qj^o of Sverdrup (1957) and Budyko (1956). (After Bryan and Webster, 1960, unpubHshed.) Positive heat transport poleward. Abscissa in degrees North Latitvide; ordinate heat transport in units 10^9 cal/day. 126 MALKUS [chap. 4 The heat expenditure for evaporation, Qe, is seen to be one of the two major ways the ocean gives off heat, the other being "back radiation", Qb [not shown in the Table; very roughly equal to {Q-\-q) — K]. Qe and Qt are of comparable size at all latitudes. Over the oceans, maximum values of evaporation and the heat expenditure for it, Qe, are observed in the subtropical high pressure belts, where the inflow of solar heat is especially great. In the proximity of the equator, the evaporation from the ocean markedly diminishes. Turbulent heat emission from sea to air, Qs, is on an annual average compara- tively small in all latitudes, averaging 13% oiQe. Its values increase somewhat with higher latitude due to a growing significance of warm currents which heat the air in the cold season, as does the ratio QsjQe, which averages 8% equator- ward of 30° and 26% poleward. In constructing a planetary radiation budget, Houghton (1954) obtained an average ratio QslQe of 43%. His global average Qst was twice that of Budyko's but since it was found as residual in a radiation balance, in which the uncertainties were comparable to the size of the residual, it is likely that the independently tabulated Qs figures of Budyko are more reliable. The results of this section have raised some vitally important questions to marine scientists. Among these are questions concerning the relative im- portance of poleward heat transport in ocean and atmosphere, the fate of the water- vapor fuel and its use in driving the air circulations, the creation of wind systems on various scales, and their role in exchange and in the maintenance of ocean currents. The quantitative heat budget of the sea surface provides an initial foundation for the pursuit of these questions, but to build it further we must consider the mean annual heat budget of the joint air-sea system and, briefly, that of the atmosphere itself. C. The Annual Heat and Water Budgets of the Ocean-Atmosphere System Using the material on the oceanic heat balance, we may now construct the joint heat and water budgets of the ocean-atmosphere system to learn further how its parts interact and affect each other. a. Heat energy budget of the system In order to analyze the joint annual heat budget, it is first necessary to formulate a conservation law analogous to (1) for a column of unit area ex- tending from the top of the atmosphere down into the ocean interior, namely, Rs = L{E-P)+Qro+Qva- (25) The same small terms have been neglected as previously, as well as storage terms in sea and air. The latter are always negligible compared to the former, which average out in an annual budget. P is precipitation in grams (or cm) per cm2 per sec, E is evaporation in grams (or cm) per cm^ per sec. Thus the term L{E — P) may be described equivalently either as the excess evaporation SECT. 2] LARGE-SCALE INTERACTIONS 127 over precipitation rate at the ocean surface, or as the flux divergence, Qvw, of water- vapor transport in the atmosphere. Qvo is the flux divergence of horizontal ocean-heat transport in cal per cm 2 per sec and Qva is the flux divergence of horizontal heat and potential energy transport in the atmosphere in cal per cm^ per sec, or very nearly the flux divergence of the transport of Cp6 {Cpd'^CpT + Agz), where 6 is potential temperature, Cp the specific heat of air at constant pressure, A the heat equivalent of work, g the acceleration of gravity and z the elevation above the ground. The compressibility of air necessitates this formulation for the heat-energy transport in the atmosphere, since vertical ascent within a column may convert sensible heat into potential energy, and conversely, depending upon the prevailing lapse rate. Rs is the radiation balance of the entire column, or the difference between the short-wave radiation absorbed and the net long- wave radiation emitted. Methods and results of evaluating it have been described in the earlier literature by Simpson (1928), Kimbafl (1930) and Baur and Phillips (1934, 1935). More recent evaluations with better and more extensive data have been presented by Houghton (1954), London (1957) and Budyko (1956), who reports the work of Bagrov (1954). Again, the greatest difficulty in these calculations resides in ascertaining the amounts of various cloud types and their radiative, reflective and absorptive properties, particularly in assessing the absorption of short- wave radiation in the atmosphere. Nevertheless, the agreement between the last three authors is apparently relatively good concerning the annual average magnitude of Rs and its dependence on latitude, particularly that between Houghton and Bagrov, as demonstrated in Fig. 13. As stated earlier (page 92ff.), all evaluations of Rs show that the ocean-air system as a whole gains radiation heat equatorward of about latitudes 38° and loses heat by radiation poleward. If the high latitudes are not to cool progres- sively with time and the low latitudes to warm up (precluded by the closely reahzed steady-state assumption (6), Introduction, page 92), specified transports in the earth's movable parts, namely sea and air, must take place. Thus equa- tion (25) states physically that in regions of positive radiation balance the excess heat energy may be carried away by sensible heat transport in the ocean and by a combination of sensible heat, latent heat and potential energy export in the atmosphere, while regions of negative radiation balance must make up the deficit by corresponding imports. Therefore, computation of Rs as a function of latitude immediately permits assessment of the total heat-energy flux divergences in sea and air together (sum of terms on right side of (25)), and by integration, enables the total heat-energy flux across latitude circles to be obtained. In the first three columns of Table I we performed this integration for the Rs figures of Bagrov (reported by Budyko) and compared them with similar computations by Houghton and London. The Russian figures and Houghton's are in nearly perfect agreement, while those of London are about 50% lower, illustrating the magnification of fairly small flux-divergence discrepancies when integrated. Houghton's and Budyko's fluxes are about 60% larger than 128 MALKUS [chap. 4 those shown in The Oceans (Sverdrup, Johnson and Fleming, 1942, p. 99, table 24) using earlier radiation figures of Kimball (1930). In order to understand how the large-scale circulations of air and sea carry out these fluxes, for the purpose of modelling circulation dynamics and energetics. kg col cm^ yr Radiation Balonce of the Earth os o Planet — I 1 1 1 1 1 r 90 N -\ 1 1 1 1 1 r Bagrov o o Houghton « ' » London Fig. 13. Heat-energy balance components of air and sea together. (After Budyko, 1956, Fig. 70.) Mean annual values in kg cal cm-2 per year as functions of latitude. Radiation balance Rg : dashed curve is determination by Bagrov (1954). Circles are values of Rg after Houghton (1954) and crosses the values of London (1957). Qyo, oceanic heat-flux divergence, from residual determination in equation (1). Qx,a^ atmospheric heat and potential energy-flux divergence, found as residual in equation (25) after assessment of L{E — P), latent heat-flux divergence, as described in text. Abscissa degrees latitude. Southern Hemisphere on right. SECT. 2] LARGE-SCALE INTERACTIONS 129 we must then be able to break down the total transports and analyze the in- dividual components in air and sea. The sea-transport divergence, Qvo, has already been obtained as residual from equation (1) by our previous heat balance for the earth's surface, for which evaporation distributions were pre- viously evaluated. Thus, to complete the heat budget in (25) we need to know the distribution of precipitation, P, and the flux divergence of sensible heat and potential energy in the atmosphere, Qia. The procedure here (and in Table I) is to start with the radiation balance, Rs, of Bagrov, then to present the best available precipitation figures and their latitudinal dependence, and finally to arrive at Qia as residual in (25) to com- pare with more direct meteorological determinations of the various atmospheric transports. We are thus following most closely the Russian development (Budyko, 1956) and presenting comparisons of their results with the more accessible western calculations where possible. It is a considerable credit to the recent advances in both meteorology and oceanography that the various independent pieces of the puzzle fit together so consistently. The average annual global energy transactions are now fairly well-known ; they both give clues to building dynamic models, and specify constraints that these models must satisfy. b. Mean annual distribution of precipitation over the oceans Adequate assessment of oceanic rainfall poses a problem. Direct shipboard measurements are very difficult to obtain reliably, and therefore extrapolation from island and coastal stations must generally be undertaken. Thus, in addi- tion to uncertainty whether coastal rainfall is a fair measure of that over the open sea, wide portions of the mid-ocean regions remain insufficiently sampled. However, since World War II, meteorological studies of oceanic and island precipitation statistics and dynamics have advanced considerably, particularly in the tropics ; they provide both the opportunity of spot-checking the extra- polated patterns and of testing whether such extrapolation is physically compatible with rainfall dynamics. Fig. 14 presents the most recent global picture of oceanic rainfall patterns (Drozdov, 1953) which was obtained by direct extrapolation between stations without any arbitrary reductions to correct for "land effect". We may compare this figure with the presentation of Jacobs, based on results by Wiist (1936). These were reduced in places by 20-30%, to agree with then existing evapora- tion figures, the required reduction being explained as a correction for presumed coastal enhancement of rainfall. The isohyetal distributions in Fig. 14, how- ever, are almost entirely similar to those of Jacobs, with patterns and maxima which are entirely superposable. A numerical comparison is made in Table VIII ; Drozdov's amounts are, as expected, almost everywhere larger than those of Jacobs and Wiist, by an average of about 20%, in better agreement with still earlier determinations by Meinardus (1934). Overall budget studies (Riehl and Malkus, 1958) suggest that the higher Russian values are the better 130 [OHAP. 4 > < O o c3 t> GO o t3 S3 p ^ (5 b o >i --^ rO P ;^ S K* o 1^ o c3 ^ Pl^ (—" (^ l*-l C O c 0 h-1 1-; o lO o IC 00 CO -M 00 o o oc- o Oi CO ■"^ '"' s o o S" iM 00 CD CO r— < o 'O >o o 05 (M 00 ^^ ^^ s to o~ J 00 00 CD o lO O CD o CD hJ hJ h-1 h-1 J 00 ^ lO o o C- Tt( ^ O O lO CO "O ■* ^-q o o ^ 05 10-^ , , , , , . O o •c o CO 00 as t^ o o o o r~ »c 00 CO ^ ^ o^ J »o o CD 00 oo 1— ( o o o lO i-H Tt* '"^ ^ •M o i^ iO~ CD M c^ 00 r-H o o o >o 00 GO (M to O 1-H ■* 1-1 hJ :^ o o' c^ Oi 'M k^ >^ ^ SECT. 2] LARGE-SCALE INTERACTIONS 131 ones, as does our increased knowledge of rainfall dynamics. Since it is now knoMii that, even in the tropics, all significant rainfall occurs in major synoptic-scale storms, it is likely that the "coast effect" on precipitation has been considerably overrated in the past. 100° 120° 140° 160° 180° 160° 140° 120° 100° 80° 60° 40° 20° 0° 20° 40° 60° 80° 100° 100° 120° 140° 160° 180° 160° 140° 120° 100° 80° 60° 40° 20° 0° 20° 40° 60° 80° 100° Mean Annual Precipitation (mm) Fig. 14. Mean annual distribution of precipitation over the oceans in mm per year. (After Drozdov, 1953.) c. The climatology of mean annual energy transactions Upon global ^ integration to obtain precipitation as a function of latitude, multiplication by L, and subtraction from Qe, we arrive at the term L{E — P) in (25). Its latitudinal dependence as used by Budyko is shown by the solid curve in Fig. 13. Qvo is entered from Table VI (dotted curve) and Qva is com- puted as residual from (25) and entered as the dot-dashed curve. We are now prepared to place a quantitative foundation under the discussion of the whole 1 Actually all computations entering. Fig. 13 were done for the whole earth and not just the ocean areas to which our presentation has been restricted. This means, for example, that evaporation and precipitation figures for land areas had also to be included and the term Q^^o ^ Table VI corrected for the ratio of ocean to whole earth's area in each latitude belt before inclusion in the graph. 132 MALKUS [chap. 4 system's operation and climatology, outlined descriptively in Section 2 of this chapter (page 92). As can be seen from Fig. 13, four basic latitudinal zones exist in each hemi- sphere, each with essentially different relationships of the heat-balance com- ponents. In the equatorial zone, which extends north and south of the equator up to 10° to 15° latitudes, the gain of heat from positive radiation balance is supplemented by a comparably large net release of precipitation heating (water- vapor flux convergence). These together assure the great export of heat by atmospheric and oceanic advection, for which the relatively narrow region from 0° to 10° latitudes constitutes the primary energy source. Fig. 13 thus demon- strates quantitatively that the equatorial trough plays its role as the atmos- phere's firebox. This is not mainly due to excess radiation received but even more largely because there the water-vapor fuel is combusted by precipitation release, an energy source even more closely tied to circulation dynamics ; the mechanisms by which it carries out this role are sought in Section 5-B, page 164. Northward and southward from the equatorial zone are the tropical regions, i.e., the locations of the atmosphere's famous easterly trade-winds which prevail throughout the equatorward sides of the subtropical high pressure cells, or "ridges". In these zones, with a positive (but diminishing with increase in latitude) radiation balance, large expenditure of heat for net evaporation is observed. In the major portion of the trade-wind regions, the loss of heat for moisture exchange (from sea to air) approaches the value of the radiation balance, and thus the input to sensible heat and potential energy advection {Qvo and Qva) is small. The fact that the trade-winds act as the fuel accumulators for the atmospheric heat engine is thus also shown quantitatively in Fig. 13. Mechanistically how they do this, and how they serve also to pump the fuel into the firebox, is examined in Sections 5 and 6 of this chapter. In the "subtropical ridge" region of 35°-40° latitudes a transitional zone is found. In this area. Fig. 13 shows that the gain and expenditure of heat in all the balance components is fairly evenly distributed and no component is numerically large. The j^oleward energy transports themselves, however, are maximum (Table I). In the high troposphere, this is the mean position of the famous subtropical jet streams, whose wave -like meanders provide the channels of poleward energy flow from the tropics. Thus the temperate atmosphere is fed with the heat to balance its radiation loss, and to store in potential energy of air-mass contrasts, a small fraction of which is released to maintain the charac- teristic cyclonic storms and restless winds of the middle latitudes. Poleward of the subtropical ridge and jet stream, all higher latitudes are regions of radiation deficit, increasing rapidly toward the poles. Energetically this zone lives on imports : from excess precipitation over evaporation, atmos- pheric advection and sea-current transports. Dynamically, it plays a very important (some workers say the dominant) role in the operation of the global circulations by concentrating and releasing the imported energy in complex and wondrous ways, producing evanescent wind systems which themselves SECT. 2] I.AROK-SCALK INTERACTIONS 133 drive sea ciirreiit and jet stream, adjusting their own imports, and thereby back-reacting upon the energy-producing circulations of low latitudes. An important implication of Fig. 13 is found in comparing the deduced magnitudes of the several heat-balance com])onents. These results suggest that, in the general heat-energy exchange between latitudes, all four terms in (25) play an important role and that neglection of any one would be a serious omission. In particular. Fig. 13 suggests that the heat-flux divergence of sea currents is comparable to that of air currents. If true, this means that "general circulation" studies must eventually include both air and sea transports to- gether, and that quite probably the latter cannot be ignored if long range weather forecasts are to succeed. The transport figures presented earlier in Table I were obtained by separate integration of the divergences in Fig. 13. The boundary conditions for total heat-energy transport were obtained using the radiation balances, Rs, of both hemispheres. The moisture transport across the equator (due primarily to the Asiatic monsoon) was obtained from Budyko's statement (reference to Zubenok, 1956) that in the Southern Hemisphere the amount of precipitation is 12 cm/ year less than evaporation. The boundary condition on sea transport was obtained as described previously and that on the atmosphere's combined sensible heat j)lus potential energy transport was found as residual and checked in integration against total air and sea transport. A test of Table I may be applied by comparing its results with what we know both qualitatively and quantitatively about atmospheric circulations and their transports. In the first place we see that the boundary between Northern and Southern Hemisphere air circulations falls naturally between 0° and 10°N, in good agreement with the mean annual position of the equatorial trough (see Figs. 22a, 23a). Latent heat is shipped into this zone from both sides and sensible heat plus potential energy is exported poleward therefrom, as confirmed by meteorological studies. Although the Asiatic monsoon and its seasonal migrations somewhat mask this function of the trade-wind region on the Northern Hemisphere side, it is revealed beautifully and to the expected magnitudes (Section 5) on the southern side, where land effects are relatively minor. Secondly, a fragmentary comparison of the deduced atmospheric transports with more direct meteorological determinations (using mean observed air motions and properties computed from meteorological data) is made in Table IX. The figures of Mintz (1955) represent the sum of sensible heat transport by geostrophic eddies and Cpd transport by the mean meridional circulation in the year 1949. The figure in the last column at latitude 15° (Palmen, Riehl and Vuorela, 1958) is 85% composed of total heat-energy transport {Cpd plus latent heat) by the mean meridional cell, with the remaining 15% estimated eddy-moisture flux. Since the year-round average would be about one-half their winter figure, or 0.6 x lO^^ cal/sec, the agreement of these totally in- dependent evaluations could hardly be better. The calculation for latitude 10° (after Riehl and Malkus, 1958) is discussed further in Section 5B. 134 [chap. 4 Table IX Comparison of Poleward Atmospheric Heat-Energy Transports by Latitude Unit : 1015 cal/sec Latitude Table I Sensible + pot. Meridional circ. (N. Hem.) r j^ energy (after Mintz, (air total) ■\ Sensible heat Air total 1955) + pot. energy 60° 0.68 0.69 0.51 50° 0.76 0.92 0.60 40° 0.77 1.02 0.53 30° 0.68 0.89 0.29 20° 0.71 0.71 0.32 15° 0.67 0.60 1.20 (winter)a 10° 0.63 0.49 0.310^ a After Palmen, Riehl and Vuorela (1958) ft After Riehl and Malkus (1958) d. The annual heat balance of the atmosphere The heat budget of the free atmosphere is now readily balanced separately. Its radiation balance, Ua, may be obtained by subtracting the radiation balance of the earth's surface, i2, from that of the whole system, Rs (Fig. 13), namely, Ra = Rs — R- (26) The radiation balance of the underlying surface is in all latitudes greater than that of the earth-atmosphere system, so that Ra is everywhere negative. This is due to the famous "greenhouse effect" : the water vapor in the atmosphere absorbs long-wave terrestrial radiation and reradiates it both downward and upward. The mean annual Ra distribution with latitude of Bagrov-Budyko is given by the dash-dotted curve in Fig. 15. It is important to note that the atmosphere's rate of heat loss by radiation varies extremely little with latitude, being everywhere about 65 kg cal/cm^ year or equivalent to a radiative cooling of approximately 0.75°C per day. London's results for Ra are not too different from these, but his atmospheric heat sink is every where greater (average 20%). While this discrepancy is not excessive in view of the uncertainties involved, it accounts almost entirely for the difference between London's and Budyko's total transports in Table I. We have seen that the earth's surface radiation balances, R, of the two authors departed little from each other and thus their deduced oceanic transports were in good agreement. However, if we accepted London's Ra instead of Budyko's, the total air transports in Table I (Northern Hemisphere) would be reduced to 63% of their presented values, implying among other things that the ocean provides nearly 40% of the critical poleward SECT. 2] LARGE-SCALE INTERACTIONS 135 transport across the subtropical ridge! Table IX, in fact, shows some indication that the air fluxes of Table I may be too large. Resolution of this vital point may be hoped for with the coming improved ways of directly measuring atmospheric radiation fluxes through the satellite program and air-borne radio- meters (see, for example, Clarke, 1959; Brewer and Houghton, 1956). Another approach to settling this question lies in determining independently each of the other terms entering the atmosphere's heat budget. The balance The Heat Bolance of the Atmosphere kg col" cm^ yr \ / \ \ / N / \^^ N / \ / Fig. 15. Heat-energy balance components of the atmosphere. (After Budyko, 1956, Fig. 71.) Mean annual values in kg cal cm-2 per year as functions of latitude. Dash-dotted curve for radiation balance, Ra, from equation (26), Figs. 10 and 13. Solid curve for precipitation warming, LP, computed from Fig. 14 with slight adjustment to achieve balance (see text). Sensible heat flux from sea, Qg, in dotted curve from Fig. 8. Atmos- pheric heat and potential energy flux divergence, Q„a, from Fig. 13. 136 MALKUS [chap. 4 equation for the atmosphere is LP-Qra+Qs + Ba = 0 (27) which states physically that the radiation losses in the atmosphere must be made up by some combination of precipitation release, transport convergence (minus Qia) and sensible heat supply from the earth's surface. Budyko's balance of this equation at each latitude is shown in Fig. 15. In addition to Ba from (26), the graph shows the latitudinal distribution of precipitation obtained from integration of Fig. 14 (as described below), the distribution of ^s (from a figure like Fig. 8 including land areas) and the —Qva found residually from (25) and shown in Fig. 13. Although the balance is gratifying and the size and latitudinal dependence of the components realistic in terms of present knowledge. Fig. 15 is still by no means a proper independent confirmation of the other budgets and the relative magnitudes of their trans- port terms. A framework for using the meteorological networks for this purpose is set up by (27), however, and the spot tests oiQva in Table IX ; its fragmentary condition is due, primarily, to insufficient radio-wind and radio-sonde measure- ments over oceans, particularly in the Southern Hemisphere, and, secondarily, to the enormous labor involved in such computations. A major conclusion is, however, quite independent of all these difficulties. Comparing Table VI and Figs. 13 and 15, we see that water exchange and its consequences is one of the most important energy transactions on our planet. Qe is one of the two major terms {R being the other) in the energy budget of the ocean surface and the major source of heat loss to the ocean. In the atmosphere, precipitation release is the primary source of heat gain at all latitudes from the equator to the arctic circle. Fortunately, the water ex- change may be assessed separately from that of energy and independently of radiation computations. e. The global water budget Water is the most important commodity on the earth ; its changes in phase distinguish this planet from all the others. The evaporation-precipitation cycle is of direct life-and-death significance to humanity, and indirectly serves to fuel the atmosphere, to modify the radiation budget of earth, sea and air, and to regulate the salinity and temperature structure of the ocean. In addition, evaluation of the global water budget provides an important computational check upon the heat-energy budget determinations just described. The recent Russian calculations permit a more complete analysis of water budgets than was previously possible, since they include assessment of radia- tion, evaporation and run -off" figures for the continents of both hemispheres (omitted for brevity from the present recapitulation). Their results for evapora- tion were obtained by averaging comparatively detailed world maps (like Fig. 7 for each month, including continental areas). These give an overall average somewhat greater than the majority of earlier calculations. While in j^receding .«?KflT. 2] LARfJE-SOAT.K INTKRACTIONS 137 works tlie value of evaporation from tlie ocean usnally ranged ])etween limits of 75-100 cm/year, the newest re^^ult sliows li:} cm/year. Since their total computed run-off from rivers into oceans gives an added water layer of approxi- mately 10 cm/year, the total precipitation over the oceans should be about 103 cm/year. This value is slightly smaller than the corresponding amounts obtained by Meinardus (1934) from Schott's maps (114 cm/year) and that calculated directly from the map of Drozdov (Fig. 14), which was 112 cm/year. However, the difference between these new figures for precipitation and sum of evaporation and run-off is only about 10%, or much smaller than the expecta- tion of earlier workers (Wiist, 1936). Therefore, Budyko, in utilizing Drozdov's results to complete the heat budgets of Figs. 13 and 15, reduced the integrated precipitation values of Fig. 14 only slightly, multiplying by 0.913. Using this reduction factor, the evaporation figures presented in Fig. 7, and the determination of continental run-off, Zubenok (1956) arrived at a water budget for each ocean separately, reproduced in Table X. The next-to-last Table X Average Annual Water Balance of the Oceans (after Zubenok, 1956) Ocean Precipitation, cm/year Evaporation, cm/year Continental run-off, cm/year , Water exchange with adjacent oceans A cm/year 10^ m3/sec Atlantic Indian Pacific Arctic 78 101 121 24 104 138 114 12 20 7 6 23 -6 -30 + 13 + 35 -0.16 -0.70 + 0.68 + 0.16 Average Gulf Stream transports 70 x 10^ na^/sec column, giving exchange of water with adjacent oceans, was computed as residual to balance the annual budget of each ocean ; a minus sign denotes a deficit which must be made up by inflow from adjoining basins. The Atlantic and Indian Oceans receive, on the average for a year, a considerable amount of water from the Arctic and Pacific Oceans. In the last column, we have made the conversion of the exchanges into actual water volume, in units of 10^ m^/sec, to compare with the transport by the Gulf Stream ; we see that the amount of water running out of the Arctic Ocean is exactly equal to the amount that flows into the Atlantic Ocean. Similar to this, the quantity of water that must leave the Pacific Ocean is closely equal to the Indian Ocean's computed water inflow. These estimates, derived from planetary water-budget considerations and confirmed by their consistency with the heat budget, stand to guide more direct oceanographic studies. They suggest critical measurements and serve to test 138 MALKUS [chap. 4 and be tested by theories of water-mass origin and age, of production of dee]) water and abyssal circulations. Furthermore, the mere removal of water by evaporation and its addition by precipitation have been shown to create sizeable hydrostatic pressure forces and surface currents, analogous to "curl of wind stress". Now that the distribution of evaporative sinks and precipitation sources is well known, the beautiful computational models of Hough (1897) and Goldsbrough (1933) can meaningfully inquire what role water exchange plays in major current dynamics, particularly in the high precipitation zones of the tropics. Evaporation and rain also alter the temperature and salinity of the surface layers, so that improved E — P figures may also give an approach to the stability and internal dynamics of the thermocline region. Comparisons of its structure produced by the given E — P in a static ocean versus the real one would permit assessment of the effects of a given advective and turbulent regime (Stern, 1960). The oceanographic consequences of these water and energy exchanges are discussed further in other chapters of this book. These sections, culminating in Table VI, and Figs. 13 and 15 answer the questions of "what?", "where?" and "how much?" concerning annual global energy transactions, and the interaction between sea and air. We can calculate the amount, origin and form of the energy entering the moving parts of the system — the planetary fluids of sea and air. As a result we have been able to estimate quantitatively in Table I (and test partially. Table IX) the average transports by circulations, and make a first endeavor to break these down between sea and air and into the type of energy transported. How these circulations are driven, how the thermal energy is converted into motion on the many scales, and how the motions in turn regulate the energy releases and transports, constitute the basic problem of geophysical fluid dynamics. It is fundamentally a nonlinear problem, or set of problems, in irreversible turbulent fluid thermodynamics, and it is an understatement to say that it has not yet been formally solved on a global scale. Only the simplest "prototype" fluid heat engines are as yet tractable to rigorous mathematical analysis (cf. W. V. R. Malkus, 1954). On the global scale, exciting beginnings of general circulation studies have been undertaken by numerical and experi- mental methods for both atmosphere (Lorenz, 1960; Phiflips, 1956; Fultz, 1956; Riehl and Fultz, 1957, 1958) and ocean (Stommel, Arons and Faller, 1958). As we shall see in Sections 5 and 6, progress in semi-theoretical modelling of the conversion of thermal energy into motion has been made for some of the component regions and scales in the atmosphere, particularly in the tropics. To progress in this direction, the mean annual picture of energy exchanges and transportation provide a beginning framework, particularly as a foundation and proving ground for steady-state circulation theories. We may inquire what average or integrated motion scheme is consistent with the energy sources, sinks, exports, imports and internal releases that have been deduced quantita- tively herein. In order to approach the vitally important fluctuations and instabilities in the air-sea system, however, we must examine the time depend- ences of these energy transactions, and how the various interacting scales of SECT. 2] LARGE-SCALE INTERACTIONS 139 motion bring these about and are themselves fuelled, braked and constrained by them and by each other. The most regular and easily approachable geo- physical time dependence is probably the seasonal cycle, since this is clearly forced ultimately by the regular and predictable cycle in solar radiation input. We next examine the system's response to this oscillating input in terms of the heat-balance components, the mean annual configuration of which has just been analyzed. D. The Seasonal March of Heat Balance and Exchange Time changes in energy sources are forced upon the air-sea system by seasonal variations in incoming short-wave radiation. These variations in solar input give rise to changes in circulation, often unstable, which in turn modify the solar input and its seasonal progression. We take our first step toward considering time changes in air-sea interaction by showing (in Figs. 16-21) the seasonal variations in Qe, Qs and R in key oceanic regions. Interpreting these in the light of Section 2 of this chapter on the overall operation of the system, we are able to deepen our understanding of exchange dynamics, its role in the major energy transactions, and in the maintenance of circulations and their fiuctuations. Fig. 16 shows the annual march of Qe, Qs and ^ in a region typical of the "firebox" : an equatorial oceanic climate (Pacific, 0° lat., 150°E long.). In this area, the radiation balance changes only slightly during the year. However, spring and autumn maxima are noticeable, shifted somewhat from the equi- noctial months (spring maximum from February to March, and autumn maximum from September to October). The heat expenditure for evaporation, Qe, is close to the radiation balance value, ranging from 5-8 kg cal/cm^ month or 160-255 cal/cm^ day (evaporation 0.3-0.4 cm/day). A turbulent heat flux, small in its absolute value (~16 cal/cm^ day), is directed from ocean to the atmosphere during the entire year. The difference R — {Qs + Qe) enables us to obtain from (1) the sum Qvo + S. This quantity is the heat exchange between the ocean surface and deeper water layers ; it is only identical with the transport divergence Qvo on an annual average, since on a shorter term basis comparable amounts of heat are commonly either being stored or removed from previous storage in the ocean column. In these equatorial regions Qvo + S develops some comparatively large positive values in autumn, when the heat gain from radiation balance considerably exceeds the expenditures for evaporation and turbulent heat emission. This surplus of heat, which is received by the water masses during autumn, is clearly transported (see Table I) from the analyzed region to higher latitudes by currents and macroturbulence, since there is found no corresponding local deficit at other seasons. The annual march of the same heat-balance components in the oceanic climates of equatorial monsoons is presented in Fig. 17 (the Arabian Sea region). Briefly, the monsoons are the results of differential heating between continents and oceans, and give rise to large-scale inflow onto the continents of 140 [chap. 4 kg-cal/cm^ month J FMAMJ JASONO Pacific Oceon, 0° Lot, I50°E Equatorial Climate Fig. 16 kg-col/cm^ month JFMAMJJ ASON 0 Indian Ocean, I5°N, 70° E Climate of equatorial monsoons Fig. 17 ,kg-cal/cm^ month J FMAMJ jaS OND Atlontic Ocean, 20°S, 30°W Tropical clirnole of ttie western periptiery of oceonic onticyclones Fig. 18 kg - cal/cm^ month -^^1 I I L. JFMAMJJASOND Atlantic Ocean, 20° S , 10° E Tropical climote ol ttie eastern periphery of oceonic anticyclones. Fig. 19 kg-cal/cm^ month J FMAMJJASOND Atlantic -Ocean, 55° N, 20° W Climale of moderate latitudes in regions of worm sea currents Fig. 20 kg -col /cm ^ month 10 "JFMAMJJASOND Pacific Ocean, 45° N, 160° E Monsoon climate of moderate loliludes Fig. 21 Figs. 16-21. Seasonal march of major heat-balance components of sea surface typical of various climatic conditions. (After Budyko, 1956, P'igs. 39-44.) SECT. 2] LARGE-SCALE INTERACTIONS 141 oceanic air in summer, and outflow of continental air over the seas in winter ; the asymmetry in the hemispheres, shown in Table I, and the large cross-equatorial flows are primarily the result of the enormous Asiatic monsoon, which domi- nates the Indian Ocean region as shown in Fig. 17. Here, the regular form of the annual radiation balance curve, with a summer maximum and a winter mini- mum, is distorted by a rapid increase of cloudiness in summer during the period of equatorial air-mass influx. An increase of cloudiness diminishes the total radiation and radiation balance in midsummer so that a secondary autumn maximum occurs. The turbulent heat emission Qs is insignificant in this region during the entire year, the result of insignificant difference between water and air temperatures. However, Qs increases somewhat during winter as is typical for monsoon climates. Expenditure of heat for evaporation, Qe, in this region changes during the year inversely to changes in radiation balance (this relation- ship is typical for the major portion of the oceans). The winter maximum of evaporation is, in this case, explained by advection of dry trade-wind air masses, and is connected with a considerable increase in saturation deficit (19). The summer Qe maximum is associated with a strong increase in wind speed during the equatorial monsoon period. As a result of the considerable increase of heat losses for evaporation in winter and summer, and of a diminished radiation balance therewith, the heat flux between the sea surface and lower water layers is directed upward {Qvo + S is negative) during these seasons, although the absolute values are comparatively small. In contrast to this, in spring and autumn great quantities of heat are transmitted from ocean surface to deeper layers. Comparison of areas (approximately that between curve R and curve Qe in Fig. 17 since Qs is so small) shows that the lower layers gain more than they lose on an annual average and thus the net Qvo must be positive, or represent an export of heat by the ocean from this region (see also Fig. 11). We thus begin to see that, even in the tropical belt, the annual march of energy transactions is vitally influenced by air and sea circulations, and is diff'erent in areas having different circulation regimes. The modifications from the mean latitudinal picture, or from that of an all-water world, are primarily produced by the distribution of continents. Because of continental barriers, the intense warm ocean currents are located at the western periphery of the oceanic anticyclones (as is described elsewhere in this book). Among other effects, this creates favorable conditions there for the largest values of Qs, the turbulent heat emission from ocean to atmosphere. As an example, we shall next analyze the region near the island of Trinidad (Southern Hemisphere, Atlantic Ocean, southeast of Brazilian Coast; to be distinguished from British West Indian island of same name). The annual march oiQe, Qs and R for this area is shown in Fig. 18. Radiation balance, under these conditions, changes in accordance with the annual march of total radia- tion, but expenditures of heat for evaporation have an o^Dposite pattern. The turbulent heat emission grows in the winter months (for the Southern Hemi- sphere) when the effect of the warm Brazilian current is strongest. At this time 142 MALKUS [chap. 4 of year the expenditure of heat for evaporation and turbulent heat emission markedly exceeds the radiation balance, that is to say, R — {Qe + Qs)=Qvo + S<^0. Therefore a considerable amount of heat must come out from storage in the deeper ocean layers. Comparison of areas in Fig. 18, however, shows that, over a year, the sea receives more net heating by radiation than it expends to the atmosphere, so that the oceanic flux divergence Qvo still averages out slightly positive (cf. Fig. 11). In contrast to the conditions just illustrated for the western periphery of the oceanic anticyclones, their eastern periphery is affected by cold sea currents and the annual march of energy transactions changes accordingly. As an example of the annual march of the heat-balance components in tropical areas at the eastern periphery of oceanic anticyclones, we will consider the region affected by the Benguela Current in the southeast portion of the Atlantic Ocean (Fig. 19). In this case the expenditure of heat for evaporation, Qe, is drastically reduced relative to that of the preceding region (Fig. 18), which is located in the same latitudinal zone. Unlike almost every other region, the turbulent flux of heat, Qs, very small in its absolute value, is negative or directed from the atmosphere to the cold ocean surface. The absolute values of Qs increase somewhat in summer, when the effect of the cold Benguela Current is most pronounced. In this region, the ocean's gain of heat from radiation balance and turbulent exchange is much larger than its losses from evaporation, and a great amount of heat energy is transmitted to the deeper layers, which is spent on heating the cold water- masses carried by the current. These expendi- tures are especially large during the summer ; we can see from both Figs. 19 and 1 1 that Qvo has here its maximal (positive) annual average. In the subtropical belt, the main features of the annual march of the heat energy transactions across the ocean surface are similar to those in the corre- sponding areas of tropical latitudes. However, annual variations of radiation balance are much more sharply pronounced, which is the result of considerable changes in the average altitude of the sun during the year. Typical annual variations of heat-balance components of the ocean surface in moderate latitudes are presented in Figs. 20 and 21. Fig. 20 shows the situation in the North Atlantic area affected by the Gulf Stream. At 55° latitude, the radiation balance of the ocean surface has a large amplitude, with pronounced negative values prevailing during winter. Here Qs and Qe are comparable. Qs is always directed from the warm ocean surface into the atmosphere, is largest for any oceanic region (comparable to that over deserts, where it has its global maxi- mum) and is much larger in winter than in summer. The expenditure of heat for evaporation, Qe, is also very large and shows a winter maximum. The ocean surface must receive a great amount of heat from deeper layers to compensate for the total expenditure from evaporation, turbulent heat flux and outgoing radiation. Fig. 20 shows a very large negative sum ofQvo + S in winter. Compari- son of areas shows that release of locally stored heat is not adequate to provide the exchange, and in contrast to the locations of Figs. 18 and 19, Qvo averages out large and negative, that is to say the powerful heat transports of the Gulf SECT. 2] LARGE-SCALE INTERACTIONS 143 Stream are drawn upon to provide a significant part of the evaporative and turbulent heat fluxes from sea to air. The annual march of the heat-balance components changes considerably when the effect of a warm current is combined with that of a monsoon climate of moderate latitudes. Such a regime is illustrated in Fig. 21, which pertains to the northwestern portion of the Pacific Ocean (southwest of the Kuril Islands). Here Qs is negative in the summer season, due to northward importation of warm oceanic air, and positive in the winter, due to outflow of cold continental air over the sea. It is therefore clear that the value of turbulent heat flux Qs represents an important quantitative index of the influence exerted by mon- soonal circulation on heat exchange. In the analyzed region, as in the previous one (Fig. 20), during the winter months the ocean surface receives heat from the deeper layers, which is here associated to a considerable extent with utilization of energy from the warm Kuroshio. In summer, however, a converse relationship obtains : the supply of heat from the radiation balance and turbu- lent heat exchange considerably exceeds expenditures for evaporation, which results in warming the upper water layers and facilitates the transmission of excessive heat into other regions by means of current transport and horizontal macroconductivity. One aim of our study of sea-air exchange was to investigate the magnitudes and functional dependence of the energy sources and sinks for the motions of ocean and atmosphere. Actually, quantitative presentation of the global balance components — their geographic distribution and seasonal variation — has emphasized the enormous back-control upon these exerted by the circula- tions themselves. We saw the effects of ocean currents highlighted by the vast difierence in annual march of R, Qe and Qs between eastern and western ocean basins, where cold and warm currents respectively prevail (compare especially Figs. 18 and 19). The even more primary control upon energy transactions exerted by air circulations has been revealed throughout, beginning with the very form of the transfer formulas. In Fig. 13, the huge energy contrast between the trade- wind zones, where the atmosphere's water-vapor fuel is accumulated, and the equatorial region, where it is condensed and exported, clearly requires a dynamic explanation. Radiative sources and sinks are insufficient clues, since the radia- tion balance of the earth's surface is maximum in the trades and the atmospheric radiative sink varies little with latitude. Fig. 13 shows that, in the trades, nearly all the ocean's excess warmth goes into evaporation, loading the atmosphere with latent heat but providing relatively little sensible warming. Mechanistically, we are told that it is the "trade-wind inversion" produced by subsidence in the poleward half of the meridional cell which restricts the upward convective pumping and release of the water vapor in the trades, giving rise to its equatorial shipment and roundabout utilization. Fig, 15 brings out the fact that the major atmospheric heat source in the equatorial trough is precipitation, a process clearly governed by the rising motion of air. But this still does not clarify whether or how the condensation of moisture 144 MALKUS [chap. 4 maintains the meridional circulation, which we have said is the outstanding feature of this part of the heat engine. To understand how the energy sources are utilized to drive air and sea currents and how, in turn, these distribute and constrain the sources, we must incorporate results of energy-budget studies into models relating forces and motions. In the next section of this chapter, two examples build the bridge between energetics and dynamics in the vital fuelling (trade-wind) and firebox (equatorial) regions of the atmosphere. 5. Heat and Water Exchange and its Role in Tropical Circulations The first step toward connecting energy exchange and the dynamics of large-scale circulations have been taken in tropical meteorology, where neces- sity, chance and invention have combined to permit isolation of tractable problems. The ultimate goal is to couple energy source to motion in a predictive manner through the hydrodynamic equations, the key link usually lying in the first law of thermodynamics, and to learn physically how the released heat produces the pressure gradients which drive the wind systems. Commonly, the foundation for such approaches has been application of the methods outlined herein to balance the heat-energy conservation law (27) for a given portion of the atmosphere or along the trajectory of a circulation branch. The crucial new step is that Qva, the atmospheric flux divergence of sensible heat plus potential energy {CpT -\-Agz, henceforth often simply called h or "heat energy") is now formulated explicitly so that it may be evaluated directly from meteorological data and broken down into its contributions from the various scales of motion in space and time. We thereby use and extend our insight into the role of the different-sized physical phenomena, such as eddy, cloud or tropical storm as energy transporters or transformers within a region ; we are able to inquire mechanistically how that portion of the heat engine performs its climatologically deduced function in the general circulation (Figs. 13-15). Usually the data available are not adequate to evaluate all terms in (27) with independent certainty, so that other equations, j)articularly (1) for the heat balance of the ocean surface, that for water conservation, and the transfer formulas, are introduced. As we shall see in the examples, the added labor is well repaid by the further insight afforded (and critical questions raised) concerning physical processes in sea and air, and their mutual interaction. A . Studies of the Energy Transactions in the Trade- Wind Zone Sea-air exchange is at its maximum in the trade-wind region and so are its direct dynamic consequences. As we saw, the latent heat acquired there is the major energy source for atmospheric motions and, as we shall see, even the four times smaller sensible heat accumulation is vital in driving this portion of the low-latitude meridional cell. SECT. 2] LAK(iK-K(!ALK INTERACTIONS 145 The trade-wind zone extends from about lO"^ to 25^ latitude in both hemi- spheres (see Figs. 22 and 23). Easterly winds with an equatorward component dominate the surface of the globe from the subtropical high pressure ridge lines all the way to the equatorial trough. In the vertical, trade-wind air is charac- terized by a layered structure : A lower moist, marginally stable convective 80° 100° 120° 140° 160° 180° 160° 140° 120° 100° 80° 60° 40° 20° 0° 20° 40° 60° 60° 40° 20° 0° 20° 40° 60° jo^ ^ Sf!l_ (b) Fig. 22. Prevailing surface winds over the oceans in winter. (After U.S. Weather Bureau. Atlas of Climatic Charts of the Oceans, 1938. Charts 3 and 27.) (a) Direction and constancy in January. Direction lines based on dominant wind arrows computed for each 5-degree square, with directional constancy as follows : => 81% and over; -> 61-80% ; > 41-60% ; > 25-40% of all winds from the quarter within which the line is a median. Solid line : mean position of equatorial trough. Dashed lines : mean positions of subtropical ridges. (b) Average wind velocity in knots — January, February and March, fi— s. I 146 [chap. 4 layer extends to a height of about 2-3 km, topped by a much drier upper troposphere. The transition zone, a few hundred meters in thickness, is the celebrated "trade-wind inversion" which performs its function in moisture accumulation by preventing upward leakage through cloud penetration. A region of rapid drying and usually of stabilization in temperature lapse rate, it dilutes away the buoyancy of the small cumuli by mixing or "entrainment" so that few of them are able to poke their towers more than a few hundred meters into the dry layer. The height of the inversion is determined by a balance between the large-scale sinking motion and localized convection; the latter 80 100 120 140 160 180 160 140 120 100 80 60 40 20 0 20 40 60 ^ -i^^^^^^^&^:^-^-/i#^ I I I I I I I L 80 100° 120° 140° 160° 180° 160° 140° 120° 100° 80° 60° 40° 20° 0° 20° 40° 60° (a) 40* 60" (b) Fig. 23. Prevailing surface winds over the oceans in summer. (After U.S. Weather Bureau, Atlas of Climatic Charts of the Oceans, 1938, Charts 9 and 29.) (a) Direction and constancy in July. Same notation as Fig. 22a. (b) Average wind velocity in knots — June, July and August. SECT. 2] LARGE-SCALE INTERACTIONS 147 gradually wins the battle as the air flows equatorward. As illustrated schemati- cally in Fig. 24, the moist layer is itself vertically subdivided. A well-stirred turbulent subcloud layer about 600 m deep is topped by the cloud layer, where the picturesque trade cumuli grow day and night, in bunches or in streets elongated parallel to the wind. The low-level trades are the world's steadiest surface wind system ; they couple a horizontally homogeneous air mass by vertical mixing to a uniform sea over 31% of the globe or 62 million square miles. Near the ground, variations in temperature and humidity are slight and synoptic disturbances, for the most part, weakly developed. Here, if anywhere, the daily flow patterns resemble the climatic mean, so that steady-state models may be attempted. Only the concentration of rainfall into two or three days per month suggests that at EAST WIND DRY, STABLE, SINKING AIR ^"^^'V "'(""^ INVERSION f 7 I * jJhts^ EVAPORATION 7 TURBULENT EDOIES ~l 5m ph SEA SURFACE Fig. 24. Schematic vertical cross-section along the path of the trade winds. (After Malkus, 1958, Fig. 1.) Typical wind speeds at various levels are indicated by arrows at the right. The moist layer deepens by about 1000 ft in 500 miles horizontal distance ; clouds are thus drawn much larger than to actual scale. NE stands for northeast and SW for south-west, denoting the trajectory orientation of the Northern Hemisphere trades. upper levels the regularity in tropical flows gives way to restlessness and fluctuations. Interestingly enough, the wind steadiness takes its sharp drop just above cloud tops, or through the inversion layer. Aloft the meridional cell shows marked longitudinal and time variations in its return, poleward-moving branch. The Meteor Expedition of 1924-26 was an epoch-making event in atmos- pheric as well as oceanographic science since it opened tropical meteorology and first revealed the role of the trades as water-vapor accumulators (Ficker, 1936, 1936a). Since then, several ship and aircraft expeditions of the Woods Hole Oceanographic Institution (Wyman et al., 1946; Bunker et al., 1949; Malkus, 1958) have explored the turbulent eddy structure near the air-sea boundary, tropical clouds and their role in energy transports, which will be incorporated further on. However, for a large-scale energy-budget study using theequations and methods of Sections 3 and 4 (pages 100-144), the data requirements exceed the scope of a single expedition and, except in rare cases, 148 MALKUS [chap. 4 are beyond the provisions of the routine weather services. To date, the two opportunities for such studies in the trades were fortuitous by-products of catastrophe. Tlie first was due to World War II, which required weather ships along the aircrcaft route of the Pacific war theater, and the second resulted from the 1954-55 hurricane invasion of the United States' eastern seaboard, which caused the establishment of an observational network ringing the Caribbean Sea. By fortunate coincidence, the Pacific weather stations (Hawaii and three up- wind ships) provided a full season's data along 2400 km of an air trajectory in a region where the trade is steady and two-dimensional. Taking advantage of this opportunity, Riehl and his collaborators (Riehl et al., 1951 ; Riehl and Malkus, 1957) were able to examine the energy transactions in a 3-km deep slice of air following the fiow along this portion of the trade current. By working out mass, water-vapor, heat and momentum budgets for each vertical sub-layer (mixed layer, cloud layer, inversion layer) separately, the mechanisms of energy exchange and transfer were identified. The trade-wind cumulus clouds were shown to be the agency building up and deepening the moist layer down- stream by their myriads of cut-off towers (see Fig. 24). Furthermore the basis for later dynamic treatment (Malkus, 1956) was laid when it was shown that the sensible heat accumulation in the moist layer was responsible for the downstream "pressure head" driving the easterly current against friction. To obtain Qs and Qe, necessary ingredients in the energetics and pressure gradient, the authors used the transfer formulas (20) and (21). Lack of oceano- graphic data prevented a separate test of these by the energy-budget method, although comparisons between joint atmospheric heat and water requirements were highly satisfactory. Their wind (pilot balloon) data gave out at about 3 km elevation, so that little could be said about the upper layers, except that local precipitation heating was inadequate to balance radiation loss. About 53-63% of the accumulated moisture was laterally exported equatorward, while most of the remainder (27% of that accumulated) was precipitated in the local cloud layer below 3 km. a. Joint air-sea energy budget study of the Caribbean region The hurricane-inspired ring of fourteen radiosonde-radiowind stations en- circling the Caribbean Sea (Fig. 25) permitted an upward extension and reapplication of the approach of Riehl and collaborators to an important portion of the trades; this opportunity was seized by Colon (1960). Here, fortunately, there were also enough oceanographic data to undertake a joint air-ocean budget. To our knowledge, this is the first time that such a joint study has been carried through to and checked by its mechanistic and dynamic implications. It will be of considerable value to all marine scientists to under- stand the approaches, difficulties and questions raised by such an inquiry, as well as to learn of its physical results in a region which mothers both the Gulf Stream and much of North American (and probably hemispheric) weather patterns. SKCT. 2] LARGE-SCALE INTERACTIONS 149 Particularly in the winter months, this area constitutes a key energy supply for the Northern Hemisphere jet streams ; its upper circulation is one of the three main high-level outflow channels from the equatorial trough zone. These consist of steady, highly concentrated bands of southwesterly "antitrade" winds at about 40,000 ft elevation (12 km or 200 mb pressure) emanating over the Canal Zone Region, West Africa and central Indian Ocean. In the Carib- bean, the low-level trades are also particularly strong and steady ; the only unsteady winds are found at an elevation of about 25,000 ft (400 mb) where the shift in direction between easterly and westerly is confined to a shallow transition layer. The three-dimensional air-flow conditions are summarized in Fig. 26 ; we see that the Mdnd vector nowhere shows appreciable rotation with height. Fig. 25. Framework of Caribbean study. (After Colon, 1960, Figs. 1 and 174.) Dashed line shows outline of ellipse boundary through which, when extended vertically, air and sea fluxes were calculated. Meteorological stations denoted by crosses. Arrows repre- sent sea currents (after Sverdrup, 1942, fig. 174); insert gives speed code. Dashed straight line is trajectory of low-level air flow. Co-ordinates in N latitude and W longi- tude degrees. Colon's meteorological study was mainly directed toward (27) and a separate kinetic energy budget. To obtain Qs and LP in (27) he introduced an additional equation for the heat balance of the sea surface, namely, Qs + Qe = R-S-Q,o (1) and the equation for atmospheric water-vapor conservation which is written L{E-P) =Qe-LP = Q,^, (28) where the term Qvtv, the water-vapor flux divergence in the atmosphere, is to be stated explicitly, broken down according to scale of motion, and evaluated using the weather station network. Qs and Qe are found from the energy budget method using (1), (2) and the radiation evaluations as outlined in Section 3 (pages 100-114), and compared with separate computations from the transfer formulas. Then Qe is substituted into (28) to obtain LP, the remaining un- measured quantity in the atmospheric balance, as residual. In these equations, the components will now be integrated over the volume bounded laterally by the ring of stations in Fig. 25. 150 [chap. 4 (a) (b) DD {-) 90 180 270 360 1 1 1 r- (0 10 20 30 V (Knots) 1 - ' 1 ' 1 ' 1 ' 2 - ^ ■ 3 - ^^^^^ 4 - ,,^ ^^"'^^ E . V . O O 6 - ^^^^^^.^^ Q. 7 - ^"x ■ 8 - Dec 1956 9 - J 10 , 1 , 1 , 1 y (d) 20 40 60 80 100 Wind Steadiness (%) RKOT. 2] LAROK-SCALE INTKRAOTIONS 151 Tlie solution from (1) for the sum Qs + Qe was carried out, using climatological data, for each month of the year, althougli the meteorological study was restricted to two particular winter months. The Caribbean Sea is particularly suited for this purpose. A small, nearly enclosed water body, it contains one well-defined sea current with entrance in the Lesser Antilles and exit in the Yucatan Channel (see Fig. 25). Plentiful oceanographic data permit direct estimates of Qro and iS in (1) and their seasonal march, as we shall see. The computation of the radiation balance R of the ocean surface is sum- marized in the first eight entries of Table XI. Monthly values of the incoming radiation at the top of the atmosphere for the latitude belt 10°-20°N were adopted from the Smithsonian Tables (List, 1951). Monthly values of atmos- pheric transmissivity were adjusted from London's (1957) seasonal results, ignoring local departures from the latitudinal average water-vapor, ozone and cloudiness structure. The insolation reaching the earth's surface was then entered in line 3, Table XI. Absorption by the sea (line 4) was calculated using a constant 6% albedo throughout the year. The net long-wave radiation loss by the sea surface was obtained for cloudless conditions from Sverdrup's (1942) graph, which gives Qbo as a function of water temperature and relative humidity at ship's deck level. The magnitudes of the actual Qb for cloudy skies were obtained from equation (5a) and London's monthly values of cloudiness in the belt 10°-20°N (line 6, Table XI). Qb appears in line 7, Table XI, and the final result for R, the radiation balance of the Caribbean Sea surface, in line 8. The absorbed short-wave radiation (line 4, Table XI) is greatest in April and May and least in December, with a range about 30% of the annual mean. Seasonal differences result mainly from variations in solar altitude and mean cloudiness ; the latter is especially important in the warm season. During June, July and August, the sun's height is about the same as in April and May but the cloudiness is much greater. Since Qb changes little (slight opposite seasonal march to that of cloudiness), the radiation balance varies in phase with the short-wave absorption. It decreases from a maximum of about 10.2 kg cal cm~2 per month ( ~ 340 cal cm~2 day~"i) in May to a minimum of 6.3 kg cal cm~2 per month (210 cal cm~2 day^i) in December, a range 42% of the annual mean. Fig. 26. Meteorological conditions over the Caribbean region in December, 1956, typical of winter season. (After Colon, 1960, Figs. 3, 4 and 7.) (a) The low-level flow patterns at the 850 mb (~ 5000 ft) pressure surface. Stream- lines are solid lines with arrow-heads. Isotachs (knots) are dashed lines. Min. denotes regions of low windspeed ; Max. denotes regions of high windspeed. (b) The high-level (upper troposphere) flow patterns at the 200 mb (~ 40,000 ft) pressure surface. Streamlines are solid lines with arrowheads. Isotachs (knots) are dashed lines. (c) Typical vertical wind-profile with height. Station depicted is Guantanamo at the southeastern tip of Cuba. Wind-speed profile solid line ; wind-direction profile dashed line. Vertical co-ordinate is pressure in lOO's of millibars. (d) Vertical profile of directional wind steadiness in per cent. Ordinate is pressure in lOO's of millibars. 152 MALKUS [chap. 4 lO i-H ^ fC lO ^ 1— 1 1— 1 C^ -* (M t(< 05 o 1 o r 05 t> CO o o 00 o o 1—1 I— I JO 1-H O Oi X w o 05 1— H o O ce o x/i O o ce w be c p a, 0 TiH C^_ CO 05 i-H (N CO 05 ■* CO r— 1 ^H CO 00 i>i lO o «o O ■* 00 + d + d d (>) ^ '^ 00 ^ LO o Tj3 05 + d + i6 d Tt* o t-; lO (N CO •4 OJ -h d + d d T)H lO iM TjH (M t- Th 05 + d + t-^ d «2 (M 05 CO CO t^ -* d + d + t-: d 00 2^ O t- f-i O t^ o + 1 CO . 5 ^ .22 .•= -g _j C ri O O -P J g g i^ cS -C l-J H I-] -d T3 tf o° tf P5 O 5 ^ T5 0-03 O a^ ^ -^ -1^ ^ ^ 03 <:> c '^ ^ 5d fi rii > cS oj c/; O* O- O* CO ■* 05 O ^ iM SECT. 2] LARGE-SCALE INTERACTIONS 153 For application of the results to the Caribbean ellipse via equation (1), it is now only necessary to multiply B by the surface area of 2.19 x 10^^ cm'-. One of the most intriguing parts of this Caribbean study by Colon was his computation of the storage S to put in (1) directly from oceanographic measure- ments. Later comparisons of the results for Qs and Qe with those of the transfer formulas and with atmospheric heat and \\ater requirements permit a more critical test of the storage determination than has been possible in previous estimations (cf. Pattullo, 1957). To evaluate S, first the annual temperature cycle and its distribution with depth to the vanishing point is required. For this purpose a tabulation of about 8000 monthly temperature soundings, averaged for one-degree squares, was compiled from bathythermograph data at the Woods Hole Oceanographic Institution. In constructing the ellipse-averaged temperature- depth distribution for each month shown in Fig. 27, a small but JFMAMJJASOND Fig. 27. Seasonal march of water temperature in Caribbean Sea from surface down to level (in meters) where seasonal cycle is assvimed to vanish. Curves at depth obtained from bathythermograph data of Woods Hole Oceanographic Institution. Amplitudes adjusted slightly to fit surface curve adopted from Fuglister's (1947) charts. (After Colon, 1960. Fig. 10.) iinjiortant adjustment was made, presumably to correct for slightly un- representative sampling in summer when the data coverage was sparsest : the annual surface temperature range would have been 5.9°F from the bathy- thermograph sample ; it was reduced to 4.6°F to agree with climatic data for the region (Fuglister, 1947; Atlas of Climatic Charts of the Oceans, U.S. Weather Bureau, 1938) and the amplitudes at depth were adjusted propor- tionally. The annual range is essentially undiminished at 15 m, reduced to 87% at 30 m, to 76% at 45 m and to 26% at 75 m. At 90 m, the seasonal cycle was assumed to have vanished ; below that it is very small and, if anything, appears to change phase. 154 MALKUS [chap. 4 The storage in this 90-m deep oceanic layer was computed from the relation S = cp^\—dv, (29) where c and ptv are the specific heat of water and its density, both taken as unity, t is time and v denotes volume. The average temperature, T, for the layer 0-90 m was determined from the depth profile for each month by the method of equal areas. Then dTjci was approximated by taking two-month overlapping temperature differences centered at each month; finally each result was multiplied by the volume of water (area of ellipse times 90 m depth of layer). The resulting seasonal march of S is shown in line 9, Table XI, expressed per unit area to compare with previous figures. The storage is zero in late September and late February. The oceanic heat content decreases rapidly in fall and early winter, while the increase in spring is more gradual. The average storage rate from peak summer to peak winter is about — 2.5 kg cal cm"^ per month ( — 85 cal cm""^ day~i) ; from winter to summer the average is + 1.8 kg cal cm~2 per month ( + 61 cal cm~2 day^i). Clearly 8 is in general not a negligible term in equation (1) and, according to these results, may amount in some months to 25-50% of the dominant terms [Qe and R). Colon's estimates of 8 compare favorably with most figures available in the literature. Gabites (1950) obtained storage values for oceanic areas by latitude belts, assuming that the surface temperature variation is maintained for the first 25 m with a steady decrease from there to zero at 125 m. His values for the belt 10°-20°N are almost identical with those in Table XI, as are those of Fritz (1958) computed from bathythermograph records. Much larger values, however, were obtained by Pattullo (1957) from the identical data as used by Colon; the reasons for the discrepancy lie in different analyses. Pattullo treated the sample temperatures as totally representative, which as mentioned showed a larger annual range than did climatic charts, and she also assumed a much greater depth of the seasonal cycle. Her rates of storage are about double those presented here. This magnitude uncertainty is probably inherent today in storage calculations. Nevertheless, the inconsistency of the higher values with transfer formula results and joint budget requirements leads us, for the present anyway, to place somewhat greater confidence in Colon's determinations. Estimation of the oceanic heat-flux divergence, Qvo, in the Caribbean is simplified by the existence of one main, well-defined current which crosses the ellipse approximately from east to west. The following equation, therefore, could be used to calculate Qro '■ Qro = C r 31 hy TydAy-C j Mhe TedAc (30) where Mhy and Ty are the horizontal water-mass flow and temperature across the Yucatan Channel ; the Mi,e and T^ denote the same ]mrameters on the east side of the Caribbean. The A's, denote the lateral areas aross the outflow and inflow boundaries respectively. The nature of the available data was such that SECT. 2] LARGE-SCALE INTERACTIONS 155 several assiini])tions and approximations were required in applying (30), namely, (i) The basic ciuTent was taken constant throughout the year. For mass flux, the transport across the Florida Straits of 26 x 10^ m^ sec"i (Sverdrup, 1942) was used, assuming that one-third occurs in the upper 90 m, thus contributing to the heat flow. (ii) The mean monthly temperatures across the entrance and exit ends were computed using the temperature depth dependence of Fig. 27 and the surface temperature profiles of Fig. 28. (iii) Vertical heat flux was ignored. Stommel's (1958) suggested upwelling (of the order of meters per year) may be shown to contribute a negligible correction to the computed Qvo. (iv) Space correlations of mass flux and temperature across the channel were neglected, as were time correlations on a scale less than a month, due to lack of data. Thus the eff'ect of all oceanic scales of motion smaller than the basic advective current and its monthly temperature variations had to be left out. With these restrictions, the resulting Q^o was calculated and appears in line 10 of Table XI. Positive heat-flux divergence in summer gives way to negative (convergence) in winter because, as shown in Fig. 28, the Gulf of Mexico III: 1 T" 1 1 1 \ \-" I East Coribbean y/ ^\ \. N / y / • \ \ / • \ \ / ^ \ \ - / • \ \ \ //- \ \ _^ /^ ^ \ ^ " \ / \ \ \ ^ ^/ \ \ \ \ ^^Z^-'\y 1 \ \ L_ 1 1 1 84 ■t 82 S 81 I *I 80 o 5 79 78 h 77 J FMAMJ JASOND Fig. 28. Seasonal march of sea-surface teinperature at entrance (East Caribbean — dashed) and exit (Yucatan Channel — solid) ends of Caribbean ellipse. Determined from Fuglister's (1947) charts. (After Colon, 1960, Fig. 11.) (current exit) is warmer than the Atlantic (current entrance) in summer and cooler in winter. Actually Qvo proves to be the smallest component in the Caribbean heat balance, so that fairly large errors in its assessment are not important. In June and September, the maximum oceanic heat export is 4-9% of R, while in November and December a maximum import of 5-6% R was indicated which, as we saw earlier, are within the uncertainty of radiation- balance computations. 156 [chap. 4 With the results in Hnes 8, 9 and 10 of Table XT, equation (1) may now be solved for the sum Q,c + Qs, the total sensible plus latent heat energy flux from sea to air. Colon made the separation between these using a Bowen ratio of 10%, a good mean flgure for the trades (see Bunker, 1960; Riehl et al., 1951). The final results for Qe and Qs from the energy budget method are given in the last two lines of Table XI. The values of Qe so obtained range from 10.0 kg cal cm-2 per month (333 cal cm-2 day-i) in November to 5.4 kg cal cm-2 per month (174 cal cm-2 day-i) in August. The corresponding evaporation rates are 0.58 cm day-i and 0.31 cm day-i, respectively, with an annual average of 161 cm per year. The seasonal march of all components of the sea-surface heat balance are plotted together in Fig. 29a. Particularly notable is the strong effect kg cal cm^ month (q) Fig, 2 0 - 2 -4 10 col/day 7 (Whole orea) g (b) kg col JFMAMJjaSOND J FMAMJ J ASOND 29. Results of Caribbean study. (After Colon, 1960, Fig. 12.) (a) Seasonal march of sea-surface heat-balance components determined by energy- budget method. Dominant components are radiation balance, R, and latent heat flux, Qg. The storage term S becomes significant in midwinter and midsummer. Sensible heat exchange, Qg, and oceanic heat flux divergence, Q^o, are small through- out. Units: kg cal cm~2 per month. (b) Comparison of total exchange Qe + Qs (Bowen ratio assumed 10%) by energy- budget method (solid curve) and transfer formulas (dashed curve). For area of whole Caribbean ellipse (2.19 x 10^^ cm^) in lO^^ cal/day, left-hand ordinate ; for unit area of sea surface in kg cal cm"'^ pgj. month, right-hand ordinate. Crosses are separate trans- fer-formula evaluations of Qg + Qg from ship data for the months of the meteorological study, namely December, 1956, and January, 1957. exerted by the storage regime upon the computed Qe, which is thereby opposite in phase to the radiation balance, R, and shows a pronounced summer minimum (cf. Fig. 18). A partial test of these results lies in whether the Qe derived from the transfer formulas, independent of radiation or storage assumptions, shows a similar seasonal profile. Transfer formula (21) was used to evaluate the latent heat flux Qe from cHmatological data {Marine Atlas of the World, Atlantic Ocean. U.S. Navy, 1955) for a small sector of the Caribbean north of the Canal Zone. The data were not adequate to calculate Qs, which was again assumed 10% Qe in the SECT. 2] l.AHCK-SCALE INTERAt'TIONS 157 o()inj)aris()n show n in Fig. 29b. Tlie transfer fornuilas relate the sea-air fluxes to local, routinely measured meteorological jmrameters. Regardless of u!i- certainties in their pro])orti()nality constants, the excellent agreement in the months of maximum, mininuim and trends can hardly be accidental. It gives, in fact, considerably increased confidence in the ability of both methods to provide a useful quantitative picture of air-sea interaction and energetics. In Fig. 29b, the magnitudes of total exchange Qs-\-Qe computed from the two methods agree within 15% except in autumn when the energy-budget method gives 30% higher transfer in October. Table XII compares the present results for mean annual evaporation with those of other authors. Table XII Comparison of Annual Evaporation Computations for the Caribbean Region Author Method Evaporation, cm/year Colon (1960) Energy budget 161 Colon (1960) Formulas— Marine Atlas (U.S. Navy) 146 Budyko (1956) Formulas— Atlas of the Heat Balance (U.S.S.R.) 138 Jacobs (1951a) Formulas — U.S. Weather Bureau Climatic Charts 124 Wiist (1936) Budget (Lat. 10°-20°N Atlantic) 146 Separate transfer formula (20 and 21) evaluations of Qs and Qe were made from synoptic ship data for the two special months of the meteorological study, namely December, 1956, and January, 1957. The Bowen ratio came out exactly 10%. The sum Qe-\-Qs was somewhat higher than the climatic means computed, as shown by the x's in Fig. 29b. Comparison of Figs. 29a and 18 is also very good, showing the same magnitudes and seasonal march of the balance com- ponents in climatically similar oceanic regions of opposite hemispheres. How- ever, it should be pointed out that the methods of Budyko in obtaining Fig. 18 and of Colon in computing Table XI (from which Fig. 29a was made) are based on the same type of radiation calculations and transfer formulas, the latter with nearly identical coefficients. Later Australian work^ has suggested somewhat larger coefficients in the transfer formulas than those of (20) and (21), while, as we saw, the storage calculations of Pattullo (1957) suggest that S could be larger (but not smaller) than found here. Clearly a larger 8 and Qe are mutually incompatible in summer unless the radiation balance in that season has been significantly (about 50%) underestimated. Therefore, a final critique of all these joint budget studies rests upon and awaits improvement in atmospheric radiation determinations, which are at present a real "bottleneck" in much of meteorology and oceanography. 1 Priestley (1959) has suggested an 88% increase in the coeflficient in equation (20) for sensible heat flux, while an unpublished work by Swinbank argues for a 30% increase in the coefficient in (21) for latent heat flux. 158 MALKUS [chap. 4 b. The heat and water- vapor budgets of the Caribbean troposphere The major purpose for which C/oloii undertook evakiation of sea-air fluxes was to construct an atmospheric energy budget, thereby to investigate the workings of the Caribbean portion of the tropical cell and the mechanisms of its energy transformations. We begin by assuming steady conditions in the air contained within the Caribbean ellipse from the sea surface to the tropopause. When we integrate equation (27) over this volume, we have, after using Green's theorem to change the flux divergence term to a surface integral, ^ LP + Qs + Ba = ( ( p{cpT + Agz)u-nda, (27a) where a is the bounding surface of the volume, 7i is the unit vector pointing outward and u is the air velocity vector. In this formulation, the source and sink terms LP and Ra now denote volume integrated values and Qs is summed over the lower surface area, so that units of each term will be in calories per second. The transport term may be further subdivided into lateral and vertical fluxes, namely, LP + Qs + Pa = {cpT + Agz)cndl{dplg)+ pt{CpT + Agz)tWtdAt JpJl JJA, pb{CpT + Agz)bWb dAb, (27b) ft where the first integral term, the lateral flux, has been transformed into an integral with respect to pressure, p, using the hydrostatic equation; c„ is the normal component of the horizontal wind, positive outward, and Z is a line element of the bounding surface. The last two integrals, the vertical upward flux through the top, t, and the bottom, b, surface of the volume, vanish when the integration is performed from the sea surface to the tropopause, where the vertical air velocity, w, may be assumed to vanish, but must be included when intermediate layers are considered separately. The water-vapor conservation equation (28) may be similarly integrated and expressed in a form suitable for use with meteorological data, namely, Qe-LP= pLqu-nda (28a) or Qe-LP = Lqcndl{dplg)+ \\ ptLqtWt dAt- \\ pbLqbWbdAb, (28b) where q is the specific humidity in grams of vapor per gram of air. First LP, the precipitation warming, is to be evaluated residually from integration of (27b) and (28b) from the surface to the tropopause and the results compared. 1 An excellent derivation of this equation directly from the law of total energy con- servation is given by Kraus (1959). SECT. 2] LARGE-SCALE INTERACTIONS 159 Qe and Qs are taken over from the oceanographic study. Ea, the atmospheric radiation sink for the vokime, is calculated using the methods and figures of London (1957). In evaluating c„, the wind data available and method of analysis impose limitations on the time and space scales of the motions whose fluxes can be included in the budget. Colon's method of handling this problem is typical of a simplifying hypothesis that can often be used in treating the tropical atmos- phere. For each of the two months, the meteorological network data were analyzed to obtain mean values of Cn and the energy parameters at twenty evenly spaced grid points around the ellipse periphery at each of ten pressure levels between 1000 and 100 mb (approximately sea-level to tropopause). Thus contributions of "time eddies" of a scale less than one month and "standing space eddies" of a size less than one-twentieth of the ellipse circumference (about 3° latitude) were ignored. The steadiness of the flow and relative rarity of synoptic disturbances in winter justify this, while the contribution of still smaller scales of motion in maintaining the balances are deducible later. The two independent precipitation estimates came out within 50% of each other for both months. Their average for December, 1956, which we use in later work, is about 3 cm per month. This is less than half the mean monthly rainfall rate for the region found by other authors. However, December is in the dry season in a climatic regime where the seasonal precipitation cycle may easily have a two-to-one amplitude about the mean (Riehl, 1954). The energy transactions, their role and mechanism, take physical life in connection with process when these heat and moisture balances are performed separately for three vertically superposed layers : surface to 900 mb, 900- 500 mb and 500-100 mb. This subdivision corresponds roughly to sub-cloud, cloud and above-inversion layers, although cloud bases and tops average more nearly 960 and 750 mb, respectively. In order to complete these balances, we need to draw on mass continuity requirements to deduce mean vertical mass flows through the 900- and 500-mb levels. Evaluation of the mass flow through the lateral boundaries of the Caribbean ellipse indicated, as expected, a net convergence near 200 mb (about 40,000 ft elevation) and divergence in the trade regime below. Continuity calls for sinking motion of several hundred meters per day throughout the troposphere, with a maximum descent of 700- 800 m day~i at the 300-mb level. As we shall see, this sinking motion proves essential in the heat-energy budget of the upper layers. The budgets by layers are summarized in Fig. 30. The Caribbean ellipse is 5% area- wise of the Northern Hemisphere trade region. The numbers in the figure when multiplied by lO^^ cal/sec give the extrapolated fluxes for the entire 10°-20° latitude belt to compare with the equatorial study to follow. When multiplied by 5 x lO^^ cal/sec they give the contribution of the elliptical cylinder actually computed by Colon. The h = CpT + Agz terms are segregated on the left side of the diagram, while the latent heat (Lq) terms are entered on the right. The solid arrows show heat and water- vapor transports by the mean 160 [chap. 4 mass flow, which in the horizontal arises from c„, the average Cn around the whole ellipse periphery. The vertical solid arrows result from the mean mass descent computed by continuity from c„. The dashed arrows are the contribu- tion to the respective horizontal fluxes from the "standing space eddy" terms. Physically, these arise simply from the fact that the low-level divergence is made up of a greater mass outflow at the western end of the ellipse than inflow c r + Agz Lq ft 1 54,000 11,50 0.07 Rod. -0 74 ■9 12 ■0 16 Precip 030 1.80 I ? 0.58 - 0 37- Precip. -0.30 1 1 016 1000 .178 ■0 09 Radiation loss 0.07 ■002 0, ■1.43 021 - 023- 3400 016 I div h =-0.86 I div Lq = 1.41 Fig. 30. Heat-energy budget of the atmosphere above the Caribbean ellipse by layers. (After Colon, 1960, Fig. 18.) Vertical co-ordinate given in pressure units on left and height (approximate) units on right. Sensible heat plus potential energy (h — CpT + Agz) terms on left; latent heat (Lq) terms on right. When multiplied by 10^5 numbers in figure give fluxes in cal/sec extrapolated for entire latitude belt 10°-20'^N. When multiplied by 5 x 10^3 they give in cal/sec Colon's actual results for Caribbean ellipse of svirface area 2.19 x IQl^ cm2. Solid arrows give contribution of mean ageostrophic mass flow. Dashed horizontal arrows give "standing" geostrophic eddy contribution. Upward directed dotted arrows are residually computed fluxes required for balance, assumed due to convection and turbulence. at the eastern end (net mass export by the lower trades) while the high-level convergence comes from an opposite difference between inflow and outflow (net mass import) in the upper westerlies. To complete the balances, radiation sinks are first apportioned between layers. An important and physically reasonable confirmation of the entire analysis results : first, the requirements in the lowest (surface to 900-mb) layer, which is mostly below cloud base, are satisfied entirely by the Qs transfer from SECT. 2] LARUE-SOALK INTERACTIONS 161 the ocean ; and, secondly, the bulk of the requirement for precipitation heating falls in the 900-500 mb layer, to which nearly all convective clouds are confined in this season. \Mthin computational accuracy, the heat deficit in the upper troposphere is balanced by the flux convergence of h and, implicitly, conversion of potential energy, ^4^72, into sensible heat energy, CpT, by the mean sinking motion. Actually, 500 mb is considerably above the mean height of cloud tops (800-700 mb) and some compressional heating by subsidence is also needed and found in the middle layer, where the radiation sink is more than twice LP. As a whole the Caribbean ellipse region exports 6.1 x lO^^ cal/day latent heat in December and imports 61% that much sensible heat ; the net export of total heat energy {Q = CpT + Agz + Lq) to other parts of the globe thus amounts to 29% of the total transfer {Qs-\-Q,€) received locally from the ocean, in excellent agreement with the Pacific work of Riehl et al. (1951). This result brings out an important and paradoxical feature of the tropical atmosphere, illustrating the complex linkage between planetary fluid dynamics and energy transformations. Examining the whole trade -wind troposphere, we see that an outside source of energy is needed to drive the circulation and offset radiational losses. This is so despite the fact that a more-than-adequate source is readily at hand in the form of latent heat. The latter is, however, exported rather than released and processed locally. Therefore, while the region as a whole exports heat energy, it is at the same time dependent upon processes elsewhere for its own main- tenance I The final completion of heat and moisture balance for each layer requires the upward-directed dotted arrows (residuals) across the 900 and 500 mb surfaces. If real, these transfers must be achieved by processes on a time and space scale much smaller than that of the mean circulation, namely eddy turbulence or convection. Similar fluxes were obtained in the Pacific trade by Riehl et al. (1951), who demonstrated quantitatively that the trade cumulus chimneys (Fig. 2) were easily able to pump up the required moisture. Since then, the Woods Hole expeditions have confirmed the existence of such fluxes. Their individual cloud measurements and convection models (Malkus, 1958) illustrate how the cumulus groups accomplish the net warming, moistening and deepening of the cloud layer as the trade-wind air flows downstream. In the Caribbean study, Colon used their results to show that the total upward energy flux through 900 mb may be achieved by normal cumulus updrafts if the mean cloudiness is 35% and 2% of the cloudy matter rises at 2 m/sec. The trade cumuli serve more purposes than to decorate postcards of the tropical atolls which they inadequately water! The link between heat source and circulation dynamics is forged when we examine the pressure drop along the trade- wind trajectory. It is, similar to the net warming, large at the ocean boundary and, also similar to the warming, vanishes at the top of the moist layer, so that the 700-mb pressure surface does not incline downstream. In fluid mechanics, the basic driving force is "pressure head" or horizontal pressure gradient; on the rotating earth, its downstream (ageostrophic) component is the intermediary by which the energy sources are 162 MALKUS [chap. 4 ■=■ e 7654 m/sec WSW 2000 1500 1000 500 0654 (km) '*— ENE m/sec 700 7 \ 800 L 900 - \ \ \ \ \ \ \ 1 \ \ 1000 i— , , ''\ 0 2 4 6 (a) 2 1 (b) (0 0 0.2 0.6 ,^6 - 10 sec Divergence ~ h~ ^ OS (d) Fig. 31. Atmospheric conditions along a trade-wind trajectory, from a study of the Pacific trades. (After Malkus, 1956, Figs. 1, 2, 4 and 5a.) (a) Vertical structure of the air along the trajectory indicated in (c). Horizontal distance given in km downstream of entrance end. Vertical co-ordinate to right in mb, to left in km (approximate). The heavy lines separate the layers described in text. The lines with arrows are trajectories while the lighter solid lines are potential temperature isopleths labelled in degrees absokite. Trade cumulus clouds are entered schematically. To the right is the profile of wind speed (along the section) at the up- stream end and to the left is the wind-profile at the downstream end. Wind speeds are in m/sec. (b) Graph showing observed net downstream warming in terms of potential temperature increase. Ad, of the air parcels, calculated as a function of height by following trajectories in (a). The dashed curve is a simple exponential used later in the theory (see equation 42a, Section 6-C, page 196) to approximate the observed curve. (c) Solid line shows orientation of trade-wind trajectory in Pacific. The Hawaiian Islands are shown at the down-wind termination, while the U.S. west coast appears in the upper right-hand corner. The lighter solid lines are mean 700-mb contours (lO's ft, first digit, a one, omitted), while the dashed lines are isopleths of gradient wind speed (m/sec, positive values have component from west) and thus are parallel to the sea- level isobars. (d) Horizontal velocity divergence (assumed equal to 8u/8s) calculated from ob- served wind-profiles for Pacific trade trajectory, averaged over whole length of section. Profile of 8u/ds shown as function of pressure on left and height on right. SECT. 2] t.ak(!K-.S(:alk interac:tions 163 used to drive the flow. In tlie atmosphere, as in tlie ocean, pressures are pro- duced liydrostatically : warm Huid columns are less dense and exert less pressure near the ground than cold ones, so that horizontal gradients in pressure are created when warm and cold columns exist side-by-side. The dynamically important role of the oceanic heat input to the trades is thus to maintain a downstream-directed pressure force by adding to the heat content of the air as it flows equatorward. Physically, then, the trade-wind heat source arises because, in the moist layer, the terms Qs + LP are greater than the radiation sink, Ba. Convection distributes and releases the added energy, so that individual air trajectories move toward higher h = CpT + Agz'^Cpd, or toward higher potential tempera- ture, as illustrated in Fig. 31a. Riehl and Malkus (1957) have shown theoreti- cally that under such conditions the pressure head is simply related to the heat source as follows : Jzjs U Apo = -K \ —dsdz (31) in a unit-width section along the flow, where the externally imposed pressure drop at the top of the heat source is zero, as it is in the trades. In equation (31), Apo is the difference in surface pressure between the end points of integration along a trajectory s, u is the horizontal air velocity, iT is a theoretically determined constant (;^1.4x 10^^ g2 cm~2 sec~2 cal~i in the trades) and the vertical z integration is performed through the depth of the moist layer. The net heat source, H (in cal g~i sec~i), is evaluated from the potential temperature increase following the trajectories (solid arrows. Fig. 31a). In the Pacific trade, with a mean wind speed of about 6 m/sec, the observed pressure drop of about 1 mb per 500 km trajectory distance is nicely accounted for using (31) and the heat source, which increases the average potential temperature of the moist layer by about 0.6°C in a day's travel (Fig. 31b), In the Caribbean winter, the slightly greater net warming is more than compensated by a doubly large wind speed. The calculated downstream pressure drop of about 1 mb per 1000 km, however, is in good agreement with climatic charts and those of Colon for the selected months. Thus the increase in heat content along the lower trades, due to exchange with the sea surface, makes a vital contribution to the balance of forces, permitting the flow-sustaining pressure head to be created locally. This portion of the meridional cell is thus a directly maintained thermal circulation. Above the level where the convection distributes the heating, the internally developed downstream pressure drop disappears and the wind flow becomes more nearly geostrophic. In the very high troposj^here, the pressure head sustaining the antitrade westerlies is created in the equatorial trough zone, from combustion of the water-vapor fuel shipped there by the lower trades. We shall next look into the operation of the "firebox". 164 MALKUS [chap. 4 B. Energy Transformations in the Equatorial Trough Zone 'Vhe poleward (subsiding) portions of the meridional cell of the tropics has been combed by research ships and airplanes, studying the air-sea boundary fluxes and the eddies and tropical clouds which effect the energy transports. A connective structure has been framed by several large-scale budget studies and the first links between energetics and dynamics forged by introduction of the heat sources into the hydrodynamic equations (Malkus, 1956). Despite deficient data, these studies have been aided by relatively simple conditions : a wdde separation in space and time scales exists between the vital physical processes, namely convective turbulence and the large-scale circulation itself, which is steady and two dimensional. Therefore, to fill in the climatological outline of the region's function (Figs. 13-15), we have at least the framework of the dynamic skeleton and the mechanistic muscles. The case is quite different in the equatorial zone, where in many longitudes the trades of opposing hemispheres clash and ascend, lifting and converting the imported water vapor. The physical situation is both more complex and far more fragmentarily explored. Little connective tissue has joined the few widely-separated synoptic, dynamic and strictly oceanographic investigations. Climatology suggests what energy conversions and transports the region must undertake, but the greater unsteadiness of the air circulation does not permit as ready translation of average magnitude into dynamic process. As the trade-wind air flows westward and equatorward, away from the steadying influence of the subtropical high -pressure ridges and toward the low pressure trough associated with the thermal equator, the average depth of the moist convective layer grows ; the mean strength of the trade-wind inversion weakens and gradually vanishes. But there is observational evidence that this occurs in bursts, or alternations between intense convective build-ups and un- disturbed trade regime, showing time fluctuations and preferred locations. Quite possibly the ascending portion of the "meridional cell" is a somewhat fictitious average over intermittency. In other words, an intermediate scale of motion, the so-called "synoptic scale" in meteorology, appears to intervene and play a vital role. As we have suggested, such an alteration in atmospheric flow implies heavy consequences for the interaction between sea and air and the feedback between their circulations. In 1958, a large-scale budget study of the equatorial region was attempted by Riehl and Malkus. Although even more severely hampered by data deficiency than similar efforts in the trades, it has laid a specific framework for further critical exploration. Moreover, even the qualitative completion of a heat- energy balance in this zone required a new and remarkable mechanistic hypo- thesis which stands to be tested by the more modern observational tools, such as the satellite programs. Experience gained in the trade-wind (and rotating dishpan) studies has shown the advantage of natural co-ordinate systems selected for the problem to be treated, rather than the polar co-ordinates customarily used. Since we SKIT. 2] LAROK-SCAI,!; 1 NTKKACTIONS 165 are intercisted hero in the e(]iiatorial troiiuh /.one, we employ a co-ordiruite system following the mean position oi' the trough, the extremes of" which are given in Figs. 22 and 23. In this way, the north south meanderings of its seasonal mean position are eliminated ; we further gain vastly more data, since each radiosonde station appears at different latitudinal distances relative to the trough line as it migrates. Not all of the needed information for complete budgets is available in this reference frame, particularly the radiation, but it is assumed that, for the latter, we can employ the values computed for latitudes fixed with respect to the mean trough position. Similar to the trade-wind studies, we shall attempt to balance the atmos- j^heric heat and water- vapor conservation equations in the forms (27b) and (28b), using (1) for the sea-surface heat balance to help determine exchange. The oceanic storage term is uncertain and may be large even in these latitudes ; in some months it has been estimated above one-third Qe by Gabites (1950) and even larger by Pattullo (1957). Lacking sufficient bathythermograph observa- tions in this belt, we restrict the heat-source calculations to the end of February and August, when storage is known to be small and the equatorial trough reaches its most poleward position. Under these conditions (1) becomes R = Qe + Qs + Qro. (la) Rather than computing B, the radiation balance of the sea surface, as done in the trade-wind studies, w^e transform (la) to a suitable form to use the radiation results of London directly. Substituting (26), Es = R + Ba, (26) where Bs is the radiation balance of the air-sea system and Ba the (negative) radiation balance of the atmosphere, we have Bs = Qs + Qe + Qvo+Ba. (lb) London has prepared tables and diagrams giving Bs and Ba as a function of latitude and season for the Northern Hemisphere. We shall use these presently to supplement the transfer formula determinations of air-sea exchange with (lb), but first let us consider the north-south migration of the trough. a. Trough position and movement in relation to energy fluxes The seasonal migration amplitude of the equatorial trough is from about 5°S to 12°N latitude, or less than half that of the sun — a strange feature which has provoked much discussion in the literature. From this it has been con- cluded, for instance b}^ Rossby (1949), that the classical explanation of the trough as a simple thermally induced j)henomenon cannot be maintained. If we now integrate London's Bs figures for the no-storage months of August and February to obtain transports (similar procedure as done for the annual figures in Table I), a very interesting insight into this behavior of the equatorial trough is provided. 166 MALKUS [chap. 4 Fig. 32 shows the result under the assumption that the August Rs distribu- tion is generally valid for the summer hemisphere and the February distribution for the winter hemisphere. Heat fluxes from summer to winter pole are pre- sented in multiples of lO^^ cal/sec, the basic "unit" used in the equatorial study. The zero line is crossed near latitude 12° in the summer hemisphere, which corresponds well to the mean equatorial trough position in July- September. Maximum transport is 0.49 units to the summer pole and 1.06 units 90 60 50 40 30 20 10 (U XI ^ 0 To Summer Pole __J I \ I L 1.0 0.8 0.6 0.4 0.2 lO'^ col/ sec 0.6 0.8 1.0 1.2 10- 20- 30 40 50- 60- 90 Fig. 32. Latitudinal heat fluxes computed from integrating London's (1957) Northern Hemisphere Rg figures for no-storage months of February and August. February fluxes assumed applicable to winter hemisphere, August fluxes to summer hemisphere. (After Riehl and Malkus, 1958, Fig. 3.) Ordinate in degrees latitude; abscissa in lO^^ cal/sec around whole latitude belt. to the winter pole, a difference of 0.57 units. In the equatorial zone, Rs is about 0.31 units in belts 10°-latitude wide in both seasons. Since the trough shifts seasonally by 17° latitude in the mean, an area yielding Rs = (17/10) x 0.31 = 0.55 units is transferred from the summer to the winter side of the trough during its seasonal migration. That amount is practically identical with the difference of heat transport to summer and winter poles ! This calculation suggests that no heat flow takes place through the equatorial trough, at least at the seasonal SECT. 2] LARGE-SCALE INTERACTIONS 167 extremes, so that its position and migration are determined largely by the distribution of heat sources and sinks, in particular by the strength of the cold source at high latitudes in the winter hemisphere. There thus does not appear to be any geometrically fixed equatorial boundary condition for the atmos- phere's general circulation. For the determinations of exchange to enter as source terms in (27b) and (28b), we should like an independent estimate of Qs and Qe from the transfer formulas and an energy budget check on their sum, at least, from (lb). Un- fortunately this is not possible because data to assess Qs from the transfer formula are lacking. For Qe, equation (21), the Atlas of Climatological Charts of the Oceans (U.S. Weather Bureau, 1938) and Bean and Abbot's (1957) surface humidity maps were used to derive Table XIII into which contribu- tions from land areas have been weighted. For the continents of the humid tropics, the same evaporation rate was used as over the oceans ; over deserts it was taken as zero. These assumptions are well justified by the continental evaporation figures reported by Budyko (1956). Using the results for Qe we may find Qs as residual in (lb) after entering London's figures for Rs and Ra, and provided some estimate of Qvo may be made. Riehl and Malkus assumed that it would be a negligible term in the ocean's heat budget. In equatorial regions where sea-temperature gradients are slight and currents zonal, it was envisaged as even smaller than in the Caribbean, where Qvo was within the computational uncertainty of the larger balance components. In the following analysis, we consider mainly the belt extending from the troughline to a distance of 10° latitude on the winter side. In this belt the solution of (lb) for Qs (neglecting Qvo) gives Qs = 0.3l{Rs) + l.00{- Ra)-0.dS{Qe) = 0.38 units (32) which implies a Bowen ratio {QsjQe) of 41% and is much larger than expected from other determinations in the tropics. If, as suggested by Fig. 13, Qvo in these regions is as much as 20% Qe, we obtain a Qs of 19 units and a more plausible Bowen ratio of 20%. Although 19 units are within the accuracy of the radiation computations, a 20% reduction in London's R— Rs — Ra is less likely in view of the agreement therein with Budyko (table VI, page 121). Furthermore the Qe's in Table XIII are in good agreement with all recent evaporation determinations i for the region and 25% greater than that of Wiist (1936) ; the Qe in (32) can hardly be much underestimated. We shall, therefore, accept a larger ratio oi QsjQe in this zone than in the trades (3-10%) as a possi- bility to be explored, particularly since a plausible mechanism exists to explain it, as we shall see. 1 To compare Qe in Table XIII with the results in Table XI we must recall that Colon considered a geographically fixed locality at 10^-20°N, which is, relative to the trough, roughly 0^-10" in summer and 10^-20^ in winter. 1G8 [CHAl". 4 m < o O ^-- H 1^ o o ■+^ .5 fS ce I— 1 C ^ 2 o O 2 jH ^" fe 2 ^ ^ ^^ cS o > Or' O g O O 03 o II ~S ^ 0; « X s r^ c c« p X 2 5t ^4 It -c S cS '^ A\ O o Cl o o ^ ^ 2 -i^ ^? 0) fi ^ g |o:| CC +-' o opt: -c S * ■'^ i >■- I .o i S? 3^ 2 SKOT. 2] LARriK-SCALE INTERACTIONS 169 b. Lateral heat and moisture fluxes To complete the lieat and moisture budgets, (27b) and (2Sb) are to be integrated vertically from the sea surface to the base of the stratosphere, so that they may be written LP + Qs+Ba= I ^^Cn{CpT + Agz}dl{dplg) (27c) and Qe-LP= ( { CnLqdl{dplg). (28c) We use the condition of no flux across the troughline itself. It should be empha- sized that this does not assume zero flux across the geographic equator ; on the contrary, asymmetri6.7 m/sec (rough surface) as shown by the dashed line in Fig. 6. Hidaka (1958) has extended their method to produce charts for the Southern Hemisphere and Indian Ocean. 182 MAI.KTJS [chap. 4 20* !»• 40* 50* 60' TO" 180* ITO- 160* 150' 140" I30» 1 20' 1 0* DO* 90" 30' " *~"^ 1 1 V 1 ^ ^ 1 L- J 1 -^ t^ TtT ■ r ^ P C -< > J/i-^ North Pacific MEAN WIND STRESS ^ ^ -^ -O V f f • \>3 ^ DYNES CM -2 k 1/ • •\ ,« ^■ ^ - ^ ^ ^ >» A^ V, Scole of Mogni 0 Values S 0 1 dy tude nes cm-2 € ^N. - - - - -> ^ - - " - - ^ ^ ^ - / ( ANNUAL f^ f ^ y^ ^^ ~^.. ■-^ - - - ' ^ - - " ' ^ ^ ■ - - \ ^ P\ T\ A * . V !• • • - • ' ' - • - /■ ' t ^ K Y r ' ' - , ' ' - ' - ' ' ' ' ' *>, - " '- ^ • / / / .\: ^ I b' ' - - • - - - - - - - ♦- -- - t^ ^ ^^ J^ x- • y • 0 V. r 'a - - - ' ^ ^ ^ " -^ " '- -- -p-- ^ ^ ^ ^ ^ y y^ y y ^ - 0 0 / Psl, i 1 i^ ^0 - - ' ' ^ ^ - -- '■ -- ^ -- ^ ^ ^ " - - - - - > . « . 0 ' ^ fy 7m^ ^ 0 ' - - ' y - - - - - - V -- - X. V \ \ V * » . < t ' / V{;?H .VT' -^• ^ • _^ ' ' - - - - ' - - - - ■^ ^ ■^ "K X - > > •^ •i> 1 120' 130' 140' ISO" 160' 170' 180' 170- 160' (50' 140" 130* I20' 110" 100* 90' 80* Fig. 38. Mean annual wind stress over the North Pacific computed from transfer formula (17). (After Scripps Institution of Oceanography, Oceanographic Report No. 14, 1948.) 120' 130* MO" ISC 160° 170" ISO' 170° ISO' ISO- 140° 130° 120' 110° 100' 90' 80' 120' 130* 140' 150' I60" 170' ISO' 170' 160° 150' WC 130' 120' 00' gc 80' Fig. 39. Mean wind stress for January over the North Pacific computed from transfer formula (17). (After Scripps Institution of Oceanography, Oceanographic Report No. 14, 1948.) SECT. 2] liAKGE-SCALE INTERACTIONS 183 IZO' I3tf l«0' 150* 160* 170' 180* 170' 160' ISO" 140* 130' 120' IIP* 100' 90' 80* Fig. 40. Mean wind stress for July over the North Pacific computed from transfer formula (17). (After Scripps Institution of Oceanography, Oceanographic Report No. 14, 1948.) 100° 90° 80° 70° 60° 50° 40° 30° 20° 10 ATLANTIC MEAN WIND STRESS DYNES CM Scale of Mognitude Volues = 0.60 dynes cm-2 100° 90° 80° 70° 60° 50° 40° 30' 10° 20° 30° Fig. 41. Mean annual wind stress over the North Atlantic computed from transfer formula (17). (After Scripps Institution of Oceanography, Oceanographic Report No. 21, 1950.) 184 MALKUS [chap. 4 ATLANTIC MEAN WIND STRESS Fig. 42. Mean wind stress for January over the North Atlantic computed from transfer formula (17). (After Scripps Institution of Oceanography, Oceanographic Report No. 21, 1950.) The computations for the Pacific and Atlantic Oceans were carried out some- what differently. In the Pacific, the basic data were the wind roses of the U.S. Hydrographic Office Pilot Charts for each month. Air densities were calculated from the mean pressure and temperature distributions. The U.S. Weather Bureau Atlas of Climatic Charts of the Oceans was used to assess the wind-speed distribution about the mean for each Beaufort number. In tropical regions (5°S to 30°N latitude), a Gaussian distribution was believed satisfactory, with a standard deviation of one-half the Beaufort interval. In temperate regions, the speed distribution proved highly skewed with an average u composed of many winds slightly less than u and a very few much greater. Histograms were therefore constructed for each Beaufort interval from the climatic charts and the mean stress for that interval computed numerically from the histogram. The resultant stress in each quadrangle was finally obtained from the frequency- weighted sum of rectangular components. Fig. 38 shows the mean annual stress distribution for the Pacific, while Figs. 39 and 40 are reproductions of their results for the single months of January and July respectively. Charts for each month and season also appear in the Scripps Report, showing vector resultant stress and corresponding charts for its north- south and east-west components suitable for stress curl computations. For the Atlantic, the basic data were the Summary of Marine Data cards of the U.S. Weather Bureau and the Monthly Meteorological Charts of the British Air Ministry, which also give mean Beaufort interval and per cent SECT. 2] liABGE-SCALE INTERACTIONS 185 100° 90° 80° 70° 60° 50° 40° 30° 20° 10° ATLANTIC MEAN WIND STRESS 20° 30* Fig. 43. Mean wind stress for July over the North Atlantic computed from transfer formula (17). (After Scripps Institution of Oceanography, Oceanographic Report No. 21, 1950.) frequency of winds in sixteen compass directions by month and by 5-degree quadrangles. The procedure here was to compute the stress from (17) and the mean wind speed in each Beaufort interval, using cz)= 1.7 x 10~3 for Beaufort force 4, where the mean speed is 6.7 m/sec, or the transition of the dashed curve in Fig. 6. The Atlantic method was, therefore, cruder than that used in the Pacific, especially since the data required that sometimes two Beaufort intervals had to be lumped together. The results for the mean annual picture are shown in Fig. 41, while Figs. 42 and 43 reproduce their Atlantic charts for January and July, respectively. b. Range of validity and representativeness of the computed stress distributions We have made numerous tests for the purpose of evaluating the computa- tional methods of the Scripps reports and the representativeness of their results, within the framework that equation (17) is basically a correct formula- tion. It should be kept in mind that this is only sound for the tropics, while in temperate latitudes any such stress determinations may be off by a factor of two due to thermal stratification. What we have attempted to analyze is the effect of assumed variation in Cd, the frequency distribution of wind speeds and the chances that, in any given month, the vector wind stress computed via (17) from actual ship measurements would resemble the climatic mean stress for that month. We have taken one location typical of the tropical Atlantic (just north of San Juan, Puerto Rico, at latitude 19° 30'N, longitude 66° W) and one 186 MALKUS [CHAP. 4 Wymon Data (a) R= 69 ot 10.8 knots (b) R = 84° at 7 knots Scole of Wind Percentages Weather Stiip "C" Pilot Chart February Av. (d) / Pilot Chort March Av. » -®— ^ R = 15 ot 16 knots R = 237 at 19 knots R = 237°ot 17 knots .^ Tq-T, (°C) Fig. 44. Conditions for test computations of oceanic shearing stress. (a-e) Wind roses : Length of arrow proportional to percentage of time wind blows from each compass direction. Barbs (one for each Beaufort number) give mean strength of wind in that directional range. Figure at center is percentage of calms. R is vector resultant wind. (a) Wind rose from Wyman research vessel in tropics (19° 30'N, 66°W) April, 1946. (b) Mean climatological wind rose for April for location of Wyman research vessel. (After U.S. Hydrographic Office Pilot Charts.) (c) Wind rose constructed from Weather Ship C data February 26-March 19, 1960. (d) Mean climatological wind rose for February for location of Weather Ship C (52° 45'N, 35" 30'W). (After U.S. Hydrographic Office Pilot Charts.) (e) Mean climatological wind rose for March for location of Weather Ship C. (After U.S. Hydrographic Office Pilot Charts. ) (f ) Histogram of air-sea temperature difference in °C at Weather Ship C, February 26-March 19, 1960. SECT. 2] LARGE-SCALE INTERACTIONS 187 typical of the North Atlantic (position of Weather Ship C, latitude 52° 45'N, longitude 35^^ 30'W). For the tropical situation, tlie wind data from the research vessel of the Woods Hole Oceanographic Institution's April, 1946, Wy man- Woodcock expedition (see Table II) were used. In this case the winds were measured every few hours in knots using carefully mounted anemometers. The resultant stress for this seventeen-day period was determined from (17) in the following ways : (i) directly from calculating each rectangular component for each observation, using the cd of the solid line of Fig. 6, and (ii) by means of constructing a wind rose for the period (Fig. 44). Resultant stress was computed from the wind rose in three different ways : first, using the Scripps method for the Atlantic (mean speed for Beaufort intervals, some lumped) ; secondly, using the Scripps method for the Pacific (normal distribution within each Beaufort interval) ; and, thirdly, using a revised version of the Scripps Atlantic method (mean speed in each Beaufort interval) with the Cd of the solid line of Fig. 6, instead of the dashed line used in all the Scripps calculations. The results are shown in Table XIV. Table XIV Ocean Surface Stress Determinations from Research-Vessel Tropical-Wind Data, Wyman Expedition 19° 30'N', 66°W April 1-4, 12-17, 22-28, 1946 p = 1.16 X 10-3 gcm-3 Surface stress Method Direction, Magnitude, ° from N dynes/cm^ I. Direct summation Solid curve Fig. 6 069 0.625 II. Wind rose Scripps Pacific Scripps Atlantic Mod. Scripps Atlantic (cj) solid curve, Fig. 6) TO = p^D~ 3.0°C, indicating instability (roughly), and only 20% where the ocean was colder than the air. Under these conditions stress estimations from (17) should be correct to within 50% and all sensible methods in Table XV agree with each other to better than this leeway. Even the use of the crew's log wind data gives a stress vector differing only 15° in direction and 15% in magnitude from the direct summation methods, suggesting the feasibility of obtaining SECT. 2] LARGE-SCALE INTERACTIONS Table XV ()ceaii Surface Stress Determinations from Weather Ship C Wind Data 52° 45'N, 35° 30'W February 26-March 19, 1960 p = 1.27 X 10-3 g cm-3 189 Method Surface stress Direction, ° from N Magnitude, dyne/cm2 I. Direct summation Solid curve, Fig. 6 Dashed curve, Fig. 6 II. Wind rose Scripps Pacific Scripps Atlantic Mod. Scripps Atlantic (cj) solid curve. Fig. 6) Winds from crew's log (in Beaufort, vising Scripps Pacific) TO = pCDUa^ Mean wind speed for period TO = pODUa^ Resultant wind for period Resultant wind for period: 015°, 7.8 m/sec Average wind speed for period: 11.3 m/sec Directional wind steadiness for period: 69% 004 006 007 Oil 004 350 015 3.3 3.65 3.7 3.4 3.0 3.0 3.65 1.3 meaningful surface stresses from ordinary merchant ship reports. The stress computed from the resultant wind is more than a factor of two down, as expected with the smaller wind steadiness. Comparison with the climatic mean stress for the period exemplifies the time variations in stress which may be expected in high latitudes. Unfortunately, the Scripps monthly maps showed no data for the location of Weather Ship C, but the appropriate wind roses are shown in Figs. 44d and e. The Scripps method gives a mean stress for February of 1.85 dynes/cm^ at 235° and for March, 1.81 dynes/cm^ at 238°. The three weeks studied thus showed stronger and more northerly wind stresses than the climatic average. That the period was ex- tremely stormy was attested to by the battered condition of the ship and reports of its crew upon return to port. Although the stress magnitude in the period selected is only twice as large as the climatic average, the directions depart by more than 120° and the vector difference exceeds the size of either stress. This is in complete contrast to the tropical situation and of considerable consequence to oceanography. A very good estimate of the representativeness of climatic stress distributions and the meteorological dynamics of their 190 MALKUS [chap. 4 production is given by the single parameter of wind steadiness. Figs. 22 and 23 showed maps of the prevaihng surface winds over the oceans, their strength and steadiness. These should be used in conjunction with mean stress maps such as those of the Scripps report. c. The global distribution of momentum exchange and its time variations The mean annual stress distribution is well illustrated by Figs. 38 and 41, which are not, however, exactly comparable due to differences in data sources and computation method. Extreme seasonal variations in both oceans are revealed by Figs. 39, 40, 42 and 43 (in conjunction with Figs. 22 and 23). The Northern Hemisphere trade- wind stress weakens considerably from winter to summer, particularly in the Pacific, while the compensating westerly drag in temperate latitudes entirely shifts hemispheres with the season. The effects of the Asiatic monsoon are pronounced over the Japan Sea in winter, with large northerly stresses occasioned by the outflow of cold continental air. The unsteady flow in that region of Fig. 22 suggests that this occurs in intermittent outbursts, probably in the polar outbreaks following frontal cyclones. Altogether, in the belt of westerlies, the resultant stress arises from a succes- sion of short-lived synoptic-scale atmospheric eddies, and the mean stress picture for a month says relatively little about the stress that might actually be working on the ocean in a given period or even over a given season. The relative constancy of the large-scale ocean currents must, therefore, be attri- buted to two factors : namely, the high inertia of the ocean's response to fluctuations in stress, which is beyond the scope of the present chapter ; and to the reliability of the easterly trade winds, which is thus of major import to ocean dynamics. The operation and steadiness of the trade system is analyzed next. C. Momentum Production and Flow Steadiness in the Trades In Section 5 of this chapter, the role of the oceanic heat source in maintaining the lower trade winds was introduced in terms of the downstream pressure drop. But we did not explicitly explore what the pressure head was used for, nor did we relate it, other than spatially, to the remarkable steadiness of the flow, which also vanishes above the inversion. In the foregoing, we have shown that shearing stress is the momentum link between air and sea. Stress is also the internal momentum linkage within the atmosphere and its height derivative is a major frictional drag term in equations of air motion. We shall now in- corporate this turbulent drag with the heating in a dynamic model of the trades and their stability. This paves the way to examining the mechanisms of momentum production and its input to the ocean at the sea-air boundary, thus completing a full cycle in the coupling of the systems. SECT. 2] LARGE-SCALE INTERACTIONS 191 a. Momentum budget along a trade-wind air trajectory We consider first the momentum budget along the Pacific trade trajectory of Fig. 31 ; the situation along the Caribbean trajectory of Fig. 25 is qualita- tively similar. In the natural co-ordinate system, s is chosen along the two- dimensional flow, u{s, z). The vertical axis, z, points upward and the normal axis, n, completes a right-handed system. In steady state, the s-component of the equation of motion is 8u du I'l dp F\ , , The frictional force per unit volume, F, is well approximated by F = ^. ,34, where the shearing stress component, tsz, is the vertical transport of s momen- tum by convective, turbulent and smaller scales of motion. Upon multiplying through by p and integrating over a volume bounded by ds, dz and of unit thickness in the normal direction n, we have pu — ds dz+ \\ piv — ds dz+ \\ -^ ds dz— I — {tsz) ds dz = 0 (35) which becomes puudz— puudz+ puwds— puwds+ {pn s —pds)dz Ju.s. jd.s. Jc Jb J [{Tsz)t-{Tsz)b]ds = 0, (36) -/■ where the subscripts "u.s," and "d.s." indicate upstream and downstream respectively ; t and b refer to the top and bottom of the layer. In equation (36), we are considering the budget of large-scale momentum, namely that of the trade-wind flow, u. The first four terms are thus import and export of this scale momentum by u itself and w, the overall sinking motion produced by the divergence field. The latter is almost entirely two-dimensional and well approximated by dujds (Fig. 31d), so that w results from the down- stream acceleration of the trade. The fifth term is the local production of momentum by pressure forces, and the sixth term is the transport divergence by the much smaller scales of motion which are parameterized here in terms of the shearing stresses they produce. At the surface, the frictional term tszq = pCoUa^ (equation 17). In the Pacific study of Riehl et al. (1951), this term was computed twice daily at each of the four observation stations and its component along s was found by multiplica- tion with the cosine of the angle between the individual wind direction and s; Cd was taken as 3 x 10~3. The horizontal transport terms were evaluated 192 [chap. 4 from the twelve-hour pilot balloon observations, but daily values of vertical motion, w, were not available. The vertical transport terms were thus computed from the average vertical motion deduced from the divergence field of Fig. 31d. As the results will indicate, this approximation is not likely to be important. Fig. 45 shows the momentum budget of the Pacific trade section : the lowest stratum, 1020-960 mb, is the sub-cloud layer. The next two (cf. Fig. 31a), 960-880 mb and 880-800 mb, include the cloud layer while the top one, 800- 720 mb, is everywhere above the inversion base. The solid arrows are the Pressure Production = 0 Pressure Production 12 Pressure Production = 26 Pressure Production = 2.7 P mb 720 Unit; 10 g cm sec" Fig. 45. Momentum budget of air flowing along Pacific trade trajectory of Fig. 31. Vertical co-ordinate is pressure in mb. Upstream end denoted by u.s. ; downstream end by d.s. Lowest layer (1020-960 mb) is sub-cloud layer. Middle two layers (960-880 mb and 880-800 mb) include the cloud layer, while upper layer (800-720 mb) is every- where above inversion base. Terms in momentum -budget (unit 10^ g cm sec-2) evaluated using equation (36) as described in text. Solid arrows are transports by mean motion. Dashed arrows are contributions of turbulent-convective ^hearing stresses; bottom one found from equation (17), remaining ones as residuals for balance. Arrow directed out of box denotes export of easterly momentum. transports by the mean motion. The production term is evaluated from the observationally known pressure field. The dotted arrows are turbulent- convective stresses. Only the bottom one, the easterly momentum input to the sea, was directly computable; it corresponds to a shearing stress of 1.8 dynes/ cm2, a rather large value due to the high choice of cd. The main physical results would not be altered by dividing it by two, more plausible in view of Figs, 6 and 40. The remaining stress terms were found as residuals to complete the balance in each layer, beginning with the bottom. There is a vertical turbulent transfer of appreciable magnitude which, as it should, changes sign at the east-wind maximum just above cloud base SECT. 2] LARGE-SCALE INTERACTIONS 193 (^930 mb, see Fig. 31a). The trades are thus transferring easterly momentum downward to the ocean and upward to the high troposphere, or as commonly said in meteorology, the trades receive westerly momentum from both above and below. The Woods Hole expeditions have related the stresses mechanistically to turbulent elements (50-150 m dimensions) in the sub-cloud layer and with the cumuli themselves, and their associated eddying in the cloud layer. The important result of Fig. 45 is that in the momentum budget the advec- tive terms are of no importance. Essentially we have a balance involving only pressure and frictional forces. For example, in the lowest layer, the pressure production force is 2.7 units, while the turbulent drag, the difference between stress through top and bottom, is —2.8 units. The pressure drop along the trajectory accelerates the air toward west-southwest, and this acceleration is counteracted by friction, so that for many purposes in illustrating basic force- momentum relations the tangential equation of motion (33) may be written : The balance of forces in the cross-stream, n, equation may be shown to be approximately that between the earth's rotation (Coriolis) force directed to the right (Northern Hemisphere) looking downstream and the normal pressure gradient force, —{llp)dpldn, directed to the left, with lateral stresses con- tributing not more than 20% of the two major terms. Thus while we have a balance of forces in the lower trade, it is quite different from the quasi- geostrophic or gradient balance often hypothesized for the free atmosphere in mid-latitudes, or from the Ekman friction layer discussed by oceanographers, where the current vector rotates with depth. Here in the lower tropical atmos- phere, the current is two-dimensional and the isobars rotate counterclockwise with height (Fig. 31c) becoming nearly parallel to the flow at the top of the moist layer. Fig. 45 brings out the important further result that, along with the downstream pressure head, turbulent friction also vanishes at the same level. Thus at the top of the convective layer, the balance of forces is similar to that found farther poleward in broad currents with little acceleration; the wind is nearly parallel to the isobars and thus is quasi-geostrophic. Aloft, only the mecKanism of travelling synoptic disturbances is available to provide momentum transports and stresses, so that it is small wonder that the flow steadiness breaks down. Therefore, the oceanic heat source plays a dual role in the loiver trades: produc- tion of downstream warming and maintenance of vertical turbulence in a layer of limited thickness. The coincident requirements on the balance of forces and the mass distribution arising from this dual role are such as to permit direct utilization of the heat gained to maintain the trades. The preceding deductions depend specifically on two characteristics of the region studied : existence of a trade inversion, and a vertical wind-profile with a curvature in the convective layer such that the turbulence must act to retard the flow (cf. Fig. 31a). This kind of profile "knee" is typical of those portions 194 MALKTTS [chap. 4 of the trade-wind belts where the meridional temperature gradient is directed poleward. The geostrophic wind is strongest at the ground and decreases throughout the trade-wind layer. On account of ground friction, however, the maximum actual wind is situated at some distance above the ground, and the vertical profile assumes the shape depicted in Fig. 31a. Thus the conclusions drawn cannot be readily applied to tropical regions in their entirety. A separate assessment of the role of the heat source must be made, as we saw, for those regions such as the equatorial trough where the instability extends through a deep layer, and to those parts of the trades where easterlies do not decrease, but perhaps even increase with height (Riehl, 1951). Nevertheless, in the some- what less complex region studied here, we may deduce some of these important features as the consequences of a simplified dynamic model. b. A dynamic model of the lower trades The features of the low-level trade-wind section are sufficiently remarkable to offer hope that we may have here a simple enough geophysical heat engine to handle theoretically, perhaps even to provide a prototype for more complex planetary thermal circulations. Fig. 31 and the natural co-ordinate system serve as the framework of the study. Taking the two-dimensionality and the layered stability structure as given, we shall attempt to relate the heating and the overall sinking motion via the steady-state hydrodynamic equations. Thus we investigate the interaction between the physically important scales of motion, namely convective turbulence and the large-scale trade flow, under prescribed constraints. It will be assumed that a uniform wind, u, enters our section at the upstream end and that u'{s, z) is the increment within the section. The other hydro- dynamic variables are similarly divided into barred quantities, referring to the observationally given properties of the current at the entrance end, and primed quantities denoting changes produced within ; these latter are the dependent variables to be solved for. The vertical velocity is of the magnitude of a primed quantity, being related to u' via the continuity equation ; both u' and w' are independent of the n co-ordinate. Since changes within the section are small compared to entrance values of the variables, this separation permits linear mathematics, which may be shown not essential to the physical results (Stern, 1956). Under these conditions, the basic equations become _ du' 1 dj)' 1 Btsz . , . w^= -=^ + =:^' 33b OS p CS p cz 1 dr>' p' -^-f = ^g' (37) p dz p du' dw' ^s dz p' T -s^-z='^ (^^^ (39) P T SECT. 2] LARGE-SCALE INTERACTIONS 195 where equations (37-39) are the hydrostatic, continuity and gas laws respec- tively. The small approximations and neglections therein have been detailed in the original paper by Malkus (1956) and justified in similar meteorological studies of flows under imposed heating (Malkus and Stern, 1953; Stern and Malkus, 1953; Stern, 1954; Smith, 1955). One more equation is needed to complete the set and eliminate between the five dependent variables u' , w' , p', p' and T'. The crucial step is the introduc- tion of the first law of thermodynamics to achieve this ; by this means the heat source is related to the motion and temperature fields. A standard meteoro- logical form of the first law, namely, is expanded and linearized, using the hydrostatic law, to obtain H _dT' ,/dT g\ er ,,^ ^ where dTldz= —y, the vertical lapse rate of upstream-end temperature, T, and — gjCp — F, the dry adiabatic lapse rate. The net heat source H (cal g~i sec~i) is the same as that in equation (31) and is to be evaluated from Fig. 31b in terms of ddjdt, the substantial time rate of change of potential temperature following air trajectories. Elimination of all other dependent variables is now readily performed to obtain a simple differential equation in w' , which is dz^ u^ CpU^T pu dz^ ' ^ ^ where 6^ ^{r—y)IT ^{Ijd) ddjdz, the given static stability at the upstream end. The latter is, from the large-scale viewpoint, independent of the distance co- ordinate, s, as is the heating function, H, so that writing (42) as an ordinary differential equation with constant coefficients (within each vertical layer) is justifiable. An important physical hypothesis underlying the formulation is the separation of scales of motion. The dependent variables to be solved for, w' and u' , are of the large, tropical-cell scale, while the heating function and stress distribution are created by the far smaller turbulent-convective scale. The latter may, therefore, to first order, be treated as "forcing function" dependent upon the vertical co-ordinate only. Although left out of the steady-state formalism, feed-back between variations in the large-scale trade and its forcing function may clearly be significant ; it is discussed later when we analyze the steadiness of the flow. Meanwhile we inquire what distribution of u' and w' are brought about when we impose the average forcing function as evaluated from the Pacific trade observations. To determine the vertical distribution of the second term on the right of (42), which involves second derivatives of tsz, would clearly be pure speculation from the kind of observations presently available. Before describing how this 196 MALKUS [chap. 4 difficulty was surmounted in the theoretical study, let us first omit this term to illuminate some general properties of bottoin-heated, two-dimensional flows. The heating function (Fig. 31b) is fairly well apjiroximated by an exponential decay with height, so that the frictionless equation may be written 'iJ^ + B^w' = Pe-'/d, (42a) where B-^ = g^lu^; P^gHolcpU^T. The decay -height of the heat source, d, is chosen from Fig. 31b to give the correct integrated heating for the layer ; T is its mean temperature. For a deep, uniformly stable air layer, bounded below by the ground, the general solution to this equation is The divergence field is prescribed as dw' du' P Since the sine function is zero at the ground, the divergence must always be jjositive there. This means a downstream acceleration of the surface wind and subsidence in the low levels, regardless of the value of B, just so long as the heat source is positive and decays with height. With weak stability, the di- vergence extends through a 1-2 km deep layer and fades out approximately with the heat source. Equations (33b) and (44) together show that the surface pressure must drop downstream. However, without friction the magnitudes are unrealistically related. The observed heating function gives much too great a divergence or, conversely, the observed divergence requires a pressure drop nearly two orders of magnitude lower than observed. Furthermore, from (44) we see that the downstream divergence is maximum at the ground and decreases monotonically upward, giving a similar-shaped profile of u + u'. Nevertheless, we see the important relation between heating and subsidence in a statically stable flow which is required to be two dimensional. A larger heating function gives greater subsidence by means of enhancing the downstream pressure drop. Thus an ageostrophic mass flow may be thermally driven and, contrary to popular preconception, does not require friction for its existence ; the latter only magnifies the pressure head and heat source required to maintain it. Realistic quantitative relations between the parameters are derived by the theory when frictional stress is introduced and (42) is solved for a two-layer model. The lower sub-cloud layer is adiabatic {6^ — 0) and the upper (cloud) layer is slightly stable, with a top boundary {w' = 0) at the inversion. In the original work, a trick was devised to deduce the total forcing function from the observations, approximating it by the difference in two exponentials in z, namely, d 2?// -^-^ + BW = -F{z) = Pe-'fd-Ge-'/^, (42b) SECT. 2] LARGE-SCALE INTERACTIONS 197 where P 0 and B^ is proportional to S^, the static stability. In the sub-cloud layer, S^'^O and so du'jds increases rapidly upward. At the base of the cloud layer, positive stability sets in suddenly. Since w' is negative, the sign of the right-hand term is rapidly changed to negative. The divergence and wind then begin to decrease with height, giving rise to the physically important "knee" in their profiles. In laterally broad, wind-driven ocean currents, a similar approach might be applied to explain the rapid decrease of current speed through the thermocline. Also of consequence to oceanography is the reliabihty of the trade-wind flow, of which this model permits a rudimentary treatment. c. The steadiness of the lower trades The pronounced steadiness of the flow has been emphasized as a significant feature of the trades. The overall dynamic stability of a Hadley-type meridional 198 MALKUS [chap. 4 cell in the low latitudes of a heated, rotating system, has been studied in experimental (Fultz, 1949) and theoretical (Kuo, 1954) investigations of simple planetary analogs. The foregoing treatment of a limited portion of the real tropical cell suggests another kind of stability which may not be unrelated. We have examined the steady-state coupling between the small-scale motions occurring within a trade section, and the modifications in the large-scale flow observed between its inflow and outflow ends ; the heat source and its convective distribution are found essential in maintaining the mean flow as observed. When either scale of motion suffers alteration, the response of the other is significant in determining the nature and amplitude of the system's fluctuations about its average condition. In the mid-latitude westerlies, disturbances of certain critical sizes, drawing upon stored energy, are able to grow in a self-accelerative fashion, and, in so doing often upset the entire structure of the flow, which thereby fluctuates so wildly that an average picture has only a very restricted meaning as a "steady state". In the tropical easterlies, by contrast, the average and daily pictures bear a closer resemblance to one another. The infrequency of intense disturbances in the lower trades, and the reliable presence of trade cumulus clouds, suggest that the interaction between small- and large-scale processes is stable and perhaps contributes to the overall dynamic stability of the tropical circulation. In the framework of the present model, therefore, we might inquire what happens if the net heat source is weakened, for example through a diminution of convective precipitation. A stable coupling of the type suggested would prevail if weakening the heat source leads to decreased subsidence. It is well known that subsidence in the trades is one of the most efifective brakes upon convection, and even a slight release of this brake results in vigorous outbursts of shower activity. Any such approach is clearly a drastic oversimplification and its formulation here is intended only to be suggestive. No simple causal relation between heating and trade flow is implied by the existing model ; it merely prescribes relations between the two which must on the average be satisfied. To deal with the complete, time-dependent behavior of the motions much larger circulation branches must be considered. The present theory can, on the other hand, suggest what other steady pr quasi-steady configurations are possible and self-consistent for the section. If such configurations may be interpreted as describing the slow fluctuations about the seasonal mean picture, we may profitably inquire whether a given one of these exhibits a structure likely to restore the seasonal mean or lead to even greater departures from it. Re-examining equation (42) and temporarily ignoring variations in gS^/u^, we see that the flow may be restored in two ways : either by a stable interaction between the whole forcing function and the motion field, or by internal re- adjustment between the two right-hand terms. In both categories, only a few of the possible rearrangements will give a consistent relation between the resulting wind-profile and stress distribution ; the remainder are, therefore, excluded. Due to inadequate knowledge of the dependence of the two SECT. 2] LAKGE -SCALE INTERACTIONS 199 right-hand terms both upon one another and upon external influences, only one illustrative calculation is presented here. Rainfall in the trade stream is the least reliable of its properties. Even casual observers have noted that a skyful of trade cumuli on some days produces plentiful showers from clouds of all sizes, while on other days with apparently similar cloud conditions no drop of rain appears. We try the hypothesis that due to deficient rainfall the amplitude of the heating function is cut by 10% without altering its height dependence or the d^Tsz/dz'^ term in (42). The solution now gives the dotted curves in Figs. 46 and 47. The boundary condition that the stress vanishes at the wind maximum and internal consistency with the wind field requires a surface stress of 2.2 dynes/cm^. The new situation is in all respects an extreme of observed configurations in the trades (Charnock, Francis and Sheppard, 1956). It does, however, con- tain restoring influences. Fig. 46a shows that sinking has been replaced by mean ascent up to 1 km, and that subsidence is much weaker than normal throughout the cloud layer. Weakening the descent in the cloud layer favors increased cloudiness and precipitation, while ascent in the sub-cloud layer aids upward transport of sensible heat, both directly and by destabilizing the already near-neutral lapse rate. Observational studies (Malkus, 1955, 1958) suggest that processes in the tropical sub-cloud layer are critically sensitive to shght amounts of subsidence or its removal. In view of this, it seems likely that the system would usually restore its heating and return toward normal even before departing so far as the conditions of the dotted curves in Figs. 46 and 47. Kraus (1959) has broadened the basis of inquiry into the relationship between sea-air exchange and the stability of tropical flows. The form of the transfer formulas suggest that the local heat-energy input from sea to air iQs + Qe) is proportional to a lower power of the wind speed than is the momen- tum loss. The rate of kinetic energy dissipation is still more wind-sensitive, being proportional to the stress times the wind speed. Following this up with an extensive statistical analysis, Kraus showed, in fact, that when an integra- tion over the whole tropical ocean area from 30°S to 30°N is made, the energy input depends upon the first power of the area-median wind speed, while the frictional dissipation is approximately proportional to its cube! Thus if the fraction of Qe + Qs converted into pressure head or the efficiency of the thermal engine is specified, there is one and only one mean equilibrium wind speed at which input and dissipation are balanced. Below that speed the trades will accelerate, and above it they will run down. Under a fixed efficiency, the situation of the dotted curves in Figs. 46 and 47, which show an abnormally large stress and small heating, probably could not maintain long as a steady state ; we have perhaps described the mechanism by which the system goes back to normal. In considering efficiency, we should recall that the heating function in equation (42) is the net residual of sensible heat sources minus sinks, namely H = Qs + LP + Ra. It will be larger for a given Qe + Qsy the greater the fraction of Qe converted into LP by condensation 200 MAliKUS [chap. 4 (Km) 3 - \ 1 2 - ]■■■■' 1 ..--••"■' // // n Jill ImB) 700 \ 1 1 . 1 800 - \ )■ 900 - // 000 1 •'' 1 1 1 / 1 1 1 20 0-20 -40 -60 -80 - IQO RISE. DESCENT 1 (m/doy) (a) ■I 2 -08 -0.4 0 04 08 12 CONV. p . 6 , DIV. ^ 10 sec"' ds (b) Fig. 46. Vertical motion and divergence profiles for the Pacific trade trajectory of Fig. 31. (After Malkus, 1956, Figs. 6, 8 and 52.) (a) Vertical motion (m/day) as function of height. Solid curve is average w calcu- lated from observations of dujds. Dashed profile computed from theory when heating function approximately equal to observed mean (Fig. 31b, dashed curve). Dotted profile computed from theory when heating reduced 10%. (b) Divergence (scale 10^ sec~l) as function of height. Solid curve average of observa- tions. Dashed curve computed from theory when heating function approximately equal to observed mean. Dotted curve computed from theory when heating reduced 10%. (km) 3 '\ 2 \ \ \ \ 1 \ "\ \ \ \ ^\ \ 1 . \. 1^. (mb) 700 -20 0 2.0 Tj, ( dyne /em ) (o) (km) - 3 0 02 04 06 08 1.0 i£ (mb/500km) (b) Fig. 47. Theoretical profiles of shearing stress and downstream pressure drop for Pacific trade trajectory of Fig. 31. (After Malkus, 1956, Fig. 7 and Tables 2 and 3.) (a) Shearing stress component tsz (vertical transport of s momentum) in dynes/cm^ as function of height. Dashed curve computed from theory for normal heating function; dotted curve for 10% reduced heating function. (b) Pressure drop along trajectory in mb/500 km as function of height. Dashed curve computed from theory for normal heating function; dotted curve for 10°'o reduced heating function. SECT. 2] LARGE-SCALE INTERACTIONS 201 and/or the less the radiation loss. The latter becomes the predominant factor as the size and time scale of the considered circulation branch becomes large. Efficiency of condensation conversion only might be considered on the synoptic scale, as done by Riehl and Byers (1958) in seeking upper limits for tropical storm development, while in the 2500 km-long trade section we would probably be unjustified in ignoring variations in Ra. Kraus' large-scale study considered efficiency variations entirely in terms of alteration in radiation sink ; in the entire tropical belt, as we saw, nearly all latent heat is converted before export. Clearly the dependence of fluid heat-engine efficiency upon scale of motion is a frontier facet of stability studies as yet barely touched by geophysicists (Spiegel, 1958; Goody, 1956). A fortunate and vastly important feature of the lower trades is their two dimensionality, which had to be treated as given in the limited dynamic model described. Also related intimately to their stability against synoptic-scale disturbances, it has so far received only a physical explanation (Riehl, 1951). To roll up the currents into waves or vortices, with lateral and tangential velocity components varying both with s and n, hydrostatic pressure fields to sustain these flow configurations must be produced; that is to say, similarly convoluted horizontal temperature fields must be maintainable in the atmos- phere, with synoptic-scale gradients in the s and n directions. In the lower trades, this is just what sea-air interaction prevents. The close coupling by exchange and vertical convection to a vast homogeneous oceanic reservoir precludes development of the necessary sharp air temperature gradients. Thus lateral eddies must recede in importance : because of lack of thermal stability over millions of square miles, the extreme smallness of oceanic temperature gradients, and the effectiveness of vertical turbulence, horizontal circulations are held to a minimum. Flow in the low levels is, therefore, quasi-uniform, particularly under trade inversions ; the air below is both shielded from any disturbing influence from high levels and contains internal mechanisms for restoring departures which may be imposed. Above the convective layer, as in most of the middle-latitudes, these restoring mechanisms and the vertical coupling to a uniform sea no longer exist and the steadiness is absent. Controlled rotating dishpan or channel experiments could profitably be undertaken to explore the range of validity of this two-dimensional property of bottom-heated currents. As the trades themselves flow equatorward, they sow the seeds of their own destruction as a stable system, by weakening the inversion lid and deepening the moist layer by convection. When, finally, "hot towers" manage to shoot through to the tropopause, deep wave- and vortex-sustaining pressure fields can be organized (Malkus and Riehl, 1960). Then synoptic-scale disturbances be- come common through the whole depth of the troposphere. In these, air-sea fluxes and their mechanisms are violently stepped up and, paradoxically, play a dominant role in disrupting the very machinery whose smooth operation they formerly maintained. It is with the exchange mechanisms themselves and their fluctuations on the synoptic and longer scales that we shall be concerned in the final sections. 202 MALKUS [chap. 4 7. Exchange Mechanisms and Fluctuations How is sea-air exchange brought about? What physical processes put sea- water into the air and impart the winds' momentum to the ocean? To find out, we must come down from the broad view of planetary energetics and climato- logy to scrutinize the small-scale chaos near the sea surface ; we must look closely at the interaction between atmospheric eddy and the foaming wave crest it strikes as it swirls through its evanescent life in the sub-cloud layer. Ocean and atmosphere are turbulent fluids and exchange between them is primarily turbulent transfer. If the boundary turbulence is suppressed, for all practical purposes exchange ceases. Fluid turbulence is one of the most chal- lenging and complex problems at the frontier of physical science. The form of the transfer formulas, derived in the 1930's, was based on then-existing models of turbulent flow, largely carried over from the engineering studies in wind timnels led by Prandtl and Von Karman. The formulas themselves, and the laws upon which they were based, contain empirical hypotheses and constants which, as we saw, must still rely on experiment for their justiflcation and evaluation. In the decade 1950-1960, exciting theoretical developments have been made (W. V. R. Malkus, 1954, 1956) in modelling fluid turbulence starting from funda- mental physical principles. With one energetic hypothesis and no disposable constants, this formulation permits prediction of the transports, gradients and motion spectra in very simplified situations in which the turbulence is produced by either imposed heating or shear flow alone under uniform, specified boundary conditions. In the real atmosphere and ocean, the eddying motion is most often driven by a combination of heating and shear flow, neither of which can be quite regarded as externally imposed in a fixed fashion. The boundary conditions are generally neither uniform nor determined independently of the flow. Thus, although the new models have not yet been extended to treat the complex geophysical situation, they do establish a language of inquiry, suggest possibly isolatable prototype problems and point to critical measurements to be made. Within this framework, we shall attempt a physical, mechanistic description of the air-sea boundary. We shall now examine the sizes, structure and behavior of the elements effecting the exchanges of heat, water and momentum, the climatology and dynamic importance of which have been outlined. Again, the trade-wind region is used as the starting point. This globally important segment of the interface is fortunately in a relatively steady state. The overall turbulent structure is reproducible when sampled over wide regions and time intervals, and the tropical ocean surface provides as nearly uniform a boundary as the atmosphere ever experiences. Numerous expeditions have provided quantitative information on the important scales of motion and their operation. After describing exchange mechanisms in the normal trade regime, we shall inquire into the nature and causes of their variations. In particular, we shall seek to learn how the exchange processes are organized and altered by SECT. 2] LARGE-SCALK INTERACTIONS 203 the larger-scale instabilities in the flow, paving the way for a more limited discussion of the fluctuations in middle latitudes and the longer period inter- action anomalies which may persist over seasons and decades. A. Exchange Mechanisms in the Trade-Wind Region a. The layer below cloud base We shall now take a closer look at the trade-wind sub -cloud layer, schemati- cally indicated in Fig. 24. The physical nature of the motions occurring is beautifully illustrated in Figs. 48 and 49, which are photographs made on the Fig. 48. Aerial photograph of smoke trail laid by stationary ship over Caribbean Sea, August 29, 1944. Sea minus air temperature, 1.2°C. Wind speed, 7.7 m/sec. Note lateral displacement of smoke to right and left of average wind direction. (After Woodcock and Wyman, 1947, Plate 2. By courtesy of N.Y. Academy of Sciences.) first (1944) Woods Hole expedition to the trades (Woodcock and Wyman, 1947). In Fig. 49a, a trail of smoke has been laid by an aircraft flying up-wind at 300 ft. Fig. 49b shows what has happened to the smoke trail 88 sec later. The predominant scale of motion causing transport of the smoke has horizontal dimensions about equal to the distance from A to B in the photograph, or comparable to the 300-ft long destroyer. Smaller scales of motion are also detectable: those of the nodules, about 50-100 ft across, and still smaller scales giving rise to the thickening of the plume in the vertical and horizontal (Fig. 48). Measurements, corrected for the smoke's own buoyancy, showed that 204 [chap. 4 (a) (b) Fig. 49. Photograph of smoke laid parallel to the wind by an aircraft flying horizontally at 92 m elevation over the Caribbean, December 19, 1944. Wind speed at 10 m, 6.2 m/sec. Sea minus air temperature, 0.2°C. Airspeed of plane, 120 knots. Elapsed time between (a) and (b) is 88 seconds. (After Woodcock and Wywnan, 1947, Plate 5. By courtesy of the N.Y. Academy of Sciences.) Fig. 50. Schematic representation of eddy motion at low levels over tropical oceans. (After Riehl, 1954a, Fig. 1.) SECT. 2] LARGE-SCALE INTERACTIONS 205 it was being transported vertically at rates of up to +1 m/sec. After another 152 sec, the smoke loop at B in Fig. 49b had descended into contact with the sea surface. Using these photographic suggestions, we may hypothesize the exchange mechanism. Let us look at a single eddy, roughly the size of the smoke loop. As illustrated schematically in Fig. 50, it first descends to the ocean surface, stays there for a little while and then rises again. Let us suppose that the air on the downward leg is about 0.5-1.0°C colder than the water, as is frequently true. Then the air will be warmed by contact with the ocean and ascend with a temperature a fraction of a degree higher than it had during the descent. It has thus received an increment to its sensible heat. One generally assumes that the vapor pressure (or specific humidity) at the ocean surface corresponds to the saturation vapor pressure (or specific humidity) at the temperature of the surface. The descending air is cooler than the water and its relative humidity is generally less than 100%, perhaps 70-80%. Thus the vapor pressure of the eddy will be lower than that of the water and a little evaporation takes place from sea to air. The rising air will carry away with it a slightly larger amount of water vapor than it brought down ; its latent -heat content has been increased. Still another type of energy exchange takes place. The air undergoes frictional braking at the surface, and also, therefore, a decrease of its kinetic energy. This decrease is so small compared to the sensible and latent heat gains that, from the energy viewpoint, it is negligible. For instance, a reduction of wind speed from 7 to 6 m/sec produces an energy decrease equivalent to about 2 cal in heat units per kg of air; a temperature rise of 0.1°C for the same mass corresponds to an energy increase of 20 cal and a moisture increase of 0.1 g to 60 cal! From the momentum viewpoint, however, the slowing down is very important. As we saw, a reduction of wind speed in the trades represents a loss of easterly momentum or, as frequently put, the air receives westerly momentum from the sea. For the ocean, in turn, this momentum input humps up the boundary into waves and sets the surface layers in motion. Thus, when the ocean temperature exceeds that of the overlying atmosphere, we should observe that air leaving the surface has a slightly higher temperature and moisture content, and a little lower wind speed than when it approached the surface. These predictions need testing by measurements, and if such sized motions are effecting the transfers, we need to know how they are maintained. Of course, it is difficult to follow an individual eddy with measuring equipment. We can, however, determine whether instruments recording temperature, humidity and wind on board a ship reveal fluctuations of these magnitudes as the air moves past ; further, we can find how these fluctuations are correlated. In 1946, a second expedition from the Woods Hole Oceanographic Institution went to the Caribbean to study trade cumuli and the layer below the clouds, this time using both an instrumented ship and aircraft (Wyman et al., 1946; Bunker et al., 1949). The normal excess sea temperature over that of the air at ship's deck level (~6 m) was found to be 0.2-0.5°C, with an extreme range 206 [chap. 4 between +2°C and — 1°C. Fig. 51a is typical of the wind, temperature and moisture traces when the sea temperature exceeds that of the air by the normal amount. Such records, now available in large quantities, demonstrate that fluctuations on the horizontal scale of 50-300 m are the rule, with amplitudes of several knots in wind speed, 0.5-1.0°C in temperature, and 1 mb in vapor pressure (~0.6 g/kg in specific humidity). These records suggest, and longer records confirm, that larger "eddies" also occur. These may have dimensions from 10-50 km. The measurements shown in Fig. 51a were made near Puerto Rico (at (a) 2 3 4 5 6 7 8 9 10 II 12 13 Air Distance (thousands of yards) -* 69 Fig. 51. Fluctuations of atmospheric parameters measured from deck of Woods Hole research vessel under varying conditions as function of air distance (wind speed times time). (After Wyman et al., 1946, vmpublished.) (a) Typical profile obtained when ship stationed in western tropical Atlantic, north of Puerto Rico (19° 30'N, 66°W). Fluctuations of wind (knots), temperature (°F) and vapor pressure (mb). Sea minus air temperature ~0.6°C. Surface wind from 040° at 14 mph ; cumulus base 1700 ft, April 23, 1946. (b) Typical profile obtained when ship stationed in eastern tropical Pacific, south of Panama Canal Zone during March, 1946. Profiles of wet-bulb temperature (°F above) and dry-bulb temperature (°F below). Sea minus air temperature minus 3.5°C. Wind speed 15 knots. SECT. 2] LARGE-SCALIi INTERACTIONS 207 19° 30'N, 66°W) ill the western Atlantic trade. However, the Woods Hole ship alst) visited the eastern Pacific Ocean, south of the Panama Canal Zone where the air is warmer than the sea. Comparison of the traces obtained there (Fig. 51b) with those obtained in the Caribbean is startling. The fluctuations have almost disappeared; temperature and wet bulb^ are nearly constant. Vapor pressure of sea and air are almost identical, and there is little or no latent heat transfer. The sensible heat flow will be downward, but since the air is stabilized from below, the energy exchange becomes extremely small at the prevailing wind speed (~15 knots). Fig. 51 should be compared with Figs. 18 and 19 which contrast the annual marches ofQs and Qe consequent upon these different air-sea boundary structures. On inspecting Fig. 51a, we observe a distinct but not perfect positive correla- tion on the 50-300 m eddy scale between temperature and moisture, and a negative correlation between these and wind speed, supporting our mechanistic picture. In the longer "eddies", temperature and moisture variations appear to be out of phase. The Woods Hole aircraft data extend the picture and bring in the missing link of vertical motion. Traverses, flying as low as 50 ft over the sea, were made while recording horizontal and vertical motion, temperature and moisture simultaneously, with a resolution of one-fifth second ( ~ 10 m distance), except for moisture which was averaged over one second ( -^ 50 m). Correlation of ascending speeds of 20-50 cm/sec with warm, wet eddies of deficient horizontal speed was established in the lowest 300-500 ft of the sub-cloud layer. Fluxes of sensible heat, latent heat and momentum have been calculated directly from these records (see Bunker, 1960) using equations (22)-(24). It was established that in all but the lowest strata, the major transports in the layer below cloud base are effected by the 50-300 m eddies described and made visible by the smoke in Figs. 48 and 49. As the fluctuation amplitudes in Fig. 51a indicate, by far the largest fluxes effected by boundary turbulence are the upward transport of latent heat and the downward transport of momentum. Temperature fluctuations are normally small and the sensible heat flux is 2-10% of the latent, only marginally adequate to balance radiation losses below cloud base. In undisturbed conditions, in fact, it frequently reverses sign in mid-layer, becoming downward above about 500 ft. Yet we have seen from Fig. 51b that the ocean surface must be some- what warmer than the air in order for the transport-producing eddy motion to occur at all ! This implies the existence of a sensible heat flow. The air rising from the sea is a little warmer than its surroundings and is thus slightly buoy- ant. A temperature excess of 0.1 °C has the same effect in decreasing air density as does a moisture increment of 0.6 g/kg. Both amounts are in the range of observed fluctuation amplitudes : it therefore follows that the sensible heat flow has at least as much influence on the stability, and therefore, on main- taining the low-level turbulence, as does the water-vapor flow with its much larger energy content. Thus the sensible heat exchange plays a major part in the 1 Change of 1°F in wet bulb temperature, T^;, under these conditions corresponds roughly to a change of 1.2 mb in vapor pressure or 0.75 g/kg in specific humidity. 208 [chap. 4 transfer, even though it is a small term on the balance sheet. Its importance lies in facilitating the evaporation and turbulent momentum exchange. There is some evidence (Woodcock and Wyman, loc. cit.) that these buoyant, low-level eddies are organized, possibly into the polygonal or long roll type configurations of Benard (1900) so beautifully illustrated by the laboratory experiments of Avsec (1939). The natural attempt of the 1946 Woods Hole expedition to relate these to trade-cumulus patterns, however, failed com- pletely and in so doing led to further expeditions and much of the material on air-sea interaction contained in this chapter. The reason for the failure is brought out in Fig. 52, which shows the typical vertical temperature and moisture structure up to the inversion. 2400 2200 2000 1800 1600- 1400 1200 X 1000 800 600 400 200 6 8 10 12 14 16 18 20 22 24 26 6 8 10 12 14 Temperature, C I Mixing Ratio, g/kg Fig. 52. Typical vertical temperature and moisture sounding of the western Atlantic trade-wind region. Made at 19° 30'N, 66°W by Woods Hole aircraft on April 27, 1946. (After Bunker et al., 1949, Fig. 58.) Dry and wet-bulb temperatures in °C ; mixing ratio (approximately equal to specific humidity) in g/kg. All are functions of height in meters. Sounding shows normal structure of moist layer up to inversion base under conditions of strong trade. Surface wind from 080°, 17 mph. Sea minus air temperature 0.5°C. Cumulus base about 2500 ft. Sounding made in clear space between cloud groups. SEC:T. 2J LARt;K-SCALE INTERACTIONS 209 Concentrating for the moment upon the layer below cloud base, we see that its most important feature is that it is almost completely mixed. Throughout its lowest two-thirds, the specific humidity falls oif (from ship's deck level) an average amount comparable to its small fluctuations, or only about 0.3- 0.5 g/kg out of a mean value of 15 g/kg. This well-mixed region has thus been christened the "homogeneous layer". Its potential temperature also is almost, but not quite, constant with elevation. Weakly unstable in the lowest strata, the temperature lapse rate becomes less than adiabatio above about 200 m g/kg Fig. 53. Typical aircraft run at 550 ft elevation over the sea in the western Atlantic trade- wind region. Aircraft flying up-wind. (By courtesy of A. F. Bunker.) Wind from 070°, 8 m/sec. February 25, 1958 ; ITN, 60°W. Sky coverage, five-tenths cumulus. Vertical velocity (cm/sec), departure of horizontal velocity from mean at the level (cm/sec), departure of temperature from mean at the level (°C), and departure of specific humidity from mean at the level (g/kg) shown as function of time. Aircraft flying at about 60 m/sec so 10-sec interval represents a horizontal distance of approxi- mately 600 m. Instrumentation was such that velocities and temperatures could be read at one-fifth second intervals. Due to slow response of humidity element, moisture values are one-second averages. elevation. Careful statistical studies of numerous profiles like that of Fig. 52 (Bunker et al., 1949) show that under normal and strong trade conditions the upper two-thirds of the sub-cloud layer are slightly statically stable in all but the upstream and poleward fringes of the trades. The effect of this stabilization upon the important scales of motion is brought out in Fig. 53, a typical aircraft run at the 550-ft level. ^ While the moisture fluctuations and their correlations with up-drafts are nearly unchanged from the low levels, the temperature fluctuations are now mainly out-of -phase with the rising motions on the 50-300 m scale as well as in the longer "eddies". This 1 A much faster-responding humidity element is now in use on the aircraft but no reduced records are yet available at the time of going to press. 8— s. I 210 MALKUS [chap. 4 suggests, but does not prove, downward heat flux, since counter-gradient heat flows are not unusual in the atmosphere (Bunker, 1956). Much more important mechanisticaUy, however, it shows that the small turbulent eddies are com- monly "overshooting" their level of zero buoyancy at least a thousand feet below cloud base, so that wind stirring is necessary to convey the water vapor the remaining distance to the condensation level. This is but one link in the now firm chain of evidence that the oceanic trade cumuli, unlike their con- tinental relatives, do not have individual "roots" in buoyant cloud-scale thermals penetrating up into them from the sea surface. In fact, the aircraft records show that small-scale turbulence is no more developed on flights just below cloud bases than on those in the intervening clear spaces. b. The origin of trade-wind cumuli The evidence that small-scale convective turbulence near the sea surface and the cumulus activity in the cloud layer are somewhat decoupled from each other was an unexpected result of the early Woods Hole expeditions. It raised a serious puzzle about the origin of the trade cumuli, the important role of which in the overall circulation we have brought out in the preceding sections. One of the most striking and suggestive features of the trade-wind clouds is their arrangement into groups, separated by comparable or somewhat larger clear spaces. Frequently, especially in the wet season, these groups consist of 50-100 km long lines oriented parallel to the flow, while on other occasions the bunching appears to be purely random (Riehl, Gray, Malkus and Ronne, 1959). The orientation into lines becomes more pronounced as the conditions for convection are more favorable, particularly when synoptic-scale con- vergence in the wind field is present, while the random bunching is more characteristic of inversion-dominated situations and the outer fringes of the trades. In both cases, the cloud groups break out in those regions where the homogeneous layer is thickened relative to its surroundings and the slightly stable "transition layer" of Fig. 52 is absent. Thus the cloud base is, within observational error, at the level of water-vapor condensation of the average sub-cloud air. Its height is readily calculated from a tephigram (or other meteorological thermodynamic diagram) by carrying low-level air upward along a dry adiabat to saturation. The depth of the homogeneous layer averages about 550 m, with variations of about 20% in space and as much as 100% in time. Extreme day-to-day ranges are about 300-800 m. The space scale of the depth variations suggest their connection with the longer-period "eddies" of Figs. 51a and 53, which appear to retain their phase throughout the mixed layer. Thus there is evidence of horizontal convergence and divergence on the cloud-group scale, so that it is still possible to seek the "roots" of the cumuli in deformations of the sub-cloud layer as a whole. When these take the form of long rolls oriented with the wind, some kind of instability to this type of convective motion is suggested, although it is not clear whether the instability is dynamic, thermal or some combination. SECT. 2] LARGE-SCALE INTERACTIONS 211 The problem is a difficult one to treat theoretically since, in fully turbulent flow, the applicability of the classical laminar stability treatments (Rayleigh, 1916; Jeffreys, 1926; Avsec, 1939; see also review by Stommel, 1947) is doubtful, and the new turbulent models have not yet been extended to such complex geophysical situations. When the cloud groups are more in random bunches, it is likely that this scale of instability is not developed. In these cases, there is evidence that the spatial variations in the ' thickness of the mixed layer are associated with, or possibly forced by, thermal inhomogeneities in the sea surface below. A theoretical treatment (Malkus, 1957) shows that the observed ocean-surface temperature anomalies of 0.05-0.3°C are adequate to produce the necessary deformations of the air flow ; the warm spots weakly simulate the effects of small heated "islands" or "mountains" in the sea! The interaction on the meso-, or 10-100 km, scale of oceanic and atmospheric processes is a frontier of marine science just barely opened in the first decade of aircraft research. Thus, trade-wind cloud groups are born in those areas where the homo- geneous layer extends to the level of water-vapor condensation. At first, small cloudlets 100-300 m across break out there in great numbers, from the wetter eddies swirled upward by the turbulent trade. The larger clouds appear to be built up by aggregations of several of these cloudlets, and the group as a whole may have a lifetime of many hours. c. The cloud layer and its transports Growing day and night over tens of millions of square miles, the trade-wind cumulus is the most common, and the most persistently studied, cloud form. The cloud layer is defined as the vertical stratum extending from the condensa- tion level to inversion base ; its thickness ranges from a few hundred meters in the eastern and poleward fringes of the trades, to about 3 km as the equatorial zone is approached. Here the cumuli themselves are the transporting elements. In the intervening clear spaces, the air is weakly descending and small-scale turbulence is entirely suppressed. Since 1946, the Woods Hole expeditions have been building up a picture of the structure of these clouds and how they interact with their surroundings (see reviews by Malkus, 1952; Malkus and Witt, 1959). An aircraft profile through a large, active trade cumulus is shown in Fig. 54 (Malkus, 1954). Photographs of the cloud are shown in Fig. 55. The buoyant updraft portion, with rising speeds of 1-4 m/sec, is restricted to small regions within the cloud much of which is descending even in the active phase shown. This phase lasts only 3-10 min, or a short fraction of the cloud's visible lifetime, so that most cumuli seen or photographed on a given occasion are in an in- active or decaying stage. We wish to inquire first whether and how these evanescent and undistinguished-looking chimneys can be capable of the enormous vertical moisture transports deduced in the budget studies (Section 5, pages 144-147). 212 [chap. 4 In that portion of the oceanic trade-wind belt from 10°-20°N, covering an area of 32 x lO^^ cm2, the evaporation is roughly 1.1 x IQis cal/sec. Of this latent heat supply, about 84%, or 1.5 x IO12 g/sec, of water vapor is being transported upward through the lower cloud layer. We shall use Fig. 52 to deduce what the trade-wind clouds would have to be like to effect this trans- port, and Fig. 54 to check the prediction. 6-28-52 U T^ =20.5 r„ = 19 6 .g = 15.4, ^ -f. =21.0 '1:'^^- ="^ " -'— ' — '— "- - " "'" % - 22.: •I 200 mi- Distance (m) Fig. 54. Aircraft profile made through the large, active trade cumulus photographed in Fig. 55. (After Malkus, 1954, Fig. 3. By courtesy of the American Meteorological Society.) Cloud profile constructed from photographs with individual aircraft runs measuring vertical motion (m/sec), solid curves; temperature (°C), dashed curves; and specific humidity (g/kg), x-ed curves superposed to scale. Environment values of wet-bulb temperature Ty^, specific humidity q, and dry-bulb temperature T^i given to right. Cloud was studied in active phase over western Atlantic trade-wind ocean under normal trade conditions on June 28, 1952. Series of six horizontal passes, in the plane of the wind from top down, completed within twenty minutes. The equation for the transfer of water vapor across a given reference level, say at 1400 m (or ~ 850 mb), is F = 2a WapaQa' ^a - 2c Wcpcqc' ^c - 2d WdRaqd ^d, (46) where F is the vertical water-vapor flux in g/sec, iv is vertical velocity in cm/sec, p is air density in g/cm^, q is the specific humidity in g vapor per g air and A is area in cm^. The subscript a refers to the actively ascending cloud portion, c to stationary or descending cloud portions, and d to the more weakly descending clear air between clouds. The superscript s indicates that the in-cloud air is saturated. Since this is an order-of-magnitude calculation, the liquid-water content (~10% of the vapor), the vertical component of the mean motion SECT. 2] LARGE-SCALE INTERACTIONS 213 (a) ;b) Fig. 55. Aerial photographs of cloud whose measured profile is shown in Fig. 54. (After Malkus, 1954, Fig. 2. By courtesy of the American Meteorological Society.) Wind blows from left (east-southeast) to right across the picture. Photographs made from 1.4 km elevation with sufficient precision to reconstruct cloud dimensions. Lower picture taken roughly two minutes after upper one. 214 MALKus [chap. 4 (~ 100 m/day or 10-^ cm/sec) and the 0.5-2% density variations between cloud and clear air are neglected, so that we may write Flpm = {Wa Aa-Wc Ac)qs-Wd AdQ, (47) where pm is the mean air density at the level, the bars denote mean values, qs is the saturation specific humidity at the average air temperature and q is its clear air value. The equation of continuity is Wa Aa-Wc Ac = Wd Ad. (48) Substituting (48) into (47) we have Wd = F/pmiqs-q) Ad. (49) All quantities for evaluation of (49) are known. i^=1.5xl0i2 g/sec, pm = 1.1 X 10-3, while Fig. 52 shows that at 1400 m the air temperature is about 16°C, so that (from tables) qs^ 13.6 g/kg and (from the right-hand curve) q is 8 g/kg. With a mean cloud cover of 35% at this level (see Table XI, line 6), MJd= 1.2 cm/sec, a value well supported by the Woods Hole measurements and deductions therefrom (Malkus, 1958; Bunker, 1959). From equation (48) Wa = {WcAc + WdAd)IAa. (50) If 94% of the cloudy matter is iliactive and descending at an average rate of 10 cm/sec, so that only 2% of the area (6% of the cloudy matter) is actively rising, Wa comes out as 2 m/sec, in conservative agreement with Fig. 54 and the other Woods Hole observations. Thus, despite their appearance, the ordinary trade cumuli are easily adequate fuel pumps ; they are raising energy more than a hundred times as fast as its rate of dissipation by all air and sea motions combined. When first photographed (Fig. 55a), the topmost tower of the cloud shown in Fig. 54 had penetrated nearly 200 m above the inversion base into the dry air. Shortly after, the tower was observed to cut off and evaporate, like the one schematically shown in Fig. 24, making its contribution to the raising and destruction of the inversion. The beginning of this process is seen in Fig. 55b, taken about 2 min after Fig. 55a. The tower has risen about 350 m and shrunk in diameter from about 670 to about 450 m, imparting its evaporated moisture to the dry surroundings. In poking through the inversion base, this cloud proved itself an exceptional trade cumulus. In a detailed study of the mechan- isms in the moist layer, Malkus (1958) showed that its observed deepening downstream is achieved if only one-tenth of the cloud matter active in mid- layer is left above the inversion base in the process of tower dissipation. In the verification of this prediction lies the key feature of trade-wind clouds. The important question to ask of them is not "Why do they grow?" but "Why do they not grow taller and more vigorously?" As Fig. 52 typifies, temperature lapse rates are statically unstable to saturated motions at least to the inversion and commonly throughout the troposphere ; air parcels rising wet adiabatically SECT. 2] LAKGE-SCALE INTERACTIONS 215 from cloud base should have several degrees temperature excess and strong up- ward acceleration throughout. Yet only one trade cumulus in ten normally reaches the inversion base ; the rest peter out ignominiously far below. A braking mechanism is clearly suggested. The identification and pursuit of this inhibitory mechanism and its implications has been the major contribution of the Woods Hole group, as is well documented in the literature. Cumulus clouds interact with their surroundings primarily by exchange of air. Drawing in drier air from outside or "entraining", they dilute away the buoyancy-producing moisture which is their life blood ; even the vigorous cloud in Fig. 54 has only a fraction of the excess warmth and water content that a non-interacting air parcel would show. The original epoch-making paper by Stommel (1947a) postulated the existence of entrainment by devising a method to calculate its magnitude from the cloud profiles of the Wyman- Woodcock expedition. Using it, we find that clouds like the one in Fig. 54 commonly incorporate a mass flux comparable to the total within their up- draft in a vertical ascent of only 1-2 km ! Temperature lapse rate thus loses its unique significance in prescribing conditions for cloud growth and the ambient moisture structure becomes critical. But the exchange is a two-way process ; the brake upon cloud growth also serves as the mechanism by which the cumuli alter their environment. Since the updrafts retain about the same size with height, an efflux of cloudy air approximately equal to the entrainment may be inferred. In this way moisture and momentum brought up from lower levels are imparted to the surroundings as the clouds shed their protuberances and erode away. Thus the moist layer accumulates its water vapor and easterly momentum is spread upward from the trade-wind maximum (Malkus, 1949, 1958). While demonstrably an adequate mechanism for moisture and momentum transport, there is real doubt whether the ordinary trade-cumulus population can fulfill the final important function of the cloud layer, namely release of precipitation heating. The evidence suggests (Garstang, 1958; Riehl, Gray, Malkus and Ronne, 1959) that all significant amounts of tropical oceanic rain fall in synoptic-scale disturbances, where convergent flow and towering cumulonimbus are found. To be sure, giant "hot towers" are rare, but under special conditions they do occur, even in the trade-wind zones. Comparison of such tower heights measured from photographs with temperature soundings (Malkus and Ronne, 1954) showed that these giants cannot be significantly diluted. Here the entrainment brake is somehow released ; the photographic study pointed to cloud diameter as the critical factor in penetration. Aided by a series of ingenious laboratory experiments (Scorer and Ronne, 1956; Wood- ward, 1959) workers in cumulus dynamics have begun to develop this relation mechanistically. The active, buoyant regions within a cloud body are called "thermals". These may take the form of one or a succession of discrete, rounded elements with an internal vortical circulation which, under particularly favorable conditions, appear to join or elongate into more or less continuous plumes with 216 MALKUS [chap. 4 hemispheric caps. Considerable success has been achieved in modelhng the discrete phase (Levine, 1959; Malkus and Witt, 1959; Ludlam, 1958). These elements turn themselves slowly inside out as they rise. Experiments and observational tests suggest that they entrain and shed roughly one -half their own mass per cycle. What is important is that the rate of turning inside out is inversely related to size and is completed in about two diameters ascent. Dilution, supposed proportional to the rate of exposure of surface area (per unit mass flux) to the dry surroundings, is thus much greater for the small bubbles than the large ones. An ordinary trade-cumulus element of 500 m dia- meter turns inside out once per kilometer rise and is diluted in this distance by the incorporation of an amount of outside air equal to about half its own mass. A 5-km cumulonimbus element, in contrast, rises through nearly the whole troposphere before doing this. If protected by a cloud body for the critical first few kilometers,! such large elements may reach 30-40,000 ft or even to the tropopause with little or no diminution of their original heat content (Malkus, 1960). The predictions of this model have been tested against photographic measure- ments in several contexts, particularly that of the hurricane. The important question of how the large elements are formed, and the relationship between synoptic-scale convergence and their occurrence, is just beginning to be studied. It is becoming uncomfortably clear, however, that even in the "steady" trades, we shall have to examine the "abnormal" situation to comprehend the normal. This means a closer examination of fluctuations in the air-sea system and its exchanges. B. Exchange Fluctuations in the Tropics The interface layers in the tropics constitute the primary energy-regulating valve for the planetary fluids of ocean and atmosphere. Since the sea is both more uniform and much slower to respond to change than is the air, we have implied throughout this chapter that it is mainly variations in air structure which close down or open up this valve. Simplifying to bare essentials, we may say that the exchange valve is opened as the air-sea temperature and humidity difference is enhanced and/or the wind speed strengthened, and is closed as these diminish. Thus we must seek to examine fluctuations in terms of the processes and scales of motion which produce alterations in these property differences : that is to say, primarily those which change the temperature, humidity and wind speed in the air overlying the ocean surface. We may now interpret the seasonal marches of Qs, Qe and momentum ex- change in this light (Figs. 18, 19, 29, 39, 40, 42, 43). In winter, the subtropical high pressure cells are strongest and in their most equatorward positions. 1 The argument of p. 175 shows that above 4-5 km the saturation specific humidity is very small due to low temperature. Thus the difference between in-cloud and ambient moisture content diminishes with elevation and the drying of a cloud by entrainment becomes ineffective in the upper atinosphere. SECT. 2] LARGE-SCALE INTERACTIONS 217 Large latitudinal temperature differences cause their axes to slope equatorward with height, superimposing westerlies aloft over most of the tropical easterlies (Figs. 22, 23, 26b and c). Pressure heads are maximal ; the average trade-flow is vigorous, importing cool, dry air rapidly into the tropics, so that exchange of all properties is greatest in the winter months. Interaction between the dis- turbances of the mid-latitude westerlies and the trades is also common from time to time, and the tropics are frequently invaded by "polar troughs" and shearlines, left over remnants of the polar front which cause strong resurgences of the trades in their rear (Riehl, 1945, 1954). Except in such disturbances, the moist layer is generally shallow and over most of the tropics proper winter is the dry season. i In summer, the subtropical high cells are farthest poleward, weakest, and their axes nearly vertical. The trades are slower but deeper and largely de- coupled from middle latitudes. The thicker moist layer is deformed by dis- turbances of purely tropical origin, such as the easterly wave and equatorial vortex (Riehl, 1945 ; Palmer, 1952). Since both zonal and meridional circulations are more sluggish, reduced property differences and wind speed combine to produce, over the sea, the minimum seasonal values of Qg, Qe and shearing stress. On the other hand, the oceanic trades have a summer rainfall maximum. Aside from the periodic fluctuations in exchange imposed by the seasonal and diurnal cycles, the two causes of the greatest alterations in air-sea inter- action are aperiodic : namely, those associated with variations in overall circulation strength, of one to several weeks in duration, and those associated with the passage of synoptic-scale disturbances which, operating on the 1-3 day scale, can create the largest disruptions of all. Unfortunately, periodic or otherwise, all of these factors are mutually interdependent, as we shall see, and their effects upon exchange superpose in a fashion scarcely likely to be linear. In the context developed by investigating these, still longer period (climatic) fluctuations will be brought up briefly in the last section. On the short side of the spectrum, meso-scale (individual cloud group, 10-100 km size and duration of some hours) variations in air-sea interaction have, as we saw, just been opened up for study (Malkus, 1957; Bunker, 1959). a. Effects of changes in overall circulation strength Within each season, there are variations in the strength and character of the trades associated with large-scale changes in the position and develojiment of the subtropical high pressure cells over major parts, or all, of a hemisphere. Periods of several weeks occur when the ridges show higher than normal central pressures and are stretched out from east to west. The pressure dif- ference from subtropical ridge to equatorial trough is then large. The trades 1 Exceptions are found in the polewartl fringes. An oceanic example is the Hawaiian region where (Usturbances of the westerly jet stream produce a winter rainfall maximum. Continental examples are subtropical west coasts, such as California and Chile, where upwelling enhances summer dryness. 218 MALKUS [chap. 4 are strong, zonally uniform and relatively uninterrupted by the invasion of disturbances. These "high index" periods alternate with those of "low index" when the subtropical ridges are weaker, elongated meridionally and often broken up into two or more feeble centers over an ocean basin. Under these conditions, disturbances are common and the trade flow is weak and disrupted. Thus, predominant speeds above 6-7 m/sec will normally be associated with high index, lower speeds with low index. Further, at the start of high index, when the surface ridge builds in the subtropics, we observe strong acceleration of the trades with flow toward lower pressure, i.e., equatorward. At times this acceleration is spectacular resulting in "surges of the trades" with speeds of 15-20 m/sec over wide portions of a tropical ocean. It is likely that the equator- ward flow is greatest during the period of acceleration. Also the rapid transport of relatively cool dry air over warm water should cause evaporation to reach a maximum at these times. High index, however, is also the period of minimum exchange of air-masses across the subtropics. After an initial surge of perhaps two to four days' duration, we can expect that the temperature difference between sea and air will again decrease, initiating the next stage of the index cycle. These oscillations were first noted by Riehl (1954a), who has emphasized their importance to the mid-latitude energy supply. Subsequently, Kraus (1959) attempted a theoretical analysis of their time-scale in terms of the delay between evaporation in the trades and precipitation in the equatorial trough. A weak link confronting further progress lies in our vast lack of knowledge concerning the stability of the easterly trades to the formation and deepening of synoptic disturbances. If it were established that the instability to dis- turbances became greater as the trade current decreased, a physically reason- able chain of events is suggested ; unfortunately, so many interacting circulation branches must be considered that it is naive to hope for a simple or rapid resolution. In itself, the observation that the trade strength may vary by a factor of two or more about its seasonal mean is significant as far as export of latent heat is concerned, particularly since the time scale appears to be a crucial one in meteorology. These variations become even more significant when we re- examine Fig. 6 and find that the mean speed, about which the fluctuations occur, is in the region of strongest slope of the Cd curve, so that amplified variations in exchange of energy and momentum may result simultaneously over wide regions. It is probably not coincidental that 6-7 m/sec is also the critical air velocity for the formation of white caps. The role of wave shape and sea spray in transfer processes has been discussed but not resolved (Munk, 1947; Neumann, 1951; Rofl, 1952). The consequences to the ocean, however, of the most conservatively esti- mated evaporation fluctuations are not negligible. Reducing it by one-half over a month retains about 3000 cal per cm^ of sea surface that would otherwise be removed by the atmosphere. If the upper 100 m share this equally, the mean temperature of this layer would become 0.3°C above normal : a small-sounding increment until we recall that it is comparable to the average difference between SECT. 2] LARGE-SCALE INTERACTIONS 219 sea and air, and thus might well affect subsequent exchange, storm formation, etc., as well as the salinity, convective structure and biological processes in the thermocline region of the sea. However, rather than indulge in the intriguing speculation to which this whole subject tempts the unwary, it is less news- worthy but more useful to lay a quantitative foundation for a series of specific questions. What actually happens to processes in the moist layer when the trade strength oscillates about its mean? The description of trade-wind structure and transports in the preceding sub- section was based largely upon the April, 1946, (Wyman-Woodcock) Caribbean expedition, which investigated a strong trade regime (average wind from 090°, 9.1 m/sec). Fortunately, a succeeding Woods Hole expedition in April, 1953, found in the same area and season a situation of low index and poorly developed trade (average wind from 106°, 5.7 m/sec). Although the data were not adequate to resolve definitively many of the vital questions, a framework of inquiry was set up in a monograph by Malkus (1958) contrasting the two sets of results. The most obvious visual differences between the weak and strong trade situations lay in the sea state and cloud structure. In the strong trade case, white caps and rough seas prevailed on all observing days, while they were absent 80% of the time in the low index situation. Apparently correlated was the development of trade cumulus clouds, which were plentiful and vigorous during the first expedition and suppressed, feeble and sparse during the second. On the days of weakest and most southerly flow, glassy seas with unprecedented clear spaces, hundreds of miles across, frustrated the aircraft observers, who sought to duplicate the cloud penetrations of Figs. 54 and 55. The sounding data showed that the reason lay in the disrupted function of the lowest air layers and suggested the stopping down of the exchange valve. While the properties of the cloud layer differed little from those shown in Fig. 52, in the low index case the mixed layer was abnormally shallow. On the poorest days for trade cumuli, its top fell about 200 m below the condensation level. In addition, it was far from homogeneous. While the mean temperature and specific humidity of the lowest kilometer were nearly identical to those of Fig. 52, their vertical stratification showed the consequences of reduced turbulence : the temperature lapse rate was slightly less stable than normal and, most important, the upward rate of moisture decrease was 3.5 times greater than in the strong trade situation. It appears that the water vapor was just not getting pumped up through the sub-cloud layer and it, therefore, accumulated in the lowest levels. This may in turn have inhibited further evaporation. Reduced upward transport resulted from weak and southerly flow. The former gave less surface roughness and diminished forced stirring. The latter, perhaps, was responsible for weakened thermal turbulence at low levels due to reduction of the sea-air temperature difference. The aircraft-measured momentum fluxes averaged about 40% of normal. In striking contrast to the suppression of trade cumuli, the April, 1953, low- index period provided one of the best displays of tropical hot towers which 220 MALKUS [chap. 4 have ever been captured for quantitative study (Malkus and Ronne, 1954). Concentrated in rows in the convergent area of a polar trough disturbance, these shot upward into the intense westerhes of the high troposphere, their tops spreading in 100-km streamers in the jet, centered at about 45,000 ft. Measure- ments from time-lapse films showed towers several kilometers across rising from a 7-10 km wide cloud body over open ocean at ascent rates exceeding 12 m/sec. Simultaneously, the nearby trade cumuli were sparse, feeble and stunted. A computation similar to that of equations (46-50) showed that, averaged over the convergent area, the net upward vapor transport by the cumulonimbi was not less than 400 cal/cm^ per day, or nearly twice the normal evaporation rate. Thus the ordinary trade cumuli seem to require a good mixed layer, wind stirring and the normal operation of exchange, but grow well in the face of the weak subsidence ( ~ 10"^ per sec ; see Fig. 46b) characteristic of the undisturbed trade, while giant cumulonimbi are dependent mainly on intermittent flow convergence. Over the open sea, the latter is associated with the passage of synoptic-scale disturbances. The frequency of these depends on the stability of the air flow, which varies with location and season. In regions where they are common, it is unlikely that the dynamics or energetics of air-sea interaction can be modelled or understood mechanistically without taking this scale of motion carefully into account. b. Effects of moderate disturbances and the diurnal cycle {i ) The nature of synoptic-scale perturbations in low latitudes The only type of tropical disturbance known to most mid-latitude dwellers is the notorious hurricane, which they would prefer remained at home in the tropics. In addition to its destructive record and distant excursions, the hurricane is the real black sheep among tropical disturbances on several other counts, the most important being its rarity. An average tropical station ex- periences ten or more synoptic-scale perturbations per month ; less than 10% of these, even in season, attain the dubious fame of a girl's name. In addition, the hurricane extends through the entire troposphere and, as any schoolchild knows, intense winds (by definition > 65 knots) rage about its center. In the latter sense, most tropical disturbances are not properly termed "storms" since they commonly exhibit weaker-than-average winds until moderate development is exceeded, nor until then do they extend throughout the troposphere, in general being confined principally to its lower or upper half. As has been brought out previously, the significant feature of the tropical atmosphere is its two-layered structure ; ordinarily the different disturbances of each layer move within it, at speeds comparable to its mean wind, and to a degree, independently of what goes on in the other. In fact, although little is known of the formal criteria, the requirement for an intense storm is its ability to lock together the layers, creating a coherent circulation which occupies SECT. 2] LARGE-SCALE INTERACTIONS 221 most or all of the troposphere. Observations suggest that this may result from an accidental superposition or from a disturbance in either layer working its way up or down. However, these attempts fail far more often than they succeed. Unfortunately neither the necessary nor sufficient conditions for deep develop- ment have been dehneated. Tropical disturbances may be defined as regions of convergent and divergent flow (order of IQ-^ sec-i) with associated patterns of vertical motion on the 200-2000-km scale (area-averaged W'^ 1-50 cm/sec). They are recognizable on the synoptic weather charts for a lifetime ranging from two days to four weeks ; the identifying deformation in the streamline field may have the shape of a wave, a closed cyclonic vortex, or merely a shear line where two parallel currents of different speed are closely juxtaposed. The frequency of each type varies with region, season and elevation, and the different forms may interact, combine or superpose vertically in a multiplicity of permutations. Despite this complexity, the outstanding effect of all types, weatherwise and physically, is that their presence means alterations in the thickness of the moist layer and thereby in the conditions for cumulus convection. In regions of low-level convergence, the inversion lid may be destroyed altogether and a deep moist layer and hot towers build all the way to the tropopause, while where low-level divergence prevails, the inversion is lowered and strengthened, and convection is suppressed. In contrast to the middle -latitude cyclone, tropical disturbances do not contain fronts, but form and live their lives entirely within tropical air (except for the escaped hurricane). Thus they are not able to feed upon stored potential energy in pre-existent horizontal temperature contrasts, but must either create their own density gradients by differential latent heat release or draw their kinetic energy and momentum from other scales of motion. The relative importance of these processes is not well known, except in deep disturbances of the wet season where the major energy source is demonstrably condensation (Riehl and Gentry, 1958; Riehl, 1959; Malkus and Riehl, 1960). This is not to imply that even here dynamic factors may not be vital triggers. In winter, when westerlies overlie most of the trades, disturbances most often move eastward and have an extra-tropical origin. They consist of shearline remnants of cold fronts, or downward penetrations of polar troughs or "cut-off cold lows" from jet-stream undulations. The latter two types are usually best developed aloft, with the ground-dweller often just seeing their effects in cloudiness and precipitation (cf. hot towers described on pp. 219-220). In summer, when the easterlies are deep and more decoupled from mid- latitude influences, tropical perturbations commonly travel toward the west, the significant ones at speeds of 10-12 knots, or somewhat slower than the trade current. The famous "easterly wave" discovered by Dunn (1940) and described extensively in the literature of the 1940's and early 1950's by Riehl (1954a) and his collaborators, is illustrated schematically in Fig. 56. An im- portant source of low-level disturbances is the equatorial trough, which in its downstream active portions often consists of a series of cyclonic vortices 222 [chap. 4 (a) (b) (c) V 7 7 miles Ahead of Wave Trough Reor of Wave Trough Direction of Wave Travel SECT. 2] LARfJE-SCALE INTERACTIONS 223 id) Fig. 56. Schematic illustrations of major features of "easterly wave" type of tropical disturbance. (After Malkus, 1958a, Figs. 2, 3, 4 and 8.) (a) Siu'face streamlines of weak-to-moderate amplitude wave, which typically moves toward west at speed of 10-12 knots or slightly slower than prevailing trade. Its direction of motion is denoted by heavy arrow. Streamlines are drawn everywhere parallel to the wind direction, so that ahead (west) of the wave trough, denoted by the heavy solid line, winds blow from slightly north of east and to its rear (east) they blow from slightly south of east. The wave trough line thus marks the wind shift. The barbs on the wind arrows denote speed, each small one representing 5 knots. The area of cloudiness and rain (indicated by triangular shower symbols) is commonly found to the rear of the trough. The subtropical high pressure cell is the bean -shaped region with "H" at its center. (b) 15,000-ft streamline pattern in typical moderate easterly wave. Note that the wave amplitude is greater than at the surface. (c) Vertical cross -section going from west (right) to east (left) through wave shown in (a) and (b). Cloud forms are shown schematically (not to scale). Winds (horizontal) are denoted by barbed lines, each short barb representing 5 knots. Winds blow/?-o??i the direction pointed by the tail, with north straight up, east to the right, south straight down and west to the left, so that winds ahead of the wave are east or north of east and winds to its rear are east or south of east. The wave trough is the heavy solid line, while the top of the moist layer (trade-wind inversion) is denoted by the light solid line. The triangles indicate showers and rain. (d) Schematic picture of a very deep easterly wave, showing streamlines and the major cloud bands. Note the closed central vortex with wind turned all the way to the southwest at its southern rim. Other svibsidiary cloud bands (not shown) are numerous throughout the rain area, which is indicated by hatching. The central surface pressure in such a disturbance may become as low as 1000 mb, with winds up to 35-40 knots. About 10-20% of easterly waves, or 1-2 per month, reach this intensity in the wet (summer) season. (Palmer, 1951), each with a trailing convergence zone filled with lines of hot towers. In the Caribbean, these frequently trigger easterly waves to their north (Fig. 57). Easterly waves disturb the wind field most at 15-20,000-ft elevation and weaken toward the surface and high troposphere, while equatorial vortices are frequently confined to the lowest 10-15,000 ft. Favorable upper-level 224 MALKUS [chap. 4 conditions are required (Riehl, 1948) for either of these to penetrate throughout the troposphere and thereby to deepen markedly. However, this whole area is a wide-open frontier and will remain so until proper three-dimensional data become available over the unexplored low-latitude oceans. Fig. 57. Schematic picture of equatorial vortex of summer season which sets off easterly wave trough (heavy solid line) to north. Surface streamlines are light solid lines with arrow-heads. Area of convergent low-level flow, with cloud bands and showers shown by stipples. (n ) Synoptic and diurnal exchange fluctuations in the equatorial trough zone A pioneering endeavor to study at close range the exchanges and vital mechanisms of the Atlantic equatorial zone was made by Garstang (1958). The Woods Hole Oceanographic Institution's small research vessel Crawford (125 ft length; 250 tons gross) was stationed, mostly hove to, at 11° OO'N, 52° 25'W from 14 August through 5 September, 1957, and accomplished a detailed set of joint oceanographic and meteorological measurements, including the heroic feat of decent rainfall records, six-hourly pilot balloons and twice daily radiosondes ! During this typical wet-season period, the easterly current was well estab- lished to heights above 35,000 ft. The top of the moist layer fluctuated around 11,000 ft, being much deeper in convergent zones of disturbances, and much shallower when unusually fair -conditions and divergent flow prevailed. The most significant result of the cruise was the marked variations in exchange, which were clearly associated with the synoptic-scale patterns, as summarized in Table XVI. The concomitant wind structure is shown in Fig. 58. The com- putations oiQs and Qe were made hourly, using transfer formulas (20) and (21). SECT. 2] LAKGE-SCALE INTERACTIONS 225 Table XVI Synoptic-Scale Exchange Fluctuations in the Equatorial Atlantic. From R.V. Crawford Cruise August 14-Septeraber 5, 1957 Period To-Ta, ?0-9a- Ua, Qs, Qe, X g/kg m/sec cal/cm2 day cal/cm2 day Disturbed : convergent and rainy (Period A) (96 obs.) ().H8 .^>.3 4.5 12.7 245 Very fair (Period B and C) (96 obs.) 0.06 4.9 6.0 1.5 302 "Normal" — all other (188 obs.) 0.22 .'5.4 5.6 5.1 310 Disturbed Undisturbed Averoge Wind Speed (m/sec) Steadiness (%) Fig. 58. Wind speed and directional steadiness profiles for Equatorial ocean at 11°00' N, 52° 25'VV determined from data of research vessel Crawford on station from 14 August-5 September, 1957. Profiles broken down into disturbed (Period A) and un- disturbed (Periods B and C) as described in text. (After Garstang, 1958, unpublished.) 226 AIALKUS [CHAP. 4 While Qe varied little throughout, we see a marked enhancement of sensible heat input in the disturbed period, and a marked suppression in the fair period, in inverse variation with the wind strength. As the table shows, the explanation lies in the changes in the sea-air temperature difference which was very large in disturbed conditions and averaged vanishingly small in fair periods. Figs. 59-61 provide increased mechanistic insight. In these figures, the more detailed breakdown into periods is as follows : Period A: 17-19 August. Ship station continually in disturbed convergent zone. Total rainfall 41.9 mm, cloud cover always greater than 75%. Average wind speed 4.5 m/sec, direction variable, sometimes westerly. Directional steadiness low. Period B: 25-26 August. Fairer than average, only three light showers. Cloud cover less than 50%. Average wind 4.8 m/sec from east-northeast. Period C : 1-2 September. Very fair period with only one brief shower. Cloud cover well below 50%. Average wind 7.2 m/sec from east-northeast. The ocean-surface temperature varies little between the periods, although it is slightly cooler in the strong-wind fair period C, than during the weaker-wind fair period B, indicating the effect of stirring. However, the dominant control upon To— Ta is produced by air temperature variations. Disturbed Period A is consistently colder (up to 3.0°F) than any other: the fairest period, C, is the warmest. Reduced insolation due to cloud cover, direct cooling by rain showers, as well as the descent from aloft of cool air associated with the condensation- precipitation cycle, all contribute to these differences which are demonstrably not advective in origin. Figs. 59-61 in conjunction with Figs. 62-65 i)ermit for the first time a quantitative picture of the diurnal cycle in exchange and transport over the tropical oceans. The average sea-surface temperature (solid curve. Fig. 60) shows a well defined diurnal variation with a spread of 0.6°C. The diurnal air- temperature range is about 1.5°C, in contrast with the 7-8°C common over islands and land-masses in the same latitudes and season. Comparison of Figs. 59 and 60 shows that there is considerable lag in the diurnal heating and cooling of the sea surface relative to the air. Whereas the minimum air tempera- ture is registered at 0500 LST, the minimum sea-surface temperature occurs at 0700 LST ; similarly, the maximum air temperature is reached at 1300 LST, while the sea-surface temperature reaches its maximum between 1600 and 1700 LST. We may deduce from this an important consequence to exchange, namely, a minimum in Tq— Ta around midday. Fig. 61 shows that the minimum consists of an actual reversal in sign, so that the sensible heat flow is from air to sea in the noon hours. Thermal turbulence should then reach its minimum. Figs. 63-65 confirm this deduction and illustrate its consequences upon cumulus convection. As astute observers in the tropics have long siis])ected. there was an oceanic cloudiness minimum around midday and maxima in the dawn and midnight hours. Our suggestion in the preceding sub-section connecting trade cumuli with active transports in a turbulent mixed layer has received direct support, since SECT. 2] LARGE-SCALE INTERACTIONS 227 Hours (LST) Fig. 59. Diurnal march of air temperature (°F) at ship's deck level during Crawford Atlantic equatorial cruise. (After Garstang, 1958, unpublished.) Time is in LST, or local standard time, four hours earlier than Greenwich Mean Time. Sea Temperature Fig. 60. Diurnal march of sea-surface temperature C^F) during Crawford Atlantic equatorial cruise. Made by careful dip-bucket determinations. (After Garstang, 1958, un- published.) 228 MALKUS [chap. 4 T Air > 7" Seo Hours (LST) Fig. 61. Diurnal march of sea-air temperature difference during Crawford Atlantic equa- torial cruise. Constructed from data in Figs. 59 and 60. (After Garstang, 1958, un- published.) _l__l I L. _l I I I I J Hours (LST) Fig. 62. Average diurnal march of sea temperature (°F) with depth (ft) during Crawford equatorial Atlantic cruise as determined by bathythermograph observations. (After Garstang, 1958, impublished.) SECT. 2] LARGE-SCALE INTERACTIONS 229 24 - K 22 - 1 \ 2.0 - / \ 1.8 - 1 \ 1.6 ■ 1 \ 14 y 1 12 ^^~~y / 1.0 ■ / .8 ■ 1 .6 - 1 .4 - .2 0 - / A / -.2 • ^^ A / -.4 - \ / \ A / -& - V \ /\ / -.8 -1.0 c - V \/ )0 ' ' i'. ' ' ' ' ,2 ' ' 1 Zi Fig. 63. Diurnal march oiQg, the sensible heat flux from sea to air, in cal cm~2 per day as computed from transfer formula (20) from ship's deck level data taken during Crawford equatorial Atlantic cruise. (After Garstang, 1958, unpublished.) Fig. 64. Diurnal march of air temperature (°F) and relative humidity (%) at ship's deck level during Crawford equatorial Atlantic cruise. Note inverse relationship. (After Garstang, 1958, unpublished.) 230 MALKUS [chap. 4 cumulus minima on both the diurnal and synoptic scales (see also curve for Period C, Fig. 61) are periods where the sea-air temperature difference has become negative, so that Qs is reversed. In analyzing the results of the Crawford cruise, particular attention was paid to the most pronounced disturbance of the period. Consisting of an equatorial vortex, which set off an easterly wave to its north (Fig. 57), it showed con- vergence of 5-10 X 10"5 sec~i in the area around its closed center. Winds in this region were light and variable, rising occasionally to 7-8 m/sec in the inter- mittent showery "squall lines". In the twenty-four hours as the center passed the ship station, 41.4 mm of rain fell, or more than half the total precipitation Hours (LST) Fig. 65. Diurnal march of low cloud cover (cumulus) in oktas during Crawford equatorial Atlantic cruise. Estimated from shipboard by trained weather observers. (After Garstang, 1958, unpublished.) of the sixteen-day period. In this convergent zone, the sensible heat input, Qs, averaged 21.8 cal cm"2 day~i or nearly twice the disturbed period mean of Table XVI. Since the corresponding Qe was 227 cal cm~2 dayi, the Bowen ratio was nearly 10%, in contrast to its value of less than 2% in the normal and fair periods. These results suggest that in the equatorial zone and quite probably throughout the tropics, the sea-air flux of sensible heat is concentrated in disturbances, where it may reach or exceed 10% of the normal Qe. In equatorial regions these disturbances are sufficiently common so that their contribution to the average heat flow should not be ignored. Mechanistically, the enhanced Qs comes about through increased sea-air temperature difference, which more than compensates the light winds near the centers of weak- to -moderate SECT. 2] LARGE-SCALE INTERACTIONS 231 perturbations. In the stronger disturbances, or true tropical storms, where central wind speeds exceed the average (see example in Fig. 56d), a still further amplification of the exchange might be expected. We shall next look at the extreme case of transfer in the tropical hurricane. c. Exchange and the mature hurricane Ocean-atmosphere interaction reaches its acme in the full hurricane. The only vicious offspring of the normally benign tropics, it draws both its life-blood and its ability to deal death from the sea. Feeding on evaporated sea-water, it pays back for its keep by boiling up the oceans into chaotic mountains and often driving their unleashed fury on shore. Wave heights exceeding 50 ft are not uncommon in mid-ocean, hurled by 100-200 knot winds, against which even huge ocean liners are helpless. At coast lines, surf damage is augmented by abnormal tides and the terrifying "storm surge" of embay ments, which may wipe out entire towns in a few hours (Dunn and Miller, 1960, pp. 206-230; Tannehill, 1938; also see Chapter 17 of this volume). Unnecessary in the general circulation, the only good blown by these ill winds is the impetus their dire necessity has given research. Through the establishment, in 1955, of the National Hurricane Research Project of the U.S. Weather Bureau and its instrumented aircraft program, more observational material is available on the interior structure of hurricanes than for any other atmospheric phenomenon. As a consequence, a physical model of this type of thermal engine is in the process of evolution, which we shall draw upon to build our description. Some of its analytical phases will be introduced subsequently to examine sea-air interaction at its terrestrial extreme. (^ ) Operation of the hurricane engine The hurricane is a thermally driven circulation whose primary energy source is the latent heat of condensation. The heating acts to estabhsh the pressure gradients which produce and maintain the furious winds. Radar and photo- graphic maps (Jordan, Hurt and Lowrey, 1960; Malkus, Ronne and Chaffee, 1961) demonstrate abundantly that latent heat is not released uniformly throughout the rain area, but that it is concentrated in spiral convective bands of narrow width oriented along the low-level inflowing air trajectories and particularly concentrated in a vicious ring of penetrative hot towers encircling the calm eye at the center. The storms vary widely in size, from the midgets of 50-100 km radius to the giant Pacific typhoons which dominate whole ocean regions more than 2000 km across. As schematically illustrated in Figs. 66 and 67, the typical hurricane consists of the following radial subdivisions : 1. An external ring of weak subsidence and divergence, 2. An "outer rain area" with pressures above or near 1000 mb and winds below or marginally at hurricane force (65 knots). 232 MALKUS [chap. 4 3. A much smaller central core consisting of: a. An "inner rain area" with pressures less than 1000 mb and hurricane force winds. b. A relatively calm, clear central "eye" containing the lowest pressure. The key feature of the full hurricane, distinguishing it from the sub-hurricane tropical storm, is the presence of a central core ( ~ 100-400 km in diameter) of raging winds and tight radial pressure gradients, of the order of 30 mb per 60 km (Fig. 67). Thus the central pressures of a moderate hurricane (maximum wind ~ 100 knots) must be around 960 mb, 4-5% below the sea-level mean. Fig. 66. Diagram showing low-level spiral trajectories (solid lines with arrow-heads) into moderate strength tropical hurricane. (In part from Tropical Meteorology by H. Riehl. Copyright, 1954. McGraw-Hill Book Co. Used by permission.) Distance co-ordinate degrees latitude (1° lat=lll km). Dashed circles are iso- chrones (labelled in hours) which give the time an air parcel travelling horizontally along the trajectories would take to reach a point 0.5° latitude from storm center, for mean moderate storm. Co-ordinate system superposed is the natural co-ordinate system used in model. The line s is everywhere tangential to the trajectories ; n is the normal to the trajectories ; ^ is the "crossing angle" at which the trajectories intersect lines of equal radius r. R denotes the radius of curvature of the trajectory. Shown also is a standard polar co-ordinate framework with azimuth tfj, radial distance positive outward. The new model relates core maintenance to mechanism, namely, cumulonimbus convection and sea-air exchange. To sustain the required pressure gradients, two coupled processes are necessary: first, a greatly magnified oceanic input of sensible and- latent heat, and, secondly, the undilute release of the latter in concentrated hot tower ascent, so that air of high heat content is pumped rapidly into the upper troposphere. The quantitative establishment of these crucial relationships was carried out in a joint analytic and observational framework (Malkus and Riehl, 1960). SECT. 2] LAR(iK-S('ALE INTERACTIONS 233 \Mth the basic assuin])tion of an undisturbed top near the tropopause, so that lateral pressure gradients are produced by density variations within the troposphere, it is readily shown that ascent and condensation, however rapid, of unmodified tropical air cannot produce a hurricane. Curve E on the tephi- gram in Fig. 68 shows the structure of the mean tropical atmosphere in the hurricane season, the environment and air source for the developing storm. If its sub-cloud air (T = 26.0''C; g=18.5 g/kg) is lifted dry adiabatically to Fig. 67. Schematic diagram of core region of typical moderate strength tropical hurricane, which is defined to consist of an "inner rain area" and an "eye" (see text). Major spii'al bands of cumulonimbus along trajectories and forming eye wall idealized. Arrow at top denotes direction of storm motion. Stippled region, marked v^, is region of maximum wind speed, frequently oriented as shown. Typical surface pressures and wind speeds indicated. Dimensions of real hurricanes vary widely and figures in this diagram should not be taken too literally. condensation and from thence moist adiabatically, the dashed curve results. This is the maximum warming of air columns achievable by converting the original latent heat content to sensible heat content. It is uniquely fixed by the initial total heat-energy content Q[Q = CpT + Agz + Lq^ {CpT + Lq)o] of the lifted air, which is approximately 84 cal/g in mean tropical air. According to the hydrostatic equation, the lowest surface pressure obtained through this ascent will be about 1000 mb (Riehl, 1954). This is a threshold value and it is interesting to note that many tropical storms (cf. Fig. 56d) reach equilibrium 234 MALKUS [chap. 4 at this central pressure. The total heat content of Jiormal tropical air raised un- dilute {without entrainment) to the level of zero buoyancy is insufficient to generate pressures substantially below 1000 mb. Formerly it was believed that descent and compressional warming in an outward-sloping eye could account for the rapid pressure drop in the inner rain area. The aircraft data have refuted this ; no large enough eye-wall slopes have yet been observed ; in most cases the hot towers grow almost vertically to great heights. A typical "inner rain area" sounding (curve D, Fig. 68) from hurricane Daisy, 1958, contains the heart of the new physical modelling and points to the existence of the extra heat source. This sounding very closely Fig. 68. Thermodynamic features of mean tropical atmosphere (curve E), wet adiabatic ascent of mean tropical surface air (dashed) and mean moderate hurricane (D, actually from observations of hvirricane Daisy, 1958) plotted on tephigram. Abscissa is temperature in °C. Slanting ticks are pressures in mb. Oi'dinate (not shoMTi) is potential temperature 6. Arrow labelled Tq is sea-surface temperature (29°C) pre- vailing for hurricane Daisy situation. Isopleths of total heat-energy content, Q, would be parallel to dashed line, increasing toward upper right. Dotted extension of lower part of curve E shows properties of surface air, if cooled adiabatically, in flowing horizontally toward center. represents the moist adiabatic ascent of surface air whose total heat-energy content has been increased by somewhat more than 2 cal/g over mean tropical air. By a hydrostatic computation we find the surface pressure associated with curve D to be 960-970 mb.i Thus we may generate low pressures by increasing the heat content of the rising air, which moves the adiabat ascended towards the upper right, or to warmer temperatures. The surface pressure goes down nearly linearly at about 13 mb per cal/g added. But is it required that the entire inner rain area rises wet adiabatically? What of the concentration in hot towers and the intervening space between the undilute cores in Fig. 67? We see from curve D of Fig. 68 that the tempera- ture excess over mean tropical air increases upward, becoming a maximum in the upper troposphere. In fact, the hydrostatic computation shows that 80% of 1 The range depends upon relative humidity and the undisturbed upper level assumed. SECT. 2] LARGE-SCALE INTERACTIONS 235 the surface pressure dro]) is produced by warming above 500 mb, and 50% by warming above 300 mb. Thus the main function of the hot towers is to grow tall and to pump large amounts of augmented heat-content air to great heights, filling the upper regions of the core. The intervening clear spaces and evapora- tional cooling which reduce middle-level temperatures several degrees below the adiabatic value change the surface pressure by only a few millibars. The structure, distribution and dynamics of the cumulonimbus towers have been discussed at length in the hurricane literature following 1958 (see especially Malkus, 1960; Malkus, Ronne and Chaffee, 1961). We shall be concerned here mostly with the air-sea interaction phase and its dynamic implications. (ii ) Evidence for the extra oceanic heat source Actually, the existence of the extra heat source in hurricanes was pointed out by Byers in 1944 and its dynamic importance was foreseen by Riehl in his remarkable textbook on Tropical Meteorology (1954). Many published records, from the early days of Depperman (1946), have proved that the surface air temperature outside the eye is constant or decreases only very slightly toward the center. If adiabatic expansion occurred during pressure reduction, on the other hand, the temperature of surface air spiralling toward a hurricane center should decrease markedly. For instance, air entering the circulation with the mean tropical sub-cloud properties should reach the 960-mb isobar with a temperature of 20.5°C and a specific humidity of 16.5 g/kg (dotted extension of curve E, Fig. 68). Because of condensation, a dense fog should prevail at the ground inward of the 985-mb isobar. But this is never observed. It follows that the potential temperature of the surface air increases along the inward trajec- tories. We also know that the specific humidity increases and that the cloud bases remain between a few hundred and a thousand feet. The surface air thus acquires both latent and sensible heat during its travel toward loiv pressure. Fig. 69 shows the variation of potential temperature, 6, and specific humidity, q, during isothermal expansion at 26.0°C deduced from observations in hurricane Daisy, 1958, and typical of the moderate-strength hurricane. We may find the heat-energy increments in the core, or as the air moves from 1000 to 960 mb, from the graph as follows : hQ = CpBd + LSq (51) = 0.24 X 2.7 + 600 x 3.2 x 10-3 = 0.65+1.9 ^ 2.6cal/g. We see that this is just the difference between the surface air in curves E and D in Fig. 68. If the latter (29 = 960 mb, T = 26.0°C, ^^ = 21.8 g/kg) is lifted dry adiabatically to the condensation level and from thence moist adiabatically, we get a curve identical to D in the high levels and slightly warmer than D up to 400 mb. We note from Fig. 69 this further remarkable fact : reading horizon- tally from a given surface pressure {ps) ordinate, the potential temperature and 236 [chap. 4 specific humidity, and lifting surface air with these properties, produces a moist adiabatic ascent curve for which the corresponding hydrostatic surface pressure is just shghtly lower than the given jps. This may be, a posteriori, obvious, but only in the framework of the model. The storm must thus be able to meet two simultaneous constraints : accelerated heat pick-up and sufficiently con- centrated release to maintain hydrostatically the surface pressure gradient necessary to ensure this pick-up. There is thus both a boundary and an internal constraint. The rarity of the phenomenon suggests that one or both of these is ?4 26 0(°A) Fig. 69. Diagram showing sub-cloud air properties as a function of surface pressure (ordi- nate) in horizontal inflow into hvirricane as deduced from observations in hurricane Daisy, 1958. Solid line shows potential temperature {6 in °A, lower abscissa). It corresponds to isothermal expansion of mean tropical air at 26'^C. Dashed curve shows specific humidity {q in g/kg, upper abscissa) (see text and Table XVIII, page 244). difficult to satisfy. To pursue this point, let us examine the air-sea relationship still more explicitly. If, and only if, the inflowing air can increase its heat content by 2-3 cal/g in the core regions can the pressure gradients for a moderate hurricane be sus- tained. The question is whether and how exchange can be augmented to this degree. Can the necessary magnitudes be deduced from the ordinary form of the transfer formulas with coefficients unchanged from normal, or must the processes of exchange be radically altered? Is the ratio of sensible to latent heat input of 34% deduced by aj^plying equation (51) to Fig. 69 necessary or SECT. 2] LARGE-SCALE INTERACTIONS 237 even reasonable? Must the exchange demand per unit sea surface be ten, one hundred or one milhon times as rapid as that in the normal trade? These questions cannot be answered quantitatively without some knowledge of the dynamic relations in the hurricane core. We need to know the radial pressure distribution to tie the pressure ordinate of Fig. 69 to radial distance ; we also need to know the inflow paths, the speed the air moves along them, and the rate at which it is being exposed to the sea surface. True, there are almost enough data available to set up these computations purely observa- tionally ; however, one loses then any predictive relationships. Secondly, analytic formulation of any part of the foregoing model permits both its testing from existing observations and the possibility of going beyond these to suggest new ones and even extensions of the approach to other problems. Since this is one of the few atmospheric circulations where a flow-dependent exchange has been explicitly connected into a dynamic model, the hurricane menace may have the beneficial facet of serving as a prototype for studying even more complex geophysical flows. {in ) A dynamic model of the loiv-level rain area The inflow into a hurricane is confined mainly to low levels. Sub-cloud air is accelerated inward along spiral-shaped trajectories ; acceleration results from excess work done by pressure-gradient forces over frictional retardation. We shall consider the dynamics of the inflow layer in a natural co-ordinate frame- work (Fig. 66) as we did in the model of the trades (Section 6-C, pages 194-201). As already clear, this heat-engine model of the hurricane has much in common procedurally and philosophically with the trade- wind model. W^e consider as basic the hydrostatic pressure gradients along the trajectories and inquire how the smaller scales of motion, namely turbulent exchange and convection, produce these ; the dynamic part of the model relates the studied scale of motion to its energy sources in a steady state via the pressure field. Thus the hurricane model has the same limitations as did the trade model. The trajec- tories are assumed given ; how they got that w^ay, and the life-cycle aspect of the phenomenon, cannot as yet be treated. For the steady-state, stationary (or slow-moving) hurricane inflow layer, the tangential and normal equations of motion are as follows : du 8u \ dp I 8tsz 1 cp . - 1 drsz ,_,, -Tf- ^ u— = --^ + --^ ^ - —sm^ + -^ (52) at cs p cs p cz p cr p cz and ^+/„=_i|P=l|?lco.^. (53) R p en p cr The natural co-ordinate system is oriented as shown in Fig. 66, superimposed on a standard polar co-ordinate system, where the radial distance, r, increases outward. R is the radius of curvature of the trajectories ; ^ is the "inflow angle" 238 MALKus [chap. 4 or the angle at which they cross circles of constant r ; / is the Coriolis parameter, and the remaining symbols have the same meaning as in the trade case. Addi- tional assumptions made so far are as follows : (1) The pressure field is radially nearly symmetrical as observed, i.e., (1/r) dpl8ip<^ Bpjdr. Since all other quantities may vary from one trajectory to the next, and thence with azimuth angle ip, this choice does not restrict the applicability of the model to symmetrical circulations. (2) The vertical transport of s-momentum by the mean motion w dujdz {w is the vertical velocity) is small compared to dulds. This assumption is valid because the vertical motion is zero at the ground and dujdz is small through the inflow layer except quite close to the ground. Vertical momentum transport by convective-scale elements is included in the shearing stress term. (3) Lateral turbulent transport of s-momentum is neglected compared to vertical transport, with the hypothesis that momentum from the air inside the hurricane is abstracted by the ocean and not diffused laterally outward by small-scale eddies. This assumption is probably weak if single trajectories or only small segments of a storm are considered, but, due to the difficulty of prescribing vertical eddy transport accurately, is not critical at present. (4) The wind direction is nearly constant with height through the inflow layer so that the shearing stress term drmldz may be omitted. We shall now substitute r/cos j8 for R, the radius of trajectory curvature in (53). This is exact for the logarithmic spiral, which is closely followed by observed hurricane trajectories (Senn, Hiser and Bourret, 1957); here cosj3 = constant. It is nearly true when the inflow angle varies only slowly along the trajectory. When |8:^20° or less, R differs from r by only about 6% for any trajectory. Let equation (52) be multiplied by cos j8 and equation (53) by sin j8. The pressure-gradient force is then eliminated by combining these equations and we have, after dividing again by cos {3, — sm B + fu tan B — u-;— = — (54) r ^ '' ^ ds p dz ^ Upon averaging vertically through the inflow layer of height Sz we obtain du p hz — sin B + fu tan B — u^ r ds = Tso = CdpqUo" (55) introducing (17) for surface stress. The symbol ~ denotes vertical averaging. We neglect the slight difference between po and p, also between uq and u since the vertical shear above anemometer level is very weak in hurricane interiors, especially over water. The important assumption in going from the right side of equation (54) to that of (55) is that the shearing stress vanishes at the top of the inflow layer, which is near the level of maximum wind. Since we shall consider only quantities averaged through the inflow layer, the symbol ~ is henceforth omitted. SECT. 2] I.ARGK-SCALE INTERACTIONS 239 Dividing out u and utilizing the definition (Hilds= —{duj'dr) sin ^, we derive tlie following first-order differential equation for the velocity u along any trajectory as a function of radial distance, r. from the storm center du dr CD r sin jS 82 ^ (56) cos j8 Since it will prove more convenient to set boundary conditions on u^, the tangential velocity component, we may obtain an equation for it by using the fact that u = m^/cos ^. namely, duj, - + C(r) r -L (57) where C{r) = — colsin ^ 8z. This simple differential equation is readily solved for u^ when C{r) and a boundary condition are specified. The results of a mean hurricane-momentum budget by Palmen and Riehl (1957) suggest that, although cd and 8z may each vary by a factor of two under hurricane conditions (increasing inward), their ratio is constant within 20%. The ratio c^/Ss; was chosen as 1.36 x 10^^ cm~i for the analysis, which permits Cd to vary from 1.1 to 2.7 x 10"^ for a range of depth of the inflow layer from 750 m to 2 km. We make the simplest choice of ^ for our trajectory calculations, namely ^ = |8i = constant for the outer rain area r > 100 km, and decreasing from there linearly to zero at r = 25 km, the assumed radius of the eye wall. Integrating (57) when C(r) = constant, we have f u^r = ±-^{l-Cr) + Cie-cr_ (53) The constant of integration Ci is evaluated by choosing an outer radius ro where the relative vorticity vanishes, that is du^ldr+{u^lr) = 0. Since Ur = U4, tan jS, Ur also satisfies this relation at ro, which thus separates the region of horizontal convergence (rain area) from that of the surrounding horizontal divergence. Choice of ro is arbitrary, but the computed structure of the rain area is not sensitive to the choice so long as ro> 500 km, a figure suggested by moderate storm data. Under these conditions u^r = /[l-(7r-eC(ro-r)]. (59) For the inner rain area, namely rE X o t3 § T 3 o a '■+3 c3 H 02 .2 00 1 H-; 0) O 1— H P-> X o CO PL, CO c6 oo e X o I O ■73 c30t-.iocniO(>ir~ooqoeoTt*ccoc» 0000000050>05050iO>OSOi05 00 -H lo ^ -^ ec CO 00^(MfC005C<5«6o5CCo6cOOCOM «Of JO 05 's*! M* >0 »C »0 >0 M ?0 (I<) ^ C<1 oooo^ca>F-H?oi>^coo6 + + + + + + + + I i I I I I I I «0;0i©i0O00OiCi <35l>-H(MCC0_, |> 1© SO ©OOCD-'t'co-^OO MCDOO(M»OOM(N OOOOOOOOr-^-a0C*5>-HOO5l>«0ff'5i— Jo ■^0505-^O^MC-<*Tj< co(M';Doieci>ico5 2222'^^oooooooooin OpOOOOOO05Q0l>«0»0-.*C0 - ri r, r- 30 20 \ 10 V \ \ 5 - \ \ '^. \ 2 1 1 1 \ °\ \ 1 1 1 r (km) (C) Fig. 70. Some comparisons of moderate hurricane model with observed hurricanes. (After Malkus and Riehl, 1960, Figs. 2, 3 and 4.) (a) Upper graph is low-level wind-profile as function of radial distance from storm center in km. Solid line computed from model (Table XVII). Circles represent peak wind speed at each radius in hurricane Carrie, 1957, while the crosses denote the radial average for the same storm. Lower graph is surface pressure as function of radial distance from storm center in km. Solid line computed from model (Table XVII). Profile with crosses observed from hurricane Carrie, 1957; two isolated circles from Daisy, 1958. (b) Upper graph is mass inflow as function of radius in km. The quantity plotted is Ur'T in knots times degrees latitude. Solid line computed from model (Table XVII). Long-dashed curve with x's computed from wind observations in hurricane Carrie, 1957 ; short-dashed curve computed from wind observations in hurricane Daisy, 1958. Lower graph is radial velocity u^ (in knots, here positive inward) as function of radius in km. Solid curve computed from model, x-ed curve measured in Carrie and short- dashed curve measured in Daisy. (c) Surface-shearing stress (dynes/cm^) versus radius in km. Co-ordinates both on logarithmic scale. Solid curve from model moderate storm (Table XVII) ; dashed curve from momentum budgets calculated by Palmen and Riehl (1957) for mean hurricane data. are predicted i for the moderate storm; thus stresses exceeding 150 dynes/cm^ are inferrable for extreme hurricanes, up more than two orders of magnitude over normal. This is probably the highest wind stress experienced by the ocean and it is, fortunately, only exerted over small regions and times. 1 The highest marine stress to date measured by the aircraft method was 5 dyne/cm^ at 500 ft in a 50-knot wind of an extra-tropical storm near Bermuda. 244 MALKUS [chap. 4 (iv) Pressure field and the oceanic heat source As just demonstrated, pressures and pressure gradients of the moderate hurricane model are similar to those observed in storms of medium intensity such as Dais3% 1958. Outward of r = 90 km, where ps^ 996 mb, the pressure field may be maintained by a mixture of air with the characteristics of the average tropical atmosphere and varying amounts of sub-cloud air that have ascended in cumulus towers. The admixture of low-level air must increase inward so that, at r = 90 km, the vertical temperature distribution becomes entirely contrclled by the moist adiabatic ascent. Inward of r~30 km or ^s~966 mb. a sloping eye wall may possibly be called upon to compute the excessive pressure drop often encountered just inside the eye boundary. Neither of these solutions can account for the pressure drop of approximately 30 mb between r = 90 km and r = 30 km. This must be related to adiabatic ascent at increasing values of Q, and we are now prepared to relate the heating quantitatively to ocean input. We shall first do this following trajectories in a new "Lagrangian" approach to exchange and then in the old-fashioned areal manner, in terms of mass, heat and moisture budgets. We first take the radial pressure field of Table XVII and deduce from it the low-level potential temperature, d, and specific humidity, q, as a function of radius from Fig. 69. These are entered in Table XVIII. The condensation level (LCL) was determined (roughly) from film measurements of cloud base in Daisy and the surface relative humidity, rh, follows. Table XIX gives the trajectory distances and travel time for air particles by radial interval computed from the data of Table XVII, and the sensible and latent heat increments experienced by these particles in calories /gram. Table XVIII Thermodynamic Properties of the Moderate Hurricane Inflow Layer r, ps, 6, T, q, rh, LCL, km mb °A °C g/kg % m 90 996 299.4 26.0 18.5 84 400 70 991 299.9 26.0 19.1 86 300 50 982.4 300.5 26.0 20.1 90 200 30 966 30L8 26.0 21.8 97 80 One should expect that, following a particle, the heat transfer from the ocean will be governed only by the differences in temperature and vapor pressure between sea and air and wind speed. The subscript p will denote calculations performed with respect to the moving particle. Denoting sensible and latent transfer by Qsp and Qep, we may postulate that Qsp = ksr,u{TQ-Ta) (61) and Qep = kepU{eo-ea). (62) SECT. 2] LARGE-SCALE INTERACTIONS 245 Table XIX Oceanic Heat Source for Moderate Hurricane Radial 8s, 8t, cpSe, L8q, ^0-%. ksp^ kfp. interval, Urn min cal/g cal/g mb 10-9 cal g-i 10-9 cal g-1 km cm~l deg-l cm-l mb-l 70-90 79.7 30 0.11 0.32 11.5 4.6 3.5 50-70 125.4 42 0.17 0.56 10.5 4.5 4.2 30-50 294.0 88 0.37 1.01 8.5 4.2 4.0 30-90 0.65 1.89 8s denotes distance along each trajectory leg, from 90 to 70 km etc., computed from dynamic model. 8t is time needed to traverse leg, also computed from dynamic model. 86 is the increment in potential temperature in 8s and 8t. 8q is the increment in specific humidity in 8s and 8t. Here e is vapor pressure and ksp and kep are the Lagrangian coefficients of turbulent exchange. Now Qsp St = Cp 86; Qep 8t = L 8q so that Cp8dl8s{To-Ta) = ksp (63) and L 8ql8s{eo-ea) = kep. (64) The ocean temperature is taken as 29.0°C as in the Daisy case (Fig. 68) so that To — Ta = ^.0°C and eo — Ca follows from Table XVIII and psychrometric tables. The resulting coefficients (last two columns in Table XIX) are constant within computational limits, hence the air trajectory computed from purely dynamic considerations is consistent with the independent constraints of equations (61) and (62). The trajectory takes a course such that physically impossible demands are not placed upon the thermodynamic interaction between sea and air. This Lagrangian approach to exchange, viewing the process by following air trajectories, may prove useful in other than hurricane situations ; it is being developed by Riehl and his students in laboratory treatments of extra-tropical cyclones. {v) Heat-energy budget of the inflow layer Up to now, calculations have followed a particle on its spiral path. Now heat flux and energy exchange between air and sea will be examined spatially. The motive is to compare the heat flux per cm^ of the ocean surface as determined here and as estimated from the ordinary transfer formulas and with the normal fluxes in the tropics. For this purpose, it will be necessary to construct a heat budget, as in the trade-wind studies. 246 MALKUS [chap. 4 Now equations (27a) and (28a) are used in the form Qs = Cp \\ pdu-nda -^ Cpy M^O (27d) and Qe = {{ pLqu-ndG = 2 Jf.Lg, (28d) where a steady state is assumed in the inflow layer ; radiation terms are negli- gible over these short times, and condensation conversion only affects the air vertically exported from the box of surface area a. As in the trades case, Qs and Qe are for the whole oceanic area below the air considered and must be divided by this area to compare with the transfer formulas. On the right sides of the expressions actually used for the calculation, M„ is the mass flux in g/sec through each face of the box considered. Radial and vertical fluxes of sensible and latent heat may be computed from the data in Tables XVII and XVIII. These fluxes and the heat sources required for balance are shown in Table XX and Fig. 71, These diagrams were derived as follows : the mass flux through each vertical face (at r = 90, 70, 50 and 30 km) was obtained from M = Ur- r'lrr 8plg, where Ur-r is taken from the dynamic calculation (Table XVII, column 5). The mass flux through the top face, AM, is the difference between the horizontal fluxes through the successive vertical faces. Mass leaving through the top of each box was assumed to go out with the mean property of the air in the box. Heat sources were calculated as residuals to meet continuity. Over the area within the core, the source is 5.38 x 10^2 cal/ sec latent heat of water vapor and 1.60 x lO^^ cal/sec sensible heat, a total of 6.98 X 1012 cal/sec. It is interesting to note that this is nearly 1% (Table I) of the entire heat energy exported from the tropics and 3|% of the poleward transport of the Atlantic ocean-current systems (Fig. 12)! It corresponds to a total heat increment to the air of 2.50 cal/g, within computational error of the result of Table XIX. Table XXI compares the sensible and latent heat fluxes in cal/cm^ per day from the budget method with those computed from transfer formulas (20) and (21). Agreement is extremely good. The most important result of Table XXI is that the ordinary formulas and coefficients are easily adequate to meet the exchange requirements of the moderate hurricane. No special demand is created during transition from trade-wind speeds of 5-7 m/sec to hurricane velocities for an increase of the transport efficiency of the energy spectrum near the ground. No impossible or difficult restriction is made when it is postulated that the lowering of surface pressure in hurricanes depends upon an "extra" oceanic heat source in a storm's interior, although it would be extremely interesting to pursue further the effects upon the ocean-surface layers of the abstraction of this much heat during hurricane passage. Comparison with Table II shows that the actual transports in the hurricane are very large indeed compared to the trades. Sensible heat pick-up is 720 cal/ cm2 per day, an increase of sixty times over the trades ; the increase in Tq — Ta SECT. 2] lah(;k-scai.k interactions 247 Tablk XX Heat and Moisture Sources in Moderate Hurricane by Energy Budget Method A. Sensible heat flux r. e'. M, AM, Mcpd', Cp6', AMcpd' km °A g/sec g/sec cal/sec cal/g cal/sec X 10-12 X 10-12 X 10-12 X 10-12 90 0 7.40 2.8 0 0.060 0.170 70 0.5 4.65 2.5 0.556 0.191 0.478 50 1.1 2.15 1.9 0.558 0.418 0.795 30 2.4 0.28 0.161 0' denotes ^ — 299.4°A and the bar indicates area averaging. B. Latent heat flux 1', km q', g/kg Lq', cal/g Lq'M, cal/sec X 10-12 Lq\ cal/g AMLq' cal/sec X 10-12 90 0 0 0 0.172 0.482 70 0.6 0.353 1.64 0.647 1.62 50 1.6 0.941 2.02 1.44 2.74 30 3.3 1.94 0.544 q' denotes q— 18.5 g/kg and the bar indicates area averaging. 0 795 0478 1 0 170 1 1 1 _ -0.161 - -0.558 - -0.556 ^ 0,398 t 0 480 Residual 0 726 0 ' 5 0 ' 7 5 1 9 Sensible Heat Flux (lo'^cal/sec) Mkm) Latent Heat Flux ( lo'^cal/sec) Fig. 71. Sensible (left) and latent heat (right) budgets for inflow layer of model moderate storm. (After Malkus and Riehl, 1960, Figs. 5 and 6.) Heat fluxes (1012 cal/sec) by boxes whose sides are cylinders concentric with storm center; Ar for each box is 20 km, height 1.1 km or 100 mb. Calculation performed as described in text and Tables XVII-XX. Oceanic sources computed as residuals to meet continuity. 248 MAIiKUS [chap. 4 o w c o < o3 O bD o s o ■1^ O- -0 o o 73 3^3 -« S o ^ fl 1— I i;s 0) > o c3 <1^ , U) o Tj< ■* (M O eo CO I> t^ O CD O O l> o t-; C>1 ^ CO CO ic o o lO CO 1 o i t- lO SECT. 2] LARGE-SCALE INTERACTIONS 249 from about 0.5 to 3.0^C accounts for only a factor of six ; the remainder is therefore due to the high wind speed. The latent heat pick-up of 2420 cal/cm^ per day is an enhancement by a factor of seven over the trades, and is practi- cally entirely due to high wind, since qo-Qa is little altered from normal (compare Tables II, XVI and XXI). Altogether, Qs + Qe, or the heat energy abstracted from unit sea surface, is about 3000 cal/cm^ per day or up by an order of magnitude in the moderate hurricane. Since the surface stress times wind speed, a measure of kinetic energy dissipation, is up by three orders of magnitude, it is suggested that in the hurricane the atmosphere has created an abnormally efficient heat engine. This point was pursued further in the original paper by Malkus and Riehl (1960). In ordinary tropical disturbances, we also noted (Table XVI) that heat exchange is enhanced above normal, and that the increase in Qs was per- centually larger. However, the increase of To — Ta in this type of situation was related to evaporation of falling rain and thunderstorm downdrafts rather than to adiabatic expansion during horizontal motion toward lower pressure. These mechanisms for lowering the air temperature need not be discounted in hurricane circulations, especially in the outskirts and formation stage, since all hurricanes form from pre-existing disturbances of the kinds described pre- viously. Clearly, however, the foregoing has demonstrated that, although the enhanced boundary input is essential, it is not the difficult constraint to meet in deepening a tropical storm to the moderate hurricane stage ; the critical restriction must be sought in the concentrated release of the latent heat in the middle and high troposphere. Nor can the enhanced pick-up be regarded as the cause of hurricane development, since it is largely the result of the high winds ; it is merely a vital part of the machinery. (vi) Maximum kinetic energy production, the Bowen ratio, and conditions for extreme storms We may use the hurricane as a prototype example of a geophysical thermal circulation by examining some of its vital relationships in the framework of the new theoretical approach to convection by W. V. R. Malkus and Veronis (1958). These authors showed that when a hierarchy of solutions exists to the hydrodynamic equations of motion, that one will be realized by the system which, under the operating constraints, maximizes the release of potential energy or the production of kinetic energy. This so-called "relative stability criterion" has been worked out entirely formally in simple examples of convec- tion, where it has also been tested by corresponding precise laboratory experi- ments. A crude application of this approach by Malkus and Riehl (1960) suggests strongly that the hurricane is aware of this principle and is a fruitful ground for creating the bridge between idealized cases, which can be treated rigorously, and the vast complexity of real geophysical, thermally-maintained flows. The solutions to the dynamic equations for the hurricane inflow layer contain a parameter C{r) = — Ci>/sin ^ hz, variations in which gave rise to an infinite 250 MALKUS [chap. 4 family of logarithmic spiral trajectories for each choice of the boundary condi- tion, ro. Using the language of the new approach, we suggest that the thermal constraints, which in nature operate on the system in addition to the dynamic laws, impose a choice between or a limit upon the range of dynamically possible trajectories, so that most or even all of these may be prevented from occurring in a real situation. The thermal constraints operate through the surface- pressure gradient along the trajectory, s, in the storm core ; the realizable dpics is restricted by the possible heat transfer at the air-sea boundary and the thermodynamics of condensation heat release in the vertical. As in the work in Section 6-C (pages 190-201) on the trades, we introduce thermal constraints by means of the first law of thermodynamics. There we wrote a standard meteorological form of this law, namely, where H is the rate of sensible heat addition per unit mass. The equation says that sensible heat added may be used to increase the enthalpy (CpT) or to do mechanical work by means of pressure gradients. In the steady-state hurricane situation, for particles moving horizontally and isothermally toward the center, (40) ma}'^ be written The well-known kinetic energy equation is obtained by multiplying equation (52) by u, du dK u dp u drsz ,„„, dt dt p ds p oz where K is kinetic energy per unit mass. The term —{ulp)dplds is the rate of production of kinetic energy by pressure forces. It is now apparent that kinetic energy is produced from the oceanic heat source at a maximum rate during isothermal and horizontal motion, since for a prescribed Qsp, dpfds is maximal when dTjdt is zero. From equations (61) and (40a), the pressure-gradient force may be related explicitly to the boundary-heat transfers Qsp = ksp{To-Ta)u = --^ (66) p cs so that --^ = ksp{To-Ta). (67) The pressure gradient along the trajectory is thus limited by the input rate of sensible heat from sea to air. A second thermal constraint must be met by the system. Since hydrostatic equilibrium is required, pressures exerted by and on sub-cloud air must be SECT. 2] LARGE-SCALE INTERACTIONS 251 consistent with the density of the air column above them. If the lapse rate is essentially wet adiabatic, we saw that the pressure fall was linearly related to the increased heat content of the rising air, or where y^lS mb per cal/g. Furthermore, ^ = U^^=Qsp + Qep. (69) Combining (69) and (68) and substituting from (62), we have --^ = ^-{Qsp+Qep) ='^[ksp{To-Ta) + kep{eo-ea)]. (70) p OS U p p To be consistent with (67), hp{To-Ta) = '^[ksp{To-Ta) + kep{eo-ea)] (71) P or -AQsp = Qep, (71a) and a relation between sensible and latent heat pick-up, or the Bowen ratio, is prescribed. When y is placed in proper units, y/p;i;0.26 and Qsp = 0.35Qep, in excellent agreement with the results of Table XXI. We see that not only must exchanges be augmented to maintain a hurricane but also the enhanced Bowen ratio is an essential feature of its machinery! This precision engine requires many finely fitted parts to operate, and as we look at it more closely its rarity in nature becomes less surprising. The foregoing analysis suggested an inverted approach to the hurricane problem. Rather than asking what exchange is required to maintain an observa- tionally realistic dynamic situation, as previously, we now reframe the question: Let the product ksp{TQ-Ta) be fixed, and let this prescription choose between the dynamically possible solutions to (57), using the relative stabihty criterion of W. V. R. Malkus and Veronis (1958). This was done by integrating (65) vertically through the inflow layer, and substituting (17) for to so that the kinetic energy equation becomes dK U 8p CD ^ ,r.tr s -_ = — ^- u^ (65a) at p OS bz or has the form, using (40a) and (61), ^ = Au-Bu\ (65b) at 252 MALKTjs [chap. 4 Adhere A and B are held fixed. We now substitute u from the sohition to the differential equation (57) as a function of C{r) and find (after some labor) the value of sin /3 which maximizes dKjdt. In the case where ksp{To-Ta) was fixed as in Table XIX. we get the now obvious result that /S must be 20"" I The method, however, permitted a rapid extension to exchange and release efficiency requirements for intense and record storms, which will only be briefly sum- marized by a physical discussion here. As we mentioned earlier, the dynamic calculation gave an intense storm with an inflow angle of about 2o\ Higher inflow angles permit stronger central winds due to reduction of trajectory distance and thereby of frictional dissipa- tion of kinetic energy. The larger the inflow angle, the closer the velocity profile with r approaches that of the constant angular momentum vortex which, without friction, would arise from any inflow angle. Ver\- high inflow angles are prevented in real situations due to thermodynamic restrictions on the reahzable pressure gradients. AATien the trajectory distance is reduced too much, the air cannot pick up and release by condensation enough extra heat energy to achieve the required gradient of Q and thus of surface pressure. To achieve record storms, with 200-knot winds, inflow angles of 25'' and central pressure at or below 900 mb. the analysis shows that .4. or core pressure gradient cpjcs, must be held at a value about twice that for the moderate storm studied. Since values of To— Ta of 5°-6T are probably excessive, it appears marginal whether normal heat-energy exchange mechanisms are adequate ; it may prove necessary to call upon slanting eye walls to sustain the pressure fields in these extreme situations. In any case, maximum development is almost surely restricted by boundary exchange since, as pursued further in the paper by Malkus and Riehl (1960), dissipation increases as the wind speed cubed, and input and condensation only linearly. It is interesting to note that their extreme storm released condensation energy more than twice as efficiently as did the moderate storm. Attempts are now being made to apply the energetic principles of relative stability to cumu- lonimbus convection in hurricanes, the other facet of pressure-field maintenance. It is hoped to relate thereby the sizes of the convection elements to large-scale dynamics, although the degree of rigor achievable is not yet clear ; the problem hes, as it did in the exchange aspect, in specification of the constraints upon the system. We have now completed our presentation of the efforts to relate air-sea exchange to the dynamics of large-scale circulations in terms of models which are at least, to a degree, analytic and begin from the basic laws governing fluid motions. These endeavors date only from the 1950's. They are plainly in the rudimentary beginning stages and are conflned to the relatively simplest and best-documented aspects of planetary flows, such as the lower trades and the mature hurricane. It is nevertheless significant that such models have been attempted, and their inquiries framed in such a way that the new developments in the fundamental mechanics of turbulent heated fluids may be used, if crudely, to guide the question-asking of those facing real geophysical problems. SECT. 2] LARt;E-S(ALE INTERACTIONS 253 Not until dynamics, energy relations and thermodynamics are tied together in some such framework will it be possible to gain further insight into the physics behind exchange climatology, in which even the most exact and massive descriptive approaches soon reach the stage of diminishing returns. Xor will it be possible to bridge the gap between the real planetary fluids, dominated by energv transfers, and driven and braked bv exchange, and the world of the theoretician, who in meteorology and oceanography has been largeh' con- cerned with purely dynamic problems. 8. Exchange Fluctuations in Mid-Latitudes and Long-Period Interaction Anomalies For the mid-latitude atmosphere, its contact with the sea surface is primarily a braking mechanism by which the momentum of the winds is transferred to the upper ocean layers, and their kinetic energy dissipated. An exception is found at continental east coasts in winter. During polar air outbreaks, sensible heat flux from sea to air reaches its all time high : arctic air-masses are modified by the resulting convection and the birth of unstable cyclones is instigated or materially aided by the resulting convergent mass flows. Actually, the most important exchange affecting mid-latitude atmospheric circulations has already been discussed, namely evaporation from the tropical oceans, which indirectly provides their maintenance. The dynamics of mid- latitude flows have been more immediately concerned with the structure of the westerly jet streams, the uneven importation of energy and momentum across the subtropical ridge, and the release of the stored potential energy by instability on the synoptic scale — the dominant feature of the motions. The products of this dynamic instability, the travelling cyclones and anticyclones and their arrangement, control the direction and speed of the air flow and, to a large extent, the air-sea property difiBrences. Thus they bring about exchange fluctuations which may even modify their own stability conditions and the marine environment to be met by their successors. The climatology of middle- latitude exchange cannot be interpreted without reference to the three- dimensional structure of these disturbances : conversely, it is becoming ap- parent that the modifying effects of even direct local exchange cannot remain ignored in treating the dynamics of extra-tropical flows and their fluctuations, from the synoptic up to the geological time scale. This whole frontier is a vast jungle of nonlinear complexity, into the tangles of which only a few brave souls have begun to beat paths. We shall confine ourselves here to several explicit studies which have attempted to frame quantitatively pursuable questions ; their shortcomings should not be judged too harshly in view of the scope and difiiculty of the problems. A. Synoptic-Scale Exchange Variations off East Coasts in Wiyiter In re-examining the climatological patterns (see especially Figs. 7, 8, 11, 21, 39-43) we see that exchanges are maximal off continental east coasts in winter. 254 [chap. 4 In the regions of the warm waters of the Gulf Stream and Kuroshio, evapora- tion is as high as in the tropics and sensible heat fiux becomes the dominant term in the energy budget of both sea and air ! Here the ocean gives back the warmth it has acquired under tropical and summer suns to wintry air-masses pouring over it off frozen ground. As implied, these fluxes are by no means steady, but occur spasmodically when the large-scale flow pattern is such that strong northwesterly winds bring the cold dry air rapidly over the warm coastal waters, usually in the outbursts following cold front passage. Manabe (1957) has studied the fluxes in a strong outbreak over the Japan Sea and contrasted it with the average situation there in winter. a. A cold-air outbreak over the Japan Sea during the winter monsoon Manabe selected for study a particularly intense cold air outbreak which occurred between 20 December, 1954, and 3 January, 1955. During this time, the average sea-air temperature difference exceeded 10 C, so that it will be especially interesting to contrast the energy budgets and exchange relation- ships with the tropical situations described earlier, particularly as the methods of investigation are very similar. Fig. 72 shows that, like the Caribbean ellipse, 130° 140° -40° Fig. 72. Framework for budget study of Japan Sea region, showing meteorological observa- tion stations. Superposed are the mean streamlines of air arriving at Wajima (Jaj)an) at each standard level up to 700 mb for period studied, namely 20 December, 1954, to 3 January, 1955. (After Manabe, 1957, Fig. 1.) SECT. 2] LARGE-SCALE INTERACTIONS 255 the Japan Sea is an enclosed basin, nearly surrounded by surface and upper air stations, and regularly combed by vessels of the Japanese fisheries. The main physical features of the air-flow and its modification in its passage across the sea, from Russia and Korea, to the west coast of Japan, are shown in Figs. 73-75. As in the trade case, the inversion base rises markedly from the upstream to the downstream end of the section, in correlation with the develop- ment of convection which has become intense off the west coast of Japan. Also similar to the trades, convective clouds are most vigorous where the wind speed and sea-air temperature differences are largest. As seen in Fig. 75, the down- stream increase in potential temperature suggests a large oceanic heat source ; however, a careful budget study including all terms in the heat balance is needed to separate this input from precipitation and compressional warming. Manabe performed a heat and moisture budget for a box the bottom area of which was the Japan Sea polygon shown in Fig. 72 extending vertically to 500 mb. Its surface area is 0.76 x lO^^ cm^, or about one-third of the Caribbean ellipse of Colon. Here equations (27b) and (28b) had to be formulated in a manner to include the important effects of time dependence and were actually set up for the computation in the following forms : LP + Qs + Ra = dhldt = lti\ \ Cn{CpT + Agz) dl{dplg) + Aj.s.[wtpt{cpT + Agz)t - Wbpb{CpT + Agz)b] (27e) and Qe-LP = dLqIdt = 2ti CnLq dl{dplg) + Aj.s.[wt'pt{Lq)t-Wbpb{Lq)b], (28e) where the bar denotes time averaging over the period, the subscript J.S. denotes the Japan Sea polygon, and the symbol 2« denotes time integration in a series of finite steps. The subscript t denotes top and the subscript b denotes bottom. The following assumptions were made : (i) Local time-dependent terms, d/dt, in both potential temperature and mixing ratio were neglected compared to advective changes and sources and sinks. (ii) Radiation, precipitation and vertical advection were computed entirely from soundings and data averaged over the whole fifteen day period, so that fluctuations and vertical eddy transports were left out. Turbulent fluxes through the 500 mb surface were set to zero on the grounds that the inversion base was always below this. (iii) The velocities in the horizontal advective terms were evaluated from the geostrophic approximation, using the contour heights of pressure surfaces with a grid containing six divisions on either side of the Japan Sea. At the surface, the horizontal advection was reduced to one-half the geostrophic in rough correspondence to the observed ratio of actual to geostrophic winds at 256 MALKUS [chap. 4 Surf. Isotachs (m/sec) 03Z 20 Dec. 1954 - I5Z 3 Jan. 1955 To iX) 03Z 20 Dec. 1954 - I5Z 3 Jan. 1955 (a) ;b) 20 Dec. 1954 5Z 3 Jan. 1955 (c) (d) Fig. 73. Summary of low-level conditions prevailing in the Japan Sea region during the cold outburst studied. (After Manabe, 1957, Figs. 3, 7 and 8.) (a) Surface isotachs (m/sec). (b) Sea-surface isotherms (°C) published by the Fishery Institute of the Tokai region. (c) Isopleths of sea minus air temperature difference (°C). (d) Isopleths of sea minus air specific humidity difference (g/kg). The sea-surface values for qg obtained as the saturation specific humidity at the temperature of the water. SKCT. 2] LARGE-SCALE INTERACTIONS 257 low-lying coastal stations. The author estimated that isobaric curvature should cause the geostrophic approximation to overestimate advection by about 13%, which leads roughly to a 20% overestimate in the net source. The time integra- tion was made in this term by computing it at twice daily intervals and averaging. (iv) The region was subdivided vertically, using Fig. 74 as a guide, into three layers : surface to 900 mb, 900-700 mb and 700-500 mb. mb 850 Inversion Base Vladivostok FukuokQ Yonogo Wajimo Akita Sapporo Wakkanoi (a) Fukuoka Yonago Wajimo (b) Akita Sapporo Fig. 74. Some vertical properties of the atmosphere during the cold air outbreak over the Japan Sea. (After Manabe, 1957, Fig. 5.) (a) Distribution of the height of the inversion base over both sides of the Japan Sea (see Fig. 72). (b) Cloud distribution along the west coast of the Japanese islands. Cu stands for cumulus, Ac for altocumulus and Ci for cirrus. Approximate coverage indicated by the fractions where obtainable. The basic steps in both budgets were to determine the total source term as the difference between horizontal and vertical advection, then to evaluate radiation and precipitation independently, finding the ocean fluxes as residuals. The vertical advection term was obtained by integrating the actual wind vectors at the surrounding observation stations throughout the period and computing the average divergence at the various levels ; thus, from continuity, the vertical profile of the mean vertical velocity over the Japan Sea during the period was obtained. The results are shown in Fig. 76. Since the wind data began to give out above 700 mb, it is not clear how the average advective transport through the 500-mb surface was obtained, although Fig. 76 and common meteorological experience in similar situations suggest subsidence of 258 MALKUS [chap. 4 the order of 5 mm/sec (500 m/day). In unfortunate contrast to the Caribbean study of Colon, a complete mass balance throughout the troposphere was not possible here, so there can be no assurance that the contribution to heat and moisture divergence from the mean ageostrophic mass flow was properly taken into account. Our own brief repetition of Manabe's calculations show that Mean Distribution of Mixing Ratio (g/kg) Surf. Kimpo mb P 200 Surf, ■ — -^ ^-360, ■7::: —340 330 ^.....-320 30( 1 ,"' S! 0 ?80- ^ / ^ ^^'' / / ! 270' / p mb 500 Surf, Kimpo mb 500 ""■"■-"•^ X \ 1.0 / \ \ \ \ N \ V V s \ ^v \ < ^■*»— .— — \ x^^ "^•^ \ ^^1.0 \ 'N. V ^\^ \ ""2.0^ S ^ "^■»» 1 — 1 ~3.0 / ^-> K.^^ > ' ^ N. '^. 850- — -1^ Surf Fukuoka Yonogo Wajima Fukuoka Yonogo Wajima Akito Sopporo Wokkonoi (03Z 20 Dec. 1954- I5Z 3 Jan. 1955) Akito Sapporo Wokkonoi (a) (b) Fig. 75. Vertical cross-sections of potential temperature and specific humidity along both coasts of the Japan Sea during the polar outbreak studied. In each case top section is before passage of air across sea, bottom section after. (After Manabe, 1957, Fig. 6.) (a) Cross -sections of potential temperature. Vertical co-ordinate pressure in mb. Potential temperature isopleths labelled in °A. Configuration along Russian-Korean side, top ; Japanese side, bottom. (b) Cross- sections of specific humidity. Vertical co-ordinate pressure in mb. Specific humidity isopleths labelled in g/kg. Configuration along Russian-Korean side, top ; Japanese side, bottom. SECT. 2] LARGE-SCALE INTERACTIONS 259 neglecting a mean mass flow corresponding to the estimated net subsidence of 5 mm/sec at 500 mb would lead to a 5% underestimate of the final Qe and a 15% overestimate in Qs. This is because sinking motion and outflow brings drier air from aloft than that exported horizontally, while, in sensible heat, it imports potentially warmer air through the top of the box than is exported at the sides. Thus the net oceanic sensible heat source may be overestimated by as much as one-third when the additive error of neglecting isobaric curvature is also included. 03Z 20 Dec. 1954- I5Z 3 Jan. 1955 Fig. 76. Vertical profile of mean divergence (10~6 sec"l) on right. Computed by averaging actual wind vectors at surrounding observation stations throughout outburst period . On the left is shown thevertical velocity profile (mm/sec) computed from the divergehcerfield and the equation of continuity. Wind data were too sparse to extend the computation above 700 mb. Note the convergence extending through a major portion of the cloud layer. (After Manabe, 1957, Fig. 12.) The calculation of long-wave radiation was performed using the radiation chart of Yamamoto, an improved version of the Elsasser chart. The distribu- tion of mean cloudiness in Fig, 74b, the average radiosonde observation and the water-surface temperature near the coast of Japan were used in the calcula- tion, of which the result is shown in Fig. 77, The radiational heating of the lowest air layers is chiefly due to the large sea-air temperature difference, a result quite in contrast to the tropics where a similar calculation by Riehl et al. (1951) showed that even the sub-cloud layer suffers a net radiational cooling of about 0.5°C per day. Manabe suggests that this radiational warming at low levels may have played a significant role in maintaining the vigorous convection over the sea. The part of Fig. 77 showing the mean vertical distribution of heating due to 260 [chap. 4 direct absorjDtion of insolation was obtained from a nomogram of Yamamoto and Onishi (1952). As cloud albedos, 78% was adopted for cumuliform and 65% for the stratus type. The work of Hewson (1943) was used to estimate transmission inside clouds. As it turns out in this case, the radiation term is a very small one in the final balance compared to the advective and Qs terms mb 500 600 700 800 Surf, I -0.1 mb 500 800 900 Solar Radiation Long-Wove Radiation Cooling or Heating due to Long -Wove Rodiotion over the Sea near Wajima Cloud layer -"^^T^- -0.5 0.5 col /g day ID °C/doy (a) Fig. 77. Results of radiation computations for mean air structure prevailing during the cold air outbreak. Vertical co-ordinate pressure in mb. (After Manabe, 1957, Figs. 13 and 14.) (a) Left : Mean vertical distribution of heating due to absorption of incoming solar radiation. Right : Mean vertical distribution of heating and cooling due to long-wave radiation. Unit : cal per gram per day. (b) Long-wave radiation computation repeated using thinner air layers. Unit: cooling or heating of the air in °C per day. which dominate. The figure for average net radiational cooling for the layer of 93 cal/cm2 per day, or about 0.8°C per day, is in good agreement with the results of other studies. Likewise the precipitation term, although uncertain, is very small. During such outbreaks of cold continental air, commonly only showers or snow flurries occur. Here the precipitation was estimated from the coastal stations and many islands to be about 1.3 mm per day, corresponding to an LP warming of 77 cal/cm2 per day for the layer. The final results of the budget study are summarized in Fig. 78 and Table XXII ; in the latter, they are compared with SECT. 2] LARGE-SCALE INTERACTIONS 261 Mean Net Flu« Divergence of Water Vopor ond Heal over Japan Sea (03Z 20 Dec. 1954- I5Z 3 Jan. 1955) cal/g, day ->- IO"'g/g of oir, day Fig. 78. Computation of net required heat and moisture source as a function of height (pressure) for Japan Sea polygon. (After Manabe, 1957, Fig. 16.) SoUd lines show the mean vertical distribution of horizontal net flux divergence of enthalpy {CpT) and latent heat in water vapor. Surface values reduced to one-half that deduced from geostrophic wind. Dashed lines show mean vertical distribution of individual increase of sensible heat energy, h, and specific humidity, q. Heat source abscissa (upper) in cal/g per day ; moisture source abscissa (lower) in 10"^ g/g of air per day. Turbulent -convective fluxes through 500-mb surface assumed zero. Air trajectories assumed parallel to 6 isopleths at 500 mb, so dh/8t there due to radiation loss only; curve continued above 700 mb to this point by linear extrapolation. Table XXII Results of Atmospheric Energy-Budget Study for Cold Air Outburst in Comparison with Two-Months Winter Average (after Manabe) Nature of source or sink Outburst Dec. 20, 1954^Jan. 3, 1955, cal/cm2 per day Average for Jan .-Feb. 1955, cal/cm^ per day Qs Qe Ra LP 1030 450 (7.5 mm) -93 77 (1.3 mm) 555 340 (5.6 mm) -83 118 (2.0 mm) 262 [chap. 4 I*— '-*0 v^ Mean daily Precip. 03Z 20 Dec. 1954 -I5Z 3 Jon. 1955 (a) (b) (c) Fig. 79. Precipitation distribution and transfer formula results for Japan Sea cold air outburst situation. (After Manabe, 1957, Figs. 15, 17 and 19.) (a) Isopleths of mean daily precipitation (mm/day) during the period, constructed from coastal and island stations. Value in upper left-hand corner with bar is area average for period. (b) Isopleths of evaporation (mm/day) for the period computed from transfer formula using Jacobs' (1951a) coefficient, which is about 25% larger than that in equation (21). Figure with bar at top left-hand corner is area average for period. (c) Isopleths of Bowen ratio for the period constructed from equation (7) and Figs. 73c and d. Area average value at top left-hand corner is very close to unity. SECT. 2] LABGK-SCALE INTERACTIONS 263 longer period winter-time average figures obtained by the same author (Manabe, 1958) by identical methods for the two months of January and February, 1955. It is seen that both sensible and latent heat transfers from sea to air are enhanced during the outburst, particularly the former, which is then nearly double the average winter figure. The budget results for several discrete periods are compared with computa- tions from the transfer formulas in Fig. 79 and Table XXIII. Periods were chosen in which no major cyclones or heavy precipitation took place in the region, to minimize uncertainty in the LP term. Manabe (1957) used the coefficients of Jacobs in the transfer formulas. We have repeated the computation in the last column in Table XXIII using equation (21) instead; the latter, with its lower coefficient, plainly gives better agreement with the budget study. The discrepancy in Qe should not be viewed with great alarm, since the wind and low-level air data for the transfer formulas were taken from coastal and island stations instead of ships. First, the anemo- meter level is usually higher than that of ships' deck, and, secondly, it is likely that some unrepresentative winds and perhaps also temperatures have been introduced by the land effect. One of the main conclusions drawn by Manabe from his work concerns the high value (2.3) of the ratio oiQsjQe obtained from the budget study compared to that predicted by the Bowen ratio, y = {CplL)[{To — Ta)l{qo — Qa)] (equation 7, page 105), which, as seen in Fig. 79c, averages more like unity. In view of the high sea-air temperature difference, he concluded that the dominant role of convective heat flow relative to shear flow invalidated the basic assumption of the transfer formulas that the exchange process and eddy transfer coefficients were identical for heat, moisture and momentum. While this is quite likely to be the case in a polar air outbreak over the sea, and other authors have found sensible heat fluxes around 1000 cal/cm^ per day in similar situations, we believe that the uncertainty in the advective terms in the sensible heat budget may have led to a sizable exaggeration of Qs in this study, and that, therefore, the point is not yet settled beyond doubt. Repetition of these studies with modern radio-wind soundings is much to be desired. In an attempt to place a sounder footing beneath his deduced transfers, Manabe also made an oceanic heat budget for the Japan Sea, using equation (1) in a manner similar to that of Colon. In so doing, he used for that sea an earlier annual budget of Miyazaki (1949), introducing a computation for the storage term S from oceanographic data published by the Japan Fishing Research Laboratory (Figs. 80 and 81). The result is summarized in Table XXIV. The storage is in excellent agreement with the average of the winter results of Patullo (1957) for the region. The sum Qe + Qs is almost exactly the same as that deduced from the average winter atmospheric budget in Table XXIII, although the time periods considered are somewhat different. Therefore, regardless of the uncertainties involved, it is clear that very large transfers from sea to air of both sensible heat and water vapor occur when cold air-masses pour off continents over the nearby warm ocean currents in winter. 264 MALKUS [chap. 4 d CO i> o •-^ o o H i2 o H t3 C/2 =4-1 o ill o OS "C eS ft s o o OhTJ 0) — 3 > S s o ^ s I s I Ph S 05 2 w ^ r- o ^ -I r-^ rn -H O ^ I -- ^ ^ rt >-j - ^ c!, '^S c lO 05 lO .-H •-5 05 CO ^ C 03 05 fC .- 00 1_ d 1. C 0) c 03 fi cS ^ o ^ O 5^1 o 1— 1 P=^ SECT. 2] LARGE-SCALE INTERACTIONS 265 Water Temperature (°C) Toteishizaki (1932-1933) Aug. Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug. Fig. 80. Seasonal march of sea-temperature (°C) distribution with depth (m) at location off northwest coast of Japan. (After Manabe, 1957, Fig. 9.) Note that during winter the water temperature is nearly constant with depth down to about 100 m, suggesting strong convection in the upper ocean layers. 0 1 2 3 4 5-6 ■7-5 Dec 7 °C deptti (m) / J / / / 1 i Fig. 81. Vertical distribution of ocean-temperature change (°C) between 5 December and 5 March averaged for the years 1935-1940 for the Japan Sea. Based on data published by the Japan Fishing Research Laboratory and used by Manabe to calculate storage term in the ocean's heat budget. (After Manabe, 1958, Fig. 7.) 266 MALKtJS Table XXIV [chap. 4 Summary of Oceanic Heat Budget for Japan Sea in Winter (after Manabe and Miyazaki) Unit : cal/cm2 per day R Qt 10 102 768 880 This leads to a rapid transformation of the continental air-masses, which begin to acquire maritime characteristics in a day or less ; the time of travel across the Japan Sea was about 24 h. It is not coincidental that this area (see Fig. 82) and the corresponding region off the American Coast are favored for the formation and deepening of wave cyclones upon the cold fronts which so frequently become stalled there. A clue 105° (a) 115" 120° 125° 130° 135° 140° 145° Fig. 82. Figure demonstrating the propensity for extra-tropical cyclones to form in the (a) Geographical frequency. The value of an isopleth at any point represents the number of (identifiable) cyclones that formed within a radius of 2.5 degrees latitude from that point in the months October through April, 1932-37. SECT. 2] LAKGE -SCALE INTERACTIONS 267 to the connection is found in Fig. 76. Its remarkable feature, which recurred in all the discrete periods reported in Table XXIII, is the convergence throughout most of the cloud layer. This is different from the uninterrupted divergence found in cases of cold air outbursts over continents (Palmen and Newton, 1951) and suggests that the cumulus convection caused by the oceanic heating markedly affects the synoptic-scale convergence field. This point has been pursued in a fascinating paper by Winston (1955), who finds the convectively- imposed ascent a key feature of a deep cyclonic development in the Gulf of Alaska. There a major wave trough began to develop in the westerlies due to planetary-scale dynamic readjustments. This induced an outflow of continental air over the warm coastal waters, which in turn brought about the violent deepening of a major cyclone ; the latter could not be predicted by the standard dynamic methods of vorticity advection. Since these explosive developments have subsequent pronounced effects upon the planetary wave patterns down- stream, an exciting and fruitful area of research has been opened, coupling exchanges, via convection and imposed vertical motions, with large-scale circulation dynamics. The research aircraft has opened another avenue for 105 110 115° 120° 125° 130° 135° 140° 145° 150° ^ttill \ V V ^. 0 100 200 100 400 500 •O 0 to 200m obov« seo lavet ^ 200 to 2000m obove teo lavtl ^Over 2000m obove sao lavtl 115° 120° 125° 130° 135* 140° 145° atmosphere over Japan Sea and Kuroshio region. (After Miller and Mantis, 1947, Figs. 1, 3.) (b) Map of cyclogenetic region off east coast of Asia showing February sea-surface isotherms and the course of the warm Kuroshio, or Japanese current, in winter. 268 MALKUS [chap. 4 studying these developments, by actually entering the developing cyclone and measuring its component processes as they interact within it. b. Exchange measurements in a young cyclone over the ocean On January 22, 1955, the Woods Hole Oceanographic Institution's research aircraft flew from the Island of Bermuda (32° 20'N, 64° 44'W) to a location in Rhode Island (41° 38'N, 71° 30'W) on the American continent. The path of the airplane (Fig. 83, inset) cut through a young wave cyclone that had generated on a cold front in the Gulf Stream area off Cape Hatteras and was accelerating over the ocean in a northeasterly direction. An opportunity was thus provided to measure the turbulent fluxes of momentum and heat (applying equations (22) and (23) respectively) and to tie them in with the structure of the cyclone and its air-masses. The work has been described in detail by Bunker (1957). Upon take-off from Bermuda the aircraft sounded the relatively cool air that had arrived earlier from the continent and was drifting eastward under the influence of the Atlantic high pressure cell. About 400 km along the flight path, increased turbulence was encountered in a zone that marked a rapid transition from the sluggish anticyclonic circulation to the "jet" of warm air that con- stituted the warm sector of the cyclone. The warm sector was so narrow that the airplane (flying at about 50 m/sec) passed from this transition zone to air with prefrontal characteristics in just a few minutes. Here a large wafer of cold air underlay the warm air ahead of the main cold front, creating an irregular temperature pattern which increased the stability of the air. As the airplane approached the cold front, the air became rougher, reaching a maximum in the frontal zone itself. The dome of cold air behind the front was exceedingly stable as it left the land and, in contrast to the Japan Sea case studied previously, remained so even after flowing over 300 km of warmer water. As a result of this, turbulence was slight and the heat flow was directed downward even at 100 m above the sea surface. All the observations and computations therefrom are summarized in Fig. 83. Fig. 84 shows more clearly the heat flux pattern through the cyclone. The ordinate is height in meters, while the abscissae have the double scale of Greenwich civil time of a given airplane observation and the approximate (within 50 km) distance from Bermuda in kilometers. The heavy lines depict the lines of constant potential temperature, 6, in degrees absolute computed from the readings of the airplane psychograph. In Fig. 83 small dots on the diagram show the location of a given temperature observation. Along the top of the section a few notes have been entered concerning clouds and weather encountered. The results of seventeen series of turbulence measurements have been entered at the appropriate time and height on the diagram according to the legend given in the center of this section. The units of the root-mean-square values are cm/sec and °C. Heat flows are expressed in meal cm~2 sec~^ where meal is 10"^ calories, and stresses are in dynes/cm^. Lastly, sea temperatures are given in the ovals at the bottom of the diagram. The values are actually the SKCT. 2J LARGE-SCALE INTERACTIONS 269 O 0) > O / _jt— — "V — ^"^ '° o/ '\ \ V:::r=.<^(-s?'J^ C_ w ^ ^ N / ^ p>-~^i/ ^ 9.' 2 O ^ tKr~ X^ \ n O • S -^ 03 3 03 rr 0 fe 3 o U ID ^^ id Ph I— ( w (N O M i-i S ^ 03 -:« 3 Oh >5 c3 -t-3 ■ - .2 S 5 ^ 9 S 2 .2 o ■3 t3 T3 CO fe M bD 2 o rrt 35 0 H -c <: 0) > 0 -i-> cc Cm 00 0 ^ tie oi 4^ bO 03 O O 4J (uu) JM6I8H 270 [chap. 4 potential temperatures of air in thermal equilibrium with the sea ; the water temperatures were obtained from the records of the R.V. Atlantis which was working from the United States east coast to Bermuda on the day of the air- craft flight and previous days. Homogeneity is the outstanding feature of the air-mass occupying the area around Bermuda and extending 300 km along the course of the aircraft. Throughout all of this region and up to 1000 m elevation, the potential tem- peratures were all within a degree of one another. The air-mass, continental in 16 0 0 I 4 00 H 22 JANUARY, 1955 m col P cm 2 -1 sec BERMUDA 0 r- 0 RHODE ISLAND 100 200 300 400 500 600 700 800 900 1000 I 100 Distance (km) Fig. 84. Heat flux cross-section for same Bermuda-Rhode Island flight. (After Bunker, 1957, Fig. 1. By courtesy of the American Meteorological Society.) Values of heat flux, water temperatures and even-valued potential temperature isopleths have been replotted from Fig. 83 to show heat-flvix pattern, computed from aircraft measurements by means of boxed formula. origin, was warming slowly ; it was, when observed, only one degree cooler than the water. The heat-flow measurements indicate a transport of about 1 meal cm~2 sec"~i (100 cal cm"2 per day) near the surface throughout the region. This is to be contrasted with flows found to be 5-10 times greater in air-masses fresh from the cold continent, as in the Japan Sea cases described. The turbulence was likewise quite moderate, receiving most of its energy from the decreasing convective activity. These observations coupled with strato-cumulus overcast give a good description of the final stages of air-mass modification over the ocean. The negative shearing stresses arise from the direction of flight and the shifting of the winds from 300° near the surface to 250° at 850 mb. Thus the SECT. 2] LAKGE-SCALE INTERACTIONS 271 component of the wind along the aircraft's flight path decreased with height, and since, with the equipment used at the time, only the stress component along the flight path could be measured, a negative component of the total stress was recorded. As no detailed wind-profiles were available for the flight, proper interpretation of the stress measurements was not possible. The existence of a turbulent zone between the air of the Atlantic anti- cyclone and the warm sector was noted 400 km out of Bermuda during the third series of observations. It was apparently non-frontal in nature ; the turbulence was presumably generated by horizontal shear and changing pressure gradient accelerations similar to those found in clear air in the vicinity of jet streams. The heat flow and shearing stresses computed in this region appeared puzzlingly large in the absence of wind data. In the warm sector of the cyclone all turbulence quantities dropped off to minimum values for the flight, as a result of the increased stability of the air moving in from the southwest. It is noteworthy that the stability overcame both the tendency for increased turbulence that accompanies greater wind speeds and the lateral diffusion of turbulent energy from the previously dis- cussed shearing zone. The sub-normal values of turbulence lasted for only two of the four observations made in the warm air. The third and fourth observa- tions were made in a cool mass of air that presumably had fore-run the main cold front and squeezed under the warm air. In this "cold nose" (distance 600- 700 km) the turbulence increased again as did the stresses, heat flow and temperature deviations. The origin of the cool air is open to question since rain was falling at the time of observation, which may have contributed to the low temperatures directly ahead of the cold front. No turbulence measurements were made during frontal crossing. Although the air there was rougher than normal, the turbulence was not violent ; Bunker (1957) estimated the root-mean-square vertical velocity to be 150+ 50 cm/sec at 300 m within this zone. The most unusual feature of the cold air-mass north of the front was its abnormally high static stability, which must have been maintained by strong subsidence. Beneath the level of the aircraft observations, great instability must have existed since the water was 6°C warmer than the air. The only turbulence observation taken in the cold air showed decreased turbulence and a downward flow of heat at 100 m elevation 80 km behind the front. The shearing stress measured was large and positive (3.1 dynes/cm^) as would be expected since the aircraft was flying directly upwind so that the total stress was recorded. The downward flux of heat at 100 m in an air-mass blowing over much warmer water requires careful consideration concerning the balance of the transfer processes at work, namely buoyant convection, mean subsidence and eddy dilBFusion. Using the method of Priestley (1954), Bunker estimated the maximum upflux of heat due to buoyant convection to be 0.4 meal cm~2 sec~i ; this was obtained assuming the mixing rates for momentum and heat to be equal and using the observed root-mean-square temperature deviations. At the same time he computed a downward flow of heat by eddy diffusion of 272 MALKUS [chap. 4 1 meal cm"" 2 sec ^ if the conservative value of 50 g cm i sec~' for the coefficient of turbulent mass exchange is placed in the heat diffusion equation (12). As the airplane gives the net heat flow in the atmosphere, it is apparent that a down- ward flux should be obtained from its records, as indeed it was. Thus the strong mean subsidence behind the front, which unfortunately could not be estimated, preserved the great stability and resulted in large downward flow of heat by advection and diffusion even under conditions of slight turbulence and a great sea-air temperature difference. The lowest air is heated by the ocean, becomes unstable and rises in small buoyant "thermals" which penetrate a short distance into the stable air. The heat transport by this convection jDrocess was, in this case, small compared to the downward flows ; this air-mass was heated more by warm stable air aloft than by the warm ocean. It is interesting to note that this wave cyclone did not deepen further : the inhibition of exchange by dynamic factors may have played a role in this failure, a subject interesting to speculate upon if it leads to specific questions for future research. This work was a small part of an extensive aircraft program led by Bunker and his colleagues to study the effects of exchange in air-mass modification over the ocean ; like all worth-while pioneering efforts it proceeds at the expense of painstaking labor in the face of obstacles which fall just short of being insuperable. B. Long- Period Variations in Sea- Air Interaction At the very beginning of this chapter we purposely set to one side long-period circulation anomalies in air and sea and their effects upon each other. For the purpose of making the climatological exchange picture meaningful, for develop- ing physical and analytical models of transfer processes and their consequences, it was assumed that, in the broad scale average, one winter or summer was just like any other in both media. In order to use the energy budget equations as a foundation for our models, such an assumption is both necessary and justifiable ; for example, secular heat storage in the sea over several years is clearly both a negligible and non-computable term in a joint budget study of the Caribbean in two particular winter months. This assumption is tacitly based on carrying to an extreme the knowledge that, compared to the fickle atmosphere, the ocean is a high inertia, infinite reservoir of heat and water, and that its circulation patterns respond to the in- put of solar heat and wind stress in such a way that its temperature structure shows only a seasonal cycle in a given region. Then it would be true that ex- change fluctuations are regulated by the sun and atmosphere and, when averaged over the seasonal cycle, exert an invariant effect upon the sea. It is clearly at this last point that the assumption breaks down. Because the upper layers of the ocean are somewhat decoupled from the abyss, so that the shallow strata above the thermocline may be moved about by the winds, or their thickness altered with little effect upon the deeper waters, and because it is the surface layers to which the atmosphere responds, a coupled instability of the SECT. 2] LARGE-SCALE INTERACTIONS 273 two fluids is possible. Furthermore, very small changes in the sea may have vast consequences upon air circulations, of which instability is the outstanding feature, as we have seen. The possibility, therefore, exists that an alteration in air circulation could, if it persists, modify the surface waters in such a way as to maintain itself or even to amplify with time. Unfortunately, this is one of the most difficult areas in geophysics to study in terms of well-formulated physical laws or under controlled conditions, or even quantitatively in any manner at all. One always has the uncomfortable feeling that the openness of the boundaries and the processes of necessity omitted may contribute so largely and nonlinearly as to invalidate almost any conclusion drawn. To estabhsh any theory with real confidence, first, three- dimensional data are necessary over huge regions, long periods and in both sea and air together. These data range in scope from solar output throughout its spectrum, down to vertical ocean motions, of the orders of meters per year through the thermocline ! Such data do not exist today in a quality or quantity even approaching in order of magnitude the necessary bare minimum. Nor are they likely to do so in the foreseeable future. Secondly, as we can now see from the restricted budget studies of carefully selected situations presented here, the terms in the equations needed to examine long-period changes are the smallest ones ; they are very tiny differences between the dominant terms. We are over- joyed if we can evaluate these latter to within 25%. Occasionally, however, even if the cautions we have raised are ignored, a broad sweep at an overly large phase of the interaction problem can by bold intuitive hypothesis suggest what specific phases may be worth isolating later. This will thus serve as an inspiration and a challenge for the necessary pains- taking and far less spectacular labors required for their proper establishment or rejection. We shall conclude this chapter by brief discussion of two such courageous efforts concerning long-period variations, each by a man whose outstanding experience-acquired intention in geophysics suggests that his visions are worth careful consideration and pursuit. a. Interacting anomalies of several months persistence Until the mid-1950's, serious studies of long-period sea-air interactions had more or less been abandoned since their vigorous pursuit by Helland-Hansen and Nansen (1920) early in the century. Recently, renewed interest has been stirred up, with impetus added by some rather obvious ocean-temperature anomalies. In particular, a fairly sudden warming (up to 4°F) in the eastern Pacific (Fig. 85) was noted, which began in the summer of 1957 and persisted for more than a year. Among others, this caught the curiosity of Namias (1959), whose experience in laying the foundations for long-range weather forecasting had led to recognition of anomalous air-circulation patterns which maintain themselves over similar intervals of space and time. His study is based on the premise that an average air-circulation picture over a decade or two is a meaningful climatological "normal" which can be related 10— s. I 274 [chap. 4 to a similarly averaged large-scale ocean-temperature pattern. He then supposes that if the normal ocean thermal structure is related to the normal atmospheric circulation, anomalous air circulations must give rise to anomalous conditions in the sea. The time scale of the variations is particularly suggestive. Short period, synoptic-scale fluctuations in air circulation, while large, change radically from one day to the next. Thus, while they may produce substantial changes in the surface waters, these are likely to be variable and conflicting, and so cancel out with time. However, "regimes" in which certain aberrant forms of atmospheric circulation persistently recur for months or longer have been recognized (Namias, 1953). Such sustained abnormal winds might alter the surface-water transports, divergence, radiative and exchange processes, Fig. 85. Observed mean sea-surface temperature anomaly (°F) for fall, 1957. (After Namias, 1959, Fig. 4.) Shading represents above -normal temperatures. creating thereby surface-temperature anomalies which might in turn help maintain the abnormal air circulation and thereby themselves. To examine the eastern Pacific warming in this framework, Namias first looked at seasonal averages of the overlying air circulations during the years 1957-58 and their departures from the long-period normals for each season. An example for fall 1957 is reproduced in Fig. 86. With the geostrophic relationship, we may deduce from these charts the prevailing winds for the season and their departure from normal, recalling that looking downstream the low pressure or height lies to our left, and speed is proportional to the crowding of the isopleths. The isobars at sea-level (or contours of the heights of the 700-mb surface aloft) thus give a measure of the total resultant or pre- vailing wind for the season, whereas the isopleths of anomaly represent just SECT. 2] LAKGE-SCALE INTERACTIONS 275 the anomalous component. Thus an anomalous component from the south may represent an increased southerly prevailing wind or may imply diminution of the northerly wind which is normally present. The most consistent abnormality of the Northern Hemisphere's general circulation during 1957 and early 1958 was low strength of the prevailing west winds of mid-latitudes, or persistent low "zonal index" of the westerlies. The subtropical anticyclones were weaker and more broken up than usual, and in particular, the east-central Pacific showed a much deeper than normal low pressure trough, which was pronounced throughout the period. This unusual trough, which during the fall of 1957 (Fig. 86) had a 190-ft negative anomaly at 700 mb, means that over much of the area east of its axis (165°W in the figure shown) the prevailing and resultant wind had a stronger south to north component than normal. West of the axis, the reverse anomalous conditions are implied. Experience suggests that these are composed of repetitive northward thrusts of air in advance of the trough and southward thrusts behind, as rapidly developing cyclones move into the Aleutian low pressure cell, one of the atmosphere's persistent large-scale "centers of action". Namias first attempted to isolate and estimate quantitatively the advective effect of these winds, to determine whether the major pattern and right order of magnitude of the sea-temperature anomalies would result. He used Ekman's empirical expression (Sverdrup, 1942) for the transport of water at 45° to the right of the wind direction, due to its stress upon the surface layers, namely, vjw = 0.0127/sin'/2 cp, (72) where v is the speed of the surface water current, w is the wind speed and 99 the latitude. This enabled him to compute surface-water displacement arrows, like those in Fig. 87a, which, when superimposed on the normal sea-surface iso- therms for the season in question, permitted computations of simple advective changes with respect to seasonal normal. The result for fall, 1957, is shown in Fig. 87b. The fact that there is some apparently real qualitative agreement with the observed patterns of Fig. 85 is suggestive that an air-sea link on this scale has been found, particularly in view of the drastic oversimplifications of the approach. As the author points out, every other term in the energy budget which would change the water temperature was set aside except advection, and even that was treated crudely by starting every time from the seasonal normal isotherms. The purely advective computation gave poorer results in summer when air circulations were more sluggish, although the water-temperature anomaly was undiminished. Namias then looked qualitatively at the divergence patterns of the induced water movements and their effects upon upwelling, possible changes in insolation and cloudiness associated with the weather patterns, and evaporative differences due to weaker winds. These could not be combined into a proper budget. Among the essential pieces of missing information was the depth to which the ocean warming extended, although there was evidence that it reached well below 100 m ; the latter rather ruled out observed insolation 276 MALKUS [chap. 4 SKCT. 2] Fig LARGE-SCALE INTERACTIONS 277 86 {on facing page). Atmospheric flow patterns and their departures from seasonal normal for fall, 1957. (After Namias, 1959, Fig. 9.) ( a) Solid lines are contours of 700-mb pressure surface in tens of ft. Dashed lines are departures from seasonal normal in tens of ft. (b) Solid lines are isobars of surface pressure in mb above 1000 mb. Dashed lines are departures from seasonal normal ; anomaly centers labelled in mb. (a) (b) Fig. 87. Computation and result of sea-surface temperature-anomaly pattern resultmg from advection by abnormal wind drag on normal surface water. (After Namias, 1959, Figs. 13 and 14.) (a) Anomalous surface-water displacements (arrows) computed from mean seasonal sea-level pressure anomalies for fall, 1957. Isotherms (solid lines) show normal sea- surface temperatures (°F) for October. (b) Computed sea-surface temperature anomalies (°F) for fall, 1957, using advective method described in (a) and text. Shaded area represent above-normal temperatures. Compare with Fig. 85. 278 MALKUS [chap. 4 changes as too small to contribute significantly. His final tentative conclusion was that the long period of warm water off the California coast resulted from a combination of atmospherically induced factors : the summer and early fall warmth associated with diminished upwelling and evaporation, with a slight contribution from increased solar radiation (fewer stratus clouds than normal) ; and the late fall and winter warmth mainly associated with surface advection of warmer water and somewhat reduced upwelling. A fascinating feature of the study was the continuity and seasonal migrations of the anomalous patterns in both sea and air, which, while remaining strong and identifiable, moved eastward from the summer through fall and winter. As summer goes into fall, two climatological phenomena are highly probable : (i) the westerlies over the Pacific increase in strength, and (ii) the trough along the east coast of Asia becomes established. These phenomena are associated with increased cyclonic activity off the Asiatic coast, incident to the monsoon outbursts we have described previously. The cyclones forming in the intense thermal gradients of this region travel toward the Aleutians as they develop. Thus the trough off Korea becomes "locked" into this position with the ap- proach of winter. The increasing westerlies then require a dynamic readjustment in the planetary wave pattern (Rossby, 1939) to move the next downstream trough farther away or toward the east. If such a hypothesis contains a germ of truth, one is led to the most important resulting question, namely, why did the anomaly centers and mean troughs have such long lifetimes? Namias suggests a coupled feed-back mechanism between air and sea. In the first place, the abnormally warm water in the eastern Pacific provides an enhanced source of both heat and moisture to aid cyclonic development by imposed convective ascent, as suggested by Winston's work. The longitudinal water contrast, by enhancing horizontal temperature gradients in the over- lying air-masses, may also assist cyclonic developments, which draw heavily on the stored potential energy in sharp thermal contrasts or fronts. In other words, incipient cyclones moving over these abnormal waters could feed on the in- creased moisture, sensible heat and temperature contrast imparted to the air, so that a more or less geographically fixed area favors cyclogenesis and pre- vailingly negative anomalies. However, the area of influence would also be affected by climatic conditions involving the general atmospheric circulations (as indicated above) and thus might move with time. That is, the air circulation responsible for the underlying temperature variations might change because of factors more potent than those produced solely by water anomalies. In fact, the shift of the latter, apparently in response to seasonal changes in air patterns, suggests that the upper ocean layers respond rather rapidly to atmospheric variations and that their anomalies have less inertia than previously suspected. Their persistence thus may signify a balance between rather rapid growth and destruction governed by dynamic processes rather than the existence of inert "puddles" of warm and cold water. An exciting support of this last suggestion, which has significant practical consequences, arises when we learn that 1957-58 was a catastrophic "El Nino" SECT. 2] LARGE-SCALE INTERACTIONS 279 year in the Southern Hemisphere off Peru (Rodewald, 1959; Wooster, 1960). El Nino is an interruption in the normal cold water upwelling off the South American coast ( ^ 20°-3°S), and a substitution of warmer and less saline waters flowing down from equatorial regions lying to the north. These waters contain fewer nutrients to support the plankton and small fish population serving as food for larger fishes and the guano birds, which emigrate or starve in great numbers, leading to disastrous consequences for the fisheries and fertilizer industries. There is evidence that El Niiio occurs in years when the trades of both hemispheres are weak^ and the equatorial trough goes farther to the south in winter than usual, relieving the southerly winds (southeast trades of the Southern Hemisphere) which maintain the upwelling, and permitting the invasion of warmer water from the equatorial counter-current (see Fig. 22a). Although they are too severely hampered by data limitations to prove any- thing definite as yet, these studies suggest lines of important inquiry to pursue, and further emphasize the need for joint oceanographic-meteorological measure- ments in the tropics on a continuing network basis, particularly in the meteoro-, logically blank southeastern Pacific. Furthermore, they focus attention upon a key question in long-period interaction, namely that concerning the nature and time interval of the response of the upper ocean layers to specified air- circulation changes. It may now be possible to make models or to seek natural situations where the air's input and the sea's reaction can be studied under more rigorously specified or controlled conditions. Namias' work also causes one to wonder how, in such cases, the system goes back to normal. His results suggest that the very existence of a definable "normal" state implies a nearly invariant large outside forcing of the system and a high internal restoring or damping influence which is rather slowly brought to bear. The former is plainly the solar seasonal cycle, while the damping role must be played by the vast reservoir of abyssal waters and their reaction with the surface layers. That one or both of these stabilizing influences is not quite perfect is implied by the existence of still longer period variations in sea and air resulting in slow climatic change. Our final discussion concerns an oceanic warming over several decades, which is about the limiting time interval amenable to even a semi- quantitative description of what actually changed and by how much. b. Variations in interaction over several decades In trying to approach interaction and exchange variations and their con- sequences over decades, we are in an even more precarious position. For example, in equation (1) for the energy budget of an ocean column, let us suppose that, in a tropical situation, all terms except Qe and S are unchanged. We find that if a 200-m deep layer is considered, an increased evaporation of about one per cent, or an increased heat loss of roughly 3 cal per cm^ per day 1 It is interesting to note that 195.3, in which occurred the "weak trade" of the western North Atlantic discussed in Section 7, page 202, was an El Nino year (only moderately intense). 280 MALKus [chap. 4 would lead in fifty years to a cooling of the column by nearly 3^C! Thus, even if all other budget terms were surely unaltered by less than this margin (which is, of course, preposterous) and evaporation were determined regularly over wide regions to the best accuracy foreseeable methods permit, it still is in- herently impossible to pin down this long trend in Qe from its shorter and larger fluctuations. It would be well for the reader to contemplate how differences of this size in Qvo or in radiation could be estimated. Furthermore, when we come to the point where energy fluxes of 1 cal per cm^ per day are important, the terms left out of equation (1) are no longer negligible. In particular, the dif- ference in heating produced by kinetic energy dissipation amounts to this much if winds are changed from only 6 to 8 m/sec. No climatological pictures of exchange like those we have presented in Sections 4 and 6 of this chapter existed fifty years ago and, although it is to be hoped that fifty years hence much better determinations will be available, it should be clear from the discussions in this chapter that the uncertainties in the present computations alone must preclude any chance of relating the ensuing climatic or oceanic structural changes in any rigorous, formal, or even demonstrably sound physical manner. Nevertheless, we have enough data today to infer the likelihood of secular changes in the decades since the turn of the century, particularly in atmospheric circulations and sea-temperature patterns. A brave paper by Bjerknes (1959) has attempted to describe these in the North Atlantic and to relate them temporally and spatially by mechanistic hypotheses. Test periods of eight years each were selected on the basis of data coverage, namely 1890-97 and 1926-33; this interval averages out short period fluctua- tions with which the Bjerknes study was not concerned. His analysis of changes in sea-surface temperature between these intervals is shown in Fig. 88. The significant features are a marked warming in the Gulf Stream region off Grand Banks, a slight cooling centered around 53°N south of Iceland, and a slight warming in the Bermuda-Sargasso Sea area. We are plagued here with the problem of inadequate and uneven sampling and perhaps also poor representa- tiveness of data, particularly in the 5°-latitude square of the intense warming, through which the shipping lanes' were shifted between the sample periods (due to the Titanic disaster). Nevertheless, more recent weather-ship data studied by Rodewald (1956) confirms a similar general pattern in the post-war years. The pressure change map at sea-level for the same periods, superimposed on average sea-surface isotherms, is shown in Fig. 89. Bjerknes used this, some- what in the manner of Namias, to deduce changes in the prevailing geostrophic wind between periods, and thence to infer what might be the alterations in surface-wind stress upon the sea via equation (17). Therefore, the geostrophic wind speed squared for each period is shown in Table XXV, computed from Fig. 89. The profile from Bermuda to Port-au-Prince is intended to give a measure of change in trade-wind strength, while that from Bermuda to Eastport SECT. 2] LAKGE-SCALE INTERACTIONS 281 TREND OF AVERAGE ANNUAL SEA TEMPERATURE "C FROM 1890-97 TO 1926-33 STREAMLINES MARK STEM AND BRANCHES OF THE GULF STREAM SYSTEM V V ao f^H ^^N Fig. 88. Streamlines of the Gulf Stream system (heavy lines with arrow-heads) and iso- pleths of temperature change (°C) of annual sea-surface temperature from 1890-97 to 1926-33. Positions of present weather ships marked with capital letters. (After Bjerknes, 1959, Fig. 1. By courtesy of the author and the publishers.) Table XXV Pressure Gradients and Corresponding Geostrophic Winds along Selected Profiles in the North Atlantic for Two Periods (after Bjerknes) 1890-97 1926-33 Profile pdiS., mb Geostroph A ic wind p diff., mb Geostroph ic wind m/sec in2/sec2 m/sec m2/sec2 Bermuda-Port-au-Prince Bermuda-Hatteras Bermuda-Eastport 3.68 -0.05 2.80 3.3 0 1.7 10.9 0 2.9 6.12 3.16 6.24 5.5 3.2 3.8 30.2 10.2 14.4 represents that in the zonal westerlies. The development of a pressure difference between Bermuda and Hatteras suggests the addition of a prevailing southerly- wind and stress component off the United States east coast between periods. Although there is unfortunately little direct wind data to confirm or modify these figures, the large changes in surface stress deduced here would be very 282 [chap. 4 interesting to apply in the Munk-Stommel model of wind-driven ocean currents, particularly since Munk's (1950) classical computations used stress consistent with the earlier pressure configuration. From these figures Bjerknes deduced that the clockwise mean wind circula- tion around the Atlantic anticyclone and its resultant stress on the sea had appreciably strengthened between the two test periods. On this basis he ex- plained the oceanic temperature changes. According to his argument, the strong secular heating recorded at the Atlantic "polar front" or Gulf Stream may be due to : (a) the increased warm water advection parallel to the front in AVERAGE ANNUAL PRESSURE 1926-331 MINUS •• ■• ■• 1890-971 ANNUAL SEA SURFACE ISOTHERMS IN °C Fig. 89. Change in mb of average annual sea-level pressure from the period 1890-97 to that of 1926-33 (full lines) and average annual sea-surface isotherms (dashed lines). (After Bjerknes, 1959, Fig 2. By courtesy of the author and the publishers.) the jet maximum ; (6) northward displacement of the front ; and (c) on the cold side of the front a possible increase of average temperature due to amplified meandering. The first test to seek in oceanographic data would be comparisons of mass transport in the Gulf Stream for the two periods. Fortunately, hydrographic sections exist from the Challenger Expedition of 1872-76 to compare with nearly identical sections from the work of Iselin in the 1930's (Iselin, 1936). Puzzlingly enough, both the positions and slopes of the important isotherms in the stream's core are, as far as can be determined, unchanged over that SECT. 2] LARGK-SCALE INTERACTIONS 283 somewhat longer interval.^ Thus no demonstrable increase in th(^ jjjeostrophic mass transport of the stream is deducible ; over such a long time, changes in How should surely have become adjusted with the pressure field (V^eronis and Stommel, 1956). However, it is impossible at the present stage of dynamic oceanography to predict, even in order of magnitude, the relation between warming in the northern surface waters and the mass flow of the stream. The heat transport by layers is not deducible from the mean mass flow, which is all that can be computed from the wind-driven current models. Nor are the mixing processes during meanders well enough studied to estimate how much warm water is given off, and where and how much is recycled in the gyre. Whether or not the mean position of the stream has shifted in this interval , or its meander pattern has changed, probably cannot be settled retroactively. In the region of tight surface-temperature gradients, a translation of the mean pattern by as little as 20 nautical miles could bring a 2° warmer isotherm to the surface at a given spot. The second part of Bjerknes' study develops an explanation of the cooling south of Iceland and the slight warming in the Sargasso Sea in terms of mass readjustments of the ocean layers to the increased wind circulations. The basis of the argument is clearly summarized in Fig. 90. It runs briefly as follows : an anticyclonic vortex in the ocean which is decreasing in intensity with depth must be of the warm core type ; in other words, the warm surface layer must have maximum thickness near the center. In that way, in fact, the anti- cyclonic winds around the Atlantic high do maintain in a permanent fashion a downward bulge of the lower limit of the warm surface water-mass. Thus Bjerknes suggests that anticyclonic anomaly winds, more or less concentric with the anticyclonic ocean currents, would add more depth to the warm surface water at the center. Inflation of the warm layer would tend to raise very slightly the equilibrium temperature of the ocean surface, because the water there would have become a little less exposed to mixing with the cold deep water. Conversely, a cyclonic current system decreasing with depth will be characterized by minimum thickness of the warm surface layer. Therefore, an increase in circulation around the normal Icelandic low pressure cell in the atmosphere would reinforce the cyclonic vortex of oceanic flow in high latitudes and make the warm upper layer thinner, thus exposing the water at the ocean surface to more mixing with the cold deep water. In attempting to pursue these deductions further, we run abruptly into the unsolved problem of water-mass production and destruction, and the small- scale turbulent or convective transfers through the thermocline, of which oceanographers are just becoming able to attempt serious models (Stern, 1960). As Stommel has so cogently pointed out in the conclusion of his book, The Gulf Stream, the lack of physical knowledge on this key point permits opposite conclusions to be drawn relating circulation strength and warming of the 1 The writer is deeply indebted to her colleague Henry Stomrriel for his generous sharing of his vast information, experience and oceanographic insight in constructing the discussion on this page. 284 [chap. 4 North Atlantic, If we make, as did Iselin (1940), the hypothesis that the pro- cesses which produce the warm surface masses of the central Atlantic water are more or less constant in time, then an increasing transport in the North Atlantic gyre, accompanied by the deepening of the thermocline in the Sargasso Seal (ag shown in the top right of Fig. 90), must be accompanied by a radial shrinkage of the whole sea-current system. Thus the Gulf Stream should shift to the south, leading to cooling at higher Atlantic latitudes! We are thus left with three comparative deductions between the two test periods which, though not provably incompatible, are not yet demonstrably otherwise, namely: Worm layer Deep water c o o >. 5 c < 19 i Jig Warm layer Deep woter INITIAL LATER STEADY STATE Fig. 90. Zonal, vertical profiles showing schematically the creation and maintenance of maximum thickness of warm oceanic surface layer under the influence of anti- cyclonic wind stress, and minimum thickness of same layer under cyclonic wind stress. (After Bjerknes, 1959, Fig. 3. By courtesy of the author and the publishers.) Full lines : sea surface and interior isobaric surfaces. Dashed line : density dis- continuity surface. Leftward displacement of center of oceanic deformation relative to air-circulation center due to variation of Coriolis parameter with latitude. (For explanation see, for example, Stommel, 1958, chap. VII.) increased wind stress and inferred faster water circulation around the Atlantic gyre, apparent marked warming in the meander region off Grand Banks, and seemingly unaltered geostrophic mass transport and density field in the main portion of the Gulf Stream. It is plain that such long-period changes, while they are valuable to identify where possible, and their origins intriguing to debate, are at the borderline of subject matter that our present tools and physical theories can relate to one another meaningfully, beyond partial quantitative descriptions, statistics, and courageous hypotheses, as in the works of Namias and Bjerknes. The latter 1 Stommel produces some weak confirmatory evidence for this occurrence. SECT. 2] LARGE-SCALE INTERACTIONS 285 serve the valuable function of suggesting what may be the crucial links and pro- cesses to isolate for intensive controlled study ; they have further opened the communication channels, mutual stimulation, and body of shared experience between meteorologists and oceanographers. 9. Concluding Remarks Two conclusions that one might be tempted to draw from a casual reading of this chapter are first that our greatest need to advance in these studies is more and better data ; and second, that air and sea are so closely coupled to each other that one cannot profitably study either medium without full con- sideration of the other. Like most generalizations in geophysics, these are only partial truths and thence can be misleading. It is hoped that the more serious reader has been led to see from the material herein the underlying reason, the same in both cases, why these two much- quoted statements, while certainly not false, are not adequate to communicate either the progress in the field to date, nor its needs in order to make further progress tomorrow. In the matter of data and observations, it is, of course, plain that more and better are needed. As we saw, many of our studies were forced either to restrict themselves with doubtful assumptions or to settle for ambiguous conclusions because of data limitations. In part, the brave new technology culminating in the glamor-gadgets such as satellites will help out, if miracles are not demanded of them, particularly in the radiation realm. However, many data are needed close to the air-sea boundary itself, in the upper-ocean and lower-air layers, on a routine basis, that cannot foreseeably be sought from vehicles in space. The day of the ocean weather ships, without which many of these studies could not have become quantitative, should not be allowed to close. Nor should the usefulness of the research vessel have even approached its prime ; much of the boundary data must be obtained from a platform containing intelligent beings, some of whom are aware of the problems, possess experience and intuition and are capable of making decisions, frequently departing from orthodox pro- cedures. The outstanding feature, however, of the data of the marine sciences is the appalling expense, not only in money but in indispensable human effort. The budget studies we have described were enabled as by-products of the large-scale routine networks, often supplemented by special expeditions. Considered from start to finish, the hours and years of work in instrument development, planning, installation, testing, measurement on station, evaluation, processing and storage of data — to say nothing of the laborious extraction from archives by the research workers whose names finally appeared as authors — would stagger the imagination, even of other physical scientists. It is, therefore, paramount, out of the vast myriad of measurements that could be made on the earth's fluids, from the atmosphere's outer fringes to the sea's bottom, to apply a selection process. Concerning what do we need more and better data the most? How is the selection to be guided? And is it primarily inadequate data which is the obstacle 286 MALKUS [chap. 4 opposing our efforts to understand, predict or control our planetary environ- ment? An honest answer to the last question must be in the negative : and therein lies the connection between the data problem and the other partly true con- clusion. The times that our studies were hampered by limitations due to data alone are, in fact after careful consideration, not so very large. That limitation is more often coupled to inadequate physical laws, or inadequate concepts, or inadequate formulation of the inquiry in the face of geophysical complexity. We are forced to accept limited "solutions" to our problems because we must ignore or inadequately parameterize processes, energy sources, boundary effects, nonlinear interactions and scales of motion that our present framework of turbulent fluid dynamics does not tell us how to include. Even if we could easily and cheaply measure anything we chose, we should often not be sure what to measure. What become worthwhile "facts" to measure depends upon expressible or visualized relationships they bear to one another; "facts" are rarely useful in isolation. Re-examining the material in this chapter upon which our interaction picture is based, namely turbulent boundary exchange, radiative processes, buoyant convection, ocean and atmosphere dynamics and stability of flows on all scales, we must conclude that further progress in our knowledge of air and sea in interaction depends essentially upon growth in the fundamental physics of turbulent, heat-driven fluids. Evaporation, for example, was shown to be one of the most significant sea-air exchariges. To determine evaporation at point A, time t, to an accuracy of n per cent becomes a definable worth-while endeavor primarily when we seek its fundamental functional relation to other processes and phenomena. Furthermore, improved data alone will hardly help us even to determine evaporation better when the transfer equations may give results 50% off due to inadequacies in their formulation of events at the turbulent boundary. Nor will improved data alone aid us to understand the heat supply and release by tropical circulations, until we construct some framework to relate their dynamic instability to convection, transfer and planetary energetics. Further data alone will scarcely help us to understand North Atlantic warming or climatic change until we develop meander theories and physical models of water-mass production, turbulent processes on several scales interacting with one another. Thus, to say that the ocean and air must be treated together in their entirety is even more futile than saying that all scales of motion in the atmosphere, from microscopic eddy to planetary jet, must be dealt with together. Quite on the contrary, in fact, we found that the more comprehensive the scope of phenomena an investigator attempted to treat, the less happy we felt with his resuJts, and the less we felt they contributed to our explicit understanding of natural phenomena and their relationships. At the risk of being reactionary in this age of scientific optimism, we must conclude that the head-on approach to the overall air-sea interaction problem is almost surely doomed to failure. Geophysicists must discipline themselves to SECT. 2] LABGE-SCALE INTERACTIONS 287 seek critical parts of it for careful study. Thus they must try to pose questions which isolate tractable features of the complexity which can be treated under relatively controlled conditions. This usually means recourse to idealized, simplified situations : situations which may be realized only in mathematical or laboratory models, but which strive to select and relate the important physical processes which govern some phases and motion scales of the natural system. Rarely, but often enough to be of inestimable value, nature herself performs an experiment under partially controlled conditions, as we have seen in some of the trade-wind cases ; one of the main skills of the earth scientist lies in being able to recognize and exploit these, using them to guide his measure- ment programs and as a framework within which to relate the results. As in all geophysical studies, we are faced with constant compromise and decision between the practical and the apparently impracticable, the extensive and the intensive effort. If the air-sea interaction picture developed here is any fair representation, then we find that, in the long run, the intensive study has the most extensive value to our understanding, and that the apparently im- practicable investigations of controlled or idealized situations are the in- dispensable foundations upon which practical developments depend. To evolve a basic mechanics of turbulent fluid motions, the sea and air around us provide stimulation, a proving laboratory, and a necessary practical motivation in terms of the benefit to humanity. But our progress in the study of natural planetary fluids can develop little farther without concurrent building of these foundations, to which a large fraction of our effort must be devoted. References Avsec, D., 1939. Thermoconvective eddies in air: Application to Meteorology. Sci. and Tech. Publ. of the Air Ministry. Work of the Inst, of Fluid Mech., Fac. Sci. Paris, No. 155. Bagrov, N. A., 1954. The planetary albedo of the earth. Transactions of the Central Institute of Prognoses, No. 35 (62). Baur, F. and H. Phillips, 1934, 1935. Der Warmehaushalt der Lufthiile der Nordhalbkugel im Januar und Juli und zur Zeit der Aquinoktien und Solstien. Gerlands Beitr. Oeophys., 41, 160-207 ; 45, 82-132. Bean, B. R. and R. Abbot, 1957. 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Slowly depositing deep-sea sediments accumulate these materials and to a certain extent preserve a sequential record of fall-out in contrast to the continents where dilution of the fall-out with similar-appearing terrestrial solids, combined with weathering processes, usually destroys any continuous record. Hence, deep-sea deposits provide excellent source areas for paleometeorological investigations provided eolian materials can be recognized. 1. Meteorology of Transport Dust-transport processes occur predominantly in the troposphere, although some transport is found in the stratosphere followed by settling to the tropo- sphere where it behaves similarly to other tropospheric fall-out. The tropo- spheric circulation is dominantly zonal with three main zones of air-mass movement in each hemisphere — the equatorial easterlies, the temperate westerlies and the polar easterlies (Fig. 1). Each of these zones of air circulation can carry dust, contributed primarily by the arid areas of the continents. Precipitation scrubbing appears to be the most important mechanism of fall-out from the troposphere (Wilkins, 1958). One of the first records of the transport of atmospheric dusts to oceanic areas was made in 1160 by the Arabian geographer Edrisi (as quoted in Radczewski, 1937a). Dust from North Africa is carried across the Mediterranean and far into Europe by winds called siroccos (Ehrenberg, 1847; Clerici, 1901, 1902; Hellman and Meinardus, 1901; Walther, 1903; Glawion, 1938). The reddish sirocco dust consists predominantly of quartz and clay, and is primarily removed from the atmosphere by the scrubbing effects of rain and snow. Brown (1952) reported volcanic dust over the Carribean to an elevation of over 4000 meters which was derived from a volcano in the Cape Verde Islands off the African coast. The ash moved across the Atlantic seaboard of the United States and was last observed over Bermuda after traveling about 7000 miles, all in the troposphere. Fall-out of this ash essentially covered the entire north equatorial and north temperate regions of the Atlantic. The strong westerly winds of temperate latitudes have carried volcanic ash from eruptions in Iceland across the Atlantic far into Northern Europe (Salmi, 1948; Thorarinsson, 1954) and ash from Chilean eruptions out into the South Atlantic (Larsson, 1937). In the Pacific the meteorological conditions are similar to those in the Atlantic, and, in spite of the larger areas, there are records of desert dust storms crossing broad expanses of ocean. [MS received August, 1960] 295 296 BEX AKD GOLDBERG [chap. 5 Dust from the Australian deserts has been observed to fall on New Zealand after crossing the Tasman Sea (Marshall, 1929). The amounts which precipitated in the dust storm of October, 1928, show a covariance with the intensity of the rainfalls and snowfalls. The principal minerals composing this reddish to buff colored dust were iron-oxide-coated quartz, clay and some mica flakes. I5°N S. Pole Fig. 1. Mean zonal wind averaged over all longitudes. Isotachs are in meters per second. (After Mintz, 1954. By courtesy of the American Meteorological Society.) The deserts of Central Asia, North China, Mongolia and Manchuria and the outwash flood-plains of the Central Asian mountains have provided enormous quantities of dust which formed the loess deposits of this region (Willis et ah, 1906-07 ; Barbour, 1927). The Asian dust storms which reach Japan (Futi, 1939) carry sediment across the Japan Sea and out into the North Pacific. Futi (1939) reports quartz and feldspars of a size of 2 to 50 microns as the principal minerals identified in observed falls. The strong westerlies blowing across the Pacific basin have the large Asiatic SECT. 2] INSOLUBLES 297 deserts as a dust source in the Northern Hemisphere, and the smaller South African and Australian deserts in the Southern Hemisphere. But, unlike the Atlantic easterlies, the Pacific equatorial easterlies have no large desert regions as a potential dust source. However, there is a continuous volcanic chain along the eastern margin of the Pacific basin in the zone of prevailing easterlies. For this reason one expects, in the Pacific, a predominantly volcanic fall-out from the easterlies and continental dust from the westerlies. In addition there are continental katabatic winds, such as the Santa Anas, which carry desert silt seaward from the California coast ; these, however, are observed only near the continents. The advent of nuclear explosions provided a means of studying the movement of atmospheric dust over long distances. A low yield explosion introduced fission products into the troposphere over Nevada in 1955. The fall-out from this explosion was observed consecutively at Paris, Cairo and Asahikawa, Japan (Miyake, Sugiura and Katsuragi, 1956). The fall-out in Japan occurred as a mud rain two weeks after being introduced into the atmosphere over Nevada. Transport had been in the jet stream. It seems obvious that the jet stream with its enormous speeds should play a major role in global transport of dust, but the first clear-cut example was the Asahikawa fall-out. The fission products were absorbed on a fine buff silt dominantly of 2 to 10 microns size and composed of quartz, feldspar, calcite, mica and clay minerals. Miyake {loc. cit.) noted the chemical similarity of this dust to Manchurian loess and suggested that it is Manchurian loess which has scavenged fission products. Actually this size range of 2 to 10 microns falls into the true aerosol range (Junge, 1957), while loess (Scheidig, 1934; Smith, 1942) is somewhat coarser and falls largely into the size range where gravitational settling predominates. Nuclear weapon tests have shown that gigaton explosions are required to lift surface dust into the stratosphere (Libby, 1956, 1956a). For this reason there is little cause to expect transport of continental dust by stratospheric winds. Volcanic eruptions, however, may be of sufficient force to raise dust well into the stratosphere. The most famous example of this was the explosion of Krakatoa which spread volcanic ash into the stratosphere where it circulated in both hemispheres for several years. More recently the advent of high flying aircraft has made possible direct observations of stratospheric dust. The eruption of Mt. Spurr, near Anchorage, Alaska, on July 9, 1953, deposited a blanket of tropospheric fall-out of volcanic ash in Alaska (Wilcox, 1959). The mushroom ash cloud reached an altitude of 70,000 ft. By July 27, ash from this eruption crossed England at an altitude of nearly 50,000 ft, well within the stratosphere (Jacobs, 1954). In 1956 a volcanic eruption in Kamchatka intro- duced ash into the stratosphere which was observed five days later over England at an altitude of about 55,000 ft (Bull and James, 1956). The low moisture content of the stratosphere inhibits precipitation scrubbing of dust, and residence times for fission products are known to be much longer than in the troposphere and of the order of a year (Feely, 1960). Fission product studies likewise show that maximum fall-out from the stratosphere comes at 298 REX AND GOLDBERG [CHAP. 5 mid-latitudes over the break in the tropopause (Feely, 1960). This occurs over the meander path of the jet stream making it somewhat unhkely that we shall be able to see a mineralogical contribution of stratospheric fall-out among the much more abundant accumulation of purely tropospheric fall-out in pelagic sediments, 2. Eolian Materials in Marine Sediments Darwin (1846) was aware of the atmosphere as a path of transport of dust from the continents to the sea and may well have been correct when he stated that "a widely extended deposit may be in the process of formation" by such mechanisms. Over the past 100 years or so, a modest literature has evolved emphasizing the importance of wind-transported materials in marine sedi- mentation. Three genetic classes of eolian transported materials are evident : (1) Extra-terrestrial materials (2) Solids of biological origins from the continents (3) Solids of inorganic origin from the lithosphere, including debris from volcanic activity. A. Extra-terrestrial Components In 1876 Murray directed attention to some rather curious spherical particles in deep-sea sediments which had a highly magnetic character. Their sizes were seldom greater than 0.2 mm and more commonly in the 30-60 micron range. These spherules normally have an inner nucleus of iron, surrounded by a mag- netic crust. The metallic phase suggested a similarity to iron meteorites and led Murray to postulate a cosmic origin for the spheres, a hypothesis later confirmed by chemical analyses (Smales, Mapper and Wood, 1958 ; Castaing and Fredriks- son, 1958; and Hecht and Patzaic, 1957). Murray attributed the oxidized coating to a combustion of the iron nucleus while passing through the earth's atmosphere ; this conclusion was experimentally tested by Castaing and Fredricksson (1958) who found that the contents of nickel, cobalt and iron were in like concentration in both the shell and metallic core. No geographic distributions of such spherules have as yet been established, but, on the basis of their concentration in deep-sea sediments as a function of depth below the sediment-water interface, Pettersson and Fredriksson (1958) suggest that the frequency of such spherules may have been higher in recent times than in the past. They further compute the yearly accruement of the spherules to the earth's surface to be of the order of 2.5 to 5 x 10^ grams annually. B. Biological Components The pioneering studies of Ehrenberg (1847) on Darwin's samples of Sahara dust collected over the Atlantic and other dust samples showed the abundance of diatom fragments in materials collected from the atmosphere. The occurrence of freshwater diatoms in marine sediments (Lohman, 1941 ; Kolbe, 1955, 1957) SECT. 2] INSOLUBLES 299 in Atlantic deep-sea cores has invited thought as to a process of wind conveyance from land. These plant remains were found in Atlantic deep-sea cores but not in the Pacific and Indian Ocean sites traversed by the Swedish Deep-Sea Expedition, according to Kolbe {op. cit.). In the Atlantic, over 60 freshwater species, belonging to various ecological niches, were found. Kolbe suggests that the transporting agency might be the "Harmattan Haze", an atmospheric disturbance resulting from the Northeast Trade Winds. Investigations preceding the work of Kolbe had found that the haze was com- posed primarily of diatom shells and their fragments. Of the 51 freshwater diatoms identified in the haze, 25 were found in Atlantic sediments. The components of the haze apparently originate in the arid, but periodically inundated, swamp districts of the Niger and its tributaries. In addition to the diatoms, the Harmattan dust also carries ashes of burnt plants resulting from prairie fires fanned by the Northeast Trades. C. Continental and Volcanic Components An initial entry into the problem of the identification of wind-borne materials in marine sediments was made by Radczewski (1937, 1937a) who was impressed by the rather common occurrences of dust falls observed as far as a thousand nautical miles from land. The rather reddish coloration in such clouds resulted from the preferential loss of the larger yellowish quartz grains to near-coastal areas. Particles found in the dusts included quartz, clay minerals, calcite, iron oxides, feldspars, with minor quantities of green hornblende, biotite, tourmaline, garnet, epidote, titanite, rutile and zircon. Radczewski was stimulated to seek out a material, diagnostic of the source area, which was a major component in the dusts and which could be readily identified in the sediments. His area of study was the Cape Verde basin and the most probable source for eolian material was the Sahara desert. The mode of transport was presumed to be the equatorial easterly winds. Quartz grains, coated with a red-brown hematite, proved to be an excellent mineral indicator, characteristic of the desert material. This material, named "Wiistenquarz", represented between 3.1 and 39% of the quartz in the 10-50 micron size range and between 5 and 22% in the 5.5-10 micron size range. These percentages decreased in those deposits farthest from shore and the highest values were found in the inshore coarse fractions. Finally, the Wiisten- quarz was found in all samples of the last glacial and interglacial stages. In the Pacific, Rex and Goldberg (1958), on the basis of a marked latitudinal dependence of quartz in surface sediment samples, suggested a windborne origin for this material. Quartz is a mineral nearly alien to the basaltic rocks of the Pacific basin, and its distribution provides a key to the transporting processes. It shows a maximum concentration (on a calcium carbonate free basis) around 30°N and somewhat less distinctly around 35°S (Fig. 2). The particles occur in the deposits as chips and shards in the silt range with 2-10 micron particles most abimdant. Highest concentrations occur in a region 300 REX AND GOLDBERG [chap. 5 farthest from land in the area north of Hawaii, where around 25% of the solid phases consist of quartz. The Stoke's law settling times of approximately two years for 10 micron particles to depths of 4000 meters (average Pacific deep-sea depth), in conjunc- tion with the above observation of the highest quartz contents being farthest from land, have suggested modes of transport other than water currents. It is very difficult to fit the observed latitudinal variations with any observed circulation of Pacific waters. A number of lines of evidence have strengthened the hypothesis of the atmospheric transportation of the quartz. These include the observed correla- tion of quartz distribution in sediments to exposed arid land areas and the location of the jet streams in both hemispheres. Also the high wind velocities of the main jet streams in the upper troposphere make them logical transporting 25 ^i 15 if lOh 5 SOUTH LATITUDE NORTH Fig. 2. The quartz concentrations and arid land areas as a function of latitude. agents. Near-surface filter-feeding plankton, collected from a number of mid- Pacific areas, all contained mineral debris, including quartz, mica and feldspars in their guts, indicating recent dust falls. Finally, X-ray diffraction analysis of the Asahikawa fall-out and pelagic red clays (on a calcium carbonate free basis) from north of Hawaii are indistinguishable in nearly all details. The mica and feldspars show a close correlation with quartz in sediment from the Pacific and they also occur in the Asahikawa fall-out. In Fig. 3 feldspar contents are seen to show linear dependences upon the quartz concentrations with one correlation under the path of the westerlies of the Northern Hemi- sphere and still another under the easterly tropical circulation. The micaceous clay mineral illite shows a distribution pattern in the Pacific somewhat paralleling that of quartz (Griffin and Goldberg, Volume 3). Illite is much more prevalent in the North than in the South Pacific and its concentration follows a latitudinal band somewhat broader and more diffuse SECT. 2] INSOLUBLES 301 than that of quartz. These observations probably are related to the much smaller size range of the clay mineral (less than 1 micron) and of atmospheric and hydrospheric fractionation processes. One might expect the quartz to be less dispersed on the sea floor, subsequent to fall-out, due to its larger size and hence much smaller residence time in both the air and oceans. An independent line of evidence as to the continental origin of the illite evolves from the work of Hurley, Hart, Pinson and Fairbairn (1959) who found, from potassium-argon dating of surface Pacific illitic clays, various ages up to hundreds of millions of years. This work clearly negates an authigenic origin of illites. The longer cores from the Pacific show marked changes in both the quartz and illite contents from modern to older strata (Fig. 4). Such curves, when combined with appropriate age-dating techniques, may provide a paleo- meteorological or paleoclimatic tool which may have recorded in the sediments past climates and/or past air movements. Feldspar % = 0.14 quartz % + 0.43 4 0^ -- 0.043 n W =26 ^ ^ % ^ r = 0.94 • rA*' c o • <:^* " 2 ■y o Q. ^y^ 3 1 „< £ ^'' s? [ Ill Feldspar % = 0.51 quartz % + 0.32 (J' = 0.045 N =12 r = 0.96 7 10 14 % quartz 2 6 % quartz (a) (b) Fig. 3. (a) All samples from the Province of Hilly Topography in the eastern Pacific north of 8°N, east of 170°W and more than 500 miles from the continent. (b) All samples containing detectable quartz and feldspar (phillipsite-free) from the Province of Hilly Topography from 8°N to 18°S more than 500 miles from the continent. Volcanic ash horizons have been of major importance as stratigraphic indicators. For example, Bramlette and Bradley (1940) used sihcic volcanic ash zones to establish a stratigraphy in Atlantic sediments. The correlation of ash beds with sources and a transport mechanism has received some attention. The difficulties in such work stem from our insufficient capability of predicting the dispersal patterns of dust. Further, quantitative approaches to the relative contributions of volcanic materials to sediments are hampered primarily because of the difficulty of recognizing degradation pro- ducts resulting from chemical and physical changes in the original material. Mellis (1954) has suggested that certain ash horizons in Mediterranean cores arose from activity of the Santorin eruptions. Kuenen (1950) and Neeb (1943) noted ash from modern eruptions of Tambora and Oena Oena in Indonesia out to distances of 300 km, which contributed up to 80% of the material in certain deposits. 302 REX AlTD GOLDBERG [chap. 5 Recently, an extensive ash deposit, extending from at least 11°N to 12°S within a few hundred miles off the coast of Central and South America has been described by Worzel (1959) and Ewing, Heezen and Erickson (1959). The ash layer, 5 to 30 cm thick over this area, is attributed to one of the possible volcanic sources, the Galapagos Islands, eastern Ecuador and Central America. Ewing et al. {op. cit.) point out that the prevailing surface winds through the area are easterly and could have transported the ash. Also, currents in the equatorial system may have been responsible for the dispersal. They suggest that, although both atmospheric and marine transport are possible, no combina- tion of such paths could have provided the uniform ash layer over the areas CAPRICORN BP 50 WSS N 124°12 W "I — I — I — I — I — I — I — I r~ 17A PEAK AREA / 10 A PEAK AREA X 4 Fig. 4. The quartz and montmorillonite/illite ratios of Capricorn 50BP as a function of depth in the core. Station at 14° 55'N, 124° 12'W. Depth 4270 meters. investigated without providing substantial amounts over an even far larger area. Certainly further work is needed to understand better this extensive ash layer. It appears evident that paleometeorology has only begun to be developed and that in the oceanic accumulations of volcanic, continental and extra- terrestrial sediments we have a means of studying long-period processes in the earth's atmosphere. References Barbour, G. B., 1927. Loess in China. Ann. Rep. Smithsonian Inst. (1926), 279-296. Bramlette, M. N. and W. H. Bradley, 1940. Geology and biology of North Atlantic Deep- Sea cores between Newfoundland and Ireland. Part 1. Lithology and geologic inter- pretations. U.S. Oeol. Surv. Prof. Paper, 196A, 34 pp. Brown, W. F., 1952. Volcanic ash over the Carribbean, June 1951. Monthly Weather Rev., 80, 59-62. SECT. 2] INSOLUBLES 303 Bull, C A. and D. G. James, 1956. Dust in the stratosphere over Western Britain on April 3 and 4. 1956. Met. Mag., 85, 293-297. Castaing, R. and K. Fredriksson, 1958. Analysis of cosmic spherules with an X-ray micro- analyzer. Geochim. et Cosmochim. Acta, 14, 114—117. Clerici, E., 1901. Le polveri sciroccali cadute in Italia nel marzo 1901. Boll. Soc. Geol. Ital., 20, 169-178. Clerici, E., 1902. Ancora sulle polveri sciroccali e suUe pallottole dei tufi vulcanici. Boll. Soc. Geol. Ital., 21, 39-41. Darwin, C, 1846. An account of the fine dust which falls on vessels in the Atlantic Ocean. Q. J. Geol. Soc. London, 2, 26. Ehrenberg, C, 1847. Passatstaub und Blutregen. Ahhandl. Konigl. Akad. Wiss. Berlin, 269-460. Ewing, M., B. C. Heezen and D. B. Erickson, 1959. Significance of the Worzel Deep Sea Ash. Proc. Nat. Acad. Sci., 45, 355-361. Feely, H. W., 1960. Strontium-90 content of the stratosphere. Science, 131, 645-649. Futi, H., 1939. On dust-storms in China and Manchoukuo. J. Met. Soc. Japan, ser. 2, 17, 473-486. Glawion, H., 1938. Staub und Staubfalle in Arosa. Beitr. Phys. frei. Atmos., 25, 1-43. Hecht, F. and R. Patzaic, 1957. Astronaut. Acta, 3, 47 (quoted by Smales, Mapper and Wood, 1958). Hellman, G. and W. Meinardus, 1901. Der grosse Staubfall vom 9 bis 12 Marz 1901 in Nordafrika, Siid- und Mitteleuropa. Konigl. Preuss. Met. Inst., 2, 1-93. Hurley, P. M., S. R. Hart, W. H. Pinson and H. W. Fairbairn, 1959. Authigenic versus detrital illite in sediments. Geol. Soc. Amer. Meeting Ahstr., 64A. Jacobs, L., 1954. Dust clouds in the stratosphere. Met. Mag., 83, 115-119. Junge, C. E., 1957. "Remarks about the size distribution of natural aerosols", in Artificial stimulation of rain. Weickman and Smith, ed., Pergamon Press, New York, 3-17. Kolbe, R. W., 1955. Diatoms from Equatorial Pacific cores. Rep. Swed. Deep-Sea Exped., 6. Kolbe, R. W., 1957. Fresh water diatoms from Atlantic deep-sea sediments. Science, 126, 1053-1056. Kuenen, P. H., 1950. Marine Geology. Wiley, New York. Larsson, W., 1937. Vulkanischen Asche vom Ausbruch des chilenischen Vulkans Quizapii (1932) in Argentina gesamelt: Eine studie uber aolische Differentiation. Bull. Geol. Inst. Univ. Upsala, 26, 27-52. Libby, W. F., 1956. Radioactive strontium fall-out. Proc. Nat. Acad. Sci., 42, 365-390. Libby, W. F., 1956a. Radioactive fallout and radioactive strontium. Science, 123, 657-660. Lohman, K. W., 1941. Diatomaceae. Geol. Biol. N. Atlantic Deep-Sea Cores, Part 3. Marshall, P., 1929. Dust storm of October 1928. N.Z. J. Sci. Tech., 10, 291-292. Mellis, O., 1954. Volcanic ash-horizons in deep-sea sediments from the eastern Mediter- ranean. Deep-Sea Res., 2, 89-92. Mintz, Y., 1954. Observed zonal circulation of the atmosphere. Bull. Amer. Met. Soc, 35, 208-214. Miyake, Y., Y. Sugiura and Y. Katsuragi, 1956. Radioactive fall-out at Asahikawa, Hokkaido in April, 1955. J. Met. Soc. Japan, ser. 2, 34, 226-230. Murray, J. and A. F. Renard, 1891. The scientific results of the voyage of H.M.S. Challenger. Deep-Sea Deposits. 1-525. Neeb, G. A., 1943. Bottom Samples. Snelliu^- Exped., 5, pt. 3, 55-265. Pettersson, H. and K. Fredriksson, 1958. Magnetic spherules in deep-sea deposits. Pacific Sci., 12, 71-81. Radczewski, O. E., 1937. ' Eolian deposits in marine sediments", in Recent Marine Sedi- ments. Parker Trask, ed., Amer. Assoc. Petrol. Geol., Tulsa, 496-502. Radczewski, O. E., 1937a. Die Mineralfazies der Sediments des Kapverden Beckens. Wiss. Erbegn. Deut. Atlant. Exped. 'Meteor', 1925-1929, B3, Tl. 3, 262-277. 304 REX AND GOLDBERG [CHAP. 5 Revelle, R., M. Bramlette, G. Arrhenius and E. D. Ooldberg, 1955. Pt'lagic sediments of the Pacific. Oeol. Soc. Amer. Spec. Paper, 62, 221-236. Rex, R. W. and E. D. Goldberg, 1958. Quartz contents of pelagic sediments of the Pacific Ocean. Tellus, 10, 153-159. Salmi, M., 1948. Hekla ashfalls in Finland A. D. 1947. Bull. Comm. Geol. Finlande, 21, 87-96. Scheidig, A., 1934. Der Loss und seine geotechnischen Eigenschaften. Dresden und Leipzig, 223 pp. Smales, A. A., D. Mapper and A. J. Wood, 1958. Radioactivation analysis of cosmic and other magnetic spherules. Geochim. et Cosmochini. Acta, 13, 123-126. Smith, G. D., 1942. Illinois Loess. Univ. III. Agr. Exp. Sta. Bull., 490, 139-184. Thorarinsson, S., 1954. The eruption of Hekla 1947-48, II, 3. The tephra-fall from Hekla on March 29, 1947. Soc. Set. Islandica, Reykjavik, 1-68. Walther, J., 1903. Der grosse Staubfall von 1901 und das Lossproblem. Naturwiss., 18, 603-605. Wilcox, R. E., 1959. Some effects of recent volcanic ash falls with special reference to Alaska. U.S. Oeol. Surv. Bull., 1028-N, 409-476. Wilkins, E. M., 1958. Precipitation scavenging from atomic bomb clouds at distances of one thovisand to two thousand miles. Trans. Amer. Geophys. Un., 39, 60-62. Willis, B., E. Blackwelder and R. E. Sargent, 1906-1907. Research in China, 3 vols. Carnegie Inst., Washington D. C. Worzel, J. L., 1959. Extensive deep-sea sub-bottom reflections identified as white ash. Proc. Nat. Acad. Sci., 45, 349-355. 6. SOLUBLES A. H. Woodcock Among the many processes and events occurring in the surface waters of the sea is the momentary trapping of air in the form of bubbles. These bubbles range in size from a few millimeters to several microns in diameter. They result in major part from meteorologically induced disturbances, such as the breaking of wind-waves, the impact of raindrops and the melting of snow and hail (Blanchard and Woodcock, 1957), («) (b) Fig. 1. (a) Composite of motion pictures showing stages in bubble collapse, jet and droplet formation. Bubble diam. 1.7 mm; time interval top to bottom frame 0.002 sec. (b) Oblique view of jet and droplets from 1.0 mm diam. bubble. Smallest of three ejected droplets 90 [i diam. Exposure 30 micro-sec. [MS received October, 1960] 305 II— S.I 306 WOODCOCK [chap. 6 In the sea these bubbles will dissolve or grow larger, depending upon their size and the degree to which the water is undersaturated or supersaturated with atmospheric gases (loc. cit.). However, we are not concerned here with the role of the bubbles in the exchange of gases between sea and air, but in their effectiveness, upon bursting, in the ejection of droplets of sea-water into the overlying air. Changes in the bubble sizes below the surface are important, however, for they will alter the eventual sizes of the droplets produced at the surface {loc. cit.). DIAMETER OF BUBBLE (^i) Fig. 2. Droplet ejection height as a function of size of bubbles bursting in sea-water and distilled water. The distilled water data are from Stuhlman (1932). High-speed photography of bursting bubbles has revealed something of the nature of the mechanism which projects several droplets of sea-water from the tip of a jet formed as the bubble cavity collapses (see Fig. 1 and Kientzler et al., 1954). The relationship between bubble size, droplet-ejection height and droplet size are shown in Figs. 2 and 3. In clean sea-water, free from surface films, the size of the top droplet and the height to which it is thrown changes very little for bubbles of nearly constant diameter. The lower drops from these bubbles show greater non-uniformity, as can be seen in Figs. 2 and 3. In SECT. 2] SOLUBLES 307 Fig. 3 it is seen that droplets ranging from about 5 [j, to 100 (j. radius are pro- duced. These droplets have fall speeds of 0.4 to 80 cm sec-i. Some of them evaporate almost completely, leaving crystalline sea-salt residues immersed in the atmosphere. Other droplets remain in the air a relatively short time, only partially evaporating before falling back into the sea. In addition to the water and dissolved salts, Blanchard (1958) has shown that these droplets carry an electric charge into the air. Organic detritus and surface-film materials are also occasionally observed and it is suspected that living organisms in the form of bacteria or viruses may also become airborne in the droplets (Woodcock, 1955 ; Zobell, 1942). 1000 2000 DIAMETER OF BUBBLES (|i Fig. 3. The size and salt content of droplets ejected by bubbles of various diameters bursting in sea-water. We are concerned here, however, with the solubles only. Among the particle sizes represented in Fig. 3, these solubles have been shown to be almost entirely "sea spray" salts. For instance Junge (1955, 1957) separated the marine aerosol particles ^ 0.8 [j, radius from those smaller. He found that the ratios of the chloride to the sulfate and chloride to sodium among these "giant" nuclei were nearly the same as those found in sea-water. He concluded that these nuclei are largely sea-salt. Twomey (1954) and Woodcock and GifiFord (1949) reached the same conclusion ; the former from a study of the chloride and phase transition point of individual particles, and the latter from the applica- tion of isopiestic and micro-titration methods to the analysis of many particles sampled by impingement. Fig. 4 shows photomicrographs of atmospheric 308 WOODCOCK [chap. 6 sea-salt particles resting upon sampling surfaces. They are shown as crystalline masses at low relative humidity and as hemispheric droplets at higher humidity. Through the use of the above physical and physical-chemical methods, the weights and the numbers of these particles in marine atmospheres have been Reluiive humidity 91% V i i zom Q^ Relative humidity 42% Fig. 4. Photomicrographs of sea-salt deposited on glass sampHng sur- faces which had been exposed to marine air about 5 m above the sea surface. Pictures selected to show some of the largest of the particles, as wet crystalline masses at low relative humidity and as entirely liquid hemi- spheric droplets at high humid- ity. SEA-SALT PARTICLE WEIGHTS (IQ-'^ g) 10° 10' 10^ 10' 10^ 10 u I I I I Mil GD 10 -nri 1 Mill! SEA-SALT |19 m-' 2.4 5.1 11.0 23.7 PARTICLE RADIUS AT 99% R.H. (fi) Fig. 5. Typical distribution curves for sea-salt particles from various altitudes in Hawaii. Air conditionally unstable. Similar distribu- tion patterns are found in marine air of Australia, New England (U.S.A.), Bermuda, the Lesser Antillean Islands and Florida. determined at various altitudes, positions and surface-wind speeds. The range of weights found largely corresponds to that produced by the jets which arise from bursting bubbles equal in size to those found in breaking waves in sea- water (Blan chard and Woodcock, 1957). SECT. 2] SOLUBLES 309 Fig. 5 illustrates one common type of variability of particle size and number distribution with altitude over the sea. Similar distributions are found in conditionally unstable oceanic air near Australia, Hawaii, the West Indies, New England (U.S.A.) and over the central North Atlantic. The particles are largely confined to the lower atmosphere which is convectively or frictionally mixed or mixing. In stable air moving over colder water the ejected particles 10^ mIO 10- 10' ^;— I — I I I 1 1 III 1 — I 11 1 II -\ ALT. ^•^t^Q 153 10= k ->§.:s;-305 — I I I Miiy SEA-SALT 16 9 12 3 8.0 0 4 N ■^; •^ 305 \ : 300 r I / - E200 _ 1 h I-,- + ? r^ioo - \ n 1 + °i 5 T.°C -I •N* '.\ % -^•j 10^ 10' 10' 10- SEA-SALT PARTICLE WEIGHTS Fig. 6. Example of the effects of the surface cooling and stabilizing of the lower air in confining the major part of the salt aerosols to the regions near the sea surface. April 30, 1948, 48° 18'N, 70° 50'W. Wind SW, force 4. PARTICLE RADIUS AT 99% R.H. ( ji ) 51 no 237 511 10° 10' lO' lO' 10* lO' WEIGHT OF SEA-SALT PARTICLES (jitiq) Fig. 7. Wind force effects upon the number and the weight of sea-salt particles in the sub-cloud layer of air over the sea. Samples taken 100 m below base altitudes of local cumulus clouds. Curves read as follows : in force 5 wind, there are about 10^ particles m~3 larger than 170 [i[ig or during force 5 winds there are 900 particles m~3 in the weight range of about 1100 to 4000 y.[ig. are confined to a relatively shallow layer of air near the sea surface, as shown in Fig. 6. Fig. 7 shows the average numbers of salt particles associated with winds of various forces in the sub-cloud layer over the sea. This relationship is probably due in major part to the wind-produced "breaking" of waves and the con- sequent trapping of air in the form of multitudes of small bubbles. (The effects 310 WOODCOCK [chap. 6 of bubbles introduced by snow or rain storms upon aerosol populations have not been adequately measured.) It is clear from Fig. 7, assuming an equilibrium or steady-state condition, that the number of the larger airborne particles is more affected by altered wind force than is the number of the smaller. Note, for instance, that a change in wind from force 3 to 7 causes an increase of 30 times in the number of 10~9 g particles, while the 10~ii g particles are increased by 3 times. The detailed physical factors producing this result are not known. It is supposed that stronger winds produce more large bubbles in the sea, consequently, more large particles in the lower air, and that these particles are more likely to mix up to cloud-base altitudes by the added turbulence associated with these winds. 25001 I 5 10 15 20 25 SEA-SALT IN AIR (lO'^gm"') Fig. 8. Average total weight of sea-salt particles as a function of altitude and wind force. All measurements made in marine air but in widely separated geographical locations. The author has determined the weights of airborne sea-salt among several hundreds of samples taken over the seas at different geographical locations, times, altitudes and surface winds. These observations are summarized on Fig. 8 in a way which shows the average variability of the sea-salt load of marine air as a function of altitude and wind force. The integrated amounts at the various levels and surface-wind conditions were derived and are expressed in the table on Fig. 8, as grams per square meter of surface. It should be pointed out that all of the samples represented in Fig. 8 were taken in marine air which had passed over many hundreds and, most often, thousands of kilometers of sea surface. Ample time for an extensive exchange of properties was available, though it is uncertain to what extent an equilibrium SECT. 2] SOLUBLES 311 existed between the introduction of salt particles into the atmosphere and their return to the sea. In each sampling area, cimiulus clouds were developing at the top of a nearly mixed layer of air which extended down to the sea surface. Through the use of averages of data from the Hawaii area of the Pacific, and making certain assumptions about the relative humidity at the top of the boundary layer of the sea and the salinity of rains, Eriksson (1959, pp. 399-400) has estimated an average rate of fallout of salt from the whole atmosphere of about 10^ metric tons per year. Somewhat more realistic estimates by Blanchard {in litt.), using mean winds from a world climatic atlas, the data of Fig. 7 and reasonable assumptions about relative humidity and the salinity of rains, produced a value of about 2 x 10^ tons per year. These are the best estimates which have been made indicating the probable rate of production at the sea surface of salt particles which remain airborne at least long enough to be transported up to local cumulus cloud-base altitudes. The maximum value (2 x 10^ tons) is the quantity of salt contained in about the upper one-tenth of a millimeter of the sea surface. There is little doubt, however, that far more salt and water is ejected into the surface air by bursting bubbles remaining airborne only a relatively short time before falling back into the sea. A study of the droplets within a few meters of the sea surface is required before one can attempt a direct evaluation of their role in evaporation. From the above estimates it is clear that the rate of production of salt particles (droplets) at the surface must exceed that derived from the salt at cloud levels by a thousand times or more to contribute significantly to the total evaporation from the seas. It should be reported that droplet production has also been observed by Mason (1957) from the shattered surface films of bursting bubbles. The salt content of the largest of these droplets was estimated to be 2 x 10~i4 g, about one fiftieth of the weight of the smallest particle considered here. From Junge's numerous impactor measurements (Junge, 1957) it is reasonably concluded that particles ^2xl0~i4g contain less than 1% of the airborne chlorides, and contribute little to the exchange of solubles between sea and air. These film droplets may be meteorologically important, however, if they are produced in great numbers. At the present time the major significance of the exchange of sea-salt particles between the seas and the atmosphere seems to lie in their role as nuclei for the formation of large cloud droplets and of raindrops (Woodcock and Blanchard, 1955), in contributing to the "cyclic salts" of geochemistry (Eriksson, 1959), and in the transfer of charge (Blanchard, 1960). This exchange is a further example of the interrelation of many oceanic and atmospheric properties and problems. Exploration and expansion of this exciting area of study and under- standing has just begun. References Blanchard, D. C, 1958. Electrically charged drops from bubbles in sea water and their meteorological significance. J. Met., 15, 383-396. 312 WOODCOCK [chap. 6 Blanchard, D. C, 1960. The electrifioation of the atmosphere by paiticle.s from bubbles in the sea. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass. Blanchard, D. C. and A. H. Woodcock, 1957. Bubble formation and modification in the sea and its meteorological significance. Tellus, 9, 145-158. Eriksson, E., 1959. The yearly circulation of chloride and sulfur; meteorological, geo- chemical and pedological implications. Part I. Tellus, 11, 375-403. Junge, C. E., 1955. Recent investigations in air chemistry. Tellus, 8, 127-139. Junge, C. E., 1957. The composition of hygroscopic particles in the atmosphere. J. Met., 11, 334-338. Kientzler, C. F. et al., 1954. Photographic investigation of the projection of droplets by bubbles bursting at a water surface. Tellus, 6, 1-7. Mason, B. J., 1957. The oceans as a source of cloud-forming nuclei. Geofis. pur. appl. 36, 148-155. Stuhlman, O., 1932. The mechanics of effervescence. Physics, 2, 457-466. Twomey, S., 1954. The composition of hygroscopic particles in the atmosphere. J. Met., 11, 334-338. Woodcock, A. H., 1955. Bursting bubbles and air pollution. Sewage Industr. Wastes, 27, 1189-1192. Woodcock, A. H. and D. C. Blanchard, 1955. Tests of the salt-nuclei hypothesis of rain formation. Tellus, 4, 437-448. Woodcock, A. H. and M. M. Gifford, 1949. Sampling atmospheric sea-salt nuclei over the ocean. J. Mar. Res., 8, 177-197. Zobell, C. E., 1942. Microorganisms in marine air. Contr. Scripps Inst. Oceanog., No. 157. 7. GASES R. Revelle and H. E. Suess 1. Introduction The total volume of the oceans is 1.37 x lO^i liters. The amount of gases in the terrestrial atmosphere, expressed in liters of gas at standard temperature and pressure, is 4.32 x lO^i liters, or about three times larger than the volume of the oceans. The solubility of gases in a liquid is in general expressed as an absorption coefficient, a, giving the ratio of the concentration of the gas in the liquid to that in the gas phase. This ratio is independent of the partial pressure of the gas and is a function of the temperature. For the atmospheric gases the absorption coefficient in water is small, of the order of a few per cent. Because of the salting out effect, it is even smaller in sea-water than in fresh water. The fraction of the atmospheric gases dissolved in the oceans is, therefore, in general small compared to their total amount present on the surface of the earth. Table I Gases in Deep Ocean Water as Compared with Their Concentration in Air Percentage of total gas on Milliliter per Milliliter per earth dissolved liter air liter water in ocean O2 210 2 to 8 0.5 N2 781 13 0.6 He 52 X 10-4 0.5 x 10-4 0.3 Ne 182x10-4 1.8x10-4 0.3 A 9.32 0.32 1.2 Kr 10x10-4 0.6x10-4 2 Xe 0.8 X 10-4 0.07 x 10-4 3 CO2 0.3 50 6000 An exception to this is carbon dioxide. This gas is present in ocean water chemically bound in the form of carbonate or bicarbonate, in amounts that exceed that of gaseous CO 2 in the atmosphere by about a factor of 60. The solubility of CO 2 in sea-water depends on its alkalinity and temperature. The greater part of this chapter will deal with the CO 2 equilibrium between ocean and atmosphere, a system about which the least has been known and on which the greatest progress has been made in the last few years (Revelle and Fair- bridge, 1957). However, it will be shown that our knowledge of the quantities involved is by no means satisfactory at the present time. Table I lists the average content of gases in deep -ocean water in comparison with the amounts present in air. The data for the rare gases He, Ne, Kr and Xe are from a single analysis by Hintenberger, Konig and Suess (unpublished) of [MS received March, 1961] 313 314 REVELLE AND SUESS [CHAP. 7 a gas sample extracted from ocean water taken at 5350 meters depth halfway between Hawaii and Christmas Island (14° 12'N, 155° 08' W) in the Pacific Ocean. 2. Oxygen and Nitrogen The oxygen content is one of the most characteristic quantities of an oceanic mass and varies between the rather large limits of 2 to 8 milliliters/liter. These variations, on which an extensive literature exists, are due to biological activity, viz. photosynthesis and bacterial oxidation of organic matter. A summary of present ideas is given by Richards (1957). The most interesting feature of the O2 distribution in the oceans is the occurrence of an oxygen minimum at intermediate depth, and its quantitative explanation includes unsolved oceanographic problems. The amounts of dis- solved nitrogen are far less accurately investigated. According to Rakestraw and Emmel (1938) these amounts show only minor variations. In general the nitrogen concentration should correspond to its solubility, although effects from biological activity may be expected locally. 3. Rare Gases The rare gases dissolved in the oceans have not yet been investigated in detail. Because of their chemical inertness, their concentration should correspond approximately to their solubilities at the respective water temperatures. However, the temperature dependence of solubility is relatively large and is greatest in the case of Xe which is more than twice as soluble in water at 0°C than at 25°C (Morrison and Johnstone, 1954). Heat transfer through the ocean surface by conductivity and radiation is presumably considerably faster than transfer and equilibration of atmospheric gases, and, therefore, one might expect appreciable deviations from equilibrium concentrations. Mass-spectro- scopic isotope-dilution techniques now make it possible to determine small quantities of rare gases accurately to within 1% without too much effort, so that such measurements may provide new possibilities to study the ocean- atmosphere interchange of gases and transfer phenomena. Even more sig- nificantly, the mixing of water masses of different temperatures might be determined by accurate measurements of the variations in the xenon/neon ratio, taking advantage of the fact that the temperature coefficient for solubility of xenon is not linear with temperature. By comparison of the xenon con- centration with that of helium in the bottom water (see below) the amount of heating of this water by heat from the earth's interior and hence its "age" could be independently determined and an estimate could be made of the rate of escape of helium from the crust and mantle. The rare gas isotopes "^He and ^^A are produced by radioactive decay. The amounts produced, however, are small. In one liter of sea-water about 30 He atoms per minute are produced from the decay of uranium and its daughters, SECT. 2] GASES 315 and about 50 ^"A atoms per minute from the decay of ^oK. To generate the ''He and ^^A dissolved in the ocean through decay of U and K present in sea- water would take 10** and 10^ i years respectively. The apparent radiocarbon age of deep sea-water (see page 317) gives an upper limit of about 10^ years for the time since equilibration with atmospheric gases has been established. Because of its low atomic weight, He escapes from the earth's atmosphere. The small amount present in the atmosphere and the oceans represents the equilibrium between the amount escaping and the amount produced by radio- active decay in the lithosphere and transferred to the atmosphere and ocean. A He-flux from the sea floor, supplying 5% of the He present in the bottom 1000 meters of sea-water in 1000 years, would correspond to the amount of He produced from U and Th in a few kilometers of sediments and suboceanic rock. It may be possible, therefore, to detect with reflned measurements a helium gradient in the bottom water of the oceans produced by the escape of helium from beneath the sea floor. The situation for argon is much less favorable because of the higher 40^ content of the atmosphere and the higher proportion of the dissolved argon to atmospheric argon. Argon, unlike He, does not escape from the atmosphere and, therefore, 40A produced from ^ok decay has accumulated in the atmos- phere during the lifetime of the earth. At the same time it is more soluble in sea- water than helium. 4. Carbon Dioxide The dissolved CO2 in the ocean and its hydrated forms of molecules or ions constitute a complex system. Equilibrium between these various components is established in sea- water, even though the rate of hydration of CO 2 is a relatively slow chemical reaction, in the sense that its time constant is of the order of seconds or minutes, depending on temperature, pH, and other variables. The rate of uptake of CO 2 by a solution is also known to be slow and dependent on many factors, which makes it difficult to compare results of laboratory experiments with conditions prevailing in nature. If CO2 is added to the atmosphere, as has happened during the past century by the combustion of fossil fuels, part of this added amount will be taken up by the ocean. The partition between the ocean and the atmosphere and the rate of partition are both important quantities because the CO 2 concentration in the atmosphere may very well be a factor in the heat balance of the earth as a whole, and notable amounts of CO 2 have been released into the atmosphere by industrial combustion of fossil fuel. However, until a few years ago, virtually nothing was known about the rate of exchange and uptake of CO 2 from the atmosphere into the sea, and estimates of the time constants involved ranged from some 10,000 years to a few days. The discovery of natural radiocarbon has now at least allowed narrowing of this range to a value accurate presumably to about one order of magnitude. The British meteorologist, Callendar (1938, 1958), believed that nearly all 316 REVELLE AND SITESS [OHAP. 7 the carbon dioxide produced by fossil-fuel combustion remained in the atmos- phere. He came to this conclusion by com])aring COq analyses of air made in the 19th century with those made more recently. Detailed statistical evalua- tions have been made by Slocum (1955) and Bray (1959) with widely differing conclusions as to the statistical significance of the available data. As was shown in a previous paper by the authors (Revelle and Suess, 1957), one can conclude from radiocarbon measurements that the rate of exchange and uptake of CO2 by the oceans must be of the order of 10 years. Arnold and Anderson (1957) arrived independently at the same conclusion. Other in- vestigators (Bolin and Eriksson, 1959; Bolin, 1960; Broecker, Tucek and Olson, 1959; Craig, 1957, 1958; Rafter and Fergusson, 1957) attempted to derive a more precise figure for this exchange rate, or more accurately, for the residence time of CO2 in the atmosphere, by evaluation of the empirical measure- ments by more rigorous calculations. All the considerations were based on the measurements of two empirical quantities: (1) the apparent i^C age of ocean ATMOSPHERE i-^lf- No MIXED LAYER 1,2 N, STRATOSPHERE TROPOSPHERE i^lf- OCEAN 58 Na BIOSPHERE 0.5 Aio -If— HUMUS 1.5 A/o Fig. 1. Carbon exchange reservoirs. N^ denotes the number of C atoms in the atmosphere. For each exchange reservoir the number of C atoms in terms of Na is shown. water and (2) the effect of industrial fuel combustion on the specific activity of atmospheric CO 2. However, our knowledge of both these quantities seems now less firm than it was thought to be a few years ago. A more detailed mathematical treatment cannot, therefore, appreciably improve the precision of the results. Fig. 1 shows schematically the various "exchange reservoirs" of carbon and their relative size, taking CO2 of the atmosphere as unity. The ocean is divided by the thermocline into a mixed layer and deep water. Craig (1958) discusses a "chain model" and a "cyclic model". The first considers transport of CO2 into the deep sea through the mixed layer [(1) in Fig. 1] ; the second also takes into account direct exchange with the deep water in polar waters [(2) in Fig. 1] where no thermocline exists. The third alternative, viz. no downward mixing through the thermocline and exchange exclusively occurring in polar waters was used by Broecker, Gerard, Ewing and Heezen (1960). Craig (1958) obtains a value for the residence time in the atmosphere of 7 + 3 years. Broecker, Gerard, Ewing and Heezen (1960) obtain a similar value and also discuss non- SKOT. 2] GASKS 317 steady-state mixing in the ocean, with ])eriods of ra])id mixing alternating with periods during which no mixing occurs. Duration of periods is assumed to be of the order of 100 years. Fergusson (1958) beheves that the residence time is less than 5 years, based on observation of local changes in oceanic ^'^C in the surface-water layers. Also Bolin and Eriksson (1959) favor a residence time of five years. Over the last five years a large number of ^^C determinations of the dissolved bicarbonate in ocean water have been carried out by the Lament Geological Observatory (Broecker and Olson, 1959), by the New Zealand Radiocarbon Laboratory (Burling and Garner, 1959) and at the Scripps Institution of Oceanography (Bien, Rakestraw and Suess, 1960). The new data, however, tend to demonstrate the uncertainties in our knowledge rather than to allow a more precise evaluation. The most important new observation is that i'*C concentrations in Pacific surface and deep waters north of the Antarctic con- vergence differ by a larger factor than in the surface and deep water in the Atlantic. Fig. 2, taken from a paper by Bien, Rakestraw and Suess (1960), illustrates the situation. The radiocarbon concentrations in the Atlantic cover a much more narrow range, indicating faster mixing. The situation in the Pacific is simplified by the fact that cold deep and bottom water is supplied only from the Antarctic. The northward movement of water at 3500-m depth can be recognized by an increase in apparent i^C-age from 1500 years at latitude 30°S to 1700 years at latitude 30°N. The fact that deep Pacific water shows a greater apparent age than Atlantic water suggests a somewhat slower exchange and a longer residence time of CO2 in the atmosphere than was assumed previously. A second new observation that demonstrates our ignorance is that the decrease in the specific ^^C activity measured in tree-rings, which was assumed to be caused by burning of fossil fuel, did not begin at about 1850 but roughly a century earlier (deVries, 1958, 1959; Broecker, Olson and Bird, 1959; Suess, 1960). DeVries suspected that this decrease in i^Q activity had something to do with the general glacial advances during the 18th century. In his opinion, an inverse correlation exists between I'^C in the atmosphere and glacial advances and retreats. With such a correlation, the effects from industrial coal combus- tion should in reality be larger than the observed decrease of 2% in i-*Q because the glaciers began to retreat again during the middle of the last century, so that a reversal of the general trend should have occurred. This reversal was compensated for by artificial fuel combustion. De Vries estimates that the total effect of fossil fuel combustion should be about 2.7%. However, the present ^''C values in the atmosphere can be obtained by a mere extrapolation of the trend observed during the 18th and 19th centuries. Although it still seems probable that the observed decrease of 2% over the past 100 years is at least in part due to combustion of fossil fuel, it is now impossible to draw any quantitative conclusions from the magnitude of the effect. There is good reason to believe that, during the coming years, a third method will become available to determine the residence time of CO 2 in the atmosphere A"C PER MILL, LAMONT NORMALIZATION -50 100 -150 -200 -250 APPARENT "C AGE RELATIVE TO STANDARD 19™ CENTURY WOOD + 50 0 -50 -100 -150 -200 250 A'X PER MILL, LAMONT NORMALIZATION Fig. 2. Radiocarbon in the oceans according to Bien, Rakestraw and Suess (1962). The figure shown here deviates in some details from the one pubUshed in the reference. The New Zealand values have been changed by 9%o according to informa- tion received from G. J. Fergusson. Also some minor changes in the La JoUa data were made corresponding to a better normalization of our standards. Upper part : A^'^C in the Pacific Ocean plotted as a function of latitude for surface- water samples (open circles indicate La JoUa measurements, cross New Zealand measurements), and for samples from approximately 3500 m depth (solid circles). Note the low values of the surface waters south of 60°S latitude, showing that equili- brium with the atmosphere has not been established. Middle part : Range of A ^^C at all depths in the Atlantic and in upwelling surface waters off California. Normalization and the conversion of A^'^C to apparent ages. Values, except that for the NBS oxalic acid standard, are normalized to a S^^C of — 25%o relative to the Chicago PDB standard. Lower part : A^'^C as a function of depth in the Pacific Ocean. Crosses indicate New Zealand measurements. Open circles show results on samples collected by J. A. Knauss, and solid circles by J. E. Reid. SECT. 2] GASES 319 and the rate of its uptake into the ocean. Since 1954, large amounts of "arti- ficial" ^-^C have been introduced into the atmosphere, primarily into the stratosphere. At present, mixing of tropospheric and stratospheric air has not been completed and at sea-level the i^C concentration in the atmospheric CO 2 is still rising. In the spring of 1960, the ^^C in the troposphere had risen to more than 30° Q above its pre-bomb 1953 value. The increase due to bomb testing was first published by Rafter and Fergusson (1957) of the New Zealand labora- tory, and has since been observed by most i^c laboratories. The level of i^C in the Southern Hemisphere lags about eighteen months behind that of the Northern Hemisphere. If and when there is a cessation of atmospheric testing, the artificial excess ^^C will, over the following years, equilibrate with that of the oceans, so that the atmospheric ^^C activity will again decrease steadily down to near normal levels. After some time required for the mixing of the stratospheric and tropospheric air-masses, the rate of decrease will correspond to the mean residence time of CO 2 in the atmosphere. At the same time the 14C in the surface water of the oceans will rise markedly. Broecker and Olson (1950. 1960) estimated the expected changes in ^^C concentration under the assumption that testing had ceased in 1960. According to these authors the atmospheric i^C activity would have reached a maximum around 1964, and average surface ocean water could have been expected to reach a maximum 14C activity, of about 15% above the present, around 1975. Because of the un- certainties in the basis of these estimates, a considerably different course of events would not be surprising. An accurate knowledge of the residence time of CO2 in the atmosphere and rate of absorption into the sea must be combined with other factors in order to predict the effect of fossil-fuel combustion upon the CO2 content of the atmosphere in future times with desired precision. In particular one has to consider the peculiar buffer mechanism of sea-water, which causes an increase in the partial CO 2 pressure of more than an order of magnitude higher than the increase in the total CO 2 concentration when CO 2 is added and the alkalinity remains constant (Revelle and Suess, 1957 ; Kanwisher, 1960). This affects not only the equilibrium distribution of CO2 between atmosphere and ocean but also the rate of CO 2 uptake in particular in the case of relatively slow mixing through the thermocline. This circumstance increases the amount of CO 2 from industrial fuel combustion in the atmosphere. In any case, radiocarbon measurements have shown that a large fraction of the CO 2 released by industrial coal combustion ^in any case more than 50%) has been taken up by the ocean. To arrive at a more quantitative figure and make more accurate predictions as to the future increase of CO 2 in the atmos- phere seems at present impossible. In order to make precise predictions of this secular increase, one will probably have to go back to the original way sug- gested by Callendar, which is simply to compare precise analytical data taken over many years. Because of their poor accuracy, the data from analyses of CO 2 in air made in the 19th century cannot be used for this purpose. A very precise series of measurements started during the IGY by Keeling (1960) will 320 REVELLE AND SITESS [CHAP. 7 certainly be useful for comparing with future data. Within the next ten years, a possible increase of a few parts per million in the CO 2 in the air will not escape empirical observation. References Arnold, J. R. and E. C. Anderson, 1957. The distribution of carbon-14 in nature. Tellus, 9, 28. Bien, G., N. Rakestraw and H. E. Suess, 1962. Radiocarbon concentration in Pacific Ocean water. Tellus, in press. Bolin, B., 1960. On the exchange of carbon dioxide between the atmosphere and the sea. Tellus, 12, 274. Bolin, B. and E. Eriksson, 1959. Changes in the carbon dioxide content of the atmosphere and sea due to fossil fuel combustion. The Rossby Memorial Volume, Rockefeller Institute Press, New York. Bray, J. R., 1959. An analysis of the possible recent changes in atmospheric carbon dioxide concentration. Tellus, 11, 220-229. Broecker, W. S., R. Gerard, M. Ewing and B. C. Heezen, 1960. Natural radiocarbon in the Atlantic Ocean. J. Geophys. Res., 65, 2903. Broecker, W. S. and E. A. Olson, 1959. Lamont radiocarbon measurements. IV. Am,er. J. Sci., Radiocarbon Suppl., 1, 111. Broecker, W. S. and E. A. Olson, 1960. Further information on radiocarbon from nuclear tests. II. Science, 132, 712. Broecker, W. S., E. A. Olson and J. Bird, 1959. Radiocarbon measurements on samples of known age. Nature, 183, 1582. Broecker, W. S., C. S. Tucek and E. A. Olson, 1959. Radiocarbon analysis of oceanic CO2. Intern. J . Appl. Radiation Isotopes, 7, 1. Burling, R. W. and D. M. Garner, 1959. A section of C^^ activities of sea water between 9° S and 66° S in the southwest Pacific Ocean. N.Z. J. Geol. GeopMjs., 2, 799. Callendar, G. S., 1938. The artificial production of carbon dioxide and its influence on temperature. Q. J. Roy. Met. Soc, 64, 223. Callendar, G. S., 1958. On the amount of CO2 in the atmosphere. Tellus, 10, 243-248. Craig, H., 1957. The natural distribution of radiocarbon and the exchange time of carbon dioxide between atmosphere and sea. Tellus, 9, 1. Craig, H., 1958. A critical evaluation of radiocarbon techniques for determining mixing rates in the oceans and the atmosphere. Proc. Second Intern. Conf. Peaceful Uses Atomic Energy, Geneva. DeVries, HI., 1958. Variation in concentration of radiocarbon with time and location on earth. Koninkl. Ned. Akad. Wetenschap., B61, 94-102. DeVries, HI., 1959. Review article on measurement and use of natural radiocarbon. In Researches in Geochemistry, Wiley Press, New York. Fergusson, G. J., 1958. Reduction of atmospheric radiocarbon concentration by fossil fuel carbon dioxide and the mean life of carbon dioxide in the atmosphere. Proc. Roy. Soc. London, A243, 561. Kanwisher, J., 1960. CO2 in seawater and its effect on the movement of CO2 in nature. Tellus, 12, 209. Keeling, Charles D., 1960. The concentration and isotopic abundance of carbon dioxide in the atmosphere. Tellus, 12, 200-203. Morrison, T. J. and N. B. Johnstone, 1954. Solubilities of inert gases in water. J. Chem. Soc, 3, 441. Rafter, T. A. and Fergusson, G. J., 1957. Recent increase in the C^"* content of the atmos- phere, biosphere and surface water of the ocean. N.Z. J. Sci. Tech., B38, 871. SECT. 2] GASES 321 Rakestraw, N. W. and V. M. Emmel, 1938. The solubility of nitrogen and argon in sea- water. J. Phys. Chem., 42, 1211. Revelle, R. and R. W. Fairbridge, 1957. Carbonates and carbon dioxide. In "Treatise on marine ecology and paleoecology". Geol. Soc. Amer. Mem., 67, 239. Waverly Press, Baltimore, Maryland. Revelle, R. and H. E. Suess, 1957. Carbon dioxide exchange between atmosphere and ocean and the question of an increase of atmospheric CO2 during the past decades. Tellus, 9, 18. Richards, F. A., 1957. Oxygen in the Ocean. In "Treatise on marine ecology and paleoeco- logy". Geol. Soc. Amer. Mem., 67, 185. Waverly Press, Baltimore, Maryland. Slocum, G., 1955. Has the amount of carbon dioxide in the atmosphere changed signifi- cantly since the beginning of the twentieth century? Monthly Weather Rev., Oct., 225. Suess, H. E., 1960. Secular changes in the concentration of atmospheric radiocarbon. Proc. Conf. Subcommittee on Nuclear Geophysics of the Committee on. Nuclear Science of the NAS-NRC, June 1960. NAS-NRC Publ. 845, Nuc. Sci. Ser. Rep. No. 33, p. 90. III. DYNAMICS OF OCEAN CURRENTS N. P. ForoNOFF During the past two decades there has been a marked increase of interest in the theory of ocean currents and dynamical processes in the ocean in general. A measure of success has been achieved in explaining certain gross features of ocean circulation in terms of steady-state theory. This success is responsible, at least in part, for spurring activity in the more difficult fields of time-dependent and convective circulations in the ocean. The classical works of Bjerknes (1898), Ekman (1905), Bjerknes and Sandstrom (1910) and Bjerknes, Hesselberg and Devik (191 1) have had aremark- ably strong influence on subsequent development of dynamical theories of ocean currents. Sverdrup, Johnson and Fleming (1942, chaps. XII and XIII) and Dietrich (1957, chap. VII) follow the classical ideas closely in their treat- ment of statics, kinematics and dynamics, including a discussion of the circula- tion theorem and the Ekman theory of ocean currents. In recent years, the Ekman theory of wind drift and gradient currents has been developed and generalized for application to the steady and time- dependent circulation in shallow seas. The theory is not considered in detail here and the interested reader is referred to Sverdrup, Johnson and Fleming (1942, chap. XIII) or to the recent and more complete summary by Felsenbaum (1956) for the details. i Shtokman (1946) recognized the deficiencies of the Ekman theory in ex- plaining steady-state ocean currents in a non-homogeneous ocean and derived equations for total transport in which the density structure does not appear explicitly. Sverdrup (1947) was able to show that, if the variation of the Coriolis parameter 2 with latitude is taken into account, a realistic distribution of transport can be deduced from surface-wind stress. Stommel (1948) showed that the circulation can be completed by an intensified meridional flow along the western boundary of the ocean provided the variation with latitude of the Coriolis parameter is retained in the equations governing the flow. These developments were followed by a more complete and rigorous application of the theory by Munk (1950), who was able to deduce the major features of ocean circulation from estimates of the actual wind stress over the ocean surface. Further developments have been restricted to modifications of the theory of formation of the intensified meridional currents along the western boundaries of the ocean. 1 Other recent papers on the Ekman theory include Ichiye (1949, 1950), Sarkisian (1954, 1957), Welander (1957), Felsenbaum (1956a), Ozmidov (1959) and Saint-Guily (1959). 2 Defined as twice the magnitude of the vertical component of the earth's rotation vector. [MS received November, 1960] 323 324 FOFONOFF [sect. 3 The steady-state theory developed in terms of total transport provides no insight into the interplay between the circulation and the processes of mixing and convection which affect the density field in the ocean. Lineikin (1955) carried out a simplified analysis of the interaction using a linearized perturba- tion model in which an initial uniform vertical gradient of density was assumed. He was able to show that currents induced by the wind weakened with depth because of the stability of the density stratification. Attempts to construct more realistic convective models are hampered by the non-linearity of both the equation of state of sea-water and the conservation equations describing the processes involved. Theoretical studies of the dynamical response of an ocean were initiated by Rossby (1938). He showed that the ocean would respond barotropically to variations of wind stress under a wide range of conditions. Rossby 's and most subsequent studies have been confined to an ocean without boundaries and in the absence of strong steady currents. The basic problem of the interaction of the time-dependent modes of response and swift steady currents, such as the Gulf Stream and Kuroshio, has yet to be solved. The lack of understanding of the interaction severely limits the applicability of time-dependent theory to the real ocean. A non-mathematical survey of modern theories of steady-state currents, convective models and time-dependent variations of ocean circulation has been given recently by Stommel (1957). He pointed out the close relationship between modern theories and some earlier investigations of tidal dynamics. A discussion of many aspects of recent theories of ocean currents is given also by Stommel (1958) in his comprehensive treatment of the Gulf Stream system. In the present discussion of the dynamics of ocean currents, the material is divided into the two major topics of steady-state and time-dependent theory. Emphasis is placed on recent developments of the theory and an attempt is made to present the concepts in generalized, but elementary, form. Some of the more complex developments are not given in detail, but the more elementary concepts are developed in full detail in order to provide a background for assimilating recent literature. Many aspects of the dynamical theory of ocean currents are not included in the present treatment, but it is hoped that sufficient topics have been discussed to provide a basis for appreciating some of the problems as well as achievements in the field of the dynamics of ocean currents. 1. Conservation Equations for Momentum and Mass In order to retain maximum simplicity and uniformity in the mathematical treatment of the dynamics of ocean currents, we shall apply the equations expressing conservation of momentum and mass exclusively in terms of a rectangular co-ordinate system. Such a system allows considerably more scope in applying elementary analytical procedures and obtaining simple results than does the more natural spherical co-ordinate system or one derived from the SKCT. 3] DYNAMICS OF OCEAN CURRENTS 325 spherical system. ^ It is clear that we must show that a result obtained in the rectangular system has its analogue in the spherical system before we can apply it without ambiguity. In many cases the translation to the spherical system is simple. However, some results can be expected to be appreciably modified by spherical geometry.'^ The change of Coriolis parameter with latitude is simulated in the rectangular co-ordinate system by assuming the parameter to be a function of the horizontal co-ordinates. To simplify analysis, the Coriolis parameter is often approximated by a constant except when differentiated with respect to a co-ordinate repre- senting latitude. Then, the derivative is assumed to have a constant non-zero value 6. If this simplification is carried out, it is referred to as the /S-plane approximation. The |S-plane can be considered to simulate mid-latitudes of the earth. In equatorial regions the Coriolis parameter approaches zero so that its variation with latitude is not small compared with its magnitude. In polar regions the derivative of the Coriolis parameter ^ approaches zero and cannot be approximated by a constant. In both cases the ^-plane approximation is poor and other techniques have to be used. The |8-plane approximation was introduced by Rossby (1939) and has proved to be a useful tool for developing new ideas on the dynamics of ocean currents, particularly in time-dependent theory. However, its use must be regarded as an intermediate step in the development of more complete and satisfactory theories of ocean circulation. In setting up the conservation equations for momentum and mass, we shall consider the ocean to be unbounded in the j8-plane, unless effects of boundaries are being considered. We assume also that gravity, g, is everywhere constant and acts in a parallel direction. To specify position in the ocean, we introduce a right-handed rectangular co-ordinate system with the origin at the mean free surface of the ocean. We shall use two systems of notation to denote the co- ordinates in the conservation equations ; the cartesian tensor notation to describe general properties of the equations and the x-y-z notation to express more detailed results and solutions. The co-oi-dinates are oriented so that xi,x points eastwards, X2,y northwards and X3,z upwards and parallel to the gravitational force. The velocity com- ponents along the co-ordinate axes are denoted by ui,u for the eastward com- ponent, U2,v for the northward component and us,w for the vertical component. Using tensor notation, 3 we can express the conservation of momentum as : 1 The main handicap in using surface co-ordinates, such as those introduced by Morgan (1956), is presented by the convergence of meridians towards the poles of the earth. The conservation equations must contain coefficients dependent on latitude in such a system and, hence, are more difficult to analyse. 2 Stommel and Arons (1960) give an interesting example of the different results obtained in examining simple flows in rectangular and spherical co-ordinates. 3 An index appearing twice in a term implies summation over all three index values. 326 FOFONOFF [sect. 3 where p is density, t time, jp pressure, Qt components of rotation corresponding to the components at the earth's surface, assumed to be functions of X2,y only, and oij the components of stress due to molecular viscosity. The symbol ^nk denotes the permutation tensor having the values e^yfc = -\-\\ii, j and k are in cyclic order, = —\ a i, j and k are in anticyclic order, = 0 if any pair, or all three, indices have the same value, and §31 is unity for i = 3, otherwise zero. The stress tensor aij can be expressed in terms of the rate of deformation of a fluid element by the motion through the relationship where /x is the molecular viscosity. Conservation of mass is achieved by requiring the velocity components and density to satisfy the equation In addition, we have conservation equations for other properties 9 of the general form dp(p ^ dpcpuj ^ _^^q (4) dt 8xj 8xj where Fj are components of flux of the property cp set up by internal forces or pressures and q is the total internal source of the property (p. Equations (1) and (3) are special cases of (4). Similar equations can be written for the con- servation of kinetic and internal energies, dissolved salts in sea-water and so on. 2. Separation into Steady and Time-Dependent Motion The complete set of equations given by (1) and (3) cannot be solVed exactly except in extremely simple cases. In order to obtain useful results from the equations, we must carry out a series of simplifications. We shall first separate the equations into a mean flow and a time-dependent flow that represents the departure at any instant of time of the flow from its mean. Because of the non- linearity of the general equations, we cannot carry out the separation into two sets of independent equations. Each set contains terms representing inter- actions between the steady and time-dependent modes of motion. By adding (3) multiplied by Ui to (1), substituting (2) for a^j and neglecting compressibility in the frictional terms, we can express (1) in the form : which is essentially identical with the general form (4). SECT. 3] DYNAMICS OF OCKAN CURRENTS 327 We can now obtain the steady-state equations by averaging (3) and (5) with respect to time. Using the definition ^ = I™ (2^ JI '' *) to denote the average of any property 93, we obtain -^^ + 2eo-.^,,^. =. _ _ _ ,^ S3, + ^ __ (6) and 5.T; = 0. (7) The time-dependent equations are obtained by subtracting (6) from (5) and (7) from (3).i Thus, the time-dependent equations are dpuj dipUjUj-pUjUj) — -7T- -^ 7— 1- ^€ij]cUj{pUk — pUk) = 8^, (p-/>)yg3.-f/x g^^g^. (8) and ^+^''"'r^' = 0. (9) We can expand the averages of products in the equations by sphtting the dependent variables — velocity components, pressure and density — into steady and time-dependent parts so that ui = Ui + u'i V = P+P' (10) P = P + P'y where u'i, p' and p' have zero averages. We can then separate the averages of products into terms expressible in terms of the averages of the individual factors and correlations between fluctuations of the factors. For example, the average pUiUj becomes pUiU} + pu'iu'j + p'u'j Ui + p'u'i U} + p'u'iu'j. Variations of density in the ocean are of the order of 0.1 % of the mean density, whereas velocity fluctuations are much larger and can be of the same magnitude as their mean. Consequently, the terms containing correlations between fluctuations of density and velocity will be siuall compared to terms containing correlations between velocity com])onents. We assume, therefore, that the 1 The time-dependent motion is deHnecl to have a zero mean. This definition is imphed in most studies of time-dependent How, l>ut it is not always exphcitly stated. 328 FOFONOFF [sect. 3 density fluctuations are not significant in the acceleration terms and that the correlations between them and velocity fluctuations may be neglected. ^ Using this assumption, we can write (6) and (7) in the simplified form ^ = 0, (12) where — pu'ifi'j, the Reynolds stress tensor, is denoted by Rij. Similarly, the time-dependent equation becomes _ du'i _^^ du'i _ , dUi dR'ij ^ ^ _ , dp' dxjdxj ct dxj where R'a is introduced for — p{u'iu'j — u'iu'j). From (11) we see that the influence of the time-dependent motion on the steady flow is represented by the Reynolds stress tensor, Ra, only. The inter- actions in the time-dependent equations are more complex. The interaction terms sej3arate into convection of fluctuating momentum by the steady flow, convection of steady-state momentum by the fluctuating flow and the diver- gence of momentum-flux variations of the purely time-dependent flow. The time-dependent equations (13) and (14) include fluctuations of all frequencies. If we are interested in variations of ocean currents with time- scales of a few days or longer, we can carry out a second averaging process using an averaging time that is short compared with the periods of the motion being considered. In this way we can distinguish between the slower variations being considered and high-frequency turbulence and wave components. If we carry out the flnite average in equations (13) and (14), we find that the only term that is altered in character is R'ij. The remaining terms enter linearly into the equations and may be replaced directly by their averages. The finite average of R'a is equal to the diflFerence between the finite and steady-state average of pu'tu'j. Fluctuations with periods shorter than the finite averaging time will contribute approximately equally to both averages and will cancel when the difference is taken. If we interpret the Reynolds stresses in the steady-state equations as representing a dissipative mechanism, we have to consider the divergence of the finite average of R'a as a source of momentum to which only the longer-period motions can contribute significantly. In other words, the time-dependent motion of a given time-scale can draw momentum 1 However, the correlation between density and velocity fluctuations may not be negligible in the other equations where velocity correlations do not appear. In particular, the correlation may not be negligible in the equation for mass continuity. SKCT. 3] DYNAMICS OF OCEAN OTTRRKNTS 329 |)ninarily from motion of longer time-scale and, conversely, must pass momen- tum primarily to shorter time-scales. The mechanism by which momentum and energy are transferred to shorter time-scales has been studied more thoroughly in connection with turbulence (Townsend, 1956). Turbulent motion constitutes an essential part of the dissipative mechanism in the ocean. Its action is complex and cannot be readily taken into account in the momentum equations. However, it is often qualitatively simulated in its dissipative action by assuming the Reynolds stresses to be proportional to the strain rate of the mean flow. By analogy to molecular friction, the propor- tionality constant is referred to as the virtual or eddy viscosity coefficient. The concept of eddy viscosity has proved useful in providing a simple dissipative mechanism, but, at best, it is a crude approximation to the action of turbulence and results obtained through its use must be applied with caution. Separate coefficients of eddy viscosity are used to estimate stresses from horizontal and vertical gradients of the mean flow. The necessity for using at least two coefficients is evident from considerations of the dimensions of the ocean, as will be shown later. We can introduce the eddy viscosity coefficients by relations of the form (Saint-Guily, 1955), T. -— ; — r ,-^^Ui ...dUj , , Rij = -pUiUj = A^0)-^+/^(O-^' (1^) where fx{i ), fx{j ) are equal to the lateral eddy viscosity /x// for i, jV 3 and to the vertical eddy viscosity /X7 for i,j = 3. It is clear from (15) that the magnitudes of jjiH, iJi-v will depend on the scale of the motion and can be estimated approxi- mately by dimensional analysis. If we substitute (15) for the Reynolds stresses in (11) and neglect molecular friction and density variations in the frictional terms, we obtain where V^ is introduced for 8^ldxi"+ d^jdxz^. 3. Magnitudes of Forces The separation of the flow into steady and time-dependent modes is accom- plished by methods closely analogous to those used in turbulence theory (cf. Townsend, 1956, chap. II). The basic difference in application of the equations is in the time- and size-scales of motion that are considered. In studying ocean circulation, we shall conflne our attention primarily to the low-frequency, large-scale current systems for which the non-linear interaction terms are negligible or capable of simplification. The relative magnitudes of the various forces and accelerations present in (11) and (13) can be compared among each other by dimensional analysis. Ocean currents are slow enough that pressure can be approximated with high accuracy by the hydrostatic equation. Consequently, pressure gradients are 330 FOFONOFF [sect. 3 produced primarily by slopes of the sea surface and variations of density within the interior of the ocean. Furthermore, the forces due to horizontal gradients of pressure are balanced primarily by the Coriolis forces. The remain- ing forces due to accelerations and friction are generally smaller than the pressure gradient and Coriolis forces, but can become important in some regions of the ocean. The magnitudes of the forces can be compared by con- verting the momentum and continuity equations to non-dimensional form such that the larger forces are of unit magnitude. The magnitude of the remain- ing forces will then be indicated relative to unity by the non-dimensional coefficients of the force terms formed by the conversion. To convert the steady-state equations (11) and (12) to non-dimensional form, we introduce a characteristic length, L, and depth, H, to describe the horizontal and vertical scales of the motion. We characterize the velocities by Uq for the mean, uq for the fluctuating horizontal components ; Wq for the mean, M'o for the fluctuating vertical component. From the continuity equations (12) and (14), we define Wq= UqHJL and wq = uqHIL. If we denote a typical slope of the ocean surface by s, we obtain pogs for the characteristic pressure gradient. We can express the balance between pressure gradient and Coriolis forces by choosing Uq so that /o?7o = grs, where /o is the characteristic magnitude of the Coriolis parameter 'IQ-^. Each variable divided by its characteristic magnitude becomes a non- dimensional variable of unit magnitude, i.e. its magnitude will range between zero and unity. i If we denote the non-dimensional variable by primes, we obtain the non-dimensional steady-state equations in the form : Ro U'i -4 dU'i dx'i U', uqY dr'ij Uol dx'j + ^Hk^'i U/c dp' dx'i Rq Re V2C7',.+ 2 d^U'i {i= 1,2) (17) dx'i Uol dx'j + Sesjk^'jU'k ,dp' dx'z dp'U'j dx'i 1 + Eli Re\ VWs+ - 2 dWs dx': = 0, (18) (19) where 1/p' = a' = po/p ~ 1 r'ij = Rijlpouo^ Ro = UolfoL, Rossby number Re = poUoLjp,, Reynolds number Fr = Uo^/gH, Froude number. 1 Strictly speaking, this will not be true unless we use maximum magnitudes as for the characteristic values. As these are not generally known beforehand, we must interpret the non-dimensional magnitudes as being near unity, that is, within an order of magnitude of unity. SKdT. 3] DYNAMICS OF OCEAN CURRENTS 331 By evaluating the coefficients Bq, Re, Fj and s for a given flow, we obtain an appreciation of the magnitudes of the forces involved and can decide which force needs to be taken into account for a satisfactory interpretation of the flow in terms of the equations. Conversely, we can examine the coefficients to determine the horizontal and vertical scales for which any given term of the equations becomes comparable to unity. For example, the ratio of the non- linear accelerations to the Coriolis forces is given by the Rossby number, Rq. If we use the observation that steady velocities in the ocean do not exceed 2 or 3 m/sec, we can make the Rossby number comparable to unity only by as- suming the current to be sharply confined horizontally {L small), or by assuming a current near the equator (/o small). At mid-latitudes, the Rossby number approaches unity for horizontal scales of 20 to 30 km for the maximum velocities given above. Such conditions are approached only in concentrated current systems such as the Gulf Stream and Kuroshio. Another way to interpret the Rossby number is to note that the ratio UolL is the characteristic magnitude of the vertical component of relative vorticity, dUzjdxx — dUijdxz. Thus, the Rossby number can be considered as the ratio of the relative vorticity to the Coriolis parameter. Hence, if the relative vorticity approaches the Coriolis parameter, the Rossby number will be near unity and non-linear accelerations will be of the same magnitude as the Coriolis forces. The Reynolds stress terms become important if the magnitude of the velocity fluctuations exceed that of the steady flow {uqJUq>\), or if the Reynolds stresses change rapidly over distances that are short compared with the characteristic length of the steady flow {dr'ijjdxj > 1). Thus, if we retain Reynolds stresses and neglect steady-state accelerations in setting up a model of ocean circulation, we are implicitly assuming that the characteristic length over which the stresses act is larger than the length for which the Rossby number is unity. Consequently, the steady flow must be an average of velocity com- ponents whose fluctuations greatly exceed their mean. The other coefficients, the Reynolds number, Re, and the Froude number, Fr, approach unity only at extremely small horizontal and vertical scales of motion. Consequently, molecular friction and vertical accelerations may always be neglected in the steady state. The vertical components of the Coriolis forces, characterized by s in equation (18), do not exceed 0.1% of the vertical pressure gradient even at the extreme velocities encountered in the ocean. Thus, the vertical balance of forces is represented to a high degree of accuracy by the hydrostatic equation «'|^+1 = 0. (20) dx 3 If eddy viscosity coefficients are used in the momentum equations instead of Reynolds stresses, their magnitudes must be related to the Reynolds stresses according to the non-dimensional equation uo\" dr'ij 1 UoJ dx'j R'e ^ >..L-' s-'u: IxhH^ dx's^ (2J) 332 FOFONOFF [sect. 3 where R'e = poUoLlfXH can be considered analogous to the Reynolds number for molecular friction. In order that frictional forces be comparable to the acceleration terms, the coefficient R'e must be of unit magnitude, i.e. fiH ~ pqUqL ~ po{UolL)L'^, where L is the horizontal scale of the flow and UqIL is a measure of the strain or deformation rate of the mean flow. Similarly, for friction due to vertical shear to become important, we must have fiv - tiH{HILY- ~ po{UolL)H\ The eddy viscosity terms will dominate over the acceleration terms if R'e is less than the Rossby number. This occurs if [xh > pofoL'^- For L of the order of 20 to 30 km, we obtain hh> 10^ g cm~i sec~i. Values of /x// reported in the literature range from 10^ g cm^i sec~i for small current systems to lO^o g cm~i sec~i for the Antarctic Circumpolar Current (Hidaka and Tsuchiya, 1953). The time-dependent equations (13) and (14) can be converted to non- dimensional form using the characteristic magnitudes already introduced for the steady-state equations. We need, in addition, a time-scale To to charac- terize the rate of variation and a density difference Apo to indicate density variations. Using these characteristic magnitudes, we obtain the time-dependent equations in the form : 1 du"i F'r Up + Rc du"i „ dU'i ^ i -r-r + u j — — dx'i Re + eijicQ'jU"ic L\2 du"i dx'j dx z' {i = 1,2) (22) L(C/o/L)T dt du"z , jj, c>u"3 „ dp" dx's dx'i P" + — Rp Up Uo dU's dx'j Up Uo dr" 3; dx'j , pouofo „, „ + — ; €ijlci'-i jU k ^pog VV3+(^ L\2 d^u"z dx'?!^ 1 Apo pp {UpIL)Tp dt' dp" dp'u'j dx'j = 0, (23) f24l where the variables depending on time are denoted by double primes and F'r is introduced for Up^l{Apolpp)gH, the internal Froude number. The derivative with respect to time in (22) is comparable to unity if the time- scale or period of the variation is of the order of a half-pendulum day (27r//o) or less. If fpTp is small compared with unity, the acceleration term will dominate over the Coriolis term and can be balanced only by the pressure gradient. For this reason, short-period fluctuations {To<27rjfo) are sometimes referred to as inertio-gravitational motion. For longer periods (T'o> 277-//o), the balance will SEl^T. 3] nVNAMU'S OF OCEAN CURRENTS 333 be primarily between pressure gradient and Coriolis forces with the accelera- tions serving as a perturbation of the motion. The interaction terms re])resenting convection by either the steady or time- dependent flow become of the same order of magnitude as the pressure gradient and Coriolis forces if the Rossby number of the steady flow approaches unity. AVe have already suggested that the Rossby number is comparable to unity in comparatively swift currents such as the Gulf Stream and Kuroshio. Con- sequently, the interactions between the steady and time-dependent modes cannot be negligible in these concentrated currents. Unfortunately, we do not have a clear understanding of the consequences of the interactions. The convection terms become comparable to local accelerations if Rq is of the same order of magnitude as I/'/qTo, i.e. if LJTq^ Uq. If we interpret L as the wave-length and Tq as the period of the time-dependent motion, we can consider LITq to be the velocity (in the sense of phase velocity). Consequently, the non-linear convection terms cannot be neglected in comparison to local accelerations if the steady flow approaches the phase velocity of the varying niotion. The divergence of momentum flux, containing the term r'a, will become important if the Rossby number for the time-dependent flow, JRoUojUo, ap- proaches unity. The Reynolds stresses enter into the time-dependent equations with opposite sign to that of the steady-state equations. Consequently, we cannot interpret r"ij as a dissipative term in the equations and cannot simulate its action in terms of eddy viscosity. In fact, the divergence of the Reynolds stresses represents the rate at which momentum is being added to the time- dependent flow from the steady flow. If (22) is averaged to eliminate high- frequency components, we see that the contributions to the average of r"i} come primarily from velocity fluctuations the period of which is greater than the averaging time. Thus, we do not have a term corresponding to "eddy" dissipation in the time-dependent equations. As in the steady-state equations, the terms representing molecular friction are negligible except at extremely small scales of motion. In the vertical component of the momentum equations, the coefficient F'r can approach unity for ocean currents. However, the factor HjL is small so that the vertical accelerations can become appreciable only at high frequencies. The vertical component of Coriolis force is small and may be neglected. Thus, the variations of pressure in time-dependent motion result primarily from variations of density and surface slope and may be adequately approximated by a hydrostatic equation except at high frequencies. 4. Steady-State Circulation Ocean currents exhibit fluctuations of a wide range of frequencies and size scales. Surveys of the Gulf Stream, for example, have revealed complex struc- ture both in space and time (Fuglister, 1951 ; Wertheim, 1954). Recent measure- ments of deep currents (Swallow, 1955, 1957 ; Swallow and Hamon, 1960) 334 FOFONOFF [sect. 3 indicate that the deep flow is also highly variable. Hence, we may conclude that the steady circulation in the ocean has meaning only as an average with respect to time or space. In addition, the ocean basins and shorelines are complex in shape and their influence on the circulation is difficult to determine in detail. Also, the exchange of momentum, energy and mass across the surface of the ocean is conditioned by intricate processes both in the atmosphere and ocean. In order to outline even the major dynamical processes that determine the behaviour of an ocean system, we have to resort to drastic simplifications of the boundary conditions and of the exchange processes that drive the circulation. Various models of steady-state circulation have been constructed to elucidate separately the effects of friction, non-linear accelerations and thermohaline processes. Attempts to combine two or more of these mechanisms in a single mathematical model have not been very satisfactory because of the analytical difficulties that are encountered. Because of these difficulties, our knowledge of the interactions between the various mechanisms is inadequate. In discussing steady flow, we shall first consider regions of the ocean that are not directly affected by boundaries. Where boundary influences are considered, the boundaries will be simplified to straight vertical coastlines and a level, or slowly varying, bottom. By this simplification, we exclude a number of in- teresting features found in coastal regions which depend on details of bottom topography and coastline. However, even with these simplifications, we cannot solve the steady-state equations in three dimensions. A further reduction to a two-dimensional system is necessary to obtain solutions of interest. This reduction is accomplished most readily and systematically by integrating the momentum and continuity equations vertically from the bottom to the surface of the ocean. We can write the general equations governing steady motion (12) and (16), using the x-y-z notation, in the form : f = -P9 (27) ^ + ^ + M' = 0, (28) dx oy cz where / is introduced for the Coriolis parameter, 2Qz, and p for mean density. On the basis of the dimensional analysis carried out earlier, we have neglected horizontal components of Coriolis force due to vertical motion in (25) and vertical accelerations, Coriolis and frictional forces in (27). The terms that have SECT. 3] DYNAMICS OF 0(1EAN CITRRENTS 335 been neglected are small for the scales of motion considered in the general circu- lation of the ocean. However, these terms are not always negligible at the extremes of the scale range and should always be examined for significance in specific applications of the equations. A. Geostrophic Currents Except near boundaries of the ocean, where sharp horizontal and vertical gradients of velocity can be developed, the opposition of the pressure gradient and Coriolis forces is the major dynamical restriction to be satisfied by the steady flow. Geostrophic currents, which are defined by assuming an exact balance of pressure gradient and Coriolis forces, can yield useful approxima- tions to the actual currents in regions removed from solid boundaries and the free surface of the ocean. The theory and application of the geostrophic velocity approximation has been discussed at length by Sverdrup et al. (1942). The theory is redeveloped here in slightly revised form to serve as an introduction to the subsequent discussion of geostrophic mass transport. Gravitational acceleration can be expressed in terms of a potential, 0, defined so that its difference at two levels is equal to the work per unit mass required to move a body from one level to the other. The co-ordinate z has been chosen parallel to the direction of action of gravitational acceleration so that 0 is a function of z only. Surfaces of constant z correspond to surfaces of constant gravitational acceleration potential (geopotential). The geopotential is related to height z by the equation d0 = g dz. (29) Thus, the geopotential at any level z is 0{z) = <^o + gdz, (30) where 0o is the geopotential at the origin of the co-ordinate system (2 = 0). The geopotential can be expressed in terms of specific volume of sea-water and pressure by means of the hydrostatic equation adP = -gdz = -d'-^'^- We have interpreted the barotropic velocity as being independent of depth and equal to the deep-water velocity. By analogy, we can term the transport components in (55) as the barotropic transports. Similarly, the baroclinic components of velocity give rise to baroclinic transports which, if we neglect non-linear terms and surface stresses, are given by : fV, = |. fU. = -| (56) The anomaly of potential energy, x, can be calculated from oceanographic data. Therefore, it is of interest to interpret the relations in (56) more fully. It can be seen from (56) that the baroclinic transport must be directed along 340 FOFONOFF [sect. 3 contours of x- Furthermore, the transport between two contours xi and ^2 is equal to (x2 — xO//' where/ is the mean vakie of the Coriohs parameter between the two contours. If the contours do not he along lines of equal Coriolis para- meter, the baroclinic flow must be divergent. For example, if the contours xi and X2 cross latitudes for which the Coriolis parameters are /i and/2 respectively, the difference in transport Ai/jg, where ^ ,, „ 7—, — sm ny. (103) ^ ^ sinhifcoTT ri=i'^i^o^ + ^) coshA:„a -^ ^ ' By following the same procedure for positive uo, we obtain TT sin koy -^ {-ly^+^ko^ cos knX . ^ smkoTT „=i w(«^o^-w^) cos A;„a where kn^ = ko^ — n^. This solution can also be obtained from (103) by replacing ko,kn by iko,ikn and changing sign. The solution is valid provided A;o is not an integer and kna not an odd integral multiple of 7r/2. As ko is large compared with unity for slow interior velocities, we can derive much simpler approximate expressions for «/» from the exact solutions. For L = 2500 km, jS = 2 X 10~i3 cm-i sec-i and wo= —5 cm sec~i, ko has the value 15.9 approximately. Thus, for the first few terms of the series in (103), kn will not be much greater than ko, and we can factor the function of x out of the series, neglecting all terms in which n is not small compared with ko. The series then consists of the first few terms of the Fourier expansion of (102). Hence, replacing the series by (102) and simplifying the resultant function by neglecting terms of order e-^o" and e~^<'^, we obtain the approximation 0 ~ _[2/_7re-fco(''-l/)][l-e-fco(a-a;)_e-*o(a+x)]^ (105) which is equivalent to the boundary-layer solution given by FofonofiF (1954). We obtain the characteristic width of the boundary region, I//7rA;o = (|wo|/iS)'/^ ~ 50 km, and maximum velocity in the eastward jet, L{P\uo\y'^^ ^ 250 cm sec-i. Solutions of (98) for Co^/o contain a jet at both zonal boundaries, except in the special case Co =fo + ^L in which the eastward jet is present at the southern boundary only. For /o < Co fo + ^L, one of the jets is westward. In all cases, the interior flow is westward. For positive values of wo, the flow appears to consist of a complex series of eddies covering the entire ocean. No eastward drift is evident in the solution. As this solution can be interpreted most readily in terms of time-dependent 354 FOFONOFF [sect. 3 theory, discussion of its features is given in the section deahng with time- dependent motion. There are several important characteristics of the inertial flow for negative values of uq that we will note before examining the two-layer ocean. The stream-function ip is symmetrical with respect to the central meridian of the ocean. Hence, u and rj are also symmetrical and v is antisymmetrical. As the equations for the two-layer ocean are basically the same as those for the homogeneous ocean, we anticipate the same symmetries in the solutions. The width of the boundary region depends only on the interior velocity and jS. It does not depend on the size of the ocean and is least for the slowest interior velocities. As in the frictional model, no eastward acceleration of the flow is present. Acceleration takes place only along the western boundary where the pressure gradient has a down-stream component. Deceleration occurs along the eastern boundary where the flow is against the pressure gradient. An eastern-boundary current can occur only if it is fed by a jet of high velocity and high relative vorticity. Because of the concentration of the flow into a narrow jet, any dissipation in the system would have a drastic effect on the flow. If relative vorticity is lost in the zonal jet not all of the stream-lines will be able to regain their original latitude and the flow will shift toward the jet. We may speculate that if friction were suddenly to act on an inertial flow with a jet along the northern boundary, the flow would cease first along the southern boundary, and the region of no motion would grow northwards as relative vorticity is removed from the jet. The inertial circulation would finally be confined to the north-west corner of the ocean before ceasing altogether. If this speculation is correct in a gross sense, we can interpret qualitatively the action of non-linear terms in the frictional model. The solutions for inertial flow indicate that no western-boundary currents can exist at latitudes for which the interior flow is eastward as the eastward flow would require the boundary current to decelerate. Thus, the boundary current can extend into latitudes of eastward interior flows only if there is an inertial recirculation so that the boundary current is not decelerated. Thus, immediately seaward of the boundary current there should be a counterflow with a westward component of velocity. On the basis of these qualitative arguments, it appears that the counterflow suggested by the frictional model is essential to the stability of the western-boundary currents. However, more detailed and rigorous analyses would have to be carried out before the conjectures given here could be sub- stantiated. A. Inertial Baroclinic Flow The non-linear momentum equations examined for the homogeneous ocean can be solved also for simple classes of baroclinic flow in a two-layer ocean by applying approximate boundary-layer techniques. The two-layer ocean model is a better approximation to the real ocean than the homogeneous model in that both barotropic and baroclinic motion can exist. Yet, the model is sufficiently SECT. 3] DYNAMICS OF OCEAN CUKRENTS 355 simple for a relatively complete analytical examination of its features to be made. In the two-layer approximation, the ocean is assumed to consist of two homogeneous layers of water of slightly different density with the lighter water overlying the more dense layer. Velocities within each layer are constant with depth and change abruptly across the interface between the two layers. Frictional stresses along the interface are neglected. In the general case, motion can exist in both layers, i.e. the flow can be a combination of both barotropic and baroclinic modes. However, if there is motion in both layers, it does not appear possible to find an interface such that the normal component of velocity is zero everywhere except in the special case of non-divergent zonal flow.i The equations are considerably simplified if either the barotropic or baroclinic mode is absent, or if the baroclinic velocities are equal and opposite to the barotropic velocities (motion in the lower layer only). The barotropic flow has already been examined for the homogeneous ocean and can be applied with minor changes to the two-layer ocean. The two remaining simple flows are basically similar to each other. If the barotropic mode is absent, the flow is entirely in the upper layer and pressure gradients in the lower layer are zero. Conversely, if the baroclinic flow is equal and opposite to the barotropic flow, the motion is confined entirely to the bottom layer. Provided we assume the bottom of the ocean to be level, we can consider these two types of flow to be "mirror images" of each other. Hence, we shall consider only the baroclinic mode. In the absence of the barotropic mode, the baroclinic velocities become absolute velocities relative to the co-ordinate system. We may write the momentum equations in the form i^Vi^V/M = -I (.07) ^ + ^ = 0, (108) OX oy where h = ^ — r]i, rn denoting the position of the interface between the two layers, and X = r%8 dp = lApgh"^ = \pg'h\ g' = {Aplp)g, (109) where Jp is the difference of density between the lower and upper layer. 1 Unfortunately, a proof that no steady interface can exist for motion in both layers has not been constructed for the general inertial system of equations with friction neg- lected. It can be given if the non-linear acceleration terms are neglected. Such a proof would enable us to assert that steady flow in both layers could exist only in the presence of processes, such as internal mixing, that alter density along a stream-line of the flow. If implicit mechanisms are postulated to allow flow across the interface, it is possible to construct a simplified theory of thermohaline circulation (Stommel, Arons and Faller, 1958; Stommel and Arons, 1960a). 356 FOFONOFF [sect. 3 The equations for the t\\'o-layer ocean can be reduced to ^uh=-^, (111) h dij where ^a/h is the potential vorticity and Q is equal to g'h + ^(m^ _|_ ^2) xhe basic difference between these equations for the two-layer ocean and (89) and (90) is that h cannot be considered to be approximately constant. By introducing a volume -transport function ijj such that ,,;, = ^, vh= -^4' (112) dy dx ^ ' we obtain relations, analogous to (95) and (96), of the type Q = Q{^) (113) ^ = -^. (114) h di/j ^ ' We assume that a solution for the transport function for the two-layer ocean can be obtained that is symmetrical with respect to the central meridian of a rectangular ocean. Such a solution requires h, x, Q, Ca, u to be symmetrical and V to be antisymmetrical as in the homogeneous ocean. Charney (1955) and Morgan (1956) have shown that the two-layer equations allow the formation of an intensified accelerating current along the western boundary. From the sym- metry of the equations, we can expect an intense decelerating flow along the eastern boundary. The solutions for the homogeneous ocean lead us to expect an eastward jet joining the two boundary currents. Thus, we anticipate solu- tions for the two-layer ocean that have several features in common with the inertial flow in the homogeneous ocean. Fofonoflf (1954) pointed out that the westward intensification of anticyclonic inertial circulation (poleward flow in the western boundary current) is more pronounced than that of cyclonic circulation. This feature is not present in the homogeneous ocean. The absence of symmetry can be seen from the equations governing Q and ^alh in the boundary region. If we assume the interior flow is westward and slow, we obtain iUlhh =filhi (115) and g'h + l{Ub^ + Vb^) = g'hi (116) along a transport line entering the western-boundary region. Solving (115) for the relative vorticity, we obtain r iff\ \"'i~"'f>)f tft = -{jb-Ji) 1 Ji ' (117) If f. {Ub^ + Vb^) SECT. 3] DYNAMICS OF OCEAN CXIRRENTS 357 If the boundary flow is northward {fb>fi), an increase in velocity yields an increase in the magnitude of relative vorticity ^b. Conversely, if the boundary flow is towards the equator (/&o-Xb = lg'{ho^-h^), (121) where ^^o is the depth of the upper layer at the western boundary {x = 0). 1 If the transport function in the interior is assumed to be a linear function of y, it is possible to show that the relative vorticity is always positive for cyclonic circulation, i.e. hb>hifb/fi- However, the relative vorticity might become negative for more complex interior flows. 358 FOFONOFF [sect. 3 Similarly, if we assume that the interior flow is westward and slow enough for relative vorticity to be neglected, the velocity will be geostrophic, so that Iu.= -g^ (122) and For anticyclonic circulation with poleward flow in the western-boundary region, we can assume that the westward drift in the interior extends to the southern boundary {y = 0). Integrating (123), we obtain (124) hdy As (121) and (124) must give the same value for the potential energy at each point in the interior of the ocean, we must have ry Xb = xbo-fh = Xi = xo-fh+^ h dy (125) for ipb = ijji. Hence, the condition Xbo = xo + ^ \%idy (126) must be satisfied by the boundary solution to be compatible with the interior solution. As xbo cannot become negative, we obtain the inequality In order to interpret the meaning of the inequality (127) more completely and to obtain explicit solutions in the boundary region, we have to specify the interior flow. A simple form of interior flow is obtained by assuming ipi to be linear with y, that is, ifji = uhy = —Uohoy, (128) where Uo is the speed of the westward flow and ^o is the depth of the upper layer at the southern boundary. For this type of flow the transport component is constant at each latitude and the westward velocity is inversely proportional to the depth. Substitution of (128) into the inequality (127) yields liAmaxl ^ g'ho^l^yma^x, (129) where «/max is the latitude at which xbo is zero. We may express (129) in the alternative form ymax2 < g'holUo^ = {Uol^){g'holUo^) - W^/Fr, (130) SECT. 3] DYNAMICS OF OCEAN CURKENTS 359 where W = v ( Uol^) is a characteristic width and Fr is the internal Froude ninnber at the southern boundary. From (130) we can see that the meridional extent of the circulation is limited. In particular, if ?/max is less than the distance between the northern and southern boundaries of the ocean, the surface layer cannot cover the entire ocean and lower-layer water will be exposed to the surface north of ?/max- Consequently, in the two-layer ocean, we can have a separation of the western-boundary current from the boundary at an interior point. For a given depth ho at the southern boundary, the northern limit of the upper layer varies inversely as the square root of the speed of westward drift. The stronger the flow, the more southerly will be the point of separation of the flow from the western boundary. The depth of the upper layer in the interior of the ocean is a function of ipt. Substituting (128) into (124), we obtain .2 , . , 2foUoho , ^Uoho (131) g g Uoho At a given latitude in the boundary region, the layer depth is related to the transport function by (121). As each transport line ipb in the boundary region must be connected to the interior transport line ipt having the same value, we can evaluate hi in terms of ht,. Eliminating the transport function from (131) by substituting from (121), we obtain hi^ = ho^-i^{hbo^-h^) + l-^&-^{h,o^-h^)^ J PUoho ^J32) = ho^ - a{ho^ - hb^) + la^b^ho^ - hb^)^ where a=folf< 1 and h = ^g' jJo^Uoho. As hb must approach hi for large values of X, a solution of (132) for hb — hi must exist. The condition for a real solution to exist is (l-a)2/a252 ^ ho^-hbo^ or 101 = Uohoy ^ g'{ho^-hbo^)l^y. (133) But from (126) we see that the equality must hold if ipi is a linear function of y. Hence hi^ = ho^-a{hbo^-hb^)+ ^.l^\~''l\Ahbo^-hb^)^. (134) We can obtain the variation of depth of the upper layer in the boundary region from (116) and (119) by neglecting Ub in comparison to Vb. As the flow is 360 FOFONOFF [sect. 3 predominantly poleward in the boundary region, the component normal to the coast will be small. Thus, we obtain the approximate equation or, solving for the derivative, ^Ah\2 Of 2 t) =f (^,-A.). (135) Equation (135) is an ordinary non-linear differential equation for hb as hi is given in terms of hb by (134). We can convert (135) to a non-dimensional form by introducing h'b = hblho, h'i = hilho and x' = xlW, where Pf is a characteristic width. The non-dimensional equations are and h'i-^ = l-a{h'bo^-h'bn + ^^^^^^{h'bo^-h'b^)^. (137) All the variables in (136) will be of unit magnitude if we choose W = {g'hol2fo^y'^K In general, the solutions to (136) will have to be obtained numeri- cally. However, as x' does not enter the equation explicitly, the solutions are readily obtainable in inverse form by numerical evaluation of the integral 2x' = f* -^;*iL^. (138) An explicit solution of (136) can be obtained in the special case amin = /o//max = i. For this case, the expression for the interior depth (137) can be reduced to h'b^ + {i+yr ^'- 2(1+2/') ' ^^^^^ where y' = ylyma\, by substituting 1/(1 -(-?/') for a and l—y'^ for h'bo^ from (126). Substituting (139) into (138) and evaluating the integral, we obtain h'b = l+2/'-[l+l/'-(l-2/'2)'/^]e-^'[2/(l+l/')]'/^ (140) Having obtained h'b, we can derive the transport function from (121) and the velocity from (135). Hence, the boundary -layer solution is complete. In the limiting case y -^ ymax, (140) reduces to h'b = 2(1 -e-^') (141) SECT. 3] DYNAMICS OF OCEAN CURRENTS 361 wliich, if we replace x' by {!/mn\ — y)IW, is also the boundary-layer solution for the eastward jet.^ The special case, «inin = J, is of particular interest because of its simplicity. It can be seen from (131) that the interface in the interior of the ocean will, in general, be a hyperbola with respect to y if the transport function is linear in y. Hence, the potential vorticity will not be a simple function of i/». For ainin = 2' the potential vorticity is constant. We can verify this feature by assuming filhi to be constant and differentiating it with respect to y. The differentiation yields i_A^=l+fj!]^ = 0 (142) hi hr Sy hi g'hi^ or g'hi^^ gV^^ Uihi = ^^- = ^^— (143) Multiplying (143) by ymax, we obtain |Wi%max| = IfAmaxI = g'ho-^ymaxlfo^ (144) But, from (133), |i/fmax| =g'ho^l^ym&x- Hence, the interior potential vorticity is independent of y only if/o = /S?/max, i.e. a=/o//max = |. We can gain some insight into the magnitudes of the variables describing the anticyclonic inertial circulation by considering a specific example. Choosing ^0 to be 400 m,/o as 0.5 x 10-^ sec-i and Apjp as 2 x 10"^ and using g= 10^ cm sec~2 {g' = 2 cm sec~2) and |8 = 2 x lO^^^ cm~i sec~i, we obtain: /max = fola = 10-4 sec-1 ^max = '2ho = 800 m 2/max = (/max -fo)!^ = 2500 km i^max = g'ho^"l^ymax = 64 X 10^ m^ sec-1 ^0 = ifjm&xlhoym&x = 6.4 cm sec-i Vb max = i'Zg'hoY'^ = 400 cm sec-i W = (25r'Ao)'V/max = 40 km. Contours of the transport function 0 (150) or -^-- + —^- < 0. (151) By integrating (151), we can see that the acceleration is zero if poWoL = {poWoL)yJo^lf^. (152) Hence, for a constant vertical velocity, the boundary flow is unaccelerated, i.e. constant, if the width of the ocean decreases with latitude as I//2. If the width decreases more rapidly, the boundary flow is accelerated and, if less rapidly, the flow is decelerated. However, if wq is assumed to vary with latitude, the stable regions could only be determined from the exact distributions of wq and L. In the extreme case, given any variation of L with latitude, it is possible to choose Wq so that (151) is either satisfied or violated at every latitude. Hence, SKCT. 3] DYNAMICS OF OCEAN CURRENTS 367 conclusions based on the application of (146) and (148) depend critically on the assumptions made about the vertical velocity. Nevertheless, it appears reason- able to conclude that strong deep-water currents are induced along some portions of the western boundaries by vertical flow in the interior of the ocean. Stommel (1957) suggested that downward flow in the deep water is conflned to relatively small regions in the North Atlantic and Weddell Sea and that, over the remainder of the world ocean, the flow has an upward component. Thus, the deep-water flow is assumed to be driven by concentrated high- latitude sources with the remainder of the ocean acting as a more or less uni- formly distributed sink. The deep water is dissipated by upward flow into the upper layers of the ocean. Assuming tacitly that the western-boundary flow can exist whether or not (151) is satisfied (as, for example, in the purely frictional boundary layer), Stommel and Arons (1960, 1960a) worked out a scheme of interior deep-water circulations connected by western-boundary currents for all of the major oceans. The interior circulation does not cross the equator according to (147). Hence, a separate circulation was assumed in each hemi- sphere. The boundary currents do not depend on the Coriolis parameter but rather on j8, which does not vanish at the equator. Thus, the boundary current is allowed to cross the equator to connect the interior flows in each hemisphere. Its stability at the equator has not apparently been examined. Using these simple arguments, Stommel and Arons were able to make estimates of the magnitude of the deep circulation and also of the length of time required to renew the volume of water in the lower layers of the oceans. Although the circulation scheme proposed by Stommel and Arons is not free from objections, some of which are given here, and has not been confirmed as yet by observations, it illustrates the far-reaching consequences of imaginative application of even the simplest dynamical arguments. In order to develop a more satisfying description of the mechanism of convective circulation, we have to introduce conservation equations for heat and salt. As in the momentum equations, we can neglect molecular transport processes in comparison with the turbulent exchange or flux of the properties. Although mixing must occur by molecular diffusion, the diffusion occurs at much shorter length scales than those considered for ocean currents. The flow of heat and salt relative to the mean current is usually approximated by diff'usion equations expressed in terms of eddy diff'usivity coefficients. Assuming that the heat content of sea-water is linearly proportional to the temperature and independent of the salinity, we may express the flux of heat Fqi as FQi = -K{i)^> (153) where K{i) is equal to the horizontal eddy conductivity, Kh, for i=l, 2 and to the vertical eddy conductivity, Kv, for i = 3. Similarly, the flux of salt, Fst, is Fsi = -D{i)^, (154) OXi 368 FOFONOFF [sect. 3 where S is salinity and D{i) is either the horizontal eddy diffusivity, Dn, or the vertical eddy diffusivity, 7)^, depending on whether i is different from or equal to 3.^ Neglecting internal sources of heat due to dissipation of kinetic energy of the motion, we obtain the conservation equations for heat and salt by sub- stituting (153) and (154) for the flux in the general conservation equation (4). Expressing the heat in terms of temperature and using the continuity equation (3), we obtain the conservation equations in the form f:+„,£^ = ^v.r+:^|!^, ,156) Ct CXj pCp pCp CXz^ where Cp is specific heat, and ^ + Uj^ = V^S + —— 156 Ot CXj p p 0x3^ We have assumed that Kh,Kv and Dh,Dv are constant in deriving (155) and (156). However, as density depends both on temperature and salinity, it is likely that turbulence will be modified at some length scales by the presence of temperature and salinity gradients. Hence, the eddy coefficients would depend on the gradients of the properties and the intensity of the mean flow. However, because of the complexity of the complete set of conservation equations, we are unable to cope with more than the simplest form of the equations. If we assume that the density can be approximated by a linear equation of the form p - pQ\i-a{T-To) + h{S-So)'\ (157) and that KnlpCp = Duip = kH, KvjpCp = Dvjp = ky, (158) we can reduce (155) and (156) to the single equation |,„,g..„VV...^, (159, From its form, we may interpret (159) as an equation for the "diffusion" of density. Clearly, the simplification of the conservation equations to (159) can only be done in the linear approximation to the equation of state. ^ We can make an alternar^ive interpretation of the diffusion equation by introducing an apparent temperature T* defined by T*_To* = T-To-{bla){S-So). (160) 1 The reader should compare the assumed form of the turbulent flux equations with those for molecular transports given in Chapter 1, eqns. (68) and (69), page 26, and in Chapter 2, eqns. (16) and (19), page 34. 2 Fofonoff (1956) has suggested that non-linearities in the equation of state of sea-water may have a significant influence on the density structure of Antarctic bottom water. SEt^T. 3J DYNAMICS OF 0(!EAN CURRENTS 369 The equation for density in terms of T* is p = poll-^'(^*-To*)l (161) and (159) is Q/Ti* a/77* P|27^* _ + „,__ = i„V^r. + i,_. (162) The boundary condition at the ocean surface for the apparent temperature is h~ = g* = -^--So{E-P), (163) cxs pCp a where q is the flux of heat across the surface, E — P the net rate of evaporation and *S^o the salinity at the surface. At a solid boundary, the flux may be taken to be zero. In the steady state, the integral of the apparent surface source Q* over the entire ocean surface must vanish. Hence, if Q* is different from zero, both positive and negative sources must be present. However, (162) yields un- stable distributions of density if Q* is negative. Thus, the diffusion equation can only be applied in regions where heating is dominant, or where net precipitation is sufficient to maintain stability in the presence of cooling. As mentioned earlier, Stommel (1957) has suggested that regions of instability, in which downward flow is present, may be confined to relatively small areas at high latitudes in the real oceans. Lineikin (1955) carried out an analysis of (159) together with the momentum and continuity equations, (25) to (28). He linearized the problem by con- sidering perturbation velocities about a state of rest in which the density increased linearly with depth. Expressing the surface- wind stress in terms of a Fourier series, he examined the rate of decay of the perturbations of velocity and density with depth for the general term of the series. Lineikin assumed the Coriolis parameter to be constant and the eddy viscosity coefficients to be equal to the diffusivity coefficients. Under these assumptions, he was able to show that the characteristic depth of penetration of the perturbations was given by fH\/E, where L is the horizontal scale of the wind stress component and E is the gravitational stability assumed in the unperturbed state. The depth of penetration is of the order of 1000 metres for horizontal scales of 100 km. For oceanic scales of the order of several thousand kilometres, the indicated depth of penetration is unreasonably large. Stommel and Veronis (1957) suggested that a more realistic depth could be obtained by taking into account the varia- tion of the Coriolis parameter. They examined simple models in which rotation was neglected altogether, taken as constant, and allowed to vary with latitude. The results obtained indicated that the depth was limited much more sharply if /S is taken to be different from zero. It is interesting to note that the depth was least in the case of no rotation. It can be seen by converting (159) to non-dimensional form that the magni- tude of the diffusion terms in comparison with the convective terms is given by 13— s. I 370 FOFONOFF [SKC'T. 3 knlLUo and kvlHWo where L and H are the horizontal and vertical scales of the motion and Uo and Wo the characteristic horizontal and vertical velocities. Major variations of density in the real ocean are generally confined to the upper one to two kilometres of the ocean and vertical velocities are estimated to be in the range 10~4 to 10~5 cm sec~i. Hence, for the diffusion terms to be im- portant in (159), but not dominant, the ratios knlLUo, kvlHWo must be of unit magnitude. Thus, in order to get non-trivial solutions of the convective equations, we must assume that ky is about 1 to 10 cm^/sec and kn is in the range 10'^ to 10^ cm^/sec^^. For eddy viscosities of the same order of magnitude, internal stresses due to gradients of the mean flow are negligible except possibly in the intensified boundary currents. Therefore, we can use the geostrophic approximation to evaluate the steady convective flow in the interior of the ocean. This approximation has been utilized recently by Robinson and Stommel (1959) and Welander (1959) to set up models of the oceanic thermocline pro- duced by convective circulation. The stresses due to velocity shear can be important only if we assume that the eddy viscosity is very much larger than the diffusivity. Such a model has been examined by Ichiye (1958) for zonally uniform flow. However, unless the ratio Avjky is of the order of 10^ or greater, the model gives unrealistically high zonal velocities. The geostrophic model of convective circulation can best be formulated in terms of the potential energy function E'j, = r Pdz^ r p {pB-p)gdz'dz + {z-Zs)[PB-hPB9{^-ZB)]- (164) For 2 = 77, the function E' ^p becomes identical with the potential energy, E^, of the water column relative to the ocean bottom introduced in (44). If the convective flow is assumed to be entirely baroclinic, both the bottom pressure, 'Pq, and the bottom density, p^, are functions of z^ only. Therefore, we can write (164) in the form E\ = E'j,o + X> (165) where x' = {pB-p)gdz' dz (166) is an anomaly of potential energy and E'po is a function of 2 only. The potential energy anomaly, x\ is defined relative to level surfaces rather than isobaric surfaces and, although similar, is not equivalent to the potential energy anomaly defined in (46). We shall not carry the primes on the potential energy functions in the remainder of this section but the diff"erence in the definitions of X must be kept in mind. A function essentially identical with (166) was introduced by Welander (1959) in the study of convective flow with no diffusion. SECT. 3] DYNAMICS OF OCEAN OUKRENTS 371 The geostrophic equations for baroclinic flow expressed in terms of x a-re and the density anomaly is 1 d^x PB-P = -^- (169) Substitution of (167) and (168) into the continuity equation (28) and integra- tion with respect to z yields .-=/.!• (170) Hence, by substituting for the density and velocity components in (159), we obtain the steady-state equation _iJXJ!]^A.3LJiL^l?x^^. V2!!x , . ^ ,171^ / By dz 8x 8z^ f dx 8z 8y 8z^ P 8x 8z^ ^ " 8z^ ^^ Sz* ^ ' ' for the potential energy anomaly. A measure of simplicity is achieved by assuming pkn and pkv to be constant. The boundary conditions on x cannot be applied at the surface of the ocean because the flow is not geostrophic near the surface in the Ekman frictional layer and is, therefore, not represented by (171). However, we can consider the upper boundary to be at a level surface below the Ekman boundary layer. Along this surface, we assume the vertical velocity and the diffusive flux, or the density anomaly, are given. These conditions can be expressed as p^-P'^' = -8^ 8^ ^^^^^ The boundary conditions in the deep water are satisfled if we require x and its derivatives with respect to z to approach zero near the bottom. For convenience in handling the bottom-boundary conditions, we can assume the ocean to be infinitely deep provided we consider only baroclinic flow. We can then require X to approach zero asymptotically for large values of z in place of the bottom conditions. As we have neglected both viscous and acceleration terms in deriving (171), we cannot satisfy lateral boundary conditions except in special cases for which one of the horizontal velocity components is zero. In the more general case, we must assume that the potential energy function is known for flow entering the region under consideration. In spite of the formidable non-linearity of (171), a start has been made in examining certain classes of approximate solutions. Welander (1959) studied 372 FOFONOFF [sect. 3 (171) for purely convective flow {kH = kv = 0). In the absence of diffusion, the equation admits arbitrary zonal flows in which both the vertical and meridional components of velocity are zero, i.e. x ^ function of y and z only. Welander then assumed x to be of the more general form P{x,y)Q[zlF{y)] and found that solutions existed for arbitrary P{x,y) \iQ{zjF{y)] was of the form e^f^^lf, where k is an arbitrary constant. For positive values of k, the convective circulation dies away exponentially with depth. The penetration of the circulation is least at low latitudes and increases polewards. The arbitrary constant k can be estimated by dimensional analysis or chosen to agree with the observed decrease of the density anomaly with depth at a given latitude. Welander showed that if the observed density distribution at the ocean surface is used to define P{x,y), the density field derived from the theoretical solution had features in common with the observed density field. In particular, surfaces of constant density were deepest at mid-latitudes. For the purely convective flows studied by Welander, the density must remain constant along a stream-line of the flow. We can obtain flows of this type that are not uniform zonally provided we do not insist that the vertical velocity at the ocean surface be zero. Instead we apply the upper boundary condition on the vertical velocity along a surface below the Ekman frictional layer and assume that the flow across the surface is absorbed by the divergence of the Ekman transport. Similarly, we assume that the flux of heat and salt into the Ekman layer is balanced by surface sources. Hence, both the divergence of Ekman transport and the surface sources are implicitly specified by the choice of interior solutions of (171). The more direct approach in which we assume that the wind stress and surface sources are given and then look for an interior solution is considerably more difficult to carry out. As yet, no solutions using the direct approach have been obtained. Before examining the model of the oceanic thermocline proposed by Robinson and Stommel (1959), we shall consider a simpler baroclinic model to introduce some of the features of convective flow in the presence of diffusion. We con- cluded that in the special case, (152), in which vertical flow is independent of a; and y and the zonal width of the ocean varies as 1//^, the flow will take place in a meridional plane without inducing a zonal velocity component. A western- boundary current in this special ocean either transports a constant volume of water along the boundary or is entirely absent. We shall now see if these results are consistent with (171) for purely baroclinic flow. Because the Coriolis parameter is present in the coefficients of the non- linear terms of (171), we cannot assume that x is entirely independent of y. However, we assume that x varies only slowly with y so that the zonal velocity is small compared with the meridional velocity, i.e. that derivatives with respect to y may be neglected in (171). We assume further that lateral diffusion is negligible in comparison with vertical diffusion. Hence, we reduce (171) to SECT. 3] DYNAMICS OF OCEAN CUBRENTS 373 For the vertical velocity to be independent of x, we must choose x of the form A{z) + B{z)x. The dependence on y will enter parametrically into the functions A{z) and B{z). A simple particular solution of the linear form in x is X = 4pokvfHxo-x)l^z, (175) where xo is an arbitrary constant. As the zonal component of velocity is zero at x = xo, we can interpret xq as the position of a meridional boundary. By differentiating (175), we obtain the vertical velocity -=/4|=-^^ (176, and the density anomaly Pb-R 1 d^x _ 8pokvfHxo-x)^ ^^^^ g dz^ g^z^ As z is negative in the interior of the ocean, a stable solution, p^ > p, occurs only for x > xq. Hence, xq must represent a western boundary. The horizontal velocities represented by (175) are The meridional flow is uniform zonally and increases with latitude. The zonal flow is westward and independent of y. Stream-lines at each level form a family of hyperbolae in the x-y plane with the flow curving in the anticyclonic sense. Although this simple solution is not of a sufficiently general form to be applied to a real ocean, it has important theoretical implications. In particular, no solutions of (174) exist in which the flow is towards the equator. Even in the unstable case, x < xo, the flow is poleward. We conclude, therefore, that flow towards the equator cannot exist as a purely baroclinic mode. The vertical velocity (176) is independent of x and y as assumed. It is pro- portional to the eddy diflfusivity, kv, and increases towards the surface where the Ekman transport must be divergent {we > 0) and the surface source, Q* positive. We can eliminate ky between (176) and (177) to obtain Substitution of the representative values: w = 4:X 10"^^ cm sec~i, p^ — p=10~3 g cm-3, a;-a:o = 4000 km, g=10^ cm sec-2, /= 10-4 sec-i and /S = 2xl0-i3 cm-i sec-i, into (180) yields a depth of 400 m. As the density anomaly decreases as z~^, the ocean will be virtually homogeneous for depths of the order of 1000 m or more. The simple introductory model that we have considered bears little re- semblance to the density structure and circulation in any real ocean. In order 374 FOFONOFF [sect. 3 to construct a more realistic model, we must introduce barotropic flow and consider a more realistic variation of density or potential energy with latitude. This has been accomplished, in part, by Robinson and Stommel (1959) in their study of the oceanic thermocline and the associated thermal circulation. They assumed that the vertical velocity did not decrease to zero in the deep water but approached a constant value Woo. Thus, for their model, (170) is replaced by It can be seen immediately from (181) that solutions with x decreasing eastwards (flow towards the equator) are possible if Wao is greater than zero. The non-zero vertical velocity in the deep water implies the presence of a convergent baro- tropic flow that supplies water to the baroclinic mode. However, Robinson and Stommel ignored the horizontal components of the barotropic flow in the equations. With this simplification, they were able to construct a surprisingly realistic model of the thermocline. The presence of an unspecified vertical component of flow in the deep water introduces an indeterminacy into the interior solution. Consequently, choosing a particular form of the interior solution is not sufficient to specify the di- vergence of the Ekman flow or the surface sources. Thus, some freedom is available in choosing the boundary conditions (172) and (173). Robinson and Stommel developed their model in terms of temperature and vertical velocity, neglecting lateral diffusion and convection by the zonal component of flow. We shall follow their notation and terminology in describing the model rather than developing their equations in terms of potential energy. The basic equations describing the thermocline model are (183) f dw ?22 /3 dv ga^ dT f 8z /2 dx (184) Clearly, we can replace the temperature T by the apparent temperature T* defined in (160) with no change of the equations or results. Thus, the model can be applied to salinity as well as temperature so long as the linear approxima- tion (157) is admitted. Robinson and Stommel looked for solutions of the form y ^ (P,i-^ ^ _1_ a^ ^ (185) poa poga dz^ w = Wo. + -^^^ = H{x)co{^). (186) SECT. 3] DYNAMICS OF OCEAN CURRENTS 375 where ^^zF{x). They assumed that the temi)erature did not vary with x at the surface and changed Hnearly with y as To+ IQ-^Tiy. By substituting (185) and (186) into (182) and (184), they found that solutions having the desired characteristics had to be of the form T = m, ^ = x-y^oj{i) (187) and I = zx-^K They considered solutions for | > 0 only, i.e. z < 0, a; < 0. However, this restriction excludes poleward flows and will therefore not be imposed. ^ We shall consider both positive and negative values of |. The basic equations were transformed to remove some of the dependence on y and to introduce units to make the terms of the equations of unit magnitude. The final equations are d^ 8& SWd& where W = {xle)y^w, ^ = {ejx)y^z, r? = 10-8?/, y=10-»fl^, e^ga^/Sp. Boundary conditions, imposed at the bottom of the Ekman frictional layer (^ = 0), are ^0 = To+10-^Tiy = To+Tirj (190) Wo= We= {xl€)y^We, (191) where Wg is the vertical velocity given by the divergence of the Ekman trans- port components. The conditions for |^| ^ go are d' ^ 0 and W -^ Woo. We shall also consider the implications of the boundary condition kv = g* - \xl€\y^Q* (192) which expresses the "apparent" heat flux across the ocean surface. Robinson and Stommel linearized (188) by replacing W and dd-ldr] by their averages over a depth interval L, where L is defined as the interval in which both functions differ by more than l/e2(^l/7) from their asymptotic values. Then, by choosing simple exponential expressions for W and & that satisfy the boundary conditions and yield the correct averages, they integrated (188) and (189) over ^ to obtain algebraic equations relating the averages. We shall use the same approximate procedure. The meridional temperature gradient, d&ldiq, varies between Ti at the surface and zero for |^| -^ oo. Hence, its average will be of the order of Tij'l. Similarly, the velocity, W, varies between We and Wx and has the approximate average 1 Robinson and Stommel introduced this restriction so that an eastern boundary could be placed at a; = 0. However, the solutions are singular at a: = 0 so that little is gained by this procediire. It is preferable to assume that the solutions apply strictly to interior regions and can be connected to the boundaries by higher-order boundary solutions. 376 FOFONOFF [sect. 3 {We+ Wx)l2. We can consider either the mean vertical velocity or the mean meridional temperature gradient, but not both, as unknown. Integration of the linearized equation yields ^^MJo 2 ^'-^^' 2 = ^ (193) and, from (189), &d\!;,\ for ^ > 0, a; < 0 JO (194) = + &d\^\ for ^ < 0, :c > 0. If W does not change sign with depth {we, w, Wod > 0), we can approximate W and & by W = lfoo + (lfe-lFao)e-2l^l/^ (195) ^ = ^oe-2l?l/^. (196) Substitution of (196) into (194) yields JO &dm = Mol2. (197) The boundary conditions become ..,'!)„ ^ -^ -^ 0 (198) and m\ _2Mo_ ldW\ ^ -liWe-Woo) \ for ^ < 0. By substituting (197) and (198) or (199) into (194), we obtain We- Woo ^ ^oi>2/4 = kv'~&o^lq*'- and, from (193), 2q* + {We+Wao)&0+^T,{We-Wo,) = 0. (199) (200) [201) If we assume that We, Woo and q* are given, we can calculate d-Q from (200) and T) from (201) provided we consider ky as known. On the other hand, if we use Robinson's and Stommel's assumption that We, &o and Ti are known, SECT. 3] DYNAMICS OF OCEAN CUKRENTS 377 we are able to estimate Wx> and q* (or L). Ths known constants assumed by Robinson and Stommel, except for ky, can be estimated with reasonable ac- curacy from observations of the real oceans, whereas Woo and g* are known only within an order of magnitude. Nevertheless, the thermal circulation is driven by the flow. Wod. and the heat flux, g*, as well as the divergence of the Ekman transports. From the point of view of the thermal mechanism, the vertical flow from the deep water is arbitrary, i.e. the barotropic mode is not controlled by the thermal circulation. For Mv> 0 and Q* > 0, we find from (200) and (183) that the horizontal flow is towards the equator if | Wao\ >\We\ and towards the poles if | Woo\ <\We\. As iWxl must exceed \We\ for flow towards the equator, upwelling motion in the deep water is an essential feature of the large anticyclonic gyres in the oceans. However, this feature is not necessarily present for cyclonic gyrals. As has already been indicated in the simple example considered earlier, poleward flow can occur as a purely baroclinic mode {Wao= Wcc = 0). It is interesting to note that if the upward flow in the deep water is sufficiently intense the baroclinic mode occurs as an anticyclonic circulation with flow towards the equator even though the Ekman transport is divergent {we > 0) and the integrated or total transport is poleward. Consequently, we can obtain estimates of Wx, relative to iVe in a region where the curl of wind stress is positive by observing whether the baroclinic flow, calculated from standard oceanographic station data, is cyclonic or anticyclonic. Another consequence of the upward flow in the deep water is that the baroclinic transport, computed from station data, is less than the total transport computed from the distribution of wind stress for cyclonic circulation in the ocean. The convergent Ekman transport (mv < 0) presents a more difficult problem. It is clear that downward flow from the Ekman layer cannot extend into the deep water as this would require an increasing vertical gradient of temperature with depth. Hence, we must assume that the downward flow decreases to zero at some depth below the Ekman layer. Below this depth, the vertical flow will be upward. The horizontal flow cannot be poleward under these conditions of vertical flow and we need only consider the case ^ > 0. Robinson and Stommel assumed that the downward flow extended to a small, but unknown, depth h and separated the ocean into two layers by the surface of zero vertical flow. They estimated the averages of W and d&ldir] separately in each layer and integrated the linearized equations to obtain relations among the averages and the other parameters describing the model. We shall follow the same procedure but will use slightly different approxima- tions in order to take heat flux into account. Anticipating that h will be relatively small and that the temperature will not change rapidly in the upper layer, we assume that the meridional gradient, Ti, does not change appreciably in the upper layer so that its average is approxi- mately Ti. The average of the meridional gradient over a depth interval, L, of the lower layer is approximated by Ti/2. The average of the vertical velocity is of the order of Wel2 in the upper layer and W^I'I in the lower layer. We 378 FOFONOFF [sect. 3 assume, further, that W and & vary approximately Hnearly in the upper layer and, as Tf = >r41-e-2(«-A)/L], (202) ^ = ^^e-2(c-/.)/L (203) in the lower layer, where d-h is the temperature along the surface of zero vertical flow iC^h). These functions are consistent with the averages chosen in each layer. We replace W and d&ldr) in (188) by their approximate averages and integrate the linearized equation separately over each layer to obtain qh*-qo* = We{&o-&h)+yTiWe (204) for the upper layer and q^* = -Woo&h + lyTiWoo (205) for the lower layer, where are the fluxes of heat at ^ = 0 and l, = li respectively. Similarly, integration of (189) yields bW for the upper layer and dVJ_ ___ + __ = (^„_^,)_^__ (208) ^^ = ^'i^-^^ >^ ^^^^^ for the lower layer. Equation (208) reduces to the relation, Wefh — —2WoolL, given by Robinson and Stommel, if we neglect the heat flux term. However, this relation implies d'h=^Q and is not consistent with our other approximations. The six equations, (204) to (209), are sufficient to determine the six un- knowns d'o, &h, qh*, h, L and Ti if go*, We, Woo and ky are assumed to be known. The equations cannot be solved explicitly for the unknowns. However, we can reduce them to two algebraic equations for h and L that can be solved numeri- cally. The two equations, in non-dimensional form, are A2(A-Ao) = Xo{h' + hi){h'^-h2^) > 0 (210) A = {l-h'^)lh', (211) where h' = hlho ~0.80 A = LolL -0.59 ho = {2kvWelqo*y^'^ -0.58 SECT. 3] DYNAMICS OF OCEAN CURRENTS 379 Lo = 2\Woo\holWe -0.78 Ao = \Wo.\%olkv{2We+\Wa,\) -0.29 hi = 2kvlWeho -1.14 A2= (- PFe 0.69. The magnitudes of the constants and the approximate solutions of (210) and (211) are given for the values: We = ^ Dcm sec~i, Woo= —2 Dcm sec~i qo* = 30 D°C cm sec-i and kv = l cm^ sec-i, where D indicates the dimensions, (°C cm sec)'/3, of (a:/e)'/3. For (x/e)^3-7x lO^D, or a; - 4500 km and e-1.3x 10-6 °C-i sec-i, these values correspond to vertical velocities of + 2.9 x 10"^ cm sec^i in the deep water and — 4.3 x 10"^ cm sec-i at the bottom of the Ekman layer and a heat flux of 4.3 x 10-^ cal cm-2 sec-^. For these values, which may be considered typical of real oceans, the surface of zero vertical flow is at 320 m and the depth interval corresponding to L is 920 m. The vertical range of temperature is 30.6°C and the meridional gradient is — 4.3°C per 1000 km. The model is very sensitive to the value assumed for kv. For example, if we use kv = 2 cm2 sec-i, the vertical temperature range is reduced nearly in half to 16.5°C, the depth of the upper layer is increased to 570 m and L to nearly 1300 m. The meridional gradient, however, is increased only to — 4.5°C per 1000 km. An increase in the vertical velocity of the deep water reduces both h and L and the meridional gradient of temperature but increases the surface temperature. At a given position in the ocean, there are nine variable parameters required to describe the model: We, qo*, ^o, Ti at ^ = 0, qn*, ^h at <^ = h, the two charac- teristic depths, h and L, and the deep-water velocity, Woo. In addition, the eddy diffusivity, ky, is unknown and must be treated as a variable parameter. There are six relations among the total of ten parameters. In principle, we can choose any four of the parameters independently and solve the equations for the remaining six. Thus, we can apply the model to the real oceans by evaluating from observations four of the more readily observable parameters and using the model to obtain estimates of the remaining six. Robinson and Stommel, for example, evaluated We, &h, h and L from observations and used relations similar to (204) to (209) to estimate Woo, &o and kv. Additional parameters that can be estimated from observations can be used to check the consistency of the model. The upward flow from the deep water is arbitrary from the point of view of the thermal mechanism except that it cannot be zero in regions where the Ekman transport is convergent. As the major part of the world ocean is covered by a convergent Ekman layer, the total upward transport can be comparable to the transports of the horizontal flows. Stommel and Arons (1960a) estimate the total upward transport to be of the order of 40-50 million cubic metres per second. The deep water lost through the thermocline is presumably replaced in relatively concentrated regions of sinking in the North Atlantic Ocean and the 380 FOFONOFF [sect. 3 Weddell Sea. The observational evidence for these sources, although indirect, is convincing. However, although we know approximately where the sources are, we do not have a clear understanding of the processes that control their strength. It is evident that the formation of deeja water is ultimately controlled by the distribution of fluxes of heat and water across the ocean surface but we do not know the details of the processes involved. The model for the con- vergent Ekman transport yields larger meridional gradients of temperature for weaker upward flow. The larger gradients may, in turn, favour an increased rate of formation of deep water. Hence, the thermal circulation may tend to be self-regulating. In this sense, the interpretation that the thermocline mechanism "demands" a certain upward flux of water from below, suggested by Stommel and Arons (1960a), may be justified. In any case, the model of convective circulation constructed by Robinson and Stommel deserves further study and elaboration. 7. Time-Dependent Motion The theoretical study of time-dependent motion in bodies of water of oceanic scale has been confined for the most part to the investigation of various types of waves that can be propagated in a homogeneous or two-layer ocean without interaction with the steady flow\ We shall, therefore, confine our attention to these two idealizations of the structure of the real oceans and consider primarily wave modes with periods greater than a half-pendulum day {'l-nlf). Waves of shorter period have been studied in greater detail in connection with tidal theory and the study of surface waves and swell. Rossby (1939) showed that the variation of the Coriolis parameter with latitude enables the ocean to execute a low-frequency wave motion that propagates westwards relative to the particle motion. These waves, called planetary or Rossby waves, provide a mechanism for transporting energy on a time-scale greater than a half-pendulum day. One of their characteristics, pointed out in the section on dimensional analysis of the momentum equations, is that the horizontal velocities in the wave are nearly geostrophic. Thus, in a sense, the waves can be considered as moving current systems. These low- frequency waves play a fundamental role in the approach of ocean circulation to the steady state and in the transient response of the ocean to variations in the driving forces that maintain the steady circulation. Our present knowledge of the time-dependent modes of response of the ocean is far from adequate. For example, we have a poor understanding of the interactions between wave modes and the steady flow. We have seen that the steady flow can be concentrated into intense currents near the ocean bound- aries. In the boundary regions the relative vorticity can become as large as the Coriolis parameter so that the Rossby number is unity. Under these conditions the interaction terms between the steady flow and time-dependent modes of flow are not negligible, and appreciable exchanges of energy between the modes may occur. However, boundaries are extremely difficult to incorporate into the SECT. 3] DYNAMICS OF OCEAN CURRENTS 381 solution of the time-dependent equations and much of the theory has been developed for an unbounded ocean only.^ The theory of time-dependent motion is presented here in a form similar to that developed by Veronis and Stommel (1956). However, their procedure of solving the characteristic equation for the frequencies is not followed in favour of the simpler procedure of solving for the wave numbers. Also, the effect of a constant steady velocity on the time-dependent motion is considered in order to interpret more fully the nature of the boundary currents already considered. A. Simplified Time- Dependent Equations We have seen from the analysis of the magnitude of the terms in the time- dependent equations that the interactions Avith steady flow depend on the magnitude of the Rossby number. We shall restrict our attention to flows with Rossby numbers sufficiently small so that we can neglect the interaction terms. Similarly, we assumed that the gradients of the correlations, represented by r'ij in (17), are small. For simplicity, we can consider these simplifying assump- tions as being equivalent to the assumption that the steady flow is entirely absent. However, such a strong assumption is not necessary and can be modified later in the light of the solutions obtained. Using the simplifying assumptions given above, we can write the time-dependent equations (13) and (14) in the form ^1-^=-^ (-) ^ + ^ + %^ = 0, (215) ex cy cz where pa is the steady-state pressure. The variation of density with time, present in (14), does not have to be considered in the equations for the two- layer ocean provided we assume thjat the vertical motion of the interface is small compared with the thickness of the layers so that boundary conditions can be applied at the mean position of the interface. We can proceed to analyse the system of time-dependent equations using the concept of barotropic and baroclinic velocity components introduced for the steady state. We denote the barotropic velocity components of the time- dependent motion by w^, Vq and the baroclinic components by Ug, Vg. The 1 Some idea of the difficulties encountered in solving the time-dependent equations for a bounded ocean can be obtained by examining the solution for constant Coriolis para- meter given by Taylor (1922). He was able to separate the motion into symmetrical and antisynMnetrical modes. These modes are coupled together if ^ is different from zero. 382 FOFONOFF [sect. 3 barotropic component is uniform with dejith and equal to the velocity in the lower layer. The baroclinic component is equal to the relative motion between the upper and lower layer. Waves in which the baroclinic component is absent will be termed barotropic waves and those in which the baroclinic component is present, baroclinic waves. i We can integrate the time-dependent equations vertically from the bottom of the ocean to the interface and from the interface to the surface. The integra- tion yields the two sets of equations du^hi r . 7 S{p-po)2 ,^,„. /)2— ^-P2>B^2 = -h2 ^ (216) ^%^2 , ^ . . . S{p-po)2 p2 o. + p2jU^n2 = -fl2 7- l^i') 8h2 ^ 8uBh2 ^ dvBh2 ^ ^ ,2i8) 8t 8x 8y for the lower layer, and pi g^ pif{vB + Vg)hi ^ -hi — (219) 8{VB + Vg)hl /■/ , M, I, ^{P-P0)l ,oOA\ ^1 + g/ +pif{uB + Ug)hi = -hi ^^^ (220) 8hi 8(UB + Ug)hi 8{VB + Vg)hi ^ 8t 8x 8y ^ ' for the upper layer, where ^i, h2 are the depths and pi, p2 the densities of the upper and lower layers respectively. The pressure in the upper layer, from (214), is P = pigil-z), po = pigC^-z), (222) where -q is the instantaneous and 7; the mean position of the free surface. Similarly, in the lower layer p = pig{-n-Zi) + p2g{zi-z), po = pig{^-zi) + p2g{zi-z), (223) where Zi is the instantaneous and Zi the mean position of the interface between the two layers. In the absence of steady flow, we can assume that rj is zero and that Zi is constant. The horizontal gradients of pressure can then be expressed as ^<^ = ..4: (224) for the upper layer and 8{p-po)2 87] 8zi —^—=p,g-+(f.,-p,)g- (226) 1 This coincides with the terminology used by Veronis and Stommel (1956), who con- sidered absolute velocity components in each layer. A barotropic component is present in both wave types. SECT. 3] DYNAMICS OF OCEAN CURRENTS 383 for the lower layer. Similar expressions obtain for the ^/-component. The presence of steady flow will not affect the difference between the instantaneous and mean pressure, p—po. However, in the presence of a steady baroclinic component, the interface zt can no longer be assumed to be level. The variation in depth of the upper and lower layer will enter the equations for the time- dependent motion through the continuity equations, (218) and (221). The influence on time-dependent motion of the variation of thickness of the layers, or, in the related problem, of the variation of bottom topography, has not been studied adequately as yet. We assume that the variations in the depths of both layers is sufficiently small so that we can replace the instantaneous depths, hi, hz, by their means. Hi and H2, except where the depths are multiplied by gravity, g. Introducing -qi for Zi — Zi to represent the deviation of the interface from its mean position and substituting (224) and (225) into the integrated equations, we obtain «'" + /,, (^ + ^) = 0 (228) dt \ dx . By for the barotropic velocity, and t-/"--^'^^ (229) ^-/"--^'^ ,230) drj /8ub dvB\ rj /^% ^^9 for the baroclinic velocity, where go={pilp2)g, g' — {Aplp)g and H = Hi + H2. The equations for a homogeneous ocean are obtained by setting g' and Hi to zero. We can study the wave modes allowed by the two-layer system of equations by assuming that the deviations of the free surface and the interface from their mean positions and the velocity components are periodic functions of time and horizontal distance. As the equations governing the motion are linear, more complex motions can be considered as linear combinations of simple plane waves. In addition to wave motion, the linear equations admit aperiodic solutions that are of importance in interpreting simple interactions with steady flow. We assume that all of the variables can be represented by functions of the general form ^ei("'<+*a;+i2/)^ where ^ is a complex dimensional coefficient that is independent of time but may vary with latitude. The coefficients a», k and I will be complex in general. For wave solutions, oj is real and is interpreted as 384 FOFONOFF [sect. 3 the angular velocity of the wave motion. The real parts of k and I are the wave numbers. Substituting for the variables in (226) and (227), we obtain for the baro- tropic mode iojtiB^-fvB^ = -ik{gor,^ + g'-ni^) (232) ftiB^ + icoVB^' = -iligov^' + G'Vi^), (^33) where the zero superscript denotes the coefficient of e*'<'"'+*-^+'^) for each variable. We have assumed that rj^ and rji^ are independent of y and x. Solving for the velocity components, wc obtain UB^ = ^^^^A9ov' + U'Vi') (234) ^BO = ^^^^(^o^o + f7'^»0). (235) Similar expressions can be derived for the baroclinic velocity components. From (234) and (235), we can see that the motion in a wave is completely specified by the momentum equations except for a relation between angular velocity and the wave numbers. If we assume that rji^ is zero and cd is real and represent the vertical velocity, ivb^, at the surface by ioj-q^, we note that the coefficients satisfy the relation luB^-kvB^ + ^^^P^ ojb' = 0. (236) Assuming further that k and I are real, we see that the particle motion lies in a plane perpendicular to the vector [I, - k, gof{k^ + l")laj{f" - co^)].! This vector is perpendicular to the direction of travel [parallel to { — k, — I, 0)] of the wave and inclined at an angle, y, to the horizontal plane, where tan y = — -p; — ■ (237) At high frequencies, co^f, the angle y approaches zero and the particle motion is in a nearly vertical plane. At low frequencies, co/, both Ao and eo approach zero. 1 Strictly speaking, cb and Cg represent wave speecis in the sense of phase velocities only at high frequencies, w>>f. At low frequencies, a» a^^ S^,^. Hence, from (250), A; = Ao + S^ > 0. Thus, both pairs of barotropic and baroclinic waves travel westwards. We shall refer to these waves as Rossby waves. ^ We note from (249) that Rossby waves occur if ^^ < J[/^-(/^-i32c2)V2] ^ ^2c2/4y2 (251) provided/ 2 |>^c. Other wave solutions occur for CO' > H/2 + (/'*- i82c2)V2] ^/2. (252) Except near oj=f, these high-frequency waves are essentially equivalent to the waves obtained assuming a constant Coriolis parameter. These have been described thoroughly by Proudman (1953). An interesting study of the propaga- tion of the high-frequency waves in the vicinity of partial boundaries has been carried out by Crease (1956). We can examine minimum wave periods and the corresponding wave-lengths 1 Rossby (1939) described the zonal barotropic wave in a study of time-dependent motion in the atmosphere. Stommel (1957) pointed out that similar waves had been studied earlier in connection with tidal theory. SECT. 3] DYNAMICS OF OCEAN CURRENTS 387 for Rossby waves by inserting into (251) the values: Hi — SQO m, 7/ = 4000 m, Aplp = 2x 10-3, g=lO^ cm sec-^ /= lO-i sec-i and /8 = 2x lO-i^ cm-i sec-i. We obtain 200 m sec-i for cb and 3.6 m sec""i for Cg. The Rossby waves will have periods in excess of Tmin., where Tmiu. = 27r/aJmax = 47r//j3c = 3.6 days for barotropic waves = 6.8 months for baroclinic waves. The wave-lengths for zonal waves for the minimum periods are 277/Ao = 4:7TOJmaxl^ = 27rc// = 12,600 km for barotropic waves = 230 km for baroclinic waves. (253) (254) Fig. 5. Variation of wave number with angular velocity for zonal barotropic Rossby and higher frequency waves. The curves are computed for c^/f^ = 0.2 and presented in non-dimensional form with k'—fk/^ and cL>' = co/f. For comparison, the relationship (broken curve) between k' and cv' is shown for uniform rotation {^ — 0) and for no rotation (k= ±co/c). The curves are symmetrical with respect to the origin. The ratio of wave-length to period yields the phase velocity of the waves. For the barotropic mode, the phase velocity is about 40 m sec-i and for the baro- clinic mode, only 1.3 cm sec^i. The relation between wave number and angular velocity for zonal waves is shown in non-dimensional form in Fig. 5. Curves near 6u//= 1 are not shown because the approximations used in deriving (244) are not valid in this region (eo -^ 00 for co — >/). The particle motion in Rossby waves is directed primarily along the crests with a much weaker component in the direction of propagation. This may be seen from (234) and (235) by forming the ratio \u\l\v\ ~ co//. The ratio is about 0.2 for barotropic waves and about 4 x 10"^ for baroclinic waves at the maximum angular velocity. The phase velocity a>/\/(A2 + v^) varies with the direction of propagation of the waves. By substituting ko cos 6 for A and 388 FOFONOFF [sect. 3 ko sin 6 for v in (249), where Icq is the wave number along the direction of travel and 6 is the angle between the direction of travel and a latitude circle, we obtain a2 = ^^o2 - 2Ao^o cos 6 + Ao2. (255) Substituting for Ao and a from (247) and (249), we obtain the approximate relation ko ko^ + iplc^) ^ ''''' for the phase velocity. Thus, we can see that Rossby waves travelling at an angle to a latitude circle will move more slowly than zonal waves of the same wave-length. However, the westward progression of a wave crest along a given latitude circle is equal to that of a zonal wave of the same wavelength, and is independent of the direction of travel of the wave. Thus, for angles of travel approaching ttJ'Z, the waves become almost stationary with the particle motion reducing essentially to a system of zonal currents. Rossby waves cannot be propagated meridionally. We can obtain the same result more directly by substituting A = 0 into (249). As Ao'^ > cr^, there are no real solutions for v and, consequently, no meridional Rossby waves for co> 0. Rossby waves for + v and — v can be combined to yield a zero meridional velocity along a single zonal boundary or a pair of zonal boundaries. Thus, Rossby waves can be reflected from zonal boundaries. The meridional motion of the reflected wave is opposite to that of the incident wave. As both waves at a given frequency travel westwards, they cannot be reflected from meridional boundaries. Nevertheless, the two waves can be combined to yield zero zonal velocities at meridional boundaries as was shown by Arons and Stommel (1956). The combination of two waves cannot be interpreted as reflection because the zonal flux of energy is not zero. In order to interpret the solutions given by Arons and Stommel, we will first consider solutions of (249) for 0-2 < 0. These are of the form k = XQ±iSv, 1= ±v — i€o, (257) where 8^= \^\a^ — v^. These solutions represent a wave-like motion that progresses westwards but changes its amplitude along the direction of travel. Consequently, the energy flux associated with the motion is divergent zonally. As the total energy must be conserved, there is a flow of energy along the crests of the waves. In order to examine this characteristic of the motion in greater detail, we formulate the energy equation for a time-dependent motion in a homogeneous ocean {g' = 0, H2 = H). Multiplying (226) and (227) by the velocity components and adding, we obtain where Eic = ^{u^ + v^)H is the total kinetic energy of a column of water of unit SECT. 3] DYNAMICS OF OCEAN CURRENTS 389 horizontal area. By introducing Ep for the potential energy, Igt]'^. and using (228), we can transform (258) to + 9HH:- + ^\=0. (259) 8t \ dx dy We can interpret gHurj and gHv-q as the components of flux of total energy associated with the motion. Taking the average of (258), we obtain du-n dvri Thus, the divergence of the mean energy flux is zero and the mean energy density at a given jDoint is constant. The mean energy flux associated with zonal Rossby waves {v = Q, 8^, = So) is proportional to ut] = igr]'- aj(Ao± So) — eof e+2^o2/ (261) mj = 0. (262) The energy flow is eastward for short Rossby waves (Ao+So) and westward for long Rossby waves (Aq — So). For both waves, the energy flow is independent of X. For the solutions in the range ct^ < o given in (257), we obtain UTj = i^grjO- ojAo— eof e±25o2;+2eo2/ (263) ^= +lgr^02|-y 3^1(^2 _^2)]e±25o.T+2eo2/. (264) These flux components satisfy (260) provided we apply the assumptions made for (244) and treat So and eo as constants. As ojAq— eo/is approximately equal to — jS/2, the zonal energy flow is westward. The meridional flow is towards the equator for + So and towards the poles for — So. For poleward flow of energy, the amplitude of the motion decreases westwards. We can see how solutions for g'^ < 0 arise by considering an ocean divided by the y-axis into two regions of different depths. We can always choose the angular velocity of the Rossby weaves in the deeper region such that ct^ < 0 in the shallower region. Solutions in the shallower region will then be of the form given in (257). The Rossby waves will not propagate into the shallower region and the energy from the waves will be transported meridionally on reaching the shallower region. The flow of energy is zonal for each zonal Rossby wave separately. However, there is meridional flow for a combination of the two waves at a given fre- quency. If we represent the sum of the two waves by 77+ + 77-, where rj+ is the 390 FOFONOFF [sect. 3 short wave corresponding to Ao + So and 17- the long wave corresponding to Ao — So, we obtain the components a»Ao— eo/ V = —■ -^- — I sm 80X + — ^ cos Soa: Qi(wt+XoX)+eaX_ (270) U-n = u+in+-\-U-r)--\--— r^ g-n+^rj-^ COS 2 SoXe^^oV (265) — ^ _grj+rj-JSo ^.^ ^ Soa;e2.o2/. (266) By choosing V = ^f2_^2)Uol{ojK-eof)g, V-' = {P-cv^)Uol{cok--eof)g, (267) we obtain the wave solution 7] = rj+ + rj- = -{Uoc^lgco^)[2aj So cos Sox + ijSsin 8oxy(-<+^o^)+eo2/ (268) u = 2iUo sin Sore e^(<^t+x,z)+.oy (269) of Taking the imaginary part of the solution, we obtain the flux components v^ = -{2Uo^c^^lgoj^)sm^SQX e^ov (271) mj = (2C7o2c2/ So/g'ojS) sin 2Soa; e^ov. (272) From (269), we can see that the solution will satisfy the boundary condition w = 0 at two boundaries, x = 0 and x = a, if Soa is an integral multiple of n. Hence, solving for to, we obtain the "eigenvalues" where n is an integer. For the first mode, n=l, the flow of energy is towards the equator in the eastern half of the ocean and towards the poles in the western half. There is a westward energy flow throughout the ocean. As no eastward flow of energy is present, the solution is not closed. Energy enters a given region in the eastern part of the ocean and leaves in the western part. For the higher modes, n>l, there are alternate regions of positive and negative meri- dional flows of energy. These solutions cannot be applied to a bounded ocean because the particle motion is not zero along any zonal plane and energy cannot be returned to the eastern part of the ocean. Rossby waves travelling obliquely to latitude circles can be made to satisfy the boundary conditions along zonal boundaries but not along meridional boundaries. We may conclude, therefore, that Rossby waves do not constitute a complete mechanism for considering periodic variations of flow in a bounded, or partially bounded, ocean. In the presence of zonal boundaries, the equations admit an additional periodic motion in the form of zonal Kelvin waves. The meridional velocity component for these waves is zero and the characteristic equation relating SECT. 3] DYNAMICS OF OCEAN CURBENTS 391 wave number to angular velocity can be found by substituting —ifkja} for I in (244). The zonal wave number is given by , ^ 2(/2-a;2) iS< 2(/2-a;2) a>2\ + ■^1 • (274) Kelvin waves move westwards along a boundary on the poleward side of an ocean and eastwards along the equatorial side. The amplitudes of the waves decrease with distance from the boundaries. Meridional Kelvin waves do not exist for frequencies in the Rossby wave range. It seems probable that a solution for time-dependent motion in an enclosed ocean could be obtained by combining Rossby and Kelvin waves together with periodic solutions of the type given in (257). Such a solution would be useful in interpreting the balance of energy flow in the ocean. Unfortunately, an explicit solution is not available and a satisfactory interpretation of the oscillation modes cannot be made. In the absence of an explicit solution, we can speculate that the energy flow in the interior of the ocean of rectangular shape is due to Rossby waves. Near the meridional boundaries, the energy is transferred meridionally by the wave-like motion given by (257). The energy is then returned zonally in the vicinity of a zonal boundary by Kelvin waves. If this speculation is justified, we could conclude that the time-dependent equations yield a flow of energy by wave motion that is similar to the flow of mass given by the steady-state equations. Periodic solutions of the time-dependent equations do not transport mass. However, we can find solutions that are not periodic in time by substituting — iAo, the phase velocity co/k is westward for the solution corresponding to — Aq-hS^ and eastward to — Aq— §^. The aperiodic solutions can be used to examine time -dependent flow from one part of an ocean to another but they are introduced primarily because of their importance in interpreting boundary -layer solutions of the steady-state equations. The time-dependent solutions that we have considered can be extended to apply to a homogeneous ocean with a level bottom in which there is a steady and uniform zonal current, Uq. In the linear approximation, the equations governing the flow are similar to the six equations (226) to (231), with d/dt replaced by dl8t+ Uq djdx. Hence, the only effect of the steady flow is to move 392 FOFONOFF [sect. 3 the waves along at the velocity, Uq. Consequently, we can apply the time- dependent solutions already examined by replacing oj everywhere by co + Uok. It is of particular interest to examine those solutions that are brought to rest (a> = 0), relative to the boundaries, by the zonal flow. As Rossby waves move westwards, they can be made stationary by an eastward flow. Replacing co by Uok in (249) and simplifying the resultant equation, we obtain k^ + v^^^lUo, (277) where we have neglected terms of the order of /^/c^^ 10~i^ cm~2 and smaller in comparison with /S/t/o~ 10"^"^ cm~2. Equation (277) is equivalent to the characteristic equation for the homogeneous solutions of the vorticity equation (101). Hence, the solution (104) consists of a steady zonal flow towards the east with barotropic Rossby waves progressing westwards relative to the water, but of such wave-lengths that the motion relative to the boundaries is zero. As indicated by (256), Rossby waves of a given wave-length have the same west- ward component of phase velocity so that they can be made stationary by a uniform eastward flow regardless of their direction of propagation. The inertial solution (104) includes a system of zonal currents that can be interpreted as a stationary meridionally-directed Rossby wave. Only the short Rossby waves appear in the inertial solution because the long waves have phase velocities much larger than Uq and cannot be brought to rest by the zonal flow. For C7o<0, the aperiodic solutions given by (276) can be made stationary. Substitution of Uok for oj in (276) yields A;2 = (^/|t/o|) + v2, (278) which is equivalent to the characteristic equation for the solution given in (103). Hence, the inertial boundary currents along the western and eastern boundaries can be considered as a time-dependent motion that propagates eastwards at the same speed as the westward interior flow. Because of its eastward motion relative to the interior flow, the western-boundary current cannot be formed if the interior flow is eastward or zero. It is possible that, if the westward interior flow weakens, the western-boundary current could not adjust in time to prevent its breaking away from the boundary. If this were the case, it would perhaps not be unreasonable to suggest that variations of the in- terior flow could lead to breaking away and re-formation of the western- boundary current yielding a "multiple-stream" structure such as that suggested by Fuglister (1951). This result does not follow from the analysis presented here although its possibility is suggested. Further investigation of the stability of the western-boundary currents could yield significant results that would improve our understanding of ocean circulation. The interpretation in terms of barotropic wave motion of the solutions for inertial circulation in a homogeneous ocean cannot be extended to inertial baroclinic circulation. The baroclinic Rossby waves move very slowly. Their maximum westward phase velocity is of the order of 3 cm sec~i. As steady velocities in the upper layers of the oceans are of the same magnitude or SECT. 3] DYNAMICS OF OCEAN CURRENTS 393 higher, their infiiience on the baroclinic waves is large. Moreover, in the presence of steady barocHnic flow, the interface cannot be level. Variations in depth of the upper layer would have significant effects on the baroclinic wave modes. If we consider, to a first approximation, that the variation of layer depth can be taken into account by replacing / by hofjh, where h is the mean depth of the upper layer at any point and ho the average depth over the region, we see that baroclinic Rossby waves cannot be generated for steady westward flow for which //A is constant. On the other hand, the effective value of the variation of hoflh may be considerably larger than /3 for steady eastward flow. Analysis of the time-dependent response of an unbounded two-layer ocean to fluctuations of wind stress has been carried out by Veronis and Stommel (1956). They considered a one-dimensional model and did not include steady flow components. They found that the response for periods of wind fluctuations of the order of one to seven weeks was principally in the form of barotropic Rossby waves. For longer periods, the baroclinic Rossby waves become in- creasingly important. The contribution to the velocity in the upper layer by barotropic and baroclinic waves is about equal for periods of about one year. For very long periods ( > 100 years), the response is purely baroclinic. Veronis (1956) calculated the proportion of energy that appears in geostrophic and non- geostrophic motions for winds of short duration. He concluded that a large fraction of the energy of the resultant motion appears as high-frequency {co>f) wave motion if momentum is added impulsively. However, if the same amount of momentum is added over a period of a half-pendulum day, only about one-tenth of the energy goes into the high-frequency waves. Thus, for wind duration of the order of a half -pendulum day or longer, the high-frequency wave modes can be neglected and the induced motion can be considered approximately geostrophic. The studies of the forced response of an unbounded ocean have given us a qualitative description of the effects of variations of wind stress. However, it appears likely that the results will be modified significantly by the presence of boundaries and components of steady flow. Further investigations of the time- dependent equations are required both to improve our understanding of the mechanism of Rossby waves and to determine in more detail the response characteristics of a partially or fully enclosed ocean. References Arons, A. B. and H. Stommel, 1956. A ^S-plane analysis of free periods of the second class in meridional and zonal oceans. Deep-Sea Res., 4, 23-31. Bjerknes, V., 1898. Uber einen hydrodynamischen Fundamentalsatz und seine Anwen- dung, besonders auf die Mekanik der Atmosphare und des Weltmeeres. Kgl. Svenska Veteitskapsakad. Handl., 31, 1-35. Bjerknes, V. and J. W. Sandstrom, 1910. Dynamic meteorology and hydrography. Statics. Puhl. Car?iegie Inst. Wash., No. 88, Part I, 1-146. Bjerknes, V., Th. Hesselberg and O. Devik, 1911. Dynamic meteorology and hydro- graphy. Kinematics. Publ. Carnegie Inst. Wash., No. 88, Part II, 1-175. 394 rOFONOFF [sect. 3 Charney, J. G., 1955. The Gulf Stream as an inertial boundary layer. Froc. Nat. Acad. Sci., 41, 731-740. Crease, J., 1956. Long waves on a rotating earth in the presence of a semi-infinite barrier. J. Fluid Mech., 1, 86-96. Dietrich, G., 1957. Allgemeine Meereskunde. Gebriider Borntraeger, Berlin-Nikolassee. Ekman, V. W., 1905. On the influence of the earth's rotation on ocean currents. Ark. Mat. Astr. Fysik, 2, 1-53. Felsenbaum, A. I., 1956. On the method of integrated mass transport and a classification of the theories of ocean circulation. Trudy Inst. Okeanol. Akad. Nauk S.S.S.R., 19, 57-82. Felsenbaum, A. I., 1956a. An extension of Ekman's theory to the case of a non-uniform wind and an arbitrary bottom relief in a closed sea. Doklady Akad. Nauk S.S.S.R., 109, 299-302. Fofonoff, N. P., 1954. Steady flow in a frictionless homogeneous ocean. J. Mar. Res., 13, 254-262. Fofonoff, N. P., 1956. Some properties of sea water influencing the formation of Antarctic bottom water. Deep-Sea Res., 4, 32-35. Fuglister, F. C., 1951. Multiple currents in the Gulf Stream System. Tellus, 3, 230-233. Hidaka, K. and M. Tsuchiya, 1953. On the Antarctic Circumpolar Current. J. Mar. Res., 12, 214-222. Ichiye, T., 1949. On the theory of drift current in an enclosed sea. Oceanog. Mag., 1, 128- 132. Ichiye, T., 1950. A note on the friction terms in the equation of ocean currents. Oceanog. Mag., 2, 49-52. Ichiye, T., 1958. On convective circulation and density distribution in a zonally uniform ocean. Oceanog. Mag., 10, 97-135. Ichiye, T., 1960. On water budget in a two-layered ocean. Oceanog. Mag., 11, 111-126. Lineikin, P. S., 1955. On the determination of the thickness of the baroclinic layer in the sea. Doklady Akad. Nauk S.S.S.R., 101, 461-464. Lineikin, P. S., 1957. On the dynamics of the baroclinic layer in the ocean. Doklady Akad. NaukS.S.S.R., 177, 971-974. Morgan, G. W., 1956. On the wind-driven ocean circulation. Tellus, 8, 301-320. Munk, W. H., 1950. On the wind-driven ocean circulation. J. Met., 7, 79-93. Munk, W. H. and G. F. Carrier, 1950. The wind-driven circulation in ocean basins of various shapes. Tellus, 2, 158-167. Munk, W. H., G. W. Groves and G. F. Carrier, 1950. Note on the dynamics of the Gulf Stream. J. Mar. Res., 9, 218-238. Ozmidov, R. V., 1959. Extension of Ekman's theory of unsteady purely drift currents to the case of arbitrary winds. Doklady Akad. Nauk S.S.S.R., 128, 913-916. Proudman, J., 1953. Dynamical Oceanography. Methuen and Co., Ltd, London; John Wiley and Sons, Inc., New York. Robinson, A. and H. Stommel, 1959. The oceanic thermocline and the associated thermo- haline circulation. Tellus, 11, 295-308. Rossby, C.-G., 1937. On the mutvial adjustment of pressure and velocity distributions in certain simple current systems. J. Mar. Res., 1, 15-28. Rossby, C.-G., 1938. On the mutual adjustment of pressure and velocity distributions in certain simple current systems, II. J. Mar. Res., 1, 239-263. Rossby, C.-G., 1939. Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action. J. Mar. Res., 2, 38-55. Saint-Guily, B., 1955. Sur la forme des equations de la dynamique des oceans en regime turbulent. Bull. d'Inform., C.O.E.C, No. 7, 295-303. Saint-Guily, B., 1959. Sur la solution du probleme d'Ekman. Deut. Hydrog. Z., 12, 262- 270. SECT. 3] DYNAMICS OF OCEAN CURRENTS 395 Sarkisian, A. S., 1954. The calculation of stationary wind currents in an ocean. Izvest. Akad. Nauk S.S.S.R., Ser. Geofz., 6, 554-561. Sarkisian, A. S., 1957. On non-stationary wind-driven current in a homogeneous ocean. Izvest. Akad. Nauk S.S.S.R., Ser. Geofiz., S, 1008-1019. Shtokman, W. B., 1946. Equations of a field of total flow induced by the wind in a non- homogeneous ocean. Doklady Akad. Nauk S.S.S.R., 54, 403-406. Stommel, H., 1948. The westward intensification of wind-driven ocean currents. Trans. Amer. Geophys. Un., 29, 202-206. Stommel, H., 1954. Discussions on the relationships between meteorology and oceano- graphy. J. Mar. Res., 14, 504^510. Stommel, H., 1956. Oceanography. Washington conference on theoretical geophysics. J. Geophys. Res., 61, 320-323. Stommel, H., 1956a. On the determination of the depth of no meridional motion. Deep-Sea Res., 3, 273-278. Stommel, H., 1957. A survey of ocean current theory. Deep-Sea Res., 4, 149-184. Stommel, H., 1958. The Gulf Stream. Univ. of California Press, Berkeley and Los Angeles ; Cambridge Univ. Press, London. Stommel, H. and A. B. Arons, 1960. On the abyssal circulation of the world ocean — I. Stationary planetary flow patterns on a sphere. Deep-Sea Res., 6, 140-154. Stommel, H. and A. B. Arons, 1960a. On the abyssal circulation of the world ocean — II. An idealized model of the circulation pattern and amplitude in oceanic basins. Deep-Sea Res., 6, 217-233. Stommel, H., A. B. Arons, and A. J. Faller, 1958. Some examples of stationary planetary flow patterns in bounded basins. Tellv^, 10, 179-187. Stommel, H. and G. Veronis, 1957. Steady convective motion in a horizontal layer of fluid heated imiformly from above and cooled non-uniformly from below. Tellu^, 9, 401-407. Sverdrup, H. U., 1947. Wind-driven currents in a baroclinic ocean; with applications to the equatorial currents of the Eastern Pacific. Proc. Nat. Acad. Sci., 33, 318-326. Sverdrup, H. U., M. W. Johnson and R. H. Fleming, 1942. The Oceans. Prentice -Hall, Inc., N. Y. Swallow, J. C, 1955. A neutral-bouyancy float for measuring deep currents. Deep-Sea Res., 3, 74-81. Swallow, J. C, 1957. Some further deep current measurements using neutrally-bouyant floats. Deep-Sea Res., 4, 93-104. Swallow, J. C. and B. V. Hamon, 1960. Some measurements of deep currents in the Eastern North Atlantic. Deep-Sea Res., 6, 155-168. Swallow, J. C. and L. V. Worthington, 1957. Measurements of deep currents in the western North Atlantic. Nature, 179, 1183-1184. Taylor, G. I., 1922. Tidal oscillations in gulfs and rectangular basins. Proc. Lond. Math. Soc, Ser. 2, 20, 148-181. Townsend, A. A., 1956. The structure of turbulent shear flow. Cambridge Univ. Press, London. Veronis, G., 1956. Partition of energy between geostrophic and non-geostrophic motions. Deep-Sea Res., 3, 157-177. Veronis, G., and H. Stommel, 1956. The action of variable wind stress on a stratified ocean. J. Mar. Res., 15, 43-75. Welander, P., 1957. Wind action on a shallow sea : some generalizations of Ekman's theory. Tellus, 9, 45-52. Welander, P., 1959. An advective model of the ocean thermocline. Telliis, II, 309-318. Wertheim, G. K., 1954. Studies of the electric potential between Key West, Florida, and Havana, Cuba. Trans. Amer. Geophys. Un., 35, 872-882. IV. TRANSMISSION OF ENERGY WITHIN THE SEA 8. LIGHT J. E. Tyler and R. W. Preisendorfer For the past ten years the physical aspects of radiative transfer within the ocean have been under intensive study by both theoreticians and experimental physicists. Probably the most significant advance during this period has been the application of diffusion theory to the interaction of light with the sea and the development of a complete theory of radiative transfer in the ocean. This work has been carried on by Preisendorfer at the Scripps Institution of Oceano- graphy, and to some extent by LeNoble at I'Ecole Superieure de Physique et Chemie. This theory exhaustively treats the phenomena of multiple scattering within a scattering-absorbing medium in which the volume scattering func- tion is not isotropic. The theory is a unique and powerful tool for studying the interaction of electromagnetic radiation with the sea, for predicting flux distributions under water, and for computing the magnitude of important effects such as the deterioration of image contrast along horizontal and inclined paths of sight. The theory has also made possible the development of new instrumentation for studying the optical properties of the sea and has clearly shown the relationships between these various optical properties. The broad physical base which has thus been established for light measurements in the sea has brought about a valuable standardization among various laboratories and has provided a foundation to which special measurements or constructs important to biology or geophysics can be directly related. 1. Physical Constructs A. Radiance The most useful construct for the study of underwater light fields is the vector construct, radiance, borrowed from the field of radiometry. Radiance is flux per unit projected area per unit solid angle in a specific direction. It is defined by the equation N = PI{A cos dQ) (1) and can be measured by means of a simple device illustrated in Fig. 1, or by means of a properly designed optical system. The radiances measured in various directions from a fixed point below the ocean surface can be exhibited in three dimensions by assembling around the point vector arrows whose lengths are [MS received June, 1960] 397 398 TYLER AND PBEISENDOBFEB [chap. 8 CALIBRATED RECORDER Fig. 1. Schematic drawing of a radiance, or Gershun, tube. The output of the photo- detector area, A, is proportional to the incident flux, P, the soUd angle, Q, and the area, A, of the photodetector. The solid angle and area are fixed instrumental constants. The angle 6 of equation (1) is here zero. Fig. 2. Radiance-distribution diagrams in the vertical plane including the sun : (a) for clear sky with sun at 56.6° altitude, depth 53.7 m ; and (b) for overcast sky, depth 55 m (sun at 40° altitude but not visible). These two diagrams have the same relative magnitude. Because of the scale used, the fixed point of observation below the surface appears to be on the perimeter at the bottom of each diagram. Actually this point is inside the diagram (cf. Fig. 19). SECT. 4] LIGHT 399 proportional to the measured radiances. Such a display is called a radiance distribution solid. The shape of this solid will vary with depth, approaching a prolate spheroid whose major and minor axes depend uniquely on the values of a and s for the medium. Preisendorfer (1959) has shown in theory that, in a medium containing only scattering centers, the limiting shape is a sphere, whereas in a medium exhibiting only absorption the limiting shape is a vertical line. It is common practice to illustrate the radiance-distribution solid by showing only those vectors lying in a particular plane of interest, for example, the plane parallel to the sun's rays. Typical radiance-distribution diagrams of this latter type are shown in Fig. 2. B. Irradiance A second important construct, also borrowed from radiometry, is irradiance, or flux per unit area. The definition of irradiance is given in the equation H = PjA. (2) If we consider not the gross flux P but the parcels of flux dP arriving from the various solid angles da> we can write dH = dPjA but from equation (1) N = dPI{A cos e do)) so that dH = N cos e doj . (3) Thus irradiance H can be obtained from radiance distribution by summation, that is H = 21N cose Aoj. (4) C. Scalar and Spherical Irradiance Scalar and spherical irradiance are the last two of the major radiometric concepts to be discussed here. Scalar irradiance gives a quantitative measure of the total radiant flux arriving at a point from all directions about the point. Scalar irradiance, in essence, is a measure of the amount of radiant energy per unit volume of space at a given point ; the individual amount coming in from each direction about the point is unimportant, only the total is of interest. Scalar irradiance can be determined if the radiance distribution is known for all directions around the point of interest. Let N{p, 6, (j>) be the field radiance at point p, arriving from the direction {d, are measured from 400 TYLER AND PREISENDORFER [CHAP. 8 some fixed reference system. Then the scalar irradiance, h{p), at point p is defined as : h{p) = r r N{p, d, cf>) dQ, (5) Je=o J0=o where dQ = sin e dd d(f>. We can obtain an analytic expression for spherical irradiance, h4n{p), in the following way : consider a small spherical collector of radius r with center at p. Then the amount P{p, 6, (f>) of radiant flux intercepted by the spherical surface from a unit solid angle in the direction {d, (f)) is (using the cosine law) Pip,d,cf>) = N{p,d,cf>) ( cosi/jdA, J hemisphere (6) where the hemisphere of integration is determined by the plane of the great circle which is perpendicular to the direction {d, cf)). The integral is easy to evaluate because it simply represents the projected area of the hemisphere on the plane of its great circle, so that P{p, d, cf>) = rrr'^Nip, d, 0). (7) The amount P{p) of flux intercepted by the sphere from all directions is : Pip) = r r p(P' ^' */•) ^^ = ^r^Mp)- J 6=0 J)= { a{p, d, 0, e', f ) N{p, d', cj>') dQ{d', cf>'), (20) Jin where the point p is now being irradiated by flux from all directions about p. An example of the calculation of N^ {p, 9, <^) in real media is given in Preisen- dorfer (1956). a{p, 6, ^, 6', cf)') is the value of the volume scattering function at point p for fight incident in the direction {6', cf)') and scattered off in the direc- tion {6, (f)). {6, cf)) and {6', (f>') are measured with respect to some fixed reference frame [in the derivation {6, 0) was taken as (0, 0) and ) has the same general interpretation as N^{6) above, but now the radiance per unit 406 TYLER AND PREISENDORFER [CHAP. 8 length in the direction of observation is generated by hght scattered into the line of sight from all directions about the point p. By property (4), a{p, 6, (f>, 6', cf)') for any pair {6, cf)), {9', (j)') is known from the determination of a{d) at point p, as defined in (17). {in ) The volume total scattering coefficient From the above arguments we now have an explicit expression for the term, s, which arose in the discussion of the volume attenuation coefficient. For this term is evidently none other than that given in (15) which, by (16), may be written s = ( a{d) dQ = 277 f" a{d) sin 6 dO. (21) J47r jo The second expression follows from facts (3) and (4). This is the volume total scattering coefficient. In non-homogeneous media, it may change from point to point but, in any event, s does not depend on the direction of the irradiating beam [facts (3) and (4)]. Closely related to s are the {volume) forward scattering and {volume) backward scattering coefficients, f and b respectively, defined by the following formulas : / = 277 \ ' a{d) sin d dd, (22) jo 6 = 277 f " a{d) sin d dd, (23) Jn/2 so that s=f+b. (24) As in the case of s, both / and b may vary with position, but they do not in any event depend on the direction of the irradiating beam. {iv) Equation of transfer From the preceding interpretations of a and N^ , it is easy to verify that the equation of transfer for field radiance (or surface radiance), Nr, in a source-free medium is expressible as : ^ = -aNr + N^. (25) The first term on the right gives the space rate of loss of Nr by attenuation ; the second term gives the space rate of gain of Nr by scattering. c. The volume absorption coefficient During the discussion of the volume attenuation coefficient a, we found that a included two distinct types of action by the medium on the beam — absorp- tion and scattering. Thus, if a and s are known, we may obtain a by subtraction : a — s = a. (26) SECT. 4] LIOHT 407 There exists another way of obtaining a. This method requires no previous knowledge of a or s. It appHes to horizontally stratified media like the ocean, and is exact, completely general, and particularly simple to use in natural hydrosols. This is the method which makes use of the divergence relation of the light field (Preisendorfer, 1957) and yields the equation : ''"^^^^^^a(Z)HZ). (27) H{Z, +) is the net upwelling irradiance measured at depth Z, i.e. H{Z, +) = H{Z, +) — H{Z, —). Here H{Z,+) is the irradiance at depth Z on a cosine collector which receives the upward moving flux. H{Z, — ) is the irradiance at depth Z due to downward moving flux. h{Z) is the scalar irradiance at depth Z, and a{Z) is the value of the volume absorption function at depth Z. Accord- ing to (27), to obtain a{Z) one performs the following operation: 1 dH{Z,+) ^(^) = h{Z) dZ ^^^^ on the measurable quantities H{Z, +) and h{Z). In the determination_of a{Z) by this method, it is clear that in order to evaluate the derivative oi H{Z, +), measurements of H{Z,+) and H{Z,-) must be made in some interval of depths about the depth Z. The volume absorption coefficient is an* inherent optical property of the medium, and has the dimension of reciprocal length. E. Apparent Optical Properties The apparent optical properties of natural waters consist at present of a set of seven quantities whose numerical values depend on the angular structure of the light field as well as on the physical composition of the water. The apparent optical properties can be obtained from four basic measure- ments ; a pair of irradiances and a pair of scalar irradiances. In each of these pairs, one member is assigned to upwelling flux, the other to down welling flux. The reason that there are precisely four such quantities stems from the con- ceptual decomposition of the flow of radiant energy in any natural hydrosol (stratified or not) into two streams — an upward flowing stream and a dow^n- ward flowing stream across each horizontal plane in the medium. The four basic irradiances are: H{Z,+) h{Z,+) (29) H{Z,-) h{Z,-). H{Z, +) and H{Z, -) are the upwelling { + ) and doivnwelling { — ) irradiance, respectively. They are induced by the up- and downwelling flux streams at depth Z. These quantities may be obtained from field-radiance measurements, or they may be measured by flat irradiance collectors. In like manner, h{Z, + ) 408 TYLER AND PREISENDOBFER [chap. 8 and h{Z, — ) are the upwellincj ( + ) ond doivmvelling ( — ) sadar irradiances, and refer to up- and downwelling flux, respectively, at depth Z. They may be obtained from field-radiance measurements. Alternatively, spherical irradiance collectors may be used to measure these quantities. A possible experimental arrangement is shown in Fig. 7. Observe that the collectors are complete spheres in each case, but the sphere that measures h{Z, — ), for example, should be shielded from the upwelling flux by some device which at the same time impedes as little as possible the interchange of flux across the horizontal plane at depth Z. In analogy to our earlier discussion of the relation between h and ^4„, we can show that the downwelling spherical irradiance, ^4„(Z, — ), actually t To Surface (o) (b) Fig. 7. Schematic arrangement for determining downwelling ( — ) and upwelling ( + ) spherical irradiance. measured by the shielded sphere shown schematically in Fig. 7, is related to h{Z, -)hy h^.{Z,-) = IHZ,-). (30) Similarly, the upwelling spherical irradiance, hi„{Z, +), measured by the other shielded sphere shown schematically in Fig. 7, is related to h{Z, + ) by h4A^,+) = lh{Z,+). (3i; The connection between h4„ and the spherical irradiances deflned above, assuming ideal shielding, is straightforward : h,„{Z) = hi„{Z,-) + h4AZ,+). (32] Furthermore, h{Z) = h{Z,-) + h{Z,+). (33) SECT. 4] LIGHT 409 a. The reflectance functions The refiectance functions are defined by : R{Z,-) = H{Z,+)IH{Z,-) (34) B{Z,+) ^ H{Z,-)IH{Z,+). The physical interpretation of R{Z, — ) is straightforward : it represents the ratio of the iipwelhng irradiance at depth Z to the downwelhng irradiance at depth Z, so that B{Z, — ) may be thought of as the reflectance, with respect to the downweUing flux, of a hypothetical plane surface at depth Z in the medium. For completeness, we have included the reflectance R{Z, +) for the upwelling stream. However, this is simply the reciprocal oi R{Z, — ). In actuality, E{Z, —) depends on the scattering properties of the entire medium above and below this level. It will also depend in part on the reflectance properties of the upper and lower boiuidaries of the medium if these are within sight of the flux col- lectors. R{Z, — ) is not an inherent property of the medium, for experiments and theory show, in general, that for a given medium and a given depth in that medium, the value R{Z, — ) changes with the external lighting conditions. In optically deep homogeneous hydrosols, R{Z, — ) varies very little with depth. Near the surface of these media, it shows relatively high variability with depth depending on the state of the surface and incident lighting patterns, but with increasing depth soon settles down and approaches a coilstant value independent of depth. B{Z, ) thereby takes on the status of an apparent optical property of the medium. Furthermore, in media that have no self-luminous organisms, R{Z, — ) behaves as reflectance should : it is never greater than one. In fact, in most natural hydrosols the values of R{Z, —) are usually found to be some- where in the neighborhood of 0.02, for green light. In media containing self- luminous organisms distributed throughout some layer, it is quite possible, however, for the values of R{Z, — ) to approach one as this layer is approached, and even become greater than one just before it enters the layer. b. The distribution functions A particularly simple means of characterizing the depth dependence of the shape of radiance distributions, without resorting to an actual measurement of the radiance over all directions at each depth, is given by the distribution functions : D{Z,-) = h{Z,-)jH{Z,-) D{Z,+) = h{Z,+)IH{Z,+). It is easily seen from the definitions of h and H that if the shape of the radiance distribution changes with depth, then D{Z, — ) and D{Z, +) wiU also change with depth ; and conversely, if the values of the distribution functions vary with depth, the radiance distributions must be changing shape with depth. It is clear from the definitions that D{Z, — ) gives an index of the shape of the 410 TYLER AND PREISENDORFER [CHAP. 8 radiance distribution in the upper hemisphere (i.e. for the downwelhng flux), and D{Z, +) does a similar job of characterizing the shape of the radiance distribution in the lower hemisphere (i.e. for the upwelling flux). Detailed experimental studies of the light fleld in Lake Pend Oreille show that both D{Z,+) and D{Z,—) exhibit relatively httle change with depth (Tyler, 1958). Furthermore, this independence of depth is found whether the external lighting conditions are sunny or overcast (see Table XIV). In addition to characterizing the depth dependence of the angular structure of radiance distributions, D{Z, — ) and D{Z, + ) play indispensable roles in the equations of applied radiative transfer theory, particularly in those equations which link the inherent and apparent optical properties of a medium. These roles will be illustrated in the course of the discussion below. c. The K-functions In this section we now discuss the quantities which characterize the in- dividual depth dependence of the up- and downwelling irradiances and of the scalar irradiance. These are called the iC-functions. The motivations for the definitions of these functions are supplied by both theoretical and experimental precedent extending back over at least fifty years of applied radiative-transfer theory. The experimental motivation for the A'-functions rests in early empirical relations of the kind Iz = he-^^, (36) which simultaneously were to characterize the depth dependence of Iz and define its depth-rate of decay, K.ln the above relation, Iz took many forms: in some studies it was downwelling irradiance, in others it was a scalar irradiance-like quantity ; in still others, its exact nature was not quite clear. It was not until 1938 (Atkins et al., 1938) that there was agreement as to what should really be measured. A plot of Iz on semi-log paper with depth as abscissa yielded —K as the slope of the straight line. K could thus be defined opera- tionally as A-=^l„^. (37) These early theoretical and experimental approaches to characterize a 7i-like optical property of natural hydrosols were not sufficiently detailed to permit precision and completeness in modern hydrological optics. In current basic research, Iz is replaced bj^ the three precisely defined irradiances H{Z,—), H{Z,-\-) and }i{Z). Furthermore, it has become necessary to distinguish not only between the magnitudes H{Z,—), H{Z,+) and A(Z), but also their logarithmic rate of change with depth. Careful measurements show that their logarithmic rates of change are generally different, and the difference far exceeds the range of experimental error. In general, semi-log plots oi H{Z, —), SECT. 4] LIGHT 411 H{Z, + ) and h{Z) also exhibit noticeable departures from linearity, especially in near-surface regions. These departures from linearity were detected in the early measurements but were not always attributed to changes in flux distribu- tion in the field surrounding the collector. The current views in hydrologic optics are such that the departures from linearity by semi-log plots of H{Z,-), H{Z,+) and h{Z) are a source of extremely useful insight into the intricate structure of light fields in natural hydrosols. The logarithmic slopes of the H{Z, - ), H{Z, + ) and h{Z) plots are defined in general as follows : K(7 +)- 1 dHjZ, ± ) 1 dH{Z) ^^^^ = -m ~d^' ^^^^ F. Some Relations Between Inherent and Apparent Optical Properties Certain relations between the inherent and apparent optical properties discussed above have been found helpful in collating the data of basic experi- mental research and have provided, in some instances, deeper insight into the "whys" and "hows" of the fine structure of the depth dependence of the apparent optical properties. The derivations of these relations may be found elsewhere (Preisendorfer, 1958). The most important of these connecting relations is the following : K(Z,-)-a(Z,-) ^<^'-> = Z(Z,+) + a(Z,+)' (*«) where a{Z, ±) = D{Z, ±)a{Z). (41) Thus (40) links together the ^-functions for irradiance, the i?-functions and the i)-functions, i.e. the main apparent optical properties, with the inherent optical property a. There are also available the following useful inequalities : a[Z,-) ^ K{Z,-) < a(Z, -) (42) or, equivalently, Similarly, or, equivalently, where a{Z) ^ K{Z,-)ID{Z,-) ^ a{Z). (43) a(Z, 4-) ^ K{Z,-^) ^ a{Z,+) (44) a{Z) ^ K{Z,+)ID{Z,+) < a{Z), (45) a(Z, ± ) = D{Z, ± )a{Z). (46) 412 TYLER AND PREISENDORFEB [CHAP. 8 The right-hand sides of all these inequalities hold without qualification. However, the left-hand side of (42) holds whenever O^K{Z, + ). The left-hand side of (44) holds whenever K{Z, -)^0. The condition O^K{Z,+) almost always holds, so that the inequalities of (42) for down welling streams almost always hold. However, the condition K{Z, — ) ^ 0 hardly ever holds, so that the left side of (44) for the upwelling stream hardly ever holds. The condition K{Z, — )^0 means that the down welling stream is constant or growing with increasing depth, a situation which occurs, if at all, only in regions of very shallow depths in the hydrosol, or in regions where there are self-luminous sources distributed throughout some layer. Some further inequalities which are helpful in checking experimentally obtained optical properties and also aid in the understanding of the mutual interactions between the up- and downwelling stream of radiant flux are : K{Z,+)R{Z,-) ^ K{Z,-) (47) or, equivalently, dHjZ,-) dH{Z,+) — IT- ^ —Jz — (^^) These relations hold for arbitrarily stratified source -free media. The same is true for : ^^^^ = R{Z, - )[K{Z, - ) - K{Z, + )]. (49) The quantities a{Z, ± ), a{Z, ± ) defined in (41) and (46) are hybrid optical properties : they are the result of simple combinations of the inherent and apparent optical properties. Equation (41) gives the volume absorption func- tion for each stream, and (46) gives the volume attenuation function for each stream. These quantities by definition do not fall directly into either the inherent or apparent class. To round off and complete the picture of the hybrid optical properties, we mention the {volume) forward scattering functions : f{Z, ± ) (50) the {volume) backward scattering functions : b{Z,±) (51) and the {volume) total scattering functions : s{Z, ± ) (52) for each stream. Detailed definitions and discussions of these quantities may be found in the references (Preisendorfer, 1957a). The hybrid optical properties play important roles in the exact theoretical discussions of the two-flow analysis of the light fields. They are also of use in collating experimental data on in- herent and apparent optical properties. Examples of such uses may be found in the references (Preisendorfer, 1958). SECT. 4] LIGHT 413 G. The Behavior of the Apparent Optical Properties at Great Depths It was emphasized repeatedly during the introduction and discussion of the apparent optical properties that they exhibit certain useful, regular behavior patterns. One of the most striking of these patterns occurs at great depths in optically deep natural waters. We briefly summarize here some of the more important of these facts. Proofs of these results, their historical background, and practical consequences are given elsewhere. (Preisendorfer, 1958, 1958a, 1958b). For simplicity, we consider an infinitely deep source-free homogeneous natural hydrosol. In actuality the results cited below hold in all natural hydro- sols in which the ratio a/a becomes independent of depth with increasing depth. In analogy to K{Z, + ), K{Z, — ) and k{Z), we can define one more iT-function. This is associated with radiance N{Z, 6, cf)) : Ki7 ft M 1 dN{Z, d, 4>) It can be shown that (i) k{Z) approaches a limit as Z -^ oo. Let this limit be denoted by kco. In symbols : kao = Hmz-^co k{Z). (54) It can be shown that kao does not exceed a. In symbols : 0 :^ ^00 ^ a. (u) For each fixed {6, ^), K{Z, 6, 0) approaches a limit as Z ^^ oo, and this limit is independent of {6, (/>). This common limit for all directions {6, «/») is A^oo. In symbols : hmz->«, K{Z, 0, (f>) = kca for all {d, (i>). {Hi) K{Z, — ) and K{Z, + ) approach limits as Z — > oo and these limits are equal to A:oo. In symbols : hmz->oo K{Z, - ) = hmz-^oo K{Z, +) = kao {iv) The distribution functions D{Z,+) and D{Z,—) approach a limit as Z -^ CO. Let these limits be denoted by D{ + ) and D{ — ). D{-) = \imz^a,D{Z,-) (55) D{ + ) = limz^oo D{Z,+). (v) The reflectance function R{Z, — ) approaches a limit as Z -> oo. Let this limit be denoted by Roo. In symbols : Rao = Hmz^oo R{Z,-). 414 TYLER AND PREISENDORFER [CHAP. 8 Then it follows from (40) that ^„ = l--''\->. (66) Property {%) shows that the depth-dependence of radiant density, or scalar irradiance in a natural hydrosol, eventually becomes exactly exponential in behavior. The value A;oo is uniquely determined by a and a. Property (u ) states that the radiance distribution eventually assumes a fixed angular structure (the asymptotic radiance distribution) at great depths. This limiting angular structure is readily found in principle ; it is independent of the external lighting conditions and depends only on the angular structure of u. Property {iv) is an equivalent assertion to (u), but now phrased in terms of the distribution functions. The quantities D{ + ) and D{ — ) are readily obtained from the limiting form of the radiance distribution functions. Properties {Hi) and {i) show that the logarithmic derivatives of irradiance and scalar irradiance eventually coincide as depth increases indefinitely. It may easily be shown that the logarithmic derivatives of h{Z, + ) and h{Z, — ) also approach A:oo as Z ^ 00. (Of course, then so do the logarithmic derivatives of A477, and h^„{Z, + ) approach A;oo as Z ^ 00.) Finally, property {v) states that R{Z, — ) approaches a fixed value as Z ^ 00, and this value is characterized in terms of koo, D{± ) and a as shown in (56). 2. Instrumentation for the Measurement of the Underwater Light Field and the Determination of the Optical Properties of the Sea In the application of radiometry to light measurements in the sea^ there are five important parameters which must be always kept in mind. 1. The directional distribution of the light being measured. 2. The bandwidth or wavelength range included in the measurement. 3. The state of polarization of the radiation. 4. The magnitude of the radiant quantity measured. 5. The direction of propogation of the radiant quantity being measured. The measurement of some optical properties like the attenuation coefficient and the volume scattering function requires artificial light which is strictly limited in its distribution, i.e. a collimated beam of light. Other quantities, like irradiance or spherical irradiance, and optical properties, like the diffuse attenuation function K, are determined for the light field as it exists in nature. Directional distribution is, therefore, under the control of the experimentalist only to the extent that in building instruments he must conform to the strict requirements of the defined quantities. Bandwidth, on the other hand, is left almost entirely to the discretion of the experimentalist. The selection of bandwidth has been a vexing problem because it is not always clear what bandwidth should be used for a specific application. SECT. 4] LIGHT 415 Even when known, it is often difficult to obtain phototube-filter combinations that will duplicate a desired bandwidth. Added to these difficulties is the fact that ocean water itself limits bandwidth (Tyler, 1959) and in great depths may entirely control the bandwidth associated with the measurement. Current thought with respect to the selection of bandwidth is divided according to the requirements of individual problems. The physical approach to the selection of bandwidth is to adopt a mono- chromatic criterion. Monochromatic measurements are the most generally useful since, once available, they can be applied to a variety of problems. A second approach to bandwidth selection has been to match the instru- mental bandwidth with the acceptance spectra or utilization spectra of the organism or image-forming system being studied. The recent discovery of the accessory-pigments effect (Emerson et al., 1957) in the photosynthetic action of certain plankton species has emphasized the importance of this approach. A third approach to bandwidth selection has been the use of a broadband sharp cut-off filter which limits the bandwidth of the instrument to that region of the spectrum where the attenuation coefficients of the water are known to be lowest. Near-surface measurements can in this way be made to correspond closely in bandwidth to the deep measurements. The state of polarization of the flux in the underwater light field is, of course, related to the state of polarization of the flux from the sky and to the subsequent scattering effects in the water. Recent measurements by Ivanoff and Waterman (1958) of the amount and direction of polarized light under- water indicate that polarization exists at considerable depth and should be given proper consideration in the design of instruments. Of the remaining two parameters, magnitude and direction, a word should be said about magnitude. In light measurements in the sea, instrument readings are related to radiant flux by a factor composed of circuit constants, which are generally designed to remain constant, plus an optical coupling factor which must remain constant for all aspects of the light field. For an irradiance collector, for example, the optical coupling is stated by the cosine law : Jq = Jq cos d. A collector plate with an error that becomes progressively worse as 6 in- creases will not yield readings which are directly proportional to irradiance for all sun angles because the direct rays from the sun, which would be the major determinant in the magnitude of the irradiance, will be weighted differently at different angles. Similarly a spherical collector will not yield readings directly proportional to the spherical irradiance for all light fields if the optical coupling factor is not a constant for every aspect of the sphere. Optical equipment should obviously be carefully designed and tested to insure a constant coupling between collector and detector. The physical quantities and properties which are most important for the 416 TYLER AND PREISENDORFER [CHAP. 8 optical documentation of the sea and of the underwater hght field are reviewed below : Radiance distribution N{Z, 9, (/>) Attenuation coefficient a Volume scattering function a{d) Irradiance H Scalar irradiance h Diffuse attenuation function K Distribution function D Reflectance function R Absorption coefficient a Total scattering coefficient s Path function N^^: Beam transmittance T The first three can be regarded as having primary importance since the remain- ing nine can be derived from them (Tyler, Richardson and Holmes, 1959). However, in the determination of these latter nine properties it is sometimes more convenient to make direct measurements, and considerable attention has been given to specialized instrumentation for this purpose. 3. Radiance Distribution The early exploratory measurements of radiance distribution underwater were made with instruments that were simple but ingenious and well designed for detecting the major features of radiance distribution underwater. The work of Pettersson (1938) and Johnson and Liljequist (1938) is of particular interest since it represents one of the major advances in the optical exploration of the ocean and focused attention on radiance distribution as an important parameter in optical oceanography. The existence of "characteristic diffuse" light (or asymptotic radiance distribution) was deduced from these early measurements. A limited amount of data was also obtained on the influence of the light field above the water surface and the rate of change of radiance with depth. Complete radiance distribution data for lake water under clear sunny and overcast lighting conditions was obtained at several depths by Tyler in 1957 (Tyler, 1960). These data are given in Tables I through XII. These data are, of course, characteristic of optically identical water under the same lighting conditions anywhere in the world. The instrumentation used by Tyler is shown in Figs. 8 and 9. This instrument employs a gyrosyn compass for azimuth position information and a servo mechanism with positive propeller drive to maintain the azimuth position of the instrument underwater to better than 1°. The zenith angle is controlled by means of a synchronous motor and event marks which appear on a synchronously driven strip chart along with the data. The angular field is 6.6°, obtained by means of a Gershun tube. 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O-^COOCOlOOiOOCDOiO^t^ o o o 00 o o CO (M o o TtiOicocicoi— l(^^l:^ OCOCiCOlOCOfM^ (M c:i>io>o-^cococococo o o o o o o o o o o o o o t^ CO OC0O (MCOfMOi-^COC^M o I— I O O t^ OS tJh (M 1— I tC ^"^ lo" t-T ^^ 00 CO ^ CO CO CO CO CO I:^OcOCD-<*CO(Nr— ii— I (M (M C^ I— I r-H o o o o o o o o o o o o o o o o o o ■«*-rt o o o o o o -^ ^ rt »-H CD -^ (M I— I p— I l> lO -^ CO CO CO CO ooooooooooooooooooo 1— i^CO-^lOCDl^OOCSOi-HC^lCOTt^lOCOt^OO 418 TYLER AND PREISENDORFEB [chap. 8 -©- o GO o <~> o 1^ (^ o O o O o o o o o {^ (^ o CO CO lO r^ ^ F-H c^ ■^ -^ CO C^4 O' CO 00 1— 1 (M lO o t— 1— 1 lO o Oi CO 00 lO ^ CO (M <^^ a^ t^ Tt< CO 1— 1 F— 1 1— 1 o o ^ ^ ^ o o o o o o o O o ^ ^ (^ o o o c^ (TO ^ CI Of) CD 00 00 lO r^ 1 — 1 r- 00 Oi CO (N th C5 CO TjH CO j I> Tt^ CO (M 1— 1 o o o o o o o Ci CO Oi CO CO 00 oo CO lO CO (N (M ^ ^ o o o o o o o O -* Oi CO CO 00 00 C^ lO CO (M C<1 ^ 1— I o o o o o o o lO t^ (M 00 Tj< 00 00 t^ lO -* CO l> CO l> Oi l> t> (M 00 lO CO (M p—H 00 >0 CO CO CM oooooooo CO-^COlOi— ITJ^QOOO (M00CO-rt^CO r^ lO t^ CO Oi CO tH (M l-H ^H I> iO -^ oo" oo" oT 1— 1 T)H~ T|H~ GO lO CO -* (M »0 ^ t^ O O ^ (M CO tJ< (N ^H ^ O tr- io CO ^H _ OOOOOOOO COCNCO00. ce r^ -^j M >. 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CO-^-rt 9 05 p CO 10 02 o GO o CO o o a; bC ooooooociincoooo COCDCOCO O CO (M r— t CO Tin t-; Oi M <^1 I> CO t)H GO CO id 00 t^ CO 10 10 10 000 o CO as CO CO lO Ci O O rt^ O CO O 10 1-H Oi CO COCXJt-'^COlOi— 1^. t— OilOCO'^li-Hr^i^ CO GO rt5 m ci CO 10 CO O 10 lO 000000 CO O O GO GO O CO O f-i (M O " c-l 01 O' o o o o CO O O O Oi CO CO CO 10 o ;~ C5 CO CO IM 1— I ^ 5 2 § ?^ ?^ q -*. ^. ^. ^. ^. 0'-^^"^'^1^^^COCOCOOCD>0 Ci r- CO lO 10 "O O O CI _^ CI CO CO -^ c» 10 o Ir- GO CO o: CO'^COC^'^fMi— If-H CO CO CO 00 Tt^ 'M I— I O TtH CD OS C^ go' r~ oi CO 10 Oi t-' CO 10 10 10 00000 OOOOsOCOp-hlo COOOOt^t— lGO-^Cit--OClCO COCOi-HOOSi—it^'^t-'"*'^' '"''"' ^ TiT c^f o =^ '^ ^' "^ ,—1 Ol > ce r^ •+3 OQ >1 H 11 <3^ O o r-i 02 !-i cS o o w h^ o 00 n < '-S H ^ .is 1^ ft «5 a c ce CO r-H p^ II O O CO ;3 CO o 00 o «5 o bC T3 00l0Cvll:^O05^HG0C^I-^0ir-;C0lOOC0G0l0-^ O O O CO CSI (N I> GO r^ lO Oi (M O I— I CO O I> -^ CO iM 1— I O O CO»OCOC0C000iOrt< CO t^ ^ ^ CO c Tt* CO CM -— I r-H O Oi 00 l> t> t- c^c^tr^r— i-HOjt;^4 lOOt^cOCO oo(M05QO-rjH-^QqiO'^ C0^Q0lOC0(M^^(^j^(M"^'(j p ^_ o id CO >*' Ci lO CO (N ■^ CO p l> CO 1-H C^ CO ai 00 t^ CO CO 00 iq "* i> t^ r-' O O O CO lO Tt^ 00 lO 1— I ^H CO 05 CO (M O I> '^ (M O ^ »0 CO t^ CO CO t^ o oq ^ P --; 20 <^. p lO 00 lo -*_ '— ii— ioo6idodco^'a5oo'i>t^t> CO CO C<1 ^H r— I 1— I o o o CO CO ■<* CO O0(M i> CO CO p CO iq t>; iq 00 lo -^^ cj '^' 1— I oi oo' t^ i>^ t-" OOOr— i-^COtMCO OCOGOCOCO OOp— i|:^C<|rHl>CppG0lO'*_ cOTti(M05COCO(Mr-H(^*_);onai-«*i-Hoio6r^r^t^' (N ■>* 00 Oi tJh I> -^ — I OOOOiOCO^CO OOi-H(M^O'MiO'^ coio-^oc^-^ir;^ CO 00 CO CO t-; t-; p t)H p p p 00 iq ■*_ t-^ O O lO (M O o6 t-^ O t-^ ■<^ CO (N 1— I 1— I 1-H CO 00 CO CO OOOOOOOK5CO 00 o X 00 o c- 00 lO »^ ^ lO iq 00 (M r-j i>. GO iq Tin CO CO lo ^H 00 Tt c^ 1— ( .^' ^ (j r—tr—ir—ti—i nr)lomr 00 iq tJH CO CO CO C t> l> I— 1 l-H 1— ( 1 — 1 00 lO CO (N 1—1 1— ( p— 1 ooooooooooooooooooo ^(MCO-^lOCOt^OOCSO-— I(MC0tJHIOCOI>00 422 TYLER AND PREISENDORFER [chap. 8 > (J < o3 c8 CO Q ^ CO ft CO o GO o CO O O C^5 O o o 00 o o o ■^3 C CO CD Oi C:(35p_;o6i-H"i>'5i^coc-; 20 (M (M ■>!* tH GO -^_ (M --; p P P pQ0-^Q0lj0oic0C006ldc0(Mi-irHf-If-Hr-Hi-!d (Ml— ii— ioiCO(Mi— I (M TtH CO O 05 lO OOCOTjHrHT— lOOlOlOOO Ot^t^i— 'Oi00iOCO»OOiiO(M'— (OOOi OC0»0-^Q0i0C0(M I> ^ (M I— I f— I I— I I— ( 1— i r— I 1— I O (MiOOOOO'— (tJHO O GOO-^fMOiiOiOOOi I>C0(35>O(M'— lOOCi O O CO I> CD Oi lO CO (M t> Tt* (M 1— I t^ 00 1—1 CO t- CO o o CD lO O 05 (M(MOOP'-ipOi 00 poipp »0 CO ^_ppp S;:r!^!2<^oa5cdo5idco O I— I Oi CO • 00 K5 00 CO O (M CO (M p (M I— H p p p O CO I— H l> (M' i-I t-^ d CD CO (M' 1-1 >- 00 lO CO I— I 1—1 o CD lO O Ci O O C^ I-! r-i d (M Oi T)H O g r^ P ^ 00 (M (>i (M • lO lO i-H-<*Qpcpi-Ht^TtHOCO CO ^ O 00 o r^ t^ tJh Csl O O 05 0qi-HC0-^_C0<>^p-Hi-Hi-H^r-(O -©- o 00 CO CO 00 00 00 id OS iw 00 id CO (Kl -rt^ t)H CO (M ^ r-H d d d »o O o o d d d d O 00 (M o CO u l-H V ;3 CO CO 0) ft o o o o 00 o CO o o 00 ^ CO <^ 1* 00 CO o (M (M (N '^ Oi Id CO - - - . C^ TtH M O O Ci t~-;T)HC0 00 t- lo ^ Ci (N ~„ ^. ^ OiOO^iOiOOO d> oqioco-^i>>(N(M 00 t-' OJ CO' d CO CO ; TtH CO (M <:^ d> CO 00 05 lO CO (M -* ,.„„-.... O TtH TfH oqTjHoqiocooqaici-Hcdcd ■<* lo -^ rt^ TtH 00 -* (M i-hC0CO001OCOC "*■ (N lO OS o ^t^Tti(M^OOOOOOOOOO »o OS o ^oooooooooo CO lO 00 I> (M CD CO Tt< 00 00 00 r-H t^ !>• 00 OS CD OS (m' (N C^ C d> <6 <6 <6 ai -e-. v ooooooooooooooooooo '-H(>lCO'^i-':iCDl>OOOSO--H(MCO'5iHiOCDl^OO 424 TYLER AND PREISENDORFER [chap. S © o 'Sd 00 ee j:i -^ Jj s o 'S CD <11 '^ -©- H > O H P5 a, o X O o o o o o o o o bC O O C C: O C O O O O' O O O O O O O O O CI ^ -H O lO _ _ C-l O O O 'M CO CO CO CO CI 'M -H CC O) lO ^ CO o o -H -ri ^ CO Tt^ CO IX) o OC lO -t o o o o o o o C5 kO CXJ OO 00 CO CO O lO ^ Ol GO 00 GO CO Ol Ol -H r-H rt -^ O O O O' o o o o O O O O O' o o o O O O O O "' -H 0-1 t- Ol I— CO CD ^ oi o CO O' o-i CO CO oi CO CO 01 O-J 1— < o o o o o o o o o o o O X' t^ lO o] t^ o o c: CO CO COF-HOilOCOCDO-HiO^OXXX X LO 't CO Ol O) -M -H — ^ 1-^ CO O O O O O' o O O O O O " O O O 'O o o .__ooo-^cox 1 '^. ^„ '^, ^„ ^, f" ^ ^ CD 01 t-' of lo' r-' irf CO ^ ^'^ "+ Ot^OCOCDCOOl-H 0 CO 01 CO CO Ol o o o o -t c: CD CO O X X X CO CO Ol Ol I— I OOO'COOOOOOOOOOCOOOO oooooO'00ot^oia2co-^o~— ^coco OOOOOiOt^^t^-H,— iCOCOt-COOCiXX Ol CD rt< CO (M Ol Ol i-H rt -^ ^H r-H ^^ CD -H Cvl -H X ^ '-H CO CO Ol Ol -H O O O C' o o o o o o 0 O O C; O F^' c: o c: oi' 01 r-^ Ci O) UO CO CO O) Ol -^ Ci' o ^ CO CO Ol OOOOiO-^t^OCDOiOCDCDCO CiXr-iooc;coco>oxco-Hc:xx t^ -f ^J ^H 000 000 000 -H lO 01 CO 01 Ol O -t^ CO CO CO C-l 00000000 oooooxr-o r^CDiM^nXOOit^ -f CO t^ 1-H 10 X t^ O CD X ^ Ol -H 1— I o o CO CO X X OJ Ol p-< ^ 000 000 o^ o__ o ^ of (m' x' ci' t-' 01 CO rt »o t^ X CO CO CO O-J "H 00000 00000 o p^ co_ x^ o x" o' '+ CO 00000000000 OOOi'OlOt^'+lOCOCOCO COlO'+OlXOlOOvlOXX ^-^ ^'- t^ 10 CO CO O) (M 0-1 -H ^ 000 o o__ o •^ X' -h' >>} CO Ol CO CO CO 000000000000000 OOOOOOXiOOikOOlXCOCOCO Ol0^C0-HOt^^0i^>0(MOXX O of -T ^' oi" 5.f t- 10 CO CO Ol 0-1 Ovi r^ -^ X c: 10 CO ^H 1— I O O' O O O' O O O O O O O O O o o OOOOOOOOOOXXXOi-^OJ OOOOOt^X— it^'^iOiiO" ' '^ tr- o o CD CO X X — I Ol CD 10 CD 10 0-1 0-1 05 0 •^ CO 0-1 0-1 0-1 -H ^H '— ^ 0 (^ 0 0 (^ 0 0 0 0 ,3 r— ^^ CO 0-1 CO ^) >o 0 I^ CO CO 0^ p 0 CO ^H 01 CO CO 0 X X 0" of X 0 -t^ CO Ol 01 01 1—1 ^^ OI —< 01 CO -t 10 CD I- X 5 — r-H 01 CO 00000 -t o --o r^ X SECT. 4] 425 X m < j3 o <: -e- bO G cC >. -^3 ^ •f4 C/2 H o u > O ft o o 00 o CO o o o o o 00 o CD o o o o o o o o O O O O CO Ci ^ Ci 00 1-H t^ o O lO CD O lO CO (N (N CDiOOiCiOt^iOr-i CO Tt< Ttl Tj^ o o oo o o o o o o o ^ ^ 00 O O ," 00 o o 00 lO CO Tt^ 00 Cooo:Oii— ic^iOi— I _ »OCDOt^CD-<*TjH-^ lOC0(M• -* ^ CCOCOOOlOCOC 00 CD Ol (M CD t^ O CD CD I— I CO O (>1 COiOOOOCOOiCDiOTjH-«^ -^OOlO-^COC^i— irti— It— ii— ( _ _oooococo»ocoooo OiOOOiiO-^""" ■■" O # Ol l> (MiOiOOiCDTtHCOfMi— I lO O -— I »o -* ■* o o o o o o o o o o o OOOiOt^CSTt^O ■*OiOio:ciO-TtHTtH 00 Ttl -^ TfH 00 •«* lO CC fo" t-" ^' t- ^ =^ COOCO-*COG^(M--H CO (M (M (M 1— I i> ■* (^^ 1— I rt ooooooooooooooooooo 1— i(MCOTf. -p ^ 7^ m H -Ir^ II O X f— ' k4 '-5 o 00 n ;=! < rO H 'C p o CI o CO Ph p o 3 M o 00 o CO o o CD O O OOOOCOlTQlOCOOO -^^ oi iq CO oi oi 00 00 o (m' id o t^ CO id •«* ■«# CO-^CCC^C<|i— li— ii— inHi— I ■^(Mi— ii-HOiCOCOiO'-HC^cOiOTjJ-^ rt< O CD CD'^COi^J(Mi-H,— ii— I,— II— I OOOOt-00(MOiTj< C0*COOTiHCO(MOOiCOail:^COTjH'* (M(N(Ni— li— I OOiOtJHCNCMi— li— li— li— II— I OOOOOOOt-OO COt^CO'^Oit^O-<*T* '^-'^i— ICOOCD-*(Mr-l C 00 iq iq p 00 rH ci 00 ^oJ^OcoOr-^cDrt''^ OiiOTftC0(Miocooo6cd '*'-<# CiCO-^COfMC^i— Ir-H^Hi— < Oi (M 05 —H 00 ■<:*< OOOOOOOCOTt^CO COOOJOiiOOOi— iiOlO ■^lO^HCOi— iCO-^CNr— iCO^hS<|i— ICOO (M(M(Ni— 1^ OlCDTJ^C0<^^. ^ ^a m H +3 II 02 O ce Qi o o i-H U © > O 1— 1 1 X c w .o o 00 iji '■^ PQ :3 < ^ H ,L3 O d ft ■c/2 O 00 o CD O o CO o O o o © T3 s +3 4^ ce CO c © © JO ^ t^: ^ c^i CO rt5 o CO 00 id CO c-i csi ^' I-H r-H ^' rH OOOCOCO-^OiCOCOt'iO -^(MiMCO-'^OOOiCOOiOOt^CDO O CO CO lO Oi iS£^t!;^|^Tfii6^coo6idco(M'clO ^■"^ojcopcoiocooiocopoor-^cDco ^ !>•' CO CO (m' CO 00 id CO CO c p p CO C;pp COtJ^C0C;PP ioioc O pi -^ © o 'bb (M r-< f— + S t^ -^^ -« Trt CO H K5 Oh O =1 O 00 o CO o o o o GO o CO o o bD -G- bC ^t-i:c>cOfOO-^coioooo>0'<*Tt* cdi>'tJhi;D'<*'X>'— iio^ooTticcc^(Mi— ii— ii— ii— I oi id O tJh O O rj? <^i i-H ,-H O O O O O O O O O oo:-^oqGO<>icocopco :J CO ■* c^ r ■ GO O Oi CO o (M (M (M o o _ - CO •* CO CO »o o -^ CO (M C^J 'do o o o o —< GO CO CO 00 lO -^ -^ o o o o o p p »0 O (M O C5 r^ >-< ci (M (M (M 1- t^ '^' oi O 05 r^ Ci CO i-H CZ) CO -^ lO O GO CO CO GO lO -* -^ CO'^CO'MC^If— I^H-H^H (M p p (M_ p p CO lO lO lO t-. p 00 o 1— 1 GO CO CO CO 00 o r-H 00 1— 1 00 CO CO oi t-' c^i CO ^" t-^ -4 (m' F— 1 1— 1 d d d d d d d d d (M (M -^^ CO (M_ (N T-H ,-H .— J 1— J c5GO(M"i^i-Ht-^T*4(M'i-Hi-Hddddddddd (M OCOa5>OCOi— lOOiO-*-^ t^C^GOGOp-Ht^TtlCOSvlCvji— ii— (1— ii—j t-'^'cvi^'i-lddddddddd CiCOCO^-^r^GOCOCO C0i0C0t1hc0'*O»OI>^00i0-*'* COCOOi-HCOGOGOOGOr— lI>lCCO(M(Mi— ii— li— i^H Oi Oi -ri^ CC 01 (M CM (M 1— I r-H t-rtH(M— i^OOOOOOOOO — iCiOiMiC^OOCOCO OCOGOGOCCCOOCOt^r-HOOiOTt^^ COlOOOOcrsOiGOi-Ht^lOCOiMM^'-i^'-H Oi Oi rt< 00 (M OO -^ (M C• lO CO M C^l f-H rH ^H 1— J 1— i^'ddddddddd ooooooooooooooooooo r— itNcOTtfiocot'GOOO^tMcO'^iocor^oo J3 c a, .-H O u *^ ti o o 11 TJ fi !m P 03 ^ "a m a, S i^ ■ CO dj 11 O Q ^ ^ . 45 05 CI, iab S bc 02 t3 ,2 C 2 C 'o C "o cS ^ ^ X w CO eg ce X CO 03 Oh s o 3 © , © © o CO CO 2 as H OJ bSj © i^ -i-i ^ ID rC © bJD^ s Eh ^ -u 03 © - © CO ;4 0) g "3 < o © Ph-^ X5 rC (V) 0 ■^ -^ H X. -M ^ c _2 .5 © 2 ° 'Z. ^ 2 c 03 H S « u O o s ^4 u 03 ■^ -p ^ ^3 c 13 © p 1 4^ © 00 © ^ 4^ a.S b) S 430 TYIiEB AND PREISENDORFER [OHAP. 8 head with two RCA 931 -A multipher phototubes each coupled to an amplifier with output proportional to the logarithm of the flux input. Measurements of radiance distribution in the meridional plane have recently been made by Jerlov and Fukuda (1960) in the stratified waters of Gullmar Fjord. In this work the instrument was guided into the water along a taut wire which also served to maintain a fixed azimuth orientation of the measuring head. The resolving power of the instrument was 7.0°. The near-surface experi- mental measurements are compared with computed results based on a simple theory that neglects multiple scattering. -REVOLVING PHOTOMETER Fig. 10. Drawing of instrument developed by Sasaki et al. (1958) for radiance measure- ments in the horizontal plane. A series of interesting studies has been made recently by Sasaki (Sasaki et al., 1958) on the influence of sky lighting and depth on the radiance distribution in a horizontal plane. The instrument used by Sasaki is shown in Fig. 10. This instrument employs a remotely indicated magnetic compass for azimuth position information. No azimuth control is provided and evidently is not necessary since the readings are taken at fixed angles from the compass heading, which is always known. The maximum radiance reading locates the vertical plane containing the sun. SECT. 4] 431 The angular field of view of the instrument is 10°, obtained by means of an optical system, and the flux is recorded by means of an RCA 931 -A multiplier phototube and a microam meter. The zenith angle is fixed at 90°. The instrument shown in Fig. 11 was developed by Ivanoff (IvanofF and Waterman, 1958), and has been used by him for measurements of the polariza- tion of underwater light. The instrument is also capable of radiance measure- ments in the horizontal plane and is of interest because of its simplicity and low cost. The instrument uses a Westaphot photocell connected to a galvanometer. Fig. 11. Drawing of instrument developed by Ivanoff and Waterman (1958) for stvidies of polarization of light under water. Flux from the horizontal direction is directed to the cell by means of a mirror which scans the azimuth continuously. Azimuth position stability depends on a rudder. Used for radiance measurements, the continuous scan will generally record radiance as a wave-form, the maximum of which would locate the vertical plane containing the sun. The length of the wave-form would establish the equivalent of 360° in azimuth and the phase difference between two successive wave-forms would be indicative of the stability of the azimuth position. The zenith angle of observation of this instrument is also fixed at 90°, 4. Attenuation Coefficient Instruments for measuring the transmittance of a fixed path length of water have been in use in oceanography for a long time. Some of these instruments 432 TYLER AND PREISENDORFER [chap. 8 have undoubtedly come close to yielding the attenuation coefficient as defined in the section on theory, others have not. Rather than describe specific instru- ments we will put down here the essential optical characteristics of an a-meter. The basic principle of an a-meter is contained in the equation T = NrINo = e-^^. PROJECT ION LENS 6 5 VOLT FIELD LAMP FILAME NT STOP CONDENSING PLEXI GLAS WINDOW 5.05 diameter beam About 200 mm PLEX IGL AS W IN DOW 8 08 diameter (a) All dimensions in millimeters EXIT PUPI L STOP 7 83 diameter filament 2 708 diameter image diameter field stop About 170 mm 711 diameter tfoter image PHOTOCELL Scattering angle max = I All dimensions in millimeters (b) Fig. 12. Optical system for measuring a, the total attenuation coefficient, (a) Details of the projection system ; (b) details of the detection system. Since transmittance {T) is obtained from the ratio of an air reading to a water reading it is important to utilize a beam of light which is not adversely affected by a change in index of refraction along its length. A suitable optical system is shown in Fig. 12. In this system the beam is cylindrically restricted to a constant cross-section by imaging the field stop at the photocell with a magnification of one between the aperture stop and the image of the field stop. A change in index of refraction SKdT. 41 433 10 II 12 13 14 15 Fig. 13. Optical system of a-meter described by Kozlyaninov (1958). between the windows will, under these circumstances, cause a change in flux distribution on the surface of the detector but, if the latter has uniform sensiti- vity over its surface, or constant coupling, no error will result. The lamp must deliver a steady flux output during the measurement period and should be run on regulated voltage. Voltage control of the light output is not desirable because the flux output of the lamp is a function of the tempera- ture of the container, which may be quite different in water than in air. Probably the most satisfactory method for controlling the light output is to provide another photocell near the lamp which monitors its flux output directly. The inherent errors associated with the measurement of a have been dis- cussed by Wills and Jones (1953) and by Preisendorfer (1958c). The major Fig. 14. Compilation of data for the total attenuation coefficient, a. [After Dawson and Hulburt (1934), Hulburt (1945) and Curcio and Petty (1951).] 15 — s. I 434 tyIjER and pbeisendorfeb [chap. 8 sources of error are perturbation of the light field as a result of the presence of the instrument and forward scattered light within the beam (which is un- wanted). Preisendorfer has estimated that, for a one-meter instrument with a 1-cm diameter beam of light, the error due to perturbation is 0.3% when it is used in water having a = 0.402/m and ao= 1.92/m steradian. An a-meter of advanced design for oceanographic work is in use by the Institute of Oceanology of the Academy of Sciences, U.S.S.R. (Kozlyaninov, 1958j. The optical system of this 1-m instrument is shown in Fig. 13. The vacuum phototube, 6, monitors the flux output from the lamp, 1. Not shown is the modulator which modulates the direct beam and also the comparison beam. The two pulsating signals thus obtained are compared by means of a balance circuit and their ratio is plotted on an EPP-09 (Russian) strip chart recorder. This a-meter has calibration filters as well as color filters, 16, which can be introduced into the beam from a remote control station. The instrument is built to withstand depths up to 200 m and, in addition, has a sample tube that fits between the windows, 9 and 10, which can be used for the measure- ment of samples taken at greater depths. A great deal of specialized information is available in the literature on the variability of the total attenuation coefficient, a, with location, depth, wave- length, time and other parameters. Fig. 14 gives the wavelength variability of distilled water after Dawson and Hulburt (1934), Hulburt (1945) and Curcio and Petty (1951). 5. Volume Scattering Functions and Total Scattering Coefficient The measurement of the volume scattering function is difficult. From equation (16) it can be seen that the measurement requires volume calibration as a function of angle as well as a determination of input irradiance. Because of the difficulties associated with these calibrations and, perhaps, because of a lack of interest in the real magnitude of the volume scattering function, there have been very few instruments specifically developed for this measurement. Some instruments and techniques which have been described recently are applicable to the problem of measuring the volume scattering function and these will be mentioned briefly. The need for an in situ type instrument with a controlled sample volume was recognized by Waldram (1945), who was interested in light scattering in the atmosphere. Waldram developed a rotating stop which operated to maintain a constant sample volume as the angle for (j{d) determination was changed. A stop of this design has been incorporated by Tyler (1958) in a nephelometer for measurements of the volume scattering function in natural waters. Tyler's instrument, shown in Fig. 15, was designed for in situ measurements and has a cylindrically restricted beam of detectivity as well as a cylindrically restricted beam of light. Stray light is controlled by internally baffled lens shades which can be seen in Fig. 15 and by the black trap which forms the background for the beam of detectivity. Experimental values of directional intensity, input SECT. 4] LIGHT 436 irradiance and sample volume are used to compute the volume scattering function between 20° and 160°. These values can then be integrated to give the total scattering coefficient (see equation 21), and the forward and backward scattering coefficients. More recently Jerlov has developed the instrument shown in Fig. 16 for studying the volume scattering function of ocean water. The detector in this instrument is fixed in a vertical position as shown in the figure. The lamp assembly can be released to turn slowly around the point, p, carrying a series of semicircular ly arranged stops with it. These stops are designed to maintain a Fig. 15. Scattering meter designed by Tyler (1958) for in situ measurements oi a{d). sample volume of j&xed size at jp. Relative values of the volume scattering function are obtained at 12 angles from 10° to 165°. Measurements of the scattering of ocean-water samples at a fixed, angle have been found useful for characterizing waters of various types, and for studying the particle distribution in the ocean. Jerlov (1953) has developed the equip- ment shown in Fig. 17 for this purpose and, in his investigations, has used the premise that a direct correlation exists between his measurements at 45° from the beam and the total scattering coefficient. The equipment consists of a modified Zeiss turbidity meter and a Pulfrich photometer set at the desired scattering angle. The volume of the sample is constant by virtue of the fact that it is contained in a glass cylinder which is immersed in water to reduce interface reflection. Measurements are obtained by visually matching the scattered light to a comparison glass illuminated directly by the source. This is done in the spht field of the Pulfrich photometer. Since the radiance of the comparison glass is 436 TYIiER AND PREISENDORFER [chap. 8 SCATTERING VOL Fig. 16. Scattering meter developed by Jerlov (unpublished) for in situ measurements of aid). WATER CHAMBER < PULFRICH PHOTOMETER CARBON ARC Fig. 17. Scattering meter developed by Jerlov (1953) for contained samples. SECT. 4] LIGHT 437 directly proportional to the irradiance input to the sample, the readings can easily be converted into values of volume scattering functions at the specified angle by calibrating the instrument. The instrument, shown in Fig. 18, was developed by Kozlyaninov (1957) and called a spectro-hydro nephelometer. This instrument has two voltage-regulated light sources (5 and 15) of which the former irradiates the contained sample 1, the other irradiates a "milk" glass component (12, 13, 14, 17) by means of t-'I'I'l 'I'hpv" 26 29 <4- Fig. 18. Scattering meter developed by Kozlyaninov (1957) for contained samples. the knife-edge mirror, 8, and the mechanical attenuating arrangement, 16. The scattered light is visually matched in the split field of the eye piece with the radiance of the milk glass. The scattering angle is changed by rotating the lamp assembly, 5, around the vertical axis marked 6. Data is obtained from 1° to 150° and can be converted to values of the volume scattering functions by calibration procedures. The instrument is equipped with six narrow band filters, 10, and is used as well for a measurements by rotating the lamp assembly, 5, to the position ^ = 0°. 438 TYLER AND PKKISKNDORFER [chap. 8 Fig. 19 gives the volume scattering function for three samples of com- mercially available "distilled" water together with the computed values of the total scattering coefficients (Tyler, 1960). Fig. 20 gives the average of 30 determinations of the relative volume scattering function obtained between Madeira and Gibraltar (Jerlov, unpublished). Many of the design features adopted by Pritchard and Elliott (1960) in their Recording Polar Nephelometer for atmospheric measurements could be directly applied to the problem of measuring volume scattering functions in the ocean. 0.1 0.01 0.001 0.0001 1 — I 1 1 1 I \ I I r h Angular width of beam 5 = 0.01045 5 = 0.00887 = 0.01587 160 200 Fig. 19. Volume scattering function and total scattering coefficients for three samples of "distilled" water, for peak wavelength 522 mjj., half-band width 56 mjj.. Anyone who is seriously considering the development of instrumentation for this purpose would do well to consult their paper. In addition to these instruments, two interesting proposals have been advanced for the direct measurement of the total scattering coefficient. The first by Beuttell and Brewer (1949), who, like Waldram, were working on atmospheric scattering, proposed a small Lambert emitter as a source which was to be viewed parallel to its surface at distance h above the surface. Their analysis demonstrates that when the absorption coefficient of the medium can be SECT. 4] 439 neglected, the total scattering coefficient s = 'I-nhNII^, where N is the measured radiance of the infinite path observed and /o is the initial intensity of the plate. Because of the relatively high absorption coefficients of water throughout the spectrum this method has limited application to the measurement of the scattering coefficient of sea-water. 1 ^0° 60° 80° 100° 120= ANGLE OF OBSERVATION Fig. 20. Relative volume scattering function obtained by Jerlov in the Atlantic near the entrance to the Mediterranean Sea for peak wavelength 465 mji.. Mean value of 30 records taken in the area between Madeira and Gibraltar. The second proposal advanced by Tyler (1957) utilizes a sample contained in a watertight glass sphere on a G.E. Spectrophotometer. Using the technique described, the value obtained from the spectrophotometer is the total fraction of scattered light from a collimated beam passing through a fixed volume. From this information the total scattering coefficient can be computed as a function of wavelength. 6. Irradiance From equation (3) it can be seen that, in order to measure irradiance, it is necessary to use a device which collects flux according to the cosine law : Je — Jo cos 6. Since photodetector surfaces do not collect flux in this manner, an optical 440 TYLEB AND PREISENDORFEB [chap. 8 collector must be provided. This usually consists of a diffusing glass or plastic disc which has been prepared or mounted so as to collect flux in accordance with the above cosine law. However, diffusing materials differ widely in their properties, and do not in general behave the same way when submerged as they do in air. Consequently design specifications are of limited value. The best procedure is to conduct a practical underwater test of the collecting properties of a plate as a function of the angle of incidence, using an axis tangent to the front surface of the plate. The source of light should be a uniform field of PHOTOCELL EDGE STOP IRRAOIANCE COLLECTOR OPTICAL FILTER Fig. 21. Suggested design for an irradiance collector using a barrier-layer-type photo- detector. collimated light which more than floods the surface being tested. In this way the plate's properties can be brought to any desired degree of perfection. Fig. 21 illustrates a flat-plate collector which exhibited a total error of 2% in irradiance when used to measure a typical underwater light field. 7. Diffuse Attentuation Function and Reflectance Function The measurement of these two properties is intimately associated with the measurement of irradiance [see equations (38) and (34)]. Photoelectric instru- ments for measuring the downwelling flux and the upwelling flux have been in use at least since 1922 (Shelford and Gail, 1922). However, measurements since that time have not all been made using a diffusing collector, and the "extinction coefficients" obtained can in no sense be substituted for the diffuse attenuation coefficient as defined herein. Atkins et al. (1938) recommended the use of diffusing glass in measurements of "submarine illumination." This important recommendation makes much of the data published since that time of value in approximating the value of K at various locations. Some of the diffusing plates used since 1938 may not have collected exactly according to the cosine law (this would mean that the value of K obtained was a function of the directional distribution of the flux in the field as well as of the properties of the diffusing plate), but the diffuse attenuation function, it will be remembered, is obtained from the ratio of two values obtained at two depths. The errors due to an imperfect plate thus tend to cancel. If a diffusing plate of SECT. 4] 441 some kind was used, deep data and data taken on overcast days should be quite comparable to the diffuse attenuation function defined in equation (38). This reasoning does not hold, however, for measurements of the reflection function or the absorption coefficient. The directional distribution in the upper hemisphere underwater is quite different from that in the lower hemisphere (see Fig. 22) and the proportionality factor, which connects irradiance with the instrument reading, will be quite different for the two cases, thus leaving a significant residual error in the value of the reflectance function if the collector is imperfect. Fig. 22. Distribution of flux from upper and lower hemispheres. Data shown is for overcast hghting at a depth of 42.8 m (from Tyler, 1960) ; lower hemisphere data has been enlarged to exhibit the difference in shape. For sunny conditions the difference in shape would be still more pronounced. An instrument suitable for the determination of the diffuse attenuation function. A', and the reflectance function, R, is shown in Fig. 23. Recently irradiance-measuring instruments of advanced design have been developed by Boden, Kampa and Snodgrass (1960) at the Scripps Institution of Oceanography and by Hubbard and Richardson (1959) at the Woods Hole Oceanographic Institute. The former instrument employs interference-type filters to isolate narrow spectral bands. Desaturation of the band because of large angles of incidence is avoided by a coUimating tube located between the filter and the irradiance 442 TYIiER AND PREISENDORFER [chap. 8 (a) (b) Fig. 23. Photocell-type instrument designed by R. W. Austui (unpublished) for the determination of K and R. This instrument is equipped with a diffusing plastic sphere which replaces the upper irradiance plate thus making it possible to determine the distribution functions {D,±) and the absorption coefficient, a. A pressure transducer, visible in Fig. 23a, gives accurate depth determination. A "deck" cell shown in Fig. 23b monitors the surface lighting. SECT. 4] 443 plate which hmits the angle of incidence on the filter to 5° (see Fig. 24). The instrument is equipped with a pressure transducer and can withstand pressures down to 600 m depth. A series of remotely controlled aperture stops and an electronic circuit, designed to yield readings proportional to the logarithm of the light flux, give the instrument a wide range. Fig. 24. Instrviment developed by Boden, Kampa and tSnodgrass (1960). The irradiance measuring head with amplifier below is on the right. The instrument designed by Hubbard and Richardson employs a similar irradiance collector but the flux is sampled by means of a submersible mono- chromator shown in Fig. 25. The data obtained is automatically corrected for the spectral sensitivity of the multiplier phototube as well as for the nonlinear dispersion of the monochromator. The recorded voltage is, therefore, directly proportional to the flux per unit wavelength. Monochromatic data for the diffuse attenuation function is given in Table XIII. 444 TYLER AND PREISENDOBFER [chap. 8 ACHROMATIC -COLLIMATING LENSES EXIT SLIT PHOTOTUBE CATHODE OPTICAL SYSTEM Fig. 25. Arrangement of the optics in the Hubbard submersible monochromator. The exit- sHt mirror covers the upper half of the collimating lens so that the incoming flux passes under it to the prism. Irradiance is measured by utilizing an irradiance col- lector in front of the collimating lens on the left. Table XIII (Half bandwidth K Location and 10m[jL) A Son TOP r ^ OWUl \j\^ Depth Depth 28-56 m 56-159 m 400 0.121/m 420 0.139 0.0902/m 39° 38'N 440 0.133 0.0820 68° 42'W 460 0.118 0.0749 480 0.115 0.0611 500 0.106 0.0527 520 0.0995 0.0545 540 0.107 0.0806 560 0.105 0.0852 580 0.110 0.0908 600 0.107 0.0933 (After Hubbard, 1958) Depth Depth 150-250 m 122-198 m 410 0.065 452 0.053 470 0.051 32° 40'N 489 0.048 117°40'W 503 0.047 541 0.064 581 0.081 (After Boden, unpublished) SECT. 4] LIGHT 446 8. Scalar Irradiance (Spherical Irradiance) The use of a spherical diffuse collector for measuring spherical irradiance was proposed by Gershun (1939). Very little use has been made of this device in underwater light measurements. In current theory, scalar irradiance is required for the determination of important optical properties such as the distribution functions and the absorption coefficient. Spherical irradiance can also be used to evaluate the iC-functions and in this role has the advantage of being insensitive to tilting — a problem encountered sometimes with a horizontal irradiance collector. Fig. 26. Instrument for determining Rj-^K and D. An instrument was described by Tyler (1955) which was used for the deter- mination of K, R and D functions in lake waters. This instrument is shown in Fig. 26. 9. Absorption Coefficient An instrument for the measurement of the absorption coefficient, a, of horizontally stratified water has been described by Tyler (1960). This instru- ment is based on the theoretical work of Preisendorfer (1957), and provides a direct method for the determination of a independent of the scattering of the water. The instrument shown in Fig. 23 has an accessary spherical irradiance collector and can be used to determine the absorption coefficient of large bodies of water. 446 TYLER AND PREISENDORFER 10. Path Function [chap. 8 No instrument has been developed for the measurement of the path function under water. 11. Data Values of the reflectance function, Rao, the distribution function, D{ — ) and D{ + ), the diffuse attenuation function, K{ — ), and the absorption coefficient, a, computed for various depths from Tyler's (1960) radiance distribution data, are given in Table XIV. The peak wavelength for this data is 480 y. and the half-band width 64 [i. Table XIV Depth, i?oo D{-) D( + ) K{-), a, m per meter per meter Overcast Sky 6.1 0.0221 1.29 2.77 0.216 18.3 0.0250 1.33 2.82 0.206 0.144 30.5 0.0266 1.33 2.77 0.189 0.119 42.8 0.0279 1.34 2.79 0.180 0.123 55.0 0.0258 1.34 2.94 0.178 Clear Sunny Sky 4.24 0.0215 1.247 2.704 0.129 10.4 0.0184 1.288 2.727 0.153 0.115 16.6 0.0204 1.291 2.778 0.174 0.118 29.0 0.0227 1.313 2.781 0.169 0.117 41.3 0.0235 1.315 2.757 0.165 0.117 53.7 0.0234 1.307 2.763 0.158 0.112 66.1 0.0190 1.308 2.947 0.154 12. Applications At the present time there appear to be three broad areas of application for light measurements in the ocean. These can be classified under the headings : A. Descriptive oceanography and other geophysical applications. B. Photosynthesis and other biological phenomena. C. Image-recording equipment. A. Descriptive Oceanography and Other Geophysical Applications The fact that certain geophysical features of the ocean are associated with characteristic optical properties has led to the use of optical measurements as a means for locating and describing these features. SECT. 4] 447 Work of this type was undertaken at least as early as 1943 by Joseph and Wattenberg (1944) who developed maps showing regional differences in light transmission of the Kattegat and of the area west of Lim Fjord. More recently, Joseph (1955) has described specialized instrumentation which he has used to develop detailed cross-sections and maps showing the correlation between important oceanographic features and the transmittance properties of the water. Fig. 27. The Coastal Survey meter designed by R. W. Austin measures both a and K. Similar work has been done by Jerlov (1953) who has developed particle distribution cross-sections from scattering measurements. Russian oceanographers are also using this technique and Kozlyaninov (1958) has described a section through the North Pacific current by trans- mittance measurements. Equipment of advanced design is being used in the United States for coastal and harbor surveys. This equipment, called a "Water Clarity Meter", was designed by R. W. Austin (1959) and is shown in Fig. 27. 448 TYLER A2SrD PREISENDOKFER [CHAP. 8 B. Photosynthesis and Other Biological Phenomena The phytoplankton productivity of various areas of the ocean and the total standing crop at any location are intimately associated with the amount of light available for photosynthesis. It is well known that an increase in micro- organisms is associated with a change in optical properties. Thus it is to be expected that correlation will be found between light measurements and nutrient concentration, or temperature. This work is reported by Clarke and Denton (page 456). C. Image- Recording Equipment In image -recording equipment like the human eye, the camera or television apparatus, a prime optical requisite for useful operation, is adequate contrast from the object. The theoretical work of Duntley (1952) and Preisendorfer {loc. cit.) clearly shows the degenerative effect of the ocean environment on object contrast as it is seen by such equipment, and provides means for com- puting contrast under specified circumstances. The methods are discussed by Duntley (page 452). 13. List of Symbols N Radiance Nq Inherent radiance Nf Apparent radiance N(p, 6, (f>) Field radiance at point p arriving from direction 6, cf) N{p, 6', (f>') Field radiance at point p arriving from direction 6' , (f)' Ni*{d) Field radiance for path of sight I at angle 6 from beam of light iV^ Path function iV^(^) Path function — radiance per unit length in direction 6 generated by scattering N*{p, 0, (f>) Path function at point p observed in direction 6, H Irradiance H{Z, + ) Upwelling irradiance H{Z, — ) Downwelling irradiance H{Z, + ) Net upwelling irradiance dH(Z, + ) Net upwelling irradiance h Scalar irradiance h{p) Scalar irradiance at point p h{Z, + ) Upwelling scalar irradiance h(Z, — ) Downwelling scalar irradiance h(Z) Scalar irradiance at depth Z h/\^T, Spherical irradiance h^nip) Spherical irradiance at point p P Flux P{p, 6, (f)) Flux at point p observed in direction 6, cf) a Total attenuation coefficient a{Z, ± ) Total attenuation coefficient for up ( + ) and down ( — ) streams of flux a Absorption coefficient a(Z) Absorption coefficient at depth Z a{Z,±) Absorption coefficient for up ( + ) and down ( — ) streams of flux SKCT. 4 J LIGHT 449 s Scattering coefficient •sl^, ± ) Scattering coefficient for up ( + ) and clown ( — ) streams of flux a Volume scattering function o(d) Volume scattering function o(p, 6. (j). 6', (f)') Volume scattering function at point p observed in direction ^, ^ ; irradiated from direction 9', cf)' f Forward scattering coefficient f(Z, ± ) Forward scattering coefficient for up ( + ) and down ( — ) streams of flux b Backward scattering coefficient b(Z, ± ) Backward scattering coefficient for up ( + ) and down ( — ) streams of flux K Diffuse attenuation function K{Z, + ) Diffuse attenuation function for upwelling irradiance K{Z, — ) Diffuse attenuation function for downwelling irradiance k(Z, + ) Diffuse attenuation function for upwelling scalar irradiance k[Z, — ) Diffuse attenuation function for downwelling scalar irradiance A'oo Limit of k{Z, + ) as Z ^> oo D{Z, + ) Distribution function for upwelling stream D(Z, — ) Distribution function for downwelling stream D( + ) Limit of D{Z, + ) as Z ^ oc D{-) Limit of Z>(Z,-) as Z-> 00 H(Z, + ) Reflectance function for upwelling stream B(Z, — ) Reflectance function for downwelling stream 7?oo Limit of R{Z, + ) as Z -^ x Jq Initial intensity Iz Intensity at depth Z Z Depth dZ Depth differential r Radius of a sphere r Path length ;■' Path length different from r 1(6) Path length 6 Angle froni forward beam of nephelometer 6 Angle from zenith d' Angle from zenith (j) Azimuth angle ' Azimuth angle Q Solid angle ^r Specific solid angle dQ(d', (j)') Specific solid angle differential Aoj Solid angle (small) da> Solid angle differential A Area A{d) Projected area S Source G Gershun tube Jg Radiant iiit(Misity of a beam at an angle d fi'om the normal. References Atkins, W. R. 0., C!. L. Clarke, H. Pettersson, H. H. Poole, C. L. Utterback and A. Angstrom, 1938. Measurement of submarine daylight. J . Cons. Explor. Mer, 13, 37-57. Austin, R. W'., 1959. \\'ater clarity meter, operating and maintenance instructions. S.I.O. Ref. Xo. 59-9. Available : L".S. Dej)t. of Commerce, Office of Technical Services Rpt. No. 147597, 109 pp. 450 TYLER AND PRBISENDORFER [CHAP. 8 Beuttell, R. G. and A. W. Brewer, 1949. Instruments for the measurement of the visual range. J. Sci. Instrum., 26, 357-359. Boden, B. P., E. M. Kampa and J. M. Snodgrass, 1960. Underwater daylight in the Bay of Biscay. J. Mar. Biol. Assoc. U.K., 39, 227-238. Curcio, J. A. and C. C. Petty, 1951. The near infrared absorption spectrum of liquid water. J. Opt. Soc. Amer., 41, 302-304. Dawson, L. H. and E. C. Hulburt, 1934. The absorption of ultraviolet and visible light by water. J. Opt. Soc. Amer., 24, 175-177. Duntley, S. Q., 1952. The visibility of submerged objects. Final Rep. Vis. Lab., Mass. Inst. Tech., 31 August, 1952. Emerson, R., R. Chalmers and C. Cedarstraw, 1957. Some factors influencing the long- wave limit of photosynthesis. Proc. Nat. Acad. Sci., 43, 133-143. Gershun, A., 1939. The light field, SVETOVOE, Moscow, 1936. Translated by Parry Moon and G. Timoshenko. J. Math. Phys., 18, 51-151. Hubbard, C. J., 1958. Measurement of the spectral distribution of light underwater. Woods Hole Oceanographic Institution, Ref. No. 58-6. Hubbard, C. J. and W. S. Richardson, 1959. Measurements of the spectrum of under- water light. Woods Hole Oceanographic Institution, Ref. No. 59-30. Hulburt, E. O., 1945. Optics of distilled and natural waters. J. Opt. Soc. Amer., 35, 698-705. Invanoff, A. and J. H. Waterman, 1958. Factors, mainly depth and wavelength, affecting the degree of underwater light polarization. J. Mar. Res., 16, 283-307. Jerlov (Johnson), N. G., 1953. Particle distribution in the ocean. Reps. Sued. Deep-Sea Exped., 3, Phys. and Chem., fasc. 3. Goteborg, 1947-48. Jerlov, N. G. and M. Fukuda, 1960. Radiance distribution in the upper layers of the sea. Tellus, 12, 348-355. Jerlov, N. G., 1960. Irradiance in the sea in relation to particle distribution. General Assembly at Helsinki, Assoc. d'Oceanographie Physique, Union Geodesique et Geo- physique Internationale (1960). Johnson (Jerlov), N. G. and G. Liljequist, 1938. On the angular distribution of submarine daylight and on the total submarine illumination. Svenska hydrogr.-hiol. Komm Skr., n.s. Hydrografi, 14, 3-15. Joseph, J. and H. Wattenberg, 1944. Untersuchungen iiber die optischen verhaltnisse im meere Mitteilungen, Des Chefs Des Hydrogr. Dienstes im OKH, 30 Seiten. Joseph, J., 1955. Extinction measurements to indicate distribution and transport of water masses. Proc. Unesco Sym. Phys. Oceanog., Tokyo, 1955. Kozlyaninov, M. B., 1957. New instrument for measuring optical property of sea water. Proc. Inst. Oceanology, J. Acad. Sci. U.S.S.R., 25, 134-142. Kozlyaninov, M. B., 1958. Hydro-optical research on the U.S.S. Vityaz. Symposium given on board the Vityaz. Kozlyaninov, M. B., 1959. Hydro-optical characteristics and methods of determining them. Trudy Inst. Okeanol. Akad. Nauk S.S.S.R., XXXV, 3-29 (1959). LeNoble, Jacqueline, Mile., 1955. Sur un application de la methode de Chandrasekhar a I'etude du rayonment diffuse dans les couches de brume. C.J?. Acad. Sci. Paris, 241, 567-569. Pettersson, Hans, 1938. Measurements of the angular distribution of submarine light. Rapp. Cons. Explor. Mer., 108, 9-12. Preisendorfer, R. W., 1956. Calculation of the path function: Theory and numerical example. Visibility Laboratory Report. Preisendorfer, R. W., 1957. The divergence of the light field in optical media. S.I.O. Ref. No. 58-41, 10 pp. U.S. Govt. Research Reps., PB-153990 (1961). Preisendorfer, R. W., 1957a. Unified irradiance equations. S.I.O. Ref. No. 58-43, 38 pp. U.S. Govt. Research Reps., PB-153992 (1961). Preisendorfer, R. W., 1958. Directly observable quantities for light fields in natural hydrosols. S.I.O. Ref. No. 58-46, 29 pp. U.S. Govt. Research Reps., PB-153993 (1961). SECT. 4] LIGHT 451 Preisendorfer, R. W., lOoSa. On the existence uf diaracteristic diffuse light in natural waters. S.l.O. Kef. No. 58-r>9, 15 pp. U.S. (i), measured at depth 2i by a radiance photometer pointed in a direction having zenith angle d and azimuth angle ^ is found to be related to the corresponding spectral radiance N{z2, d, (J)) at depth Z2 by the approximation N{z2,d,cf>) = N{zi, d, cl>) exj) {-[K{z, d, )ldr = -K{z, 6, cf>) cos d N{z, d, (j)), (2) where r cos 6 = zi — zi, and if the equation of transfer for spectral field radiance is written dN{z, d, cf>)ldr = N^ {z, d, ■) - a{z)N{z, d, cf)), (3) and if the equation of transfer for the apparent spectral radiance, t^i^t, d, (f>), of the visual target is written dtN{z, e, cf>)ldr = N^ {z, 6, ) - a{z)tN{z, 6, ), (4) [MS received July, 1960] 452 SECT. 4] TTNDEUWATER VISIBILITY 453 wherein the depth of the target 2^ = 21 and the depth of the observer 2 = 22, then (2), (3) and (4) can be combined and integrated throughout the ])ath of sight to show that the apparent spectral radiance of the target. iNr{z, 6, (/>), is related to the inherent spectral radiance of the target, t^oizi, 6, ), by the equation tNr{z, e, ) = tNoizt, e, ) exp { + K{z, d, )r cos ^} [ 1 - exp { - a{z)r + K{z, d,(f>)r cos d}], (5) wherein the first term on the right represents the residual image-forming light from the target and the second term represents radiance contributed by the scattering of ambient light in the sea throughout the path of sight. If the target is seen against a background of inherent spectral radiance, bNo{zt, 6, (f>), the apparent spectral radiance, bNr{z, 6, (f>), of the background will be given by an equation identical with (5) after replacing the presubscripts t by b. This equation and equation (5) can be combined with the defining relations for inherent spectral contrast, Co{zt, 6, ^), and apparent spectral contrast Cr{z, 6, (f)), which are, respectively, Coizt, d, ) = [tNoizt, e, ct>)-bNo{zt, e, cf>)]lbNo{zt, d, ), and Cr{z, e, «/.) = [tNriz, d, )-bNr{z, 6, )]lbNr{z, 6, ). When this is done, the ratio of inherent spectral contrast to the apparent spectral contrast is found to be Coizt, 6, cf>)ICr{z, d, ) = 1 - [iV(2,, e, (f>)lbNo{zt, e, <^)][1 - exp {a{z)r - K{z, 6, cf>) r cos d}]. (6) In the special case of an object suspended in deep water, bNo{zt, 6, ) = Co{zt, e, )r cos d}. (7) When the observer's path of sight to an object seen against a background of water is horizontal the apparent spectral contrast is Cr{z) = Co(2)exp{-a(2)r}, (8) which indicates that the apparent contrast is independent of azimuth and depends only on the total attenuation coefficient, a(2), for image-forming light. For many practical purposes it is a sufficient approximation to assume K{z, 6, (f)) to be independent of direction and to be of the same magnitude as the irradiance X-functions described in the previous chapter, i.e. to assume K{z, 6, (f>) = k{z, - ) = K{z, - ) = K{z, + ) = K{z) in all of the foregoing equations, indicating thereby that the reduction of contrast in image transmission through water is virtually independent of azimuth. The principles described by the foregoing equations were first discovered in the course of experiments with an 454 DUNTLEY [chap. 9 underwater telephotometer by Duntley (1949, 1950). The data provide verifica- tion of the contrast reduction equations and demonstrate the practical vahdity of the approximation (Duntley, 1951; Duntley and Preisendorfer, 1952; Preisendorfer, 1957). 2. Inherent Contrast The inherent spectral contrast, Co{zt, 9, ), of objects under water presents a far more intricate analytical problem than does the contrast reduction effect discussed above. To be rigorous, all of the reflectance and gloss characteristics of both the target and its background must be known, and the three-dimensional configuration of target and background must be taken into account with respect to the underwater radiance distribution which irradiates their surfaces. No practical general procedure for meeting these requirements is available but research directed toward this goal is in progress. Two common and important special cases are, however, easily treated : (1) an object which appears as a dark silhouette, wherein the inherent contrast is —1, and (2) a horizontal matte surface of known submerged reflectance, wherein the inherent spectral contrast is controlled by the downwelling and upwelling spectral irradiances H{z, — ) and H{z,+) (Duntley, 1960). 3. Sighting Range Most underwater sighting ranges are so short that the visual angle sub- tended by ordinary objects is sufficient to make the exact angular size of the object unimportant. Underwater sighting ranges are, therefore, usually con- trolled by the contrast transmittance of the path of sight, i.e. by water clarity. This is not true of very small objects (e.g. small pebbles, grains of sand, etc.) nor is it true when semi-darkness prevails because of depth or low solar eleva- tion. Nomographic charts for predicting underwater sighting ranges for objects of any size from data on a{z), K{z), depth, solar altitude, target reflectance, bottom reflectance, etc. have been prepared by Duntley (1960) on the basis of equation (6) and visual threshold data by Taylor (1960). Application of the nomographic visibility charts to a wide variety of under- water visibility problems in many kinds of natural water has resulted in the following useful rules-of-thumb : 1. Most objects can be sighted at 4 to 5 times the distance ll[a{z)-K{z) cos 6]. 2. Large dark objects, seen as silhouettes against a water background, can be sighted at the distance 4/a(z) when the path of sight is horizontal. (This rule can be used by swimmers for estimating a{z).) 3. In some natural waters a{z) = 2.1K{z); in such waters the downward sighting range of most objects = | the horizontal sighting range of large, dark objects. SECT. 4] UNDERWATER VISIBILITY 455 Exceptions to these rules-of-thumb are common. They seldom, for example, apply to a white Secchi disk in clear water, even if the disk is observed by a swimmer ; this is because of the high inherent contrast of the white disk. No simple conversion exists between Secchi-disk data and the sighting ranges of other objects. The nomographic charts, however, provide correct sighting ranges for Secchi disks and other objects under virtually all circumstances. References Duntley, S. Q., 1949. Exploratory studies of the physical factors which influence the visibility of submerged objects. Proc. Armed Forces-Nat. Res. Council Vision Comm., 23, 123. Duntley, S. Q., 1950. The visibility of submerged objects I. Proc. Armed Forces-Nat. Res. Council Vision Comm., 27, 57. Duntley, S. Q., 1951. The visibility of submerged objects II. Proc. Armed Forces-Nat. Res. Council Vision Comm,., 28, 60. Duntley, S. Q., 1960. Improved nomographs for calculating visibility by swimmers, (natural light). Bureau of Ships Contract NObs-72039, Rep. 5-3, Feb. Duntley, S. Q. and R. W. Preisendorfer, 1952. The visibility of submerged objects. Final Rep. Visibility Laboratory, N5ori 07864, Mass. Inst. Tech., Aug. Preisendorfer, R. W., 1957. A model for radiance distributions in natural hydrosols. Scripps Institution of Oceanography, SIO Reference No. 58-42. Taylor, J. H., 1960. Visual contrast thresholds for large targets. Part I. The case of low adapting luminances. SIO Reference No. 60-25, June 1960. 10. LIGHT AND ANIMAL LIFE G. L. Clarke and E. J. Denton Light is important to marine organisms in relation to photosynthesis, vision and other vital processes. Since the growth of green plants is the first step in the ecological cycle of the sea, just as it is on land, the supply of energy for photosynthesis by means of the penetration of sunlight into the water is of critical importance, directly or indirectly, to all life throughout the entire ocean. Due to the fact that a minimum light intensity of about 1% of noon sunlight is required for sufficient photosynthesis for growth, green plants are limited to the upper layers of the sea, the exact depth depending upon the transparency (see Chapter 13). Many animals can use their visual powers at light intensities as low as 10~io of sunlight, or even less, and such reactions as photokinesis, phototaxis, phototropism and photoperiodism may also be elicited by intensities far below that required by photosynthesis, and hence at very much greater depths. Quantitative information on the condition of light in all parts of the ocean is thus necessary for an understanding of the factors controlling the occurrence and activities of the various types of marine life (Clarke, 1954, chap. 6). The aspects of light having ecological importance in the sea are : the intensity (or ambient light flux), length of day, quantity of light energy received per 24- hour period, rate of change of light intensity in time and space, cyclic changes (such as those of the day, month and year), spectral composition (including special effects of ultra-violet radiation), angular distribution (including direc- tion of maximum flux and degree of diffusion) and polarization. The control of many of these aspects originates above the surface (such as the change in flux due to the rising and setting of the sun), but for other aspects it originates within the water (such as changes in diffusion due to a turbid stratum), or is profoundly modified by the water (such as spectral changes due to selective absorption). Direct measurements of ambient light flux have been made at a variety of places and under various circumstances using watertight photometers lowered into the sea. Values obtained indirectly by extrapolation from measurements of water samples brought into the laboratory have been less reliable because of the difficulty of, flrst, detecting minute differences between small samples of clear sea-water, secondly, allowing for changes in the water sample after being removed from the ocean, thirdly, allowing for differences caused by the walls of the measuring tube and the use of an artificial light source, and, lastly, determining the effects of scattering and absorption in overlying water strata for samples coming from great depths (Clarke and James, 1939). No one instrument used at sea can provide information on all the optical properties of the water of interest to the biologist, but measurements have been obtained of the average extinction rates through the use of a type of photometer containing a light-sensitive element behind a flat, horizontal receiving window, usually covered with a transluscent diffusing disc and some- times also by a colour filter (Angstrom et at., 1938). Observations are made of the [MS received June, 1960] 456 SKUT. 4] LUJHT AND ANIMAL LIFIO 457 Hiix ])er unit area from sunlight plus skylight above the water surface, below the surface, and at all desired depths in the water to the limit of the instrument. The vertical extinction coefficient, k, is found from the equation ///o = e~^'^ (where /o = intensity ^ at one depth, / = intensity^ at a second depth, L — vertical distance between depths in metres, and e = 2.7). Many of the photo- meters used for the measurement of light flux have broad spectral sensitivities. The range of some covers the whole of the visible spectrum and this also in- cludes the spectral regions of most significance to marine organisms (see below for animals and Chapter 13 for plants). The sensitivity of other photometers has been narrowed for the investigation of particular regions of the spectrum as discussed elsewhere in this chapter (see also Tyler, 1960). Photometers employ- ing photovoltaic cells, which can respond to intensities as low as 0.1% of sunlight, or slightly lower, are useful for light measurements in the upper water strata (to about 220 m in the clearest water) and particularly in relation to photosynthesis. Photometers with photomultiplier tubes are sensitive to intensities as low as 10~'^ (jiW/cm^ (about lO"^^ of sunlight), or slightly lower, and hence may be used for light measurements at great depths and the study of the photic reactions of animals (Clarke and Hubbard, 1959). When a photometer, sensitive to the visible spectrum and the near-ultra- violet, is lowered into optically homogeneous water, the rate of reduction of light is found to be higher near the surface. This is due to the rapid loss of radiation near the limits of this portion of the spectrum in the upper water layers. At greater depths the spectrum has been narrowed by the water to the most penetrating wave-lengths. Since this effect becomes small below a depth of about 30 m in the clearest water and at shallower depths in less transparent water, photometers with a broad spectral sensitivity may be used for measuring the extinction coefficient of the remaining portion of the spectrum at greater depths. Photometers with a narrowly limited spectral sensitivity or field of view, or with other special design, have provided additional information on the optical conditions in the sea. Extensive studies of extinction rates, angular distribution, and colour in a variety of water-masses around the world have been reported by Jerlov (1951) and he has proposed a classification of sea- waters on the basis of their optical properties. Recent investigations of the penetration of ultra-violet radiation into the sea have been reported by Lenoble (1956). Coastal waters have generally been found to have their maximum trans- parency in the yellow-green portion of the spectrum (500-600 mjj.) as is also true of most fresh waters. Curves showing the transparencies of representative natural waters based on the average extinction coefficient for this component of daylight are given by Clarke (1954, fig. 6.8). Depths at which daylight is reduced to 1% of its surface value usually range from 10 to 30 m (A; = 0.46 to 0.15) in coastal waters, as compared to about 100 m (A; = 0.046) for clear oceanic 1 The word "intensity" refers to the flux collected by the diffusing disc described above. If this disc is designed to collect according to the cosine law, then the "extinction coeffi- cient" will be numerically the same as the "diffuse attenuation coefficient" described in Chapter 8. 458 CLARKE AND DENTON [OHAP. 10 regions and to values of less than 3 ni (A; =1.5) in very turbid harbours or estuaries. The clearest ocean waters, particularly those in the warmer areas of the world, have their maximum transparency in the blue portion of the spectrum (400-500 mfi.). Measurements with the bathyphotometer in the Atlantic Ocean and in the Mediterranean Sea have revealed the presence of optically uniform water of high transparency {k = 0.03-0.04) at depths below the first few hundred metres and extending at least to 700 m in some instances. Equally transparent water probably occurs at still greater depths and, perhaps, to the very bottom of the ocean, but exact determination of transparency has not been possible in the deeper strata due to the fact that below about 700 m the presence of luminescence from marine organisms has interfered. However, some detectable light from the surface was recorded down to a depth of 800 m in the Mediter- ranean (Clarke and Breslau, 1959) and to 950 m in Caribbean waters (Clarke, unpublished data). (See Fig. 1.) Measurements with the bathyphotometer have shown that at times and depths at which only weak illumination is received from the surface (that is, at night or at deeper levels during the day) bioluminescence contributes a significant portion of the total light present. In all localities thus far studied in the Atlantic (Clarke and Hubbard, 1959), Pacific (Kampa and Boden, 1957), and Mediterranean (Boden and Kampa, 1958; Clarke and Breslau, 1959), at least some bioluminescence was recorded at every depth investigated at night, and during the day, at all depths below the level at which light from the surface interfered. Measurements in the slope water south-east of New York showed that the frequency of luminescent flashing at night reached a maximum of over 160 per minute at 100 m, with a secondary maximum of about 90 flashes per minute at 900 m, and dropped to a minimum of about 1 flash per minute at 3750 m. During the day flashes appeared above the ambient light penetrating from the surface at about 400 m and increased in frequency with depth to over 1 10 per minute at 900 m. The flux received from the brightest flashes was above 10~3 ptW/cm^ or about 10^'^ x surface light with bright sun. Flashes ranged from less than 0.2 sec to more than 1 sec in duration, with light from overlapping flashes lasting sometimes for more than 10 sec. Measurements in the Mediterranean indicated the occurrence of a larger number of the brightest flashes (some over 10~2 fj,W/cm2) but the frequency of flashing was generafly less than in the Atlantic. In Phosphorescent Bay, Puerto Rico, sustained light levels of above 10^^ j^W/cm^ were produced by continued agitation of the water which contained large populations of dinoflagellates (Clarke and Breslau, 1960). A great many kinds of marine organisms are known to emit luminescent light, either spontaneously or when stimulated, and it is difficult to determine which types are chiefly responsible in any given situation. Some macroscopic forms have been photographed by means of the luminescence camera which is triggered by the animal's own flash (Breslau and Edgerton, 1958). However, in the Mediterranean study of Clarke and Breslau (1959), since only a few SECT. 4] LIGHT AND ANIMAL LIFK 459 photographs of animals larger than 1 cm were obtained from more than 1300 exposures, it appears that most of the flashing was produced by very small organisms, perhaps chiefly by unicellular forms. Up to the time of writing, relatively few studies of bioluminescence have been made in the deep sea, but LIGHT INTENSITY (^W/m^) 10"' 10"' Fig. 1. Schematic diagram to show the penetration of sunhght into the clearest ocean water (A; = 0.033) and into clear coastal water (A; = 0.1 5) in relation to minimum in- tensity values for the vision of man and of certain deop-soa fishes. The approximate minimum values for the attraction of Crustacea (Nicol, 1959), colour vision in man, and for phytoplankton growth arc indicated as well as the range of intensity of bioluininescence in the sea. The penetration of moonlight and the approximate intensities of upward scattered (?/) sunlight and moonlight in the clearest ocean water are also shown. Since light penetrating deep into the sea is confined to a narrow band of wave- lengths, the values given for colour vision represent the approximate inaximuin depths at which a blue hue would be observed. present evidence is that in many situations the light produced locally by living organisms provides a luminous flux as strong as, or stronger than, the daylight penetrating from the surface. Bioluminescence providing continuous light or individual flashes of intensities above the threshold for animal perception has been recorded at all depths investigated to 3750 m. The light intensity, fre- quency and duration of the flashes have been shown to vary greatly and indicate the presence of very different populations at various depths. The strength and varied pattern of the light from oceanic organisms, coupled with 460 CLABKE AND DENTON fCHAP. 10 the high transparency of the water, suggest that bioluminescence is probably of great importance in the ecological relations of the deep sea, Bioluminescence is not always independent of light and we may give as an interesting example the behaviour of the dinoflagellate, Gonyaulax polyedra. This organism has an endogenous diurnal rhythm in which phases of bright and feeble luminescence alternate. This rhythm is normally synchronized with the natural cycle of night and day. During a phase in which its capacity for luminescence is high (normally night time) illumination diminishes this capacity. During a phase in which its capacity for luminescence is low (normally daytime) the effect of illumination is to enhance the luminescence during the subsequent "bright" phase. The relationships between luminescence, photosynthesis and light in Gonyaulax polyhedra are discussed by Sweeney, Haxo and Hastings (1959). Animals produce luminescence either intracellularly or extracellularly and sometimes by controlling the activity of luminous bacteria. Sometimes the flashes emitted are given by small photophores, sometimes they cover the whole animal, whilst sometimes a luminous secretion is discharged into the sea. But whatever the source of the flash, it is usually blue or bluish green in colour, other colours of light being rarely emitted. Furthermore, although this light extends over a considerable part of the spectrum, it is confined to much narrower bands of wave-lengths than either daylight (at the surface of the sea) or the light given by artificial sources of light such as a tungsten lamp or a common fluorescent tube. In, for example, Myctophum punctatum (a lantern fish) more than 80% of the luminescent energy lies between 450-550 my.. Since the known visual pigments of deep-sea animals absorb blue and green light very efficiently, we can usually get an approximate idea of how eff'ective a lumines- cent flash will be for vision by assuming that all its energy is concentrated at the wave-length of maximum emission. Nicol (1958, 1960), in the course of a most extensive study of the physiology of bioluminescence, has compared the energies emitted by a number of animals, and, for long-lasting flashes, he gives values ranging from about 1 x 10"^ to 2 x 10"^ jxW per cm^ of a receptor surface placed at 1 m from the luminescent source and normal to it. Such energies are trivial when compared with daylight (the solar constant is 0.135 W/cm2) but they would generally be well above the visual threshold for the dark-adapted human eye looking at a small source of light. The combined luminescence of countless organisms near the surface, or concentrated at some other depth, might sometimes act as an extended source of light. However, luminescent light sources, unlike penetrating daylight, are usually small sources ; the luminous flux which they give on a receptive surface will decrease, not only because of the absorption and scattering of light by sea- water, but also as the inverse square of distance as the light spreads out on leaving the source. The visual problems of deep-sea animals are, then, to detect both penetrating daylight and the flashing lights of other animals and to use both daylight and luminescent light to see the objects around them. SECT. 4] LIGHT AND ANIMAL LIFE 461 There is no certain knowledge of the depths at which animals can see by daylight. Observers in a bathyscaphe could detect daylight when looking horizontally and could see light reflected by objects down to 600 m (Dietz, 1959). The fully dark-adapted human eye with a pupil of about 0.5 cm^ can detect about 10"^ ^W/cm^ of retina (A 510 m[ji) when looking at 5-sec flashes of a uniform circular field subtending 47° at the eye. The area of retina covered by such a broad field will be 1.5 cm 2 so that, at the pupil, the flux will be 3 X 10-8 [jt,W/cm2 (data taken from Pirenne, 1956). Recent direct measurements by Clarke (unpublished data) show that, in the very clear waters of the Brown- son Deep, 50 miles north of Puerto Rico, daylight will be reduced to this intensity at a depth of about 880 m 1 [a depth slightly less than this was given by a small extrapolation of measurements obtained in the Mediterranean off Monaco (Clarke and Breslau, 1959)]. In the eye of a deep-sea fish, the aperture is greater, the eye media are more transparent, and a greater fraction of the light striking the retina is absorbed by the retinal photosensitive pigments than in the human eye. There are, therefore, good reasons for believing that deep-sea fish will be able to respond to lower intensities of light than we can. Denton and Warren (1957) have dis- cussed this problem and suggest that deep-sea fish might be expected to be between approximately 10 and 100 times more sensitive than man. If a fish were 100 times more sensitive than man, Clarke's data shows that it would, in the Brownson Deep, be able to detect daylight (in the absence of luminescent fight) at a depth 120 m greater than man. If the water below 880 m had been as clear as the clearest stratum measured, the extra depth would have been 200 m. In the human, visual acuity rises very rapidly as the light intensity is in- creased above absolute threshold (Pirenne quoted by Pirenne and Denton, 1952). When looking at a broad field of hght only a few times brighter than the absolute threshold of vision, the fingers of a hand held in front of the eye can easily be seen. A fish wifl, therefore, probably begin to discriminate surrounding objects quite well by the light of day only a short distance above the depth at which it can just detect daylight. If the two eyes have the same aperture, then, in detecting the broad field of penetrating daylight, the illumination on the retina of a small fish (or cepha- lopod) eye will be equal to that on the retina of a fish with a big eye. The big eye may be able to sum the responses of light over a larger area of retina, yet the advantage of a big eye may not be very great in these circumstances. To detect a small flashing light, a large eye which can collect light over a large area of pupil and concentrate it on to a small area of retina is clearly better than a small eye, just as a large telescope is better than a small one at detecting distant stars. In the sea we do find animals, particularly squid, with very large eyes indeed. The eye of one giant squid measured 37 cm in diameter (Cooke 1 The estimate of 950 m by Denton and Warren (1957) for the absolute limit to which the human eye could detect daylight is too high, being based on the assumption of too high a value for the transparency of oceanic water. 462 CLAKKE AND DENTON [OHAP. 10 cited by Beer, 1897) and the eyes of deep-sea animals, particularly those in the "twilight zone" are generally very large with respect to the size of the animal. Now the human eye is very sensitive to small flashing lights and the distances at which man can detect luminescent animals will probably give us a good indication of the kinds of distances over which deep-sea animals can see one another. Nicol (1958) has made estimates based on the known energies emitted by luminescent animals and the known sensitivity of the human eye under the best conditions. He gives, for example, the maximum possible distance at which the human eye could just detect the ctenophore, Beroe, as 120 m, if the sea-water had a transmission of 99%/m, and 41 m if the sea-water had a trans- mission of 90%/m. Observations of an artificial source of light lowered into the clear waters of the Brownson Deep (transmission 97%) at night have shown, however, that in more natural and difficult conditions of observation, a small light about 1000 times brighter than Beroe could only just be seen at 56 m by a man under water and wearing a face mask using foveal vision (Clarke, un- published observations). The distance at which a light source can be distinguished depends not only on the intensity of the flux reaching the eye, the region of the retina on which the image falls and the state of adaptation of the eye, but also on the degree of diffusion of the image by scattering and the degree of interference of other light flux present, particularly that from the surface. In the test at sea just described, the intensity of upward scattered light was somewhat greater than 10~4 [jtW/cm^ at a depth of 1 m. At greater depths interference from scattered surface light would be correspondingly less (and discrimination would thus be easier), but at any given depth an eye looking upward would receive about 100 X more intense ambient flux than a downward directed eye (down to the depths at which perceptible light penetrates). Therefore, under actual conditions and particularly near the surface, the distances at which animals can distinguish the flashes of other animals may be considerably less than those calculated by Nicol for men under the best conditions. Further observations on the visibility of small light sources are greatly needed. The eyes of fishes and cephalopods differ from those of land vertebrates in that the cornea plays practically no role in the formation of the image (in the human eye, for example, the principal refracting surface is the curved air-hquid interface of the cornea). In the fish or cephalopod eye the image is formed by a spherical lens which is so beautifully contrived that the retinal image is almost free of both spherical and chromatic aberrations. This seems to be achieved by continuously varying the refractive index of the lens from 1.53 (that of almost pure protein) in the centre of the lens to approximately that of sea-water at its periphery (Pumphrey, 1961; Fletcher, Murphy and Young, 1954). The lens in the fish eye is not generally stopped down by the iris, which usually merely prevents light from passing by the side of the lens. The fish eye has, therefore, a very large aperture (//0.8) and, since the lens can form very good retinal images, it is well adapted for detecting very weak and small sources of light. The lens shows one other adaptation to depth ; in the surface forms where SECT. 4] LIGHT AND ANIMAL LIFE 463 the ej^e will be exposed to some near-ultra- violet light the lens usually contains pigments absorbing the far blue and the near-ultra-violet light ; in the deep-sea forms the lens is transparent to wave-lengths down to about 310 mji, (Denton, 1956; Kennedy and Milkman, 1956; and Motais, 1957). In vertebrates, two kinds of receptors, the rods and cones, are found in the retina. The rods are used for night vision and the cones for day vision. As we might expect, the eyes of deep-sea fish contain only rods (Brauer, 1908). These rods have been shown moreover to contain golden coloured photosensitive pigments, absorbing maximally at about 480 my. and especially suitable for absorbing the blue light which penetrates best into the ocean and is most commonly emitted by luminescent animals (Denton and Warren, 1957 ; Munz, 1958 ; Wald, Brown and Brown 1957). The rods are very long and the amount of pigment they contain is so high that often over 90% of the light striking the eye is absorbed in one passage across the retina (Denton, 1959). These golden retinal pigments differ from those of coastal fish, which are generally red in colour, as well as from those of freshwater fish, which are generally purple (Wald, 1945). In the spectral absorption curves of their retinal pigments the deep-sea fish form a very homogeneous group. We find that the freshwater eel even before it leaves fresh water to migrate to the deep sea changes its retinal pigment from one with the purple colour, characteristic of a freshwater fish, to one with the golden colour, characteristic of the deep-sea fish (Carlisle and Denton, 1959; Brown and Brown cited by Wald, 1958). Such golden coloured photosensitive pigments are not found only in fish. Kampa (1955) showed that the euphausid eye (this is much more difficult material to work with than the vertebrate eye) contained such a pigment ; her work has recently been extended by Fisher and Goldie (1958). As the spectral composition and distribution of light change with depth, so the animals are found to change in colour. In the upper waters whilst fish are often dark on their dorsal surface and silvery on their ventral surface, many animals are transparent or tinted with blue. As we go below the photosynthetic zone, silvery fish become numerous and there is an increase in the numbers of reddish, dark and black animals whilst, at greater depths, dark colours in- cluding red and black predominate. Since the light in the deej) sea is pre- dominantly blue, pigments which absorb and so do not reflect blue light will appear black no matter what other colours they reflect ; thus, the deep-sea prawn Sergestes, which appears bright red in daylight, will be inconspicuous in the deep sea. The behaviour of marine animals depends very much on stimulation by light and some of the most striking examples of this are furnished by the diurnal migrations which animals of many phyla undertake — migrations which, in some oceanic animals, extend over several hundred metres. It is generally thought that animals which would find the upper waters unsafe in the bright light of day, hide in the depths of the sea and move into the food-rich upper layers at night time. Hardy (1953) has suggested that vertical migrations into 464 CLARKE AND DENTON [CHAP. 10 lower layers of the sea, where the currents differ from those at the surface, may help to distribute the animals horizontally. Russell (1927) and Gushing (1951) have made important reviews of the very extensive literature dealing with this subject and have particularly well described the work done on animals living in coastal waters. Typically animals are thought to be deepest in the daytime, to ascend at dusk, to sink or become more scattered during the night, and to rise again at dawn before descending for the new day. One hypothesis is that animals simply move so as to remain at an optimal illumination. The night "fall" could then be explained by the animals moving more randomly when the light intensity is very low and below the optimal value at all depths. This is a useful but generally too simple hypothesis. Clarke and Backus (1956) have found that deep scattering layers (the sound is almost certainly scattered by animals) sometimes rose before sunset at a rate greater than that needed to remain in a region of constant light intensity ; sometimes these animals moved into light one hundred times brighter than that existing at the level which they had kept in the middle of the day. Again, Southern and Gardiner (1932), studying crustaceans in Lough Derg, have found that an upward movement brought about by the disappearance of light after sunset may be continued in complete darkness. We may emphasize, however, that although other stimuli, e.g. pressure and temperature, may affect the vertical migrations of animals (Moore, 1958), these are more closely correlated with the strength and rate of change of penetrating daylight than with any other stimulus. There have been vigorous attempts to account for the complex behaviour of animals in natural light in terms of a few simple responses. After some early difficulties about nomenclature, the following terms which are taken from Fraenkel and Gunn (1961) (largely after Kuhn) are now widely used and their definition will give a good idea of the kind of analysis which is attempted. The term phototropism is now restricted to the bending responses of sessile animals and plants. The term photokinesis is used to describe a response in which velocity and rate of change of direction of an animal are changed with change in intensity of light regardless of the direction from which the light comes. Photokinesis can lead to apparently directional behaviour and we may give as a simple example that of an animal which is stationary in a dim light but moves quickly in a bright light ; such an animal will, by purely random movements in the light, eventually find a dark place where it comes to rest. Responses by which animals can orientate themselves with respect to the direction of the light are known as phototaxes. These may be subdivided as follows : klinotaxis when an animal with directionally sensitive receptors makes bending movements, successively samples the light reaching it from different directions, and so can orientate itself with respect to the direction of the light ; telotaxis when an animal can move directly towards or away from one of a number of light sources ; and tropotaxis when a bilaterally symmetrical animal moves to maintain an equal stimulation of paired receptors. Two other taxes have been given the special names of dorsal light reaction and compass reaction. In the first of these, an animal tends to orientate itself with its dorsal SECT. 4] LIGHT AND ANIMAL LIFE 465 surface towards the light ; in the second, the animal can maintain some definite angle between the direction in which it moves and the direction of light incident on it. A single animal may display responses of several different kinds, and it is a formidable task to analyse the complex of responses which make an animal undertake extensive diurnal vertical migrations. These migrations are more- over modified in the different seasons of the year and within one species by maturity, sex and brood (Russell, 1933; Moore, 1958), Indeed, lunar and seasonal changes in light intensity and changes in the length of day may greatly affect the general behaviour of animals, determining, for example, the periods at which they breed (Korringa, 1957). Not least of the difficulties in experimentation is that of the scale of vertical migration, which is very often of tens of metres and sometimes of hundreds of metres. One ingenious approach is that of Harris and Wolfe (1955). They put the freshwater crustacean, Daphnia magna, into a column of water which con- tained Indian ink, and in which light intensity changed rapidly with depth. In response to changes in the intensity of light incident on the top of the column of inky water, the Daphnia were shown to make vertical migrations and to distribute themselves in patterns very similar to those found for many marine animals in the sea. It was found that the duration of the cycle of vertical migration was related only to the variation in overhead light intensity and that a whole cycle of a "diurnal" migration could be compressed into a few hours. At the lower levels of illumination used, the Daphnia had always an orientation such that, on making swimming movements, they would rise, and at these levels their activity increased with light intensity. In the dark the animals were inactive and, being denser than sea-water, passively sank (the "midnight fall") whilst on increase in intensity they became more active and automatically rose towards the surface (the "dawn rise"). This is an example of a photokinetic reaction, for an increase in light intensity produced an up- ward movement of the animal no matter from what direction the light came. At higher light intensities, the animals gave phototactic responses sometimes swimming for brief alternating periods first towards, and then away from, the direction of the light (the net movement depending on which of these periods was the longer) and sometimes, at close to the optimal light intensity, swimming horizontally with the dorsal side towards the direction from which the light came (a dorsal light reflex). Harris and Wolfe (1955) showed also that, at the higher levels of illumination, the simple mechanism just described was modified so that whilst in response to slow changes of light intensity the animals moved to remain at an optimal intensity, their position was little affected by rapid changes in illumination such as might be produced in nature by a cloud going over the sun. Many responses of animals to polarized light have been described and great interest has been aroused in the possibility that such responses may help animals to navigate in the sea (Waterman, 1958 ; Jander and Waterman 1960). One great difficulty has been that of deciding whether animals could detect 16— s. I 466 CLAKKE AND DENTON [CHAP. 10 changes in the plane of polarization of light or whether they simply responded to changes in the pattern of scattered and reflected light which changes in polarization had produced. It is now thought that some of the responses which had previously been described were responses to changes in the pattern of light (Baylor, 1959). The work on animals has greatly stimulated work on the polarization of light in the sea (IvanoflF, Jerlov and Waterman, 1961). Ivanoff and Waterman (1958) have shown that, in general, linear polarization is maximal near the surface, where it may in clear water attain 60%, and di- minishes rapidly in the first 10-40 m and after that decreases slowly to some equilibrium value. They conclude that the submarine linear polarization arises through scattering, for the degree of polarization decreases as the dififuseness of underwater light increases, i.e. decreases with depth, with cloudiness of the sky, with turbidity and at the wave-length of greatest penetration. In this brief survey we have principally described experiments dealing with the open oceans rather than with those made in coastal water, for it is in the study of the deep sea that the ratio of the newer knowledge to the older is probably most great. We have, even so, only been able to indicate with a few examples the considerable progress which has been made in recent years in our study of the relationships between animals and light in the sea. References Angstrom, A., W. R. G. Atkins, G. L. Clarke, H. Pettersson, H. H. Poole and C. L. Utter- back, 1938. Measvirements of submarine daylight. J. Cons. Explor. Mer, 13, 37-57. Baylor, E. J., 1959. Is polarized light a valuable navigational aid? International Oceano- graphic Congress, preprints 178-179. Beer, Th., 1897. Die Akkomodation des Kephalopodenauges. Arch. Ges. Physiol, 67, 541- 586. Boden, B. P. and E. M. Kampa, 1958. Lumiere, bioluminescence et migrations de la couche diffusante profonde en Mediterranee occidentale. Vie et Milieu, 9(1), 1-10. Brauer, A., 1908. Die Tiefseefische. II. Anatomischer Teil. Wiss. Ergebn. 'Valdivia\ 15, 1-266. Breslau, L. R. and H. E. Edgerton, 1958. The luminescence camera. J. Biol. Phot. Assoc, 26, 49-58. Carlisle, D. B. and E. J. Denton, 1959. On the metamorphosis of the visual pigments of Anguilla anguilla (L.). J. Mar. Biol. Assoc. U.K., 38, 97-102. Clarke, G. L., 1954. Elements of Ecology. Ch. 6. Light. Wiley, New York, 534 pp. Clarke, G. L. and R. H. Backus, 1956. Measurements of light penetration in relation to vertical migration and records of luminescence of deep sea animals. Deep-Sea Res., 4, 1-14. Clarke, G. L. and L. R. Breslau, 1959. Measurements of Bioluminescence off Monaco and Northern Corsica. Bull. Inst. Oceanog. Monaco, No. 1147, 1-31. Clarke, G. L. and L. R. Breslau, 1960. Studies of luminescent flashing in Phosphorescent Bay, Puerto Rico, and in the Gulf of Naples using a portable bathyphotometer. Bull. Inst. Oceanog. Monaco, No. 1171, 1-32. Clarke, G. L. and C. J. Hubbard, 1959. Quantitative records of the luminescent flashing of oceanic animals at great depths. Limnol. Oceanog., 4, 163-180. Clarke, G. L. and H. R. James, 1939. Laboratory analysis of the selective absorption of light by sea water. J. Opt. Soc. Amer., 29, 43-55. SECT. 4] LIGHT AND ANIMAL LIFE 467 Gushing, D. H., 1951. The vertical migration of planktonic Crustacea. Biol. Revs. Cambridge Phil. Soc, 26, 158-192. Denton, E. J., 1956. Recherches sur I'absorption de la lumiere par le cristallin des Poissons. Bidl. Inst. Ocdanog. Monaco, No. 1071, 1-10. Denton, E. J., 1959. The contributions of orientated photosensitive and other molecules to the absorption of whole retina. Proc. Roy. Soc. London, B150, 78-96. Denton, E. J. and M. H. Pirenne, 1952. Accuracy and sensitivity of the human eye. Nature, 170, 1039-1042. Denton, E. J. and F. J. Warren, 1957. The photosensitive pigments in the retinae of deep sea fish. J. Mar. Biol. Assoc. U.K., 36, 651-662. Dietz, R. S., 1959. 1100 meter dive in the bathyscaphe Trieste. Limnol. Oceanog., 4, 94-101. Fisher, L. R. and E. H. Goldie, 1958. The eye pigments of a euphausiid crustacean Meganyctiphanes norvegica (M. Sars). 15th International Congress of Zoology. Section 6 — Paper 35. Fletcher, A., T. Murphy and A. Young, 1954. Solution of two optical problems. Proc. Roy. Soc. London, A223, 216. Fraenkel, G. S. and D. L. Gunn, 1961. The orientation of animals. Dover Publications Inc., New York, 376 pp. Hardy, A. C., 1953. Some problems of pelagic life. In Essays in Marine Biology being the Richard Elmhirst Memorial Lectures, Edinburgh. Oliver and Boyd. Harris, J. E. and Ursula K. Wolfe, 1955. A laboratory study of vertical migration. Proc. Roy. Soc. London, B144, 329-354. Ivanoff, A., N. Jerlov and T. H. Waterman. 1961. A comparative study of irradiance, beam transmittance and scattering in the sea near Bermuda. Limnol. Oceanog., 6 (2), 129-148. Ivanoff, A. and T. H. Waterman, 1958. Factors, mainly depth and wavelength, affecting the degree of vmderwater light polarization. J. Mar. Res., 16, 283-307. Jander, R. and T. H. Waterman, 1960. Sensory discrimination between polarised light and light intensity patterns by arthropods. J. Cell. Comp. Physiol., 56 (3), 137-160. Jerlov, N. G., 1951. Optical Studies of Ocean Waters. Rep. Swed. Deep-Sea Exped., 3, Phys. & Chem. fasc. 1, 1-59. Kampa, E. M., 1955. Euphausiopsin, a new photosensitive pigment from the eyes of euphausiid crustaceans. Nature, 175, 996-998. Kampa, E. M. and B. P. Boden, 1957. Light generation in a sonic -scattering layer. Deep- Sea Res., 4, 73-92. Kennedy, D. and R. D. Milkman, 1956. Selective light absorption by the lenses of lower vertebrates and its influence on spectral sensitivity. Biol. Bull. Woods Hole Oceanog. Inst., Ill, 375-386. Korringa, P., 1957. Lunar Periodicity. Treatise on Marine Ecology and Paleoecology, Geol Soc. Amer., Memoir 67, Vol. 1, 917-934. Lenoble, J., 1956. Etude de la penetration de I'ultraviolet dans la mer. Ann. Geophys, 12, 16-31. Moore, H. B., 1958. Marine Ecology. Wiley, New York, 493 pp. Motais, R., 1957. Sur I'absorption de la lumiere par le cristallin de quelques poissons de grande profondeur. Bull. Inst. Oceanog. Monaco, No. 1094, 1-4. Munz, F. W., 1958. Photosensitive pigments in the retinae of certain deep sea fishes. J. Physiol., 140, 220-235. Nicol, J. A. C., 1958. Observations on luminescence in pelagic animals. J. Mar. Biol. Assoc. U.K., 37, 70&-752. Nicol, J. A. C., 1959. Studies in luminescence. Attraction of animals to a weak light. J. Mar. Biol. Assoc. U.K., 38, 477-479. Nicol, J. A. C., 1960. The biology of marine animals. Pitman, London, 707 pp. Pirenne, M. H., 1956. Physiological mechanisms of vision and the quantum nature of light. Biol. Revs. Cambridge Phil. Soc, 31, 19^241. 468 CLABKE AND DENTON [CHAP. 10 Pumphrey, R. J., 1961. ' Concerning Vision ' from The Cell and the Organism. Essays pre- sented to Sir James Gray. Cambridge Univ. Press. Russell, F. S., 1927. The vertical distribution of plankton in the sea. Biol. Revs. Cambridge Phil. Soc, 2, 213-262. Russell, F. S. 1933. On the biology of Sagitta. Observations on the natural history of Sagitta elegans Verrill and Sagitta setosa J. Muller in the Plymouth area. J. Mar. Biol. Assoc. U.K., 18, 147-160. Southern, R. and A. C. Gardiner, 1932. The diurnal migrations of the Crustacea of the plankton of Lough Derg. Proc. Roy. Irish Acad., B40, 121-159. Sweeney, B. M., F. T. Haxo and J. W. Hastings, 1959. Action spectra for two effects of light on luminescence in Gonyaulax polyedra. J. gen. Physiol., 43, 285-299. Tyler, J. E., 1960. Radiance distribution as a function of depth in an underwater environ- ment. Bull. Scripps Inst. Oceanog. Univ. Calif., 7, 363-412. Wald, G., 1945-6. The chemical evolution of vision. Harvey Lect., Ser. 41, 117-60. Wald, G., 1958. The significance of vertebrate metamorphosis. Science, 128, 1481-1490. Wald, G., P. K. Brown and P. S. Brown, 1957. Visual pigments and depths of habitat of marine fishes. Nature, 180, 969-971. Waterman, T. H., 1958. Polarized light and plankton navigation. Perspectives in Marine Biology. Ed. by A. A. Buzzati Traverso, 428-450. University of California Press. 11. OTHER ELECTROMAGNETIC RADIATION L. N. LlEBERMANN 1. Introduction The theory of propagation of electromagnetic energy in the sea is contained completely in the remarkable electromagnetic equations given by Maxwell in 1873. Surprisingly the large amount of experimentation with undersea electro- magnetic propagation, particularly during World War II and the post-war era, has contributed negligibly to our basic understanding over that given by Maxwell in his theoretical tour de force ; experiments have simply confirmed principles which were already known and inherent in Maxwell's equations. As is well known, Maxwell's equations predict the propagation of electro- magnetic waves in the form of a periodically varying electric field and magnetic field in the form E = Eo exp (icut — yz) (1) H = //o exp {icot — yz). The quantity y, termed the propagation constant, is given immediately from Maxwell's equations in the form y = iio\/{ejx — ia^la)). (2) The quantities e, /z and a are electromagnetic properties of the medium, which are discussed in detail immediately below. Inasmuch as these are always real quantities the propagation constant, y, can be complex. It is seen from (1) that the real part of y leads to attenuation of the electric and magnetic fields with propagation of the wave. 2. Electromagnetic Properties of Sea-Water The physical significance of the characteristic constants of the medium are as follows : the quantity a is the conductivity of the medium ; /x is the permeability ; and e is the dielectric constant. Table I gives values for these constants for typical media. Table I O, €, mhos/m farads/m Dry soil 0.015 lOeo Sea -water 4 78€o Copper 5.8 X 107 eo = 8.35x 10-12 \_MS received July, 1960] 469 470 LIEBERMANN [CHAP. 11 It is noted that sea-water has a conductivity more than 300 times that of dry soil. On the other hand, it can hardly be considered a good conductor compared with copper. The dielectric constant, e, is readily obtained for the insulators ; it is rather difficult to estimate for good conductors such as copper. The perme- ability, fi, is practically identical in all substances except the ferro-magnetic materials. The first term under the radical in (2) is often termed the displacement current ; the second term arises from the conduction current. Clearly, in sea- water the conductivity is sufficiently large that the conduction current is dominant except at very high frequencies. When the conduction current is of the same order of magnitude or is comparable to the displacement current, the propagation constant, y, will become small. This phenomenon would appear to occur in sea-water, utilizing the above values in the table, at frequencies in the vicinity of a kilomegacycle (10^ c/s). Actually the frequency at which this transmission "window" occurs is considerably higher because the effective conductivity increases at sufficiently high frequencies ; associated with the rotation of the polar water molecules is an additional contribution which be- comes increasingly important at high frequencies. The electromagnetic "win- dow" actually occurs at much higher frequencies, namely, in the visible light range. Hence the familiar penetration of visible light in the sea is a direct consequence of electromagnetic theory. 3. Propagation through Sea-Water At lower frequencies the displacement current can be neglected and the propagation constant, y, assumes the simpler form y = \/(cT/Aaj/2)(l+i). The distinction "low frequency" or "high frequency" is purely an academic one, for the low-frequency approximation given immediately above is applicable to sea-water up to frequencies including the microwave range (IQi'^ c/s). As stated above, the real part of y gives the attenuation ; thus the equation often appears, S = V(2/a;/xa), where S is defined as the "skin depth." The quantity 8 is the distance at which the field intensity is diminished by 8.6 dB. For sea-water, S = 250/'v// meters. For example, at 1 mc/s, 8 is j m ; at 100 c/s the "skin depth" is 25 m. Hence there is little possibility of significant electromagnetic propagation through sea-water except at low audiofrequencies. The imaginary part of y is related to the velocity of propagation given by V = V(2cu/c7/x). (3) It is seen that the velocity of propagation of sea-water, unlike that of electro- magnetic propagation in air, depends on frequency ; as in the case of attenua- tion, velocity increases as oj'/^. Both the variation of velocity with frequency SECT. 4] OTHER ELECTROMAGNETIC RADIATION 471 and the attenuation are summarized in Table II. Note the curious result that, at 1 c/s, the electromagnetic and acoustic velocities are practically identical in sea-water! This table also gives the thickness of sea-water through which an electromagnetic wave will be attenuated 40 dB, this being a reasonably practical limit of penetration. Table II Propagation of Electromagnetic Waves through Sea-Water Frequency Penetration Velocity distance ( — 40 dB) in sea- water, m/s 1 c/s 1160 m 1.8x103 100 c/s 116 m 18x103 1 mc/s 1 m 2 X 106 104 mc/s 4 mm 2.7 x lO^ 4. The Effect of the Sea-Surface on Electromagnetic Propagation It was the intention of the preceding section to demonstrate that the at- tenuation of electromagnetic waves in sea-water is sufficiently high that penetration beyond 100 m even at low audiofrequencies is practically im- possible. In spite of this assertion a seemingly contradictory experiment can AIR DIPOLE Fig. 1 . vSurface-wave propagation from a submerged dipole. be performed : if a horizontal electric dipole is submerged below the sea- surface and excited with audiofrequencies, it will be found to radiate an electromagnetic field at considerably greater distances than is shown in Table II. This apparently paradoxical propagation arises from the effect of the sea- surface. The path of an electromagnetic wave between submerged source and receiver is not a straight line through the medium but rather vertically to the sea-surface and then horizontally along the interface. This form of propagation is known as a "surface wave." Fig. 1 shows schematically the radiation path 472 LIEBERMANN [CHAP. 11 from an antenna submerged in the sea. Note that the large attenuation resulting from propagation through the conducting sea-water occurs only during the vertical portions of the path. The theory of this curious surface -wave propagation was first treated by Sommerfeld (1909). Banos and Wesley (1960) have recently given an accurate and exhaustive treatment of this subject with extensive references to the literature. The various formulae for the three electric and three magnetic- field components are rather lengthy and will not be given here. However, as an example, the electric field in the vertical direction, Ez, is given below j^ ^ cos ^ I I . ,^. The quantity h represents the combined depths of the receiver and radiator ; p is the dipole strength and p the horizontal range. Note the exponential attenuation with increasing depth ; this justifies the assertion that the radiation initially travels vertically up to the surface. In addition the field component attenuates as the inverse square. Hence attenuation, even after the wave has reached the sea-surface, is still considerably more rapid than long-distance radio transmission in air, where attenuation is inverse first power. In all experiments with the surface wave the radiating electric dipole must be horizontal, for only the horizontal component of the electric field leads to the surface wave, the vertical components exerting a negligible field at large distances. A . Reflectivity of Electromagnetic Waves from the Sea-Surface Nearly all of electromagnetic radiation originating in the air above the sea is reflected from the sea-surface. For example, at frequencies below the micro- wave range (i.e. less than lO^^ c/s), the reflection coefficient magnitude, F, is approximately unity, or perfectly reflecting at normal incidence. Reflectivity can be somewhat less at other angles of incidence. The dependence of reflecti- vity on the angle of incidence is given by the equation r =-{s-i)i{s + \), (5) where (1 + 0 2a j cos Q Q being the incident angle, measured from the vertical. It is seen from the above equation that the reflectivity of sea-water depends on its conductivity. Conductivity in turn depends on the temperature (salinity variations will be neglected in this discussion). This suggests the possibility that the reflectivity of electromagnetic waves might be used to measure surface temperatures from an aircraft. It can readily be shown from the above equation SECT. 4] OTHER ELECTROMAGNETKl RADIATION 473 that reflectivity depends most sensitively on conductivity for grazing angle of incidence. At this angle Arir = AalGx'2, that is. the percentage change of reflection coefficient is of the same order as percentage change in conductivity. The percentage change in conductivity is given by Aa 0.19 JT _^ The numerical value, 0.19, is the activation energy for sea-water. Hence a 1°C change in temperature will result in a 2.4% change in the conductivity of sea-water. A reflectivity change of 2.4% is readily observable without elaborate instrumentation . B. Radar Reflection from the Sea-Surface Two types of sea waves are primarily responsible for the back scattering of ultra-high frequencies (radar). First, there are the highest amplitude gravity waves. These are obviously effective scatterers simply because most of the wave energy is concentrated in these waves. Secondly, there are surface waves whose wavelengths happen to be close to "resonance" with the incident radar wavelength ; because radar wavelengths are short, usually in the vicinity of 3 cm, these waves will generally be capillary or very short gravity waves. Consider first the well-developed sea waves : the phase velocity of these waves, providing a moving target, will generate a Doppler shift in the scattered radiation. For example, a gravity wave of wavelength 22 meters with velocity 600 cm/sec will be responsible for a Doppler shift given by AF — 2AvjX — 1200/A. Hence 3-cm radar will be shifted in frequency approximately 400 c/s. This frequency shift is readily detectable with present equipment. In general this frequency shift will be broadened by 10 to 20% by the effect of the particle velocity in these well-developed waves. The wavelengths of sea waves which "resonate" or give maximum back scattering are obtained from the familiar optical grating formula. sin i3-f sin 6 = Xjd, where d is now interpreted as the scattering wavelength on the sea-surface. For maximum back scattering at grazing incidence (sin ^ = sin ^= 1) the relation between radar wavelength. A, and capillary wave, d, is X=2d. This result may be interpreted as meaning that, of the broad spectrum of capillary waves ever present, only those waves satisfying this "resonance" relationship will be effective scatterers. For example, 3-cm radar will "resonate" with sea waves of wavelength 1.5 cm. These are capillary waves whose velocity is 24 cm/sec ; this results in a frequency shift of 32 c/s. Hence the statement can be made that 3-cm radar will always exhibit a Doppler modulation of 32 c/s in sea return. 474 LIEBEBMANN [chap. 11 5. Natural Electromagnetic Radiation or "Noise" in the Sea The upper Hmit of the natural electromagnetic noise spectrum in the sea is obviously limited by the attenuation of electromagnetic radiation in sea-water. It was shown above that the possibilities of penetration above 100 c/s are poor, and this frequency shall be arbitrarily taken as the upper limit of electro- magnetic noise. In the frequency range 5 c/s to 100 c/s world-wide lightning storms provide the dominant source of electromagnetic noise. The level of noise arising from this source is surprisingly constant. This is because there are always more than 100 lightning strokes per second averaged over the whole world, day or night. Because electromagnetic radiation in this frequency range propagates with low attenuation, distant lightning storms, originating over a large fraction of the earth, generate a rather constant output of noise. > CP 20 "" ■■ -^^ ^ ^ 4 ■"**^ 40 ■* '^-^: :., -"V '-- - .._ KCl 1 001 0.1 10 100 FREQUENCY (c/s) Fig. 2. Spectrum level of natural electromagnetic noise. Below 5 c/s lightning ceases to make an important contribution to the noise. The origin of this noise in the band 5 c/s to 0.001 c/s remains unknown, but correlation with sun-spot activity is sometimes established. In addition to a general background level, there are at least two types of sinusoidal damped oscillations which may persist for many cycles and also impulsive noise signals of shorter duration. These signals are generally intermittent occupying only a fraction of the measurement time. Fig. 2 represents a compilation of data on the spectrum of natural geo- magnetic noise. It is seen that the spectrum level diminishes with increasing frequency approximately according to the relation /~i- 2. Virtually all the data given are for the high-amplitude, highly intermittent noise. Almost nothing is known about the steady noise background. The data are predominantly for the horizontal component. The conductivity of sea-water tends to damp the vertical component of the magnetic field so that only the horizontal component is of major importance within the ocean. The amplitude is strongly latitude dependent, increasing as the auroral zone is approached. However, in the northern latitudes, where the total field direc- tion is nearly vertical, the noise is much diminished by the effect of induced SECT. 4] OTHER ELECTROMAGNETIC RADIATION 475 currents in the sea. In the lower latitudes, the total magnetic field is pre- dominantly horizontal and eddy currents have little effect. References Bancs, A. and J. Wesley, 1960. Horizontal electric dipole in a conducting half-space. U.S. Office of Technical Services PB 152676. Sommerfeld, A. O., 1909. XTber die Ausbreitung der Urellen in der Drahtlosen Telegraphie. A7in. Phys., 28, 665. 12. SOUND IN THE SEA P. ViGOUREUx and J. B. Hersey 1. The Nature of Sound Sound is produced by changes of jDressure in elastic media. If, for instance, the pressure at a point in a large volume of water is momentarily increased by the explosion of a small detonator, the pressure change is communicated in all directions giving rise to a spherical wave travelling with a velocity which depends on the elasticity and the density of the water ; an observer in the water can hear the disturbance as the wave passes him. Strictly speaking the word "sound" is restricted to those components of a disturbance which are audible. The slow pressure changes due to tides or to sea waves, for example, are not referred to as sound. Also waves that have travelled long distances from earthquakes or heavy explosions contain frequencies which lie partly below and partly overlapping the frequency range of sound. These are treated elsewhere in this text. But it will be convenient here to include under "sound" all elastic disturbances of frequency higher than, say, 20 c/s. True, the upper limit of audibility is some 20 kc/s and higher frequencies can be "heard" only by frequency changing, but the similarity of the methods of generation and of the laws of propagation makes it profitable to treat the whole range together. The upper limit is in any case fixed by Nature, for the absorp- tion increases so much with frequency that even if sound energy of thousands of megacycles per second could be generated, it would be absorbed by the water before reaching the detector. The change of pressure due to the wave at any point is called excess pressure or acoustic pressure ; it is accompanied by movement of the particles of water at the point, and the velocity of this motion is called particle velocity. The treatment of sound propagation is greatly simplified if it is assumed that the excess pressure is small compared with the static pressure. This condition is always fulfilled at great distances from even relatively powerful sources like explosive chemicals. At close range, however, the assumption would be in- correct and a more complicated treatment is necessary. 2. Propagation of Sound in Water A. The Wave Equation The relation between acoustic pressure, time and position in a homogeneous elastic medium is obtained from the wave equation, , (3) where p is the density. [MS received July, 1960] 476 SECT. 4] SOUND IN THE SEA 477 The velocity of propagation, c, depends on the way pressure changes with density ; it is given by c2 = {dpl8p)s, (4) where the letter S denotes an adiabatic process. As absorption of sound by water is very dependent on frequency, it is con- venient when studying propagation to consider waves of one frequency only, afterwards, if necessary, combining a number of waves to represent the dis- turbance. Indeed, one is often interested in sound of a pure note produced by a harmonic generator, or in that component of a sound which is accepted by a tuned detector. In all such cases equation (1) can be simplified to (^2 + V2),^ = 0, (5) where k is called the propagation constant and has been written for cojc, oj being the angular frequency. As is always the case with wave propagation, the length A of the wave and its frequency / are connected by the relation A/ = c, (6) so that k is equal to 277/A. B. Velocity The velocity of sound in sea-water has been given in several variants of the form c = co,35,o + ^cs + Act + Acp + Acstp, (7) where Co,35,o is the velocity at 0°C, 35%o salinity and atmospheric pressure; Act, Acp and Acs are respectively correction terms for temperature, hydrostatic pressure and salinity; finally, Acstp is a^ correction term for simultaneous variation of the three properties. A formula due to Kuwahara (1939) is given by Horton (1957); reduced to metric units it is c- 1 399 +1.3LS + 4.592i- 0.0444^2 + 0.182/i. (8) More recently Del Grosso (1952), Greenspan and Tschiegg (1955) and Wilson (1960) have measured the velocity of distilled water at t = 30°C,^ = 1.033 kg/cm2, and Del Grosso (1952) and Wilson (1960) have measured sea-water of 35 %„ sahnity under the same conditions, as tabulated below. Distilled Sea-water water (35%o) Del Grosso 1.509.6 Greenspan 1509.44 Wilson 1509.66 1546.3 1546.16 478 VIGOUREUX AND HERSEY [CHAP. 12 Wilson made 581 measurements of the velocity of sea-water at temperatures from —3° to 30°C and for five salinities between 33 %o and 37 %o. These were fitted to MacKenzie's (1960) formula, which is based on Kuwahara's tables. Of this work Wilson reports : "this equation resulted in a standard deviation from the mean of the differences between the computed data and the measured data of 0.29 m/sec. The equation was consequently changed and only the terms resulting from the differential analysis of fourth order polynomials were re- tained." The resulting terms from equation (7) are given below. co.35,0 = 1449.22 Act = 4.6233^- 5.4585 X 10-2^2 -1- 2.822 X 10-4^3 - 5.07 x lO-^i^ Acp = 1.60518x10-^+1-0279x10-5^2 + 3.451 X 10-9p3 _ 3.503 x 10-12^4^ Acs = 1.391(>S- 35) -7.8x10-2(^-35)2, and Acstp = (^-35)(- 1.197x10-2^ + 2.61x10-4^ -1.96X 10-7^2 _ 209 X 10-6^0 +p{ - 2.796 X 10-4^ + 1.3302 x 10-5^2 - 6.644 X 10-8^3) +p2( _ 2.391 X 10-'?« + 9.286 X 10-10^2) _ 1.745 x lO-i^^^. where p is in kg/cm2, S in parts per thousand (%o), and c in m/sec. Hays (1961) made seven sound- velocity profiles in the Mediterranean Sea in conjunction with seven hydrographic stations, during which many determinations of temperature and salinity were made. He used an instrument designed by Tschiegg (see Tschiegg and Hays, 1959). He has compared his measured velocities at the same depths as the salinity /temperature measurements with velocity values computed by Wilson's formula. The differences between the two show a nearly linear increase with depth, with the computed velocities less than the measured by about 0.2 m/sec at 700 m and by about 0.5 m/sec at 2200 m. At shallower depths than 700 m the differences are much larger and less systematic, probably owing to horizontal gradients in the water through which the ship was drifting during the observations. At greater depths than 2200 m the difference decreases to about 0.4 m/sec at 2800 m. The systematic difference between Wilson's formula and Hays's measure- ments suggests that further investigation is needed to assure an increase by a factor of ten in accuracy from the work of Matthews and Kuwahara. Neverthe- less, the discrepancy is small ; of the same order as the differences between the laboratory results for distilled water of Del Grosso, Greenspan, and Wilson. At the time of writing, the velocimeters of Greenspan and Tschiegg (1955) have been used under severe conditions at sea as well as in the laboratory. They have proven to give reproducible results and trouble-free service. By now several tens of velocity soundings have been made in the deep ocean. Never- theless, for some time to come acousticians will be dependent on measurements of salinity and temperature as a function of depth. SECT. 4] SOUND IN THE SEA 479 G. Impedance The product, pc, of the density and the velocity of propagation gives for plane waves the quotient of the acoustic pressure and the particle velocity. It is called characteristic impedance and is the measure of an important property of the medium in that it controls, among other things, the passage of sound normally across the common boundary of two media. For plane sinusoidal waves the intensity, i.e. the power through unit area perpendicular to the direction of propagation, is given by P = Ipcvo'^ = Po^l2pc, (9) if po and vq denote not the instantaneous values but the maxima of acoustic pressure and of particle velocity. D. Attenuation a. Geometric spread Conservation of energy requires that, if the sound energy radiated from a point inside an unbounded homogeneous medium is not dissipated, the energy flowing through unit area perpendicular to the direction of propagation shall be inversely proportional to the square of the distance from the source of sound. This spherical spread results in a rapid decrease in intensity. In terms of acoustic pressure it is expressed by the formula por = C, (10) where r is the distance from the source of sound and C is constant for any given source. In very deep water this law is, for moderate ranges, a good approxima- tion of what actually happens. If, on the other hand, propagation took place from a point between the two parallel plane boundaries of an otherwise unbounded medium, the intensity would be inversely proportional to the range, and there would be cylindrical spread given by pory^ = C. (11) This law would apply to propagation in shallow seas of uniform depth if reflection from the surface and the bottom were perfect. b. Absorption Another cause of decrease of intensity is the gradual conversion into heat of the energy of the wave. Viscosity, thermal conductivity and inframolecular processes all play their part in producing this attenuation, which is, in general, much larger than the value predicted on the assumption that viscosity is the sole cause. In liquids other than mercury, thermal conduction has a negligible effect, so for the sea there are only viscosity and inframolecular processes to consider. They both produce an exponential decrease of intensity which it is 480 VIGOUREUX AND HERSEY [CHAP. 12 usual to indicate by a coefficient referred to amplitude so that, for plane waves, intensities at points a distance x apart in the direction of propagation are in the ratio of 1 to exp ( — 2aa:;). (For practical purposes the decibel per unit length is often used to express attenuation, which is then 10 times 2a log e or 8.686a.) For sea-water at 10°C the absorption (Horton, 1957) in decibels per kilo- yard at / kc/s is given to a good approximation by the formula •^ - + 0.000 275/2, (12) 4100 +/2 in which the first term represents the relaxation in dissociation of salts in solution (Liebermann, 1948, 1949) and the second is due to viscosity and to relaxation in compressibility. The salt mostly responsible appears to be mag- nesium sulphate, and dissociation and hydrolysis of both ions are involved. Almost all over the range, the absorption decreases with increase of tempera- ture, the reduction at low frequencies being as much as 50% for 17°C. Sound energy can also be scattered by small perfectly reflecting particles uniformly distributed throughout the water. Although in this ideal case the total energy is not thereby decreased, the intensity in the direction of propaga- tion is reduced exponentially as with absorption. The most important scatterer is the air bubble, which also absorbs sound, especially at resonance. Bubbles are sometimes found just below the surface, having been entrained by sea waves ; they are also liberated from air in solution in the sea by the reduction of pressure at propellers of ships. If a shi23 lays a wake across a sonar beam, the sound is often almost completely prevented from going through ; the bubbles in the wake absorb some and reflect the rest, giving rise to a strong sonar echo (Horton, 1957). Turbulence and finely divided vegetable and animal matter also scatter, but, as with bubbles, the effect is local and variable. E. Reflection at Surface and at Sea- Bed The main difference between air and water as carriers of sound is that the characteristic impedance, pc, of water is some 3700 times that of air, which means that the "mismatch" between the two media is very great. For this reason very little sound crosses the free surface either way ; most of it is re- flected. At normal incidence, which is the most favourable direction for trans- mission across the surface, the ratio of transmitted to incident energy is ^^ {pc)w^ /{pc)a {pc)a V (pc) ^ (13) 1000 A wave reflected at the surface may reach the sea-bed, at any rate in not too deep water, and if the sea-bed also reflects, the process recurs and propagation takes place in hops between the surface and the bottom. In the following development the sea-bed will be regarded as unconsolidated sediment without rigidity, i.e. as being incapable of transmitting shearing SECT. 4] SOUND IN THE SEA 481 forces. This property is commonly found, but not everywhere. Where the sea- bed consists of sohd rock a more general treatment is required (e.g. see Ewing, Jardetzky and Press, 1957). If p, c and pi, ci denote the density and velocity of propagation in the sea and in the ground below (Fig. I), for the present assumed homogeneous and of Fig. 1. Reflection of sound at boundary between elastic media. infinite extent, a wave, p, of phase a>t + kz cos d — ky sin d incident at an angle 6 to the normal is in general split into two waves. One, pr, of phase cot — kz cos 6 — ky sin Q, is reflected at an angle 6 on the other side of the normal ; the other, pt, of phase a>^-|- A:i2; cos ^i — ^'ly sin ^i, enters -the ground at an angle di, where (Rayleigh, 1878), (14) sin 01 Cl sin d c Vr ■(?- cot ^1^ 1 Ipi cot ^1 V COtd)/\p cot^, Pt _ P -'-fl /pi cot^i\ \ p cot ^ / (15) :i6) As in general Ci is greater than c, the transmitted wave is deflected slightly away from the normal until a critical value, sin-i (c/ci), of 6 is reached which makes sin di unity, after which there is total reflection. In these circumstances (15) takes the form Pr V cos ^ + 7 1 pC •^\ C2 plCl pc COS 6 Jci^ sin^ d :i7) Thus Pr has the same amplitude as p but its phase is cot — kz cos d — ky sin 6 + 2x, where [(ci/c)2sin2^-l]V'2 tan Y = (piCi/pc) cos 6 (18) 482 VIGOUKEUX AND HERSEY [CHAP. 12 Corresponding to (17) there is a "bound wave" ptip = 2 cos ;(;e*^i[(ci/c)- sin^ e-ljViz (I9) of phase ojt — ky sin d + x- Owing to the exponential term in (19) this wave is rapidly attenuated with depth below the sea-bed, along which it propagates with a velocity c cosec 6 equal to Ci at the critical angle and less than ci for all greater angles of incidence. There is no transmission of energy into the lower medium but a diffraction across the interface associated with and bound to the reflected wave (Officer, 1958). For all angles of incidence less than sin~i (c/ci), however, (15) is real and gives for the ratio of reflected to incident energy Apicot di j [pi cot 01^ 2 / y p cot 6 I \p cot 6 (20) On the provisional assumption made above that the bottom of the sea is homogeneous and infinite, the proportion of energy lost on reflection is the second term of (20) taken with the positive sign. F. Propagation in Shallow Water In shallow water the sound wave is continuously reflected from the surface and the bottom, and there is rapid attenuation for angles of incidence less than the critical angle even if the second term of (20) is small, so that propagation is impossible even at moderate range. But for angles of incidence greater than the critical angle, reflection is total and sound is propagated to great ranges. Propagation at any one frequency is, however, restricted to a few discrete angles of incidence, as can be seen by considering the phases of the two waves. If, for simplicity, we flrst assume a perfectly rigid sea-bed (pc= oo) and a per- fectly "soft" air above the surface {pc = 0), the wave p incident on the sea-bed (Fig. 1) has phase ojt -\- kz cos d — ky sin 6 and the wave pr reflected from it has, since x of (18) vanishes, phase cut — kz cos d — ky sin 6. Since at the surface the pressure must vanish, these two waves differ in phase by n or more generally (271— 1)77- if w is a positive integer ; thus, if the depth be ^, 2A;^cos 6 = {2n-l)TT ^cos^ = (n-i)A/2. (21) This equation shows that n, called the "order" of the "mode" of propagation , cannot have all integral values, since it cannot exceed | -1- 2^/A. In fact if the depth t, is less than A/4 the waves are not propagated. In practice the air above the surface is a sufficiently good approximation to a perfectly soft medium, although there is attenuation due to the small amount of energy which finds its way into it, but the sea-bed is not infinitely rigid and SECT. 4] SOUND IN THE SEA 483 reflection, although total, occurs with a phase change 2x given by (18); a reasoning similar to that used in arriving at (21) then gives 2A;^cos^-2x = (2w-l)7r Ceos9=(™-i+^)^. (22) Since x is itself a function of d, it is not possible to obtain a simple formula for the angles corresponding to the various modes, but if x/tt and (2^/A) cos d be plotted against 6, the several values of d which satisfy (22) can be determined graphically, and those not less than the critical angle correspond to the modes which are propagated. Since neither the sea-bed nor the surface is perfectly smooth and horizontal, there is some scattering of the energy, and the fronts of the incident and reflected waves do not make equal angles with the vertical. There is some loss due to this effect but multiple scattering also transfers energy from one mode to another. Even without this transfer the modes are not all equally attenu- ated because the small loss at the surface decreases as the angle of incidence increases. Hitherto we have considered a single frequency ; if, however, the source of sound has a continuous frequency spectrum, as, for instance, noise of ships or the noise of an explosion, waves are propagated at all angles of incidence greater than the critical angle. Also, as absorption of sound increases rapidly with frequency, at long distances from the source, the spectrum is more and more shifted towards the low-frequency end. To all the effects mentioned above the cylindrical spread of energy which takes place in shallow water adds its own reduction of intensity with range. The simple treatment above applies to plane waves only and can be extended to the calculation of intensity in the vertical. The problem is more complicated when energy is radiated from a point transmitter at some depth zi below the surface to a receiver at depth 22. Pekeris (1948) and Ide et at. (1954) show that the expression for the velocity potential then contains sinusoidal factors depending on zift, and 22/^, and another factor giving the relative strength of the various modes. The expression for the phase shows that the propagation constant in the horizontal direction y is, k sin 6. The phase velocity, F, or velocity with which phase is propagated along y, is thus c cosec d. When instead of a single fre- quency there is a continuous frequency spectrum, for instance that of a sonar pulse, 6 changes with oj and so does the propagation constant. In this case an important characteristic of propagation is the "group" velocity, U, which is the frequency with which the spectrum is propagated; it is the value of dyjdt corresponding to a stationary value of the phase and is given here approxi- mately by jj = ^ ^^/^^ (23) sin d{dojldd) -f CO cos ^ 484 VIGOURETJX AND HERSEY [chap. 12 Since the relation between oj and 6 is not simi3le, calculation of U from (23), (22) and (18) is complicated. Pekeris (1948) shows that U is equal to ci at the frequency corresponding to the critical angle ; the phase velocity, V, is also C\ for that angle. As 6 increases, U decreases more rapidly than V , goes through c, and reaches a minimum, after which it tends towards c sin d ; at grazing in- cidence, i.e. at infinite frequency, V and U are both equal to c. O. Velocity Gradients Since salinity and temperature of the sea can vary with depth as well as locality, the velocity of propagation of sound varies with depth, as can be seen from (8), not only because of the change of pressure but also because of changes in salinity and temperature. The component of gradient of velocity in a direc- tion perpendicular to the ray causes the ray to bend away from the region of high velocity towards that of low velocity ; thus, as a rule, sound rays in the sea are not straight but curved. As, however, the gradients are small, the curva- tures are small, and in shallow water it may not be necessary to take them into VELOCITY RANGE SURFACE I I- Q. ^ UJt D Fig. 2. Sound rays from wide-angle projector when velocity decreases with depth account ; but in the deep ocean, where long ranges are reached before a ray arrives at the sea-bed, the angle of incidence can differ considerably from that which the ray makes with the vertical at the source of sound, and this effect must be allowed for when calculating the intensity as a function of range or depth. In these calculations it is in general permissible to assume that over the ranges of interest the velocity is everywhere the same at the same depth, so that the variation with depth is the same everywhere. In such circumstances repeated application of Snell's law (14) to slices bounded by horizontal planes shows that if c is the velocity and 6 the angle which a ray makes with the vertical, and Co, ^o the values of these quantities at some starting point, e.g. at the source of sound, c cosec 6 = cq cosec ^o- (24) It is thus possible, by proceeding in steps, to trace a ray leaving the source at any angle with the vertical. It is instructive to consider in the first instance two cases of propagation affected by velocity. The first is the case when velocity decreases with depth part of the way and afterwards remains constant right to the bottom. In this case (Fig. 2) the rays are eventually bent downwards and, apart from some SECT. 4] SOUND IN THE SEA 485 scattering, no sound reaches beyond the grazing ray ; thus detection by echo fails for an object beyond that ray. However, unless the frequency is so high that the energy is all absorbed before the sound reaches the bottom, each ray is reflected with some loss, or no loss, according to the angle of incidence and the nature of the sea-bed. It then proceeds upwards along a path which is the mirror image, about a perpendicular to the sea-bed, of its downcoming path, as in Fig. 2, where the sea-bed is horizontal. Thus the grazing ray reaches the surface again horizontally, whereas the ray which had been emitted in a horizontal direction becomes horizontal again at the depth of the source, and the process goes on repeating itself as long as geometric spread, scattering and absorption do not reduce the intensity to a trivial value. If the emitted beam of sound is narrow, i.e. if the angle between the extreme rays is small at any rate in the vertical, there are at depths of the order of that of the source a number of regions in which the intensity is negli- gible, alternating with others in whiah it is appreciable. Even if the beam is VELOCITY RANGE SURFACE ZJ. Fig. 3. Sound rays from directional source when velocity increases with depth. wide or even if the source emits fairly uniformly in all directions, as for instance an explosion does, the rays reaching the bottom at small angles of incidence, like 3 and 4 in Fig. 2, are reflected with some reduction and may not carry sufficient energy to fill the "shadows", so that the intensity near the surface may even then go through maxima and minima instead of decreasing uni- formly with range. A second simple case can occur when the sea has been well mixed by a warm, moist wind, and the weather turns fine, calm and cold. The temperature near the surface is then less than it is lower down, and the rays are bent up- wards (Fig. 3). After reflection they travel in a path concave upwards and are reflected again and again, thus hugging the surface and never reaching the bottom if the original beam is emitted in a horizontal or nearly horizontal direction. There is little loss and the sound can carry many miles although an object a few fathoms below the surface, even if it is at close range, may never receive any sound and may, therefore, fail to be detected by the echo method. Even a purely isothermal layer, formed by mixing and extending from the surface to a few fathoms below it, produces a similar though less marked effect, since velocity is then controlled only by pressure and, therefore, increases slowly with depth according to (8). 486 VIGOUIIEUX AND HERSEY [chap. 12 From these simple cases we pass to the more general one when the tempera- ture, which almost always affects velocity to a greater degree than salinity or depth, is fairly uniform over the first 50 to 1000 fathoms, depending on locality and season, then decreases with depth perhaps by some 10 to 20°C in the next 200 fathoms, after which it decreases only very slightly with depth right to the bottom. The region of rapid change is called the "thermocline". There velocity and temperature profiles have approximately the same shape (Fig. 4) VELOCITY RANGE SURFACE Fig. 4. Sound rays from directional source above a thermocline. and rays emitted by a source near the surface are bent downwards and eventu- ally reach the bottom where reflection occurs as in the first example, although those rays emitted nearly horizontally may keep close to the surface if the velocity gradient above the thermocline is due to pressure only. Finally, if beyond a certain depth temperature is constant or decreases very little, as often happens in deep water, there is a depth after which the velocity, which at first decreased with depth because of decreasing temperature, starts increasing, since the effect of pressure, or depth (formula (8)), eventually pre- dominates. In this case the rays may be bent upwards and become horizontal again before they reach the bottom. Indeed, rays emitted in directions near to horizontal from a source at a depth where the velocity is a minimum are continually returned to that same depth, since they are convex upwards above it and concave upwards below. As these rays reach neither the surface nor the bottom there is no loss by reflection and almost no loss by scattering, which occurs mostly at those two boundaries. The geometrical spread in this case is cylindrical and the loss due almost solely to absorption. As this loss is low at low frequencies, a disturbance rich in low frequency — for instance, the explosion of a mine — can be detected thousands of miles away by a detector at the same depth, usually a few hundred fathoms. In 1952, U.S. Navy Sofar stations at Point Sur and Point Arena, California, gave remarkable records of the eruption of the Myojin submarine volcano, 8600 km away (Dietz and Sheehy, 1954), about 200 nautical miles south of Tokyo. In March, 1960, shots (200 and 300 lb of TNT) fired at the axis of the SECT. 4] SOUND IN THE SEA 487 Sofar channel from Vema, research ship of the Lamont Geological Observatory of Columbia University, and Diamantina of the Australian Navy at 33° 13'S, 113° 43'E, approximately the antipodes of Bermuda, were detected appreciably above the background noise of geophones mounted on the bottom of the slope of Bermuda. (Mr. Carl Hartdegen of the Columbia University Geophysical Field Station at Bermuda directed the receiving observations ; shots were fired by Dr. John Nafe.) H. Propagation in the Sea-Bed Sound propagation in the sea-bed is the concern of exploration seismology as well as sound. It will be treated more extensively in Volume 3. I. Scattering of Sound in the Sea The term scattering is applied to reflection of waves by an object. If a beam of plane waves of sound in water impinges on a sphere of material whose impedance differs from that of water, energy is reflected or scattered uniformly in all directions. The term is usually taken to include not only the energy deflected from the direction of the beam but also the forward scatter or signal, and the back scatter or echo. The signal which would have been carried by that part of the beam intercepted by the object is thus reduced to the forward scatter, and, if there are numerous obstacles, the signal can be considerably attenuated. The body of the sea does not normally contain scatterers capable of seriously reducing the intensity of beams of sound. Bubbles, fish, vegetation, dis- continuities in temperature, turbulence, all have some effect, but except in special circumstances it is true to say that scatter by the body of the sea is negligible compared with that by the surface and the bottom. Although sea waves are often very large compared with the wavelength of sound used for submarine signalling, back scatter from the surface is not considerable because the sound beams are nearly horizontal at the surface. Moreover, as they are in general bent downwards by temperature gradients, scatter by the surface occurs only at close range and possibly at a limited number of other regions. The result is that in the deep ocean, even in rough weather, long range sonar apparatus is remarkably free from disturbance by back scatter, and the echo from a distant object is not masked by "surface re- verberation". On the other hand the bottom of the sea is a powerful scatterer, especially when the sound beam reaches it at an angle and when the scale of the irregu- larities is large. Scattering by the bottom does not, however, normally impair reception of sound from a distant source because the signals, especially when they are short, reach the detector before the scattered energy. But things are very different in echo detection by sonar apparatus, which may receive re- verberation from the bottom and the surface before, during and after reception of the echo. Reverberation comes from an area of the bottom or surface pro- portional to the range R, so the ratio of echo intensity to reverberation intensity 488 VIGOUKEUX AND HEKSEY [CHAP. 12 is proportional to IjR provided the degree of roughness of the bottom is such that the scatter does not depend on the angle of incidence. In many cases back- scatter decreases as the angle of incidence increases, with the result that the ratio of echo intensity to reverberation intensity is roughly independent of range over a considerable portion of the range. An argument similar to the one used above indicates that the less serious body or volume scatter causes the echo/reverberation ratio to vary as IjR^ (Horton, 1957). Reverberation is thus a factor in limiting the range at which echoes can be detected. In shallow water it may well be the limiting factor, but in the deep ocean, noise originating from the sea usually sets the limit to the range of detection. J. Fluctuation of Sound Even when the emitter delivers a perfectly steady signal to the water, as measured a few feet away, the signal received at long or moderate range is subject to random fluctuations of intensity. This phenomenon, akin to scintilla- tion of stars or fluctuations of radio signals received via the ionosphere, is due to time variation of inhomogeneities of the medium. The surface, for instance, is in continual motion, swell changes the angle of reflection, and there may be turbulence in the body of the water especially near the surface ; also air bubbles move before disappearing by solution or at the surface, and there are local variations of temperature. All these effects cause the signal to fluctuate. An idea of the way in which the fluctuations depend on the range, the wave- length and the mean size of the irregularities can be obtained by supposing that the velocity, c, of propagation of sound waves varies from point to point accord- ing to the law c = Co[l+an{x, y, z)], (25) where a is the r.m.s. value of the variations of velocity and n{x, y, z) is the variation of velocity whose r.m.s. value has been normalized to unity (Mintzer, 1953). The autocorrelation function of the variations is supposed to be of the form exp { — r^ja'^), where a is the mean size or "scale" of the inhomogeneity. If the range and the propagation constant are denoted by R and k, it is found that if kR and ka are both much larger than unity, the variance of the ampli- tudes of short pulses is cr2 = TTy^k'-a^aR. (26) If, on the other hand, the frequency is so low that ka is much less than unity, although the range is large enough for kR to be still much greater than unity, a^ = ny^k^a^a^R. (27) For a continuous signal the values above must be multiplied by a factor greater than unity. SECT. 4] SOUND IN THE SEA 489 3. Noise At any point of the ocean there are, in addition to sound waves due to the signal under observation, changes of pressure covering a very wide frequency range. This "noise", dealt with at length by Horton (1957), is due in part to movements of water, in part to marine fauna, in part to man-made devices, and in part to vibrations of the ground. The quasi-periodic pressure variations due to tides and to swell can be left out of consideration because their fre- quencies are lower than what we have agreed to call sound. Much of the noise in the deep open sea seems to be due to impact of wavelets and collapse of cavities trapped just below the surface when wave crests break to form "white horses". As soon as the temporary reduction of pressure forming the cavity is over, it collapses producing a pressure change of a type such as to give the continuous spectrum proper to "cavitation", of level decreasing by 6 dB per octave. The number of cavities is very large since the whole surface of the sea contributes. The intensity is nevertheless finite under the reasonable assump- tion that the contribution from any area is proportional to the cosine of the angle between the normal and the line joining it to the point of observation. If z is the depth and a the pressure absorption coefficient, the intensity is proportional to At low frequencies a is negligible and this integral is equal to unity, which means that the intensity is independent of depth. That is in fact what is found in practice. Even at high frequencies the decrease is small for moderate depths : at 32 kc/s, for instance, a is approximately 8 dB per kiloyard and at a depth of 300 ft the intensity would be only about 1 dB less than near the surface. This result thus agrees with the statement by Horton (1957) that, for depths down to 300 ft, neither the magnitude nor the frequency characteristic of water noise depends on the depth at which the observation is made. "The quality of the noise does, however, vary somewhat with depth. As the hydrophone is lowered the sounds of individual waves, which can be separately identified near the surface, merge into a more nearly continuous sound. At the same time there is a noticeable decrease in short-time variations from the average noise level" (Horton, 1957). Another aspect of water noise is more difficult to understand : measurement has established that the spectrum of water noise is pretty well a straight line from 100 c/s up, decreasing with frequency by 5 dB per octave down to the point where it merges with the thermal noise of the equivalent electric resistance of the detector. That it is due to the motion of the surface of the sea is well demonstrated by its dependence on sea state (Fig. 5) ; but experiment shows that cavitation noise from any other source whatever decreases at the rate of 6 dB per octave, and this result is consistent with the assumption that the pressure near the time of collapse of a cavity varies exponentially. It is 490 VIGOTJREUX AND HEESEY [chap. 12 not clear why water noise should be relatively richer in high-frequency components. The noise spectrum illustrated in Fig. 5 is modified in coastal waters where waves breaking on the shore and local currents bring their contribution; nevertheless the change of character is surprisingly small. A much more noticeable change is produced by animal life in some coastal regions where the water temperature is 15°C or above. Small fish called croakers can cause as much noise again as sea noise of state 1, at frequencies below 2 kc/s, and snapping shrimp will readily produce a similar increase at frequencies up to 20 kc/s or more. Curves given by Horton (1957) are reproduced in Fig. 5; the shapes apply to the calmest sea conditions, but the volume, of course, depends on the number of the congregation . Since World War II biologists have learned much more about the characteristic sounds of many soniferous marine animals. These findings are discussed in Chapter 14, "Sound Production by Marine Animals". -80 Euj ^ o > .>~, 26 FEB, 1954 SUNSET 18J0 BELOW TRANSOOCen i 4iiK^"gteagifei?.«tei^tea^i^.iiffiip.^^ y^ 0' 10 :-2o 30 S 1649 + 4 (OUEEN) TIME Fig. 5. A 12 kc/s echo-sounder record made by lowering the echo-sounder transducer to a point midway between the surface and a deep scattering layer and holding it in this position as the layer migrated up past the transducer (see text). The increase and then decrease with time of individual scatterers may be noted as well as their upward progress. The sinuous shape of the echo sequences is due to the rolling of the ship and the consequent rise and fall of the transducer. (After Johnson, Backus, Hersey and Owen, 1956.) Generally these layers make a pronounced diurnal vertical migration but some do not. There appears to be no correlation between midday depth and con- spicuous features of the temperature-depth profile although the level to which these layers rise at night may be limited by abrupt increases in temperature. Although the typical deep scattering layer appears on the record of a surface echo-sounder as a stripe of numerous, diffuse, jumbled echoes, the scatterers can be individually resolved when the range to the layer is made short by SECT. 4] SOUND SCATTERING BY MARINE ORGANISMS 505 lowering the transducer to a position just above the layer. This is so because the scatterers have a relatively high scattering cross-section and are sufficiently few within the near part of the water volume insonified by the echo-sounder. The result of such an observation, just prior to sunset (Johnson, Backus, Hersey and Owen, 1956) are shown in Fig. 5 in which a transducer was lowered to a depth midway between the surface and a deep scattering layer. The echo- sounder was held in this position and the scattering layer migrated past on its way towards the surface. Examples of deep scattering layers recorded by surface echo-sounders are shown in Figs. 6, 7 and 8. We may describe the deep scattering layers of the western North Atlantic, as recorded by 12 kc/s echo-sounders, as representative of observations in open, deep ocean areas. About 150 records have been examined for making these DEPTH (fathoms) 0 ) 500 1000 1500 1 » 4^00 5 minutes Fig. 6. The sunrise descent of deep scattering layers recorded by a 12 kc/s echo-sounder in the eastern Pacific off northern Chile. A layer, which appears to have remained at depth throughout the night, is shown near 300 fathoms. generalities. Here one finds two principal deep scattering layers. The shallower of these has an average midday depth (midway through the layer) of 240 m, the extremes of the variation being 185 and 405 m. The average thickness of this layer is about 75 m. The deeper of the two layers has an average midday depth of about 500 m, the extreme values being 405 and 590 m. The average thickness of this layer is about 130 m. Both of these layers show the familiar diurnal vertical migration although there are non-migratory elements in both, especially in the deeper layer. Like- wise both layers often show a splintering during the vertical migration such that as many as four separate elements may be formed from what appeared to be one layer prior to migration. The record of ascending and descending layers may also be complicated by the sudden appearance or disappearance on the 506 HERSEY AND BACKUS [chap. 13 DEPTH 0 (fathoms 100 200 ct ^no Uj :v~ 1 ^^ ■ o ( o X [ i rj. ^" 10 20 40 RESONANT FREQUENCY fn (kc/s) 100 200 400 Fig. 13. Theoretical curve and experimental values of total damping constant of resonant air bubbles in water. The points are taken from the work of seven experimenters using various methods of determination. (After Devin, 1959.) By courtesy of C. Devin, Jr., The George Washington University and The David Taylor Model Basin. scattering frequency will vary as P'/^. Of course, as it returns to shallow depth it must vent or absorb gas from the swim-bladder to maintain approximately neutral buoyancy. Kanwisher and Ebeling (1957) point out that the swim- bladders of migrating lantern-fishes, caught at the surface at night, have a high percentage of oxygen, indicating that they have taken on oxygen at depth. Nevertheless they regard it unlikely on physiological grounds that these fishes maintain neutral buoyancy by changing their gas content during migra- tion (see page 535). A second possibility is that the fish allows its "bubble" to compress and expand with descent and ascent. (This assumption implies that the fish can tolerate being "heavy" at maximum depth.) Under such an assumption, the factor IjR in (18) varies as P'/», from which we may write approximately fifo = (P/Po)^/« SECT. 4] SOUND SCATTERING BY MARINE ORGANISMS 519 since p changes very slowly with depth. Generally, large migrations in depth amount to considerable change in temperature as well. The ocean temperature almost everywhere decreases with increasing depth. The effect of temperature on 1/i? is approximately (1/T)'^ Usually the temperature change will have the effect of reducing the frequency change by a small amount (order of 2% for a change of 20°C). The swim-bladders of bathypelagic fishes are not generally spherical but are more nearly like prolate spheroids (Marshall, 1951). Furthermore they probably do not behave strictly as "free" bubbles but are constrained by the body of the fish. Scattering by a prolate spheroidal bubble can be shown to have a resonant frequency which varies as P'/^ if the bubble size and shape remain constant. If the gas content remains constant and the bubble is free to com- press in all directions, then the resonant frequency varies as P'l^, as for the spherical bubble. However, if the bubble is constrained so that it will not shorten its major diameter, but will compress by becoming "slimmer", then the resonant frequency varies very nearly as P. This latter possibility seems plausible because the structure of the fish leaves it freer to compress in this way.i Unfortunately no experimental studies have been made of single speci- mens over a sufficiently great frequency range to permit the evaluation of simplified theoretical models. Smith (1954), Gushing (1955) and Gushing and Richardson (1955) have measured scattering cross-sections over limited frequency ranges. At the moment we can only hope that such work will be extended both to lower and higher frequencies. B. Scattering by Large Groups Volume scattering in the ocean is due to the contributions of many in- dividuals. In some instances individuals can be resolved, especially when they are close to source and receiver (see Fig. 5), but generally the observed volume reverberation is a "smear" of the contributions arriving simultaneously from many individuals. A common practice in volume-reverberation observations is to use the same transducer both for projecting the sound and receiving the returning reverberation. The transmission paths must be reciprocal for scatter- ing from one object. Since it is commonly assumed that single scattering over- whelmingly predominates, back-scattering is assumed to be measured when a single transducer (or two very close together) is used. A convenient method of measuring scattering is to radiate the sound in a succession of pulses each of which is short compared with the interval between pulses. The transducer receives back-scattered energy between pulses which has come from scatterers farther and farther from the source as time-after-pulse increases. It seems unnecessary to describe in detail the apparatus employed 1 The remarks in this paragraph are based on a study by Melvin Steinberg of the University of Massachusetts (manuscript in preparation) who has studied the variation of resonant frequency for a "free" jarolate spheroid, and also a "free" spheroid enclosed in a thin, flaccid membrane. 520 HERSEY AND BACKUS [CHAP. 13 since it is similar to echo-sounding or echo-ranging apparatus discussed else- where in this book. It is expected that since the pulse of sound forms an ex- panding spherical shell of constant thickness, it will generally be scattered from many objects in the water at times such that their scattered waves arrive simultaneously at the receiver. The receiver senses the appropriate sum of these independent waves. The usual treatment of the problem is based on this assumption, i.e. that at any instant scattering from many individual objects is superposed. Since these individuals occupy a volume it is convenient to define a scattering coefficient, m, of a unit volume, analogous to the back-scattering cross-section of an individual. Let us now take the transducer as the center of a spherical co-ordinate system (r, 6, (f)) and further let us assume that the energy emitted by the source per unit solid angle per second in the direction {6, ^) is described by F{r, d, (f)), where t is measured from the beginning of a pulse. The intensity of volume reverberation at the receiver at time t can be shown to be related to the volume scattering coefficient of the medium, m{r, 6, 0), through the follow- ing integral equation, /W=f nr.e^)Mr.e.me.*)^y_ ,21) jvolume r^ r^ where b{d, ) describes the directional properties of the transducer as a receiver and r = {cl2){t~r), (22) where c is the velocity of sound in sea-water in cm/sec. The following assumptions are implicit in equation (21). (1) The sound velocity in the medium is constant and not materially affected by the scatterers. (2) The definition of m given above has the effect of averaging the con- tributions of the individual scatterers. (3) A volume element begins to scatter sound at the instant it is insonified, and stops as soon as it is again "in the dark", i.e. there are no time lags — no storage of energy in the scatterers. (4) Scattering cross-sections are small enough for us to neglect multiple scattering as well as attenuation of the outgoing beam due to scattering. (5) The average reverberation intensity at the receiver is equal to the sum of the average intensities due to individual scatterers. This is a "randomizing" assumption to average out interference effects. (6) The backward-scattering coefficient, m, is independent of the direction of incidence of the beam. In general, one must solve or invert equation (21) to determine volume scattering coefficients from measured reverberation intensities. Reverberation of a sinusoidal wave-train radiated from a "searchlight" or piston-type trans- ducer (i.e. a directional source having one radiation lobe, the main lobe, much stronger than all others) and received by the same transducer was analyzed by SECT. 4] SOUND SCATTERING BY MARINE ORGANISMS 521 staff of the University of California Division of War Research (NDRC, 1946a). The echo-sounder is such an instrument ; its main lobe is directed vertically downward. The directional radiating and receiving properties are considered identical, that is, F{t, 6, ) can be written F{t) b{d, (f)) and the integral equation becomes : F{t) m{r, e, cf>)b^{d, 4>) m = dV. (23) f volume f ^ As noted earlier, observation with echo-sounders has shown that scatterers are commonly horizontally stratified ; that is, w is a function of depth only. Also, the transducer commonly has axial symmetry, b{d, (f)) = b{d). Thus (23) becomes F{t) m{r cos e)b^e) m - volume f' r^ dV or I{t) = I f" r^" :^ m{r cos e)bHe) sin 6 dr dd d. 'o Je=o J0=o r- Performing the ^-integration I{t) = 277 Jo Jo " -^ m{r cos 9) b^d) sin d dr dd or substituting (22) ^' c J.=oJo {t-rf -: (t — r) cos 6 b^e) sin d dr dd. (24) (25) (26) (27) For a sinusoidal pulse beginning at t = 0 and ending at t = to (i.e. a "pulse length" to), we assume a form p—po sin ojr; ^ po^sm^2n{fr-rlX) ^^^ pc (28) where pQ is the instantaneous pressure in dynes/cm^ at range r= 1 m. Then '<'* = ^^Jo Jo JTW^ " - [t — r) cos 6 62(61) sin d dr dd. (29) The usual choice of pulse length is such that to<^^ so that, for performing the T-integration, we may take r = ctj'2. Then ^ 477xl04|o^^^ p Ic ^^^ \^^^ ^.^^ ^ ^^ ct^ 2pc Jo \2 / Where m is regarded independent of position, ^^^^ 477Xl04;P02 ^(0 = — — TT- rom or I{t) = ct^ 2pc 77 X 104^90^ r bHd) Jo sin d dd -P ^ Torn 62(^) sin d dd. (30) (3i; (32) 522 HERSEY AND BACKUS [CHAP. 13 So far as we are aware the integral of (30) has not been evaluated or approxi- mated for a piston transducer, while for that of (31) numerical integration has been performed (NDRC, 1946a). They report measurements of w showing peak values close to 3 x 10"'^ m~i at depths of about 400 m in the frequency range from 10-80 kc/s. Machlup (see Machlup and Hersey, 1955) has treated the problem of back- scattering of the omnidirectional shock wave from a nearby explosion to both omnidirectional and piston or searchlight receivers. F{r) represents the energy flux per unit solid angle of the shock wave ; b-{d) reduces to b{d) since the explosion is considered omnidirectional; and, finally, if we assume r-^t so that r = ctl'2 is a sufficient approximation, then the r-integration can be performed. The total energy in the shock wave can be written /■*oo E = 4:77 \ F{t) dr. (33) Then equation (27) reduces to Ec r"/2 I{t) = — m{r cos d) b{9) sin 9 dd. (34) Treating the receiving transducer as a freely vibrating circular piston in an infinite rigid baffle (Morse, 1948), m = 2Ji{k sin 6) k sin 6 (35) where Ji = Bessel function of order one, k = irdjA, d — diameter of the piston, and A = c//= wavelength of the sound in the transmitting medium. Machlup (Mach- lup and Hersey, 1955), seeking a simpler analytic function than (35), found b{e) = cos« {6), (36) where n= {k^l2) — 1, to be an adequate approximation to (35), where side lobes are not important. Machlup chose b{6) = cos" 6 dehberately so that equation (34) could be put in the form I{t) = ^ - \ ' m(r cos d) oos^ d d cos d (37) 4 r^ Jo (a Fredholm equation of the first kind), which has a solution (38) Id m(z) — r- ^ ' z^ dz Yc 2"+'^(2) where Using the relationship dy ^y d (log y) dx X d (log x) SECT. 4] SOUND SCATTERING BY MARINE ORGANISMS 523 equation (38) may be re-written ^ ' Ec ^ ' d\ogz ^ ' Early in the 1950's reverberation from deep scattering layers was found to be strongly frequency-dependent (Hersey, Johnson and Davis, 1952). Ac- cordingly the total scattered energy must be analyzed within bands of fre- quencies small compared with the scale of the frequency dependence, and yet broad enough and so designed that their response time, which controls the effective pulse width t, preserves T<^t. The total energy, Et, oi the shock wave was computed by Machlup from the empirical formulae given by Arons (1954) P = 2.16(104)(lf/3/P)i-i3p.s.i. 0 = 58(10-6)lf'/3(lf'/3/i2)-o.22sec, where P is the peak pressure, ® is the decay time of the shock wave, which is approximately exponential, W is the weight of TNT in pounds, and R is the mean range in feet of a region for which E is sought. It is Et = 4:77 \ F{t) dr - 477 i?2 — ^-^r/e ^^ J pc Jo pc 2 The filter transfer characteristic was taken to be Y{w) = bl[b + iico-wo)], where co is angular frequency, ojq the "center" angular frequency of the filter, and the half-power (3-dB-down) points are cuo ± b. That is, the bandwidth of the filter is 6/77 cycles per second. The total energy, E, of equation (38) may be considered the fraction of the total energy that would be passed by the filter. This fraction, A/^Et, is computed as the square of the output voltage of the filter for an input having the same time dependence as the shock wave. Mach- lup's approximation (Machlup and Hersey, 1955) is where a = b. 1 \2 tan 8 = {-^ — b\ OJQ, coo = Stt/q. Then E of equation (38) may be written E = A/^Et' 524 HERSEY AND BACKUS [CHAP. 13 The signal available for analysis is the output voltage of the transducer. This electrical signal may be fed to filter circuits or it may be recorded, for example, on magnetic tape. Let it be e{t). Then remember that I{t) is given by m = pHt)ipc. Now e{t) = Bf^pit), where B/^ is the factor of proportionality between the acoustic pressure of a wave, arriving along the direction of greatest sensitivity of the transducer, and the open circuit voltage generated by the transducer. Therefore, m = eHt)lpcBf^^. J'hus, remembering that z = ctl2, equation (38) becomes The quantity e{t) is measured by comparing recordings of the reverberation with recordings of a "white-noise" calibration signal introduced across a resistor in series with the transducer. Other quantities are given or measurable ; thus m{z) is obtainable with (40). Machlup and Hersey (1955) used this result to cpmpute the reverberation due to several different hypothetical distributions of scatterers. They also reported the results of computations of a series of oscillographic recordings made directly at sea from the transducer through band-pass filters. A number of tape recordings made since 1953 have been analyzed recently by the method of equation (40) ; these results, which are pertinent to the general problem of frequency dependence, are now discussed. 5. Sound- Scattering Observations The data on which this discussion is based were collected in the areas in- dicated in the index chart of Fig. 14. The observations used for these analyses were made with a small charge (| pound of T.N.T. or 2| pounds of tetratol) as the sound source fired at shallow depth (three feet or less). The hydrophone was in most instances the QBG transducer. Fig. 15 shows directional properties at 5 and 27.5 kc/s and the broad-band receiving response of this transducer along its principal axis. The preamplifier was the "Suitcase" amplifier (see Hersey, 1957) and the recorder was a Magnecord or Ampex magnetic tape recorder used for high-speed direct recording. For examination of frequency behavior, the tapes were played to a Kay Electric Company Sonagraph or Vibralyzer. These instruments present the spectrum of a transient sound as relative blackening of a spark-type recording paper in which the plane co-ordinates are frequency and time (Koenig, Dunn and Lacy, 1946) (Fig. 16). The samples of scattered sound are analyzed over SECT. 4] SOUND SCATTERING BY MARINE ORGANISMS 525 and over at different gain settings ; the outline of the blackened areas of the resulting recordings give contours of equal wave-analyzer response (Koenig and Ruppel, 1948). Fig. 17 shows a sunset sequence of observations analyzed in this manner. For analysis by the method of equation (40), the tapes were played to an oscilloscope through Rayspan Filters (heterodyne filters with a constant bandwidth of 100 c/s) and photographed. A. Frequency Dependence of Sound-Scattering in the Sea Hersey and Backus (1956) noted that reverberation from certain scattering layers exhibits a frequency variation which is related to depth migration (i.e. Fig. 14. Index chart showing sources of data used for analysis of frequency-dependent characteristics of deep scattering layers. change in hydrostatic pressure) in such a manner as to suggest that the scat- terers are highly compressible. In accordance with this hypothesis, particular attention has been paid to the frequency behavior of the reverberation from scattering layers. Within the frequency range of the equipment (100 c/s to 32 kc/s) we find that reverberation from deep scattering layers has the following frequency-dependent characteristics : (1) At a given time, layers appearing at different depths exhibit strong peaks of scattering intensity at correspondingly different frequencies. These intensity peaks are of the order of 10 dB or more above background. (2) At a given time each layer exhibits peak scattering, in all but extremely questionable cases, at only one frequency band within the range under examina- tion. These peak frequencies have been found to range from about 25 kc/s down to about 2.5 kc/s. 526 HERSEY AND BACKUS [chap. 13 (3) For most layers the peak frequency increases with increasing depth during the course of a diurnal migration. This phenomenon, which may be referred to as "frequency migration", has been found to vary widely with different layers and in different parts of the ocean. Layers which have been observed in the western North Atlantic fall loosely into three categories with respect to peak frequency. These may be identified as "low-frequency", meaning about 3-5 kc/s, or the lowest frequency at which peak scattering has been found; "high-frequency", referring to the highest frequencies at which peaking has been closely examined, or about 20 kc/s ; and "mid-frequency", meaning frequencies between these extremes, or about 15 kc/s. The peak frequency of all layers varies diurnally in some manner so that the identifying frequencies just mentioned must be regarded as median values for convenience of description. Fig. 18 shows peak frequency and corresponding depth versus time relative to sunset for a typical migration in waters south of Nova Scotia north of the Gulf Stream (area A of Fig. 14). In these waters depth migration and 330° 340° 350° ( 3° 10° 20° 30° 320° \^--^^ dB RELATIVE TO MAXIMUM K. ^ 3 10° \yy\ \ l—^ 0_/ / ^;< \y\ 300° /\ /\\ VJp2 vV 290° \\ 280° /""""~~yC^"NOvv\j %u//>rJ\r' _Ur 2 70° 2 60° 4^ 2 50° ^ 2: UJ 0 50 100 150 200 250 300 350 400 450 500 550 Ji'^. Fig. 17. A sequence of sunset observations showing scattering as a function of depth and frequency. Contours of equal sovmd level are two decibels apart, with lightest areas denoting highest levels. Time of day of each observation is indicated in large numerals above the upper right-hand corner of each record. The observations were made in area A of Fig. 14; the low-frequency part of the 1906 record appears, with an ex- panded frequency scale, in Fig. 16. 18— s. I 530 HERSEY AND BACKUS [chap. 13 bubble hypothesis, this is conflicting evidence. Either the depth behavior or the frequency characteristics of this layer must be, in fact, otherwise than our analysis methods indicate ; perhaps both are. It should be noted that the frequency scale of our analysis was insensitive to small changes in the region between 3 and 6 kc/s. This is because the frequency scale was linear and designed to cover a band from 0 to 30 kc/s. In addition to the problem of recognizing frequency migration, the correct interpretation of depth migration is difficult at these lower frequencies. We 100 200 300 400 \ \ \ •^ K A V V E B -40 -20 ' <-20 -40 SUNSET E ^ ft OP ■^ 8 PEAK ->^ '■-■w>' / -40 -20 *20 +40 SUNSET TIME IN MINUTES RELATIVE TO SUNSET AREA A- SOUTH OF NOVA SCOTIA AUGUST 1954 Fig. 18. Depth and peak frequency versus time of observation relative to sunset of the principal scattering layers in area A of Fig. 14 (cf. Figs. 16 and 17). For the low- frequency layer (B), the figure shows the depth of the greatest intensity of scattering ("peak") and the apparent depth where scattering begins ("top"). have always used a weakly directional receiver (see Fig. 15a). If the layer is patchy, which we believe it is, such a transducer will not clearly define the limits of the layer with each observation, since it will be sensitive to side echoes from nearby patches. The layer can be better defined if the sampling rate is high enough to identify side echoes, as in echo-sounding. Unfortunately the explosive technique has limited the sampling rate to about one observation every two minutes. Quite apart from problems of discriminating variations of frequency in the layer and discriminating against side echoes, there is yet another cause for SECT. 4] SOUND SCATTERING BY MARINE ORGANISMS 531 ambiguity. Higher-frequency scattering layers, which clearly migrate both in depth and frequency, may during their migrations occupy depths above, and over-lapping with, the daytime depth of the low-frequency layer and further- more may scatter sound over a frequency band including that of the low- frequency layer (see Fig. 17). This phenomenon may produce an illusion of migration in the low-frequency layer. B. Measurement of Volume Back-Scattering Some scattering-layer observations in the western North Atlantic have been analyzed by the method of Machlup and Hersey (1955) [see above, pages 522- 524, especially equation (40)], by which the echo intensity (function of time) is translated into volume back-scattering coefficient (function of depth). The DEPTH (m) 300 400 600 800 1000 -70 'e 2 -80 0.2 0.3 0.4 0.6 0.8 TIME AFTER SHOCKWAVE (sec) Fig. 19. Volume back-scattering coeflficient versus time (depth), plotted on a logarithmic scale, measured at 15 kc/s ; an example where m in the scattering layer is large compared with surrounding volumes. computation was carried out for frequencies from 3.5 kc/s to 21.5 kc/s. The peak values of m range from about 10"^ m-i to 10-^ m-i. Fig. 19 shows a graph of m versus time after the shock wave, for a case where m in the layer is high relative to surrounding volumes. These values of m vary over about the same range as those reported by NDRC (1946a) except that smaller peak values of m have been found in the more recent study. The problem remains to relate measurements of volume scattering to in- formation about individual scatterers. When the individual scatterers are distributed more or less at random and when their scattering cross-sections, (t/477-, are a small fraction of the total area insonified, then the back-scattering due to a number of scatterers may be calculated by adding independently the scattering intensity due to each scatterer. Therefore, the total volume 532 HERSEY AND BACKUS [CHAP. 13 back-scattering coefficient, m{z,f), measured at depth z and for frequency/, can be expressed as follows, MzJ) = IMz)^' (41) where ai{f)l4:7T is the scattering cross-section of the i-th type of scatterer at frequency/, and ni{z) is the number of scatterers (per unit volume) of this kind at depth z. Clearly, in order to obtain population densities, ni{z), from measure- ments of volume back-scattering, ni{z), it is necessary to obtain as much information as possible about the scattering properties of each individual. From estimates of individual scattering cross-sections one then uses the above relation to estimate population densities from the measured volume scattering. Thus far, to our knowledge, no one has attempted to cope with estimates other than those based on the assumption of a single kind of scatterer, all members of the scattering layer having identical cross-sections. Euphausids and fishes with swim-bladders can be considered to some profit. As mentioned above, an upper limit for the probable scattering cross-section of a euphausid, ct/477~ 2 x 10~i2 jn2^ can be derived from Bridgman's measurement of the compressibility of oil from certain euphausids. From equation (41) the population density implied by m~10-9 to IQ-^ m-i varies from 500 to 500,000 individuals per cubic meter. Available information on the density of euphausids is sum- marized by Gushing and Richardson (1956). This shows that, with the excep- tion of the occasional unusual situation (in which a few hundred individuals per cubic meter have been observed), euphausid populations are much below the above requirements, being more the order of 10-15 per cubic meter of water. Nevertheless, these authors correlate net hauls with echo-sounder recordings of near-surface reverberation to suggest that euphausids (about 200 per cubic meter) are the scatterers in their observation. Using the value of CT/47r above it is easy to show that euphausids as densely packed as 200 per cubic meter could easily be recorded at shallow depth by echo-sounders operating at 10 kc/s. Assuming that the layers for which we have computed values of m consist of bubble scatterers, we have estimated the number of individuals required for peak values of m. The back-scattering cross-sections at resonance were obtained by computing the resonant-bubble radius, R, for the indicated frequency, taking the theoretical damping constant, §, from Fig. 13, and substituting in the formula a/47r = R^I8^. Both our results and those quoted in Machlup and Hersey (1955) suggest that the population density in layers showing a resonant frequency is not much greater than the count, made by Johnson et al. (1956), discussed above. The greatest density we have so far computed is 1.7 x 10~2 scatterers per cubic meter. SECT. 4] SOUND SCATTERING BY MARINE ORGANISMS 533 When the individual scatterers are not distributed at random, such as might happen in schools offish, the possibility exists that extra strong back-scattering may occur at selective frequencies which could depend on the spacing of the scatterers. Such circumstances may explain the observations of very low- frequency scattering that have occasionally been reported (e.g. Burling, 1956) and perhaps also those of Ellis and Winterhalter (1956). The present authors have recorded strong scattering in the shallow water of Vineyard Sound, using the seismic profiler (to be described by Hersey in Volume 3) with a narrow band- pass filter setting centered at 250 c/s. Simultaneously we observed the same scattering group with a 40 kc/s echo-sounder. Although we did not sample this scattering group, we have found on other occasions that ones of similar appear- ance on the echo-sounder record are schools of fishes of a foot or two in length. 6. What is "The Deep Scattering Layer" ? We cannot avoid a summary statement giving some tentative answer to the now time-worn question, "What is 'the deep scattering layer'?" In proposing that bathypelagic fishes with gas-filled swim-bladders might be the source of the observed sound-scattering, Marshall (1951) lists four requirements which an organism must meet before it can be considered seriously in this role. (1) The organism must be a widely distributed one. (2) The organism must be shown to exist in concentrations at the appropriate depth during the daytime. (3) The organism must show pronounced powers of diurnal vertical migration. (4) The organism must have the ability to refiect sound. Specifically, the product of the organism's scattering cross-section and its population density must lead to reverberation levels in agreement with those observed. Numerous studies, but particularly a number cited here made in connection with the scattering-layer problem itself, show that several types of animals fulfill the first three requirements. Conspicuous among these animals are bathypelagic fishes, euphausids, and other crustaceans — those of the sergestid type, for instance. Probably many other animals, about which good information is presently lacking, will also be found to meet these requirements — squids, for example. The acoustic scattering cross-section of fishes, crustaceans, squids and other sorts of "fleshy" animals without air-bladders are subject to estimate well within a factor of 10 for cases where some such animal of similar size-to-wave- length ratio has been measured. This includes animals from about 2 to 30 cm in length at frequencies from 10 to 30 kc/s. The acoustic scattering cross-sections of fishes or other animals ^ containing gas bubbles are subject to accurate estimate (within about 20%) if the size and shape of the gas bubble at one atmosphere of pressure are known. 1 In addition to fishes, a few other bathypelagic animals enclose gas bubbles ; for instance, certain siphonoiDhores and the cephalopods, Spirula and Nautilus. These animals inhabit that part of the water column in which deep scattering layers are observed but very little is known about their distribution, vertical migrations and abundance. 534 HERSEY AND BACKUS [CHAP. 13 The scattering cross-section of euphausids, and other such small animals without air-bladders, is sufficiently low as to make it highly improbable that they are the scattering agents in deep scattering layers (as we have argued earlier). It is only in rare cases that they occur with the necessary density. The larger air-bladderless animals mentioned (large crustaceans, squids, fishes and others), as well as animals with air-bladders, all have scattering cross-sections sufficiently large to make the population requirements consonant with observed or, at least, probable populations. This is to say that observations of volume back-scattering coefficient (m) are not helpful in deciding which of these larger animals comprise deep scattering layers. Even though the data are imperfectly analyzed, spectrum analyses of broad band scattering at a number of points over the western North Atlantic show that scattering layers there do have resonant peaks, with one peak to each 1 yer in the frequency range examined (1-30 kc/s). This indicates, but does not finally demonstrate, that the principal constituents of these layers are fishes with gas-filled swim-bladders. Positive identification of the scatterers will have to be accomplished by a combination of acoustical and other observational techniques in which the animal or its image is captured during the act of sound- scattering, 7. Ideas and Miscellaneous Observations A. Submarine Illumination and the Vertical Migration Two studies have been made in which the movements of deep scattering layers during their vertical migration have been compared with direct measure- ments of submarine illumination. In one of these studies scattering layers over the San Diego trough were observed with a 17.5 kc/s echo-sounder and with explosive sound sources, while light measurements were made with a submarine photometer. In this study the layers exactly followed the movements of certain isolumes save for a lag at the beginning of the evening ascent (Kampa and Boden, 1954). Somewhat different results were obtained in a similar study using a submarine photometer and a 12 kc/s echo-sounder in deep water off the coast of southern New England. A scattering layer, whose midday depth was about 380 m, split into two elements near sunset, both of which moved surface wards more rapidly than certain isolumes, and during their migrations reached levels that were illuminated 100 times more brightly than their midday level (Clarke and Backus, 1956). B. Water Temperature and Deep Scattering Layers That the midday level of deep scattering layers cannot be correlated with any conspicuous feature of the temperature-depth profile has already been noted. However, Moore (1950) observed during hours of daylight shoalings and deepenings of a principal deep scattering layer, which accompanied shoalings and deepenings of certain isotherms as a ship passed through cold SECT. 4] SOUND SCATTERING KV MARINE ORGANISMS 535 and warm bodies of water in the region of the (lulf Stream. It was not possible to say in these cases, however, that the same animal populations were repre- sented in the two water types. The change in ambient temperature experienced by a layer during the course of its diurnal migration may be considerable. A layer studied by Clarke and Backus (1956) in deep water south of New England moved up in the evening from a depth- of 345 m, where the temperature was about 7°C, to near 40 m, w'here the temperature was about 17.5°C, thus experiencing a change of more than lO'^C in a period of about 1| h. If this had an effect on the activity rates of the scatterers, as it should, it was not manifested in rate of the vertical migration, which was, as is usual, virtually uniform throughout the post-sunset period. C. Swim-bladder Function and Deep Scattering Layers One objection which has been continually raised to the hypothesis that fishes with gas-filled swim-bladders are the principal constituents of deep scattering layers is the difficulty with which the contents of the bladder would be managed during the extensive vertical migration, especially if the hydro- static function of the bladder is to be realized, i.e. if the fish is to maintain neutral buoyancy at both deep and shallow level. During the evening ascent gas must be continually absorbed. (The swim-bladder in these bathj^pelagic fishes is of the closed or physoclistous type ; that is, there is no opening between the bladder and the gut which might allow •the fish to vent gas.) During the morning descent gas must be rapidly generated to fill the shrinking chamber. That such fishes be capable of absorbing and generating gas at change-of-depth rates observed for migrating scattering layers has been questioned on physio- logical grounds (Kanwisher and Ebeling, 1957). However, Marshall (1960), in an extensive and fascinating account of the swim-bladder in bathypelagic fishes, offers ample anatomical and biological argument for the use of swim- bladder as a hydrostatic organ in vertical migrants of the upper 1000 m of the ocean. In such fishes both the gas-absorbing and gas-generating structures of the swim-bladder are very elaborately developed. The alternative in such a vertically migrating fish is to be at neutral buoyancy at near-surface level and simply to let the swim-bladder gas compress as the fish migrates to a depth where the animal would be negatively buoyant. In certain eastern Pacific myctophids, Kanwisher and Ebeling {op. cit.) observed (from the composition of gas in the bladder) that oxygen had been secreted at depth. These authors ask how the fish could have secreted the right amount of gas at depth for achieving neutral buoyancy at shallow level. That this is what these fishes do seems improbable and it is more parsimonious to reason that adjustments in the volume of gas are made during the upward migration. We have determined the relationship between peak sound-scattering fre- quency and depth during the vertical migration of deep scattering layers in three instances (see p. 525). The relationship that would obtain were the swim- bladder and its gas passively responding to changes in ambient pressure 536 HERSEY AND BACKUS [CHAP. 13 [FIFo= (P/Po)^''^] has been observed during both a sunrise and a sunset migra- tion. This seems to imply that retention of the swim-bladder is worthwhile even though the fish is at neutral buoyancy only at the top of its depth range. The relationship that would obtain were the swim-bladder contents being adjusted to maintain neutral buoyancy [P/Po = (P/Po)'"^] has been observed during a sunset migration. Thus it seems that some bathy pelagic fish is able to absorb swim-bladder gas at a rate which maintains its neutral buoyancy during an ascent of 200 m in about 1 h. It is probable (though we have not yet demonstrated it) that this animal can effect the converse operation during the sunrise descent. Since some of the steps in the arguments that these per- formances are physiologically improbable are extrapolations from the func- tionings of surface-living fishes, it seems proper to us that the problem now be returned to physiologists for further consideration. D. Deep Scattering Layers and Energy Distribution in the Ocean Scattering-layer study suggests that a very sizeable portion of the macro- scopic animals in the upper 1000 m of the water column engage in an extended diurnal vertical migration. The effect of these migrations in distributing energy downward from the euphotic zone throughout the upper 1000 m must be considerable and has not been generally appreciated. As the animals in these layers are identified and their populations are measured, estimates of this energy flow can be made. Since the speed and direction of currents may vary widely throughout the upper 1000 m of the water column, the horizontal transports of energy due to these migrations may also be important considera- tions, E. Extending Acoustical Observations At present there are instrumental limitations (especially in transducers) which have restricted studies of the frequency dependence of sound-scattering in the sea to the band from 1 or 2 kc/s to around 30 kc/s. Observations made in this band indicate that these studies could be pursued profitably down to the region of a few hundreds of cycles and up to the neighborhood of 100 kc/s. The most satisfactory means of generating broad-band sounds at present is with small explosive charges. It is evident, however, that many studies of the frequency-dependence of sound-scattering could be better done with devices generating complex sound signals repetitively, such as those recently come to use in seismic studies (see Volume 3). It only remains to make these devices efficient enough sound-generators so that they may be used with such relatively weak reflectors as scattering layers. Broad-spectrum scattering observations could be extended to many parts of the oceans and the generalizations drawn from comparing them should be useful in delimiting the faunal provinces of the upper 1000 m of the high seas. SECT. 4] SOUND SCATTERING BY MARINE ORGANISMS 537 F. Sound-Scatterers as Markers of Water Boundaries Aggregations of scatterers often accumulate at the boundaries between waters of different properties. These aggregations may occur at such places for a number of reasons. Since the i^hysical properties of the water are changing rapidly with depth at such levels, the probability that an organism may find an optimum set of conditions here is somewhat increased. Furthermore, an animal may be '^trapped" at such a level, forced downward by its negative response to light, let us say, but limited in its downward movement by a sudden decrease in temperature. Finally, animals may simply come to rest in such places at the very level at which they find themselves neutrally buoyant. In any event aggregations occur at such points and they may be used to map the boundaries which they mark. Some success has been had, for instance, in relating the distribution of sound-scatterers to the distribution of Atlantic and Mediterranean water-masses in the Strait of Gibraltar (Frassetto, Backus and Hays, in press), and Weston (1958) has described certain properties of the summer thermocline in the North Sea from echo-sounder records of scatterers accumulating there. Such a technique should be useful in studying the properties of internal waves. From Fig. 4a there is a suggestion that a wave is travelling in the region of the temperature inversion shown in Fig. 4b. References Anderson, V. C, 1950. Sound scattering from a fluid sphere. J. AcouM. Soc. Amer., 22, 42&-431. Anderson, V. C, 1953. Wide band sound scattering in the deep scattering layer. Scripps Inst. Oceanog. Ref. 53-36, 1-35 (unpubhshed manuscript). Arons, A. B., 1954. Underwater explosion shock wave parameters at large distances from the charge. J. Acoust. Soc. Amer., 26, 343-346. Backus, R. H. and H. Barnes, 1957. Television-echo sounder observations of midwater sound scatterers. Deep-Sea Fes., 4, 116-119. Baker, A. de C, 1957. Underwater photographs in the study of oceanic squid. Deep-Sea Res., 4, 126-129. Balls, R., 1948. Herring fishing with the echometer. J. Cons. Explor. Mer, 15, 193-206. Balls, R., 1951. Environmental changes in herring behaviour : a theory of light avoidance, as suggested by echo-sovinding observations in the North Sea. J. Cons. Explor. Mer, 17, 274^298. Barham, E. G., 1957. The ecology of sonic scattering layers in the Monterey Bay area. Hopkins Marine Station, Stanford University, Tech. Rep. 1, x-|- 182. Batzler, W. E. and E. C. Westerfield, 1953. Sonar studies of the deep scattering layer in the North Pacific. U.S.N. Electron. Lab., Rep. 334, 1-22. Boden, B. P., 1950. Plankton organisms in the deep scattering layer. U. S. N. Electron. Lab., Rep. 186, 1-29. Breslau, L. R. and H. E. Edgerton, 1958. The luminescence camera. J . Biol. Phot. Assoc, 26, 49-58. Breslau, L. R. and H. E. Edgerton, 1959. The interruption camera. Woods Hole Oceanog. Inst. Ref. No. 59-27, 1-9. Burling, R. L., 1956. Some unusual reflections of sound in the ocean. Geophysics, 21, 765-770. Clarke, G. L. and R. H. Backus, 1956. Measurements of light penetration in relation to vertical migration and records of luminescence of deep-sea animals. Deep-Sea Res., 4, 1-14. 538 HERSEY AND BACKUS [CHAP. 13 Clarke, G. L. and C. J. Hubbard, 1959. Quantitative records of the luminescent flashing of oceanic animals at great depths. Limnol. Oceanog., 4, 163-180. Clarke, G. L. and G. K. Wertheim, 1956. Measurements of illumination at great depths and at night in the Atlantic Ocean by means of a new bathyphotometer. Deep-Sea Res., 3, 189-205. Cousteau, J.-Y., 1954. To the depths of the sea by bathyscaphe. Nat. Geog. Mag., 106, 67-79. Craig, R. E., 1955. Appendix 2, in Hodgson and Fri6riksson, 1955. The use of echo- sounder in fish-location — a survey of present knowledge, with notes on the use of asdic. Rapp. Cons. Explor. Mer, 139, 1-49. Gushing, D. H., 1955. Some echo-sounding experiments on fish. J . Cons. Explor. Mer, 20, 266-275. Gushing, D. H., F. Devoid, J. C. Marr and H. Kristjonsson, 1952. Some modern methods offish detection: Echo sounding, echo ranging and aerial scouting. F.A.O. Fisheries Bull., 5, 95-119. Gushing, D. H., A. J. Lee and I. D. Richardson, 1956. Echo traces associated with thermo- clines. J. Mar. Res., 15, 1-13. Gushing, D. H. and I. D. Richardson, 1955. Echo sounding experiments on fish. Min. Agric. Fish., Fish Invest., Lond., ser. 2, 18, 1-34. Gushing, D. H. and I. D. Richardson, 1956. A record of plankton on the echo-sounder. J. Mar. Biol. Assoc. U.K., 35, 231-240. Devin, G., Jr., 1959. Survey of thermal, radiation and viscous damping of pulsating air bubbles in water. J. Acoust. Soc. Amer., 31, 1654r-1667. Dietz, R. S., 1948. Deep scattering layer in the Pacific and Antarctic Oceans. J. Mar. Res., 7, 430-442. Duvall, G. E. and R. J. Christensen, 1946. Stratification of sound scatterers in the ocean. J. AcoiLSt. Soc. Amer., 18, 254. (Abstract of a paper presented at the thirty-first meet- ing of the Acoust. Soc. Amer., 1946.) Ellis, L. G. and A. G. Winterhalter, 1956. Unusual reflection events in offshore seismic work. Geophysics, 21, 755-764. Eyring, G. F., R. J. Christensen and R. W. Raitt, 1948. Reverberation in the sea. J. Acoust. Soc. Amer., 20, 462-475. Frassetto, R., R. H. Backus and E. Hays, in press. Sound-scattering layers and their relation to thermal structvire in the Strait of Gibraltar. Deep -Sea Res. Herdman, H. F. P., 1953. The deep scattering layer in the sea: Association with density layering. Nature, 172, 21b-21&. Hersey, J. B., 1957. Electronics in oceanography. Advances in Electron. Electr. Phys., 9, 239-295. Hersey, J. B. and R. H. Backus, 1954. New evidence that migrating gas bubbles, probably the swimbladders offish, are largely responsible for scattering layers on the continental rise south of New England. Deep-Sea Res., 1, 190-191. Hersey, J. B., H. R. Johnson and L. G. Davis, 1952. Recent findings about the deep scattering layer. J . Mar. Res., 11, 1-9. Hersey, J. B. and H. B. Moore, 1948. Progress report on scattering layer observations in the Atlantic Ocean. Trans. Amer. Geophys. Un., 29, 341-354. Hodgson, W. G. and A. FriSriksson, 1955. Report on echo-sounding and asdic for fishing purposes. Rapp. Cojxs. Explor. Mer., 139, 1-49. Hunt, F. v., 1954. Electroacoustics : The Analysis of Transduction, and its Historical Background. Harvard Monographs in Applied Science, Number 5. Harvard Univer- sity Press, Cambridge, U.S., 1954. Johnson, H. R., R. H. Backus, J. B. Hersey and D. M. Owen, 1956. Suspended echo- sounder and camera studies of midwater sound scatterers. Deep-Sea Res., 3, 266-272. Johnson, M. W., 1948. Sound as a tool in marine ecology, from data on biological noises and the deep scattering layer. J. Mar. Res., 7, 443-458. SECT. 4] SOUND SCATTERING BY MARINE ORGANISMS 539 Kampa, E. M. and B. P. Boden, 1954. Submarine illumination and the twilight move- ments of a sonic scattering layer. Nature, 174, 869-871. Kanwisher, J. and A. Ebeling, 1957. Composition of the swim-bladder gas in bathypelagic fishes. Deep-Sea Res., 4, 211-217. Kanwisher, J. and G. Volkmann, 1955. A scattering layer observation. Science, 121, 108- 109. Koenig, W., H. K. Dunn and L. Y. Lacy, 1946. The sound spectrograph. J. Acoust. Soc. Amer., 18, 19-49. Koenig, W. and A. E. Ruppel, 1948. Quantitative amplitude representation in sound spectrograms. J. Acoust. Soc. Amer., 20, 787-795. Machlup, S., 1952. A theoretical model for sound scattering by marine crustaceans. J. Acoust. Soc. Amer., 24, 290-293. Machlup, S. and J. B. Hersey, 1955. Analysis of sound -scattering observations from non- uniform distributions of scatterers in the ocean. Deep-Sea Res., 3, 1-22. Marshall, N. B., 1951. Bathypelagic fishes as sound scatterers in the ocean. J. Mar. Res., 10, 1-17. Marshall, N. B., 1960. Swimbladder structure of deep-sea fishes in relation to their syste- matics and biology. Discovery Reps., Vol. XXXI, 1-222. Moore, H. B., 1950. The relation between the scattering layer and the Euphausiacea. Biol. Bull, 99, 181-212. Morse, P. M., 1948. Vibration and Sound. McGraw-Hill, New York. (Second ed.). Morse, P. M. and H. Feshbach, 1953. Methods of Theoretical Physics. McGraw-Hill, New York. NDRC Div. 6, 1946. The physics of sound in the sea. Summary Tech. Rept. 8, 1-566. NDRC Div. 6, 1946a. Principles and applications of underwater sound. Summary Tech. Rept. 7, 1-295. Peres, J. M., 1958. Trois plongees dans le canyon du Cap Sicie, effectuees, avec le bathy- scaphe F.N.R.S. Ill, de la Marine Nationale. Bull. Inst. Oceanog. 1115, 1-21. Raitt, R. W., 1948. Sound scatterers in the sea. J. Mar. Res., 7, 393-409. Rayleigh, Lord, 1945. The Theory of Sound. Dover, New York. (Second ed.). Smith, P. F., 1954. Further measurements of the sound scattering properties of several marine organisms. Deep-Sea Res., 2, 71-79. Sund, O., 1935. Echo sounding in fishery research. Nature, 135, 953. Tchemia, P., 1950. Observations d'oceanographie biologique faites par I'aviso polaire Commandant Charcot pendant la campagne 1948-1949. II. Observations sur la D. S. L. faites a bord du Commandant Charcot (campagne 1948-49). Bull, inform. Com. cent. oceanog. etude cotes, 2, 7-20. Tchemia, P., 1952. Quelques considerations sur I'etat actual du probleme de la couche diffusante profonde (D. S. L.). Bull, inform. Com. cent, oceanog. etude cotes, 4, 403-415. Trout, G. C, A. J. Lee, I. D. Richardson and F. R. Harden Jones, 1952. Recent echo sounder studies. Nature, 170, 1\-12. Tucker, G. H., 1951. Relation of fishes and other organisms to the scattering of under- water sound. J. Mar. Res., 10, 215-238. Univ. Calif. Div. War Research, 1943. Volume reverberation: scattering and attenuation vs. frequency. Rept. U50, 1-14. Univ. Calif. Div. War Research, 1946. Stratification of sound scatterers in the ocean. Rept. M397, 1-13. Univ. Calif. Div. War Research, 1946a. Forward scattering from the deep scattering layer. Rept. M398, 1-21. Univ. Calif. Div. War Research, 1946b. Studies of the deep scattering layer. Rept. M445, 1-10. Weston, D. E., 1958. Observations on a scattering layer at the thermoclLne. Deep-Sea Res., 5, 44-50. 14. SOUND PRODUCTION BY MARINE ANIMALS W. E. ScHEViLL, R. H. Backus, and J. B. Hersey 1. Introduction Because sound is rapidly and efficiently transmitted by water, hearing is often of greater utility than the other senses to active, wide-ranging marine animals. Although the transfer of information by light is practically instan- taneous, vision is useful only at ranges up to a few tens of meters in the clearest of surface waters because light is rapidly attentuated in water. Besides, it is night half the time. By using the chemical senses (taste and smell), exceedingly low concentrations of various substances are detected by many aquatic organisms, but the rate of transmittal between source and sensor in these systems is dependent on mixing processes, which in the ocean proceed at velocities of a few meters per second at most. The lateral line system of the elasmobranchs and bony fishes appears to be a turbulence detector which, while highly directional, operates at very short range. It is by hearing that marine animals get information from great range and with little time delay. ^ Given effective hearing, the development of sound-producing mechanisms seems a logical way in which to extend the usefulness of this sense. That is, while the simple awareness by a species of the sounds of predators and prey is of survival value, there are other ways in which sound may be put to use if it can be purposefully self -generated. It is the intention in this chapter to consider the sounds produced by marine animals — what these sounds are like, how they regulate the lives of the animals which make them, how these sounds have been and might be studied, and how they can be used by oceanographers to elucidate other problems. We are mainly restricting ourselves to the use of sound by marine animals in their natural environment, with only occasional allusions to laboratory studies of captives. This is not to underrate the contributions that such investigations may make to our understanding of the field problems, but is because we are not yet certain how to separate the artificial influences of captivity. 2. History As has often been pointed out, it was well known to the ancients that certain animals made sounds under water. The writings of Aristotle (fourth century B.C.), and subsequently those of Athenaeus and Pliny, describe sounds 1 The "absorption coefficient" of underwater acoustics is analogous to the ''extinction coefficient" of underwater optics, IJIq = e~^K where Iq is the intensity at a point of observa- tion, I is the intensity at a second point of observation further removed from the source by the distance I (by one meter in our examples), e is the base of the natural logarithm and k is the coefficient. Optical extinction coefficients vary from about 7 x 10~2 (clear oceanic water) to about 3.5 x 10"^ (turbid coastal water) for that part of the spectrum best trans- mitted. The absorption coefficient for sound of five kilocycles per second is about 6 x 10~5. Thus the rate of absorption of light by sea-water is about 1000 to 5000 times greater than that of sound of mid-frequency. [MS received Septetnber, 1960] 540 SKCT. 4] SOTNI) I'KOnrCTION n\ MAKINK ANIMALS 541 from tlic three gr()U])s of marine organisms which liave commonly been recog- nized in modern times as lieing soniferous — the cetaceans, the bony fishes, and the crustaceans.' Locating scliools of fish by underwater Hstening with the unaided ear is an ancient fishing practice still in use today in southern Asia (Westenberg, 1953: Kesteven, 1949; and Rand, 1952). Talking fishes are naturally fancy-capturing and many accounts of western European explorers and early settlers of the 16th. 17th, and 18th centuries refer to such wonders. John Smith in 1623, William Penn in 1685, de Bienville, founder of New Orleans, in 1699, and John Lawson, British surveyor of North Carolina, in 1714, did so from the waters of eastern North America. About the middle of the nineteenth century zoologists began to pay attention to the sounds of fishes, and in the next few decades a considerable literature grew, comprising lists of sound-making species, word descriptions of their sounds, anatomical investigations of sound-making structures, and some speculation about the function of these sounds. The papers of Dufosse (1874), Sorensen (1884), Smith (1905), Tower (1908) and Greene (1924) are representa- tive of the best of these efforts. Since many of the sounds noted were made by fishes under duress, it was generally concluded that sounds were mainly of a defensive nature. This popular thesis still remains to be demonstrated. Crusta- cean sounds were largely neglected, probably because they are harder to hear with the unaided ear and less distinctive than fish sounds. Although reported by whalers and other mariners, cetacean sounds received little attention by the naturalist because of the practical difficulties of listening near porpoises and wiiales, but also because of the false deductions of comparative anatomists who felt that the lack of vocal cords in the cetacean larynx was ample demonstration of the muteness of these animals. The time of World War II may be reckoned another important way -point in this study. The great increase in the amount of underwater listening and the attendant refinement of equipment, coupled with the practical necessities of understanding certain noises related to marine animals, all in connection with military oj^erations, greatly advanced this part of marine biology. Most im- portant of all, perhaps, was the introduction of the problem to a group of scientists — the acousticians — who had not considered it before. Thus Horton (1957) says: "although this fact [that certain marine animals make sounds] had long been common knowledge to fishermen and biologists, its announce- ment to w'orkers in the field of acoustics was greeted with some astonishment". In any event, it was during the war that underwater animal sounds were first 1 Recently sound-production has been attributed to a fourth group of niarine animals — the sharks. Shishkova, a Russian fishery technologist, reports "rumbling" sounds from a captive spiny dogfish, Squalus acanthias, as it seized its food (Shishkova, 1958). The spectrum of these sounds is noted as having frequency components from 500 to 1600 c/s, which does not match well with the verbal description of the sounds. We have made limited observations of this sj^ecies for the purpose of verifying these data, but have heard only adventitious sounds. Another Russian worker reports a wide variety of sounds recorded in nature which are attributed to the same species (Azhazha, 1958), but it is apparent from his account that he was hearing some species of porpoise. 542 SCHEVILL, BACKUS, AND HERSEY [CHAP. 14 measured and the techniques of the acoustician introduced to the marine biologist. Good progress has been made during the years since the war in describing identified sounds in physical terms, but the few most interesting studies con- cern the functions of these sounds in the lives of the animals making them. Although the descriptive catalog of sea-animal sounds is far from complete, it is in the use of these sounds to the animal and in their use to the marine eco- logist in pursuing other problems through them that the excitement of the next few decades in this field lies. 3. Instrumentation Instrumental aids to underwater listening are wholly developments of the past century. So far as we can discover they have been used for studying sounds of marine animals only since about 1940. Just as microphones have been developed to convert sound in air to electrical signals, so hydrophones have been developed for the same purpose in water. The question is often asked : do sounds reproduced in air by earphones or loudspeakers sound as they would under water were we to hear them directly? We are familiar with the same problem respecting microphones, and we have pursued it relentlessly in our current enthusiasm for "hi-fi" (high fidelity) sound systems. The hydrophone requirements are essentially the same. Sound waves in air or water are oscil- latory wave motions and variations of pressure of the medium. The hydrophone must convert the motion or the pressure variation into an electrical signal which is accurately proportional to whichever effect it is sensitive to. Since the motions and pressure variations are proportional to each other it is immaterial which effect is chosen. Once the electrical signal has been generated it may be amplified and fed to earphones, a loudspeaker or a recorder. If the amplification and conversion of the electrical signal is faithfully achieved, then the resulting sound will sound as it would have under water. A second well-worn question is : how loud is a sound under water? Since our natural environment is not under water, we cannot truthfully answer from personal experience. Loudness is partly a function of the receiver and partly of the sound. Sounds of marine animals can greatly exceed the local ambient noise level, that is, they form a sharp contrast with their background. The pistol prawn (snapping shrimp) is known to stun its prey with a sharp sound. Animal sounds of the open ocean carry long distances under the right condi- tions; thus the sounds of the echoing fish (see Griffin, 1955, pp. 411-412) must have travelled nearly three miles, and were still well above background. Where soniferous animals congregate, sound intensities over a hundred times the average background have been observed (in this they are just like people). On the other hand, open ocean listening in calm weather often sounds quiet (even with the gain of the amplifier turned high), and one can only conclude that there is great variety in the loudness of underwater sound just as there is in the sounds of our own daily life. SECT. 4] SOUND PRODUCTION BY MAKINE ANIMALS 543 The practical design of hydrophones is different from that of microphones because water is so much denser and so much less compressible than air. Also the electrical parts must be waterproofed, and must be capable of withstanding high hydrostatic pressures without structural damage or loss of sensitivity. But the sensitive structures are the same. These are described elsewhere in this book. All electronic systems for observing or recording underwater sounds from animals of the sea, whether at sea or in aquaria, consist of a hydrophone, a pre- amplifier and a means of reproducing the sound as sound in air or as a graph of wave motion or pressure versus time. Earphones or loudspeakers have been used for immediate sound reproduction, while various recorders, disc, wire, or magnetic tape, have been used for "storing" the sound for later reproduction. Equipment available for graphical representation of sounds includes the various oscillographs, sound level recorders, and wave analyzers (see Chap. 12; also Beranek, 1949, chap. 12, for extended discussion of methods and analysis instruments). The hydrophone, pre-amplifier, and the recorder or reproduction instrument each have a limited range of amplitude and frequency (or response time) over which they perform faithfully. The central problem of instrument design is to fit the range of faithful response to the needs of the investigation. Thus far we have depended mostly on instruments designed for other purposes, particularly those intended for sounds within the human audible range and those intended either as naval underwater detection devices or as development aids to the latter. After the war observations at sea continued to be made with surviving military hydrophones. The Brush AX-58 hydrophone was popular because it was designed for low self-noise in the human audible range ; its self-noise is somewhat less than the ambient noise in open sea at sea state 0 (Hersey, 1957). Commonly these observations consisted of long listening vigils from small research ships or small craft that had been silenced as much as possible by lying to and turning off" ship's power, the electronic equipment needed for listening and recording being operated on batteries. Some disc recordings were made, but were discontinued in favor of tape recording by 1950. Analysis of the transient sounds thought to be of animal origin continued to depend on subjective judgment of the quality of the sounds and what are best described as gropings with oscillographic recording. These latter never proved out- standingly useful because the presumed animal sounds were often not more intense than the accompanying broad-band background. Furthermore, oscillo- scope photography in that day was cumbersome and was simply not available during the more significant early observations. All in all, the few years prior to 1950 brought many intriguing first observations, nearly all of which were made with inadequate recording and analysis equipment. Since 1950 only a few individual scientists have maintained strong interest in observations at sea. Much of their work has continued to be limited by inadequate hydrophones. High quality hydrophones have not yet been designed 544 SCHEVILL, BACKUS, AND HERSEY [chap. 14 specifically for studying animal sounds ; many research programs continue to depend on the AX-58, its successor the Brush AX- 120, or similar audio- frequency transducers. Within their limited frequency range (see Fig. 1 ; this and Fig. 2 are calibrations by the U.S. Navy Underwater Sound Reference BRUSH AX-58 HYDROPHONE (OPEN-CIRCUIT VOLTAGE) 1 A ^^— -^^^ ^ t1 1 ^^ \ FREQUENCY (kc/s) Fig. 1. Response of a representative Brush Development Co. AX-58 hydrophone. (After Hersey, 1957, Fig. 8b. By courtesy of the Journal.) MASSA AX- 40 HYDROPHONE (OPEN - CIRCUIT VOLTAGE) < o o < FREQUENCY, (kc/sl Fig. 2. Response of a Massa AX-40 hydrophone. Laboratory) they have excellent signal-to-self-noise characteristics, but they do not respond uniformly to frequencies higher than 9 kc/s. The Massa AX-40 designed by Massa for Schevill in 1952 has a broad high-frequency range (Fig. 2), but unfortunately it is rather insensitive and has a bothersome response peak which limits its usefulness. More recently designed hydrophones respond uniformly over a much broader range of frequencies, but they are so insensitive sp:ct. 4] SOUND PRODUCTION BY MAKINE ANIMALS 545 compared with their inherent self-noise that they are virtually useless for studying many animal sounds. However, several new hydrophone designs are being introduced, and scientists interested in this field can hope for smaller, more sensitive hydrophones having a broader frequency response. Amplifiers have long had more faithful response than hydrophones, ear- phones, loudspeakers or recorders. In fact there is little excuse for using amplifiers that limit the effectiveness of the ensemble. Dow's Suitcase amplifier (Hersey, 1957) is representative of several early post-war amplifiers specially designed for use with hydrophones for detecting very broad-band underwater sounds. If possible, hydrophones and amplifiers should be selected so that their self- noise is considerably less than the lowest anticipated values of ambient noise. + 10 ■20 ■n ~ ~~ "" " f / 1 AMBIENT NOISE IN THE SEA I / • ■• / t\ov^^ ^ «« ^ \ / - ^s-^^^ ^ -*^ \J . c; ^.^^^ ^ r \\\ ■^^V k * ^ f^ '■ IM — ' ^ 100 10,000 100,000 1000 FREQUENCY (c/s) Fig. 3. Self-noise of an AX-58 — Suitcase system compared with ambient noise at sea state 0. (After Hersey, 1957, Fig. 10. By courtesy of the Journal.) For example, at low sea states the ambient noise of the 10 kc/s band of the AX-58 and similar hydrophones is of the order 0.5 dynes/cm^. The self-noise of the amplifier should be considerably lower than the equivalent input voltage : say, 10~6 volts. Fig. 3 shows Dow's estimate of performance of the AX-58 and Suitcase system. Radio telemetering buoys fitted wdth hydrophones have been used in various studies of natural sounds of the sea (e.g. Snodgrass and Richards, 1956; and Hashimoto, Nishimura and Maniwa, 1960). These offer attractions such as reducing the expense and effort of shipborne observations and removing the ship and the people from the immediate scene, thus reducing the possibility of abnormal behavior of the animals. A. Recorders and Monitors We know so little about animal sounds of the sea that there is seldom any reason for limiting the recording bandwidth, except to be as faithful as possible over as wide a frequency band as possible. This is not easily done with com- mercially available equipment. The magnetic tape recorder is now virtually the 546 SCHEVILL, BACKUS, AND HERSEY [CHAP. 14 sole recorder employed at sea for preserving sounds for playback. Neither direct nor FM (frequency-modulated) recording provides distortion-free playback ; the student is well advised to learn their limitations. FM recording is limited to frequencies below roughly 5 to 10 kc/s ; it is especially useful for frequencies below 100 c/s. Direct recording is generally (but not inevitably) of poor quality below 100 to 150 c/s, but can provide good recordings to 100 kc/s. Wave forms are relatively faithfully preserved in FM recordings, whereas phase distortion is characteristic of direct recording, even though uniform amplitude response is achieved for sinusoidal signals over a broad frequency band. Direct tape recorders are available in bewildering variety of size and quality. Lightweight, small portable tape recorders operated on dry batteries are commonly limited to frequencies between 50 or 100 c/s and 6 or 8 kc/s. Still portable but bulky professional models are available that will perform equally well from 50 or 150 c/s to 15, 30 or 100 kc/s. Wide bandwidths are achieved in part by very fast tape transport speeds (60 inches per sec or more). The wide -band recorders have been comparatively bulky and generally not suited for use in small boats. However, they have been used successfully in small research ships. One cannot but hope that the space -saving of solid-state circuitry will soon provide us broad-band, small tape recorders. They are sorely needed for small boat work or other techniques to permit the observer to close range on suspected soni- ferous animals. Earphones must be selected carefully for the intended research. Some designs emphasize low, others high, frequencies. For work to be done from ships rather than small craft, a high-quality loudspeaker may be used in quiet spaces. As with tape recorders, it is well worth the investigator's time to know the limitations of this part of the gear ; the possible distortions caused by the wrong earphones may prevent the observer from detecting subtle components of great significance in the ambient sounds of the sea. The best recorder is of no use if the observer has decided there is nothing of interest to record. B. Analysis of Sounds Sounds may be analyzed by measuring their intensity, their variation in time, or their frequency composition (also generally a function of time). The objective of the research should determine the method, but since this field of inquiry has not generated new methods until recently, its devotees have been forced to choose the best available from related fields. The methods of industrial- noise analysis have not proved useful excepting for the study of sounds generated by large groups, as the snapping shrimp or croakers. Other animal sounds of the open ocean are commonly heard from single individuals or from small numbers. Further, the sounds divide roughly into three categories : short transients spaced many times their own duration in time (clicks or ticking sounds), sequences of short transients closely and regularly spaced (creaking or grating sounds), and prolonged sounds containing discrete frequency com- ponents (squeals, grunts). Thus the methods of transient and vibrational analysis are indicated. SECT. 4] SOUND PRODUCTION BY MARINE ANIMALS 547 C. Spectrum Analysis By far the most use has been made of visible speech analysis equipment such as the Kay Electric Company's Sonagraph and Vibralyzer, These were the commercial outgrowths of the original sound spectrograph developed at the Bell Telephone Laboratories. They have been used in many different studies of animal sounds. Both instruments present a time and frequency analysis of transient or steady-state wave-forms as a graph in which time is the abscissa, frequency the ordinate, and the intensity is shown by relative blackening of the paper (see Figs. 4 and 6-10). These spectrograms give a pictorial representation of sound which satisfies many of the requirements for definitive description of a transient sound, especially the means for comparing and contrasting different sounds. The in- tensity display is qualitative, except that the dividing line between recording 0.3 0.4 . r/Me isec) Fig. 4. Sound spectrogram of squeal and clicks of Delphinus delphis (saddleback porpoise), made on a Kay Electric Co. Vibralyzer. and not-recording is reproducible and can be used to determine one contour of equal intensity. By re-playing the analysis at gain settings according to a desired pattern, the whole sound can be presented in contours of intensity on a map of time and frequency (see Hersey, 1957, fig. 19). For many animal sound investigations, great refinement of this sort is not needed ; we have seen that the contour technique has proved very useful in scattering-layer studies. Another function of these instruments, known as "sectioning", presents the average intensity spectrum over a short time near a selected instant. The principal drawback of this type of instrument is its program ; the sound is recorded on a magnetic drum and must be re-played for each setting of the wave analyzer throughout the spectrum. This is a tedious occupation requiring 5 min to analyze sounds from 2.4 to 24 sec long. In recent years several manu- facturers have been making matched sets of magnetostriction or crystal filters which all receive the sample at once. With these filters, analyses can be com- pleted in the time of the original sound while providing the same information 548 SCHEVILL, BACKUS, AND HERSEY [chap. 14 as the instruments just described. An instrument of this type has been recently completed at Woods Hole using one hundred Rayspan filters (Raytheon Mfg. Co.) and a multichannel recorder (Alden Products Co.). A recording is shown in Fig. 5. It is to be compared with a vibralyzer analysis of the same sound shown in Fig. 4. O 0.1 0.2 0.3 0,4 0.5 0,6 0.7 0 ' TIME (sec) Fig. 5. Sound spectrogram of Delphinus delphis calls including those of Fig. 4 made on the W. H. O. I. Rayspan-Alden combination. Inside the broken line is the part of the record represented by Fig. 4. The fre- quency response of the tape playback used for Fig. 5 is different from that of Fig. 4, accounting for some of the difference between the two spectrograms. D. Instruments for Identifying Sound Makers at Sea Methods and instruments for locating, tracking, and finally closing range on sound-making animals in the open sea is the most pressing need in this field. Real progress toward identification of sound makers will await such develop- ments. Versatile and convenient directional hydrophone arrays are needed ; they must be coupled with analyzers instantaneously presenting significant information about the chase. Probably these will have to operate from com- paratively small craft having very quiet propulsion. Furthermore, identification means deliberate visual inspection or capture of the animal, probably both. This appears a formidable instrumentation problem. Nevertheless, many elements of such systems have been used for other purposes in the past decade. Directional high-frequency echo-ranging equipment has been used to locate sound sources such as whales or porpoises. The many sharp transient sounds heard in the ocean, only a few of which have been identified, might well be located by travel time difference measurements from a suitably disposed SKOT. 4] SOUND PRODITCTION BY MARINE ANIMALS 549 array. Local circumstances of the searcli, as well as the character and spectrum of the sound, must determine the choice of directional array. The most versatile equipment now available includes the several echo-rangers used in European fisheries. Some of these are trainable in vertical angle as well as in azimuth. They are especially attractive because they are designed for use in small fishing boats (examples are Simrad, Atlas, Basdic and others). 4. Identification of Source Once the instrumental preliminaries are over, there is little trouble in hearing marine sounds, unless one has had the bad luck to strike an acoustic desert. In many regions, however, the listener will be rewarded by sounds. By what are they made? Many non-biological sounds are readily attributed (waves, ship noises). The biological sounds are not always as easy, and are often weaker than the listener's own ship noises. In some cases the sound sources have long been known ; these are usually animals from shallow depths, and include commercial fish (e.g. grunts, croakers) and such familiar creatures as snapping shrimp or pistol prawns, which make sounds that may be heard by the unaided ear, with the animal nearby or in the hand. These are a very small minority of the known marine sounds, and many others will continue to be quite difficult to identify. Most marine animals remain immersed and are, therefore, not easily noticed by human observers. ^ Among the minority that are readily seen are the air- breathing marine vertebrates : reptiles, birds and mammals, the latter including the sirenians, pinnipeds, and cetaceans. Of these, only the cetaceans have been demonstrated to make underwater sounds, although the above-surface sounds of pinnipeds have been heard under water. The identification of cetacean sounds with a particular cetacean rests upon the visual identification of the whale at the time the sounds are heard, the likelihood of error diminishing with the number of observations (sometimes a specimen must be collected for identifica- tion). There is, none the less, the lurking likelihood that one may be listening to unseen animals while watching others. With fish the difficulties are appreciably greater. Many of the smaller ones may be kept in aquaria (likewise the smaller cetaceans) ; the tendency of many animals to behave and phonate differently in captivity presents difficulties, since it is often difficult to elicit natural responses. An example of the dangers of identifying sounds without careful investiga- tions or good luck is the recent report by Azhazha (1958), just cited, on trawling in the Norwegian Sea. It appears from this report that all his experiences occurred during the hours of darkness. Without any supporting evidence that squalids can in fact make sounds, he attributes the sounds, to him unfamiliar, 1 Some, of course, may be seen by divers, but unless the diver is equipped with under- water listening gear, he is likely to notice only the loudest of underwater sounds. This is because of man's well-known lowered hearing acuity by bone-conduction (as when im- mersed), and also because most underwater breathing apparatus is very noisy. 550 SCHEVILL, BACKUS, AND HERSEY [CHAP. 14 to these small sharks {Squalus acanthias). We have over several years listened in the presence of sharks, including Squalus acanthias, without hearing any- thing except occasional crunching of food. As nearly as one can tell from Azhazha's description, the sounds that he correlates with poor herring catches are very like those that we, from repeated experience, associate with the smaller odontocetes, and we therefore suppose that he was reporting porpoise calls. 5. Purposeful and Adventitious Sounds Two classes of sounds generated by marine animals have been noted and described. These have often been called "biological" and "mechanical", or "purposeful" and "adventitious" sounds, or more simply "sounds" and "noises". However, as will be seen, it is not always easy for us to make this distinction. In the first case the sounds are generated by the body of the animal alone — generally by structures which have been modified for sound production. The sounds thus produced may be assumed to be of survival value to the individual making them, and this has been demonstrated in a number of cases. In the second case the sounds are usually generated by the contact of the animal with its surroundings. Sounds made by the motions of the animal through the water or over the bottom and those incidental to feeding, foraging, and other activities are placed in the second category. In certain fishes with swim-bladders, furthermore, sounds may be produced accidentally through sudden motions of the animal, particularly bending, which may cause the swim- bladder to vibrate. Such sounds should be distinguished from purposeful resonation of the swim-bladder by drumming muscles or other means. Adventitious sounds, since they may inadvertently attract attention, probably most generally affect their maker adversely by causing competition if not predation. However, adventitious feeding sounds in the swell-fish Spheroides maculatus attract others of this kind (Breder and Clark, 1947) and it might be supposed that these sounds are of value to the species in enabling them more effectively to exploit a food source. Such sounds have been described by Tokarev (1958) in the silverside, Atherina hepsetus, the jack, Trachurus trachurus, and the maenid, Smaris smaris. in the Black Sea. He states that each species makes characteristic feeding sounds, and, furthermore, that these sounds vary according to the nature of the food. Tokarev argues that these sounds are important in forming aggregations of fish. That these adventitious feeding sounds are of value to the species is perhaps supported by the fact that in certain marine animals purposefully produced sounds of a characteristic nature are made in the presence of food and at that time only. This has been observed in certain pomacentrid fishes (Fish, 1948) and in the porpoise, Tursiops truncatus (Wood, 1954). Although naturalists have generally supposed that the submerged swimming of marine animals is essentially silent, fishermen who locate schools of fish by underwater listening with the unaided ear have long attributed certain sounds to such a source (Westenberg, 1953). These sounds are usually likened to SECT. 4] SOUND PRODUCTION BY MAitlNE ANIMALS 551 distant wind or surf. Recently Russian fishery biologists have recorded and described sounds of this sort from schools of Trachurus trachurus and Atherina hepsetus (Shishkova, 1956 and 1958 ; Tokarev, 1958). Shishkova (1958) describes the sound of a moving school of T. trachurus as most intense in the band from 40 to 80 c/s, weaker in the band 100-320 c'/s, and beginning at 400 c/s growing again in amplitude until the band 1600-4000 c/s is reached, when the sound diminishes uniformly to the upper limit of the analysis at 16,000 c/s. Fig. 6 is the spectrogram of the swimming sound of a school of the anchovy, Anchoviella choerostoma, from the unpublished data of James M. Moulton, and shows the steady sound of the swimming school of several million individuals, as well as the emphatic sudden veering of the entire school. What significance these sounds may have in the lives of the animals remains to be seen. The possibilities of exploiting such sounds commercially is obvious. 4 • 1 E mSSt l"""^™" 1 1 '"~~""^""] , C ) 0.5 1 1.5 2 r/M£ (sec) Fig. 6. Sound spectrogram of swimming sounds of a school of Anchoviella choerostoma, showing the steady swimming motions against the low background, punctuated with two sudden "by-the-flank" ("veering") motions. (By courtesy of J. M. Moulton.) Recently Japanese workers have experimented with the transmittal of the swimming sounds of fish schools (of the yellowtail, Seriola quinqueradiata and others) by sonobuoy from ocean sites to shoreside stations (Hashimoto, Nishi- mura and Maniwa, 1960). Data from sonobuoys placed in fish traps showed a positive correlation between the occurrence of sounds attributed to the swim- ming yellowtails and the amount of the catch. The paper implies that sonobuoy surveys would be useful in selecting sites for fish traps. Swimming sounds of other marine forms, such as cephalopods, turtles, pinnipeds and cetaceans, have yet to be demonstrated. Nevertheless, it is occasionally asserted that the "tail beat" of cetaceans is audible. As far as we know, these assertions are founded upon uncritical reports by inexperienced observers (cf. the authorities quoted by Fish, 1949, pp. 33-34). This is not to say that such sounds may not yet be demonstrated, as in the case of certain fishes described above ; it is just that, to our knowledge, the only adventitious 552 SCHEVILL, BACKUS, AND HERSEY [CHAP. 14 sounds ascribable to cetaceans are splashings at the surface (of course some of these must be classed as purposeful, like "lobtailing" — whacking the surface with the flukes from above). The vocal sounds are ordinarily louder. 6. Sound-Producing Mechanisms A. Fishes Considerable attention has been paid to the sound-producing mechanisms of fishes and crustaceans, but relatively little is known about them in whales and porpoises. The mechanisms in fishes are generally of one of two sorts, or a combination of the two. The loudest sounds originate with the swim-bladder, which is made to resonate, the process being somewhat analogous to drum- beating. This is commonly accomplished by the contraction of special "drum- ming" muscles which are applied to the surfaces of the swim-bladder as in Prionotus and Opsanus as well as in some sciaenids, zeids, macrourids and many others (Jones and Marshall, 1953). In still others the musculature causing the resonance of the swim-bladder is not highly specialized, but simply the normal axial musculature in a most intimate contact with the swim- bladder. In the trigger-fishes, Batistes and Metichthys, the body-wall under the pectoral is exceedingly thin and the swim-bladder is beaten directly by rapid vibrations of the fin (Moulton, 1958). In some fishes the interior of the swim- bladder is intricately partitioned in a way as to suggest that it is an adaptation to sound-production, as in Prionotus carolinus (Fish, 1954). The second sort consists of mechanisms producing stridulatory sounds by the rubbing of one hard part against another. In most cases stridulatory sounds are made by rubbing the teeth together. The teeth may or may not have obvious specializations for sound production. In the filefish, Monocanthus hispidus, the teeth are so modified, the upper median incisors are ridged ; sound is also produced by the rapid elevation and lowering of the first dorsal spine, which grates against its articulating surface (Burkenroad, 1931). In other fishes, elements of the pectoral girdle may be rubbed against one another. In still others, parts of the vertebral column are used in such a way. In some cases stridulatory sounds are amplified by the resonant swim-bladder, as in the Haemulidae or, as they are sometimes known, grunts (Burkenroad, 1930), in which the anterior end of the swim-bladder lies against the pharyngeal teeth — the primary sound-producers in this case. Other modes of sound produc- tion in fishes may be discovered. In the goby, Bathygobius soporator, and the blenny, Chasmodes bosquianus, the exact manner of sound generation remains unknown, but the observed faint sounds appear to result from the spasmodic escape of water out of the gill openings (Tavolga, 1958, 1958a). B. Crustaceans In marine crustaceans, sounds are made by stridulation or, in some cases, by rapping one hard part against another. In the langoustes, Palinurus vulgaris and Panulirus argus, a very elaborate stridulatory mechanism has evolved SKOT. 4] SOUND PKOnUCTION BY MARINE ANIMAI.S 553 (Dijkgraaf, 195;"); Moultoii, 1957). This includes a ridged membrane on the basal portion of the antenna and a toothed surface on the shell of the head near the base of the antenna. There are mechanisms for engaging and keeping engaged the two rough surfaces as the antenna is raised. In the so-called snajDping shrimps (family Alpheidae), which are famous as noise makers, the sound is produced as the movable "finger" of the enlarged "claw" is brought down agahist the immovable "thumb" (Johnson, Everest and Young, 1947). This apparently simple arrangement is actually quite elaborate. The "finger" is fitted with a knob which fits into a groove on the "thumb". The base of the "finger" is provided with a small suction cup, which, when the "finger" is raised, meets another suction cup on an immovable portion of the "claw". Considerable muscular force is required to break this connection, so that the mechanism is, in effect, spring-loaded. C. Cetaceans Fishermen and some other seamen from remotest antiquity have been aware that the smaller cetaceans produced audible sounds. With the develop- ment of civilization this awareness has faded, until in 19th-century Europe most zoologists confidently called the cetaceans mute, discounting sailors' tales of squealing porpoises on the insufficient grounds that the animals lack vocal cords. This is an anatomical fact, yet sounds may be produced in the larynx, as has been repeatedly demonstrated, not merely by the animals themselves, but also by men pumping air through dead cetacean larynges [two such recent experiments were by Eraser and Purves {in litt., 1953) and Schevill and Law- rence (MS., 1949), in all cases on small delphinids]. The cetacean larynx has been described rejDeatedly (e.g. Murie, 1870, 1873; Turner, 1872; Boenninghaus, 1902; Hosokawa, 1950; Kleinenberg and Yablokov, 1958; etc.). The considerable difference between mysticetes (baleen whales) and odontocetes (toothed whales) is refiected here. It will suffice for our purposes to point out that, although vocal cords as such are lacking, there are projections and thin membranous parts in the laryngeal cartilages which may be made to vibrate as air is passed through the larynx. (The sounds resulting from the artificial excitation do not duplicate the animal's vocaliza- tions, but are still not entirely unlike some elements of the natural phonation.) To confine ourselves for the moment to the odontocetes, we are concerned with two basic sounds, a whistle-like squeal of limited frequency range and an impulsive click of broad (almost "white") frequency range. The squeal is what is usually heard as air-borne sound in non-instrumental listening. A number of biologists (e.g. Matthews, 1950) have supposed that all or some of the cetacean sounds are formed at the nasal orifice (blowhole), presumably because one often sees a stream of bubbles issuing therefrom. Our experience has been that a squeal always accompanies the bubbles, although the converse is not true. Since the squeal is the most intense of the two types of sound, it has seemed to us that it requires the passage of more air through the sound source than does a 554 SCHEVILL, BACKUS, AND HERSEY [CHAP. 14 click, and it is consequently not remarkable that air is usually, but not always, exhausted from the blowhole when the porpoise squeals. (The reader should remember that the larynx is intranarial, and that the mouth is cut off from the respiratory tract.) But the idea that the squeal ("whistle") originates at thp lips of the blowhole is much weakened if one tries whistling under water ; this is our chief reason for not calling the porpoise's squeal a whistle. However, captive odontocetes, notably Tursiops and Globicephala, have been observed with the blowhole out of water and obviously vibrating while somewhat chirping sounds are heard. We tend to discount these performances as not bearing directly on underwater sound, although such competent observers as K. S. Norris of the University of California at Los Angeles disagree with us. In any case, we must consider his and others' suggestions, as yet unpublished, that the larynx may not be the only part of the respiratory tract that may produce sound ; although it has not been proved, some sounds may be produced inside the nasal passages. Mysticete sounds are evidently of quite different character (they have usually been described as "moans", and sometimes as "screams"). We know of no direct evidence as to the actual source of these sounds, and so continue to suppose that this is the well-developed larynx. 7. Spectra of Sounds The spectra offish sounds generally have their limits between 50 and 5000 c/s, with most of the sound energy concentrated in the region between 100 and 800 c/s. The sounds produced by resonation of the swim-bladder are, in general, lower in tone than those produced by stridulation. Swim-bladder sounds may have components from 50 up to 1500 c/s, but characteristically show principal frequencies in the region of 100 to 300 c/s. Stridulatory sounds, on the other hand, may have components from 50 to 800 c/s or more, but typically are loudest in the region from 500 to 2000 or 3000 c/s. While new data will undoubtedly affect these numbers, it seems reasonably sure that the point made is a valid one. The following data give some idea of sound-pressure levels observed near fishes : croaker {Micropogon undulatus) chorus around hydrophone, 63 dynes/ cm2 (overall level in the band from 100 to 10,000 c/s; Knudsen, Alford and Emling, 1948) ; about 20 kinds of fishes, mostly one to two feet from receiver, examined individually, 3.5 to 50 dynes/cm^, highest octave level (Fish, 1954); a toad-fish {Opsanus tau), a few inches from the hydrophone, about 275 dynes/ cm2 in the octave band centered on 200 c/s (Dobrin, 1947). Tavolga estimates that the sounds of the courting male of Bathygobius soporator attain a pressure of 0.1-0.2 dynes/cm2 about 6-8 in. from the hydrophone (Tavolga, 1958), and that those oi Chasmodes bosquianus reach about 0.5 to 0.8 dynes/cm^ (Tavolga, 1958a). The latter sounds were just above the noise of his system and could not be detected when the hydrophone was more than a foot from the source. It can readily be seen that fish sounds vary widely in intensity. SECT. 4] SOUND PRODUCTION BY MARINE ANIMALS 555 Moulton (1958) gives spectrograms and other details of sounds of fishes of nine famihes, including swim-bladder drumming and dental stridulation, etc. (Fig. 7 is from this paper; Fig. 8, also Moulton's, is from Backus, 1958). SECONDS Fig. 7. Sound spectrogram of tooth stridulation, hy Car anx hippos. (After Moulton, 1958.) UJ 4.0 3.0 2.0 I.D 0.2 l» f* \Up i«'t • }j tt P HHiiiiiiH 0 0.4 0.8 1.2 1.6 2.0 TIME (sec) Fig. 8. Sound spectrogram of swim-bladder vibration of Prionotus sp. (By courtesy of James M. Moulton, from Backus, 1958.) Best studied of crustacean sounds are those of the "snapping shrimp". The spectrum of this noise is quite broad (about 500 c/s to 20,000 c/s), and sound- pressure levels of about 0.02 dynes/cm^/one-cycle band have been observed for choruses of these animals. Typical peak pressures for single snaps of Alpheus and Synalpheus were about 200 dynes/cm^, with peaks to 1000, at one meter 556 SCHEVILL, BACKUS, AND HERSEY [CHAP. 14 (Everest, Young and Johnson, 1948). The "rasping" sound of the langouste, Panulirus argus, has components from 40 c/s to 9000 c/s, with most energy at 800 c/s and in the band from 2500 to 7700 c/s. The "slow rattle" covers fre- quencies between 500 c/s and 3300 c/s, with most energy at 600 c/s (Moulton, 1957). The spectrum of cetacean sounds is in general very broad, reaching right across the scale of such measuring instruments as we have used, say, from less than 30 to about 200,000 cycles per second. This highest figure (Schevill and Lawrence, 1953, p. 163) is probably not biologically significant, being very likely ultrasonic even to the porpoises which produced it, but it is cited to remind us that there may be acoustic energy beyond the reach of our ordinary instruments. Many of these do not reach above 20 kc/s, and so hear at least as well as humans ; at Woods Hole we have sometimes employed systems that were useful to 30 kc/s. Most of the energy of cetacean sounds is within this band ; it is not known how high these animals hear. One experiment (Schevill and Lawrence, 1953) indicates that Tursiops, at least, responds readily to frequencies as high as 120 kc/s. We have, unfortunately, no good figures for the intensity of cetacean sounds. There are probably of the order of a hundred species, or kinds, of cetaceans. Of these only about a dozen have so far been firmly demonstrated to be soni- ferous, although a good many more figure in unscientific accounts most of which appear to allude to respiratory noises heard at the surface through the air. (These adventitious sounds are only rarely heard under water even by sensitive equipment.) Tomilin (1955), it is true, reports hearing the underwater sounds of many more species than we have, but gives no particulars either of the sounds or of the means by which he heard them. Accordingly, we confine ourselves to the species of which we have particulars, and await further details from him. The sounds of the mysticetes (baleen whales) are very poorly known, only Euhalaena and Megaptera having been reliably recorded. The rather sonorous "moans" and strident screeches of Euhalaena (right whale, Nordcaper) have been repeatedly recorded in Cape Cod waters ; Fig. 9 is a spectrogram of one of these calls, showing that the fundamental is below 200 c/s. Megaptera (hump- back whale) also concentrates on low frequencies, albeit a little higher, 300 to 400 c/s. We know of no other instrumental data, although older authors (e.g. Aldrich, 1889, pp. 32-35, including "Kelley's Band") have given vivid subjective accounts. The sounds of the odontocetes (toothed whales) are better known. In general they appear to consist of whistle-like squeals and impulsive clicks. The clicks may be made singly or in bursts at a repetition rate as high as 400 per second (Schevill and Lawrence, 1956); at these various rates sounds of different characteristics are produced, which have been described as "creaks", "barks", "snores", etc. by various authors (e.g. McBride and Hebb, 1948; Kritzler, 1952 ; Wood, 1954). These clicks cover a wide band of frequencies, and may be called at least "pale grey" if not actually "white". It is true that in our analyses SOUND PRODUCTION BY MARINE ANIMALS 557 i^^in jS£fe ■A 0.3 T/M£ (sec) 0.4 Fig. 9. Sound spectrogram of call of Euhalaena glacialis (right whale). they exhibit a preponderance of energy in the lower middle part of the spectrum, often between about 4 and 12 kc/s, but this appearance is largely due to the uneven response of the recording system (hydrophone — amplifier — recorder), and is not necessarily true of the sound made by the whale. Fig. 10 shows such a spectrogram of Physeter (sperm whale) clicks. This whale uses clicks ap- parently exclusively and with greater intensity than other odontocetes ; although we have listened to many sperm whales during the last three years, we have never heard a squeal, only these powerful clicks. Fig. 4 includes clicks of Delphinus delphis, the familiar oceanic porpoise (or "common dolphin"). The odontocete squeals are quite different in character, being sharply limited in frequency and considerably extended in time. Fig. 4 is a spectrogram of a squeal of Delphinus delphis. This sound is sufficiently intense to be heard in favorable circumstances by the unaided human ear in air (McBride, 1940; Kullenberg, 1947; Fraser, 1947), and is what led arctic seafarers to call Del- phinapterus (white whale) the "sea canary". *MktA,',%m.jn'immi&iJit • t0i ftulttti >\ ' a $'^'hwk^ *« tut 0.2 r/M£ (sec) Fig. 10. Sound spectrogram of clicks of Physeter catodon (sperm whale). 558 SCHEVILL, BACKUS, AND HERSEY [CHAP. 14 The frequency range of the squeals is not the same in all species, though several of the few we know are not easily distinguished on this account. Globice- phala (see Hersey, 1957, p. 265, fig. 11) has not only a lower squeal than Delphinus, for example, but sings a markedly different song. 8. Functions of Sound A . The Sexual Functions It has long been supposed that certain underwater sounds made by fishes have a sexual significance. This supposition grew originally from observations of that most noisy family of fishes, the Sciaenidae or drums, in which increases in sound-production were noted at times of spawning. Indeed, in five out of seven common genera of sciaenid fishes of northeastern America {Pogonias, Sciaenops, Cynoscion, Leiostomus, and Bairdiella), it is only in the male that the sound-making apparatus is developed. This is presumptive evidence that the sounds of these fishes have a sexual function. (In the sciaenid genera Menticirrhus and Micropogon the sound-producing mechanism is altogether absent in the first case and is present in both sexes in the second.) Such observa- tions in nature have been reinforced by observations in aquaria. Goode (1888) cites data from the old New York aquarium in which specimens of the sea drum, Pogonias cromis, produced sounds as male chased female in the spring. More recently, Fish (1954) has found that the squeteague, Cynoscion regalis, held in aquaria are more easily stimulated to sound-production during the approach of the breeding season than at other times. A sexual function is probable for one class of sounds emitted by various species of toad-fishes of the genus Opsanus (Fish, 1954; Tavolga, 1958b). In Opsanus tau males establish a territory and guard a nest in which successive females spawn. The male makes a loud "boat-whistle-like" sound at this time, but ceases when his milt is spent (Fish, 1954). In the sea-robins (genus Prionotus) the period of most active sound-production appears to be correlated with the time of sexual maturity or its approach (Fish, 1954; Moulton, 1956). While such data as the foregoing do suggest sexual functions for certain sounds it is not clear in most cases exactly how a given sound serves to accom- plish the union of the sexes. Sound might serve to bring widely separated individuals of opposite sex near one another, it might serve for sexual recogni- tion in a group already assembled, or it might be used at stages in the courtship or spawning procedures. It is from the last category that come the least equi- vocal observations. In the so-called purring gourami, Trichopsis vittatus, both sexes produce a sound during the prespawning behavior (Stampehl, 1931 ; Beyer, 1931 ; Reickel, 1936). Clicking sounds are produced by both male and female sea- horses {Hippocampus spp.) during the preliminaries to spawning and especially during the act itself, when "loud and continuous" sounds are made as the female deposits eggs in the brood pouch of the male (Dufosse, 1874; Fish, 1954). In the filefishes, Cantherines pulles and Alutera punctata, the large dorsal SECT. 4] SOUND PRODUCTION BY MARINE ANIMALS 559 spine is moved rai)idly back and forth by males as they display before the female (Clark, 1950). This is presumptive evidence that sound is a part of the sexual behavior of these species, since sound production by dorsal spine stridulation is known in the related filefish genus Monocanihus (Burkenroad, 1931). The most exhaustive studies in this connection are those by the animal behaviorist Tavolga (1956, 1958, 1958a) of the goby, Bathygohius soporator, and the blenny, Chasmodes hosquianus. In Bathygohius the establishment of territory and the courtship of the female by the male proceeds by a complex set of visual, chemical, and auditory stimuli, of which low grunts emitted by the male are an essential part. These grunts appear to be stimulated by the release into the water of an ovarian fluid by the female and in turn stimulate generalized activity on the part of the female which is oriented by visual stimuli received from the male. Other males may also be attracted to the scene of sound- production. A similar function for similar sounds was determined in Chasmodes. In the cetaceans certain sounds undoubtedly have a sexual significance. Captive Tursiops make peculiar high-pitched yelping sounds during the mating season. These are heard when a female which is being actively courted deserts her partner, but cease when she returns to him. Her return may be a positive reaction to the sounds (Wood, 1954; Tavolga and Essapian, 1957). Sexual motives have not yet been attributed to the sound production of strictly marine crustaceans, so far as we are aware, but in the littoral Uca pugilator, a fiddler crab, low-frequency drumming is associated with a sort of dance, called "beckoning", to which the male is incited by the appearance of the female (Burkenroad, 1947). B. The Defensive and Offensive Functions Although many marine animals, notably cetaceans, seem habitually to be- come silent when hunted or molested, many others produce sounds when they are caught, roughly handled, or otherwise disturbed. And largely on the basis of this behavior, it is supposed that some sounds are of defensive use to the individual. Sound production may accompany other acts which seem to be more clearly defensive in nature. Thus in the burrfish, Chilomycterus spinosus, and in the swellfish, Sphoeroides nephelus, a "whining scrape" is made by grinding the incisors together during or after inflation (Burkenroad, 1931). In sculpins of the genus Myoxocephalus and in the flying gurnard, Dactylopterus volitans, drumming noises are made as the flsh spreads its spiny gill-covers (Sorensen, 1884; Nichols and Breder, 1927). In the sea-robin {Prionotus sp.) a low grunt accompanies the erection of the spiny fins (Moulton, 1956) and similar observations have been made for the toad-fish, Opsanus tau (Hildebrand and Schroeder, 1928). In the langouste, Panulirus argus, sound-production accompanies the violent abdominal contractions of the restrained animal, which bring. into play the various spines and sharp edges of the exoskeleton (Moulton, 1957). 560 SCHEVILL, BACKUS, AND HERSEY [CHAP. 14 A clear example of how sound may be used offensively is found in the snap- ping shrimps of the genus Alpheus, which have been extensively observed in aquaria by MacGinitie and MacGinitie (1949). The loud snap, earlier described, is made when small fish or other organisms pass the burrow of the shrimp. Animals of sufficiently small size are actually stunned by the concussive sound and then are dragged into the burrow to be consumed. Presumably animals so large as not to be stunned are repulsed by the act. C. Other Communicative Functions Sounds are used for manifold other purposes. For instance, Lindberg (1955) has described apparently intimidative sounds of Panulirus interruptus when on the verge of conflict. For similar purposes, jaw-snapping by dominant Tursiops has been described by Wood (1954) and Tavolga and Essapian (1957). Other examples are given by Backus (1958). Wood (1954) and others have cited cases of mother and child porpoises calling to each other when separated. D. Orientation by Sound It has long seemed reasonable to suppose that marine animals use sound in orientation, hearing being so much more useful, as we have said, than the other senses. Nevertheless, beyond this general supposition and the realization that communicative calls have orientational uses (flocking, station-keeping, etc.), there is much speculation but little knowledge. That little knowledge is almost all about a few individuals (mostly captives) of one species of porpoise, Tursiops truncatus (McBride, 1956; Schevill and Lawrence, 1956; Kellogg, 1958), which evidently makes wide use of echo-location in food-finding and obstacle-avoidance. Since the sounds so utilized are the varied series of clicks described above (page 556) as common to all odontocetes the sounds of which are known, it has seemed justifiable to assume that they all utilize these sounds in echo-location. Much more needs to be learned, even for this one species. Although mysticete whales and many fish are soniferous, it is yet to be shown that they use sound in this particular way, although there is no question but that they might. Backus (1958, p. 196) has cited from various authors instances in which cave fishes have maneuvered successfully without visual clues, but, unfortunately, also without proof that they were in any way soniferous. There has been much speculation on the possible orientationally acoustic function of the lateral line system of fishes, which in its structure and disposition suggests an array of listening organs. The latest thinking (T. J. Walker, un- published, and J. Kuiper, unpublished) appears to be that instead of being actually acoustic, it is rather a turbulence ("flow-noise") sensor, somewhat analogous to Ogawa and Shida's suggestion (1950) of such a function for sinus hairs on the faces of whales. SECT. 4] SOUND PRODUCTION BY MARINE ANIMALS 561 Likewise undemonstrated is any possible role of sound in navigation, either on a local scale (beyond what has been said of Tur stops) or in the long migra- tions that many marine animals are known to make. 9. Hearing as Related to Sound Production In all cases where sounds produced are for intraspecific communication (including, of course, echo-location, which is talking to oneself), they must be heard by the species producing them. In cases of interspecific communication (for defensive and offensive functions) it would not seem essential that the producer of sounds hear them. A list of fishes in which hearing has been demonstrated, together with the frequency ranges observed, has recently been compiled by Lowenstein (1957), and it is apparent that the spectral distributions of sounds so far studied principally lie within the limits of fish hearing, which in general lie between 50 c/s and 1 to 3 kc/s, except in ostariophysan fishes which may hear up to 5 to 13 kc/s. (There have been few, if any, studies of hearing in sound-producing fishes, but doubtless the studies made in other species can be safely extrapolated to include the soniferous sorts.) It is not yet apparent what the correlation is, if any, between the acuity of hearing and the ability to produce sound. It has been suggested that, while many fishes possessing accessory hearing organs are apparently silent (for instance, the clupeids which are mute and have the swim-bladder connected to the ear), others may compensate for relatively poor hearing by the production of loud sounds (as in the toad-fishes, sea-robins and sciaenids). On the other hand, one can find numerous examples among ostario- physan fishes (all of which have accessory hearing organs) in which the ability to produce sound is well developed. The bottle -nose porpoise, Tur slops truncatus, hears sounds in the range from 150 c/s to about 120 kc/s (Schevill and Lawrence, 1953). Hearing limits have not been determined for other cetaceans. Hearing has been little studied in the crustaceans. Jahn and Wulff (1950) say that : ". . . although insects probably comprise the only group in which it is highly developed, it should not be assumed that phonoreception is completely unimportant in the life of other arthropods." Special sound-producing mech- anisms in crustaceans indicate that hearing does play a role in the lives of these animals, unless the sounds prove to be solely defensive in nature (which at present seems not to be so). 10. Eliciting and Suppressing Marine Animal Sounds Some attention has been paid in recent years to the possibility of influencing the movements of fishes by means of man-made sounds. In general fishes quickly accustom themselves to a strange sound after an initial "startle" reaction (Moulton and Backus, 1955). There have been, however, some interesting observations in connection with the effects of foreign sounds on the sound- producing behavior itself in soniferous fishes. Small explosives detonated at the 19— s. I. 562 SCHEVILL, BACKUS, AND HEKSEY [CHAP. 14 sites of croaker (Micropogon undulatus) choruses caused them to cease for periods of several minutes (Loye and Proudfoot, 1946). Moulton (1956) found that he could incite sea-robins {Prionotus spp.) to call by projecting crudely imitative sounds into the water, and suppress their calls by transmitting other sounds. Ta Volga (1958) elicited positive reactions from both male and female of the goby Bathygobius soporator in response to sounds similar to those which the male emits during the courtship. These imitations included the roughly similar sound of an unrelated species as well as purely artificial sounds including the projection of a tape-recording of Tavolga's voice saying, "ugh-ugh". Moulton's and Tavolga's observations thus indicate low discrimination by the fishes in the particular situations with which these workers were dealing. 11. Exploitation of Marine Animal Sounds by the Oceanographer Having identified patterns of sound-production with their animal sources, the ways in which oceanographers can use this information in elucidating the lives of soniferous animals and in probing still other problems seem limitless. Fish and Mowbray (1959), during studies of sound-production by Bahaman fishes, suggested that the common toad-fish there (genus Opsanus) was a species different from the known Atlantic kinds because its underwater sounds were different. Taxonomic investigation employing the usual criteria confirms this. A simple example of the type of problem that can be solved, once a sound is identified with its source, is that of the local distribution of the sound-producing animal. Thus Johnson and co-workers describe, through sound surveying, the disposition of snapping shrimp in the Point Loma area of southern California. (Johnson, Everest and Young, 1947 ; Johnson, 1948.) Such a study trans- cends the immediate issue of the distribution of the soniferous animal itself. The distribution of the sound determines the limits of the particular habitat in which the animal lives and so may be used to ascertain the distribution of silent species living in the same habitat or even the distribution of certain physical parameters of which the habitat is composed. Thus to Johnson and co- workers it seemed feasible to determine the distribution of certain commercial sponges from the distribution of snapping shrimp noise. Furthermore, their survey showed that maxima of sound pressure due to the shrimp fairly repre- sented the distribution of hard bottom in the study area. Moulton (1958) has had similar successes in a survey of the sounds of the squirrel-fish, Holocentrus ascensionis, and the Nassau grouper, Epinephelus striatus, in the Bimini area. The more one learns about a certain sound the grander the investigation one can pursue through it. During warm months in certain areas along the coast of the mid- Atlantic states, loud choruses of sounds are observed which have been attributed to the croaker, Micropogon undulatus, a species of drum in which, as mentioned above, both sexes are soniferous (Dobrin, 1947 ; Loye and Proud- foot, 1946; Johnson, 1948). Beyond its seasonal character both annual and diurnal fluctuations in the sound have been noted. Moreover, a comparison of SECT. 4] SOUND PRODUCTION BY MARINE ANIMALS 563 May and July measurements made in 1942 in the Chesapeake Bay area showed a drop in peak frequency of the choruses from about 700 c/s to about 225 c/s. It was suggested that this change was caused by fish growth and the resultant decrease in resonant frequency of the then larger swim-bladder (the sound source in this species). Such a frequency shift, however, would necessitate a doubling in the fish's size, and, since this does not occur, changes in the com- position of the croaker population through migration is likely the correct explanation (Marshall, 1954). In this connection the observations of the fishery biologist Wallace (1941) are of great interest. He finds that male croakers become sexually mature at a smaller size than the females do and, on maturing, leave the Bay. Thus he observed a 50 : 50 sex ratio in June but a proportion of males to females of 35 : 65 in August. It is obvious that if the significance of this sound and the physical factors determining its spectrum were understood, much could be learned of the population dynamics of this important fish. References Aldrich, H. li., 1889. Arctic Alaska and Siberia. Rand, McNally & Co., Chicago-New York, 234 pp. Azhazha, V, G., 1958. Akuly vosprinimayut i izdayut ul'trazvuki. Rybnoye Khoz., 34, 30-32. Backus, R. H., 1958. 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Dijkgraaf, S., 1955. Lauterzeugimg und Schallwahrnehmung bei der Languste {Palinurus vulgaris). Experientia, 11, 330-331. Dobrin, M. B., 1947. Measurements of underwater noise produced by marine life. Science, 105, 19-23. Dufosse, A., 1874. Recherches sur les bruits et les sons expressifs que font entendre les poissons d'Europe et sur les organes producteurs de ces phenomenes acoustiques. Ann. Sci. Nat. (5), 19, art. 5, 53 pp. ; 20, art. 3, 134 pp. Everest, A. F., R. W. Young and M. W. Johnson, 1948. Acoustical characteristics of noise produced by snapping shrimp. J. Acoust. Soc. Amer., 20, 137-142. Fish, M. P., 1948. Sonic fishes of the Pacific. Pacific Oceanic Biol. Proj., Tech. Rep. 2, 1-144 (Woods Hole Oceanographic Institvition, unpublished). Fish, M. P., 1949. Marine mammals of the Pacific with particular reference to the produc- tion of underwater sound. Woods Hole Oceanographic Institution, Ref. 49-30, 69 pp. (unpublished). 564 SCHEVILL, BACKUS, AND HERSEY [CHAP. 14 Fish, M. p., 1954. The character and significance of sound production among fishes of the western North Atlantic. Bull. Bingham Oceanog. Coll., 14, art. 3, 109 pp. Fish, M. P. and W. H. Mowbray, 1959. The production of underwater sound by Opsanv^ sp., a new toadfish from Bimini, Bahamas. Zoologica, 44, 71-76. Fraser, F. C, 1947. Soimd emitted by dolphins. Nature, 160, 759. Goode, G. B., 1888. American fishes. Faulkner and Allen, Philadelphia, xv + 496 pp. Greene, C. W., 1924. Physiological reactions and structure of the vocal apparatus of the California singing fish, Porichthys notatus. Amer. J. Physiol., 70, 496-499. Griffin, D. R., 1955. Hearing and acoustic orientation in marine animals. Deep-Sea Res., suppl. to 3, 406-417. Hashimoto T., M. Nishimura and Y. Maniwa, 1960. Detection of fish by sonobuoy. [Japanese with English summary.] Bull. Jap. Soc. Sci. Fish., 26, 245-249. Hersey, J. B., 1957. Electronics in oceanography. Advances in Electron. Electr. Phys., 9, 239-295. Hildebrand, S. F. and W. C. Schroeder, 1928. Fishes of Chesapeake Bay. Bull. U.S. Bur. Fish., 43, 1-388. Horton, J. W., 1957. Fundamentals of sonar. U.S. Naval Inst., Annapolis, xiv + 387 pp. Hosokawa, Hiroshi, 1950. On the cetacean larynx, with special remarks on the laryngeal sack of the sei whale and the aryteno-epiglottideal tube of the sperm whale. Sci. Reps. Whales Res. Inst., Tokyo, 3, 23-62. Jahn, T. L. and V. J. Wulff, 1950. Phonoreception in C. L. Prosser (edit.) Comparative animal physiology. W. B. Saiuiders Co., Philadelphia, 471-501. Johnson, M. W., 1948. Sound as a tool in marine ecology, from data on biological noises and the deep scattering layer. J. Mar. Res., 7, 443-458. Johnson, M. W., F. A. Everest and R. W. Young, 1947. The role of snapping shrimp (Crangon and Synalpheus) in the production of under-water noise in the sea. Biol. Bull., 93, 122-138. Jones, F. R. H. and N. B. Marshall, 1953. The structure and function of the teleostean swimbladder. Biol. Revs. Cambridge Phil. Soc, 28, 16—83. Kellogg, W. N., 1958. Echo ranging in the porpoise. Science, 128, 982-988. Kesteven, G. L., 1949. Malayan fisheries. Malaya Publishing House, Singapore. [Non,vid.] Kleinenberg, S. E. and A. V. Yablokov, 1958. O morphologii verknikh dykhatelnykh putyei Kitoobraznyk. Zool. Zhurn., 37, 1091-1099. Knudsen, V. O., R. S. Alford and J. W. Emling, 1948. Underwater ambient noise. J. Mar. Res., 7, 410-429. Kritzler, H., 1952. Observations on the pilot whale in captivity. J. Mammal., 33, 321-334. Kullenberg, B., 1947. Sound emitted by dolphins. Nature, 160, 648. Lindberg, R. G., 1955. Growth, population dynamics, and field behavior in the spiny lobster, Panidiriis interruptus (Randall). Univ. Calif. Pubis. Zool., 59, 157-247. Lowenstein, O., 1957. The acoustico -lateralis system, in Brown, M. E. (edit.) The physio- logy of fishes. Vol. 2, Behavior. Academic Press, New York, 155-186. Loye, D. P. and D. A. Proudfoot, 1946. Underwater noise due to marine life. J. Acoust. Soc. Amer., 18, 446-449. McBride, A. F., 1940. Observations on captive porpoises. Proc. Florida Acad. Sci., 4. (1939), 282. McBride, A. F., 1956. Evidence for echolocation by cetaceans. Deep-Sea Res., 3, 153-154. McBride, A. F. and D. O. Hebb, 1948. Behavior of the captive bottle-nose dolphin. Tursiops truncatus. J. Comp. Physiol. Psychol., 41, 111-123. MacGuiitie, G. E. and N. MacGinitie, 1949. Natural history of marine animals. McGraw- Hill, New York, London, Toronto, ix + 473 pp. Marshall, N. B., 1954. Aspects of deep sea biology. Philosophical Library, New York. 380 pp. Matthews, L. H., 1950. Comments on W.H.O.I. phonograph record of Delphinapterus leucas. Proc. Zool. Soc. London, 120, 451. SECT. 4] SOUND PRODUCTION BY MARINE ANIMALS 565 Moulton, J. M., 1956. Influencing the calling of sea robins (Prionohis spp.) with sound. Biol. Bull., Ill, 393-398. Moulton, J. M., 1957. Sound production in the spiny lobster Panulirus argiis (Latreille), Biol. Bull., 113, 286-295. Moulton, J. M., 1958. The acoustical behavior of some fishes in the Bimini area. Biol. Bull., 114, 357-374. Moulton, J. M. and R. H. Backus, 1955. Annotated references concerning the effects of man-made sounds on the movements of fishes. Dept. Sea and Shore Fisheries. Maine, Fish Circ, 17, 7 pp. Murie, J., 1870. On Risso's grampus, G. rissoanus (Desm.). J. Anat. Physiol., 5, 118- 138. Murie, J., 1873. On the organization of the caaing whale, Globicepkalus melas. Trails. Zool. Soc. London, 8, 235-301. Nichols, J. T. and C. M. Breder, Jr., 1927. The marine fishes of Xew York and southern New England. Zoologica, 9, 1-192. Ogawa, T. and T. Shida, 1950. On the sensory tubercles of lips and of oral cavity in the sei and fin whale. Sci. Reps. Whales Res. Inst., Tokyo, 3, 1-16. Rand, C, 1952. Crocodiles and human radars. The New Yorker, 25 Oct. 1952, 122-132. Reickel, A., 1936. 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En physiologisk og comparativ-anatomisk undersogelse. Thaning, Appels, Copenhagen. Stampehl, H., 1931. Ctenops vittatus (der Knurrende Gurami). Bl. Aquar. Terrarienk., 42, 394-395. [Non vid.] Tavolga, M. C. and F. S. Essapian, 1957. The behavior of the bottlenosed dolphin {Tur- siops truncatus). Mating, pregnancy, parturition and mother-infant behavior. Zoologica, 42, 11-31. Tavolga, W. N., 1956. Visual, chemical and sound stimuli as cues in the sex discriminatory behavior of the gobiid fish. Bathygohius soporator. Zoologica, 41, 49-64. Tavolga, W. N., 1958. The significance of underwater sounds produced by males of the gobiid fish, Bathygohius soporator. Physiol. Zool., 31, 259-271. Tavolga, W. N., 1958a. Underwater sounds produced by males of the blenniid fish, Chasmodes bosquianus. Ecology, 39, 759-760. Tavolga, W. N., 1958b. Underwater sounds produced by two species of toadfish Opsanus tau and Opsanus beta. Bull. Mar. Sci. Gulf and Caribb., 8, 278-284. Tokarev, A. K., 1958. O biologicheskikh i gidrodinamicheskikh zvukakh, izdavayemykh rybami. Trudy VNIRO, 36, 272-279. Tomilin, A. G., 1955. On behavior and sonic signalling in cetaceans. (In Russian.) Trudy Inst. Okeanol. Akad. Nauk S.S.S.R., 18, 28-47. Tower, R. W., 1908. Production of sound in drumfishes, the sea-robin and the toadfish. Ann. N.Y. Acad. Sci., 18, 149-180. 566 SCHEVILL, BACKUS, AND HEBSEY [CHAP. 14 Turner, William, 1872. An account of the great finner whale (Balaenoptera sihhaldii) stranded at Longniddry. Trans. Roy. Soc. Edinburgh, 26, 197-251. Wallace, D. H., 1941. Sexual development of the croaker, Micropogon undulatus, and distribution of the early stages in Chesapeake Bay. Trans. Amer. Fish. Soc, 70 (1940), 475-482. Westenberg, J., 1953. Acoustical aspects of some Indonesian fisheries. J. Cons. Explor. Mer., 18, 311-325. Wood, F. G., Jr., 1954. Underwater sound production and concurrent behavior of captive porpoises, Tursiops truncatus and Stenella plagiodon. Bull. Mar. Sci. Gulf and Caribb., 3, 120-133. V. WAVES 15. ANALYSIS AND STATISTICS D. E. Cart WRIGHT 1. Introduction The mathematics of periodic and other analytically-defined waves in a heavy fluid bounded by a free surface has been developed over more than a century ; the simpler properties are well known and the theory can be found in works on hydrodynamics (e.g. Lamb, 1932; Stoker, 1957; Coulson, 1944). Nevertheless, methods of analysing the ever-changing random pattern of humps and hollows continually present in the sea have been evolved only during the last fifteen years or so. Previously it was thought that the randomness was beyond analysis, that the best one could do was to assign average values for wavelengths, directions, etc. and to apply to these the classical laws derived for periodic motions, usually with but moderate success. Even Chapter XIV of "The Oceans" (1942) does not go far beyond this stage. Apart from the beneficial use of modern recording instruments, recent progress owes a great deal to modern research on the statistical analysis of random noise, developed by telecommuni- cation engineers. The study of sea waves has thereby developed into a combina- tion of time-series analysis with a sort of statistical geometry, though tied as firmly as possible to the basic laws of hydrodynamics. The fundamental newness of this approach is that the variables are treated not as analytical functions but as stochastic processes, definable only in terms of probabilities. Despite its apparent vagueness, this has been found to be the only way of coherently expressing the disordered oscillations of the sea ; it has not only explained many observed features but has already been successfully applied to important engineering problems connected with the sea, such as the motions of ships. As is to be expected, the material is scattered among a number of scientific papers, and the following pages are an attempt to give a connected account of the most relevant results. If we defer for a moment an exact definition of what we mean by a "spectrum" of sea waves, and accept intuitively that it represents the distribution of "energy", or mean-square oscillation, over a scale of frequency. Fig. 1 can be taken as a typical overall picture of the average distribution of energy present in an unfiltered recording of sea-surface level above a fixed point. The spectrum is seen to extend over a wide range of frequencies from the smallest ripples (capillaries) of some cycles per second to meteorological changes of a cycle in several hours. There are also long-term variations in sea-level extending over years, due to climatic and geological changes. Various zones of interest are [MS received December, 1959] 567 668 CARTWKIGHT [chap. 15 indicated, and these can mostly be isolated by using recording instruments sensitive only to a limited band of frequencies, or by using filters, digital or otherwise, in the subsequent analysis. In this general discussion of statistics we shall assume that some such filtering has been applied, so that we have only to deal with a spectrum which can be considered zero below a certain frequency and which converges suitably on integration to infinity. This avoids certain difficulties of minor importance which would otherwise detract from the main argument. As such the analysis will apply equally well to instrumental records from almost any part of the spectrum, though its most important application is to wind waves, which have received most attention in the literature. Excep- tions are the tides, which have line-spectra with well-defined phases, and transient phenomena, such as storm surges or waves from instantaneous explosions or eruptions. Other methods are available for dealing with such wave forms. The special problems associated with various bands of the wave spectrum are discussed in subsequent chapters. ^ 100 > IQ-' i 10-^ u 10 ^ >- fr 10" * 0.1 I 10 100 1000 10000 FREQUENCY, c/ks Fig. 1. Spectrum of sea-surface level above a point at La Jolla, California. (After Munk et al., 1957, Fig. 2.) SWEL . SEA -V SUF iGES SURF 1 \>^ \^ \ CAP LLARIES ^^\ 2. Fundamental Equations of Wave Motion We take rectangular axes x and y in the mean water surface and z vertically upwards. For irrotational motion, which we shall always assume to hold (and which is always valid in linear approximations), the velocity potential, {x, y, z, t), must satisfy the equations (Lamb, 1932) : dx^ dy^ 8z^ 0, (1) (2) where p is the pressure and p the density, assumed constant. (For non-uniform density see Chapter 22.) Since we are not here concerned with problems of generation, we may take the boundary equation to be ^ = 0 at the free surface, z = t,{x, y, t), and d(f>ldz = Q at a horizontal or very deep bottom, z= —h. (See SECT. 5] ANALYSIS AND STATISTICS 569 Chapter 21 for the effect of capillarity, neglected here for simplicity.) There is also the kinematic surface condition ^ + ^ + ^^ + ^^=0, , = ^. (3) dt dz dx dx dy 8y ' If the wave amplitude is small enough for the second-order terms in (2) and (3) to be ignored, then the following periodic solution is well known : , - ag cosh k{z + h) . , _ i ■ n (p = vrji, — ^^^ y"^'^ ^*-*^ " + y^ ^'^^ V — at+ e), (4) (J cosn fCih where 6 is an arbitrary direction of propagation, e an arbitrary phase angle, and the circular frequency a ( — 27T/period) is related to an arbitrary wave number k ( = 27r/wavelength) by the equation CT^ = gk tanh kh. (5) For practical purposes (5) is usually replaced by the simpler asymptotic form a" — gk when the depth is greater than about a quarter wavelength; alter- natively, if the depth is very small compared with the wavelength, we have the non-dispersive limit, (CT/^")= phase velocity = -y/grA. From (4) we obtain the surface elevation z = 0 which is a sinusoid of amplitude a with infinitely long crests parallel to the line y = xtaind, and the first-order horizontal and vertical velocity perturbations and pressure perturbations given by _^, _^, _^, and - ^ dx dy dz 8i respectively. Though this periodic solution is often used for rough calculations on sea waves, it is clearly unrealistic on account of its infinitely long crests and con- stancy of amplitude and frequency. A generalization of (4), also much used in analytical work, follows from Fourier's Integral Theorem, by means of a double integral over (0<^-oo 1 1 SECT. 5] ANALYSIS AND STATISTICS 573 It should here be remarked that although statisticians regard the Gaussian distribution as intuitive, there exist very few published demonstrations of its fitness. BirkhoflF and Kotik (1952) actually concluded from the available evidence that the normal distribution was inappropriate, and later acceptance is apparently based on some rather inconclusive examples shown by Pierson (1952) (see also Putz, 1954). The writer has attempted to fit normal distributions to sets of 2000 digital ordinates taken from records of wave height and slopes. Three examples are shown in Fig. 2. There is no obvious systematic departure from the normal curve, but, in fact, the y^ test for goodness of fit fails signifi- cantly in each case. However, satisfactory fitting has been found for distribu- tions which depend on the normal law (e.g. Cartwright and Longuet-Higgins, 1956), and there exist several other cases of agreement with average values calculated with the same assumption. On the whole, it appears safe to regard the normal distribution as a useful if not wholly accurate working rule. Fig. 2. Gaussian probability distributions fitted to sets of 2000 ordinates from ocean-wave records. Left to right; wave height (feet), up-wind wave slope (radians), cross-wind wave slope (radians). The vertical ordinates are of percentage of population per step. Of the various formulations described above for a linear statistical model of sea waves, we shall adopt the notation (6), at the risk of offending pure statisti- cians. Our reasons are that its physical meaning is clear, it embodies all the observed properties of stationarity, ergodicity (i.e. equivalence of time and space averages), and differentiability, and also because its mathematics is basically simple and easily handled by the average mathematical oceano- grapher without requiring knowledge of very specialized statistical techniques. 4. Properties of a Wave System in Terms of its Directional Energy Spectrum It is well known (Lamb, 1932, p. 369) that the mean wave energy per unit area of water surface, which in deep water is half kinetic and half potential, is given by the mean value of p^^2 oyer all x, y, t. On squaring the series (6), all the cross products average out to zero, and each squared term contributes a mean value Ipgan^ to the total energy. Thus, on comparison with (7), we see that pgE{k, 6) represents the mean energy per unit area contributed by wave components {k, 6) per unit increment of k and 6. This justifies the term "energy 574 CARTWRIGHT [CHAP, 15 spectrum", though the factor pg is usually dropped for convenience, so that E{k, 6) has dimensions L^. By definition E{k, 6) is essentially positive and we assume, as mentioned in the introduction, that it tends to zero for large k in a, suitably convergent manner. For some purposes it is more convenient to use the energy density with respect to frequency and direction, or with respect to component wave numbers u = k cos 6, v = ksin6, yielding directional energy spectra E'{a,d), E"{u,v), respectively, with the relations E'{a, d) = {dkjdo)E{k, d), E"{u, v) = k-^E{k, d). E"{u, v) is most useful for definining the statistical properties of the wave surfaces ^{x, y, t) (Longuet-Higgins, 1957), particularly in terms of the moments 1*00 1*00 rripq = uvv9E"{u, v) du dv. (10) J-OO J -00 As a rule most statistical properties depend only on a small finite number of these moments, regardless of whether moments of higher order exist or not. We have already seen that moo represents the total "energy" per unit area, or strictly, the mean square deviation of the surface height (either in time or in space) from its mean level, which from (6) is obviously zero, (mio/moo) and (moi/wioo) are the co-ordinates of the centroid of the spectrum, so that the wave system may be roughly described as having a mean wave number k = ■\/{miQ^ + moi^)lmoo propagating in the mean direction ^ = tan~i (moi/mio). The second moments about the centroid, ju-ii = (miiWoo-wioWoi)/moo (11) /X02 = (wo2Woo-moi2)/moo, could be used to describe a sort of inertia ellipse, definining the mean square spread of the spectrum. A more meaningful expression is derived by con- sidering the intersection of the wave surface with a vertical plane inclined at d to the a:-axis. It is easily shown (Longuet-Higgins, 1957) that the number of zero-crossings of ^ per unit distance in this plane, which is proportional to the root-mean-square wave number along the section, is [(W20 cos2 6 + 27)111 cos 6 sin ^-|-mo2 sin^ ^)/moo]'^S which is maximum and minimum in two perpendicular directions given by tan 2^p = 2mii/w2o — Wo2. (12) We may call the value of dp which gives the maximum the "principal direction" of the waves, for obvious reasons, and the ratio of the minimum to the maximum - m20 + Wi02-[(W20-W02)2-f 4Wii2]'/2 /'^ = ^20 + nio'i + [(m2o - Wo2)^ + 4mii2]'/2 (13) SECT. 5] ANALYSIS AND STATISTICS 575 the "long-crestedness" of the wave system. If y2 = o we have m2oWo2 = Wii2, and clearly the waves have infinitely long crests propagating in the direction dp (or TT-\-dp), while if y2= i the waves are very short-crested, in fact isotropic, the spectrum having circular symmetry about the origin. For sea waves, of course, y^ always lies well inside these two limits. If the spectrum is fairly concentrated about its centroid, it can be shown that y2<^ 1 and approximates to the mean square angular deviation of the spectrum about Op. The expression 'm2o cos2 6 + 2mii cos 9 sin 6 + mo2 sin^ & fX20 cos2 ^ + 2/xii cos 6 sin d + [xo2 sin^ d (14) can be shown (Longuet-Higgins, 1957, p. 373) to be a measure of the mean number of waves in a "group" in the direction 6, most significant of course when 6 = dp. By partial differentiation of the series (6) we see that m^o and mo2 are also the mean square component surface slopes, d^/dx, dt,jdy, while m\\ is their CO variance. That is, /an 2 /an 2 idi di m2o = (^j , mo2 = [jy) , mn = (^ ^ where the bars represent mean values over time or space. If, however, we expand the series for ^{d^jdx), say, equal arguments 0w occur only in the form cos (f)n sin (f)n, which have zero mean values. Thus the surface is uncorrelated with its slopes, and similarly with its first time derivative. In general the variances and covariances of any pair of spatial derivatives of ^ are either zero or of the form ± nipq, where p + q is even, and variances and covariances of measures involving time derivatives of ^ are zero or of the form + mpq^^\ where p + q is odd and the suffix r refers to moments of the function a^E"{u, v). It is important to note that odd-order moments must involve time derivatives, without which it is impossible to distinguish between E"{u, v) and E"{ — u, —v) or between E{k, 6) and E{k, d + ir). In virtue of the random phases, €n, oi the components of the series for ^ and formally similar series giving its space and time derivatives, any group of such quantities has a joint normal probability distribution, expressed entirely in terms of the variances and covariances between the quantities. For example, , . . . , 1 ( ii^ W0262-2mii^26 + W20M ,,.. P(6,l2.6)-(,^)3,^^^,,^^,,exp|-^^^ ^^ 1, (15) where Az = W20W02-Wii2, and a great many other such distributions could be written down similarly. 576 CAKTWBIGHT [CHAP. 15 All these probabilities are independent of time and space over the zone for which the wave system is assumed stationary, provided t, and its derivatives are measured simultaneously at the same point. Quantities taken at fixed spacings or time intervals have similar distributions, but will not be discussed until the next section. Probability distributions of many quantities describing the geometry of the surface can be evaluated from the normal distribution of the derivatives of ^ (Longuet-Higgins, 1957), but are mostly outside the scope of this chapter. We shall, however, mention a few t3rpical results involving only moments of order 2 or 4. The distribution of absolute slope a is p{a) = aZJa"'^' exp {-a2(wi2o + mo2)/4J2}/o{a2(m2or*^02)/4/l2}, (16) where Io{x) is the Bessel function Jo{ix). The mean length per unit area of contour of height ^ is 7r-i(mo2 + W2o/moo)'/^ exp (-^2/2moo)(l +y2)-'/2E{(l-y2)-/.), (17) where E is an elliptic integral and y^ is defined by (13). The distribution of curvature is important when considering the nature of light reflection in the wave surface. In Longuet-Higgins (1958), it is shown that while the "mean curvature" J = dmdx^+emdy^ has, as is easily seen, the normal distribution p{J) = (27ri))-'/^e-'^'/2^, D = m4o + 2m22 + mo4, the "total curvature" dx^ dy^ \dx By} determining the intensity of reflections and the sizes of images, is far from normal. The actual distribution of Q is sharply peaked at the origin and skew, and involves an awkward integral evaluated by Catton and Millis (1958). However, it depends only on moments of order 4. Other properties (derived in Longuet-Higgins, 1957) involve the spatial densities of maxima and minima (whose sum, by an important theorem, is equal to the density of "saddle points"), the velocities of zeros, of contours, and of points with a given gradient, and various properties of the wave "envelope" (an important concept when the spectrum is narrow). All these are expressed in terms of moments nipq and mpq^^'> up to a finite order. 5. Estimating the Directional Energy Spectrum We have just seen that, given the spectrum E{k, 6) or E"{u, v) of a wave system, many statistical properties of the surface can be deduced in terms of the moments of the spectrum. Conversely, from statistical analysis of various SECT. 5] ANALYSIS AND STATISTICS 577 types of measurement of the surface, we may deduce values for some of these moments, or groups of moments, which may themselves give sufficient informa- tion. In theory the complete set of moments of a function determines the function uniquely, and Longuet-Higgins (1957), in fact, suggests a method of obtaining an approximation to E"{u, v) by estimation of m^pq and rripq up to finite order. However, this method does not seem very practicable, since high- order moments are sensitive to instrumental noise, and the most practical methods are those based on estimation of either the autocovariance function or the angular harmonies of E{k, 6). The autocovariance can be defined by i/j{X, Y, r) = Ux, y, mx + X, y+ 7, t + r), (18) the bar denoting a mean over all x, y and t, although averaging over t alone with fixed values of x and y is also satisfactory. From (6) it can be deduced that ^(X, Y, t) is also expressible in the form Too /*« ijiiX, Y, t) = E"{u, v) cos {uX + vY- or) du dv, J-co J-co (19) as is well known. In some types of measurement (for example, Cote et al., 1960) only instan- taneous values of i,{x, y, t) can be measured, from which one can evaluate j/((Jl, Y, 0) by averaging over many values of a; and y. From this can be obtained the even function F{u,v) = \{E"{u,v)-\-E"{ — u, —v)] by means of a Fourier transform (Khintchine, 1934) : 1 /'oo /^oo F{u, v) = -—\ MX, Y, 0) cos {uX + vY) dXdY. 477'^ j-oo j-ao This may be sufficient if it can be assumed that E"{u, v) is zero over two adjacent quadrants (that is, all the wave energy is being propagated within + 90° of a single direction), but for a complete estimation of E"{u, v) we also need the odd function G{u, v) = \{E"{u, v) — E"{ — u, — v)]. This may be obtained by using appropriate values of t : G{u, v) = -i- P I i/r(X, Y, TTJ'Ia) sin {uX + vY) dX dY , (a - aiu, v)), 477^ j-00 j or by measuring 8^1 dt and forming the covariance function 0'(X, Y, 0) = U^, y, t){clctK{x + X, y+ Y, t) = g{u, v) E"{u, v) sin {uX-^vY) du dv, from which G{u,v) = -—4 : r U'iX, Y, 0) sin {uX + vY)dXdY, 47T^a{u, V) j-00 j ^ since cr( — w, —v) = a{u, v). We then have, of course, E"{u, v) = F{u, v) + G{u, v). 578 CART\^TIIGHT [CHAP. 15 The above is somewhat an ideahzation of the practical problem of estimating E"{u, v), since very specialized equipment borne by aircraft is needed to make sufficiently extensive surveys of l,{x, y, t), and all proposed methods (see dis- cussions by Longuet-Higgins and Pierson follo\Wng Munk et ah, 1957) have their limitations. More conventionally, we have to deal with a small number of fixed detectors recording continuously in time. Suitable anah^ical methods for such records are mostly due to Barber (1954, 1957, 1958, 1959). The first stage is to perform a cross-spectral analysis with respect to frequency between the various records ; this, in effect, filters the complex covariances between pairs of records into conveniently narrow bands of frequency. We may briefly explain the cross-spectral process \>y denoting a pair of simultaneous related records derived from our wave system by Lit) = y Zra cos (cT„f-l-A„-he„) n M{t) = 2 m„ cos {(Jnt + [Xn + en), (20) n where A„ and [Xn are phase angles depending on the type of detectors and their position relative to the wave system, and e« are as usual random. Then the cross-spectral components of these two records are : Cii{a)Sa = 2 ¥n^> Cmm{cr) Sa = 2 |w„2, da da Cim{cr) So- = 2 pK^ra COS (A„ — /x„), Qim{cr) Sff = 2 l^w^w sin (A^ — /Xn). 6ct da Cii and Cmm are the energy spectra mth respect to frequency alone of the individual functions, while Cim, Qim are their "co-spectrum" and "quadrature spectrum", sometimes called the real and imaginary parts of the cross-spectrum, and respectively equivalent to the covariance of L and M with and without a phase shift of 7r/2. In practice, as sho\^Ti in section 7 of this Chapter (page 584), we obtain weighted sums centred on the interval ha according to the type of filter used, and not a simple sum as indicated in (20), but the principle is the same. The cross-spectral estimates divide the directional spectrum into narrow rings of nearly constant k = k{a), and for any one of these rings we are con- cerned therefore only with a function of 6, E{k, d) = Ek{d) say. It is convenient to expand Ek{B) in a Fourier Series : Ek{e) = lAo{k)+ 2 {Ar(k) cos rd + Br{k) sin rd), r=\ where Ar{k) -f iBr{k) = i I '" Ek{d) €*'•« dd. (21) SECT. 5] ANALYSIS AND STATISTICS 579 If we iioM consider the records of ^(f) from any two detectors placed along the .r-axis with separation d, viz. L{t) = C(0, 0, 0 = 2 «« cos {cTnt-en), n M{t) = l,{d, 0, <) = 2 «n COS {ont-dkn COS On-^n), n the cross-spectral components at wave number k can be expressed as Cilia) = Cmm{ 0 at ri evenly spaced instants, which gives good results, and also derives the useful approximate formula I 277i\^o F{T)di ijj{r)jilj{0) = cos where F{t) = r p{r) dr, P{t) being the distribution of zero intervals and rm its median. 582 CARTWRIGHT [chap. 15 The heights of the waves themselves as measured by l^{t) are often measured as a quick and obvious way of estimating the total wave energy, mo = moo- Opinions differ, however, as to the most suitable definition of a "wave height". One may consider the total wave height, or difference in height between a crest and the preceding trough, or, alternatively, the heights of maxima above or minima below the mean level ^ = 0. Examples of both types are shown in Fig. 3. The former has intuitive appeal, and it can be shown (Longuet-Higgins, 1952) that when E{g) is fairly concentrated about a single frequency, so that ^t) has the appearance of a pure sine wave with slowly varying amplitude, the total wave heights 2a have the "Rayleigh distribution" p{a) = (a/wo) e-''2a-/mo. (27) A3 A4 Bl B2 B3 B4 Fig. 3. A trace of an actual wave record showing some selected typical values of ^,;j, height of maximum (Al-4, single arrows), and 2a, total wave height (B 1-4, double arrows). Note that A2 is negative, and that for B2 and B4, 2a ^^m- In some conventions for measviring total wave height, B2 and B4 would be ignored. If it is not assumed that the spectrum is narrow, then it seems to be almost impossible to derive a theoretical expression for p{a), though (27) fits many ocean-wave records fairly well (Watters, 1953). On the other hand, by consider- ing the joint normal distribution of ^t), t,'{t) and !l,"{t), the exact distribution of the heights of maxima ^m can be derived for any shape of spectrum. In the notation of Cartwright and Longuet-Higgins (1956), it is p{-q) = (277)-'4ee-'/^''V + (i_e2)>/.^e-y.^2 r va-(')'-/e ,-i/.^2 e-y^^ dx (28) where -q = ^m/Wo'^S e2 = (moW4 - W22)/WoW4. The parameter e is a measure of the spectral width relative to its mean frequency, and is most easily obtained from the relation No^INi^ = l-e2. (29) Curves of ^^(-17) for a range of values of e from 0 to 1 (its entire range) are shown in Fig. 4 ; the lower limit e = 0 corresponds to a very narrow spectrum, and gives p{r}) = rje~y-'^ (rj'^O), 0 (17 < 0), again a Rayleigh distribution. Note that in general p{r]) admits negative values of i,m, of proportion [| — 1(1 — e^)'/^] to the total, while 2a is essentially positive. The distribution (28) has been successfully applied (Cartwright and Longuet-Higgins, 1956) to wave records with values SECT. 5] ANALYSIS AND STATISTICS 583 of € from 0.20 to 0.67, including some cases where 2a does not fit the distribu- tion (27). Since the minima of ll,{t) also have the distribution (28) with reversed sign, the mean value of 2a, 2a say, is given exactly by 2a = 2U = 2mo'^^ = [27rmo(l - e2)]'-, Equation (27) gives 2a = (277mo)'S which is consistently too high by the factor Ni/Nq. This discrepancy is to some extent reduced by ignoring waves below some arbitrary size in measuring 2a (Pierson, 1954), but the exact relation between the mean value so derived and the mean obtained by counting all crests and troughs is not easily defined. In any case the differences become less important when e <^ 1 . Fig. 4. The family of probability distributions of maxima (28). The case e = 0 is a Rayleigh distribution, and the case e = 1 is a Gaussian distribution. A very sensitive recording of (,{t) may show high-frequency ripples, which by increasing W4 relative to the other moments may give a value of e only a little below 1. In this case, the mean of ^m is not a good measure of mo, since it contains the factor {l — e^y-^, but a better measure is the mean square ^m^, which has the expectation mo (2 — e^). Equation (27) gives a'^ = 2mo, but the exact relation between ^^^ a^id a^ is unknown. Other quantities used as proportional measures of mo'^ are the mean of one- third highest and one-tenth highest wave heights. The former is often called the "significant wave height", though with no real justification. If 2a is used for wave heights, then the above quantities can be shown, from (27), to have the expectations 4.004mo'- and 5.091wo'- respectively provided e is small. The corresponding factors for t,m are very nearly half these values for e ^ 0.3, but decrease slowly with increasing e. A similar statistic, even more easily obtained, is the largest of N wave heights. The ratio of its expected value to mo-'^ increases with N but very slowly. For wave heights 2a and values of N greater than about 20, this ratio is very nearly, for small e, 2'/^{\n N)y^ +2yy{h\ N)-y^, 584 CARTWRIGHT [CHAP. 15 where y is Euler's constant 0.5772. The same expression halved, and with iV(l — e2)'/2 (i.e. approximately half the number of zeros) put in place of N, applies to the maximum value of ^r«- Cartwright (1958) shows that the standard error of the estimate is approximately 0.64/ln N times its expectation, and also works out formulae for the second and third highest waves (which have smaller standard errors) and the reduction in the effective value of N when successive wave heights are highly correlated, as in the case of a very narrow spectrum. 7. Spectral Measurement A. Variability of Spectral Estimates In the preceding sections it has been tacitly assumed that the energy spec- trum can be evaluated to any required accuracy provided proper measurements are made. In practice we can only make statistical estimates of the spectrum the expectations of which are close to E{k, 6) or E{g), but which have a degree of variability depending on the quantity of data used. This variability was first seriously considered by Tukey (1949). We shall now briefly discuss the main criteria of variability, referring principally to the case of a single variable with spectrum E{a). Estimates of directional energy spectra have similar properties but are more complicated, and depend on the method used. Goodman (1957) derives useful results for the variability of cross-spectra. Consider first an estimate of the total spectral energy mo, derived from a record t,{t) of duration T. E{T)total = ^ J^ CHt) dt. This quantity has expectation mo, and it can be shown (Rice, 1945; Tucker, 1957) that it has a "coeflicient of variation", C.V., defined as (Standard errors expectation) 2 given by C.V. = 2, r E^a) da Tmo^ (30) provided T is long enough to contain more than about ten waves. If E{a) is constant over a band of width aw and zero elsewhere, (30) gives C.V. = 277/crM,T, showing that the narrower the spectrum the longer duration of recording is required to obtain a specified level of variability. Further, since the standard error decreases only with T~'/2, the duration has to be greatly increased to improve matters considerably. For other spectral shapes, (30) gives less simple results, but it is convenient to express C.V. in general as 'l-rrjaeT, where o-g is an "equivalent spectral width", and is very roughly proportional to the standard width (moW2 — mi2)'/2/mo. Exactly similar considerations apply when only a filtered portion of the SKCT. 5] ANALYSIS AND STATISTICS 585 total energy in E{a) is estimated. The standard method of evaluating E{g) is in fact to operate on ^(/)(0 ^t ^T) by a process whose result has expectation Too 8((7) = f{a'-a)E{a')da', where /(a>) is a filtering function of total integral unity and is large only within a fairly narrow band centred at a> = 0. The C.V. for £(ct) is 1^ p{) depends on the method of analysis, analogue or digital. In analogue methods, l,{t) is converted to an electrical voltage (greatly speeded up) which is fed through a tuned circuit. The characteristic curve of the power output of the circuit is thenf{a>), and is varied by altering the components of the circuit. Chang (1954) discusses the criteria for the choice of various types of tuned circuits. To cover the whole spectrum, ll,{t) may be fed through a number of circuits in succession, each tuned to a different frequency, or as in the case of a photo-electric analyser used at the National Institute of Oceanography (Barber et al., 1946; Tucker, 1956) a single filter is used and the time scale of t,{t) varied. The latter method actually gives the Fourier series amplitudes of ^(^)(0 ^t^T) spaced in frequency by o- = "Ztt/T. The square of any one of these amplitudes has C.V.= 1, but the sum of squares of n consecutive amplitudes has C.V.= 1/n and gives a fairly good approximation to a square-topped filter with coe = 27TnlT (Tucker, 1957). The Fourier series method (often called periodogram method) can also be applied to digital records of (,{t), but if the series is long the computing time becomes prohibitive. It is more economical to compute first the autocovariance 0(t), for m equal steps of t, and then the cosine periodogram of these m+1 586 CARTWRIGHT [CHAP. 15 values. If, for example, we have ^t) for^ ^ = 0, S, 28, . . . nS, we compute (electronically) for^ = 0, 1, 2, ... w, I 9=0 5=0 9=0 J which is (25) modified for an artificial non-zero mean of ^(0- The periodogram operation on ipipS) yields ^(S) = ^{'A(0) + (-l)^) has large oscillations which tend to zero rather slowly as |a»| increases. A much better filter is obtained by taking ^''lil = J^ (^-1)77 wS mS P^\^i + ie ip + ^Y' mS which gives „ w8 sin mojS lOe — 277 m(oS[l — (ma)8/77)2] 3m8 This is a somewhat broader filter whose oscillations are, however, negligible outside a>= ± 27T/m8. (For other filters, and in fact for a full discussion of the problem of spectral measurement, see Blackman and Tukey (1958).) The same shape of filter applies to cross-spectral estimates between two signals ^i{t) and ^2{t), though the variability is more complicated (Goodman, 1958). In analogue form one process is to measure the power of the sum ^i(i) -f- ^2(0 after passing through various filters. This is an estimate of {Ei{or) + E^i^) + 2[E i{a)E 2{or)]y2 cos 6{(r)] where d{u) is the phase angle between the signals. To obtain the sign of 6, a phase increment can be introduced to one of the signals. In digital analysis the filtering procedure is similar to that outlined above ; the co-spectrum is obtained from the cosine periodogram and the quadrature spectrum from the sine periodogram of the co variance function of l,i{t) and ^2(0> with 2) ranging from —m to +m. 8. Second- Order Approximations to Energy Spectra^ Finally we consider some approaches that have been made to account for the neglected second-order terms in the basic equations of wave motion (section 1 It is necessary that S be less than half the smallest time of oscillation of l,{t) to avoid the "aliassing" effect (Blackman and Tukey, 1958). 2 Since this section was written, the analysis of non-linear interactions in random wave systems has been considerably extended by Phillips (1960, 1961). Interested readers should also study the papers by Hasselman, Phillips, and Tick, and relevant discussions, presented at the National Academy of Sciences Conference on "Ocean Wave Spectra", Easton, Maryland, U.S.A., May 1961. SECT. 5] ANALYSIS AND STATISTICS 587 2 of this Chapter). Higher approximations to a purely periodic wave, such as those of Stokes and Gerstner, are not much help in dealing with random wave patterns, but the basic method of taking the linear spectrum as first approxima- tion and substituting this into the neglected terms of the equation of motion, to obtain a better approximation, is still valid. This procedure is worked out by Tick (1959) for a unidirectional wave pattern recorded at fixed points. If we drop the terms involving y in equations (1), (2) and (3), and eliminate ^, the following surface condition is obtained for 2 = 0, correct to second order : d^ dcf> _ 8^ dcf> 8(f> 8^ 8cf> 8^cf> 1 / 8^ 8^ 8^ 8(t>\ 'W~^~8z ~ '82^~8i~ 'dxdxdt~'8z'8zJt~g\8zJtW^8z8t^^j ^ ' With the right-hand side put equal to zero, we can develop a linear model of random waves as described in section 3, with an energy spectrum Ei{a) of surface height measured at a point. Substitution of the latter into equation (32) then yields a spectrum Ei{a) + E2{(y), where ^2(0-) = — {G + 2aj)^Ei{G+co)Ei{aj)dco + —((J^-2cTco + 2co^)Ei{a-cjo)Ei{co)daj. (33) Jo r The first of these two integrals is usually much smaller than the second, which is greatest for frequencies a little over twice the peak frequency of Ei{a), while both are usually small compared with Ei{a). Longuet-Higgins (1950) also obtained a double-frequency effect for the second-order pressure variations in deep water due to standing-wave com- ponents in the spectrum. The importance of this effect is that, though as a rule negligible at the surface compared with the first-order pressure, the second- order pressure variations persist to great depths while the former decrease exponentially and ultimately become negligible. It was shown that the principal effect consists of an integral of the product of spectral densities at wave numbers opposite in sign, with twice their associated frequency. A spectrum was worked out by Kierstead (1952). If we disregard second-order effects in waves at the surface, non-linear terms may still be introduced by the recording apparatus. An important example studied by Tucker (1959) is the signal from an accelerometer moving with the wave surface but aligned normally to it and therefore not truly vertical. The spectrum of the doubly-integrated signal is to first order the true spectrum of wave height E\{a), but when second-order terms are included, an error spectrum is introduced, given by M<^)=^Vt- co\a-ojYE^{ay)E^{a-co)doi, [^i( - co) = ^1(0;)], (34) where R2^ = {A2^ + B2^)IAq'^^\ is a parameter depending on the angular spread in the directional spectrum E{k, 6), as defined in section 5 of this Chapter. 688 CARTWKIGHT [CHAP. 15 The principal part of 62(0) occurs at low frequencies, where it can cause swell superposed on a wind-wave system to be obscured. It should be remarked that with a freely floating recorder of this type the second-order effects derived by Tick for a fixed recorder (above) do not apply, and. are in fact considerably reduced. Other types of spectra produced by non-linear instrumental response are considered by Rice (1945). References Barber, N. F., 1946. Measurements of sea conditions by the motion of a floating buoy. Admiralty Res. Lab. ARL/103.40/N2/W (unpublished). Barber, N. F., 1954. Finding the direction of travel of sea waves. Nature, 174, 1048-1050. Barber, N. F., 1957. Correlation and phase methods of direction finding. N.Z. J. Sci. Tech., 38, 416-424. Barber, N. F., 1958. Optimum arrays for direction finding. N.Z. J. Sci., 1, 35-51. Barber, N. F., 1959. A proposed method of surveying the wave state of the open ocean. N.Z. J. Sci., 2, 99-108. Barber, N. F. and F. Ursell, 1948. The generation and propagation of ocean waves and swell. Phil. Trans. Roy. Soc. Londofi, A240, 527-560. Barber, N. F., F. Ursell, J. Darbyshire and M. J. Tucker, 1946. A frequency analyser used in the study of ocean waves. Nature, 158, 329-332. Birkhoff, G. and J. Kotik, 1952. Fourier analysis of wave trains. Ch. 25 of "Gravity Waves". Nat. Bur. Standards Circular, No. 521, 221-234. Blackman, R. B, and J. W. Tukey, 1958. The measurement of power spectra. Dover Pubis., New York. Cartwright, D. E., 1956. On determining the directions of waves from a ship at sea. Proc. Roy. Soc. London, A234, 382-387. Cartwright, D. E., 1958. On estimating the mean energy of sea waves from the highest waves in a record. Proc. Roy. Soc. London, A247, 22-48. Cartwright, D. E. and M. S. Longuet-Higgins, 1956. The statistical distribution of the maxima of a random function. Proc. Roy. Soc. London, A237, 212-232. Catton, D. and B. G. Millis, 1958. Numerical evaluation of the integral ■"ioo (\a^ + a^ — lyVi e-"'" da. 277 J -I Proc. Cambridge Phil. Soc, 54, 454r-462. Chang, S. S. L., 1954. On the filter problem of the power-spectrum analyser. Proc. I.R.E., 42, 1278. Cote, L. J., J. O. Davis, W. Marks, R. J. McGough, E. Mehr, W. J. Pierson, J. F. Ropek, G. Stephenson and R. C. Vetter, 1960. The Stereo Wave Observation Project. New York Univ. Meteorological Papers, 2. Coulson, C. A., 1944. Waves (3rd Edn.). Oliver and Boyd, Edinburgh. Cramer, H. E., 1946. Mathematical Methods of Statistics. Princeton Univ. Press, U.S.A. Davies, T. V., 1951. The theory of symmetrical gravity waves of finite amplitude, I. Proc. Roy. Soc. London, A208, 475. Doob, J. L., 1953. Stochastic Processes. John Wiley, New York. Ehrenfeld, S., N. R. Goodman, S. Kaplan, E. Mehr, W. J. Pierson, R. Stevens and L. J. Tick, 1958. Theoretical and observed results for the zero and ordinate crossing problems of stationary Gaussian noise with application to pressure records of ocean waves. New York Univ. Goodman, N. R., 1957. On the joint estimation of the spectra, co-spectra and quadrature spectra of a two-dimensional stationary Gaussian process. Sci. Paper No. 10, Eng. Stat. Lab., New York Univ. SECT. 5] ANALYSIS AND STATISTICS 589 Ichiye, T., 1950. On the theory of Tsunami. Oceanog. Mag., 2, 83-100. Kampe de Feriet, J., 1958. Statistical mechanics of two-dimensional waves with finite energ\\ D. Taylor INIodel Basin. Washington, Report 1230. Khintchine, A., 1934. Korrelationstheorie der stationare stochastischen Prozesse. Math. A7}n., 109, 604-615. Kierstead, H. A., 1952. Bottom pressure fluctuations due to standing waves in a deep two- layer ocean. Trans. Amer. Geophys. Un., 33, 390-396. Lamb, H., 1932. Hydrodynamics (2nd Edn.). Cambridge Univ. Press. Longuet-Higgins. M. S., 1950. A theory of the origin of microseisms. Phil. Trans. Roy. Soc. London, A243, 1-35. Longuet-Higgins, M. S., 1952. On the statistical distribution of the heights of sea waves. J. Mar. Res., 9, 245-266. Longuet-Higgins, M. S., 1957. The statistical analysis of a random moving surface. Phil. Trans. Roy. Soc. London, A249, 321-387. Longuet-Higgins, M. S., 1958. The statistical distribution of the curvature of a random Gaussian surface. Proc. Cambridge Phil. Soc, 54, 439-453. Longuet-Higgins, M. S., 1958a. On the intervals between successive zeros of a random function. Proc. Roy. Soc. London, A246, 99-118. Marks, W., 1954. The use of a filter to sort out directions in a short-crested Gaussian sea surface. Trans. Amer. Geophys. Un., 35, 758-766. Munk, W. H., F. Snodgrass and G. Carrier, 1956. Edge waves on the Continental Shelf. Science, 123, 127-132. Munk, W. H., M. J. Tucker and F. E. Snodgrass, 1957. Remarks on the ocean wave spectrum. Ch. Ill of Naval Hydrodynamics. Nat. Acad. Sci. Washington, Publ. 515, 45-60. Phillips, O. M., 1960-1961. Unsteady gravity waves of finite amplitude. I. The elementary interactions. II. Local properties of a random wave field. J. Fluid Mech., 9, 193- 217; 11, 143-155. Pierson, W. J., 1952. A unified mathematical theory for the analysis, propagation ayid refraction of storm -generated ocean waves. Pts. I and II. New York Univ. Pierson, W. J., 1954. An interpretation of the observable properties of sea waves in terms of the energy spectrum of the Gaussian record. Trans. Amer. Geophys. Un., 35, 747-757. Pierson, W. J., 1955. Wind generated gravity waves. Advances in Geophys., 2, 93-178. Academic Press Inc., New York. Putz, R. R., 1954. Statistical analysis of wave records. I.E.R. Univ. of California Ser. 3. Issue 359. Rice, S. O., 1945. Mathematical analysis of random noise. Bell System Tech. J ., 23, 282-332 (1944) and 24, 46-156 (1945). Rosenblatt, H., 1957. A random model of the sea surface generated by a hurricane. J . Math, and Mech., 6, 235-246. Stoker, J. J., 1957. Water Waves. Interscience, New York-London. Tick, L. J., 1959. A non-linear random model of gravity waves I. J. Math, and Mech., 8, 643-652. Tucker, M. J., 1956. The N.I.O. wave analyser. Ch. 13 of Proc. 1st. Conf. Coastal Eng. Instr., Berkeley, California. Tucker, M. J., 1957. The analysis of finite length records of fiuctuating signals. Brit. J. Appl. Phijs., 8, 137-142. Tucker, M. J., 1959. The accuracy of wave measurements made with vertical accelero- meters. Deep-Sea Res., 5, 185-192. Tukey, J. W., 1949. The sampling theory of power spectrum estimates. Symp. on applica- tion of autocorrelation analysis to physical problems, p. 47. Woods Hole, Mass. Watters, J. K. A., 1953. Distribution of height in ocean waves. N.Z. J. Sci. Tech.. 34, 408-422. 16. LONG-TERM VARIATIONS IN SEA-LEVEL J, R. ROSSITEE, 1. Introduction The variations of sea-level dealt with in this Chapter can be said to fall into two categories, periodic and secular. In the first category the lower limit of period may be arbitrarily defined as the day, thereby eliminating the diurnal and shorter-period oscillations of classical tidal theory but retaining the long- period tides ; the upper limit is restricted only by the span of data available for study. Secular variations in sea-level will be found no matter how short a span of time is taken, and in view of the dominating tidal effects found in all oceans and the majority of seas the minimum span of time adopted is the day. In this Chapter, therefore, it is sought to discuss variations in level represent- ing time units of a day upwards, the most commonly used units being the day, the month and the year. Despite the increasing attention being given to the subject, it cannot be said that any revolutionary discoveries have been made during the last few decades by research workers, though certain trends in the types of research undertaken may be discerned. Fewer attempts are being made to examine and explain variations at individual stations in favour of regional and global investigations involving data from many stations. The greatest difficulty encountered by those engaged in this type of work lies in the markedly uneven distribution of tide gauges throughout the oceans and seas ; as might be expected most gauges are concentrated in areas such as European, North American and Japanese waters. Under the stimulus of the International Geophysical Year (1957-58) a great effort has been made to improve the situation, with considerable success, but much still remains to be done. 2. The Determination of Mean Sea-Level The raw material for mean sea-level research is a continuous record of heights of tide, accurately observed at fixed intervals of time, and referred to a stable bench mark. No apology is made for stating these rather obvious facts. Experience has repeatedly shown that much labour and money has been, and continues to be, wasted in taking observations without scrupulous regard to the above requirements. Modern demands of different scientific disciplines and of industry in this field appear to be for longer series of observations, at more stations and to a higher degree of accuracy, and it is a regrettable fact that only by great efforts made now will data conforming to these standards be available for research workers a generation hence. As remarked in the introduction, the astronomical tidal constituents with periods up to and including the diurnal species must first be eliminated from the observations. This may be achieved by various methods, the simplest being a direct numerical averaging process, usually applied to hourly heights of tide. More sophisticated numerical processes are frequently used, employing filters which discriminate as required against any given species of tide. The best [MS received May, 1960] 690 SECT. 5] LONG-TERM VARIATIONS IN SEA-LEVEL 591 known of these are probably those of Doodson (1928) and Groves (1955). Integration of a tide chart by planimeter is also used by some authorities. In order to avoid the labour involved in the preceding methods, "mean tide level" is sometimes computed instead of mean sea-level, being defined as the average of the observed high and low waters. For all tidal waters, however, mean tide level must be considered a poor substitute for mean sea-level in that the short-period tidal contributions are not adequately eliminated. All the fore- going methods provide daily mean values of level, and Rossiter (1958) has given a resume of their efficiency in reducing contributions from the shorter-period tides. An account is also given of a labour-saving filter which operates on heights at 3-hourly intervals with an efficiency equal to the averaging of 24 hourly heights. Monthly and annual values derived from the daily means are generally obtained by simple averaging, and the results for a world-wide network of stations are published at regular intervals by the Permanent Service for Mean Sea Level in the Publications Scientifiques series of the International Association of Physical Oceanography. Table I Maximum Contribution from Certain Tidal Constituents, Expressed as Percentages of Their Amplitudes, to the Means for (a) a 30-day month and (6) a 365-day year Mg N2 Ki Oi M4 Me Msf (a) 0.055 0.209 0.267 0.401 0.058 0.060 1.55 (b) 0.035 0.005 0.000 0.072 0.023 0.008 1.27 M2 : principal krnar semidiurnal tide, period 12.42 hours. N2 : lunar elliptic semidiurnal tide, period 12.66 hours. Ki : lunar declinational diurnal tide, period 23.93 hours. Oi : lunar declinational diurnal tide, period 25.82 hours. M4 : lunar quarter-diurnal tide, period 6.21 hours. Mg : lunar sixth-diurnal tide, period 4.14 hours. Msf: fortnightly tide, largely arising from shallow water theory, period 14.8 days. It has been customary to determine monthly and annual means to 0.001 ft or 1 mm according to the units employed, but this tends to give a false impression of accuracy. Table I illustrates the maximum contributions ^ of certain tidal constituents, expressed as a percentage of their amplitudes, to the mean of a 30-day month and a 365-day year, using a simple average of 24-hourly heights to the day. It will be seen that for a moderate amplitude (5 ft) of M2, the principal lunar semidiurnal constituent, there will be maximum contributions 1 These contributions will be associated with a cosine term, the argument of which will, in general, take up all values between 0 and 2tt. 592 KOSSiTER [chap. 16 to a monthly mean and an annual mean of the order of 0.003 ft and 0.002 ft respectively. If the gauge record can be read to an accuracy of 0.05 ft, random errors in reading the charts will give standard deviations for monthly and annual means of about 0.0019 and 0.0005 ft respectively. Much larger errors can quite easily be introduced, however, through inadequate maintenance of the gauge as, for example, by incorrect setting of the recording pen by 0.05 ft for a few months at a time. It therefore appears that, unless the most careful attention is paid to the maintenance of tide gauges and the reduction of their records, monthly and annual means can only be considered accurate to about 0.01 ft (3 mm) and 0.003 ft (1 mm) respectively. 3. Causes of Variations in Sea-Level By definition sea-level is the height of the boundary between sea and air, measured relative to a fixed point (bench mark) on land, and hence is susceptible to changes with time in A. The distribution of oceanographical factors, B. The distribution of climatological factors, C. The distance of the fixed point from the centre of the earth. To these must be added D. The long-period astronomical tides. Let us consider these four main causes separately, where possible. A. Oceanographical Factors It is well known that surface gradients must exist in all bodies of water on a rotating earth as a result of the fundamental dynamical relationships existing between gradient currents, density currents and isobaric surfaces, but the time variations of these phenomena in the open oceans cannot be studied in much detail in the absence of adequate observations of current, temperature and salinity in depth and in time. Only on continental shelves and in marginal seas does this seem feasible at present. Nevertheless, within recent years much valuable work has been done on seasonal variations in sea-level due to time changes in temperature and salinity, by comparing observed values with computed "steric" levels. Steric changes are defined as those resulting from changes in density of a water column without change in its mass and are produced, for example, when wind-driven water of low density replaces an equal weight of denser water. This is essentially a hydrostatic concept, with the underlying assumption that a reference level exists in the water at which density is independent of time, since the mathe- matical equation used is Aa dp, 'Pa where z is the steric level, pa the atmospheric pressure, po the pressure at the 1 p ~ 9 Jp SECT. 5] LONG-TERM VARIATIONS IN SEA-LEVEL 593 reference level and Aa the anomaly in specific volnme. Despite uncertainties in determining the reference level, such investigations are highly profitable, particularly when they are conceived and carried out on a grand scale. Indeed, it is highly doubtful whether such an investigation would be successful in representing steric sea-level variations at one station by means of locally observed density unless the steric variations were of considerable magnitude. This follows from the works of Pattullo et al. (1955) for a world-wide network of stations, and of Lisitzin, Polli, Hela and others for more limited regions; these clearly show that seasonal variations in steric sea-level can be considered regional phenomena, varying little over quite wide areas (see Pattullo in Vol. 2). In this context a pertinent question is whether a thin layer of surface water suffering large density changes is more efficacious in producing changes in steric level than either a thick layer of deeper water suffering smaller density changes or the seasonal incursions of bottom water into lower latitudes. The answer almost certainly depends on the particular area under consideration. The English Channel and the North Sea are rather shallow areas of strong tidal mixing ; it might be hoped, therefore, that surface densities would reflect quite closely the density structure in depth in so far as annual variations are con- cerned. Contemporary investigations by the writer for stations in these areas, using 19 years of observations, show no correlation between surface layer densities and observed annual means of sea-level. This may mean that in these regions either the properties of the surface layers are not paramount in deter- mining steric levels, or that year-to-year variations in steric level are insignificant. One further direct oceanographical factor affecting sea-level in high latitudes is the damping influence sea-ice has upon the vertical movement of the water surface. No estimate appears to have been made of its magnitude for long-term variations, though it is likely to be rather small. B. Climatological Factors The last section indicated the indirect effect of a meteorological agency (the wind) in changing sea-level through the mechanism of steric changes. The more direct effects on sea-level of atmospheric pressure and the tractive force of the wind have long been studied. The statical law of barometric influence in which the sea acts as an inverted barometer, rising approximately 1 cm for a decrease in air pressure of 1 mb, and vice versa, was first examined by Gissler in 1747 for the Gulf of Bothnia, and since by many others for other regions. The greatest difficulty experienced in verifying the law has arisen from the presence of the much larger effect due to the tractive force of the wind, and whilst the theoretical law is invariably assumed as a sufficiently close approximation to the truth, a conclusive confirmation is still lacking. The outstanding point at issue concerns the minimum period of time necessary for the inertia of the water to be overcome in order that the statical law be obeyed. Doodson (1924) has shown that for day-to-day variations in sea-level the theoretical statical coefficient is approached, for British waters, to within 15%. 20— s. I. 694 ROSSiTER [chap. 16 In consequence of the seasonal variations in total air pressure over the oceans,! however, the statical law can only be satisfactorily examined on a global scale. For empirical, statistical investigations where the individual effects of air pressure and wind stress do not need to be separated, the method first used by Witting (1918) in his study of Baltic storm-surges can be most effective. Instead of using observed wind velocities and directions, atmospheric pressure gradients in two mutually perpendicular directions can be used. This may be simplified further by using observed air pressures at three stations chosen so as to form roughly an equilateral triangle, and Doodson (1960) has used this technique to study variations in annual mean sea-level in the English Channel, the North Sea and the Cattegat. The data were made to give a best fit to the linear expression Z-Z = ar{Br-Br), (1) where Z denotes annual mean sea-level, Br the annual mean air pressure at a point r, and bars denote average values taken over a period of 27 years. The maximum individual contributions to sea-level from pressure and wind in- dicated by this type of formula during the years examined were found to be 35 mm for Newlyn, which is on the continental shelf, and 106 mm for Esbjerg, which is situated in shallow water and hence responds more violently to meteoro- logical influences. At Esbjerg the standard deviation of the annual means was 51 mm ; that of the residuals, after removing the computed contributions, was 21 mm. One noteworthy feature of Doodson's work is that, in the absence of long series of records at some stations, he experimented with monthly instead of annual means of his variables and found that the results from 36 monthly means approximated very closely to the results from 27 annual means. The piling-up of water due to wind stress, most apparent in shallow water and in the presence of land barriers, is generally accepted to be a quadratic function of the relative velocities of surface wind and surface water, but also subject to a roughness parameter ; it is essentially a phenomenon most ap- propriately discussed theoretically elsewhere (see Section 2 and Chapters 3 and 15). In passing, it should be mentioned that although the linear expression (1) can only be an approximation to the fundamental quadratic law of friction, for long-term mean values it provides remarkably high correlation coefficients (see Doodson, 1924, in respect of daily mean values). In the past, storm surges have been profitably studied by empirical, statistical methods ; the advent of the electronic computer has made possible a more theoretical approach by way of numerical integrations in time and space involving viscosity and bottom friction. It may well be that such a treatment could find a useful application to the study of seasonal variations in sea-level due to winds and air pressure, for the problems are fundamentally the same, differing only in their scales of time. 1 "From 1012 mb in December to 1014 mb in July, due principally to a shift in air mass towards Siberia in winter." (Pattullo et al., 1955, p. 101.) SECT. 5] LONG-TERM VARIATIONS IN SEA-LEVEL 595 Before leaving the subject of wind stress, mention must be made of the effect of sea-ice as a mitigating factor. Lisitzin (1957) has shown from data for the Gulf of Bothnia that this effect can have appreciable seasonal significance, and, as the incidence of sea-ice can vary considerably from year to year, it must also contribute to annual variations in sea-level. Time changes in the total water content of the oceans due to precipitation and evaporation do not appear to have been seriously studied, presumably because of lack of data. Our ignorance of the oceanic water budget has been emphasized by Munk (1958), when expressing the hope that International Geophysical Year observations will do much to improve the position. For a partially enclosed sea, excess or defect of one of these variables over another will, to a large extent, be balanced by flow in or out of the channel leading to the ocean ; changes in sea-level will thus tend to be spread over the whole surface of the oceans. Wyrtki (1954) has discussed variations in the water content of the Baltic and the water exchange with the North Sea, taking these and other factors into account. Variations in sea-level on a much longer time scale than those considered hitherto are almost certainly dominated by changes in the ice coverage of the Antarctic, Greenland and, to a much lesser extent, the Arctic. The phrase "glacial eustasy" is used by geologists to describe changes in the total volume of water on the earth's surface due to the grox^th and decline of glaciated areas, the more general theory of eustasy embracing precipitation, evaporation, movements of the ocean bed, the accumulation of sediment in the oceans, etc. The main evidence for glacial eustasy comes from geological sources as, for example, studies of old coastlines. Flint (1947) and Zeuner (1946) have provided general reviews of the subject. Various estimates (see Young, 1953) suggest that at the maximum of the fourth glaciation of the Pleistocene, sea-level was of the order of 90 m lower than at the present time. It is possible that sea-level everywhere could increase by something of the order of 30 m if all contemporary ice and snow were to melt. This is necessarily a crude estimate as our know- ledge of the thickness of the Antarctic ice-cap is meagre ; one of the projects of the International Geophysical Year (1957-58) has been to form a clearer picture of this ice-cap using seismic sounding techniques. Although a rise of 30 m is an improbable occurrence in the foreseeable future (unless nature is assisted by man's surplus of hydrogen bombs), the rate of rise in some of the more low-lying maritime countries must give rise to anxiety. It is estimated that the probability of storm-surges exceeding danger level would be doubled on the east coast of England and trebled on the west coast by a rise in mean sea-level of 0.5 ft. Qualitative climatological evidence is able to indicate trends which must have brought about changes in global sea-level in more recent times (see, for example. Brooks, 1949), such as the "Little Ice Age" between the 16th and 19th centuries, but only with the comparatively recent aid of meteorological instrumentation have trends been observed with reasonable accuracy. One example is the considerable increase in world temperatures during the first 40 years of the present century. 596 KOSSiTER [chap. 16 The longest series of sea-level observations, at Swinemiinde in the Baltic, were only commenced in 1811. Direct evidence of the most suitable kind is, therefore, strictly limited. However, as glacial eustasy is of significance in certain geophysical problems (see section 5 of this Chapter), such results as have been obtained are of interest. Three major attempts have been made in recent times to compute secular variations in level ; all have used data from as many tide-gauge stations as possible, in the form of annual means. But whereas Gutenberg (1941) and Egedal, Disney and Rossiter (1954) used data for as many years as possible, Munk and Revelle (1952) limited their investigation to the consecutive decades 1905-15 and 1915-25. Both Disney and Rossiter showed that simultaneous readings should be used when possible, as secular changes are by no means constant in time at any one place, but to do so can impose limitations on the amount of data that can be studied. Ignoring the results for North America and Scandinavia, the regions most affected by postglacial uplift of the land (see below), Gutenberg averaged his results for all areas to arrive at a figure of 11 cm per century rise in sea-level during recent decades. Munk and Revelle concluded that 10 cm per century must be considered a maximum value for this era. These estimates will be referred to later (see page 605). C. Changes in the Distance of the Tidal Bench Mark from the Centre of the Earth Reference has been made immediately above to one of the main causes of movements in the "fixed" reference mark for sea-level measurements, that of isostatic adjustment in the earth's crust following deglaciation. It is generally agreed that isostatic compensation need not be complete, immediate or continuous. Nevertheless, there can be little doubt that in terms of hundreds of years isostatic compensation is responsible for a considerable postglacial up- lift. Moreover, changes in loading of the earth's surface when persisting over thousands of years produce plastic flow in the earth's crust. Whilst direct methods exist for determining some of these movements, such as observations of lake levels and precise spirit levelling over large land areas at intervals of time of the order of 50 years (see, for example, publications of the Finnish Geodetic Institute), the possibility exists that observations of sea-level can also be used to determine them. An independent estimate must first be made of the effects of glacial eustasy, and amongst others Gutenberg (1941) has attempted this for North America and Fennoscandia. Fig. 1 illustrates his results for Fennoscandia. Apparent variations in sea-level may also arise from sudden movements in the earth's crust, but if, as is most often the case, they are of local origin, they can easily be detected by comparisons with readings from other gauges. Lastly must be mentioned the yielding of the crustal layers of the earth to the tide-generating forces. These are most conveniently discussed below. SECT. 5] LONG-TERM VARIATIONS IN SEA-LEVEL 597 10° 15' 20" 25" 30" 35° 65" 60' 55' Uplift during past 7000 years, in meters Present rate of uplift, cm per century — — Stations data available Fig. 1. Postglacial uplift in Fennoscandia. (After Gutenberg, 1941, Fig. 3.) D. The Long-Period Astronomical Tides Table II contains a list of the tides of longer period than a day. The theo- retical amplitudes of their equilibrium forms have been taken from Doodson's harmonic development of the tide-generating potential (Doodson, 1921). 598 ROSSITEB [chap. 16 Table II The Equilibrium Form of the Long-Period Tides Speed, Period, Amplitude, expressed Name of tide Constituent ° per mean mean in cm. Common solar hour solar days factor is (J — sin^ A), where A is latitude Liinar fortnightly Mf 1.0980 13.661 6.29 Luni-solar fortnightly Msf 1.0159 14.765 0.56 Lunar monthly Mm 0.5444 27.555 3.34 Solar semi-annual Ssa 0.0821 182.621 2.94 Solar annual Sa 0.0411 365.242 0.48 Nodal — -0.0022 18.62 years see eqn. (2) The question as to whether long-period tides will obey the equilibrium law has been open to doubt, partly due to the presence of land- masses on the face of the earth (Laplace's equilibrium theory postulates an ocean covering the entire earth), and partly because it has long been known that steady tidal motions can exist (tides of the "second class") which are not related to the tide- generating forces (Lamb, 1932, §214). Consider first the effect which land-masses may introduce to produce the so- called "corrected" equilibrium tide. Darwin and Turner (1886) showed theo- retically that, for long-period tides in general, if the equilibrium form is proportional to (i-sin2 8)(i-sin2A), where S is the declination of the moon above the equator and A denotes latitude, then the "corrected" equilibrium tide would be proportional to (^ — sin2 8)(sin2 Aq — sin^ A), where Ao is a latitude (possibly imaginary) determined by the geographical distribution of land-masses. Numerical integration of the area of dry land by Turner led to a value of approximately 34° for Ao, suggesting that land-masses have only a minor influence upon the distribution of long-period tides. Consider next the tides of the second class. Their possible existence prevents the assumption of the equilibrium tide. Darwin (1886) pointed out that Laplace had simply assumed that, owing to the action of dissipative forces, the equili- brium law was valid; Darwin argued that the frictional forces arising from these currents would be infinitesimally small. Now Bowden (1953) has shown that the friction affecting any tidal constituent is not simply a function of the current associated with that constituent, but depends upon the product of that current with the total current ; Proudman (1960) has used this fact to show that this product is so much larger than what was assumed by Darwin and Lamb SKCT. 5] LONG-TERM VARIATIONS IN SEA-LEVEL 599 that tlie resultant friction is such as to leave only the forced long -period tide, which must be of the (corrected) equilibrium form. The requisite amount of friction depends upon the period of the constituent, and Proudman has further deduced that, for the validity of the law, the period of the constituent must be long compared with 27r/A;, where k is a parameter of friction. For shallow water, 2nlk has its lower limit (4.6 days) and for oceanic waters its upper limit (5.1 years), making it appear certain that the nodal tide will obey the law. It is less certain that the solar annual tide (So) and the solar semi-annual tide (Ssa) will do so. The theoretical amplitudes of the equilibrium tide must be corrected for the influence of a yielding earth. The earth tides (or body tides, to distinguish them from the loading tides referred to in the next paragraph) are generated by the gravitational attractions of sun and moon, and since they too are of equilibrium form they reduce the water tide by a factor ^ ; ^ is dependent upon the rigidity of the earth. This deformation of the earth is responsible for a secondary effect, however, since it exerts an attractive force on the water bulge and thus augments it by a factor k; k is dependent upon the distribution of density within the earth. The two pure numbers h and k are the well known Love numbers. These considerations lead to the relationship Observed tide = (l+k — h) Equilibrium tide. Tomaschek (1957) has given a comprehensive account of the various methods used to estimate the quantity l+k — h. It is of interest to note that, despite the uncertainty surrounding the existence in nature of the equilibrium forms of Mm and Mf , observations of these tides give estimates for l+k — h which are in reasonable accord with the figure deduced (0.7) from methods independent of tidal theory. It should be mentioned here that in general the fortnightly tide Msf cannot be used for these determinations as it may contain a sizeable contribution from the interaction, in shallow water, of the principal lunar and solar semidiurnal tides M2 and S2. Nor can the Ssa and Sa constituents be used, as the results of harmonic analyses of tidal data at ports all over the world reveal that they are dominated by the seasonal variations of meteorological and oceanographical origin discussed in sections A and B. A fuller treatment of these last two variations is given in Pattullo, Vol. 2. One further contribution is made to the observed long-period tides (and to any other species of tide) by the yielding earth, known as the loading tide. The effect of the tidal distribution of water in tilting the earth's surface has been shown, principally from earth-tide measurements, to extend a remarkable distance inland from continental coastlines. Before leaving the subject of the long-period astronomical tides, it is desirable to comment upon the tidal cycle having a period of 18.6 years, that of the revolution of the moon's nodes ; the tide associated with this period is possibly the most quoted and the least known of all the tidal constituents. The theo- retical existence of this tide is frequently used as an argument for taking 19- yearly means, or taking a 19-year span of observations when examining data 600 ROSSITER [chap. 16 for variations of many kinds, yet its distribution and magnitude cannot yet be said to have been determined analytically. The development of the tide- generating potential indicates the uncorrected equilibrium form to be given by 18.5 (sin2 A-i) cos N mm, (2) where A denotes latitude and N is the mean longitude of the moon's ascending node. The above expression includes the factor 1+k — h, taken as 0.7. Antici- pating the analytical results presented in section 4 of this Chapter, it is necessary to demonstrate that perturbations of the nodal tide can exist under certain circumstances. The non-linear terms occurring in the hydrodynamical equations governing tidal motion give rise to higher species of tides ; especially is this the case in shallow water. The particular case of a closed rectangular basin has been examined by Proudman (1952, §145), and in general it may be stated that a single constituent (such as M2) will generate its first harmonic (M4) and a "constant" term. It can be shown from the following greatly simplified con- siderations that the "constant" term in fact contains a truly constant term plus a slowly varying term with a period of 18.6 years. In the Admiralty Manual of Tides (Doodson and Warburg, 1941) it is demon- strated that the quarter-diurnal tide is approximately proportional to the square of the semidiurnal tide. Now, (M2)2 = (/of M2)2^2cos2(F + W-g) = (/of M2)2iy2|i + cos2(F + w-g)}, (3) where/ and u are the nodal terms, H and g are the harmonic constants of M2. If M4* be the amplitude of that portion of M4 given by the second term in the bracket in equation (3), then the first term is given by (/ of M4)M4*, since/ of M4 = (/of M2)2. Now, the/of M4= 1 —0.074 cos iV, and hence the above suggests a truly constant contribution to mean level equal to M4* and a perturbation of the nodal tide amounting to O.O74M4*. This would affect the amplitude but not the phase of the equilibrium nodal tide. Similar perturbations would be generated from many of the other primary tidal constituents. M4* cannot be determined from an analysis of observed tides, but for the special case of a standing oscillation in a narrow gulf Doodson (1956) has obtained results which suggest that the truly constant contribution equal to M4* can be ap- preciable. It must therefore be admitted that the accompanying nodal per- turbation could be of the same order of magnitude as the equilibrium tide. The generation of nodal perturbations is not exhausted by the foregoing remarks. Certain tidal constituents can interact by virtue of particular proper- ties of their angular speeds and nodal factors. Such a group is M2, Ki and Oi, but it must be admitted that no estimate can be made of the magnitude of such perturbations. It must, therefore, be concluded that observational data cannot be expected, in general, to provide confirmation of an equilibrium nodal tide, even if it is corrected for the presence of land-masses and a yielding earth. SECT. 5] LONG-TERM VARIATIONS IN SEA-LEVEL 601 4. The Analysis of Observations Variations in sea-level, both of a periodic and a secular nature, are in many cases of great interest to scientific disciplines other than oceanography. It behoves the oceanographer, therefore, to be prepared to speak with some authority upon the information that can be extracted from his observations of sea-level. This requires that he use the best possible methods of treating his data numerically and, of the utmost practical importance to all concerned, provide an estimate of the accuracy of his findings. An excellent example of this arises in the analytical treatment of annual mean heights of sea-level to determine the relative magnitudes of the eifect of barometric pressure and wind stress, secular variations and the nodal tide. Previous attempts to determine the last two mentioned quantities have all suffered by ignoring the contributions of wind and pressure, as will be shown. Let Zt ^ /{Bt, x) + cT + a cos OT + b sin OT + cf){T), (4) where Z is the annual height of observed sea-level, T is time in years, /{Bt. x) is a term representing the contributions from variations in air pressure with time and distance (this includes wind effect), c is the secular variation, and the trigonometrical terms represent the nodal tide, 6 being the annual increment of — 19°. 33 per year. <^(T) represents contributions to Zt from all other causes. Due to the strong correlation between the terms cT and b sin dT, the secular variation and the nodal tide cannot be computed by independent processes. Thus it may be shown that, if c is computed directly from Z using the method of least squares, the answer will contain a contribution amounting to approxi- mately 0.16 from the nodal tide; conversely, if b is computed directly from Z by harmonic analysis, the answer will contain a contribution of 5.7c from the secular variation. Using least squares with the three variables T, sin 6T and cos 6T, complete separation can be assured, and this has been done for stations in European waters for the years 1940 to 1958 inclusive. Table III illustrates the results. Although there is good agreement between stations for the quantity c, and for the amplitude R{ = \/[*^ + ^^J) ^^^ t^^® phase lag g of the nodal tide, the latter is some 90° in advance of the equilibrium value of zero, and the amplitude is many times larger than equation (2) would indicate. It will be seen below, however, that regional agreement is no proof of the reality of the results. Consider the probable accuracy of these calculations. If each value of Z is subject to a random error such that the standard deviation of Z is a, it is shown in standard text-books on statistics that the standard error of the amplitude is a\/{2ln) and that of the phase lag is {alR)\/{2ln), n being the number of observations. The standard deviations of Z are given in Table III, and it will be seen that they vary from 24 mm at Tregde (Norway) to 79 mm at Furuogrund (Sweden). 1 For an average value of a equal to 40 mm, it follows that with 1 These values also contain contributions from the secular variations. 602 ROSSITEB [chap. 16 1^ < 'T3 — ' 05 *s3 'r' ^ ce *- >i O !=1 TO O k2 o ^^ 03 § a O 4^ > Oi 1=1 t4 o (S -l-i Ti cr! 03 > r^ M "^ 03 a> •- ^. =3 ^ o g fl u £ a> P* tM c ^^ 9 ^ OCDff0(NOQ0O-HTj*<35Oe0 ■*OfCC5OTjHO-<*f0f0G0 05 00 10 --H --< t^ t^ GO 05 C5 ■^ fO -* "Xi QO I> t-^ CO »o -^ 05MOi>«oos?ocd-<*oco'i-HQ0 0500^HiMO»C-H0500-H^C<| ccooo6 (M(MC<|iM(Ni— ii— If— ( I— 1^ <35o6o6'-h05«0;0-<*GOtjH05»C lO»CTtC^HfccD05Tt*-^oo ooo^ocoo^oio^ec I I I ^ Ji J^ o o 01 c3 0) p^ o o Is 0) °3 >. CC 05 10 GO 06 (N 000 I i ffl &s Q m M SECT. 5] LONG-TERM VARIATIONS IN SEA -LEVEL 603 w = 19 the standard error of R will be 13 mm and that of ^ will be 75° (assuming the theoretical value of R to be 10 mm). In order to reduce the standard error of gr to the order of 20°, it is necessary to have the standard deviation of the sea- level data reduced to 1 1 mm ; the standard error of R will then be about 4 mm. It was shown earlier that carefully maintained gauge records, accurately reduced, could provide annual means to an accuracy of the order of 1 or 2 mm, so that we may ignore the contributions from random observational errors and slightly imperfect elimination of the shorter-period tides. To make further progress we must either increase n by taking consecutive spans of 19 years, or first eliminate, as far as possible, the term /{Bt, x) in equation (4). As n appears within a square root sign in the expressions for the standard errors, ^ the more effective of these two approaches is the latter ; it is certainly the more instructive, and has been attempted for the stations listed, by determining the coefficients Ur in equation (1). For the stations 10 and 11 in the North Sea the pressure data for the triangle De Bilt-Stornoway-Bergen were used, and for stations 1 to 9 the triangle Warsaw-Bergen-Haparanda was combined with them ; for Newlyn the triangle De Bilt-Sciilies-Stornoway sufficed. Substituting the computed values of ttr into equation (1) for each station enables /(^r, x) to be calculated, and some examples of Z and the residuals Z —/{Bt, x) are given in Fig. 2. The standard deviations of the residuals are shown in Table III, and these are more nearly uniform over the region than were those for Z, being of the order of 20 mm. The high correlation from station to station between the original means is clearly shown, arguing regional meteorological influences modified, at some stations (e.g. Esbjerg), by local conditions. An extension of the method to include the effects of Atlantic weather, by using pressures at Reykjavik, Sable Island (Canada) and Lisbon, has produced no detectable decrease in the residuals at any station. We must, therefore, look to other causes for an explanation of these residuals. Table III contains the values of secular variation and nodal tide computed in the same fashion as before, but this time using the residuals of sea-level. In all cases the secular variations are appreciably altered, in some cases being reversed in sign. The amplitudes of the nodal tide have been greatly reduced, and are closer to the equilibrium values ; the phase lags, however, are still far removed from zero degrees. The differences between the results obtained before and after removing /{Bt, x) are, of course, direct results of the effect of pressure distribution. The question remains as to whether these differences arise from corresponding real effects in the pressures, or from random varia- tions. So far as the writer is aware, there is little reason to expect a secular variation in atmospheric pressure, and even less to suggest a measurable nodal tide ; it seems more reasonable to suppose that the figures given at the foot of Table III are contributions from random variations. The equilibrium form of the nodal tide is neither disproved nor confirmed by 1 To reduce the values of standard deviations from a to ^a would require n to be 171 years. 604 ROSSITER [CHAP. 16 the evidence given above, and, in view of the amount of scatter in the residuals and the relatively shallow depths of water in the areas considered combined with the remarks made on page 600, this is perhaps not surprising. It is barely possible that some orderly picture of the distribution of the nodal tide may emerge from a study of carefully selected data; these should preferably be from deep-water gauges, and should be first treated so as to remove as many as possible of the effects of water-density and air-pressure variations. The use of a smoothing process to reduce the standard deviation of the observations before Newlyn (England) 1940 Fig. 2. Deviations of annual mean heights of sea-level. as observed, — — • — — after eliminating meteorological effects. analyzing for the nodal tide must be viewed with suspicion; Jeffreys (1939, p. 248) has shown that whilst smoothing can reduce the scatter of a series of observations, it also introduces a spurious and undesirable correlation between individual members. So far as secular variations are concerned, the preceding analysis strongly suggests that future investigations will profit greatly if prior consideration is given to the elimination of contributions from the apparent nodal tide and of random contributions from air-pressure variations. SECT. 5] LONG-TERM VARIATIONS IN SEA-LEVEL 606 5. Some Geophysical Implications of Long-Term Variations in Sea-Level Lack of space precludes anything other than a cursory discussion of the geopliysical problems affected by long-term variations in sea-level. Nor is any atteni})t made to give a comprehensive bibliography of papers dealing with these problems. Such papers as are mentioned have been selected either because tliey are of recent date and emphasize the contribution made to the subject by sea-level changes, or because they aflFord useful guides to the subject by virtue of their references. A. Eiistatic Changes, the Length of the Day and the Variation in Latitude Astronomical observations have long revealed the existence of irregularities of an apparently random nature in the speed of angular rotation of the earth (Spencer Jones, 1939). Their magnitude is such that the standard deviation of the annual average of the length of the day during the past century is of the order of 1.6 msec. One possible source of these anomalies is the change in the moment of inertia of the earth about its axis due to the redistribution of matter on the earth's surface following periods of glacial eustasy. A period of deglaciation can have a three-fold effect : (i) by increasing the volume of water in the oceans, thus increasing the moment of inertia and, in consequence of "the conservation of momentum, increasing the length of the day, (ii) an effect opposite in sign to (i) due to isostatic compensation of the land freed from the weight of ice, (iii) tilting of the instantaneous axis of rotation following the redistribution of matter, resulting in a change of the observed latitude of a point on the earth's surface. The most recent evidence appears to be that eustatic changes could only account for a small fraction of the observed irregularities in the length of the day (over the past century). It will be sufficient here to quote from three papers on these subjects; by Munk and Revelle (1952 and 1952a) and by Young (1953). Ignoring isostatic compensation, two of these papers present independent calculations indicating that a rise in sea-level of 1 cm would in- crease the length of the day by 0.06 msec and 0.1 msec respectively. Young's estimate was made on the basis of a simple model of the ice-covered areas of the globe, whilst Munk and Rcvelle's was from a rather more detailed model. Estimates of the maximum possible eustatic changes in sea-level have been referred to on page 596, and the value of 10 cm per century, to quote Munk and Revelle, "cannot have changed the length of the day by more than 0.6 msec; the observed decrease equals 3.8 msec, and would imply a lowering of sea-level of 63 cm." This does not signify, of course, that the effect of eustatic changes on a geological time scale can be neglected in their contribution to changes in the length of the day. Indeed the same authors (1952a) have observed that 606 KOSsiTEK [chap. 16 Daly's estimate of 5 m eustatic lowering of sea-level during the last few thousand years leads to a decrease of 30 msec in the length of the day for a rapid growth of ice-caps ; this is in accord with the 27 msec derived from ancient observations of eclipses. Munk and Revelle have also put numerical values to the wandering of the north pole of rotation, and hence to the variation of latitude, caused by glacial eustasy. They find it can be a sensitive indicator of the source of melted ice, 1 cm rise in level displacing the pole "by | ft towards Chicago, or 2.8 ft towards Greenland, according to whether the ice melts on Antarctica or Greenland". Astronomical evidence of a rather doubtful nature suggests a movement of 10 ft towards Greenland over 25 years at the beginning of this century, in accordance with a eustatic rise of 10 cm/century stemming from Greenland ice ; recent confirmation of this shift has been found by Markowitz {in litt.) after re-analyzing 50 years of latitude observations. B. The Pole Tide Secular and apparently irregular changes in latitude have been referred to on page 605, but a regular movement of the pole of rotation was predicted by Euler on theoretical grounds in 1765. The period of the Eulerian nutation should be 10 months. "In 1891 a variation in latitude was discovered by Chandler, but its period was 428 days instead of the expected 305 days. The explanation was soon given by Newcomb (1892): Euler's theory applies to a rigid earth ; for the actual case, the elastic yield of the solid earth and the fluid yield of the oceans have to be taken into account, and these increase the period of the free nutation from 10 to 14 months." The above quotation is taken from the introduction to a paper by Haubrich and Munk (1959) who have used the results of a spectral analysis of sea-level observations to search for the pole tide, that oscillation of the sea which is theoretically a direct consequence of the Chandlerian nutation and the eccentri- city of the earth. Their theory indicates that the equilibrium form of the pole tide, when corrected for a yielding earth, should have an amplitude of the order of 5 mm. Thus, as with the search for the nodal tide, the greatest difficulty in determining the pole tide comes from the unwanted, usually random varia- tions, or "noise", and this is a point to which perhaps insufficient attention has been given in previous attempts to confirm the pole tide. Haubrich and Munk find that there is a weak response, barely above noise level, corresponding to the 14-month tide in the average monthly sea-level data for 11 stations. The data for some of the stations, however, when examined separately, show no peak at 0.84 cycles per year. Its frequency is well identified, but, from a com- parison with a spectral analysis of the corresponding astronomically observed variations of latitude, the mean amplitude is twice that expected from the equilibrium theory. It is unfortunate, therefore, that this comparison cannot be used to throw further light upon the values of the Love numbers. SECT. 5] LONG-TERM VARIATIONS IN SEA-LEVEL 607 C. Eustatic Changes, Tidal Friction and the Secular Acceleration of the Moon A consequence of any variation in the angular speed, oj, of the earth is that the moon's mean orbital speed, n, will be changed. It is known (Jeffreys, 1959, §8.04) that an apparent secular acceleration, v,^ of the moon will result from a deceleration in the earth's rotation according to the formula dn n do) "" ^ ~di~ZW ^^^ Eustatic changes may be involved in this phenomenon in two ways, by altering the earth's moment of inertia as discussed in A, and by altering the amount of tidal friction. Consider the orders of magnitude involved. Combining equation (5) with his results quoted in section A, Young (1953) suggests that a rise in sea-level of 1 cm/century would, if uncompensated, produce an average secular acceleration of 2 in. /(century) 2 by changing the earth's moment of inertia. In consequence of the classical paper by Taylor (1919), later amplified by Jeffreys (1920), numeri- cal values have been given to the retardation of the earth's rotation by tidal friction. The long-accepted conclusion is that nearly 70% of the friction arises in the shallow Bering Sea. As its depth is some 50 m, and as the amount of bottom friction generated by a tidal current, for a given velocity, is inversely proportional to the depth, a mean rise in sea-level over the Bering Sea of 1 cm would decrease the bottom friction by only 0.02%. The assumption that the velocity of the tidal stream would itself be inversely proportional to the depth would only double the above estimate ; nor would the increase in the frictional area resulting from such a general rise in level make any appreciable change in the energy dissipated. Estimates of that part of the moon's secular acceleration which cannot be accounted for by gravitational theory, deduced from astronomical observations, are most difficult to interpret ; especially is this so for ancient observations of equinoxes and eclipses. Nevertheless, a secular acceleration of the order in 5 in./(century)2 can probably be accepted. Murray (1957) has succeeded in separating the tidal and non-tidal contribu- tions to the acceleration from both ancient and more recent observations, and his important results suggest that the rate of dissipation of energy through tidal friction is nearly three times greater than Jeffreys proposed. He also shows that the tidal contribution is now only half what it was 2000 years ago. The most recent review of these problems and the uncertainty surrounding the various solutions proposed has been written by Munk and Macdonald (1960). D. Oceanographic Levelling A revival of interest in the possibility of using the mean sea surface as a levelling instrument, independently of the more orthodox methods of precise 1 As observed from the earth, and relative to the stars. 608 ROSSiTER [chap. 16 spirit levelling, has recently developed from a need to confirm the findings of the Unified European Levelling Network. This network, which for scientific purposes has crossed political and geographical boundaries in an attempt to integrate the levelling nets of individual countries, has produced provisional results indicating that sea-level is something like 0.5 m higher in the northern Baltic than in the Mediterranean. Here one is concerned with the deviations of observed sea-level from a geopotential surface, from place to place, but before these can be ascertained it is necessary to utilize a long series of records at many gauges, and to eliminate from them as many of the time variations as possible. Many of the results given in section 4 of this Chapter originate from an investigation of this type, and it will be seen that if equation (1) is formed for each of a pair of stations, one can obtain the mean difference in level between their respective bench marks for a tmiform atmospheric pressure over the region concerned. Doodson (1960) has given tenative values for certain stations, and has also commented upon the feasibility of attaining, observationally, a fundamental reference plane in the sea (the geoid). Whether a similar treatment can be applied so as to allow for the semi-permanent surface gradients caused by differences in the density of sea-water rests largely upon whether sufficient observational data of temperature and salinity are available. A theoretical statement of the problem has been given by Vantroys (1958). The amount of material assistance the oceanographer can offer the geodesist must be limited by the standard of accuracy the geodesist demands. There is a particular levelling problem, however, where the geodesist must rely much more upon the oceanographer. Levelling connexions across even quite narrow bodies of water provide formidable obstacles to spirit levelling. Such a body of water is the Straits of Dover; Cartwright (1960) hopes to demonstrate that, with the aid of current observations from the Straits and sea-level observations from opposite shores, it is possible to carry a level across the 21 miles of water to an accuracy of the order of a centimetre. 6. Conclusion In this chapter an attempt has been made to provide a broad outline of the subject, but it should be read in conjunction with that on seasonal variations in level (Pattullo in Vol. 2). Although a certain emphasis has been placed upon the influence of winds and air pressure, it is not suggested that these factors necessarily predominate in all seas and oceans to the exclusion of, for example, density variations, though in shallow seas this is highly probable. In deep untrammelled waters the density factor is possibly more important. Mention has not been made of any periodic variations other than the astro- nomical and pole tides. Oceanographical literature, like meteorological litera- ture (Shaw, 1928, 2, p. 320), contains many examples of the search for periods such as the sunspot cycle and even the fourth harmonic of Gabriel's cycle of 744 years ; as with the nodal and pole tides, however, the oscillations are too small to be verified conclusively with existing data. SECT. 5] LONG-TERM VARIATIONS IN SEA-LEVEL 609 References Bowden, K. F., 1953. Note on wind drift in a channel in the presence of tidal currents. Proc. Roij. Soc. London, A219, 426-446. Brooks, C. E. P., 1949. Climate through the ages (2nd edn.). Ernest Benn, London. Cartwright, D. E., 1960. A proposed method, due to J. Crease, of levelling between France and England. Assoc. Geod. Bidl. Geod., 55, 97-99. Darwin, G. H., 1886. On the dynamical theory of the tides of long period. Proc. Roy. Soc. London, 41, 337. Darwin, G. H. and H. H. Turner, 1886. On the correction to the equilibrium theory of tides for the continents. Proc. Roy. Soc. London, 40, 303-315. Doodson, A. T., 1921. The harmonic development of the tide-generating potential. Proc. Roy. Soc. London, AlOO, 305-329. Doodson, A. T., 1924. Meteorological perturbations of sea-level and tides. Mon. Not. Roy. Astr. Soc, Geophys. SuppL, 1, 124-147. Doodson, A. T., 1928. The analysis of tidal observations. Phil. Trans. Roy. Soc. London, A227, 223-279. Doodson, A. T., 1956. Tides and storin surges in a long uniform gulf. Proc. Roy. Soc. London, A237, 325-343. Doodson, A. T., 1960. Mean sea level and geodesy. Assoc. Geod. Bull. Geod., 55. In press. Doodson, A. T. and H. D. Warburg, 1941. Admiralty Manual of Tides. H.M.S.O., London. Egedal, J., L. P. Disney and J. R. Rossiter, 1954. Secvxlar variation of sea level. Assoc. Phijs. Ocean., Publ. Sci. No. 13. Flint, R. F., 1947. Glacial geology and the Pleistocene epoch. New York. Groves, G., 1955. Numerical filters for discrimination against tidal periodicities. Trans. Amer. Geophys. Un., 36, 1073-1084. Gutenberg, B., 1941. Changes in sea level, postglacial uplift, and mobility of the earth's interior. Bull. Geol. Soc. Amer., 52, 721-772. Haubrich, R., Jr., and W. Munk, 1959. The pole tide. J. Geophys. Res., 64, 2373-2388. Jeffreys, H., 1920. Tidal friction in shallow seas. Phil. Trans. Roy. Soc. London, A221, 239-264. Jeffreys, H., 1939. Theory of probability. Clarendon Press, Oxford. Jeffreys, H., 1959. The Earth (4th edn.). Cambridge Univ. Press. Jones, H. Spencer, 1939. The rotation of the earth and the secular acceleration of the sun, moon and planets. Mon. Not. Roy. Astr. Soc, 99, 541-558. Lamb, H., 1932. Hydrodynamics (6th edn.). Cambridge Univ. Press. Lisitzin, E., 1957. On the reducing influence of sea ice on the piling-up of water due to wind stress. Soc Sci. Fennica, Comment. Phys.-Math., 20-27. Munk, W. H., 1958. The seasonal budget of water. Geophysics and the IGY, Geophys. Mon. No. 2, Amer. Geophys. Un., 175-176. Munk, W. H. and G. J. F. Macdonald, 1960. The rotation of the earth. Cambridge Univ. Press. Munk, W. H. and R. Revelle, 1952. On the geophysical interpretation of irregularities in the rotation of the earth. Mon. Not. Roy. Astr. Soc, Geophys. SuppL, 6, 331-347. Munk, W. H. and R. Revelle, 1952a. Sea level and the rotation of the earth. Amer. J. Sci., 250, 829-833. Murray, C. A., 1957. The secular acceleration of the moon, and the lunar tidal couple. Mon. Not. Roy. Astr. Soc, 117, 478-482. Newcomb, S., 1892. On the dynamics of the earth's rotation, with respect to the periodic variations of latitude. Mon. Not. Roy. Astr. Soc, 52, 336-341. Pattullo, J., W. H. Munk, R. Revelle and E. Strong, 1955. The seasonal oscillation in sea level. Sears Found. J. Mar. Res., 14, 88-156. Proudman, J., 1952. Dynamical Oceanography. Methuen, London. 610 KOSSITEB [chap. 16 Proudman, J., 1960. The condition that a long period tide shall follow the equiUbrium law. Geophys. J., 3 (2), 244-249. Rossiter, J. R., 1958. Note on methods of determining monthly and annual values of mean water level. Intern. Hydrog. Rev., 35, 91-104. Shaw, N., 1928. Manual of meteorology. Cambridge Univ. Press. Taylor, G. I., 1919. Tidal friction in the Irish Sea. Phil. Trans. Roy. Soc. London, A220, 1-33. Tomaschek, R., 1957. Tides of the solid earth. Handb. Phys., 48, 775-845. Berlin. Vantroys, L., 1958. Note sur I'utilisation de la surface libre des mers dans les operations de nivellement. C.O.E.C. Bull. Inf., 10, 14-18. Witting, R., 1918. Hafystan, geoidytan och landhojningen utmed Baltiska hafvet och vid Nordsjon. Bull. Soc. Geog. Finlande, Fennia, 39, 45. Wyrtki, K., 1954. Schwankungen im Wasserhaushalt der Ostsee. Deut, Hydrog. Z., 7(3/4), 91-129. Young, A., 1953. Glacial eustasy and the rotation of the earth. Mon. Not. Roy. Astr. Soc, Geophys. Suppl. 6, 453-457. Zeuner, F. E., 1946. Dating the Past. Methuen, London. 17. SURGES P. Groen and G. W. Groves^ 1. Introduction A. Definition of Surges From the point of view of the spectrum of sea-level, surges may be charac- terized as being sea-surface disturbances with dominant periods ranging roughly from 10^ to 10^ sec, or from 1 to 10^ h, thus falling between the tsunamis and the lower frequency astronomical tides. From the causal point of view, surges are, in this treatment, understood to be phenomena of atmospheric origin, with exclusion of disturbances of crustal and of astronomical origin (tsunamis, lunar and solar tides). It is, of course, also possible that a surge in a certain restricted sea area or bay originates from a travelling disturbance of the sea elsewhere (in the same way as tides in marginal seas are induced there by the oceanic tides), so that it is, in a way, of marine origin, but its first cause will still be atmospheric if the primary disturbance is of atmospheric origin. Only surface surges will be dealt with here. B. Separation of Surges from Astronomical Tides Separating the "surge" or the "atmospheric effect" in a sea-level record from the astronomical tide is made fundamentally difficult by the fact that these phenomena are dynamically non-linear : even the definition of what is tide and what is atmospheric effect in the variation of sea-surface height offers a prob- lem. In many cases a simple subtraction of the predicted astronomical tide from the recorded heights will not work because of coupling effects, which make the difference obtained in that way show more or less pronounced secondary oscillations with tidal periods. A formal analysis of these coupling effects is briefly explained later (page 625). One of them is particularly simple, physically, viz. a time shift of the astronomical tide caused by an increased depth of the water through which the tide travels. This effect is particularly important in very shallow sea areas ; it may be eliminated by giving the pre- dicted astronomical tide curve the proper time shift before subtracting it from the tide record. Various practical ways of extracting the true "surge" from a tide record have been described by (among others) W. F. Schalkwijk (1947), R. H. Corkan (1950), G. W. Groves (1955), A. R. Miller (1958) and J. R. Rossiter (1959). C. Classification From the point of view of local time sequence of sea-surface heights, a classification of surges may be found in the extent to which the dominant period of a surge is determined by its cause, the extremes being, on the one 1 Of this chapter, sections 1, 3A and 3B have been written by P. Groen, sections 2 and 3C by G. W. Groves. [MS received June, 1960] 611 612 GROEN AND GROVES [CHAP. 17 hand, (i) surges in which the periods are mainly determined by the variation of the atmospheric action involved (wind stress, j)ressure difference) ; and, on the other hand, (ii) surges in which tlie periods are mainly determined by the properties of the body of water that is affected, the role of the atmospheric agent being in this case merely to "asj^irate" or to "excite". (i) The former will be the case if (1) the atmospheric action oscillates with a very narrow period spectrum, or a line spectrum, the surge then being a simple forced oscillation with prescribed period(s), or (2) the atmospheric action shows a very slow variation, which means that its time scale Ta is very large in comparison with a period Tb characteristic of the reacting system, the latter being at every moment in quasi-equilibrium with the atmosphere. driving force response Fig. 1. Idealized cases of time variation of atmospheric action (the time direction is from left to right) and of resulting surges (damped oscillation). (ii) The other extreme will be realized if the atmospheric action shows a variation in time which either (1) has the character of a random noise with a very broad Fourier period spectrum, giving rise to a quasi-stationary sea- surface oscillation with a period Tb that is characteristic of the body of water involved, or (2) is aperiodic with a time scale Ta-^ Tb, such as is illustrated by the examples shown in Fig. 1, the resulting effect being then a damped oscilla- tion. The time scale, or period, Ta, characteristic of the atmospheric action, is defined locally. If the atmospheric system that causes the surge is moving across the sea area involved and if its life time is long enough, we may write Ta = AjCa, (1) Ca denoting the velocity of displacement of the system and A its horizontal extent in the direction of the displacement. The period Tb is understood to be characteristic of the body of water involved in so far as it is determined by its geometry (and, to some degree, by a factor of energy dissipation in the water). If the horizontal extent of the body of water SECT. 5] SURGES 613 does not exceed that of the atmospheric system causing the surge, T^ is simply the period of a free seiche of the water in the direction in which the atmospheric action (wind stress, pressure difference) works. If, however, the extent of water affected does exceed the atmospheric system horizontally, Tt may be defined by n = AjCb, (2) Cb being the velocity of propagation of long (or "tidal") waves in the sea area under consideration and A having the same meaning as before. The above scheme of extreme cases (i) and (ii) is more or less ideal. Reality will, in general, lie somewhere in between. In particular, the two periods Ta and Tb may even be nearly equal, Ta ~ Tb, in which case a sort of resonance occurs. In the case of Ta and Tb being given by formulas (1) and (2), the condi- tion for resonance becomes Ca ~ Cb. A less sophisticated classification of surges may be established by dis- tinguishing between (1) surges occurring in an at least partly enclosed sea area (or a lake) and affecting it at any moment more or less as a whole, and (2) surges of the running-wave type, travelling over a sea area which is large in comparison with the atmospheric disturbance involved. The former type is the simpler one for mathematical treatment, especially if at the boundaries where the sea area under consideration meets the ocean outside, simple boundary conditions may be supposed to hold. Moreover, surges of this type have more extensively been studied, since damaging storm surges have especially been frequent in such partly enclosed sea areas, e.g. like the North Sea. Damaging storm surges of the running-wave type are mainly confined to those caused by tropical cyclones. D. Treatment A treatment of the dynamics and forecasting of surges of atmospheric origin will be given which follows the last mentioned classification. Besides, a more statistical treatment of surges, as "long" period variations of sea-level, is presented, including results of work on spectral analyses and correlations with weather. 2. Description The mechanism of surges varies considerably between the high- and the low- frequency portion of the spectrum we are considering. At the low-frequency end, sea-level is often in equilibrium with the disturbing forces and the principles of statics apply. At the other end, surges with periods of only a few hours are almost always governed by dynamical considerations; i.e., the inertial forces are important. For example, let us consider a typical continental shelf of constant slope whose edge is 100 m deep and 100 km from shore. The gravest mode of a standing barotropic wave has a period of about 7 h (four times the \/{gh) travel time across the shelf). Thus we might expect that the day-to-day variation would be in equilibrium with the disturbing forces, while surges of 614 GROEN AND GROVES [chap. 17 the order of 7-h period or less would not. The gravest modes of many gulfs and semi-enclosed seas have periods of the order of a few hours, and similar con- siderations apply. A. The Day-to-Day Variation of Sea- Level Observation of surges at the low-frequency end is simplified by the fact that their frequencies and those of the tide do not overlap. Consequently, to elimi- nate the tide one can use an averaging technique (such as described by Groves, 1955) as well as subtraction of the predicted tide. The former has the advantage that an identical scheme can be used at all places, the computations are simpler, SEA LEVEL (Cj,) -Of San Froncisco Fig. 2. Simultaneous plots of sea-level along the California coast. Dates correspond to 0000, 120°W meridian time. The dashed portions of the curves indicate uncertain data. "C51" refers to the 51-ordinate averaging scheme used (Groves, 1955). (After Groves, 1957. By courtesy of the American Meteorological Society.) and one is not limited to ports for which predictions have been made. The non- linear difficulties discussed in section 1-B are usually not serious. The main limitation of the averaging technique is that waves whose periods are two days or less are considerably attenuated. A series of such sea-level records was examined by Groves (1957), from which Fig. 2 was taken. Simultaneous three-month records of sea-level at ports along the California coast are shown. There is considerable coherence between adjacent ports, and some even between the northernmost (San Francisco) and SECT. 5] SURGES 615 southernmost (San Diego), separated by a distance of about 800 km. The period encompassed by the records was a typical fall-winter in that region — there were no exceptionally violent storms. Records such as these in other parts of the world indicate a general pattern : The over-all amplitude of the sea-level variation is greater (1) in regions of stormy weather, (2) along the edge of continents (rather than at islands), and (3) where the continental shelf is wide and shallow. Fig. 3 shows a comparison between sea-level and (inverted) atmospheric pressure for two of the ports. The vertical scales are such that, if the variable atmospheric pressure were the only disturbing agent acting and if the sea San Francisco Son Diego NOV. I 1952 FEB 1 1953 Fig. 3. Two comparisons of sea-level and inverted atmospheric pressure. (After Groves, 1957. By courtesy of the American Meteorological Society.) surface were in hydrostatic equilibrium at each instant, the sea-level and pressure plots would be identical. Actually, they are strikingly similar but not identical. The difference arises either from transient effects or from other agents, the latter being the more probable. The sea-level minus the atmospheric pressure has been called "corrected" or "adjusted" sea-level, plots of which are shown for the same stations in Fig. 4. Considerable reduction in over-all amplitude has been achieved, and some coherence between adjacent stations has been lost. This may be because the hydrostatic response to atmospheric pressure does not depend on bottom topography, whereas the horizontal wind- stress effect, remaining in the corrected sea-level plots, does. Now let us consider the effect of horizontal wind stress. The surface wind is seldom constant throughout a whole day — the higher frequencies predominate. Nevertheless there is an appreciable wind stress contribution to the day-to-day 616 GROEN AND GROVES [chap. 17 variation. The wind stress is so much more effective in shallow water (see section 3 of this Chapter) that the local wind in the vicinity of the port is usually more important than the integrated effect over a large region of ad- jacent ocean. The effect of even steady winds is difficult to evaluate owing to the importance of the local topograj^hy. But some general conclusions can be drawn from comparison of wind and sea-level records. Onshore winds raise sea- level. Longshore winds raise sea-level if the land lies on the right of the wind vector (in the Northern Hemisphere). On the edge of continents the corrected sea-level does not appear to be in equilibrium with the wind, but rather reflects CORRECTED SEA LEVEL TG ond AP - Son Francisco Fig. 4. Simultaneous plots of corrected sea-level along the California coast. "TG" and "AP" indicate the stations from which the tide data and the atmospheric pressure data were taken. (After Groves, 1957. By courtesy of the American Meteorological Society.) a time-integrated wind effect. That is, the sea-level extremes tend to coincide with the wind reversals, and not with the wind extremes. A possible explanation for this is that the response to the wind stress may have a time constant appreciably longer than a day. This is true in the case of response to longshore winds, and also if the response to onshore and offshore winds is mainly baroclinic ; i.e. reflects changes in the density structure of the water rather than a direct wind set-up (see Groves, 1954). Longard and Banks (1952) have observed such wind-induced changes of density structure along an open coast. At islands there is apparently no obvious relation between sea-level and the local wind. Sea-level may be related in a simple manner to the divergence or curl of the wind stress, but these cannot be inferred from wind observations at a single station. It may be useful to consider variation of sea-level at islands in SECT. 5] SURGES 617 terms of forced planetary waves of the ty])e discussed by Veronis and Stommel (1956). In the equatorial region of the Pacific there is, at times, a regular oscillation of sea-level associated with waves in the trade-wind regime (Groves, 1956). It should be mentioned that phenomena other than atmospheric ones influence the day-to-day variation of sea-level : thermal expansion of the water, long- period tides, etc. Variation in temperature can give rise to sea-level changes of the order of centimeters. Long-period tides, especially the lunar fortnightly, account for much of the coherence between stations in Fig. 4. The largest anomalies in the day-to-day variation seldom exceed a meter. The surges which do the most damage by inundating coastal areas are associated with the transient response of the sea surface. B. Transient Surges We shall now consider the high-frequency end of the surge spectrum, in which the sea surface is not in equilibrium with the disturbing forces. The usual procedure is to take a tide record, subtract the predicted tide, and to consider Allonlic City N-J. iTroced from onginol tide gage records [predicted tide curve is superficial J Fig. 5. Observed and predicted tide at Atlantic City, Sept. 14-15, 1944. (After Harris, 1959. By courtesy of the U.S. Department of Commerce Weatlier Bureau.) the remainder as the surge. Essentially one neglects the difficulties of non- linearity discussed in section 1-B of this Chapter and assumes that an accurate prediction is available. Even if a prediction is available for the very locality under study, there are errors, and so the "surge" will always contain tidal periodicities to some extent. The surges caused by tropical-type hurricanes are among the most intense observed. Regions where a wide, shallow shelf is combined with the frequent occurrence of hurricanes are particularly susceptible to damaging surges. Such conditions exist along the Atlantic and Gulf coasts of North America, where surges have been extensively studied by Harris (1956, 1957a, 1958, 1959), Kajiura (1958, 1959), and Reid (1956, 1958). Fig. 5 (taken from Harris, 1959) shows a predicted and observed tide curve for Atlantic City during the hurricane 618 GROEN AND GROVES [chap. 17 of September 14-15, 1944. The hurricane's center passed about 50 km to the east over the shallow, broad shelf. Fig. 6 (taken from Redfield and Miller, 1957) shows the corresponding surge at Atlantic City and at other ports along the same coast. These records are typical of hurricane-produced surges in this region. Redfield and Miller (1957) consider the records as consisting of three successive stages : the forerunner, hurricane surge and resurgences. The forerunner is a slow, gradual change in water level, beginning several hours before the arrival of the storm. The coherence between neighboring ports in the forerunner stage is Fig. 6. Surge of September 14-15, 1944, along the Atlantic Coast of the United States. Vertical scale is in feet, abscissa is in Eastern Standard Time. The arrows indicate the time of nearest approach of the hurricane's center, distance in miles and direction of the tide gauge from the storm track. (After Redfield and Miller, 1957. By courtesy of the American Meteorological Society.) usually good. Winds over a more extended region than the hurricane proper are probably important in this stage. If the longshore motion of the hurricane is "upcoast" (toward the right along the coast, facing the land from the sea), the forerunner is observed to be a rise in water level ; if the longshore motion of the hurricane is "down-coast" the forerunner is a gradual fall in level. The hurricane surge is the sharp rise in water level that occurs approximately when the hurricane center passes near the port. The duration of this phase is usually short — 2.5 to 5 h — but peak water levels of 3 m to 4 m have been observed. The coherence between adjacent ports is not so good, indicating that the strongest winds in the small region of the hurricane proper are responsible. SECT. 5] 619 Fig. 7 {continued overleaf). 620 GROEN AND GROVES [chap. 17 Fig. 7. Three synoptic charts of sea-surface disturbance heights during the storm-surge of 31 January- 1 February, 1953, in the North Sea. Disturbance heights are given in meters. The peak water level usually occurs considerably to the right of the hurricane's track, and the region of high water extends further to the right than toward the left. (These remarks refer to the Northern Hemisphere.) The resurgences are oscillations occurring after passage of the hurricane and the hurricane surge. A good example is the Atlantic City record in Fig. 6. They can be particularly hazardous as they are often unexpected, arriving after the storm appears to be subsiding. If the phase of the astronomic tide is right, one or more of the maxima may be higher than the original hurricane surge itself. The resurgences are due to more-or-less free ijiotion of the water, not under much influence of the hurricane once they are generated. Munk et al. (1956) attribute them to a "wake" of waves in the trail of a hurricane (analogous to a ship's wake) progressing along the coast. The period would be that of a free edge wave, whose phase velocity is equal to the velocity of motion of the hurri- cane (see page 640). Observations of many trains of resurgences seem to be in agreement with this hypothesis. But Kajiura (1959) points out that there are large differences in the period between ports separated by less than a wave- length of the supposed free-edge wave, and attributes the resurgences to a free onshore-offshore standing wave on the shelf. Other types of meteorological disturbances and other environments give rise to surges of varied characteristics. Surges in the open ocean have been observed at islands. The amplitude is seldom large at islands rising sharply from the deep ocean. Islands in shallow regions can experience large surges, such as the damaging one at Wake Island, January 17-18, 1953. SKCT. 5] 621 Certain particulars of the behavior of surges in gulfs and semi-enclosed seas are discussed on pages 626-640, dealing with the dynamics of such surges. Here we confine ourselves to giving two illustrations in Figs. 7 and 8. Fig. 7 shows three synoptic charts of the sea-surface disturbance heights caused by the disastrous storm surge of 31 January and 1 February, 1953, in the North Sea (after P. Oroen, see Koninklijk Nederlands Meteorologisch Fig. 8. The "twin" surges of the North Sea in December, 1954, as recorded at the Hook of Holland; the dashed line represents the equilibrium effect. Instituut, 1960). These charts show, among other things, a counterclockwise turning of the main direction of the lines of equal disturbance height, which is caused by the Coriolis force. Fig. 8 shows the "twin" surges of the North Sea in December, 1954, as recorded at the Hook of Holland. This was a case of resonance. Further details are discussed on page 637. 3. Dynamics and Forecasting A. Fundamental Theory The mathematical theory of surges starts with the hydrodynamical equations of motion. For the horizontal acceleration components these are : 'dUx dt du dUx dx cu, dy dUa 8z du y CUu _ + „,_ + „,_ + „,_ dp dq drxx , ^ryx drzx dp 8q drxy dryy drzy where /=2cu sin 9 = Coriolis parameter, g = potential of astronomical tide forces, Txx, ryx, etc. = components of turbulent stress tensor, while the other symbols have their usual meaning. Although lateral stresses may be of importance for the currents associated with surges (especially in narrow sea areas), stress components other than tzx 622 GROEN AND GROVES [CHAP. 17 and Tzy will not be taken into consideration in the present theory. Therefore, Tzx and Tzy will simply be written as tx and Ty, together forming the horizontal vector T. Assuming static equilibrium in the vertical, we have Jz ^x,yP = ^x,yP0 + gp0^x,y^+ gV x, y p{z') dz' , where po = atmospheric pressure at the sea surface, t, = height of the sea surface above zero level, and V^;, 2/ = horizontal component of the gradient operator. The above equations, together with the equation of continuity and the boundary conditions, also form the basis of the theory of wind-driven sea currents, which is dealt with extensively elsewhere in this book (Section 3). For the present theory of surges we shall concentrate our attention mainly on the phenomena in more or less shallow sea areas and shall neglect differences of density (/> = const.), thus confining ourselves to quasi-barotropic systems of motion. Furthermore, we shall make the problem a two-dimensional one by integrating the equations of motion vertically. For this purpose, the acceleration terms may now with a sufficient degree of accuracy be approximated by 8ux — dUx — dUx dux and dUy dUy dUy dUy where the bar over a symbol means the vertical average of the quantity con- cerned. Integrating them vertically from the bottom, z= —h, to the surface, z = ^, and dividing the result by the depth, H = h + t„ we obtain : d/ Ux\ ^ Ux d / U„ , et, 1 dpo dq Tax - Tbx .ON ^ 8x p 8x 8x p{h + C) 8t\h + CJ h + C 8x\h + C N Uy\^'Ux8IU_^ , 8t\h + ^J h + C 8x\h + t, Uy 8 / Ux\ h + r,8y\h + i) fUy h + C Uy 8 1 Uy\ fUx 81 1 8pQ 8q Tgy-Tby h + C ^ 8y p 8y 8y^ p{h + Q' ^ ^ where U = udz={h + Qu = volume transport, J-h Xa = surface stress vector ( = the stress exerted by the atmosphere on the sea), Tb — bottom stress vector ( = the stress exerted by the sea on the bottom). SECT. 5] SURGES 623 Besides the equations of motion we have the equation of continuity, whicli after vertical integration yields 8lh dUy ^ _dC dx^ dy ~ dt' ^ ' Finally we have, at boundaries of the sea area considered, boundary condi- tions of two sorts : (i) along a closed boundary (coast) the normal component Un of the volume transport vector U vanishes, U n — 0 ; (ii) along an open boundary ^ or C/^ or a quantitative relation between them is given. As for the zero level of z, it is supposed that 2 = 0 is the sea-surface level in the case of no astronomical (tidal) and atmospheric effects being present, so that l,{x, y, t) may be considered as a disturbance of sea-surface height. Of the surface and bottom stress vectors the first one is mainly determined by the wind velocity W (defined at a certain height above the sea surface). Strictly speaking one should use the relative velocity W — uo, where uo denotes the surface current velocity, which might be expressed by means of U and h-{-t,; we shall, however, discard this detail here and assume Xa to be roughly proportional to W^ x air density, the proportionality factor being dependent on the vertical stability of the air mass and also, to some degree, on W itself. The bottom stress t^ must be expressed by means of the dependent variables U and I, and of Xa in order to make the two-dimensional equations accessible to mathematical treatment. The physically most simple current model for this purpose is one which uses a vertically constant eddy viscosity (see, e.g., Welander, 1957). That this assumption fails, however, is most easily seen in the case of zero volume transport (U = 0) and negligible transverse currents (%;^0, Ta2/ = 0). As is well known, the stress exerted by the bottom would in this case, under the above assumption, become 0.5 x the surface stress, whereas in reality, in turbulent flow, it is mostly less than 0.1 x the surface stress (see e.g. Schal- kwijk, 1947; Hunt, 1956; Weenink, 1958). We shall denote the bottom stress present in this case (U = 0) by t^'^^ and assume (see Bowden, 1953) Tft(O) = — mTa(w<|l). If a volume transport is present, tj, may in fair approximation be written as the sum of Tb^°^ and a stress that is roughly proportional to pu'^, or : T. = -mT, + ^^^^, (6) where the proportionality factor s may still depend somewhat on the depth (e.g., a formula with s proportional to (A-f^)-'/^ has been used by Freeman, Baer and Jung, 1957) and on the degree of turbulence (influenced by wind and tidal currents). This formula presupposes a vertical current profile which, 624 GROEN AND GROVES [CHAP. 17 because of relatively small depth and sufficiently developed turbulence, is little influenced by the Coriolis force. So, in equations (3) and (4), we have Ta-Tb = (l+w)Ta- ^^^ • (7) The non-linearity of the equations makes it very difficult to solve them adequately and makes the separation of an astronomical effect and an atmos- pheric effect problematic. There are three ways of approaching the problem of integrating a system of non-linear equations. First, one may use a procedure of direct integration such as the method of integration along characteristics (see Schonfeld, 1951 ; Freeman, 1954, 1957). Secondly, we may have recourse to numerical integra- tion, approximating the differential equations by suitable difference equations. (This method has been applied by Doodson, 1956.) Thirdly, we may first linearize the equations and then set up an iterative process by which the non- linear equations are solved in successive approximations, as follows. In the dynamical equations we split all the terms which are non-linear into a linear part and a non-linear residue. For instance : dtyh+c) h dt hdt\h+C The most difficult to linearize are the bottom stress terms. We may, however, introduce a suitable average value Um of U, which may be a function of x and y, and then write sUV sUmV , ^. _./ 1 1\ s{U-U„,)V If we now write all non-linear residues on the right-hand side of equations (3) and (4), together with all known terms, we obtain : = {l+m)p-^Tax-hp-^^-^-h^ + X{^, U,, Uy), (8) = {l+m)p-Way-hp-^^-h^+Y{C, U,, Uy), (9) where X{i„ Ux, Uy) and Y{t„ Ux, Uy) stand for the sums of all non-linear residues, as defined above. The solving procedure is now as follows. We first neglect X and Y in the above equations and, by solving the equations thus simplified, find first-order approximations ^ along that boundary is known. If C is given along the open boundary, either as a known function <^b{s) of place (s) along the boundary or as a constant, as in equation (13), this means that the tangential derivative (i/cs along the boundary is given, or: k — + i— = — - — = r]{s), ex cy (is where ]c = dxfds, l = dylds and ry(s) is a known function of 5. By (22) and (23) this condition becomes a condition for of the following general form : A-|.x| = m ,32) where A', L and F are known functions of s (the place coordinate along the boundary), K and L being constants if h is constant along the boundary. If the open boundary is a narrow opening which acts only as a "leak", we have a boundary condition of the type of equation (14) where the transport Tab through the opening may now be written : Tab so that we have = f Unds = (f>i,-(f)A, (33) Ja <^B = (^A + A(^*AB-^**), (34) where ^a may be put equal to zero, if we like. As an example we take a model sea as shown in Fig. 9. Along the line CD, where the sea communicates freely with the deep open ocean, it is supposed that ^ = 0 (see section 3-B-a), which means that condition (32) becomes: f^ + rh-'^^ + p-Wx ^ 0 alongCD, (35) ex <^y where the a:-axis is assumed to have the direction of CD. At the opening AB condition (34) holds, ^*ab and ^** having been defined at the end of section 3-B-a. The current field may be seen as consisting of three components : (a) a current "caused" by curl t, according to the first term of the right-hand member of 632 GROEN AND GROVES [chap. 17 (29), (6) a current "caused" by Vh, according to the second term of the right- hand member of (29), and (c) a current "caused" by the leak, according to (34). Following Weenink and Groen (1958), these component current fields may be called "curl t current", (f>a, "bottom slope current", ^^ and "leak current", (f)c, respectively, and may be formally defined by splitting up the right-hand member of (29) and the boundary conditions (30), (34) and (3o) as follows : '^{cf>a) = p-^h curl T, <^a(DEA) = a{BC) = 0, f^ + rh-^^ = 0 along CD; dx ^{{,, respectively, while the remaining j^art l,c is brought about by ^c and may, therefore, be called the "leak effect" : ^U = ^a + ^&+^c- The physical mechanisms underlying the idea of these different sorts of currents are immediately clear for the "curl t current" and the "leak current". The idea of the "bottom slope current" may be explained in the following way. Suppose the depth of the sea represented in Fig. 9 decreases from west (left) to east (right) and a uniform wind blows from the north. Then the static wind effect t,T will increase from A to E, so that, since no wind stress is working in that direction, a current will run in the opposite direction, giving rise to an anticyclonic circulation in the sea area. This is the bottom slope current. We shall not enter into details of various mathematical techniques of com- puting the wind-induced current field from the above equations. Weenink (1958) has worked out various examples of such current fields for the North Sea. Weenink, with a view to the practice of forecasting wind-surge heights of SECT. 5] SURGES 633 the North Sea, used inhomogeneous wind field models composed of a number of homogeneous subfields. Workers of the Mathematisch Centrum, Amsterdam, on the other hand have computed wind effects in geometrically simple basins under linear wind fields (Veltkamp, 1954; Lauwerier, 1959). The effect of the current field upon surge heights is twofold, according to equation (28) : a Coriolis-force effect which gives a raising of the sea-levels to the right-hand side, a lowering to the left-hand side of the current, and a friction effect which adds a sloping down in the downstream direction of the path of integration used in defining the current effect according to (25), (27) and (28). (Cf. Fedorov, 1956.) c. General case The linearized, time -dependent equations of motion are written as follows : -^ = -/[ j X U] - rh-^V -ghV^ + p-^x - hp-^ Wpo, (36) where, as before, t = (1 + ni)'Za. If the bottom stress term is kept in the quadratic form, we have ^= -fUxV]-sh-WV-ghWUp-''T-hp-^Vpo. (37) The continuity equation is I = -VU. (38) The boundary conditions are those of section 3-B-a. The most simple case of non-equilibrium is the case of a free oscillation, without pressure differences or wind stresses acting and without non-stationary boundary actions (external surges). The result is a damped wave or "seiche", the possible periods of which depend on the geometry of the basin, its bound- aries (whether closed or open), the effectiveness of friction and the direction in which the body of water has been brought out of equilibrium. The mathematical theory of such oscillations will not be dealt with here. Undamped seiches with geostrophic (Coriolis) effects and damped seiches without geostrophic effects have for a long time been studied theoretically (cf. Proudman, 1953, ch. XI and XIV; Saito, 1949; Harris, 1954; Reid, 1957; Hofsommer, 1958). Inclusion of both friction and Coriolis force makes the problem much more difficult, mathematically. This problem has been treated by van Dantzig and co-workers (1958, 1959). Kinematically, the local variation of elevation at a certain point under a simple free oscillation may be described as a damped sinusoid ^ = ^(0)e-«< cos 27TtlTo for t > h, (39) satisfying the equation i + 2a§ + 6=? = 0. (40) 634 GKOEN AND GROVES [CHAP. 17 where the period of free oscillation Tq may be written '^ (41) We shall return to these formal relations later on. {i ) Wind surges Formal solutions of equations (36) or (37) and (38), including varying wind- stress, in combination with the boundary conditions of section 3-B-a, have only been obtained for the very simplest models (see e.g. Lauwerier, 1957, 1959a; Hofsommer et al., 1959). Numerical techniques of approximating solutions by stepwise integration, starting from given initial conditions and using appropriate difference equations instead of the differential equations, have been developed by various authors and are discussed briefly in section 3-B-d in connection with the forecasting of wind surges. A method of successive approximation by an iterative process, proposed by Groen, has been described by Weenink and Groen (1958). It does not reckon with predetermined initial conditions, so that any free oscillation may be super- posed on the solution thus obtained. The method is briefly as follows. In sea areas of the sort we are especially dealing with in this section 3-B, the quasi- equilibrium state corresponding to and varying with the varying wind-stress field may in many cases be looked upon as a first approximation to reality, as we have seen in section 3-B-b. Writing equation (36) (without the term with V^o) as follows: i?(U,VO-fp-iT = -^, (42) and writing the equilibrium wind effect and current field as ^o, Uo, we have, apparently, ^(Uo,V^o) + p-iT = 0, (43) VUo = 0. (44) We can now proceed to a next approximation, ^i, Ui, by substituting ^o and Uo in the left-hand members of (36) and (38) : Q{VuV^i) + p-^x = ^, (45) VUi = -^' (46) the boundary conditions being as before. An advantage of this method of approximation lies in the fact that no time-derivatives of unknown variables occur in the equations to be solved. (It bears some resemblance to the approximation of the ageostrophic wind component by means of the so-called isallobaric wind.) SECT, 5] SURGES 635 Subtracting (43) and (44) from (45) and (46) respectively and writing Ui = Uo + Uoi, we obtain : ^(Uoi, V^oi) = -^^ (47) VUoi = -~ (48) The boundary conditions for ^oi, Uoi are similar to those for ^o, Uo and for For finding Uoi one eliminates ^oi from (47) in the usual manner, by dividing by h and taking the curl of both members of the resulting equation, thus obtaining an equation for U which is analogous to (28) but for the absence of t and the presence of dVoldt. Any boundary condition applying to ^oi is transformed into a condition for Uoi by means of (47) ; the resulting condition is analogous to (32). If the sea has constant depth, it appears that the equation for Uoi derived from (47) determines the field of curl Uoi, so that the problem reduces to the well-known problem of finding a vector field Uoi if div Uoi and curl Uoi are given, together with the boundary conditions. Further approximations can be obtained by formally replacing in the above equations ^o, Uo, ^i, Ui by ^i, Ui, ^2, U2, respectively; and so on. As has been said before, the solution thus approximated is a solution on which any free oscillation may be superposed if this should be required by special initial conditions. A qualitative feature accompanying many wind surges in bays and partly enclosed seas, such as the one shown in Fig. 9, is the phenomenon of the round- going maximum. If the wind blows from the side of the main opening of the sea, the water flowing inward from the opening during the rising stage of the surge will experience a Coriolis force which causes an extra piling up on one side of the area (the west side in Fig. 9). During the falling stage of the surge, water flows back toward the opening, giving by the Coriolis force an extra piling up on the other side. Thus it has often been found in such cases that the local time- maximum of surge height travels counterclockwise along the surrounding coasts. For the North Sea, for example, this phenomenon has been described by R. H. Corkan (1950), J. and M. Darbyshire (1958) and G. Tomczak (1958) and it has been confirmed by the experience of sea-level-height forecasting services. In charts with iso-lines of disturbance height, it appears as a backing of the iso-lines in the course of time (see Fig. 7). Locally, the wind effect may, in those areas where the equilibrium effect ^0 can be used as a first approximation, be written as the sum of ^o(0 ^^^ ^ 636 GROEN AND GROVES [CHAP. 17 "perturbation term", C^^'>{t), which, of course, depends on the development in time of the whole wind field, but which in many cases is chiefly determined already by the development of ^o(0 ^^ the place under consideration. Elaborate studies by Schalkwijk (1947) and Weenink (1956) and a routine practice of forecasting sea-level heights along the Netherlands coast have indeed shown this to be so, practically. It was found that a fair approximation of ^^^>(0 is given by ^(i)(^)~ ^o{t— §) — ^o{i), so that one may write ao = ^o(^-s)+^<2)(^), where 8 is a time lag and ^<'-)(0 is a secondary correction term. At the Hook of Holland, for example, the time lag for the North Sea wind effect is, according to Schalkwijk {loc. cit.), 2 to 3 hours, on an average (see also Haurwitz, 1951, who studied this effect theoretically). One may refine the use of time lags by writing t,o{t) as the sum of contributions from different sub-areas of the sea and using different time lags for the different sub -areas. Fig. 10. Schematical example of the difference between equilibrium wind effect (dashed line) and actual wind effect, as functions of time. Physically, this time lag is a complex phenomenon. In places where the Coriolis force has a negative effect when the water is rising and a positive effect when the water is falling, as on the right-hand side of Fig. 9 (in the case of a longitudinal surge), it adds to the time lag. In places, where the opposite occurs, it diminishes the time lag. The secondary effect ^<2)(f) is mainly an oscillatory after-effect, following a rise or fall of the equilibrium effect. It results mostly in "overshooting" and "undershooting" of maxima and minima, as is illustrated by Fig. 10, which gives an illustration of both the time lag and the secondary effect C^^'>{t), the dashed line representing ^o(0 ^^^ ^he full line ^t). The amount of overshooting or undershooting depends upon the preceding rate of rising or falling of the equilibrium effect ^o and upon the development of ^o following the moment of its maximum or minimum. In the case of a very steep rise of ^o immediately followed by a steep fall, the "overshooting" of the maximum may even be negative, i.e. the equilibrium maximum is not attained in that case. A free after-oscillation (such as is seen on the right-hand side in Fig. 10) has a damping rate depending on the degree of turbulence in the sea, which in its turn depends on the tidal currents and on the wind. For the North Sea, for instance, Schalkwijk {loc. cit.) found the time needed for the amplitude (at the Hook of Holland) to become halved to be 34 h if the mean wind velocity SECT. 5] SURGES 637 was 10 m/sec and 21 h if it was 20 m/sec. The maximum oscillation period of a surge of the North Sea (in the length direction) is about 36 h. The above somewhat rough analysis of the local relation between C{t) and ^o(0 niay be formally refined by means of a mathematical model which is developed from equation (40) by introducing on its right-hand side terms representing the driving forces, in such a way that in the stationary case one obtains ^ = ^o : A simplification of this model, obtained by putting c = 0, was applied by Weenink (1956) to the "twin" storm surges of 21-24 December, 1954, in the North Sea, as recorded at the Hook of Holland. He found for that case (putting c = 0) the values a = 0.06h-i and 6 = 0.20 h"!. According to (8) this would correspond to a period of free oscillation To = 33 h (without damping it would be 31 h). The resonance period Tr, i.e. the period which a harmonic oscillation of ^0 must have in order to give a forced oscillation of ^ with maximum ampli- tude, is somewhat different from To, Tr = 277/v'(62-2a2). With the above values of a and b we have Tr=34.5h. The time interval between the two successive storm-surges was 36 h in that case, so that there was nearly complete resonance, a fact which also appears from the large time lag (6h). While the equilibrium effect oscillated with an amplitude of 1.5 m, the first actual maximum was 0.3 m higher and the second one 0.55 m higher than the corresponding equilibrium maximum. If a third storm of equal strength had followed with the same time interval, the third maximum would hardly have been higher. This is a consequence of the fact that damping was strong during this prolonged storm period. (See Fig. 8.) (n ) Air-pressure surges The effect of a changing atmospheric pressure disturbance has long ago been studied by Proudman (1929, 1953) and more recently, among others, by Harris (1957a) and Platzman (1958). The theory of this effect shows a complete formal analogy with the theory of forced tides, as is seen from equations (3) and (4), where p~^ Vpo and Vg play similar roles. Proudman showed that the effect of an atmospheric disturbance travelling with velocity V over a semi-infinite canal-shaped body of water of depth h (undisturbed), if friction is neglected, is inversely proportional to 1 — V^/gh. So, resonant coupling will occur when V = s/igh). Such resonance has been observed, e.g., on 26 June, 1954, in Lake Michigan. This case has been described by Ewing et al. (1954), Harris {loc. cit.) and Platzman {loc. cit.). Of course, an atmospheric pressure effect will in general be accompanied by a wind effect and much of what has been said about wind surges applies also to air-pressure effects. In most cases the latter will be outweighed by the wind effects. 638 GROEN AND GROVES [CHAP. 17 (iii ) External surges An external surge is a free wave penetrating into the sea region under consideration from an open boundary, where it is induced by a changing sea- level disturbance of the adjacent open sea. Typical examples have been des- cribed by Corkan (1948) for the North Sea and by Rossiter (1959a) for the English Channel. The theory of external surges is very similar to the theory of co -oscillating tides. Mathematical treatments have been given by Goldsbrough (1952), who neglected the Coriolis force, Proudman (1954), who obtained solutions repre- senting damped Kelvin- and Poincare-waves, and Crease (1956). The last author considers a system of waves approaching a semi-infinite barrier. The waves have transverse accelerations balancing the Coriolis force. The effect of the barrier is to form a shadow zone on its lee side, but the transverse accelera- tions on the edge of this zone cause Kelvin-waves to propagate at right angles with the direction of the original waves, into the region behind the barrier. The amplitude of these Kelvin-waves depends on the ratio of the given wave period to the length of the pendulum day. Thig mechanism may account for the properties of certain external surges running into a partly enclosed sea. d. Forecasting The main problem in forecasting sea-surface surges is to find the sea-level disturbance heights from the wind field. In most cases the effect of atmospheric pressure is of much less importance, except in those cases which are near resonance ; most techniques of practically forecasting sea-level heights account for it by taking the static pressure effect ( — 1 cm per millibar of atmospheric pressure difference) multiplied by a statistical reduction factor of the order of 0.5 (e.g. Schalkwijk, 1947, p. 54). Some methods (Corkan, 1950; Leppik, 1952) use the idea of a travelling sea-level disturbance by incorporating in the forecast for a certain place the disturbance observed at some other place a certain time interval before. The various methods used for computing wind effects in partly or wholly enclosed bodies of water from the wind field may be classified into three groups, according to the main line of approach underlying a method, which may be (1) an empirical approach, (2) a semi-empirical-semi-theoretical approach, (3) a theoretical approach. Methods of the first class use direct empirical relations between the wind effect at a certain place and the mean wind or the mean pressure gradient over the sea region considered or over a few different parts of the region, either at about the same time (for effects of nearby areas) or some time before (e.g. 3-9 h, in the North Sea). These methods mainly suppose a quasi-stationary develop- ment, non-equilibrium effects being only statistically (climatologically) in- cluded in the formulas used. To this group belong the methods of (among others) Corkan (1948, 1950) and Rossiter (1959) for the east and south coasts SECT. 5] SURGES 639 of Great Britain, of Tomczak (1952, 1953) for the German Bight, of Verploegh and Groen (1955) for the Dutch Wadden Sea, of Saville (1952) for Lake Okee- chobee, of Miller (1958) for the coastal waters of New England, and of Donn (1958) for some places on the east coast of the U.S.A. Methods of the second class, based on a semi-empirical-semi-theoretical approach, are those of Schalkwijk (1947) and Weenink (1958) for the southeast coast of the North Sea and of Hunt (1956) for shallow lakes, like Lake Okeechobee. The Schalkwijk- Weenink method makes explicit use of the idea of the equilibrium effect as distinct from the actual wind effect. From the basic observational material the effects due to the non-stationarity of the wind fields involved were eliminated in a more or less heuristic way so as to give the true equilibrium effects. From these data, partly by direct correlation and partly by theoretical calculation, graphs and formulas have been derived which give the equilibrium effects at various places as functions of the gradient-wind vectors over a number of sections of the North Sea (thus taking inhomogeneities of the wind field into account) and over the Channel. The influence of air-mass stability on the relation between the gradient-wind velocity and the wind stress on the sea surface is also taken into account in this method of computation. From the equilibrium effect the actual effect to be expected is then found by applying a time lag and secondary corrections, which account for such effects as "overshooting" or "undershooting" (resurging) and after-oscillation (see section 3-B-c). For a concise description of this method see Groen (1961). Hunt's method for shallow lakes divides the lake into a number of "cells" formed by a set of wind fetches and a number of their orthogonal trajectories. The number of cells to be chosen is such that in each cell fairly homogeneous conditions prevail as to depth and wind. In each cell the slope of the lake sur- face is computed by assuming equilibrium within the cell. If now a provisional line of zero disturbance is assumed, the heights along each fetch strip can be determined. Any lateral height differences thus found are smoothed out by some suitable procedure, which accounts for the current effect (see section 3-B-b), the Coriolis force being left out of consideration. The provisional line of zero disturbance and the elevations found are corrected until the condition of zero mean elevation for the whole lake is satisfied. Wholly theoretical approaches, finally, have been proposed by Kivisild (1954), Hansen (1956), Welander (1957), Fischer (1959) and Svansson (1959). These methods come down to numerical integrations of the basic equations of section 3- A, viz. the equations of motion and the continuity equation, either in their vertically integrated form (Kivisild, Hansen, Fischer, Svansson) or in the form of an integro-differential equation based on the well-known fact that, in the absence of lateral stresses, the vertical velocity profile is uniquely determined by the local time-histories of the wind stress and of the surface slope (Welander), See also Lauwerier (1960) and Welander (1961). For the bottom stress the quadratic form may as well be used here as the hnearized form. 640 GROEN AND GROVES [CHAP. 17 The numerical techniques to be used here approximate the differential equations by difference equations, the space and time intervals of which have to be chosen in such a way as to guarantee sufficient computational stability. We shall not enter into further technical details of these methods here. C. Surges along an Open Coast Let us first consider the nature of surges in the open ocean far from a coast. It is instructive to take an idealized region of the ocean the mean surface of which is a plane. Except in the very shallowest regions of the ocean the bottom stress Tb is very small and can be neglected, and the surface displacement t, is very small compared to the depth h. Then, linearizing equations (3), (4) and (5), and neglecting the tidal potential, gives ^+f\ixV + ghV:,,yt, = l{-hV:c,yP0 + Ta) (50) ct p Vx,2/-U + ^ = 0. 8t Now let us first consider the case of constant depth. Either U or ^ can be eliminated from these equations. Eliminating the former, we obtain ^/^V..,2__/2 ___._, (51) where F is the forcing function, given by 1 F = P h yx,y^P0 + ^x,y''Ca +f I CUrle Ta dt 0 (52) A solution for free barotropic waves is obtained by setting F equal to zero. One such solution is a progressive wave form t, = A cos [k{x cos 6 + y sin 6) — at], where 6 is an arbitrary direction toward which the wave form is propagated, and k and a are related according to a2 = ghk^+f^. (53) It is seen from (53) that |a| is always greater than/; i.e. this type of wave motion can occur only if its period is less than half a pendulum day. The orbital motion associated with this surface-wave form does not lie in a vertical plane perpendicular to the crests, as it would on a rotationless earth. For forced waves it is interesting to compare the effects of atmospheric pressure and horizontal wind stress. The horizontal forces arising from the gradient of the atmospheric pressure are conservative ; i.e. there is no curl. The wind stress, on the other hand, does have a curl, which exerts an integrated effect on the water motion. If steady winds were to blow, the forcing function would increase without limit, and so would the elevation and volume transport. But this results from having neglected dissipation. In the real ocean, friction SECT. 5] SURGES 641 or viscosity limits the magnitude of the currents and surface elevation. From (52) it is seen that the influence of depth is to make the pressure terms relatively more important in deep water, the wind-stress terms more important in shallow water. Coastal areas adjacent to large shallow regions are particularly susceptible to wind-generated surges. If a storm pattern moves over the water surface with speed approximately that of a free wave, a large amplitude results. For example, if F = A cos (kx — at), it is found from (51) that A cos {kx — at) ^ ^ a'^-p-ghk^' If G and k have values corresponding to a free -wave solution, then the de- nominator is zero and there is no solution. If a storm began to move in a straight line with free-wave speed, it would be found from this theory that the amplitude of the forced surface wave would increase with time without limit, owing to the neglect of dissipation. Even though the conditions postulated here are not fulfilled in nature, very large surges are sometimes generated in this way. The free-wave velocity in the deep ocean is greater by orders of magnitude than the speeds of storms, but in shallow water there is sometimes a good "match". The destructive surge in Lake Michigan on June 26, 1954, may have resulted from this effect (Harris, 1957). Attempts to take the earth's curvature into account have resulted in the "beta-plane" theory. The effect of curvature is neglected in the equation of continuity, but brought in by considering the Coriolis parameter as dependent on y (distance northward), with dfldy = ^. Both ^ and/ (when not differentiated with respect to y) are taken as constants. Cartesian coordinates are retained, but the situation is now anisotropic, as the earth's rotation introduces a pre- ferred direction. Essentially, the ocean surface is taken to be a parabolically- curved plane, of small north-south extent, tangent to the surface of the rotating earth and osculating along a meridian. The additional complication makes it practical to consider only the simplest of cases. For free waves travel- ling in the east-west direction, there are two classes of motion : the ordinary inertio-gravitational waves and the planetary, or Rossby waves (Veronis and Stommei, 1956). The former can move in either direction and their charac- teristics are only slightly modified by the curvature. The planetary waves, on the other hand, are propagated only westward, and are associated with water motion largely geostrophically balanced. These waves are dispersive. There is a wavelength of minimum period; waves longer or shorter than this wave- length have longer periods. T5rpical periods and wavelengths may be of the order of several days and several hundred kilometers, respectively. But surges are usually observed along coastlines, which greatly complicate the theory. A whole new class of free-wave motions, called edge waves, is intro- duced by a coast. The simplest mathematical way of introducing a coast is to take an infinitely long vertical wall, in which case the edge waves can be 642 GROEN AND GROVES [CHAP. 17 obtained by simple reflection of the ordinary sinusoidal waves at sea. A more complicated model, but one much more satisfying from the standpoint of similarity with a real coast, is an infinitely long beach of constant slope. In this case we deal with the half-plane x>0 with the boundary condition Ux{0, y, t) = 0, and set h = sx. Let us consider again a flat ocean with waves of small amplitude and use the linearized equations (3) and (4). Let us look for free edge-wave solutions of the form Ux = X{x) exp [i{ky + cTt)], Uy = Y{x) ex-p[i{ky + at)], C = Z{x)exip[i{ky + at)], considering k to be positive and a to be either positive or negative. Then the functions X, Y, Z, must satisfy iaX-fY + gsxZ' = 0 fX + iaY + igsxZ = 0 (54) X' + ikY + iaZ = 0 with -X'(O) = 0. The functions X and Y can be eliminated from (54) yielding where . = '-^^f± (56) gs a The only solutions of (55) giving finite Z for all x > 0 are the functions Z = Aqn{2kx), (57) where q-n is the Laguerre function (see Reid, 1958, or Eckart, 1951) of order n. The boundary condition is also satisfied by (57). But (57) is a solution only if K = {2n+\)k. (58) Equations (56) and (58) define a relationship between k and o- from which the longshore phase velocity, c — a/k, and the group velocity, dajdk, can readily be obtained. This relationship is cubic in a ; the three roots correspond to distinct modes for each order n. The order of the Laguerre function determines the onshore-offshore wavelength (which varies with distance from shore) ; the higher the n the shorter the wavelength. Greenspan (1956) has shown that storms with usual dimensions will excite the fundamental or zero order to a greater extent than the higher orders. Reid (1958) has shown that for the zero order only two modes are physically significant. The corresponding roots are oi= -U-Vigsk+ip) ^2= -U+Vigsk+y^). In the case of a rotationless earth, /= 0 and the two modes have frequencies given by CT= ± \/{gsk). That is, the waves are propagated along the coast in SECT. 5] SITRCiES 643 either direction with the same phase velocity. With rotation of the earth, the two frequencies differ in magnitude by /, and hence the phase velocities for waves (of the same wavelength) moving to right or left along the coast differ by //A* for the zero order. The effect of the earth's rotation becomes important, as we would expect, only for waves whose period becomes appreciable as compared to that of half a pendulum day. If the resurgences observed along the east coast of the United States are related to a "wake" of edge waves trailing a hurricane according to the hypo- thesis of Munk et al. (1956), edge waves would be formed whose phase velocity would equal that of the speed of propagation of the hurricane along the coast. Hurricanes in this region frequently progress northward, parallel to shore, over the shelf. This direction corresponds to the positive y direction, and so cti, which has the negative value, gives the appropriate frequency. The frequency can be obtained from the first equation of (59) as a function of the phase velocity, c = — aijk : 1-1=7+/- Let us consider as an example the surge at Atlantic City, shown in Fig. 6 of section 2, caused by the hurricane of 14-15 September, 1944. The velocity of propagation of the storm northward along the coast was about 16 m/sec. Setting this equal to c and putting s — 5.0 x 10~^ and /= 0.94 x 10"^ sec~i gives 7^ = 4.4 h and L = 250 km, as compared to the observed period of about 5.6 h. The most severe handicap of this theory, as applied to surges on a gently sloping shelf, may be the omittance of the bottom stress. The method of taking the bottom stress into account described in section 3-A would give a substantial improvement. Freeman et al. (1957) have successfully reproduced some observed storm tides along regular coastlines from wind observations by making some rather surprising assumptions : they neglect shoreward transport, longshore surface slope, field acceleration and divergence of transport (allowing the continuity condition to be violated). References Bowden, K. F., 19.53. Note on wind drift in a channel in tlie presence of tidal currents. Proc. Roy. Soc. London, A219, 426-446. Bowden, K. F., 1956. The flow of water through the Straits of Dover related to wind and differences in sea level. Phil. Trans. Roy. Soc. London, A248, 517-551. Corkan, R. H., 1948. Storm surges in the North Sea, Vols. 1 and 2. U.S. Hydrographic Office, Misc. 15702, Washington, D.C. Corkan, R. H., 1950. The levels in the North Sea associated with the storm disturbance of 8 January 1949. Phil. Trans. Roy. Soc. London, A242, 493-525. Crease, J., 1956. Long waves on a rotating earth in the presence of a semi-infinite barrier. J. Fluid Mech., 1, 86-96. Dantzig, D. van, 1958. Free oscillation of a fluid in a rectangular basin. Math. Cent. Amsterdam, Rep. TW 49. Dantzig, D. van, 1959. Einige analytische Ergebnisse iiber die Wasserbewegung in einem untiefen Meere. Z. angew. Math. Mech., 39, 169-179. 644 GROEN AND GROVES [CHAP. 17 Dantzig, D. van and H. A. Lauwerier, 1960. The North Sea problem I. Koninkl. Ned. Akad. Wetenschap. Proc, A63, 170-180. Darbyshire, J. and M., 1958. Storm surges in the North Sea. Deut. Hydrog. Z., 11, 124^129. Donn, W. L., 1958. An empirical basis for forecasting storm tides. Bull. Amer. Met. Soc, 39, 640-647. Doodson, A. T., 1956. Tides and storm surges in a long uniform gulf. Proc. Roy. Soc. London, A237, 325-343. Eckart, C, 1951. Surface waves on water of variable depth. Marine Physical Laboratory and Scripps Institution of Oceanography, Wave Rep. 100, Ref. 51-12. Ewing, M., F. Press and W. L. Donn, 1954. An explanation of the Lake Michigan wave of June 26, 1954. Science, 120, 684-686. Fedorov, K. N., 1956. [Water heights and currents during the catastrophic storm in the North Sea in 1953.]Izvest.Akad.NaukS.S.S.R.,Ser. Geoftz.,No.4:, 437-451 (inRussian). Fischer, G., 1959. Ein numerisches Verfahren zur Errechnung von Windstau und Gezeiten in Randmeeren. Tellus, 11, 60-76. Freeman, J. C, 1954. Two-dimensional storm tides in shallow water. Assoc. Intern. Oceanog. Phys. Proc. Verb., No. 6, 206-208. Freeman, J. C. and L. Baer, 1957. Pseudo-characteristics. Trans. 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Wind effect on shallow bodies of water with special reference to Lake Okeechobee. Inst. Hydraul., Kungl. Tekn. Hogsk. Handl., Stockholm, Nr. 83. Koninklijk Nederlands Meteorologisch Instituut, 1960. Meteorologische en Oceanografische aspecten van stormvloeden op de Nederlandse kust. Rapp. Delta-Commissie, deel 2, Den Haag (English summaries). Lauwerier, H. A., 1957. Exponential windfields. Math. Cent. Amsterdam, Rep. TW 42. Lauwerier, H. A., 1958. Free motions in a rotating sea which has the form of a semi- infinite strip. Math. Cent. Amsterdam, Rep. TW 46. Lauwerier, H. A., 1959. Wind effects in a rectangular gulf. Math. Cent. Amsterdam., Rep. T:\N 55. Lauwerier, H. A., 1959a. A contribution to the theory of storm surges of the North Sea. Math. Cent. Amsterdam, Rep. TW 57. Lauwerier, H. A., 1960. The North Sea problem II-V. Koninkl. Ned. Akad. Wetenschap. Proc, A63., 266-290, 339-354, 423-438. Leppik, E., 1952-53. Windstau und Windsunk an der Siidkiiste der Nordsee und in den Tidegebieten der Ems, Weser und Elbe. Mitteilungen der Wasser- und Schiffahrts- direktion, Hamburg. Longard, J. R. and R. E. Banks, 1952. Wind-induced vertical movement of the water on an open coast. Trans. Amer. Oeophys. Un., 33, 377-380. Miller, A. R., 1958. The effects of winds on water levels on the New England coast. Limnol. Oceanog., 3, 1-14. Munk, W. H., F. Snodgrass and G. Carrier, 1956. Edge waves on the continental shelf. Science, 123, 127-132. Platzman, G. W., 1958. A numerical computation of the surge of 26 June, 1954, on Lake Michigan. Geophysica, 6, 407-438. Proudman, J., 1929. The effects on the sea of changes in atmospheric pressure. Mon. Not. Roy. Astr. Soc, Oeophys. Suppl., 2, 197-209. Proudman, J,, 1953. Dynamical Oceanography. Methuen, London. Proudman, J., 1954. Note on the dynamics of storm-surges. Mon. Not. Roy. Astr. Soc, Geophys. Suppl., 7, 44-48. Proudman, J., 1955. The propagation of tide and surge in an estuary. Proc. Roy. Soc. London, A231, 8-24. Redfield, A. C. and A. R. Miller, 1957. Water leVels accompanying Atlantic coast hurri- canes. A7ner. Met. Soc, Met. Monog., 2, No. 10, 1-23. Raid, R. O., 1956. Approximate response of water level on a sloping shelf to a wind fetch which moves towards shore. U.S. Army Beach Erosion Bd. Tech. Mem. No. 83. Raid, R. O., 1956a. Modification of the quadratic bottom-stress law for turbulent channel flow in the presence of surface wind-stress. A. & M. Coll. of Texas, Dept. Oceanography, Tech. Rep. 2, Ref. 56-27T. Reid, R. O., 1957. Forced and free surges in a narrow basin of variable depth and width : a numerical approach. A. & M. Coll. of Texas, Dep. Oceanog. and Met., Tech. Rep. Ref. 57-25T. 646 GROEN AND GROVES [CHAP. 17 Reid, R. O., 1958. Effect of Coriolis force on edge waves (I). Investigation of the normal modes, J. Mar. Res., 16, 109-144. Rossiter, J. R., 1959. A method for extracting storm surges from tidal records. Deut. Hydrog. Z., 12, 117-127. Rossiter, J. R., 1959a. Research on methods of forecasting storm surges on the east and south coasts of Gt. Britain. Q. J. Roy. Met. Soc, 85, 262-278. Saito, Y., 1949. A solution to the oscillation of lake water generated by wind. J. Met. Soc. Japan, 27, 20-25. Saville Jr., Th., 1952. Wind set-up and waves in shallow water. U.S. Army Beach Erosion Bd. Tech. Mem., 27. Schalkwijk, W. F., 1947. A contribution to the theory of storm surges on the Dutch coast. Koninkl. Ned. Met. Inst., Med. Verhandel. Ser. B, no. 7. Schonfeld, J. C, 1951. Propagation of tides and similar waves. Thesis, Staatsdrukkerij, Den Haag. Schonfeld, J. C, 1955. Tides and storm surges in a shallow sea. Rapport C.S.D. 55-3-N, Rijkswaterstaat, Den Haag. Staatscommissie Zuiderzee, 1926. Verslag van de Staatscommissie inzake hogere water- standen tijdens storm als gevolg van de afsluiting van de Zuiderzee. Landsdrukkeri j , Den Haag. Svansson, A., 1959. Some computations of water heights and currents in the Baltic. Tellus, 11, 231-239. Tomczak, G., 1952. Der Einfluss der Kustengestalt und des vorgelagerten Meeresbodens auf den windbedingten Anstau des Wassers Deut. Hydrog. Z., 5, 114-131, 277-284. Tomczak, G., 1953. Die Einwirkung des Windes auf den mittleren Wasserstand der Deutschen Bucht vom 15. Februar bis 6. Marz 1951. Deut. Hydrog. Z., 6, 1-17. Tomczak, G., 1954. Der Winstau- und Sturmflutwamdienst fiir die Deutsche Nordseekiiste beim Deutschen Hydrographischen Institut. Deut. Hydrog. Z., 7, 35-41. Tomczak, G., 1958. Nachweis einer fortschreitenden Welle in der slidlichen Nordsee. Deut. Hydrog. Z, 11, 97-109. Veltkamp, G. W., 1954. De invloed van stationaire windvelden op een zee van op delen constante diepte. Math. Cent. Amsterdam, Rep. TW 24. Veronis, G. and H. Stommel, 1956. The action of variable wind stresses on a stratified ocean. J. Mar. Res., 15, 43-75. Verploegh, G. and P. Groen, 1955. The effect of the wind over the eastern part of the Dutch Wadden Sea on the height of high water at Delfzijl, Eems estuary. Koninkl. Ned. Met. Inst., Wetenschap. Rapp. 55-009. Weenink, M. P. H., 1954. Bijdrage tot de theorie der opwaaiing in een baai of randzee bij een inhomogeen windveld. Koninkl. Ned. Met. Inst., Met. Rapp. Stormvloed 1 Feb. 1953, le vervolg, 40-57. Weenink, M. P. H., 1956. The "twin" storm surges during 21st-24th December 1954. A case of resonance. Deut. Hydrog. Z., 9, 240-249. Weenink, M. P. H., 1958. A theory and method of calculation of wind effects on sea levels in a partly-enclosed sea, with special application to the southern coast of the North Sea. Koninkl. Ned. Met. Inst., Med. Verhandel. no. 73. Weenink, M. P. H. and P. Groen, 1958. A semi-theoretical, semi -empirical approach to the problem of finding wind effects on water levels in a shallow partly-enclosed sea. Koninkl. Ned. Akad. Wetenschap. Proc, B61, 198-213. Welander, P., 1957. Wind action on a shallow sea : some generalizations of Ekman's theory. Tellus, 9, 45-52. Welander, P., 1961. Numerical prediction of storm surges. Advances in Geophys., 8, 316- 379. 18. LONG OCEAN WAVES W. H. MUNK 1. Introduction "Long waves" liere refers to i:)eriod.s between swell and tides, that is, from about I minute to 12 hours. This covers ten octaves. The most conspicuous thing about waves in this frequency range is their absence. The energy^ con- tained in the swell and tides is each of the order lO^ cm 2 ; the energy contained in the entire intermediary range of frequencies is of the order of 1 cm^. Once every year or two, a "tsunami"' (or "tidal wave", or "seismic sea wave") produces energies of 10^ cm^ in this intermediary range, and the long-wave activity may remain above background for a week. Tsunamis are associated with submarine volcanic eruptions, earthquakes or landslides, whereas the background activity is meteorologically induced. To a large extent the interest in long waves derives from the occurrence of occasional tsunamis. "Seiching" in harbours and bays may affect safe anchoring conditions, and is another source of interest. "Surges" or "storm tides" induced by severe storms are typically associated with periods in excess of those of the ordinary astronomic tides, and accordingly they are considered as a separate topic (Chajiter 17). Nevertheless, the long-wave background to be described here has many of the aspects of micro-surges, and the subject cannot be so neatly separated. 2. The Instruments Observation of the intermediary frequencies puts exacting demands on instrumentation and analysis. This is the penalty for working in a sjjectral valley. The first requirement is to reduce the waves and swell by means of a suitable low-pass filter. The traditional way to accomplish this is the Kelvin tide-gauge (Fig. 1). This is still the principal source of observational evidence concerning long waves. An obvious improvement is to add a high-pass filter to reject the tides as well, thus making the long-wave recorder a band-pass filter peaked in the intermediary frequencies. 2 One simple way of accomplishing this is shown in Fig. 1. Another obvious improvement is to replace the orifice (which responds non-linearly) with a capillary, as shown. Nearly all shore- based long-wave recorders are elaborations on this simple theme (Munk, Iglesias and Folsom, 1948; Van Dorn, 1956 and 1960; Snodgrass, 1958). In general, pneumatic filtering is better adapted to this frequency range of a few cycles per hour than is electronic filtering. Temperature effects are a serious 1 Here used in the sense of the "mean-square elevation". Actually the energy per unit area is pg (mean-square elevation). (See Cartwright, Chapter 15.) 2 A low-pass filter only (e.g. a tide-gauge) is a more satisfactory long-wave instrument than a high-pass filter only (a wave recorder with a slow leak). This is because it is easier to detect tiny high-frequency wiggles superposed on low frequencies than tiny low- frequency undulations underlying high frequencies. The distinction must be physiological : the eye responds to the rate of change of the ordinate rather than to the ordinate itself. [MS received October, 1960\ 647 648 [chap. 18 problem but can be avoided by placing the transducer into a barrel of sand (first suggested by Judith Munk) or burying it in the sea bottom. Off-shore recording of long-period waves has been accomplished by means of an absolute pressure transducer on the sea bottom with a dynamic range of 10^ : 1 (Snod- grass et al., 1958). Here the output has been recorded in digital form, and all filtering is accomplished by numerical methods. To recorder Air capillary Float Water capillary -/^ Pressure ~y-y gauge Orifice Fig. 1. Schematics of tide-gauge (left) and long-wave recorder (right). The tide-gauge records the water level in a well which is shielded from (high-frequency) ocean waves by an orifice. In the long-wave recorder the top of the well is capped, and the pressure differential between the entrapped air and atmosphere is recorded. Tides are removed by a tiny capillary leak. 3. The Spectrum Fig. 2 shows a typical background record from a Snodgrass long-period recorder peaked at 1 c/ks (period 1000^), with half-power points at 0.1 c/ks 1500 1530 1600 1630 1700 Fig. 2. Sample wave record, taken on 23 February, 1956, off Camp Pendleton, California, at a depth of 20 ft. The vertical scale is in chart inches. (After Snodgrass, 1958, Fig. 5.) (10,000^) and 10 c/ks (100^) respectively. The effect of swell is barely discernible as a thickening of the record trace (ordinarily a fine line) to about 0.1 chart inches. The high frequency wiggles of the thickened trace are associated with SECT. 5] LONG OCEAN WAVES 649 "surf beat", whereas the pronounced undulations (unusually active) of about 2 cycles per hour are a "shelf resonance". A drop in the mean level by about 1 chart inch in two hours indicates a falling tide. This sample record of back- ground activity illustrates very nicely our subject matter and its frequency boundaries : from swell to surf beat to shelf resonance to tide. Surf beat and shelf resonance are more prominent in the instrument output than swell and tides ; the filtering has succeeded in inverting a spectral canyon into a spectral hump. This is a sensible procedure, no matter how sophisticated the subsequent analysis is to be. For a further discussion, the representation in terms of power spectra ^ is as convenient here as it is for ordinary ocean waves. Miles of wiggly curves in the Cycles per kilosecond Fig. 3. A typical spectrum for Camp Pendleton, California. The two curves designate the spectra of surface elevation at distances of 8000 and 13,000 ft from the beach, respectively, and corresponding water depths of 20 and 100 ft. time domain are condensed to a few simple traces in the frequency domain. All evidence regarding phase (which is not reproducible) is suppressed, and the resulting information concerning the distribution of energy in the fre- quency domain is stable and reproducible. The distortion arising from instru- mental filtering (or lack thereof) is explicit. Fig. 3 shows a typical spectrum for southern Cahfornia. The trace marked "20 ft" corresponds more or less to the sample record in Fig. 2. Allowance has been made for instrumental response factors. The Snodgrass instrument measures pressure on the sea-bed; ac- cordingly the high frequencies are suppressed relative to a record of surface elevation. The spectrum in Fig. 3 has been corrected for this attenuation in 1 See Chapter 15. Most of the spectra in this discussion were made by digital methods. A cookbook of numerical recipes is contained in Munk, Snodgrass and Tucker (1959). 660 MUNK [chap. 18 accordance with classical theory. Actually the correction is completely negligible except at the highest frequencies included in the figure. Fig. 3 is based on a composite of many measurements in the Camp Pendleton area (see Munk, Snodgrass and Tucker, 1959, charts 2.12, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3, 4.4, 5, 10.1, 10.3). The peak at 0.8 c/ks and the troughs at 0.5 and 2.3 c/ks are found on all analyses. This peak will be discussed under "shelf resonance". The broad humps between 10 and 20 c/ks are not nearly so well established. They appear to depend on the depth of water, and accordingly on the distance off-shore (see "surf beat", section 4). The sharp peak at 47 c/ks (period 21^) is typical of a long-period swell from the south. New peaks appear once or twice a week at about 40 c/ks, then increase in frequency by 5 to 10% per day, until they disappear into the high background at about 125 c/ks. The wandering peaks are associated with storm systems in the South Pacific and Indian Oceans (Munk and Snodgrass, 1957). The dispersive shifts of these frequency peaks is in marked contrast to the fixed spectral signatures of the long waves. Fig. 3 is to serve only as an idealized description of the background activity. On any given day the spectral densities may be up or down by an order of magnitude (although the relative spectrum is surprisingly invariable). The spectral structure varies from place to place, particularly with regard to shelf resonance. Finally it must be stated that the terms "surf beat" and "shelf resonance" are by no means generally accepted ; these names imply causes that may not be borne out by future work. 4. Surf Beat A typical frequency is a cycle per minute. The existence of relatively promi- nent waves in this frequency range was inferred by Munk (1949) and Tucker (1950) from a visual examination of low-frequency wave records. The waves were attributed to the fluctuating height of groups of sea and swell waves breaking on the shore, hence the name "surf beat". Tucker and Munk both found a clear relation between the amplitudes of the surf beat and of the ordinary waves, approximately 1 to 10. A similar result was obtained by J. E. Dinger for wave records from Barbados. In all these cases the wave recorders were in quite shallow water. During the fall of 1959, Snodgrass, Miller and Munk obtained several hundred successive spectra (not published) from a wave recorder at 100 m depth on the exposed (western) shore of San Clemente Island, 60 miles off-shore from the coast of southern California. In no instance do we find a spectral peak of surf-beat frequency. Rather, the spectrum is flat at about 10~2 cm^/c/ks for frequencies below the swell and above the shelf resonance. This suggests that the surf-beat activity is characteristic of shallow water. A re-examination of all .available spectra did, in fact, indicate a decrease in the spectral activity between 10 and 40 c/ks with increasing depth of water, as indicated in Fig. 3. Munk (1949) suggested that the surf beat may consist of long waves generated in the surf zone by variable "radiation pressure" of the incoming breakers, and SECT. 5] LONG OCEAN WAVES 651 then traveling seaward at a speed \/{gh). Conservation of energy flux i'^a^X^igh)) would lead to a dependence of amplitude on depth according to Green's law : a'^h"^*. This hypothesis is false. The new observation indicates a much more rapid decrease with depth than is consistent with Green's law. Furthermore, by using an array of recorders, we can now estimate the direction of the surf-beat waves : it is shoreward, not seaward. The only safe conclusion at this time is that surf beat consists of some non- linear interaction between the ordinary waves and the low-frequency waves. Tucker's observation of a phase correlation between the surf beat and the envelope of the ordinary wave record points in this direction. Whatever is responsible for the generation of the "difference-frequencies" (Tick, 1958), the important terms appear to be dependent on depth and one may assume that the dimensionless number (a/^) plays the essential role. 5. Shelf Waves Fig. 4 shows the low-frequency portion of the long-wave spectrum in more detail. The Camp Pendleton peak to the very left of Fig. 3 now appears as a broad hump of relatively high frequency as compared to other stations. Cycles per hour 4 100 0.01- 1 2 Cycles per kilosecond Fig. 4. Background spectra at Mar del Plata, Argentina ; Acapulco, Mexico ; Camp Pendle- ton, California ; and at Lahaina Wharf, Maui, Hawaii. Acapulco shows a very sharp peak at 2 c/h (Munk and Cepeda, 1961). At Maui the spectrum is complex (all peaks are reproducible) with the lowest frequency peak at 1 c/h. At Mar del Plata (Inman et al., 1962) the spectrum is of still lower frequency, with the fundamental at 0.3 c/h ; the energy density is a hundred times above that at the other stations. The occurrence of high and long-period "seiches" over the broad Argentinian shelf is well known. 652 [chap, is Fig. 5 shows simultaneous spectra at two stations, 100 m and 1500 m oif- shore respectively, from the sheltered (eastern) shore of Guadalupe Island off the coast of Mexico (Munk, Snodgrass and Tucker, 1959). The off-shore recorder shows three broad peaks with reproducible fine structure! The on-shore record shows the identical fine structure. The peaks (here included under shelf waves) actually cover the frequency range previously allotted to surf beat. There are two reasons for this : (i) the extremely narrow steep shelf off Guadalupe can be expected to lead to shelf resonance at relatively high frequencies (as we shall 0 10 20 Cycles per kilosecond Fig. 5. The spectrum at Guadalupe Island, Mexico. Top : spectra at an on-shore instrument (depth 21 ft, 100 m from shore) and an off-shore instrument (depth 372 ft, 1500 m from shore). Center : Coherence between on-shore and off-shore records. Bottom : Phase lead of the shore-based record relative to the off-shore record. (After Munk, Snodgrass and Tucker, 1959.) demonstrate) ; (ii) on the sheltered side of the island, waves (and hence surf beat) were virtually absent, thus permitting the relatively low-energy shelf peaks to rise above the surf-beat background. The two most obvious features to be discussed are : the frequency of the peaks and their sharpness. The latter are conveniently portrayed by the dimensionless parameter Q, defined as the ratio of the central frequency to the width at the half-power points ; alternately the energy amplification at resonance is Q^. Parameters referring to the fundamental peaks are summarized in Table 1. SSCT. 5] LONG OCEAN WAVES 653 Table I Shelf Wave Parameters Guadalupe Camp Acapvilco Maui« Mar del Island Pendleton Plata Observed frequency and ham d width of fundamental node To, h 0.043 0.35 0.50 1.0 3.1 /o, c/ks 6.5 0.80 0.57 0.28 0.09 Q 10 2 14 4 2 Dimensions of Shelf s' 0.06 0.02 0.02 0.014 0.00073 A, km 1.5 4.6 6.5 50 150 H',m 90 92 130 700 110 s 0.33 0.05 0.32 0.13 0.018 B, km 22 14 11 25 200 H, m 3700 800 3700 4000 3600 s/s' 5.5 2.5 16 9.2 25 ViH/H') 6.4 2.9 5.3 2.4 5.7 Computed Values /o (eqn. 1) 2.5 0.82 0.69 0.21 0.03 Q 12 (3) 10 (4) 11 "' Taking the dimensions of the Kealaikahiki channel. It is possible that the resonances are associated with the Auau channel between the islands of Maui and Lanai. A. Frequency The table is in order of descending values of the frequency of the lowest spectral peak. Values vary from 6.5 c/ks (period 0.043 h = 2.5 min) over the narrow, steep shelf of Guadalupe Island to 0.09 c/ks (period 3.1 h) over the broad gentle shelf of Argentina, We have implied here (and earlier) that these spectral bands are governed by the dimensions of the shelf. The most simple- minded model is that of a standing wave with an antinode at the shore line, and a node at the edge of the continental shelf {x = A, Fig. 6). For the funda- mental node the period is four times the travel time from a; = 0 to a; = ^ . Let h{x) be the depth and C = \/{gh) the phase velocity. Then To = 4 /o- To dx 0 W) ^ VJgH') SA dh = 4 0 Vgh VgilH') g_ H' A (1) Any of the three forms of the equation (1) proves useful. Note that the resonance would be the same for a shelf of width A and constant depth IH'. The calcula- tion presumes a coastal inclination, s', over the shelf, and neglects the bottom 654 MUNK [chap. 18 topography beyond the shelf, as if there were a precipitous cHfiF at the edge of the shelf. We have derived the formulae for the case of normal incidence on a two-step topography (as shown in Fig. 6), consisting of a continental shelf of width A and constant inclination s', and a continental slope of width B and inclination s. The derivation is based on patching the known solution for constant slope (Lamb, 1932 p. 276). Details are cumbersome and uninteresting. The result is where j8 depends in a complicated manner on the ratios s'js and H'jH. Thus the frequency is increased in the ratio 4j8/7r over the result (1). For the case s' <^s and H' <^H, we obtain 4^/77= 1.54. Fig. 6. Idealized bottom profile. The continental shelf extends from .^ = 0 to x^A, with depth h increasing linearly from 0 to H'. The continental slope extends from a; = ^ to x = A + B, with depth increasing linearly from H' to H. The computed values of /o according to (1) are given in Table I. In three cases the agreement with the observed peaks is misleadingly good ; in the other two cases the values differ by factors of 2 and 3, respectively. One can improve the agreement by various devices. Thus the correction factor 4|8/7r helps because it is relatively large (close to 2) for the two discrepant cases (Guadalupe Island and Mar del Plata) where the ratio H'/H is relatively small. Furthermore, one has considerable freedom in fitting the actual bottom topography into a "two step" straight jacket. It does not seem profitable to pursue such refinements. The only valid point is that for the station here considered an observed varia- tion of /o by two orders of magnitude is consistent with the assumption that these spectral peaks are the result of resonance over the continental shelf, somewhat in the manner of resonance in open (though tapered) organ pipes. B. Sharpness of Peak Here the results are even more uncertain than in the case of /o. The "observed" values of Q in the third line of Table I are a compromise between using the spectral width Af—foQ~''- and the energy amplification Q^ as a basis for estimating Q. The two-step topography yields as a solution Qo = l.QSViHIH') SECT. 5] LONG OCEAN WAVES 655 for the peakedness of the fundamental node, provided s' <^s and H' -^H. This approximation is certainly not justified for Camp Pendleton and Maui ; for those two cases Q was evaluated numerically. The comparison of computed and observed values indicates : (i) an order of magnitude agreement for localities other than Mar del Plata ; (ii) at Camp Pendleton the inclinations over the shelf and beyond the shelf are both small and their contrast is also small (0.02/0.05). In addition, the relatively small depth of 800 m of the Continental borderland (as compared to the deep sea) yields a relatively small ratio of HjH'. The two circumstances conspire to yield a small Q ; (iii) at Maui the break in slope occurs at relatively large depth {H' = 700 m), and the depth contrast is relatively small," hence Q is small ; (iv) at Guadalupe Island the ratio of inclinations is large, and at Acapulco very large (associated with the drop-off into the Acapulco Trench). This leads to large values in ^ ; (v) at Mar del Plata the observed Q is small and the computed Q is large. As an excuse, one might mention that the theory presumes a dependence of depth on off-shore distance only, and no variation along shore. Over the broad Argentinian shelf, the variations in the two directions are of the same order. Another point is that the peaks appear to be superimposed on a physically distinct monotonic spectrum which fills the troughs and reduces the apparent Q (Inman et al., 1962). C. Off-Shore Coherence The center and bottom displays of Fig. 5 show the coherence (or phase- stability) between the on-shore and off-shore records, and the corresponding phase relation.! For very low frequencies the two records are in phase. At about 12 c/ks the phase reverses, and for frequencies just above 12 c/ks the instruments are out of phase. There are similar abrupt changes in relative phases at 17 and 22 c/ks. The interpretation might be as follows: the lowest frequencies correspond to the largest wavelengths and the nodal line is far off- shore, beyond both instruments. The two records are then in phase. Increasing frequencies are associated with a diminishing distance from the beach to the nodal line. Presumably the nodal line "passes" the off-shore instrument at 12 c/ks so that for frequencies just above this value the two instruments are situated at opposite sides of the nodal lines and accordingly 180° out of phase, as observed. But it should be stated that an attempt to account quantitatively for the successive reversals did not lead to any clear-cut results. We have presented this figure as an illustration of the powerful tool provided by cross- spectra in the diagnosis of features in the power spectra. D. Long-Shore Coherence At this point a theoretical problem needs to be introduced which is funda- mental to our subject. So far we have discussed the total reflection of long waves which are trigonometric in deep water (beyond x = A + Bm Fig. 6). The 1 For a detailed discussion of "coherence" and "phase", we must refer to the original paper. 656 [chap. 18 fact that we treated only the case of normal incidence is of no great con- sequence ; the derivation is easily generalized to allow for oblique incidence. In the presentation of Fig. 7, the space between the ordinate and the diagonal is devoted to the shelf waves (or "continuum" or "leaky" modes). At one extreme is the normal incidence, corresponding to infinite wavelength (and hence zero wave number n) along shore. The resonances discussed in section 5-A of this Chapter correspond to the intersection of the lines marked "max" with the frequency axis. For incidences other than normal, the resonance frequencies increase somewhat, but not much. At the other extreme is the case of glancing incidence ; that is, in deep water, h~H, the waves travel in a direction parallel to shore, and the crests are normal to shore. Between normal and glancing incidence there is a critical case indicated by the dashed line : for incidence Fig. 7. Schematics of the /, /i-diagram. Here / is the frequency, and n the wave-number component along shore. more nearly glancing, the wave amplitude at the shoreline is less than in deep water, whereas for incidents more nearly normal there is shoreward amplifica- tion. This part of the/, ?i-diagram has been labelled "continuum", for every point on this diagram is a possible solution to the wave equation, and corresponds to some combination of frequency and angle of incidence. The other part of the diagram corresponds to the discrete spectrum (or normal modes, or eigenvalues, or edge waves, or "trapped" modes). The only possible combinations of fre- quency and wave number are along a series of distinct curves, corresponding to the fundamental and various harmonics. The discrete spectrum is characterized by the fact that, in deep water, the wave amplitude diminishes with increasing distance from shore, and essentially vanishes at a distance of a wavelength, SECT. 5] LONG OCEAN WAVES 657 277/n, from the outer edge of the continental slope. The jth harmonic is charac- terized by J nodal lines, that isji values of .r in the interval x^O to x = A + B for which the wave amplitude vanishes. The first theoretical discussion of edge waves is due to Stokes (Lamb, 1932, p. 446) who derived the solution for the fundamental mode. Ursell (1952) extended the solution to harmonics and demonstrated their existence in the laboratory. For periods approaching 12 pendulum hours the effect of the Coriolis force is important (Reid, 1958; Kajiura, 1958), and the solution merges with that of a Kelvin edge wave which depends essentially on the earth's rotation (Crease, 1955). From an inspection of the power spectrum alone it is impossible to decide to what extent the spectral density is associated with the continuous or discrete part of the spectrum. Cross-spectra between records at variable long-shore separation could resolve the problem in principle. Recordings at La Jolla and Oceanside, 38 km apart along the coast of southern California, are in phase and coherent for frequencies belpw 1 c/ks (Munk, Snodgrass and Tucker, 1959, chart 10.2), whereas for the case of edge waves some measurable phase dif- ferences are to be expected. This favors shelf waves. On the other hand there have been isolated instances resembling edge-wave activity. Resurgences following hurricanes traveling northward along the American east coast (Redfield and Miller, 1957) give the appearance of an "edge-wave wake" (Munk, Snodgrass and Carrier, 1956; Greenspan, 1956) but this interpretation has not been universally accepted. Edge waves along the Great Lakes have been described by Ewing, Press and Donn (1954), Donn and Ewing (1956) and Donn (1959). Cross-spectra between a coastal station and an island station 100 km offshore indicate comparable energies in the continuous and discrete parts of the spectrum (Snodgrass et al., 1961). Fig. 7 is, of course, highly schematic, but it serves to illustrate what we believe to be the principal classification of the subject of long waves, namely the separation into trapped and leaky modes. The chief weakness of the presentation lies in the assumption of straight parallel contours, i.e. h = h{x) extending to infinity along the ?/-axis. If the shelf were bounded by protruding headlands at y— —\D and y— -\-\D, then only discrete values of n are permissible, namely n = (2Z))-i, 2(2i))-i, 3(2i))-i, . . . The trapped modes now consist of a series of points. For the actual continental shelf, the roughness of the coastline must be associated with scattering of trapped energy into the leaky modes, and vice versa, and the distinction loses sharpness. We concluded that the relative contribution of the discrete and continuous parts of the spectrum to the observed long-period activity is in doubt ; the existence of sharp spectral peaks probably favors the continuum to be the predominant contributor. E. Discussion Some remarks concerning the causes of the shelf waves need to be made. The spectrum ashore may be radically shaped by the shelf resonances. Still, that does not explain what caused the oscillations in the first place. 22— s. I. 658 MUNK [chap. 18 It may occur to the reader that the long-period oscillations in sea-level may- be the surface manifestation of internal wave activity (see Cox and La Fond, Chapter 22). The two phenomena cover about the same frequency range. Furthermore the computed surface amplitudes associated with observed internal waves are not negligible as compared with measured surface amplitudes. Nevertheless any attempt to correlate simultaneous measurements has met with failure (Munk, Snodgrass and Tucker, 1959, p. 318). Internal waves may still be an important extended source of long-wave activity, but the two phenom- ena are not convincingly correlated at any time and place. In some instances the sea-level responds like an inverted barometer to fluctuations in atmospheric pressure (Donn, 1958; Van Dorn, 1960a; Gossard and Munk, 1955). The pressure fluctuations may be the surface manifestation of internal waves in the atmosphere ; or again they may be associated with line squalls. Ewing and Press (1953) have attributed the world-wide tide-gauge disturbance accompanying the eruption of Krakatoa to the atmospheric pressure wave associated with this great event. But here again, as in the case of internal waves, one finds that in the usual situation there is no obvious co- herence between simultaneous records of sea-level and atmospheric pressure. Again this does not exclude the possibility of atmospheric pressure as an extended source in the sense that the observed waves are the cumulative result of random pressure impulses over the entire ocean surface. Inasmuch as the propagation velocities of long waves in the atmosphere and ocean are not well matched (1 100 km h-i versus 600 km h-i), this would represent an ofi'-resonance flux of energy from one medium to the other. F. Waves in Ports The occurrence of undesirable oscillation in ports is usually referred to as "seiching". There can be no doubt that the resonance characteristics of the port itself are the chief factor in determining these oscillations. In most of the literature it is assumed that it is the direct action of wind and pressure on the water surface inside the port that is responsible for exciting the resonances. It seems more likely that excitation through the mouth is the predominant factor. This certainly must be true in the case of harbour seiches accompanying severe tsunamis. A review of this subject has been given by Wilson (1957). From a practical point of view it would seem desirable to determine the spectra outside the harbour along the lines of Fig. 4, and to choose depth and lateral dimensions of the harbour so as to "detune" it from the shelf resonances. For regions with sharp outside spectral peaks this procedure might reduce the amplitude inside the harbour by an order of magnitude ; for regions with broad spectral bands, the advantage to be gained is minor (Miles and Munk, 1961). 6. Tsunamis Because of the great damage caused by severe tsunamis, there have been many general accounts. We shall mention only a few highlights. One of the best SECT. 5] LONG OCEAN WAVES 659 documented accounts is that of the tsunami of 1 April, 1946, associated with an earthquake in the Aleutian trough (Shepard, MacDonald and Cox, 1950). An interesting discussion of waves caused by the eruption of a submarine volcano is due to Unoki and Nakano (1953). We have already referred to the findings of Ewing and Press (1953) that some of the distant tide-gauge disturbances follow- ing the eruption of Krakatoa were induced by atmospheric waves. This historic eruption was the subject of a fascinating report by the Royal Society (Symons, 1888). A rock-slide associated with the Alaska earthquake of 10 July, 1958, resulted in a long wave whose crest rose up to 1700 feet (!) above mean water level (Miller, 1960). Nakano (1955) has given an account of tsunamis from atomic explosions. The study of tsunami waves differs from that of the background in one Fig. 8. Tide record of 5 November, 1952, at Pago Pago, Samoa. important aspect : there is a definite (impulsive) beginning to the disturbance. Fig. 8 shows the initial arrival of the Kamchatka tsunami at Pago Pago, Samoa (Zerbe, 1953). The relatively small size of the initial crest is typical of many tsunamis. In this particular case the arrival time can be measured to the nearest few minutes ; in many instances the initial onset is not nearly as clear cut. Arrival times have been computed assuming propagation at a speed of ■\/{gh) along the great circle route between epicenter and station (see, for example, Zetler, 1947). In an ocean basin of variable depth the wave fronts propagate according to Fermat's principle, and the rays are not precisely great circle routes. Nevertheless the great circle approximation ought to be a good one, and in fact computed and observed travel times are generally in accord to within a few per cent. 660 MUNK [chap. 18 If the oceans were of constant depth and of infinite extent, the disturbance following the initial arrival would be of very short duration. This is because the waves are almost non-dispersive. For any (angular) frequency to the group velocity is given by V = {gh)y^[l-h{co%^l9)+...] and for waves of 5-min period at a depth of 4 km this equals roughly V — 0.9\/{gh). Thus the group velocity is diminished by 10% relative to waves of infinite length, and the travel time is increased accordingly. If the initial arrival took 10 h, waves of 5-min period should arrive 1 h later. Relatively little energy goes into waves of periods shorter than 5 min, and the entire disturbances should be a matter of hours. The foregoing estimates are confirmed by a more rigorous examination of the appropriate initial value problem (Prins, 1956). In fact, the disturbances following a major tsunami are noticeable for a week. The amplitude decays to 1/e of its value once every twelve hours (Munk, 1961). What is the cause of the prolonged oscillation? Trapping of tsunami energy into slow-edge waves along the continental shelves is a possibility. The scatter of tsunamis by sea mounts and other bottom irregularities would lead to reverberations. Probably a more important factor is multiple reflections at the continental boundaries, and in some instances it has been possible to identify second arrivals with definite refiected paths (Cochrane and Arthur, 1948; Shimozuru and Akima, 1952). In all events the subsequent stages of tsunamis resemble in their complexity the day-to-day background oscillation (except for being higher), and the power-spectral analysis is enlightening even though we are not dealing with stationary time series. Fig. 9 shows comparative spectra of a tsunami and of background activity at Acapulco. The tsunami spectrum is two to three orders of magnitude above background. The outstanding feature is the principal peak at 0.57 c/ks in both spectra. A smaller peak at 1.5 c/ks is also reproduced whereas a third peak at 1.15 c/ks is found only in the tsunami spectrum. In general it is found that the spectra of different tsunamis at any one station look alike, whereas one tsunami at different stations has no reproducible spectral features. The inevitable conclusion is that tsunami records are governed principally by the bottom topography near the recording station, and not by the character of the source. In the case of stations in harbours, the deep-sea spectrum is viewed through the additional "harbour filter", and the distortion is even more extreme. Some efforts to remove the effect of harbour resonances have been made by Rikitake (1949). Perhaps the use of analogue models (Ishiguro, 1959) can lead to realistic appraisals of the response function of harbours and bays. An alter- native (and perhaps preferable) scheme is to install stations on small steep islands with resonances (if any) well above the frequencies characteristic of the tsunamis. The latter point of view has been pioneered by Van Dorn (Van Dorn and Donn, 1960). He has demonstrated that, for waves the length of which equals the circumference of a circular island, the wave height will be 50% higher SECT. 5] LONG OCEAN WAVES 661 or lower on the incident and lee shore, respectively, as compared to the un- disturbed wave height. For smaller islands the distortion is even less. Largely as a result of Van Dorn's and Donn's efforts, two dozen long-period wave records were in operation during the International Geophysical Year, most of these on islands. The station at Wake Island obtained a striking record of the tsunami of 9 March, 1957. During the first two hours the frequency increased wdtli time in accordance with the expected role of dispersion. The maximum wave height was 1 ft. In contrast, at Hawaii (with roughly the same epicentral distance) the height reached 15 ft. The spectral similarity between the tsunami of 9 March, 1957, from an earthquake, with tsunamis from atomic tests pre- viously recorded at Wake Island permitted the authors to estimate the total 0.5 1.0 Cycles per kilosecond Fig. 9. Spectra of background activity and of the tsunami of 28 July, 1957, at Acapulco. energy of the wave train: 10^2 ergs. The seismic energy had been estimated at 1023 ergs. Following the devastating tsunami of 1 April, 1946, a tsunami warning system has been organized by the U.S. Coast Geodetic Survey. The basic difficulty is that it has not been possible from the seismic record of an earth- quake to determine whether or not a tsunami has been generated, and recourse has to be made to the inspection of tide records near the epicenter. The observed fact is that two earthquakes which are seismically undistinguishable with regard to location and energy can generate tsunamis whose amplitudes differ by an order of magnitude! Ewing, Tolstoy and Press (1950) have noted a correlation between the T phase on seismograms and the occurrence of tsunamis, and have suggested that this might be useful in removing some of the 662 MUNK [chap. 18 uncertainty. The T phase had previously been identified, with the propagation of a compressional wave in the deep sound channel of the oceans (the SOFAR layer). The difficulty in the warning system may be a reflection of our lack of knowledge of whether tsunamis accompanying earthquakes are the result of vertical displacements along submarine faults, or of submarine slumping induced by earthquakes. Gutenberg (1939) has argued for slumping, largely as a result of the tsunami from the Atacama earthquake of 11 November, 1922, for which the fault movement occurred inland. A similar instance occurred off Suva, Fiji, in 1953, and the evidence presented by Houtz (1960) is quite con- vincing. On the other hand, slumping in the Scripps Canyon, La Jolla, California, produced no noticeable disturbance in the output of a tsunami recorder only a few thousand feet distant. The relative role of slumping and faulting needs to be studied further, but it seems likely that both processes are capable of generating tsunamis. References Cochrane, J. D. and R. S. Arthur, 1948. Reflection of Tsunamis. J. Mar. Res., 8, 239-251. Crease, J., 1955. Long waves on a rotating earth in the presence of a semi-infinite barrier. J. Fluid Mech., 1, 86-96. Donn, W. L., 1958. The microbarovariograph. Trans. Amer. Geophys. Un., 39, 366-368. Dorm, W. L., 1959. The great lakes storm surge of May 5, 1952. J. Geophys. Res., 64, 191-198. Donn, W. L. and M. Ewing, 1956. Stokes' edge waves in Lalie Michigan. Science, 124, 1238-1242. Ewing, M. and F. Press, 1953. Tide gauge disturbances from the Great Eruption of Krakatoa. Tech. Rep. Seism. No. 27, CU35-53-AF 19(122)441 Geol., 1-14. Ewing, M., F. Press and W. L. Donn, 1954. An explanation of the Lake Michigan wave of 26 June 1954. Science, 120, 684-686. Ewing, M., I. Tolstoy and F. Press, 1950. Proposed use of the T phase in Tsunami warning systems. Bull. Seism. Soc. Amer., 50, 253-266. ossard, E. and W. H. Munk, 1955. Gravity waves in the atmosphere. Q.J. Roy. Met. Soc.,G 81, 484-487. Greenspan, H. P., 1956. The generation of edge waves by moving pressure distributions. J. Fluid Mech., 1, 574-592. Gutenberg, B., 1939. Tsunamis and earthquakes. Bull. Seism. Soc. Amer., 29, 517-526. Houtz, R. W., 1960. The 1953 Suva earthquake and Tsunami. Geol. Surv. Dept. Fiji Rep. No. 61. Inman, D., W. H. Munk and M. Balay, 1962. Spectra of low-frequency ocean waves along the Argentine shelf. Deep-Sea Res., 8, 155-164. Ishiguro, S., 1959. A method of analysis for long-wave phenomena in the ocean using electronics network models. I. The earth's rotation ignored. Phil. Trans. Roy. Soc. London, A251, 303-340. Kajiura, K., 1958. Effect of Coriolis force on edge waves. (II) Specific examples of free and forced waves. J. Mar. Res., 16, 145-157. Lamb, H., 1932. Hydrodynamics. Cambridge Univ. Press. Miles, J. W. and W. H. Munk, 1961. Harbor paradox. J. Waterways and Harbors Division, Proc. Am. Soc. Civ. Eng., 87, 111-130. SECT. 5] LONG OCEAN WAVES 663 Miller, D. J., 1960. The Alaska earthquake of July 10, 1958: giant wave in Lituya Bay. Bull. Seism. Soc. Amer., 50, 253-266. Munk, W. H., 1949. Surf beats. Trans. Amer. Geophys. Un., 30, 849-854. Munk, W. H., 1961. Some Comments regarding diffusion and absorption of Tsunamis. Bull I.U.G.G., in press. Munk, W. H. andHerminio Cepeda, 1961. Sea level spectra, 0.3 to 1.5 cycles per hour, at Acapulco and Salina Cruz. An. Inst. Geofis., Univ. Nac. Auto. Mexico (in press). Munk, W. H., H. W. Iglesias and T. R. Folsom, 1948. An instrument for recording ultra- low-frequency ocean waves. Rev. Set. lustrum,., 19, 654-658. Munk, W. H. and F. E. Snodgrass, 1957. Measurements of southern swell at Guadalupe Island. Deep-Sea Res., 4, 272-286. Munk, W. H., F. E. Snodgrass and G. Carrier, 1956. Edge waves on the continental shelf. Science, 123, 127-1S2. Munk, W. H., F. E. Snodgrass and M. J. Tucker, 1959. Spectra of low-frequency ocean wave. Bull. Scripps. Inst. Oceanog. Univ. Calif., 7, 283-362. Nakano, M., 1955. Atomic energy and the oscillation of sea level. Res. in the effects and influences of the nuclear bomb test explosions, 401-407. Prins, J. E., 1956. Characteristics of waves generated by a local surface disturbance. Inst, of E7ig. Res., Wave Res. Lab., Univ. Calif., Ser. 99, Issue 1. Redfield, A. C. and A. R. Miller, 1957. Water levels accompanying Atlantic Coast hurri- canes. Met. Monog., 2, 1-23. Reid, R. O., 1958. Effect of Coriolis force on edge waves. J. Mar. Res., 16, 109-144. Rikitake, T., 1949. The form of Tsunami-waves outside a bay as inferred from the motion of bay-water. Earthquake Res. Inst., 29, 272-282. Shepard, F. P., G. A. MacDonald and D. C. Cox, 1950. The Tsunami of April 1, 1946. Bull. Scripps Inst. Oceanog. Univ. Calif., 5, 391-528. Shimozuru, E. and T. Akima, 1952. Reflections on the Tsunami of December 21, 1946. Bull. Earthquake Res. Inst., 30, 223-230. Snodgrass, F. E., 1958. Shore-based recorder of low-frequency ocean waves. Trans. Amer. Geophys. Un., 39, 109-113. Snodgrass, F. E., W. H. Munk and M. J. Tucker, 1958. Off-shore recording of low- frequency ocean waves. Trans. Am,er. Geophys. Un., 39, 114-120. Snodgrass, F. E., W. H. Mtmk and G. R. Miller, 1961. Long-period waves over California's continental borderland. J. Mar. Res., 19, in press. Sjrmons, G. J., 1888. The Eruption of Krakatoa and Subsequent Phenomena. Trubner and Company, London. Tick, L. J., 1958. A non-linear random model of gravity waves. Res. Div.-Col. Eng., N.Y. Univ. Tucker, M. J., 1950. Surf beats: sea waves of 1 to 5 min. period. Proc. Roy. Soc. London, A202, 565-573. Unoki, S. and M. Nakano, 1953. On the Cauchy-Poisson waves caused by the eruption of a submarine volcano. Papers Met. Geophys., 4, 139-150. Ursell, F., 1952. Edge waves on a sloping beach. Proc. Roy. Soc. London, A214, 79-97. Van Dom, W. G., 1956. A portable Tsunami recorder. Trans. Amer. Geophys. Un., 37, 27-30. Van Dom, W. G., 1960. A new long-period wave recorder. J. Geophys. Res., 65, 1007-1012. Van Dom, W. G., 1960a. A low-frequency recording microbarograph. J. Geophys. Res., 65, 3693-3698. Van Dom, W. G. and W. L. Donn, 1960. Long waves. Ann. I.G.Y., 10, Chap. V. Wilson, B. W., 1957. Origin and effects of long period waves in ports. XIX Intern. Navig. Cong., Sect. II, Comm. 1, 1-49. Zerbe,' W. B., 1953. The Tsunami of November 4, 1952, as recorded at tide stations. U.S. Dept. Com., Coast Geodet. Surv. Spec. Publ. No. 300, 1-63. Zetler, B. D., 1947. Travel times of seismic sea waves to Honolulu. Pacific Sci., 1, 185-188. 19. WIND WAVES N. F. Barber and M. J. Tucker 1. Kinematics of Waves Sometimes at sea one finds the whole sea surface moving in long parallel undulations at intervals of perhaps 200 metres. This simplicity is in striking contrast to the confusion present in the storm regions where waves are generated. The organized motion of a simple regular train of waves in deep water is pictured in Fig. 1, which represents a vertical section through the water in a DIRECTION OF TRAVEL Fig. 1. In a progressive wave in deep water the water particles move on circular orbits (very nearly) and rise in succession to become the crest of the advancing wave. The surface particles crowd together near the crests and are pulled apart in the troughs. plane perpendicular to the long crest-lines of the waves. All the water particles move on circular orbits in this vertical plane. All particles take the same period of time to complete one cycle of their motion but they do not all reach the top of their orbits at the same instant ; there is a progressive delay so that water particles on the surface rise to the top of their orbits in succession and momen- tarily form the crest of one of the advancing waves. On the crests the water is moving forward in the direction of wave advance. Subsequently it moves downward as the wave passes, moves backward while near the bottom of its orbit in the wave trough and then rises and begins to move forward as the next wave overtakes it. Below the surface, the motion of the particles grows less with increasing depth, and at a depth D it is only a fraction, exp { — 'IttDJL), of that at the surface, L being the distance between crests, i.e. a "wavelength". The factor is very closely | for a submersion of one-ninth of a wavelength, \ for two-ninths of a wavelength and so on. The profile of the wave shown in Fig. 1 has crests somewhat narrower and troughs somewhat wider than a true sinusoid. The departure from sinusoidal form is more marked with steeper waves, that is waves whose height is a greater [MS received June, 1960] 664 SECT. 5] WIND WAVES 665 fraction of their length. It appears from hydrodynamical theory that the height of waves cannot exceed about one-seventh of their length (Michell, 1893; Wilton, 1913). At this limiting stage the crest of the wave develops a sharp ridge with water sloping at 30'^ to the horizontal on either side (Lamb, 1945, Art. 250). Further energy added to the wave would cause water to spill out of the sharp crest and the "white caps" seen under a strong wind probably develop in this way. In practice it is unusual to meet waves the height of which exceeds one-tenth of their wavelength (Sverdrup and Munk, 1947). Precise discussions of waves in terms of a velocity potential can be found elsewhere (Lamb, 1945, chap. IX; Stoker, 1957). Here it is sufficient to point out an elementary argument which may assist the reader in recovering the formulae relating the period wavelength and velocity of deep-water waves. Everywhere on the undulating surface the water experiences horizontal and vertical accelerations and the water surface arranges itself always at right angles to the "false vertical", the combination of these accelerations and the force of gravity. Now if it is true that each surface particle moves on a circular orbit of diameter H (the w ave height crest to trough) in a periodic time T, the acceleration towards the centre is "Itt'^HJT'^. When the particle is halfway up its orbit (as point A in Fig. 1) this acceleration is purely horizontal and it combines with gravity to give a false vertical inclined to the true vertical by an angle whose tangent is But if the wave is sufficiently low to have a profile that is nearly sinusoidal with height H and wavelength L, the water surface at a point half-way between crest and trough is inclined to the horizontal by an angle whose tangent is ttHJL. Equating these two expressions gives the wave equation, L = gT^l2n. (1) Since the velocity of advance C is necessarily equal to LjT, the equation can also be written in the manner L = 2TTC^Ig (2) G = gTl^TT. (3) Numerical values for some typical ocean waves are listed in Table I. Table I Typical Sea Waves Tj'pe Period, Wavelength, Velocity, Group velocity, sec m m/sec m/sec Ground swell 15 350 23.4 11.7 SweU 10 156 15.6 7.8 Ocean waves 7 76 10.9 5.5 In anchorages 3 14 4.7 2.3 666 BARBER AND TUCKER [chap. 19 Theory suggests some slight increase in velocity and consequent increase in length for waves of a given period if the waves are very steep (Lamb, 1945, Art. 250). The velocity is expected to increase by a factor {1 +Tr^H^I2L^); ^ = height, iy = wavelength. Formulae (2) and (3) are good enough, however, for most purposes. In some cases one is concerned not with the speed of advance of the wave crests but with the speed of advance of the larger region of wave-disturbance, as, for instance, when one attempts to predict the arrival of large "swell" from a distant storm. The two speeds are not the same. This can be seen from the rather artificial example given in Fig. 2. If two wave trains of slightly different wavelengths are present at the same time, there are some regions in which the crests of one coincide with the troughs of the other so that the combined waves are small, and other regions where the waves agree in position and the combined waves are large. Exact agreement may occur at two crests such as those TWO WAVE TRAINS B 6L M TRAVEL COMBINED WAVES |*_L -M Fig. 2. Illustrating the "group" behaviour. Wave trains having a slightly different wave- length add to give a system of "groups", or regions where the waves are high. In- dividual waves run forward through these groups, becoming high as they enter a group and losing height again as they leave. The groups themselves advance at a slower speed. labelled A. Immediately in the rear of these, the wave crests are out of step by a small distance BL, the difference in wavelength, but the longer wave is over- taking the shorter one because of the difference in velocities of the wave trains, SO. After a time interval hLjhC these two crests will coincide. During this interval the wave crests will, of course, have advanced a distance C hLfhC but relative to the waves ; the point of exact agreement between the two trains, that is the region of greatest combined wave activity, will have lost ground by one whole wavelength, L. So the group advances at a speed smaller than that of the wave crests themselves, in fact at a speed G, where G = C-LdC/dL. Equation (2) shows the connexion between C and L for the kind of waves being discussed : gravity waves in deep water. On differentiating this formula the group velocity given above is found to be just half the velocity of the waves ; G = hC = gTJ^rr. (4) SECT. 5] WIND WAVES 667 When one is able to create artificially a succession of, say, four or five waves on otherwise calm water, it is easy to see how new wave crests continually arise in the rear of the group, travel forward through it and then die away as they move out of it in front. The group itself, that is the region of water where the waves are high, also advances, but at a lower speed. The wave group is the region possessing the wave energy and, unless this energy is dissipated in some way, its progress can be followed over considerable distances and the wave group has a permanence that is not shared by individual wave crests. In- dividual waves at sea usually grow up, travel, and die away in much less than 30 sec, and if they attain a white cap it is only for a very few seconds. But the spreading of wave energy from a storm centre, that is the progress of wave groups, has been traced over great distances. A train of waves has considerable energy. This comes partly from the water motion (kinetic energy) and partly from the water having been lifted up from the troughs into the crests (potential energy). The average energy per unit area of sea surface is where p is the density of water, g the acceleration of gravity and h"^ the mean square elevation (or depression) from the undisturbed sea-level. If the waves form a sinusoidal wave train whose height (crest to trough) is H, the average energy per unit area is IpgH^ Wavelength, water depth and frequency do not enter into this formula. If a group of waves travels over the surface of otherwise undisturbed water, the energy in the w^ave motion is evidently being handed on from one region of water to the next. Even where there is no very clear division of waves into groups the energy may be thought of as advancing through the water at the group velocity. A moderate ocean swell for instance, perhaps 2 m high when in deep water, has an energy of 5 x 10^ ergs/cm^ of sea surface. If the period is 10 sec the group velocity in deep water is 7.8 m/sec and energy is transmitted at the rate of 3.9 x 10^ ergs sec~i cm^i of crestline. If the swell reaches a coast this energy is nearly all spent in turbulence in the surf zone. It approximates to 40 kilowatts per metre length of shoreline. 2. The Description of a Complicated Wave Pattern : the Wave Spectrum This subject is treated in detail in Chapter 15, but is so fundamental to the study of waves and to the understanding of this section, that it is felt worth- while to include a brief discussion of it here. A particular record of natural waves (for example, a record of the height of the water surface at a vertical pole) is complicated and very local in time and space. A record from the same pole at a different time or from another pole some distance away at the same time, will bear no resemblance to it in detail. In fact, the detail in such a record is random in character and unpredictable. 668 BARBER AND TUCKER [CHAP. 10 However, the waves on a stormy day are obviously different in some systematic way from those on a calm day, and some way must be found of describing this difference. It is, of course, the statistical properties of the wave pattern which are significant, and the most obvious ones to use are average wave height, period and direction of travel, defined in some suitable way. Most early methods of prediction aimed at predicting these properties, and for many engineering purposes a knowledge of these is sufficient. They do not contain all the sig- nificant information about the wave system, however, and more detailed information is sometimes required. It has been found in practice that the most generally useful way of describing the wave pattern is by its "power spectrum", and from this any of the other statistical properties may be derived. The concept of the wave power-spectrum is based on the assumption that the wave pattern may be regarded as a simple superposition of a large number of low, sinusoidal, long-crested wave trains such as is described in section 1 of this Chapter, each component having a different period and a different direction of travel. The interference of these components, sometimes reinforcing one another and sometimes cancelling one another, produces the complicated wave pattern observed. If 8E is the sum of the energies per unit area of the sea surface of all those component wave trains whose angular frequencies ( = 27r/ wave period) lie between ct and a+8a and whose directions of travel lie between 6 and 6 + hd, then one may choose to describe the spectrum by a spectral density function E'{g, 6) defined by : E'{a, d) = limit as Sct -> 0 and hO -> 0 of S^/(8ct hd). This is sometimes known as the two-dimensional spectrum of the sea surface since it can be pictured as a contour plot using a and 6 as co-ordinates. Some writers prefer to use wave number rather than wave frequency. There would seem to be some advantage in using polar co-ordinates rather than rectangular ones. In practice, S^" is taken as the mean square elevation of the sea surface ; strictly speaking, the energy is pg SE per unit area. In both theory and practice, E'{a, 6) is found to be continuous with frequency and direction ; that is, there are in effect an infinite number of infinitely low wave trains differing infinitesimally in frequency and direction. For many purposes the direction of travel of the component wave trains is either not important or is too difficult to measure. In these cases the "one- dimensional" spectrum is used. This is defined as E'{a) = f " E'{cy, 6) de. (5) This is the frequency spectrum of the output of a recorder which records the height of the water surface above a fixed point on the sea-bed. Up to the present time, practical methods of wave prediction give this spectrum, since knowledge of the directional properties of sea- waves is scanty (see section 10 of this Chapter). SECT. 5] WIND WAVES 669 From a knowledge of E'{a) the "significant wave height" and "significant wave period" used by engineers can be derived. For studies of ship motion, the two-dimensional spectrum E'{a, 6) is required. 3. Theories of Wave Generation by Wind Several mechanisms have been suggested by which the wind can generate waves on water, but at the present time work is concentrated on two of these which between them can probably account adequately for the generation of waves at sea, and these will be discussed here. The reader is referred to a review of the subject by Ursell (1956) for an account of the other possibilities. The two mechanisms are : (a) The deflexion of the wind as it blows over the wave-profile causing dynamic pressure differences which can feed energy into the waves. (b) The turbulence in the wind causing a moving pattern of pressure fluctua- tions which can generate waves without reaction of the weaves on the wind. If a simple long-crested wave train is travelling over a water surface with no wind blowing, the pressure difference in the air between crest and trough can be shown theoretically to be twice the static pressure difference. The simplest way of looking at this is as follows. In the water, the dynamic pressure changes just cancel the static ones, to give constant pressure at the surface (neglecting the pressure differences in the air, which are small compared to those in the water). The motion of the air particles follows a similar pattern to that of the water j^articles, but "upside-down", so that the dynamic pressure differences add to the static pressure differences. When the wind blows over the waves, a further set of dynamic pressure differences is introduced. If the flow^ were laminar and friction-free, it is evident that all these pressure differences would be in phase with the wave profile and therefore feed no energy into the wave system : every place where a pressure is acting on a down- ward moving surface is balanced by a corresponding area where the same pressure acts on a surface moving upward. However, in the presence of viscosity or turbulence, out-of-phase pressure differences are introduced which can put energy into the waves. From the observed rate of growth of waves under a wind, it is possible to estimate that the magnitude of the out-of-phase pressures should be approximately -g^ of the amplitude of the in-phase pressures. This means that the amplitude of the resultant pressures is very nearly that of the in-phase component, but that the phase of the pattern is changed by about 2° relative to the wave profile. The measurement of the amount of energy fed into the waves therefore presents a severe instrumentation problem which has not so far been solved, though such a measurement has been attempted (Longuet- Higgins, Cartwright and Smith, in press). Jeffreys (1925) was the first to attempt to calculate this energy transfer. He assumed an eddy on the lee side of the waves, so that by the theory of turbulent flows, pressure differences proportional to {V-C)^ will be set up between G70 BARBER AND TUCKER [CHAP. 19 corresponding areas on the windward and lee slopes of the waves, where V is the wind velocity and C is the phase velocity of the waves. He was not able to calculate the absolute value of the pressure difference, but had to introduce an arbitrary constant, s, which he called the "sheltering coefficient". Equating the energy fed into the waves to the energy lost by dissipation due to viscosity in the water, he was able to calculate that there would be a net gain of energy by the waves when C(F_C)2 ^ 4vg{p-p')lsp'. (6) Here, v is the kinematic viscosity, p and p' are the densities of water and air respectively. This formula allows calculation of the least wind speed which can generate waves, and comparing this with observation, Jeffreys deduced that s = 0.27, a value which is reasonable physically. Further deductions do not, however, agree with observations. This type of theory has been considerably refined and elaborated, particularly by Russian workers (Shuleikin, 1956; Krilov, 1958). In some cases the dif- ference in tangential stress (wind drag on the surface) between the crest and trough of a wave is also taken into account. However, most theories of this kind contain unknown constants equivalent to the sheltering coefficient s, which have to be determined by comparison with actual wave measurements. This is unsatisfactory, since it is impossible to calculate from first principles whether the processes postulated can feed an adequate amount of energy into the waves. Miles (1957, 1959) has overcome this difficulty. By considering the air flow over the waves, he is able to calculate the amplitude and phase of the air- pressure fluctuations in terms of the characteristics of the wind-profile, with no unknown constants. He can thus calculate the energy transfer from the wind to the waves, and can test his theory by comparing this with observations of the rate of growth of waves. He finds order-of-magnitude agreement. He is also able to work in terms of a real sea consisting of a complete spectrum of wave- lengths. The second type of mechanism has recently been studied by Phillips (1957, 1958). He resolves the moving pattern of pressure fluctuations in a turbulent wind into "long-crested" sinusoidal components, and some of these components have a relationship between wavelength and velocity which is the same as that of a free wave on the water surface. These components will build up waves by resonance. Thus, if the spectrum of the turbulent pressure fluctuations in the wind is known, he can predict the wave spectrum. His results explain qualita- tively many observed properties of sea waves (see also, for example, Phillips, 1958a), but there is at present some doubt as to whether the theory can give correct quantitative results. The difficulty here is mainly the lack of adequate knowledge of the turbulent pressure fluctuations in wind over the sea (see, for example, comments on Phillips's 1958 paper in Chapter 21). An outstanding feature of Phillips's theory is that it yields a theoretical spectrum for the wave pattern in both frequency and direction of travel. It seems likely that a complete theory of the generation of waves by wind SECT. 5] WIND WAVES 671 will have to take account of both these mechanisms. Possibly Phillips's theory" can account for the initial generation of waves, whereas a theory such as that of Miles can account for the subsequent growth. i Even if a satisfactory theory for the energy transfer from the wind to waves can be produced, it will be necessary to know much more about how waves dissipate energy before satisfactory predictions of real wave patterns can be made. In Jeffrey's energy balance quoted above (6), for example, the kinematic viscosity appears, which in practice will represent a turbulent viscosity and will almost certainly be dependent on wavelength and wind speed (Groen, 1954). In any case, it seems unlikely that a simple conception of turbulent viscosity will suffice, since steep waves probably lose most of their energy in "white horses", the foaming crests seen on the sea during strong winds. 4. Wave Prediction The fundamental theory of the generation of waves by wind has not yet reached the stage where it can usefully be applied to practical wave prediction, so that this at present relies on formulae which are to a large extent empirical, though some theory may also be involved. The first empirical formula was a very simple one published by Thomas Stevenson (the father of Robert Louis Stevenson) in 1864, and another, based on visual observations, was given by Vaughan Cornish in 1934. These formulae were not satisfactory, however, and the military necessity of the 1939-45 war started considerable research aimed at improving methods of wave prediction. The growing military and civilian importance of this subject has ensured continued effort. The systems for wave prediction developed during the war are due to Suthons (1945) and Sverdrup and Munk (1947). Both methods are still in limited use, that due to Sverdrup and Munk having been modified by Bretschneider (1959). At the time these were developed, frequency analysis had not been applied to sea waves, and it was necessary to use some suitably defined mean wave- height H and period T, and to find relationships between these and the wind speed V, duration (the time for which the wind has been blowing), fetch (the distance over which the wind is blowing), etc. Systems of this type have also been developed in Russia (for example, Krilov, 1958; Shuleikin, 1959). With the development of frequency analysis of sea waves, and the realization of the theoretical and practical importance of frequency spectra, these methods have been to a large extent superseded in the Western World by methods which predict the wave spectrum, from which the other parameters may be derived. All methods begin by considering waves on deep water, and in this case Sverdrup and Munk were able to argue that one might expect a relationship 1 Since this account was written, more theoretical and experimental work has been done on this theory, though the basic principles remain unchanged. Readers are referred to Cartwright (1961) for a summary of this work. In this paper he also summarizes other recent work on the wave spectrum. 672 BARBER AND TUCKER [chap. 19 between the wave steepness HfL {L being the wavelength) and the ratio of wave phase-velocity C to the wind speed V (which they called the "wave age"). Observations supported this, and gave an empirical curve for the relationship which formed the basis for their method of prediction. Neumann (1953), developing a method for predicting the frequency spectrum of the waves, had insufficient measured spectra. He therefore worked with the apparent period T between two successive crests passing a fixed point, and the corresponding apparent height H defined as the mean difference in elevation between these crests and the trough in between. He argued that such a wave is due to the fortuitous superposition of all the spectral components with periods near T, and that the largest value of H observed for a given value of T is, therefore, a measure of the spectral density near T. Following Sverdrup 0.2 i0.05 Observations Long Branch Wave Records May 3, 1948 May 5, 1948 [October 6, 1948 I October 7, 1948 «\ xj.,^*, *\5-x\^ y i^BED PRESSURE 0 6 0.4. G Go GROUP ^ VELOCITY y/ y ^ / 0 3 J y / "fvELOCITY, C Co v 0 2. y ^ y^ j LENGTH. L \ 0)5 y HEIGHT RANGE ^ / /^ \ 0.10 _/ \ 0 08. / y \ 006 .^DEPTH \ 0 04. VICTUAL WAVELENGT H \ 0 03- y^ \ 0.02. y ^ / /^ 0,01. y 0 002 0004 02 04 0.8 0.01 " ' " 0.1 RATIO WATER DEPTH TO DEEP WATER WAVELENGTH, hJLo Fig. 8. If waves travel from deep water to shallow water their frequency is unaltered but other characters change. The curves show how, starting from deep water on the right of the graph, the wave velocity, wavelength and group velocity all grow less as the wave enters shallower water towards the left of the graph. The accompanying increases in wave height and wave steepness are calculated assuming the waves to be incident normally on a straight coast (no refraction) and assuming that the waves remain low enough to have a sinusoidal profile. The orbits of particles on the surface become more and more elliptical, as indicated by the decreasing ratio of wave height to "range" or horizontal excursion. The fluctuating water pressure on the sea-bed is negligible in deep water but in shallow water tends to equal the water height directly above. Some curious refraction effects have been observed on the very exposed Californian coast near La Jolla (Munk and Traylor, 1947). Two branches of a submarine canyon here approach the coast quite closely and, because swell can travel along them more quickly, the water being deeper, the wave energy is refracted towards the shallower waters ; consequently the height of surf can vary greatly along this coast. Fig. 7 illustrates these results. SECT. 5] WIND WAVES 081 The changing characters of the waves as they enter shoaling water are summarized in Fig. 8. After an initial small decrease in height the waves begin to grow higher, their speed decreases and the orbits of the water particles gradually lengthen into ellipses so that the forward and backward motion is greater than the wave height. These changes are dependent on the ratio of the depth of water to the wavelength which the waves had where the water was deep. It is worth remarking therefore that in a given depth of water, near the surf zone for example, swell of low frequency will have grown relatively more than the wind waves and will be more obvious there than in the open sea. The curves of Fig. 8 apply only to quite low waves passing over gently sloping beaches. They are given by an approximate theory which merely adjusts the wave heights everywhere to make the rate of forward transport of energy the same at all stages in the wave's f)rogress. More recent theoretical treatments have found exact expression for waves striking steep beaches or overhanging cliffs, subject again to the assumption that the actual wave height is always small. A digest of these treatments is given by Stoker (1957, chap. 5). 7. The Surf Zone As waves enter into gently shoaling water, their group velocity decreases and, to maintain the same forward flow of energy, the waves grow higher. They ultimately reach a limiting form in which the crests are ridge-like. Pro- gress into still shallower ground leads to turbulent water being spilled from the (a) (b) ^^^y^-^^^y..^^^ Fig. 9. A spilling breaker (a) and a plunging one (b). crest and thereafter the wave continuously dissipates energy in this way as it advances. This is one style of breaking. It is often followed by a second stage in which the whole wave crest falls forward (Fig. 9). On steeper beaches, the spilling stage may never develop ; the energy is then dissipated very quickly in a single forward plunge. The surf zone is a region in which waves tend to be steep and in which non- linear processes become very important. Consequently it is the region least amenable to theoretical treatment. If the beach is sufficiently steep there may 682 BAKBER AND TUCKER [CHAP. 19 be no breaking at all, the waves being reflected back towards the open sea. The style of breaking or reflection depends in some way, not fully understood, upon the steepness (ratio of height to wavelength) of the waves in deep water and upon the slope of the beach. Tides are long low waves and may be said to be reflected from all coasts unless the bore which can develop over extensive littoral flats is to be thought of as a breaking tide. Low swell on the other hand may need a beach slope of perhaps 1 : 5 to reflect it without breaking. Experi- mental or theoretical studies of breaking waves are discussed by Stoker (1957, page 351), but the topic is a difficult one. As a rough guide to plunging breakers, it may be said that they usually develop where the undisturbed depth is a little greater than the height of the breaker. Spilling breakers can start in much deeper water. Though it was not mentioned in section 1 of this Chapter, it is known (Lamb, 1945, Art. 250) that steep waves produce some overall forward transport of water. When waves steepen as they pass into shallow water and reach the surf zone, they drive increasing quantities of water forward towards the shore. This water may tend to return seaward at special places, determined perhaps by the contours of the covered beach. Such "rips" carry water back through the surf and are a danger to bathers, but the current usually disappears just sea- ward of the surf zone (Shepard, Emery and LaFond, 1941). Groups of high waves can bring in extra amounts of water and the fluctuating depth of the water near the beach is often quite evident. The seaward flow of this water, during intervals when the surf is low, can produce at intervals of two to five minutes low surges that are transmitted out to sea and can be observed in deeper water by sensitive instruments (Munk, 1949; Tucker, 1950). These surges have been called the "beat of the surf". Beach material is continually in motion where the water is shallow. Sand ripples develop everywhere on the submerged beach as a result of the backward and forward motion of the water, but tend to be erased as the beach becomes exposed (Bagnold, 1947). Where long swell penetrates into shaflow water, the waves induce a forward transport of water, most especially in a thin layer next to the sea-bed (Longuet-Higgins, 1953). Such waves, therefore, drive material shoreward and build up the beach. The reverse is the case with short steep waves built up by a local wind. Their forward transport takes place mainly in the surface and the compensating drift near the sea-bed tends to carry beach material out seaward and the beach is eroded. Waves that approach a shore obliquely tend to carry material along it and it is commonly found that sandy beaches arrange themselves so that their contours are at right angles to the mean direction of the incoming waves. On a coastline inclined to the prevailing waves or swell, beach material is continually in progress along the coast. Groynes do not permanently arrest this drift but they serve to build up the beach level until sand can pass the groynes. The danger in introducing any large barrier which would arrest the drift of sand is that it may starve beaches lying further along the coast and lead to erosion. The problem in designing many harbours is that sand must be allowed to cross SECT. 5J WIND WAVES 683 the harbour mouth and yet it must not be allowed to accumulate there and block the channel. For a full discussion of coastal engineering problems the reader is referred to the Proceedings of the Conferences on Coastal Engineering, 1950 to 1954. A recent review and bibliography of the topic is given by Silvester (1959). 8. Ships and Waves The study of ship motion in waves provides at the present time one of the major incentives to wave research. In this section, sufficient account of the subject will be given to explain why this is so and to show what type of in- formation about waves is required. For more detailed information, the reader is referred to the Proceedings of the Symposium on the Behaviour of Ships in a Seaway, held at Wageningen in 1957 (see note at beginning of references), to a monograph by Korvin-Kroukovsky (1958), or to a brief survey by Cartwright (1958). A ship encountering a storm at sea first loses speed due to increased re- sistance, then, as the storm gets worse, the motion gets dangerous and engine power has to be reduced. A small ship may have to "heave-to", making no progress at all. An idea of the importance of the loss of time involved can be gained from the fact that it has been found economic to alter the routes of ships to avoid storms, even though this may considerably increase the number of miles covered. The U.S. Navy Hydrographic Department have been routing ships in the Atlantic and Pacific in this way, and estimate a mean saving of 14 h in time and about S2,000 per passage. Apart from the loss of speed, storms cause considerable damage, and, of about 300 ships of one sort or another lost at sea each year, quite a high proportion must be lost by wave damage. It seems likely that with more knowledge of the factors governing ship motion, ships could be designed which are safer, more comfortable and capable of maintaining higher speeds in storms. Rolling can nowadays be greatly reduced by the use of stabilizers, but no successful stabilizer for pitch or heave has been designed, and these remain the most important motions to consider. Useful results can be obtained by considering the ship response to be linear, though non-linear effects are im- portant when the amplitude of motion becomes large, and have to be taken into account when more accuracy is required. In the linear case, the total motion in any mode (e.g. pitch) is assumed to be the linear superposition of the responses to the individual components in the wave spectrum, taking account of frequency and direction. (It is convenient to think of the wave spectrum as being composed of a large number of discrete components, though in fact the spectrum is continuous : see section 2 of this Chapter.) The response -etf a ship to a component wave depends on two factors. The ship will have a resonance at a particular frequency, typically about 1 cycle in 5 sec for the pitching of a medium-sized ship, though often with very high 684 BABBER AND TUCKER [CHAP. 19 damping. Secondly, the hydrodynamic exciting force depends on the wave- length of the wave, and for pitching in head seas is usually greatest when the wavelength is about 1.25 the length of the ship. For a stationary ship, the component wave whose frequency is that of the ship resonance usually has a wavelength considerably below that giving maximum hydrodynamic force, but when the ship steams into the waves, the "Doppler" increase in the fre- quency of encounter means that the resonant wave has a lower frequency relative to a fixed point, and hence a longer wavelength and greater exciting force. In a storm, it will probably also lie in a region of the spectrum where the energy is higher. Thus, a ship heading into a storm can experience dangerously large motion, which can be reduced by slowing down. The phase of the motion relative to the waves is also important, and governs whether the ship will "slam", for example. It is apparent that to calculate the total response of the ship, the wave spectrum is required in both frequency and direction. The wave information required for this field of research may be summarized as follows : (1) Measurement of the directional wave spectrum and of instantaneous wave height during experiments on the correlation of ship motion and per- formance with wave conditions. (2) Knowledge of wave spectra under various meteorological conditions, to allow them to be reproduced in model experiments, and to allow calculation of the behaviour of ships (when the theory has been adequately developed). (3) Knowledge of the statistics of wave conditions on the shipping routes of the world, to allow calculation of the optimum compromise between the various factors affecting the economic running of ships. 9. Methods of Observation and Analysis — Methods Taking No Account of Direction of Travel Many laboratories have attempted to design instruments which will measure and record the wave spectrum directly (for example. Barber, 1949 and 1954; Valembois, 1955), but none of these has been used to any extent in practice. This is because there is little incentive to overcome the considerable practical difficulties involved. In the early days of spectrum analysis it was thought that an immediate knowledge of the wave spectrum might be useful for storm detection in areas where there is little meteorological coverage. Detection of long-period swell shows the existence of a storm, and the rate at which its lower period-limit changes gives the distance from the recording station (see section 5 of this Chapter). However, the information becomes available rather late and measurement of the direction of the storm is difficult, so that this method has not been used in practice. At present, there is no application which requires an immediate knowledge of the wave spectrum. Thus, it has proved more con- venient to record the waves and to perform any required analyses later. For research purposes, the recent trend has been towards instruments SECT. 5] WIND WAVES 685 recording digitally on 5-hole teleprinter tape which can be fed directly to computing machines, though an analogue record is nearly always taken at the same time as a check. Several of these are in use, but have not been des- cribed in print. Digital recording allows very high resolution, and to take advantage of this there is a tendency towards f.m. measuring heads. These produce a frequency dependent on the wave variable being measured : the number of cycles in a fixed time is counted electronically, and this count recorded digitally. Such systems do not depend on cable characteristics, and a useful resolution of as high as 10~6 of full scale can be obtained. In the most advanced wave recording system of this type which has so far been described, the counted number is printed out and has later to be punched on cards by hand (Munk et ah, 1959). ^ Though designed for studying low-frequency ocean waves, it has produced some fascinating results concerning swell (Munk and Snodgrass, 1957). 2 Most practical methods for measuring waves at the present time measure the height of the water surface above a fixed point, or some related quantity. They are thus non-directional recorders and allow^ the calculation of the non- directional wave spectrum, that is, taking no account of direction of travel of the waves. Measurement of the directional spectrum is more difficult, but methods are being developed, and, in particular, that starting with measure- ments of the pitch, roll and heave of a small buoy can be regarded as established for practical use (see below). A . Wave Recording from a Shore Station The measurement required is the height of the surface of the water above a fixed point. Where a suitable structure, such as a pier, is available or can be built, there are several types of instrument which can be used. The most straightforward consists of a small float in a tube which has long vertical slots cut in it, so that it acts merely as a guide and the water level inside is always the same as that outside (Wemelsfelder, 1955). The movement is recorded mechanically. Electrical methods of measuring the surface elevation are numerous, but most fall into three classes: (1) the water closes a series of contacts on a vertical pole ; (2) the resistance between two wires varies with the water level up them ; (3) the capacitance to earth of an insulated wire, passing vertically through the water surface, varies with the water level. For examples of these, the reader is referred to surveys of wave recording instruments by Snodgrass (1951) and Draper (1961), and to the Proceedings of the First Conference on Coastal Engineering Instruments (see references). In many cases it is either not possible or not economic to build a structure suitable for such instruments, and the measuring head has to be laid on the sea- bed. In these circumstances, an inverted echo-sounder has sometimes been 1 It is understood that a tape-punch has since been fitted. 2 A detailed account of digital techniques is given by Cartwright, Tucker and Catton (in press). 686 BARBER AND TUCKER [chap. 19 used but has several disadvantages : it records the distance to the nearest point on the sea surface, and may thus miss steep crests ; it loses echoes when the water is aerated, which often happens during storms ; it can also lose echoes when a swell is passing over an otherwise calm sea, since there may be no wave facets pointing in the correct direction to reflect sound to the transducer. The alternative principle, which is the one usually used, is to measure the pressure on or near the sea-bed as the waves pass overhead. The theoretical relationship between the amplitude P of the pressure fluctuations (measured in equivalent head of water) and the amplitude A of the surface elevation can be derived by 1.30 1.26 1.22 1.18 1.14 1. 10 1.06 1.02 0.98 0.94 1.15 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0. Theoretical attenuotion coefficient P/A 0.9 1.0 Fig. 10. The ratio of theoretical to actual pressure-change amplitudes on the sea-bed plotted against the theoretical attenuation factor. (After Draper, 1957, Fig. 3. By courtesy of the Journal.) slight extension of the theory given in Lamb's Hydrodynamics, for example, but seems to have been first quoted in the required form by Seiwell (1948). It is PjA = cos,h.k{h — z) I cosh kh, (13) where h is the depth of water, z is the depth of the measuring instrument, and k = 27T/wavelength ; also a^ = gk tanh kh, where o- = 27r/wave period. If the recorder is on the sea-bed, z = h and therefore P/A = sech kh. (14) Various attempts have been made to check this relationship experimentally, but good agreement has never been found. Draper (1957), for example, finds SECT. 5] WINO WAVES 687 discrepancies of at least 16% under some circumstances (Fig. 10) and these appear to be too large to be explained by experimental error. This is a most important problem from the engineering point of view. An important potential source of error in pressure -type wave meters was brought to our attention by R. L. Wiegel : the orbital motion of the water due to waves is disturbed by the presence of the measuring head, and considerable dynamic pressures can be set up. The design of the measuring head must be such as to minimize these. Pressure-type wave meters may be self-contained (e.g. Valembois, 1955) but these have been found to be unsatisfactory, since even the best instruments develop faults, and, on recovery, it is occasionally found that enough records have been lost to spoil a year's recordings. Electrical measuring heads designed to be as simple and reliable as possible and connected to shore by a cable are, therefore, preferred : if anything goes wrong, it is usually immediately obvious and can probably be quickly corrected. Ways of measuring pressure electrically are also numerous, and the reader is again referred to the summaries of wave recording methods by Snodgrass (1951) and Draper (1961) and to papers in the Proceedings of the First Conference on Coastal Engineering Instruments (1955). B. Wave Recording on the Deep Sea The wave recorders described above are designed for mounting on a fixed platform. On the deep sea no such platform is accessible and different tech- niques are necessary. FLAT BUOY WATER LEVEL UP POLE IS MEASURED SUP-FACE OF EQUAL PRESSURE Fig. 11. Suspended pressure -meter and pole-and-drogue-type wave recorders. One class of recorders uses the still water below the action of the waves as a reference. A drogue (for example, a large horizontal flat plate) may be suspended from a pole-type surface buoy designed so that its lift changes as little as possible as the waves pass (Fig. 11). The drogue thus effectively prevents vertical motion of the buoy, and the wave motion relative to this may be measured by one of the techniques described above. Alternatively, a wide flat 688 BARBER AND TUCKER [CHAP. 19 buoy which rises and falls with the wave surface may have a pressure meter suspended from it (Fig. 11). If the pressure meter is deep enough to be below the action of the waves, it will measure pressure fluctuations corresponding to its vertical displacement. Variations of these principles are possible. In one American instrument, the pole and drogue is replaced by a single long articu- lated buoy (Farmer et al., 1955 : the theory given in this paper has since been revised, see Chase et al., 1957, p. 73). In a Russian instrument seen by the authors, the pressure gauge of the second type was replaced by a propeller device measuring the integral of the vertical velocity. In general, instruments of this class are not very satisfactory. They tend to be difficult to handle from a ship, particularly in rough weather. If the device is attached to the ship by a cable, great care must be taken that the cable be kept slack. The buoys may be tilted by a strong wind, or currents may drag the suspended drogue or pressure meter to one side. If the longest wind-wave components (about 25 sec period) are to be effectively measured, long vertical suspensions at least 100 m are required. Thus, the present tendency is away from this class of instrument. The second class of deep-sea wave recorders measure the vertical acceleration of a buoy on the sea surface (Dorrestein, 1957). The spectrum of wave accelera- tion A' (a) is simply related to the height spectrum by A'{a) = a^E'{a). (15) If only the spectrum is required, this conversion can be done after A' {a) has been computed. However, if a wave-height record is required, the output of the accelerometer must be integrated twice either electronically before recording, or digitally after recording. In principle, the vertical acceleration can be integrated mechanically in the buoy, but the authors know of no successful instrument which does this. Ideally, the axis of the accelerometer should be kept truly vertical (by means of a gyroscope, for example), but this is somewhat difficult and expensive and in most instruments the accelerometer is either hung in gimbals to form a short-period pendulum, or fixed to a flat float so that it remains perpendicular to the local water surface. The water surface sets itself perpendicular to the direction of the vector sum G{t) of the vertical and horizontal accelerations of the water particles and gravity because, since it is effectively a constant pressure surface, there can be no component forces parallel to it. A short- period pendulum in a small buoy also sets itself in the direction oiO{t), so that the two methods of mounting are equivalent. Thus, the magnitude of G{t) is measured and is given by \G{t)\^ = {g + z)^ + x^ + r, (16) where g is the acceleration due to gravity, x, y and z are the three components of the acceleration of a particle in the water surface. Expanding for \G{t)\ to the second approximation gives \G{t)\ =g + z + {r^ + y^^)l2g. (17) SECT. 5] WIND WAVES 689 Here g is balanced out, z is the term required and {x + y)l2g represents the error due to the method of mounting. If the wave spectrum is known, the spectrum of the error term can be calcu- lated (Tucker, 1959). Its total energy over the range of wave frequencies is of the order of 2 or 3% of the wave energy, but (in terms of wave height) it rises steeply at low frequencies and presents a problem if the acceleration is in- tegrated (Fig. 12). 10^ 0.3 0.4 0.5 0.6 0.8 0.15 0.2 0.3 0.4 0.5 0.6 Angular frequency (rodians/sec) 30 20 15 12 10 Wove period (sec) Fig. 12. Wave spectra computed from Neumann's formula for an equilibrium wave system, and the spectrum of the errors introduced by measuring the waves using a buoy containing an unstabilized accelerometer whose output is integrated twice. (After Tucker, 1959. By courtesy of the Journal.) Tucker has described an instrument which combines measurement of the vertical acceleration of a ship with the pressure on its hull (a full description is given in Tucker, 1956, and a shorter description in Tucker, 1956a). In principle, the pressure at a point on the ship's hull gives the height of the water surface above this point, and this is added to the vertical displacement measured by an accelerometer and double-integrator system (Fig. 13). The pressure gauge is mounted as close to the water line as is possible without its emerging as the shijD rolls (typically 10 ft below the water line), and the accelerometer is mounted close to the pressure unit. A single measuring head 23— s. I 690 BARBER AND TUCKER [chap. 19 would thus not measure short waves approaching from the opposite side of the ship, and the height of waves approaching from its own side would be increased by reflexion. Two measuring heads are therefore mounted on opposite sides of the ship and their output is averaged. The response of the instrument drops for waves of shorter wavelength owing to the pressure units having to be mounted considerably below the water line. The shape and motion of the ship makes calculation of this effect difficult, but Korvin-Kroukovsky (1955) has found theoretically that, for short waves, the pressure fluctuations P at a point on the hull of a ship (in equivalent head of water) should be related to the surface amplitude A by approximately P = ^ exp ( - 2kz), (18) where A; = wave number = 27r/ wavelength, and z== depth of the point below the water line. Fig. 13. The principle of operation of the shipborne wave recorder. (After Tucker, 1956a, Fig. 1.) Cartwright (in press) has compared the spectrum measured by a shipborne wave recorder in a ship steaming at 9 knots with that measured using a buoy, and finds that the above expression expresses the response fairly well, but that better agreement is obtained if the factor 2 is increased to about 2.5. In spite of this uncertainty in the high-frequency response of the instrument, shipborne wave recorders have been widely used on account of their ease of operation, and the fact that they measure the waves at the position of the ship, which is convenient for ship-motion research. C. Measurement and Analysis of Wave Records The principles and statistics of the measurement and analysis of wave records have been discussed in Chapter 15. They will be discussed here briefly from a physical point of view and with emphasis on practical methods. The simplest form of analysis is to measure a mean wave height and mean wave period. The "classical" method is to count the number N of crests in a record, and then measure the mean height Hy^ of the highest NjS waves, the SECT. 5] WIND WAVES 691 height being the difference between the crest and the preceding trough. This is called "the significant wave height". "The significant wave period" is the mean time between the preceding and following troughs for each of these waves. This apparently arbitrary method was adopted because the smallest waves in a record often appear to be unimportant. For example, if a large swell is present, a local wind may produce short low waves which break up the contour of the swell and thus reduce the measured height if the mean of all crests is taken, even though the total energy has obviously increased. Taking the mean of the highest third of the waves reduces this kind of effect. Other measures have some advantage. The most fundamental measurement of wave height is the root-mean-square deviation of the record, Hr.m.s., which is proportional to the square-root of the wave energy. This is tedious to measure directly from the record without special equipment, however, and is rarely used. The height of the highest wave in a record, Hm&x, is a more reliable measure of wave height than one would expect, is easy to measure, and is the parameter which is most important to a civil engineer designing coast defence works. The average period of the wave crests, Ti, and of the crossings of the mean water-level line in an upward direction, To, are two other useful measures of wave period. They are related by the formula (TilTo)^ = 1-62, (19) where e is a parameter which is a measure of the width of the power spectrum. No adequate theory is available relating the values of Hy^ and ^max to Hr.m.s. for records with large values of e, but when e is small (in practice less than 0.3) the following relationships apply ^ Hy^ = 4.004^r.m.s. (20) ^max = [2.82 (loge i\^)'/^ -f 0.82 (loge iV)-'/^]^r.m.s. (21) valid for N > 20. All these measurements are subject to random errors which become smaller the longer the wave record. In practice, a record containing between 50 and 100 waves seems to have been accepted as a reasonable compromise between accuracy and recording time, typically giving random errors of the order of 20% (r.m.s.) in wave height. Frequency spectra may be measured by analogue or digital methods. In principle, all methods are equivalent to passing the wave signal through a filter which passes components with frequencies in a certain range, and then measuring the energy (mean square amplitude) of the output from the filter. In practice, the pass-band of the filter does not have sharp limits, and the 1 In referring to Chapter 15, section 6, note that N is the number of crests in the whole record, Nq is the sum of the number of upward and downward zero crossings per second, Ni is the sum of the number of crests a7id troughs per secotid. Thus, Tq = 2/Nq and Ti = 2/Ni. Also, Hfrn.s. =Wio*'2. 692 BARBER AND TUCKER [CHAP. 19 amplitude of the components is weighted by a "filter-function", or response- curve, which is maximum at the centre frequency and is negligibly small out- side certain limits. The "band-width" of the filter is usually defined as the frequency difference between the points at which the response falls to half (in terms of mean-square amplitude). If L is the length of the record in seconds, it can be shown to be completely represented by the sum of harmonic components with frequencies separated by IjL c/s. This is, therefore, the limit of useful resolution of the analysing system, and is fundamental to a finite recording. In "periodogram" methods of analysis, the amplitude of these individual harmonic components is measured, but they show enormous random fluctuations (the standard error is 100%) and have to be smoothed to reduce these before they can be used. Other methods use a filter with a comparatively wide pass-band in the first place. The advantage of the periodogram method is that it gives the maximum possible resolution, and the amount of smoothing permissible without loss of detail can be estimated by inspection. The wide-band methods have the advantage of reducing the amount of human time and thought necessary, but occasionally significant detail may be lost. Either type of analysis may be performed by analogue or digital methods, but the present trend is towards the latter. The reasons for this seem to be : (1) A digital computer will usually give either the correct answer with a high accuracy, or, if a fault occurs, an answer which is obviously wrong. (2) An analogue computer is a specialized piece of equipment which is not readily adaptable. Thus, it requires a high capital outlay which may be wasted if ideas change (for example, if higher accuracy is required). Suitable digital computer programmes are usually available (but can be quite expensive if one has to be developed specially), and most laboratories have access to a digital computer and its associated equipment. On the other side of the balance, if an analogue machine is available, an analysis can be performed quickly and cheaply, which can be a major advantage if many routine analyses have to be performed. Analogue methods start with the record on photographic paper (e.g. Barber et at., 1946) or on magnetic tape (e.g. Pierson and Chang, 1955). The photo- graphic methods have the advantage that the record can be seen by eye, and records can be prepared by hand if necessary. Magnetic tape methods seem to be more flexible ; in particular, it is easier to vary the length of the record. The records are run through a device which converts them to an electric signal at 103 to 104 times the recording speed, and passed through a filter whose output is recorded. For periodogram analysis, the filter has to be extremely narrow (see Tucker's description, 1956b, of a later version of the Barber et al. machine). For wide-band analysis, Chang (1954) has calculated the optimum shape of the filter function. To sweep the frequency range, the rate of reproduction may be varied, as in the Barber et al. machine, or the frequency of the filter may be varied, usually by a heterodyne system. SECT. 5] WIND WAVES 693 Periodogram analysis on a digital computer follows the standard method for extracting Fourier harmonics, which can be found in most elementary text- books of physics. The computing time is large, however, because of the necessity for computing the sine and cosine multipliers and because of the large number of multiplications involved. This time can be greatly reduced by clever pro- gramming, and a programme, due to J. Watt, is available for "Deuce" com- puters which is remarkable in this respect. The alternative wide-band method due to Tukey (1949) using correlation is usually faster, and gives directly the smoothed spectrum which is usually required, so that it is more commonly used. This forms in the first place a "lag correlation", /./(i/^^^2_i)>i (1) and 'L{t)^lR{t)^ = {\iry^'^-\)H^^rxz^-\Y^. (2) (1) and (2) are very useful as the actual amplitudes of the waves are not re- quired and the sensitivities of seismographs tend to vary over a period of years. 712 DARBYSHIRE 6 and 7$ [CHAP. 20 Fig. 8. Variation of R{t)jL{t) with distance. (After Iyer, 1959.) SECT. 5] MICROSEISMS 713 The writer successfully tracked a storm in 1951 by using this technique (1954). The correlation coefficients between the components were found by an analogue computer, described by Tucker (1952), the two records being con- verted into a black and white form and placed one above the other along the surface of a perspex semi-cylinder. Two strips of light scan both records at speed, one strip to each record, the distance between the strips being adjustable. The whole arrangement is viewed by three photo-electric cells so that variations in height on the records are converted into an electrical output. This is squared and integrated. The final result gives the correlation and this varies as the distance between the strips is varied. The maximum value is taken and the strip separation then gives the phase difference between the two records. This computer has since been improved by Iyer by incorporating band-pass filters in it so that the correlating can be confined to waves contained within a narrow period range such as 5-6 sec, 6-7 sec, etc. This proved to be extremely useful when there w^as more than one source of microseisms, as the effect of the two sources could then be distinguished if the two sets of waves generated had different period ranges. Iyer altered the original assumption that Rayleigh waves and Love waves come from the same source to that of assuming the Love waves to come equally from all directions. With this assumption, equation (1) becomes tan e = (l/r^,2_ i)y./(i/^^^2_ 1)%. (3) Iyer tracked several storms by this method, allowing for the effect of refraction by using the diagram shown in Fig. 7. He found that the estimated bearings of the source consistently lagged behind the known bearing of the storm centre so that, in the case of a north-moving storm, the generating area would be to the south of the centre. This would be expected if the wave interference theory is correct. Iyer was also able to calculate R{t)IL{t) for the various examples. His results are shown diagrammatically in Fig. 8. It can be seen that the ratio decreases as the distance of the source from the seismograph station is in- creased. This implies that energy is progressively converted from the Rayleigh- wave type to the Love-wave type as the microseisms travel on from the source. He also found the mean value of the Rayleigh constant to be 0.73, independent of the period and very near the theoretical value of 0.68 for simple Rayleigh waves. Iyer's conclusions that Love waves emanate from all directions do not agree with those of Gutenberg and Benioff (1956) who, by using strain seismographs, could distinguish directly between the two kinds. Gutenberg concluded that Rayleigh and Love waves come from the same direction. In this case the microseisms, however, had almost certainly been caused by the reflection of swell off the Pacific Coast of America so that the microseisms had not travelled over a long ocean path. 6. Instruments Analysis of microseisms is a lengthy procedure even with an analogue com- puter and it is advantageous to be able to analyse the waves as they are 714 DARBYSHTRE [CHAP. 20 recorded. The conventional type seismographs used hitherto gave very little or no electrical output biit this is not the case with modern instruments. The strain seismograph used by Gutenberg and designed by Benioff (1955), has already been mentioned. Very sharply tuned seismographs have also been used by Donn, Ewing and Press (1954) to help to locate different microseism sources. In Great Britain, a new type of seismograph has been developed by Tucker (1958). This has a high magnification (up to 18,000) and a flat response from 1 to 10 sec period. It consists of a pendulum of 1.6 sec natural period, and the displacement of the bob relative to its mean position is measured by a transducer which gives a 4000 c/s signal balanced to zero output in the mean position. This signal is amplified and rectified taking account of phase, giving Fig. 9. Photograph of correlation meter. (After Iyer and Hinde, 1959.) a voltage proportional to the bob displacement. This voltage is fed back to a coil attached to the bob and with one of its sides in the field of a permanent magnet, the sense is such that the force on the coil reduces the defiection. By arranging suitable circuits in the feed-back part, the period and characteristics of the pendulum system can be altered at will. Both E-W and N-S components are incorporated in the same model. A vertical-component seismograph works on a similar principle, the pendulum arrangement is supported in a horizontal position by a "zero length" spring. Instruments have also been devised to analyse the microseisms quickly. Iyer and Hinde (1959) have published an account of a simple instrument for doing this. It consists essentially of a moving coil movement with a needle SECT. 5] MICROSEISMS 715 which makes contact both with a series of studs attached to counters and with a series of strips which only cover half the range of movement of the needle (see Fig. 9). From the number of counts in a given time for each stud, an estimate of the r.m.s. value for each component is found. The period is found by the movement of the needle over the half strip so that a circuit is made or broken at every zero crossing. The correlations are worked out by using two of these movements so arranged that a circuit is made when the needles are on the same side on both and broken when they are on different sides. The long strip which is split at the centre is used for this purpose. The correlation co- efficients can then be worked out by the formula given by Tomoda in Japan (1956): r = sin [7r(n+ — 7i-)/2(w+ + ?!-)], where w+ is the number of counts when the needles are on the same side (i.e. both positive or both negative) and n- is the number when the needles are on opposite sides. It follows that, from the proportion of time the current is switched on to the total time w+/(w+ + ?i_), r can be calculated. These instruments and the seismographs are portable and it is proposed to set up mobile stations and combine the tripartite principle with the correlation technique, as with this method there is no limit to the length of the side of the triangle and distances of 100 miles should be possible. 7. Other Work Some work with mobile stations has already been done by Bernard (1952) in France in investigating the variation of microseisms from one place to another. Work has also been carried out in other countries. The late W. M. Jones (1947, 1949) related microseism amplitudes and periods with movements of storms past New Zealand. Work has also been done by 0. A. Jones and his collaborators at Queensland, Australia (1947). In Australasia, storms are usually isolated and so it is easier to study their effect. Thus Upton (1956, 1956a) has been able to deduce a formula for the attenuation of microseisms with distance. He found it to be of the form aocl/J?'-, a form suggested by Longuet-Higgins (1953); a similar formula was found by the writer (1957) for sea waves. A great deal of work has been done in recording microseisms in Antarctica. Imbert (1954) studied those recorded in Adelie Land. There, because of pack ice, they could not be generated near the shore and those recorded had to originate in the deep sea. Microseism activity could be associated with the onset of barometric lows and a relation was found between the microseism amplitude and the speed of the lows and the rate of rotation of the winds. A detailed investigation of the wave heights and periods to be expected in the storm area supported the wave interference theory. Similar conclusions on Antarctic studies during the I.G.Y. have been reported recently by Eppley (1959). 716 DABBYSHIRE [CHAP. 20 Hatherton (1960) has analysed microseisms recorded at Scott Base during the International Geophysical Year. He found that during January and February when the Ross Sea is clear of ice but the outer ocean is still covered, the microseisms were of short period, 1 to 3.5 sec, and the maximum amplitude varied as T^-^. Later on during March and April, when all the sea is free of ice, longer-period microseisms of 4-10 sec are recorded and the maximum amplitude varies as T^-". The shorter ones can be related to the shorter-period sea waves generated within the Ross Sea area. The longer ones are due to swell from the neighbouring ocean, and the different relation between amplitude and period is attributed to the selective period attenuation of microseisms in crossing the continental shelf. The two-to-one period ratio between microseisms and sea waves appears to hold in all cases. The work of Bath in Scandinavia has already been mentioned. Besides his conclusions on the effect of cold fronts crossing the coast, he also showed that the microseisms in Scandinavia were local in origin and not coming from the Atlantic. An examination of Fig. 7 suggests that this is due to the effect of refraction. A completely different approach to the subject has been made by Nanney (1959) who has published work to show that the incidence of microseisms is slightly correlated with that of earthquakes. A good deal of microseismic work has been done in the U.S.S.R. An interest- ing paper was published by Savarensky, Lysenko and Komplanetz (1958) on microseisms recorded at Raibach on the shores of Lake Issik-Kul. This lake is about 200 km long and 50 km wide at the widest part. The amplitude of short period microseisms (1.5-3 sec) increased very rapidly with increase in local wind speed, particularly when this was westerly. A west wind could produce high waves on the lake which would be reflected off the steep rocks lying round the lake. There is a lag of about 9 h between the time of maximum microseism intensity and maximum wind intensity and this can be explained by the time required for the wind to build up the highest waves and form a stationary wave system. The two-to-one period ratio appears to hold. It was also possible to estimate the reflection coefflcient of wave energy from the rocks. It is about ^0". Rykunov and Prosvirnin (1958) have studied microseisms from distant storms and have tracked storms in the Atlantic and off the coast of Scandinavia from stations ranging from Murmansk to the Black Sea. They find that the direction finding technique is improved by taking into account refraction, as was done by Darbyshire. Prosvirnin and others (1959) also found that for stations near Scandinavia there are two azimuths of maximum amplitude for the line joining the record- ing station to the microseism source, one being normal and the other tangential to the Scandinavian coast. This effect is attributed to the continental shelf and the mountain ranges, which tend to be ahgned normal to the coast. These conclusions have been checked and confirmed by model experiments. I SECT. 5] MICROSEISMS 717 References Banerji, S. K., 1930. Microseisms associated with disturbed weather in the Indian Seas. Phil. Trans. Roy. Soc. London, A229, 287-328. Bath, M., 1949. An investigation of the Uppsala microseisms. Met. Inst. Kungl. Univ., Meddel., No. 14. Bath, M., 1950. The microseismic importance of cold fronts in Sandinavia. Ark. Geofys., 1, 267-358. Bath, M., 1951. The distribution of microseism energy with special reference to Scandin- avia. Ark. Geofys., 1, 359-393. Bath, M., 1953. Comments on the paper by F. Press and M. Ewing, 'The ocean as an acoustic system'. Symposium on microseisms, Sept., 1952. Nat. Acad. Sci., Nat. Res. Council, 113. Benioff, H., 1955. Advances in Geophys., 2, 220-274. Bernard, P., 1941. Etude sur I'agitation microseismique et ses variations. Ann. Inst, Globe, 19, 1-77. Bernard, P., 1952. Microseismes a Saint-Michele-de-Provence. Ser. A, Trav. Sci. Bur. Centr. Seismol., Fasc, 18, 83-89. Carder, D. S., 1955. Microseisms at Bermuda. Trans. Amer. Geophys. JJyi., 36, 843. Carder, D. S. and R. A. Eppley, 1959. The microseism programme of the U.S. Navy. Terminal Report, U.S. Dept. of Commerce, Coast and Geodetic Survey, April, 1959. Cooper, R. I. B. and M. S. Longuet-Higgins, 1951. An experimental study of the pressure variation in standing water waves. Proc. Roy. Soc. London, A206, 424-435. Darbyshire, J., 1950. Sea waves and microseisms. Proc. Roy. Soc, London, A202, 439. Darbyshire, J., 1954. The structure of microseismic waves. Proc. Roy. Soc. London, • A223, 96. Darbyshire, J., 1955. Refraction of microseisms at island stations. Mon. Not. Roy. Astr. Soc, Geophys. Suppl., 7, 147. Darbyshire, J., 1957. Attenuation of swell in the North Atlantic Ocean. Q. J. Roy. Met, Soc, 83, 351. Darbyshire, J. and M. Darbyshire, 1957. The refraction of microseisms on approaching the coast of the British Isles. Mon. Not. Roy. Astr, Soc, Geophys. Suppl., 7, 301. Deacon, G. E. R., 1947. Relation between sea waves and microseisms. Nature, 160, 419- 421. Dinger, J. E. and G. H. Fisher, 1955. Microseism and ocean wave studies at Guam. Trans. Amer. Geophys. Un., 36, 262-272. Donn, W. L., 1951. Frontal microseisms generated in the Western North Atlantic Ocean. J. Met., 8, 406-415. Donn, W. L., 1951a. A comparison of microseisms and ocean waves recorded in Southern New England. Col. Univ. Tech. Rep. on Seimology, No. 21. Donn, W. L., 1952. Cyclonic microseisms generated in the Western North Atlantic Ocean. J. Met., 9, 61-71. Donn, W. L., 1952a. An investigation of swell and microseisms for the hurricane of Sept. 13-16, 1946. Trans. Amer. Geophys. Un., 33, 341-344. Donn, W. L. and M. BlaLk, 1953. A study and evaluation of the tri-partite seismic method of locating hurricanes. Bull. Seism. Soc Amer., 43, 311. Donn, W. L., M. Ewing and F. Press, 1954. Performance of resonant seismometers. Lamont Geological Observatory, Tech. Rep. No. 36. Eppley, R. A., 1959. A review of microseisms and their relation to ocean waves. Preprints, International Oceanographic Conference, New York, Sept., 1959, 749. Gherzi, E., 1923. Etude sur les microseismes. Notes de Seismologie, Obs. de Zi-ka-wei, No. 5, 1-16. Gherzi, E., 1927. Houle et microseismes sur la cote de Chine. Notes de Seismologie. Obs. de Zi-ka-wei, No. 8, 1-12. 718 DARBYSHIBE [CHAP. 20 Gilmore, M. H., 1946. Microseisms and ocean storms. Bull. Seism. Soc. Amer., 36, 89-119. Gilmore, M. H., 1953. Amplitude distribution of storm microseisms. Symposium on microseisms, Nat. Acad. Sci., Nat. Res. Council, Sept. 1952, 20-55. Gutenberg, B., 1912. Die seismische Bodenruhe. Diss. Gottingen u. Gerlands Beitr. Geophys., 11, 314-353. Gutenberg, B., 1915. tjber mikroseismiche Bodenruhe. Phys. Z., 16, 285. Gutenberg, B., 1921. Untersuchungen iiber die Bodenruhe mit perioden von 4s-10s in Europa. Veroffentl. Z. Intern. Seism. Assoc, Strassburg, 1-106. Gutenberg, B., 1952. Microseisms, microbaromes, storms and waves in Western North America. Trans. Amer. Geophys. Un., 34, 161-173. Gutenberg, B. and H. Benioff, 1956. An investigation of microseisms. Calif. Inst, of Technology, Division of Geological Sciences. Contribution No. 761. Haskell, N. A., 1951. A note on air coupled surface waves. Bull. Seism. Soc. Amer., 41, 295-300. Hatherton, T., 1960. Microseisms at Scott Base. Geophys. J. Roy. Astr. Soc, 3, 381. Hoehstrasser, U. and R. Stoneley, 1961. The transmission of Rayleigh waves across an ocean floor with two surface layers. Communication No. 40, Association de Seismologie et de Physique de ITnterieure de la Terre. U.G.G.I., 12 ieme Conf., Helsinki, 1960. Imbert, B., 1954. Rapports scientifique des expeditions polaires fran9aises. S. IV. 3, 10, fasc. 2, 1-10. Inouye, W., T. Hirona and G. Murai, 1954. Microseisms and surf. Geophys. Mag., 25, 175-183. Iyer, H. M., 1958. A study on the direction of arrival of microseisms at Kew Observatory. Geophys. J. Roy. Astr. Soc, 1, 32. Iyer, H. M., 1959. Composition and propagation of storm microseisms. Ph.D. thesis, University of London. Iyer, H. M. and B. J. Hrnde, 1959. Microseismic research at the National Institute of Oceanography. Nature, 183, 1558-1560. Iyer, H. M., D. Lambeth and B, J. Htnde, 1958. Refraction of microseisms. Nature, 181, 646. Jones, O. A., 1947. The detection and tracking of cyclones off the Queensland coast. Austral. J. Sci., 10, 43-44. Jones, W. M., 1947. New Zealand microseisms associated with the storm of 14-16th Feb., 1947. N.Z. J. Sci. Tech., 29, 142-152. Jones, W. M., 1949. New Zealand microseisms and their relation to weather conditions. 7th Pacific Sci. Congress, 1949. Lee, A. W., 1932. The effect of geological structure upon microseismic disturbances. Mon. Not. Roy. Astr. Soc, Geophys. Suppl., 3, 83-104. Lee, A. W., 1934. Further investigation on the effects of geological structure upon micro- seismic disturbances. Mon. Not. Roy. Astr. Soc, Geophys. Suppl., 3, 238-252. Lee, A. W., 1935. On the direction of approach of microseismic waves. Proc Roy. Soc London, A149, 183-199. Leet, L. Don, 1934. Analysis of New England microseismic waves. Beitr. Geophys., 42, 232-245. Longuet-Higgins, M. S., 1950. A theory of the origin of microseisms. Phil. Trans. Roy. Soc London, A243, 1-35. Longuet-Higgins, M. S., 1953. Can waves cause microseisms? Symposium on microseisms. Nat. Acad. Sci., Nat. Res. Council, Sept. 1952, Publ. 306, 74-93. Longuet-Higgins, M. S. and F. Ursell, 1948. Sea waves and microseisms. Nature, 162, 700. Macelwane, J. B., 1946. Origin of microseisms. Science, 104, 300-301. Miche, M., 1944. Mouvements ondulatories de la mer en profondeur constante ou decrois- sante. Ann. Ponts et Chaussees, 114, 25-87, 131-164, 270-292, 396-406. SECT. 5] MICROSEISMS 719 Nanney, C. A., 1959. Evidence for correlation between microseims and earthquakes. N.R.L. Report 5237, U.S. Naval Laboratory, Washington, D.C. Press, F. and M. Ewing, 1948. A theory of microseisms with geological applications. Trans. Amer. Oeophys. Un., 29, 163-174. Press, F. and M. Ewing, 1951. Ground roll coupling to atmospheric compression waves. Geophysics, 16, 416-430. Press, F. and M. Ewing, 1953. The ocean as an acoustic system. Symposium on micro- seisms. Nat. Acad. Sci., Nat. Res. Council, Sept. 1952, 109. Prosvimin, V. M., T. A. Proskuryakova, L. N. Rykunov and E. F. Savarensky, 1959. The influence of the Scandinavian relief on the propagation of microseisms. IX and XII Sections of the I.O. Y . Program [Glaciology 14 - » T : . v^ ;->-(-:.'; • I'- * - - : :-s^-^^_ Fig. 7. Depth of 64°F isotherm measured with isotherm follower from 1500 h, 23 September, to 1500 h, 24 September, 1959, off Mission Beach, California. the lower ones were probably only random fluctuations. It was found that 50% of the internal waves were higher than 5.6 ft. b. Wave period The frequency distribution for the duration of 1061 internal waves is shown in Fig. 9. Waves with periods of less than 2 min were excluded. Fifty per cent of all waves longer than 2 min had periods greater than 7.3 min. WAVE HEIGHT (FT) WAVE PERIOD (MIN) [■: :-:t>:-:-:-:-{:i:':-:":-:-:-:-L-:-:-::-:-:-i 0 4 8 12 16 20 24 Fig. 8. Composite frequency distribution of internal wave heights over 2 ft (for 1958 and 1959 combined). '0 4 8 12 16 20 24 Fig. 9. Composite frequency distribution of internal wave periods over 2 min for all data. 738 liAFOND [chap. 22 c. Speed The speed of internal waves was determined by measuring vertical oscilla- tions simultaneously in three locations (UflFord, 1947a; Lee, 1961a), and deduced from the movement of their associated sea-surface slicks (see page 746). Time-lapse films of surface slicks off southern California, ^ showed that internal waves moved toward shore at speeds of 0.11 to 0.6 knots. Other measurements from anchored ships with range markers indicated an average speed of 0.31 knots. More recent measurements averaged 0.27 knots in 60-ft deep water. d. Direction The shape of internal waves varied Avith shoreward movement and with the refraction as they moved into shallower water. Nearly all internal waves pro- ceeded from a west to southwest direction at the Mission Beach location. e. Currents Relative currents in the open sea are usually computed from the distribution of mass. Current along an isobaric surface is essentially a function of the geo- potential, or dynamic slope of the isobaric surface. If motion is taken to be negligible at a particular depth, or an isobaric surface to be level, the dynamic slope of the upper isobaric surface can be determined from variations of specific volume along the isobaric layer. The current at the upper surface, relative to any possible current at the lower surface, can thus be established (LaFond, 1951). Under the influence of internal waves, however, the average vertical specific volume above a reference level located below the thermocline will change and cause a considerable difference in the calculation of current. Short-period variations and the depth of the thermocline are difficult to treat, though long- tidal-period oscillations have been considered for serial stations off the California coast (Defant, 1950). A tidal influence on the computation of relative currents was also suspected from an examination of mid-Pacific temperature data taken near Bikini Atoll (LaFond, 1949). Here a large, transitional layer separates the relatively light surface water from the heavier, deeper water. An internal wave causing a change in the depth of the transitional layer would materially change the vertical mass field. Internal waves are not wholly random, but appear to fall in a cyclic pattern. The general trend is shown by dotted lines on the temperature curves (Fig. 10). The high phases fall about 12 h apart and have nearly the same period as the tide. The greatest changes in temperature, as well as in salinity, occur at 700 ft below the surface ; above 400 ft, the changes are small. At 900 ft below, the changes in temperature, indicative of vertical fluctuations, are somewhat 1 Mission Beach, La JoUa and San Diego Bay. SECT. 5] INTERNAL WAVES 739 smaller than at 700 ft. The decrease in the magnitude of temperature variations below 700 ft indicated that the effect of internal waves on the vertical mass field was greatest at depths of less than 700 ft. From such closely spaced observations, currents were computed by a simpli- fied procedure. By use of a temperature-salinity relation, the specific volume anomaly 8, was converted from a function of temperature, salinity and depth to a function of temperature and depth only. The numerical values of lO^S over the temperature-depth range experienced were computed. H L H L H L H JULt 8 : i JULIf 9 iJULY 10 jlZOO 1600 i 2000 0000 0400 i 0600 1200 1600 : 2000 0000 0^0 Fig. 10. Fluctuations of sea temperatvires at 100-ft intervals from 400 to 900 ft, derived through repeated bathythermograph observations over a period of 40 h near Bikini Atoll. Then, by a simple numerical integration over the depth of the water column, the dynamic-height anomaly, 10^ AD, was found. The resulting values presented an irregular pattern that can be attributed largely to internal waves. To show the effect of internal waves upon 10^ AD, and in turn on current calculations, the dynamic-height anomalies from one day of repeated bathy- thermograph observations were obtained. The data were averaged and plotted (Fig. 11). The resulting fluctuations in dynamic height anaomalies amount to as much as 0.08 dynamic meters in a few hours. Seiwell (1937) showed that internal waves will change the distribution of mass along any vertical structure, and will, therefore, cause the geopotential height of the free surface, relative to a given isobaric surface, to vary periodi- cally. For the Atlantis Station 2639, Seiwell found that, because of internal waves, the Variation of geopotential height of the free surface, relative to the 740 [chap. 22 2000-d.b surface, reached a value of 8.45 dynamic centimeters. Even more spectacular was a 14.5 dynamic centimeter variation with time noted at Snellius Station 253a. The variation demonstrated the importance of internal waves in the calculation of currents. Thus it is evident that erroneous conclusions from hydrographic observations can be drawn unless the effect of internal waves on the distribution of tempera- ture, salinity, currents, etc., is considered. Defant (1950) pointed out that certain time spacings of hydrographic stations would reduce the relative errors caused by internal waves, and developed an equation of time spacing between observations when the period of the internal tide is known. HIGH TIDE HIGH TIDE LUNAR HOURS 1200 1.24 1.25 • • A0^^*AD,3 ■•. • • OBSERVED --"^ ■ — ■ — ~--^ ^-^ ^^ ^ ..■••■■ / \ •.,'•'. V N^\ / • \^ /^ y "^ ^/ ..•••■■"'* . ' ^'"^"■'S^-j • ■•■. ■ — ^"^^^ --^ • • Fig. 11. Variations of dynamic-height anomaUes (0/305 dynamic meters) jilotted with reference to lunar time, showing the calculated diurnal {ADo^) ; semi-diurnal {AD\2) ; and the resultant {AD2^ + AD12) lunar cycles. f. Sound transmission Internal waves affect the transmission of sound through water. Sound is refracted by the vertical (and horizontal) sound velocity gradients, which, in turn, depend on the strength of the thermocline and the angle at which the sound rays intersect it. With an undulating thermocline, caused by internal waves, the sound rays intersect at different angles. The refraction and sound-focusing effects can be calculated by applying Snell's law, but it is a very tedious process. However, a theoretical sound transmission problem was solved by means of a high-speed UNI VAC computer (Lee, 1961). In this problem, a three-layered ocean was used for a theoretical two- dimensional study of an underwater sound-intensity field in the presence of an internal wave. The internal wave (heavy lines in Figs. 12 and 13) and the sound- velocity structure were idealized to simplify machine computation, but both were representative of summer conditions off the southern California coast. Sound travels at a constant speed in the top layer. The sound velocity gradient in the second and third layers was — 4.8 ft sec~i ft^^ and — 0.6 ft sec~i ft~i, respectively. SECT. 5] INTERNAL WAVES 741 Fig. 12. Diagram of rays in the sound medium, which has an internal wave on the thermo- cline. Sound was emitted from a directional source through an angle of ± 8°. The level dropped from 60 dB (an arbitrary reference level was used) at one foot from the source along a horizontal ray (^ = 0) to one -tenth of full strength along ^±8°. All acoustic ray paths passing through the layers were refracted, depending upon their angle of approach and the velocity discontinuity of each interface. For this problem the rays were traveling in a plane parallel with the direction of propagation of the internal waves. Total reflection was assumed at the sea SOUND INTENSITY FIELD (dB) 0. 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 DISTANCE (FT) I I >I5 r -110*15 1 D5-»I0 E 10»-5 r.'.'.MO* -7.5 Fig. 13. Sound level in the medium with an internal wave on the thermocline. The dB reference level is that corresponding to a sound level of 60 dB at 1 ft from the direc- tional source (X = 0, Z—10) along the horizontal. The field is contoured at intervals of 2.5 dB. 742 LAFOND [chap. 22 surface, and all sound energy reaching the bottom was absorbed. This repre- sentation was an ideal situation, but it approximated the natural sea-velocity structure more closely than any previously considered. Even in this medium a great deal of computation was required for the multiple refractions and reflec- tions for each 0.1° ray. The computation of the acoustic energy refraction by internal waves is shown in Fig. 13. The acoustic intensity of sound, as coinputed for each 10-ft interval of range and each l-ft interval of depth, is shown by the shaded zones. The sound intensity (dB reference level) is that corresponding to a ^ound level of 60 dB at one foot from the directional source (depth 10 ft, distance 0 ft) along the horizontal. Above the thermocline the sound intensity decreased approximately as the square of the distance from the source. Below the thermocline refraction focused the sound rays as they passed through the internal wave into alter- nately high- and low-intensity zones. The divergence and convergence of the rays was directly related to the sinusoidal nature of the internal waves. In this problem, there was one high- and one low-intensity zone below the thermocline for each interval of one wavelength. The width of the high- intensity zones decreased with range, and the opposite was true for the low- intensity zones. The variation with range was mainly caused by waves near the source acting as a barrier to those at greater distance, i.e. progressively fewer rays struck those waves increasingly distant from the source. The internal wave in the above problem caused horizontal changes in the sound intensity of 22 dB over distances of less than one internal wavelength (300 ft) at 400 to 700 ft in range. On the other hand, intensity changes in a medium without the internal wave are about 5 dB within 300 ft at the same range, and no intermittent zones of high and low intensity occur. Thus internal waves play an important role in underwater sound transmission. g. Other related motions In the sea, internal waves appear to take the form of progressive waves. In lakes and partially closed bodies of water, standing waves are found. The nature of progressive waves between two liquids of different densities is described by Lamb (1945). The theoretical water motions associated with this simple pro- gressive wave are shown in Fig. 14. The fine arrows represent the streamlines of particles. In the sea, in addition to the vertical oscillations, other evidence of this circulation has been demonstrated. First, in the study of lateral shear motion, direct observations of vertical dye streaks in the water become distorted by the shear at the thermocline (Fig. 15). Secondly, other evidence of this motion is shown by surface currents and other surface phenomena (LaFond, 1959a). For example, in the Bay of Bengal, the surface showed long streaks of alter- nately rough and smooth water. The ripples were 6 to 8 in. high and some streaks extended to the horizon, varying in number from two or three to at SECT. 5] INTERNAL WAVES 743 '/ /^/y/7y ///////////////// ^ ////////// // Fig. 14. Simple progressive internal wave between water of two densities. The large arrow at the top shows the direction of the wave. The water motion streamlines are shown by the small arrows and the most common location of the slick by the heavy bar. B Fig. 15. A. Dye marker being dropped vertically. B. Its deformation, after a few seconds, as a resxilt of differential movement in the water column above and below the thermo- cline. 744 LAFOND [CHAP. 22 least 10. Individual streaks or bands varied in width from an estimated 75 to 600 ft. Their orientation was always parallel to the coast, which is the tendency of the prevailing drift. As a ship cruised through the bands, it was strongly set to the right or to the left, depending on whether it was in the rough or smooth band. Similar pheno- mena have been observed off southern California. By determining the thermal structure, it was found that the internal wave crest occurred directly under the roughest zone. From the surface currents, and the thermal structure and its progressive motion, it was concluded that a shallow internal wave was causing the phenomenon. According to Lamb (1945), the vertical displacement at the interface of a two-layer density system, p and p', is given by 7] — a cos {kx — at) and the horizontal velocity of flow, u', in the upper layer is u' = — {a I h')c cos {kx — at), where h' = average thickness of the upper layer, a = amplitude of wave at inter- face, and c = wave velocity at interface. If there is no appreciable flow in the y direction (normal to propagation) and no appreciable transport of surface water in the direction of propagation, the same volume of water per unit width must pass over the trough. The speed of flow in a horizontal direction must be inversely proportional to the thickness of the upper layer Z'. Z' = h' — a cos (kx — at). This indicates that u' is maximum, and in the opposite direction of propa- gation to c, when h' approaches a and the phase angle (kx — at) is zero. In shallow internal waves, the motion over the crest is strong. The water formerly passing over the trough is now funneled through this constriction. If the crests are very near the surface, the speed of flow is increased. The funneling of water over the crest, and the reduced speed just beyond, are believed to cause the turbulence and ripples at the surface. When an internal-wave thermal boundary is near a sea floor, a similar action ensues. If this occurs, the maximum turbulence will be under the trough but the maximum speed will be in the opposite direction of wave propagation. In shallow water, the internal wave direction is shoreward, and thus the maximum speed near the bottom will be off-shore. The funneling of water through the constriction created by the trough and bottom, always in an off"-shore direction, is undoubtedly a contributor to the off-shore movement of sediment. Internal waves near the bottom in deep water can also move sediment and form ripple marks. SECT. 5] INTERNAL WAVES 745 h. Relation of internal waves to slicks Sea-surface slicks, which often represent visible evidence of internal waves below, are seen as streaks or patches of relatively calm surface water sur- rounded by rippled water. The absence of wavelets in a slick gives it a glassy appearance in contrast to the adjacent rippled water (Fig. 16). From most angles, a slick appears brighter than its peripheral area in day- time because a smooth surface reflects the sky more than a rougher one. At night, when ambient light may exist, slicks contrast with adjacent, rippled water because their unruffled surface is less susceptible to surrounding reflec- tion. Surface slicks appear darker in full sunshine when the visual angle is such that light is directly reflected toward the viewer. This is because slicks do not produce the glitter that radiates from mutual reinforcement of reflected rays on a contiguous rippled surface. Fig. 16. Sea-surface slicks off Mission Beach, California. Slicks have been studied in oceans, bays and lakes (Dietz and LaFond, 1950 ; Woodcock and Wyman, 1946 ; Forbes, 1945). According to such investiga- tions, slicks are generally present when the wind has enough force to ripple the water, but not enough to cause whitecaps (Beaufort force 3, i.e., 3.4 m/sec). Slicks frequently assume the shape of broad, web-like connecting bands, and they occasionally appear as isolated patches. In shallow ocean over a con- tinental shelf, slicks are often contoured as long bands, more or less parallel with the coast. Near shore, a wider band may develop just beyond the breaker zone. Some slicks have been discovered on surfaces above kelp beds. During a 1958 study of slicks and internal waves, it was found that slicks were present about 10% of the time. During the periods of observation, 105 slicks were recorded. The duration of a single slick, as it passed any point, was from 0.35 to 5 min, the average being 1.3 min. 746 liAFOND [CHAP. 22 03 8 T3 SECT. 5] INTERNAL WAVES 747 The occurrence of visible slicks is contingent upon proper wind, lighting, sufficient organic matter on the water, and the nature of internal waves. The concentration of surface film depends on the interrelation of internal wave height and period. The average depth of the internal wave and its relation to water depth also influences the type of circulation, and thus has a bearing on the formation of slicks. A surface slick was sometimes observed over the trough of the depression in the thermocline. On other occasions a slick wandered to a position nearly over the crest of a wave. However, in 85 out of 105 cases the slick was on the descend- ing thermocline somewhere between the crest and the following trough (Fig. 17). This relationship is believed to be the result of water circulation created by internal waves. The significant motion is a surface convergence over the trailing slope of the internal wave. Although the maximum expansion of the surface layer Avas over the trough, the slicks were normally found at the active surface con- vergence zone. i. Relation to tide Many observers, notably Helland-Hansen and Nansen (1909), Defant (1932), Uiford (1947), LaFond (1949), Rudnick and Cochrane (1951) and Arthur (1954), have noted that internal temperature fluctuations sometimes have a nearly tidal periodicity. Tidal period and phase relationships have, therefore, been compared with internal waves and sea-temperature structure measured in various waters around the world (LaFond and Rao, 1954). Haurwitz (1954) has questioned whether these observations refer to strictly periodic components of internal observations. Unless a long series of measure- ments are available, it is impossible to distinguish periodic from less regular variations if there are irregular fluctuations (as there always are in observations of internal motions). On the basis of Haurwitz's criterion there are only a few places in the ocean where it has been possible to show with fair certainty that tidal periodic internal waves exist. The most remarkable example is due to Reid (1956), who has found lunar, semi-diurnal fluctuations of large amplitude off" the coast of California (Fig. 18). While it is not certain to what extent periodic variations are present, there is no doubt that large variations of temperature of a quasi-periodic nature close to the semi-diurnal frequency are present in the sea. Below we shall discuss an example of quasi-semi-diurnal oscillations which clearly implies that the driving forces are related to the tide. Wide fluctuation of internal and surface tide phases were established by the observations of Lee and LaFond (1960) from the NEL oceanographic tower. The depth of an isotherm, recorded for seven consecutive days, was plotted with reference to the phase lag of observed high surface tide (Fig. 19). This observation showed that the relationship may change phase daily, but 748 [chap. 22 14 15" Date (Oct. 1950) Fig. 18. Lunar seini-diurnal fluctuations of large amplitude off the coast of California. 15 HIGH TIDE + 1 + 2 +j + 4 t 5 + 6 + 7 + e + 9 4 10 + II + 12 H ^ i 1 1 1 V — 1 1 I 1 1 1 25 P ^ ^ I 1- P-, 35 1 o 1 1 45 / 1 1 , A — 55 1 ^ <" Fig. 19. Relation of the depth of an isotherm in the thermocline with reference to the phases of the tide off Mission Beach in 60 ft of water. 1.0 AUTO-CORRELATION, ff^ 0.8 -\ 0.6 QT 0.4 \ HALF TIDAL \ CYCLE TIDAL CYCLE 0.2 \ ' 0 -0.1 1 N. 1 ^ ^^ -0.3 V 1 1 1 1 1 1 1 1 1 1 1 1 1 1 .1 1 1 0 2 4 6 8 10 12 14 16 PHASE LAG (h) Fig. 20. Autocorrelation of depth of (isotherm) thermocline and its relation to periods for continuous 7 -day observation. SECT. 5] INTERNAL WAVES 749 from auto -correlation studies of the same' data, a tidal relation was definitely disclosed. It was found that a significant correlation of wave depth occurred at 6.4 and 12.8 h, which were the exact lengths of semi diurnal tidal periods (Fig. 20). This implies that the spectrum of Mave depth is sharply peaked at the lunar semi-diurnal frequency. The fact that the phase changes then may indicate that the waves are generated at a distance from the observing station and suffer various phase lags in traveling from generator to receiver. j. Relation to basins and lakes It is likely that standing internal waves are present in bays, basins, and even in lakes. Such waves are frequently related to the size and character of the lake. In a two-layer rectangular bay of constant depth, the period of oscillation of a free standing internal wave is (Wedderburn, 1909; Sverdrup, Johnson and Fleming, 1942) uip-p') n _{pih)+{p'ihr where n is the number of standing wave nodes. An investigation conducted in the Gulf of California (February and March, 1939) indicated that the wavy character of the isobaric surfaces might be due to the presence of a standing internal wave with a period between 6.3 and 7.65 days and probably closer to the former value (Sverdrup, 1940). The wave would be of the first order with reference to the vertical axis (the vertical displace- ment vanishing at the surface and bottom only) and with three nodes inside the Gulf would be of the fourth order with reference to the horizontal axis. Within a standing internal wave, the horizontal velocity component reaches a maximum near the nodes, so that only large-sized particles in the water would settle in their vicinity. On the other hand, anti-nodes are characterized by small horizontal currents, which would permit small particles to deposit on the bottom. From the deposits in the Gulf of California, Revelle (1939) found that the sediment varied in a regular manner along a north-south direction, corresponding to an internal standing wave with three nodes. From the de- posits, Revelle concluded that a standing internal wave of the fourth order was not an isolated phenomenon but appeared to be of common occurrence in the Gulf of California. Munk (1941) theoretically examined the Gulf of California to determine whether any of the free internal-wave periods corresponded to the observed internal-wave period. In this analysis the equation {loc. cit. p. 41) was extended to take into account the geometric shape of the Gulf and the variation of density. It was found that there were two periods, one of 7 days and the other of 14.8 days. The observed distribution of density indicated that the first-order wave (7 -day period) was dominant, but that the presence of the second-order wave (14.8-day period) was not excluded. Munk's theoretical examination fully confirmed Sverdrup's interpretation of observations in the Gulf of California. 750 [cHAr. 22 (a) 30 JUCE - I JULY (b) //••-- ------ Fig. 21. (a), (b) Diurnal oscillation of temperature and probable currents in Sweetwater Lake. SECT. 5] INTERNAL, WAVES 751 Internal waves have been observed in closed bodies of water (Mortimer, 1952). In the spring the heating of the surface layer, mixing by wind, and differential currents at the thermocline divide the upper and lower parts of the water column with a sharp temperature gradient. Under the influence of wind the upper layer becomes distorted and lighter water accumulates at the leeward end of the lake. This results in thinning of the surface layer, and it occurs at the windward end of the lake (Sonar Data Division, Univ. of Calif,, Division of War Research, 1945). The wind causes a slow circulation to be set up that entails a movement of surface water in the same direction as the wind and a current at the thermocline in the opposite direction. The counter flow is initially in the upper layer, but some water just below the thermocline also moves windward. The slope of the distortion depends upon the force and duration of the wind. An equilibrium can be established wherein the wind stress balances the other forces. If the wind stops, the currents will reverse, and the thermocline may return to its former level or be tilted in the other direction by oppositely directed wind, aided by the momentum of the return flow. None of the currents have been measured directly, but have been inferred from the change in temperature structure and other properties of the lake. A study was made of an internal wave caused by a diurnal wind in Sweet- water Lake.i The prevailing westerly winds start about 1000 h and end around 1700 h. The shifting winds create a standing internal wave some 20 ft high (Fig. 21a, b). Such a major change in the distribution of mass makes studies of the heat budget difficult. The change in heat content of the vertical water column is caused more by the diurnal internal wave than by the daily radiation from sun and sky. ^ Near San Diego, California. PART II C. S. Cox 4. Differential Equations The equations governing the motion of waves through initially still water on a rotating sea have been treated extensively by Love (1891), Fjeldstad (1933), Groen (1948) and Eckart (1960). Let ijj{z) exp[i{kx — cot)] (1) represent the vertical displacement of water particles from an equilibrium condition in free waves (Fig. 22). Then the first order equation is I d / di/j pdz\Pdz}^Hco^-Q^ ifj = 0 where p is the undisturbed density, ---m-W' (2) (3) ////////, '//////■■'//// Fig. 22. The z axis is directed vertically upward. \\}{z) is the amplitude of internal oscilla- tions. is Vaisala's frequency (Chapter 2, Eq. 35), and Q is the inertial frequency equal to twice the angular velocity of the earth times the sine of latitude. Boundary conditions are dx\s gk^ .Q2 ifj = 0 at the surface and at a level bottom. ijj = Q 752 (4) (5) SECT. 5] INTERNAL WAVES 753 A. Limiting Frequencies Internal waves have maximum amplitude below the surface ; therefore, at this depth, d^ip/dz^ must be of opposite sign to ip. Neglecting a term p-'^{dpjdz) {difjjdz), which can be shown to be negligible for ocean water (Groen, 1948), one finds from equation (1) that the ratio of i/j" to ip is given by —k^{N^ — cxj^) {co" — Q")~^. Hence, internal waves can exist only if Q<\a>\S2)-'/^ ^ ri2(T) ex^{2rrifT) dr, where Si and S2 are the energy spectra of yi and 2/2 respectively : /'OO >S'i(/) = 4 rii(T) cos (27t/t) dr, jo and a similar expression holds for >S'2. With these definitions, 6 is the phase lead 756 cox [chap. 22 at frequency / of y2 over 2/1 and R represents the degree to which a single constituent of y\ remains at a constant phase difference with respect to a similar constituent of 2/2. Perfect coherence is indicated by i?= 1, no coherence by R = 0. The estimated values of R shown in Fig. 23 are based on limited records and, therefore, are subject to statistical errors. Even short sections of completely non-coherent records can give estimates of R greater than zero. The 95% confidence limit for estimates of R (when the true value 's zero) is shown dashed in the figure. The estimates of 6 are obviously meaningless if the true value of R is zero. Therefore estimates of 9 should be discarded unless the estimated value of R is larger than the confidence limit. An interesting feature of the recorded temperature oscillations is that there appears to be no statistically significant coherence between the oscillations at 50 and 500 m (estimates of phase shift are therefore completely unreliable). As an explanation the author considers the possibility that the oscillations about the seasonal and permanent thermoclines (Fig. 24) are not closely coupled. In this case the observations at 50 and 500 m, which are near the top of the respective thermoclines, would be poorly correlated. But this could only happen if no single mode of internal motion were dominant. For example, if only the first mode were present, observations at all depths (along a vertical line) would be in phase. Similarly, if any other single mode were dominant, the relative phase of the oscillations would be fixed. Little is known about the distribution of internal waves by mode. It would be surprising, however, if any but the first mode were dominant since higher modes involving higher shear would be expected to be more easily damped than the first. Furthermore, the low phase velocities of high modes would be expected to make them easily perturbed and destroyed by irregular propagating conditions in the sea. Another cause of reduced coherence between horizontally separated observing points will be treated more thoroughly in the next section. When internal waves of a single mode arrive from a wide variety of directions, the phase relations between observing points become variable and the coherence reduced. Coherence generally decreases with both increases of angular beam, through which the waves come, and separation (in wavelengths) between points of observation. The extreme case occurs for an angular beam 360° broad, that is, isotropic radiation. Then the coherence is unity at zero separation and drops to zero at a separation of 0.38 wavelengths (Fig. 27). The horizontal separation between thermometers off Castle Harbor was 1.5 km. A rough estimate of the phase velocity of nth mode internal waves is Cn={hln7T){N^ — cu2)'2, where h is the depth of water and N the mean value of Vaisala's frequency. Appropriate values are A = 200 m and A'^^^ 13 radians per hour, leading to a low frequency phase velocity of SAn~'^ km h~i. The co- herence would then be expected to be large for frequencies below 1.5 km/ (0.38 X 8.4^-1 km h.-^) = 0 Aln-^ c/h. The observations show no appreciable coherence even at much lower frequencies. We are forced to conclude either that there are many interfering modes of internal waves or that the temperature fluctuations are not due to free internal waves at all. SECT. 5J INTERNAL WAVES 757 6. Internal Waves and Turbulence Internal motions in the sea are not necessarily due to free internal waves but may also be due to forced waves or have the character of less well organized motions such as convection of turbulent eddies past the measuring apparatus. Indeed, the distinction between internal waves and turbulence is an arbitrary one, for internal waves, if grown too large, must break and dissipate their energy into eddying motion, while turbulent motions which involve vertical oscillations must be coupled to internal waves, particularly when the turbulent eddies become too weak to turn over. It is possible, however, to make a practical distinction on the basis of the transport of energy. Turbulent eddies, whether due to convection of heat or shear instability of ocean currents, transport energy in the form of kinetic energy of rotating water-masses. The speed with which the energy moves is, therefore, limited to the speed of the flow which transports the masses. Internal waves, on the other hand, transport energy at a group velocity which is neces- sarily dijfferent (and usually much greater) than the speed of the ocean currents within which they travel. Unlike turbulent eddies, waves would be expected to preserve coherence over a considerable distance. A . Coherence of Isotherm Fluctuations at Separated Localities A distinction between fluctuations due to internal waves and other less regular causes can, therefore, be made only if observations are made at more than one position in the sea. UflFord (1947) appears to have been the first to make observations of this sort. He studied internal waves by making repeated lowerings of bathythermographs from bow and stern of a ship and from three separated ships. In all cases, involving a total of a few hours of observing time, he found similar fluctuations at each station with a phase shift appropriate to the phase velocity of internal waves. Much longer series of isotherm-depth fluctuations have been measured at three positions on and near the NEL oceanographic tower by use of isotherm followers. Two days of data have been analyzed by statistical methods (Fig. 25). The sea bottom is practically level at 18 m depth within the observing region and featureless off-shore. Therefore it is to be expected that the waves which were incident from seaward would be uniform statistically at the three stations. The observed spectra of depth fluctuations were the same (within expected statistical fluctuations) at each station ; only their mean value is shown in Fig. 25. The spectrum shows the same monotonic decrease with frequency as the temperature spectrum from Castle Harbor. (At frequencies above 0.4 cycles per minute [c/min] the spectrum becomes flat. This presumably shows the effect of a random recording error of 0.12 m r.m.s.) Vaisala's frequency in the water column had a maximum of 0.65 c/min near the surface, decreasing to 0.15 c/min near the bottom. The analysis extends up to 0.5 c/min but significant coherence is detected only at frequencies well below the mean value of Vaisala's frequency. Under these conditions the phase 758 [CHAP. 22 y\. ^■\ 1.0 ^.— '"•-v.. ■■s.^ '- N.,^ 0.1 \ "'^•'---. 0.01 0.25 CYCLES PER MINUTE Fig. 25. Internal waves observed off Mission Beach, California, 5-7 Aug., 1959. Top panel : Vaisala's frequency as a function of depth. Second panel : mean spectra of isotherm follower oscillations. Third panel : coherence R with horizontal line showing 95% confidence level. Bottom panel : phase shift 0. Small dots : i?21 and 021- Large dots : i?32 and 6>32. Open circles: R^i and 6>3i. Insert: Plan of observing positions 1, 2, 3. Wave normals are incident at angles with respect to the lines connecting positions i, j. i I SECT. 5] INTERNAL WAVES 759 velocity becomes almost independent of frequency (for frequencies well above the inertial period) and one expects the phase shift between stations to be linear with frequency. The observations show just this relation. From the observed phase shifts, the mean direction of approach (ai2= — 1.5°) is easily found. The coherence was generally high at low frequencies and decreased at higher frequencies. The cause of this behavior is threefold: (1) Internal waves do not always come from the same direction ; when the observing stations are many wavelengths apart (i.e. at high frequencies), interference between waves with different directions of approach will vary the phase difference between stations. This reduces coherence. (2) The phase velocity of internal waves of a single frequency varies from time to time because of changes of mode, changes of density structure in the water, changeable tidal currents and effects of finite amplitude of waves. Again, the main effect is to make the phase shift at high frequencies variable and the coherence is reduced. (3) Irregular fluctuations due to turbulence. We shall estimate each of these effects separately, B. Relation of Coherence to Beam Width Suppose vertical displacements ^i{t) and ^2(0 due to internal waves of a single mode are observed by two observers situated at (0, 0) and {x, y). The relations between the two auto-correlations, the cross-correlation and the directional energy spectrum may be calculated by methods indicated in Chap. 15, page 578 and equation (6), provided the waves move without refraction, generation or decay. Let E{f, 99) represent the directional energy spectrum (see Chap. 15, page 571) which gives the relative energy of waves as a function of frequency and the direction of propagation, 99, measured from the x axis. Then one finds that the coherence R and phase lead of ^2 over ^1 are given by R exp(i^) — S~'^ E{f, cp) exp[ — i(\.2 the coherence oscillates with decreasing amplitude. The opposite extreme occurs for a narrow pencil beam of radiation. When A(p<^ 1 then i? exp(i^) = {Acpy^ eyi^Y — ixk{\ — \(p^)'\coQ{ykq))dcp-\-0{q)^) Jo = {Arpy'^A eyi^{ — ixk), where A = eji^{\ixkcp'^) cos {ykcp) dcp jo can be evaluated in terms of Fresnel integrals. Observed values are shown in Fig. 27 compared to computed values which take into account the reduction in coherence associated with finite angular SECT. 5] INTERNAL WAVES 761 beam width of the waves. In the computations it has been assumed that the phase velocity is 22 cm sec~i, independent of frequency, and that the center of the beam is directed at angle ai2= —1.5°, in accordance with the observed phase shifts shown in Fig. 25. The beam width Acp is assumed independent of frequency. The observed coherences are all smaller at frequencies between 0 and 0.05 c/min than is consistent with any value of beam width. By comparison of observed coherence between points 2 and 3 for frequencies between 0.05 and 0.2 c/min, an upper limit to the beam width of Acp = 0.4 radians is indicated. Using this value for Acp the computed coherence between points 1 and 2 scarcely decreases with frequency, while the observations show a decrease to "noise level" at /= 0.17 c/min. Therefore processes other than the finite beam width are required to reduce E12. C. Reduction of Coherence by Variable Phase Velocity The phase velocity of internal waves will be variable if the density gradient changes, if the waves are carried on variable currents, or if the wave steepness is large and variable. Let c be the mean phase velocity and Sc its variation from the mean. Then the change of phase difference 86 between points separated by L, due to changes of phase velocity, is less than or equal to 277/Z|c-i-(c + Sc)-i|. The observed phase velocity at low frequencies was 22 cm sec~i. It may be estimated that |Sc|<10 cm sec~i from all causes. Therefore 8^^180° for fL ^0.25 cm sec~i. The coherence between observing points 2 and 3 separated by 153 m would, therefore, not be much affected for frequencies well below 0.1 c/min. D. Reduction of Coherence by Turbulence If turbulent motions are superimposed on the internal waves they can reduce coherence at any frequency. If the spectrum due to turbulence and waves are respectively St and Sw, then it can readily be shown that coherence is reduced in the ratio Sw{St+Sw)~^ if, as seems reasonable, the turbulence at the two points of observation is not correlated. Since there is no frequency limitation in this relation, the reduced coherence at low frequencies can be due to this cause. The observations give i? = 0.75 for/< 0.05 c/min. From this value we estimate StlStv=0.3. E. Summary The high frequency reduction of coherence of isotherm depths observed at stations 2 and 3 could be partly due to the arrival of internal waves from a 762 cox [chap. 22 variety of directions. The half angle of the beam of directions cannot, however, be larger than 0.4 radians. The coherence observed between other pairs of stations drops off with frequency more rapidly than such a beam of directions would permit. Changes of phase velocity of a few centimeters per second by various causes are fully capable of causing the additional decrease of coherence. At frequencies below 0,05 c/min, neither the directional properties nor changes in phase velocity of internal waves can account for the observed coherences. By elimination, one is forced to conclude that irregular motions, perhaps associated with turbulence, are responsible. Part of the irregular motion can be contributed by instrumental noise in the thermocline followers, but the contribution of this noise appears to be one hundred times below the observed spectral density at low frequencies, while the observed coherence indicates that the non-wavelike fluctuations have a spectral intensity one third as large as that of the internal waves. The demonstration that fluctuations of isotherms are non-coherent within a comparatively small number of wavelengths, and that the coherence even at a small fraction of a wavelength is well below unity, emphasizes the importance of making an unambiguous distinction between internal waves and other, less coherent motions. The origin and nature of the latter are obscure. References Arthur, R. S., 1954. Oscillations in sea temperature at Scripps and Oceanside piers. Deep- Sea Res., 2, 107-121. Defant, A., 1932. Die Gezeiten und inner Gezeitenwellen des Atlantischen Oceans. Wiss. Ergebn. Deut. Atlant. Exped 'Meteor', 1925-1927, B7, Heft 1, 318 pp. Defant, A., 1950. Reality and illusion on oceanographic surveys. J. Mar. Res., 9, 120-138. Dietz, R. S. and E. C. LaFond, 1950. Natural slicks on the ocean. J. Mar. Res., 9, 69-76. Eckart, C, 1960. Hydrodynamics of oceans and atmospheres. Pergamon Press, London. Ekman, V. W., 1904. On dead water. Sci. Results, Norweg. N. Polar Exped., 1893-96, 5, 1-152. Fjeldstad, J. E., 1933. Interne Wellen. Geofys. Puhlikasjoner, 10, 3-35. Forbes, A., 1945. Photogrammetry applied to aerology. Photogrammetric Eng., 2, 181-192. Groen, P., 1948. Contribution to the theory of internal waves. Koninkl. Ned. Met. Inst, de Bilt. No. 125 Med. en Verh., 2, 11. Haurwitz, B., 1954. The occurrence of internal tides in the ocean. Arch. Met., A7, 406-424. Haurwitz, B., H. Stommel and W. H. Munk, 1959. On thermal unrest in the ocean. Rossby Memorial Vol., Rockefeller Inst. Press, New York, 74-94. Helland-Hansen, B. and F. Nansen, 1909. The Norwegian Sea. Rep. Norweg. Fish. Mar. Invest., 2, Kristiana (Oslo). Kullenberg, B., 1935. Internal waves in the Kattegat. Svenska Hydrog. Biol., Komm. Skrifter, Ny Ser. Hydrog., 12. LaFond, E. C, 1949. The use of bathythermograms to determine ocean currents. Trans. Amer. Oeophys. Un., 30, 231-237. LaFond, E. C, 1951. Processing ashore. U.S. Hydrog. Office Pub. No. 614, 1-114. LaFond, E. C, 1959. How it works — the NEL oceanographic tower. U.S.N. Inst. Proc, 85, 146-148. LaFond, E.G., 1959a. Sea surface features and internal waves in the sea. Indian J. Met. Geophys., 10, 415-419. SECT. 5] INTERNAL WAVES 7G3 LaFond, E. C, 1959b. Slicks and temperature structure in the sea. U.S.N. Electronics Lab. Rep. 937. LaFond, E. C, 196L Isotherm follower. J. Mar. Res., 19, 33-39. LaFond, E. C. and P. Rao, 1954. Vertical oscillations of tidal periods in the temperature structure of the sea. Andhra Univ. Mem. Oceanog., 5, 109-116. Lamb, H., 1945. Hydrodynamics, 6th Ed. Dover, New York. Lee, O. S., 1961. Effect of an internal wave on sound in the ocean. J. Acottst. Soc, 33, 677-680. Lee, O. S., 1961a. Observations on internal waves in shallow water. Limnol. Oceanog., 6, 312-321. Lee, O. S. and E. C. LaFond, 1963. On changes in isotherm depth in shallow water off San Diego. Submitted to J . Mar. Res. Love, A. E. H., 1891. Wave motion in a heterogeneous heavy liquid. Proc. Land. Math. Soc, 22, 307. Mortimer, C. H., 1952. Study of internal waves in Lake Windermere. Phil. Trans. Roy. Soc. London, B236, 355-404. Munk, W. H., 1941. Internal waves in the Gulf of California. J. Mar. Res., 4, 81-91. Reid, J. L., 1956. Observations of internal tides in October 1950. Trans. Amer. Geophys. Un., 37, 278-286. Revelle, R. R., 1939. Sediments of the Gulf of California. Bull. Geol. Soc. Am,er., 50. Richardson, W. S., 1958. Measurement of thermal microstructure. W. H.O.I. Ref. No. 58-11. Rudnick, P. and J. D. Cochrane, 1951. Diurnal fluctuations in bathy thermograms. J. Mar. Res., 10, 257-262. Seiwell, H. R., 1937. Short period vertical oscillations in the western basin of the North Atlantic. Contrib. No. 137, W.H.O.I. (Papers Phys. Oceanog. Met., Mass. Inst. Tech. and Woods Hole Oceanog. Inst., 5, 1-44). Shand, J. A., 1953. Internal waves in Georgia Strait. Trans. Amer. Geophys. Un., 34, 849-856. Sonar Data Division, Univ. of Calif., Div. War Research, 1945, Thermal structure of Sweetwater Lake. UCDWR No. M434, File No. 01.95. Sverdrup, H. U., 1940. The Gulf of California. Assoc. Oceanog. Phys. Proc. -Verb., No. 3, 170-171. Sverdrup, H. U., M. W^. Johnson and R. H. Fleming, 1942. The Oceans, their physics, chemistry and general biology. Prentice-Hall, New York. Ufford, C. W., 1947. Internal waves in the ocean. Trans. Amer. Geophys. Un., 28, 79-86. Ufford, C. W"., 1947a. Internal waves measured at three stations. Trans. Amer. Geophys. Un., 28, 87-95. Wedderburn, E. M., 1909. Dr. O. Pettersson's observations on deep water oscillations. Proc. Roy. Soc. Edinburgh, 29, 602. Woodcock, A. H. and T. Wyman, 1946. Convective motion in air over the sea. Ann. A\F. Acad. Sci., 48, 749-776. Zeilon, N., 1913. On the seiches of the Gullmar Fjord. Svenska Hydrog.-Biolog. Koynm. Skrifter, 5. Zeilon, N., 1934. Experiments on boundary tides. Goteborg VetensksamJi. Handl. Folj. 5, B3, No. 10. 23. TIDES W. Hansen 1. Introduction Newton was the first to give a physical explanation of oceanic tides. Later Bernoulli, Laplace, Hough, Airy, Kelvin, Darwin and Poincare set up a classical theory of tides, the aim of which was to understand qualitatively and quantitatively this natural phenomenon. To some extent this theory was completed by the work of Proudman and Doodson. The theory is of special importance since observations are only available from coastal areas. Thus the theory may be helpful in gathering information about the tides in the open sea. This theory is based on the hydrodynamical differential equations, consisting of two equations of motion and the continuity equation. The task is to develop mathematical solutions of this system, if possible in the shape of analytical terms. The problem becomes more simple if the system is a linear one, and the tide generating forces are developed in series consisting of harmonic terms. The frequency a as well as amplitudes and phases are known from astronomical data. The potential V of the tide -producing forces or the equilibrium elevation | = - Vjg with gravitational constant g is given by : 1 = 21" cos {a^t-K^) = 2 I"'! cos (7,^ + 1., 2 sin a^t. (1) V There are three groups of periods determined by the frequency ct. They are long-period, diurnal and semi-diurnal tides. The principal harmonic terms are : M2: principal lunar semi-diurnal constituent, frequency 28.98 degrees per mean solar hour. S2: principal solar semi-diurnal constituent, frequency 30.00 degrees per mean solar hour. K2 : luni-solar declinational semi-diurnal constituent, frequency 30.08 degrees per mean solar hour. Ki : luni-solar declinational diurnal constituent, frequency 15.04 degrees per mean solar hour. Oi: lunar declinational diurnal constituent, frequency 13.94 degrees per mean solar hour. As a result of tidal observations it has been established that, normally, the most important harmonic term is the lunar semi-diurnal constituent, M2. In order to avoid mathematical difficulties, theoretical investigations prefer the K2 — or, in the case of diurnal tides, the Ki — constituent. Doodson (1921) has given the complete tidal potential (see also Bartels, 1957). A more detailed representation of tides has been given by the following authors : Defant (1957, 1961), Doodson (1958), Lamb (1932), Proudman (1952), Sverdrup et al. (1955) and Thorade (1931). [MS receivedJuly, I960] 764 SECT. 5] 76^ 2. The Hydrodynamic Equations and Their Application to Tidal Problems The elevation of the tide above the mean sea surface, ^, and the components of tidal currents, u, v, are also set up by harmonic constituents: ^ = ^0 cos{at — K) = ^1 cos at + ^2 sin at ~ ^e"*"' u = Uq cos {at-Ku) = ui cos at + U2 sin at ~ we-'''^ V = Vq cos {at — Kv) — vi cos at + V2 sin at ~ ?;e~*'"^ Q-iat = cos o-^ + ^ sin at; i = -y/ — 1 . Amplitudes and phases, for instance f and /<:, are called harmonic constants ; normally they are derived from long-term records of sea-level. These records are also used in the derivation of tidal currents. The introduction of these kinds of harmonic constituents simplifies the mathematical treatment of tidal prob- lems by means of the hydrodynamical differential equations. These are 8u 8u 8u . „, , 8(^ — i) dv dv 8v . „, ^ 8U-i) (3) in cartesian, and 8u ,^^ - 2co sin m-v + B(^) +1 cosm-— U-i) = 0 8t ^ a ^ 8\ T^ + 2a} sin q)-u + R^f) +-■ — (^ - 1) = 0 Ct ^ a 8(p ^ ' 1 ^i bt a cos -[(A + ^)w]+— [(/t + |)vcos99] - 0 (4) in spherical co-ordinates without convective terms. In these equations it is assumed that the components of velocity, u, v, are mean values from the sea floor to the surface, and furthermore that the density is constant in the ocean. Neglecting the convective term and the variation of ^ compared with mean depth h and using the relations R(x) = r-u. Riy) ^ r-v. (5) the equations become linear. Equations (2), when introduced into the system of differential equations, 766 HANSEN [chap. 23 make it possible to eliminate the time variable. Three partial differential equations with only space derivatives result: {-iG + r)u-fv + g— (l-l) = ( ^_ia + r)v+fu + gy (i-l) = 0 \ (6) The next step is to eliminate the components of velocity, u, v. The following is an equation of sea-level -[/2 + (r-io-)2]| = 0 0 (7) , 1 ^f (dh h 8x \8x f 8h\l 8^ (8h ^ f 8h\ ia r — ia 8yj h 8y \8y r — ia 8xJ ' gh{r — io 8^ 8^ A corresponding equation is found for spherical co-ordinates. This partial differential equation is of the second order and of elliptic type. That means, ^ is uniquely determined in the interior of an area if on the boundary | or the derivative of f or a combination of these functions is known. There are numerous methods for solving this type of equation. The solutions depend on the shape and depth of the ocean and on the parameters of the equation. Only in very simple cases is it possible to give the solution in analytical terms of well- known mathematical functions. In principle these solutions have to be ascert- ained for each ocean or sea separately. The amount of work increases with the complication of the area under consideration. For this reason, in the classical theory of tides, the investigations are restricted to geometrically simple basins. These are : an ocean covering the whole earth, oceans bounded by two parallels of latitude and oceans bounded by meridians. In order to get some mathematical simplifications, it was assumed that the depth was a function of latitude and/or longitude. Considerable complications in these equations arise from the Coriolis force. (This force also complicates the calculations of the tides in a rectangular basin with constant depth.) Elevations of tides and components of tidal currents, u,v, are only representable by infinite series, as has been shown by Taylor (1920). An analytical solution is only known for a rotating basin bounded by a circle (Lamb, 1932). The elevation $ is represented by Bessel functions. The results of this classical theory do not give information about the tidal oscillation in an actual ocean or sea. It may be possible to approach the solution of this problem by using modern electronic computers, although this method has not yet been applied. There are now a number of methods allowing one to tackle this problem in a more direct manner ; these are not restricted to the simplifications of the classical theory which demand the linearity of the hydrodynamic equations. SECT. 6J TIDES 767 Within the present Hmitations tidal research received a remarkable stimula- tion from the work that has been done in connection with seiches. This special kind of oscillation in Lake Geneva had been observed and discussed by Forrel. Chrystal (1904) gave a hydrodynamical explanation of this kind of wave, and his theory was in good agreement with observations. Sterneck (1920) and Defant (1919) recognized that a transition and extension of this theory to tidal prob- lems should be possible and successful. To avoid mathematical difficulties, they discussed only elongated channels of variable depth, width and cross-sections. The essential idea of these scientists was not to attempt to find an exact mathematical solution of the problem but instead to concentrate all efforts toward an approximate solution of numerical type. This was done in the following manner. In the equation of motion the derivative with respect to the space variable x was replaced by finite differences : the continuity equation was integrated with respect to the same variable. In effect, therefore, the channel was divided into a number of intervals each with constant depth and width. The numerical computation off and u was possible without any difficul- ties. Defant applied his method to a large number of elongated seas, and the results he obtained are in good agreement with observations. The application of these numerical methods, based on finite differences, appear in principle to be a renunciation of an exact mathematical solution. But in this connection it should be remembered that the hydrodynamical differential equations are derived by similar processes from difference equations (Lamb, 1932). So far the idea of replacing the differential quotients by finite differences seems to be useful and correct. But it should always be kept in mind that, before applying this difference method, it is necessary to prove the degree of approximation and its numerical stability. In this respect there are still uncertainties. It therefore seems desirable to start from the hydrodynamical equations. Collatz (1955) shows by means of simple examples that the transformation of differential equations into difference equations should be done cautiously. The difference methods can be applied to general problems ; for example, to those concerned with shallow or stratified waters acted upon by tides, wind stress, or other forces. It can be expected that in the future these methods will be used to a greater extent in problems of practical dynamical oceanography, since modern electronic computers now exist to overcome the large amount of calculation. Beside the exact mathematical and the difference methods, there are a number of other proposals to solve the problems of tides in oceans and seas. These problems are usually made easier by linearizing the hydrodynamical equations, but in all cases this is still not sufficient to obtain analytical solutions. Sometimes it is convenient to start with the exact differential equations and at a later stage to continue with numerical methods. Another procedure consists of determining the exact solution for small areas of constant depth and com- bining these into a more general solution for an extended area. Altogether, in the treatment of the tidal problem, there exists a large number of possibilities combining exact and numerical procedures. For example, the characteristic 768 [chap. 23 method applied in a wide field of aerodynamics should be mentioned. Likewise hydraulic or electric models are occasionally used for tidal investigations, especially in estuaries. In any case the importance of each theory should be measured by the possibility of reproducing observed tides and tidal currents in oceans and seas. The final problem of oceanic tides may be formulated as follows. The tides and tidal currents in the actual ocean have to be computed as a whole without using any tidal observations. For this computation only the well-known tidal generating forces, the distributions of depth, and the shape of the coastline along which the normal component of velocity is assumed zero are available. The efficiency of such a theory can be proved by means of tidal observations. It seems necessary for the future to solve this problem completely. To begin with, the work should be concentrated on more simple problems as they arise in adjacent and tributary seas, in estuaries or in special parts of the oceans, taking into account all work which has been done on this subject. In any case it is necessary to gather information on tides along those parts of boundaries which are not coastlines ; usually there is a complete lack of observations, and data must be estimated. 3. Tidal Observation In numerous coastal locations, on islands in the ocean, in open seas and in inlets, bights and tidal rivers all over the world, the elevation of sea-level caused by tidal forces is recorded. These observations are analyzed for each station, assuming that the sea-level ^ can be represented by a series of simple harmonic functions with frequencies a, which are known from the tidal poten- tial F [equations (1) and (2)]. The determination of harmonic constants is made by national authorities. The results are sent to the International Hydrographic 1/4 depth 1/2 depth O 20 40 60 cm/sec 3/4 depth Bottom Lot. = 54" 16' N; Long. = T 46' E. Fig. 1. Observed tidal currents in the neighbourhood of Heligoland. The figures on the arrows give hours before or after the moon's transit of Greenwich. SECT. 5] 769 5m depth 15m depth 30m depth 50m depth 75m depth lOOm depth 125m depth Lat. = 48°28'N; Long. = 8° 51' W. Fig. 2. Observed tidal currents at the entrance to the Enghsh Channel. The figures on the arrows give hours before or after the moon's transit of Greenwich. 160 140 120 100 80- 60- 40 20 0 Western German Bight Lat.= 54°l7'Ni Long =6° 13' E Deptti 8m Inner Germon Bight Lal.= 54"'l7'N; Long =7° 14' E. Deptti 8m River Elbe Lat.= 54''N; Long.= 8''25'E. Deptti 8 m Fig. 3. Velocity and direction of tidal currents recorded at three stations in the German Bight. Velocity in centimetres per second; direction in degrees. 770 HANSEN [chap. 23 cm /sec -t -S ■* -2 0 ' ? - • « *8 ** *2 0 -2 -* -6 -» .vr Om- * Vkl ■ ■ W- \ ■ S\" ■ \\ H 10 s»* 9 ,- ■ i A >>i^ N 30m- - w- sN ^ ^ ■ ^ ^[ ■ - s ?*^ ^ 10 . a ♦tf *4 "I -4 -6 -« • < " e CJ i^ ^ N 500 m: V ^ •i.. : ; [ \ \ 7 ; \ \ \ \ : : ^* ^ k — ^ V > i -0 '■ : \ 1 \ 1 \" ^ ^ \> 1 ; \ > *>-^ v- J'O: a ': '12 H - \ '" N 2500m': ': ; ,1 : ': ■^s if : ':W- \ -0 ; \s : n i^ ; ': 70 ; \ : \ ,,, s ; -8 ■12 -16 -20 *20 i-lS *12 ■16 -12 -B -* Fig. 4. Ellipses of tidal currents (M2 constituent) in the South Atlantic 'Meteor'' Anchor Station No. 36. Lat. = 28° 8'S., long. = 19° 21'W. (After Defant, 1932, Fig. 78.) Bureau in Monaco ; this institution publishes current hsts of all these harmonic constants (at present from about 3000 places). The presently available data refer to harbours at the continental coast and on islands. From the open oceans no harmonic constants are available. A small number of observations is known from the shallow waters of marginal seas, but SECT. 5] TIDKS 771 o o H W X! J .t^ 1 rl be O C O 05 00 00 C^ t- id lO CD l6 ^' lO t^ ■* 00 1— ( -^ (^j 05 CD 00 fO CD oi CD aO (M O 00 C^l »M Tti CO 00 00 r-^ r^ id t^ -"t lO 'M CO CD lO »d id "d id CD c^ o CO CD CO (M C^ CO CO -H ,^ Cq -H CO O Tt< O ■* CO lO rt (M ■<#•<* -H Tf< CO 00 ic r^ CD CD CO ■>* ic ■* 10 CS '^7^ SS 03 03 0 03 ^J- 03 ^ . cS X t* c 0^ 0 0 -fcj cS CD C 03 0 eg ^ 3 0 .^ >. o fe (1h Ph J fq J 772 [CHAP. 2.3 normally these are not very valuable. The data from coastal areas are not all of the same degree of accuracy. Sometimes the duration of observation is too short, in other cases they are disturbed by local effects and, therefore, they are not representative for the open sea (Table I contains several harmonic constants from the Atlantic Ocean). The same is true for observations of tidal currents where there is only a very small number of observations from the open sea, chiefly performed by research vessels. In adjacent and marginal seas the situa- tion is somewhat more favourable ; in the North Sea in particular there are a considerable number of observations of tidal currents. The following figures demonstrate the behaviour of tidal currents from surface to sea bottom in the neighbourhood of Heligoland and in the English Channel. Fig. 3 contains velocity and direction of tidal currents at three stations in the German Bight approaching the Elbe estuary. In Fig. 4 the elhpses of tidal currents from anchor station Meteor No. 36 ((p = 28° 8'S, A= 19° 21'W) in the South Atlantic are drawn up. These figures show, consequent upon the influence of friction and Coriolis force, a variation of tidal ellipses with depth. Table II contains information about tidal currents in the deep Atlantic Ocean. Table II Observed Currents in the Deep Sea Maximum Velocity Observed velocity of the of Vessel Station Depth, maximum semi-diurnal residual no. m velocity, tidal current, current, cm/sec cm/sec cm/sec Meteo7- 254 2000 25 5 12 Meteor 241 2000 13 3 5 Meteor 147 2500 16 9 0 Meteor 36 2500 19 10 1 Meteor 288 3000 8 — 4 Maximum velocity of the diurnal D 1000 11 4 tidal current Armauer Hansen 7 E 1000 12 9 3 C 1000 — ■ 3 — Neutrally-buoyant up to 12 10 — float (Swallow) 1500 SEPT. ")] TIDKS 773 4. Tidal Charts Tidal charts give information about the geographical distribution of harmonic constants. Normally the very important M2-constituent is represented; K2, Ki and Oi are also sometimes given. In these charts the co-range lines (2-^^=: constant) and the co-tidal lines (/c^/cr^ = constant) are drawn. Occasionally there are also charts with lines of constant |i,^ and constant ^2,^. The range, that is the difference between high- and low-water, is 2^^, time of high-water is t = K^jay. The elevation of sea-level at time ^ = 0 is ^1, ^ and ^2, v is the value at time t = {Kv + Trl2)lay. The tidal current expressed as a vector with components u, v runs in a tidal period through an ellipse which is determined by wi, vi, uo, V2. Major and minor axes, direction of these axes and time to of extreme velocity of this ellipse are : (1/\^2)(W1- + U2- + Vi^- + V2- ± [(Wi2 + W2- + Vr + V2")- - 4(^1^2 - ^l2Vl)-]'^}'K tan 28 = 2{'UiVi+uoV2)l{ui" + U2~ + vi" — V2"), tan 2(7X0 = 2{uiU2 + viV2)l{ui~ + vi'^ — U2^ — V2"). The tidal charts are characterized by amphidromic points, in which the range becomes zero and the co-tidal lines converge. In the same way there are special points on those charts where the elements of tidal current are ellipses. If the minor axis becomes zero, the current is an alternating one ; if minor and major axes are the same, the tidal current is circular. Charts with these kinds of elements of tidal ellipses seem only to exist for the Irish Sea (Bowden, 1955) and for the North Sea (Hansen, 1952). Fig. 5 gives the direction and amount of the major axes of the semi-diurnal M2 tidal current ellipses in the North Sea. Fig. 6 contains the direction and amplitude of tidal currents for the Irish Sea. Atlases of tidal currents published by national authorities for navigational purposes normally contain direction and velocity of tidal currents for every hour before and after a certain initial time, for instance high water in a harbour, or the moon's transit of the Greenwich meridian. In most simple cases co-tidal and co-amplitude lines are constructed solely with the aid of observations on the coasts and on islands. This kind of inter- polation normally does not allow the distribution of depth to bo taken into account. Especially abrupt variations of depth are found along the continental shelf and these may influence the development of tides and tidal currents in large extended areas. In such cases it is difficult to decide to what degree coastal observations are representative of the open ocean. A similar question arises in connection with oceanic islands. Here too the tides may be influenced by the depth in the neighbourhood. As there are no observations, it is necessary to consult theoretical work to get some idea of what may happen in such cases. Thorade (1926) examined the behaviour of waves passing the continental slope; Proudman (1925) investigated the de- formation of tidal lines caused by islands, capes, bights, etc. In some of the 774 HANSEN [CHAP. 23 50° 6° 8° 10° 52° 14° Fig. 5. Direction and amount of the semi-diurnal M2 tidal current ellipses in the North Sea. SECT, nj Fig. 6. Arrows give the direction of maximum flood stream; numbers on full lines give amplitude of tidal currents at springs in knots in the Irish Sea. (After Bowden. 1955, Fig. 16. By courtesy of the Controller of H.M. Stationery Office.) existing tidal charts these types of results are evaluated. To complete the in- formation which may be available on tides in the open sea, it is clearly desirable to use observations of tidal currents. Proudman and Doodson (1924) developed a method and applied it to the North Sea. Later on, Thorade (1935) applied it to the current observations by Meteor in the South Atlantic. It is important to bear in mind that the draughting of tidal charts based solely on coastal data becomes more and more hypothetical with increasing width of the ocean. This may be the reason that, in the open ocean, most of the 776 [CHAP. 23 Fig. 7. Elevation of sea-level caused by semi-diurnal M2 tide at the time of the moon's transit of Greenwich. ^1 in cm. SECT. 5] 777 50" 6° Fig. 8. Height of sea-level caused by semi-diurnal M2 tide three lunar hours after the moon's transit of Greenwich. ^2 ^ cm. 778 [chap. 23 authors give only co-tidal lines. In marginal and adjacent seas of smaller area, co-tidal and co-range lines have been drawn for all principal tidal constituents. Defant (1957) published a list of most of the tidal charts constructed for the sea. Those best known by observation are the tides in the North Sea, in the 100 Fig. 9. Co-tidal and co-range lines for constituent M2. Numbers on the full lines give the time of high water after the moon's transit of Greenwich. The numbers on the broken lines give the range in metres. (After Hansen, 1956.) English Channel and in the Irish Sea. For these areas co-range and co-tidal charts have been drawn. Charts with constant ^1 and ^2 for the North Sea are given in Figs. 7 and 8. The tides in the Mediterranean, especially in the Adriatic, in the Red Sea, Persian Gulf and Gulf of Mexico and in the Baltic and a number of marginal seas in the Indian and Pacific Oceans had been investigated. SKCT. 5 1 TIDES 779 Co-tidal lines have been sketched for all oceans ; a general survey is given by Doodson (1958). The most recent chart is given by Villain (1951). World charts with co-range lines do not yet exist. The tides of the Atlantic are discussed in many papers ; besides coastal observations there are some deep-sea current measurements. Defant (1932) treated the Atlantic as a one-dimensional channel and compares the results of his computations with observed data. Fig. 9 con- tains co-tidal and co-range lines for the Atlantic. This has been the first attempt Fig. 10. Co-tidal and co-range lines for constituent K2. The numbers on the full lines give the time of high water in hours, which are one twelfth of the period, the time origin being on the standard meridian ^here the latitude is 32". The numbers on the broken lines give the amplitude H in cm. (After Fairbairn, 1954, Fig. 6.) to determine the tides of the open ocean by help of difference methods using the harmonic constants on the coast. Fairbairn (1954) has given co-range and co-tidal lines for the Indian Ocean north of the equator (Fig. 10). 5. Classical Theory Laplace extended Newton's theory of tides considering inertia as well as the rotation of the earth and, by this, he developed a dynamical theory which interprets the tides to be an oscillation of water-masses. Laplace considered the diurnal tide Ki. Later Hough (1897) expanded Laplace's theory. Airy (1842) treated the tides in zonal canals which go round the world or are of finite length. Goldsbrough (1913, 1914) treated tides in oceans bounded by parallels : a polar sea, zonal canals and a small canal, limited by parallels. When attempting 780 [chap. 23 exact solutions of these problems considerable mathematical difficulties im- mediately arise. The results are interesting in so far as they make clear the influence of the geometry and of the depth upon the development of the tides. A comparison of theoretical solutions with the observed oceanic tides is in- appropriate because of the considerable deviations between the shape of the models compared with the existing oceans. Fig. 11. Co-tidal and co-range lines of Ki constituent in an ocean bounded by meridians 180° apart. (After Doodson, 1935, Fig. 3.) ooooo o Fig. 12. Co-tidal and co-range lines of K2 constituents in an ocean, mean depth 2.75 miles, bounded by meridians 180° apart. (After Doodson, 1938. Fig. 21.) Fig. 13. Semi-diurnal tide K2 in a quartre of an ocean 60° wide, bounded at 62.2"N and S. (After Rossiter, 1958a, Fig. 5.) Important progress was made by Proudman and Doodson (1927), who treated the oceans bounded by meridians on the non-rotating and on the rotating earth. Doodson made calculations from a great number of models and found out that the positions of the co-tidal and co-range lines depend very much on depth. In Fig. 11 the diurnal constituent Ki and in Fig. 12 the semi- diurnal constituent K2 are shown for an ocean which covers a hemisphere. JSKCT. 5J TIDES 781 The tides of tlie K2-constituent of an ocean which is bounded by parallels of longitude separated by a distance of 60° and by the parallels of latitude 62.2°N and S were evaluated by Rossiter (1958) using the difference method (Fig. 13). In principle, when using this method, it is possible to take into consideration the exact form of the coasts and depth distribution. Poincare (1910) developed an elegant mathematical theory of the tides Avhich is principally applicable to the oceans ; practical applications of this theory are not kno^n. 6, Numerical Methods for Ascertaining the Tides and Tidal Currents. Boundary- Value Problems Until recently the classical theory could not be applied to the true oceans. Now, however, considerable scientific and practical interest exists in ascertain- ing the tides of the sea quantitatively. This is especially true for the tides of the open oceans, but nevertheless knowledge of the tides in marginal and adjacent seas and estuaries plays a great part in the work of coastal engineers. Therefore, it is to be understood that experiments have been undertaken to solve these problems without using the methods of the classical theory. As already mentioned, Defant was one of the first to renounce mathematically exact solutions of the hydrodynamical equations and content himself with numerical approximations. Defant's idea of replacing the differential equations by difference equations is, therefore, particularly efficacious, because this principle can be applied to the more general and complicated conditions found on the Earth's surface. The special method of Defant is characterized in the one- and two-dimensional cases by the elimination of the time f with the help of a time factor e"'"'. Starting A\ith the linearized hydrodynamical equations, there result in the case of a one-dimensional channel two equations from (6), the equation of motion and the equation of continuity, d {-ia + r)u + g—{^-^) = 0 ■ (8) -i(jBs^+ ^{Qu) = 0. B is the width and Q is the cross-sectional area, both functions of x and |. In a two-dimensional case there are three equations, two equations of motion and the continuity equation as written down in equations (6). In these systems the velocity components may be eliminated, and the result is one equation for f in each case d ial — ia + r)B^ + g t- ox dx = 0 (9) 782 HANSEN [chap. 23 and, for two dimensions, equation (7). These differential equations are of the second order. This means that | is uniquely determined in an area G (Fig. 14) if the value of ^ is prescribed on the boundary C or Ci and C2. From this the possibility arises of computing the tides in a channel, closed at both sides, or in a landlocked ocean or sea without any aid from observations. The boundary condition requires the normal component of velocity at the boundary to be zero ; or, expressed in another way, the direction of current must be parallel to the direction of the coastline. In this sense, only the ocean, lakes and a few adjacent seas, as for instance the Baltic, are landlocked. In all other cases a sea has some communication with neighbouring seas, so that it is necessary to set up artificial boundary lines C2 as shown in Fig. 5. Now the tides are uniquely determined within the area O by prescribing the sea-level | on C2, while the normal component of velocity must be zero on Ci. But in this connection difficulties arise, as there is a lack of observations in the open sea, and, there- fore, it is highly desirable to record the elevation of sea-level in such areas. (a) (b) Fig. 14. Scheme of (a) landlocked ocean: C boundary, O open ocean; and (b) ocean partly bounded by coastal boundary C\ and partly limited by an artificial boundary C^- Tackling the tidal problem in such cases, it becomes necessary to estimate these values on the artificial boundary C2. Sometimes it is possible to get in- formation to some extent by evaluating observations of tidal currents, as, for instance, in the northern entrance to the North Sea. A number of mathematical theories exists for solving the above mentioned systems of differential equations (Courant and Hilbert, 1924). A numerical method is briefiy described below ; this is a generalization of Defant's method. Problems of this type are sometimes named boundary- value problems. As the computation in elongated channels gives rise to no difficulty, it seems to be sufficient to discuss only the two-dimensional problem. The sea G under consideration (Fig. 5) is covered by a grid with mesh-size I. The boundary of this grid should approximate the boundary Ci of the sea as accurately as possible. On the other hand the number of grid points increases with decreasing mesh-size I. The final choice of mesh-size I should consequently be made taking into account the desired degree of approximation and the amount of SECT. 5] 783 computation which can be undertaken. To get a uniquely determined solution of tlie problem in all grid points on the boundary, the elevation of sea-level ^ is prescribed. Now in each grid point the differential equation is replaced by an equation of finite differences : {x,y)-^-^f{x,y) Yyfi^^y) f{x + l,y)-f{x-l,y) 21 f{x,y + l)-f{x,y-l) 21 (10) ^2 ^2 fix + 1, y) +f{x - 1, y) +f{x, y + l) +f{x, y-l) -m^,y)- On introducing these terms into the differential equation (7) or (9) a system of linear equations arises. The number of equations is equal to the number of unknown values and in principle there are no mathematical difficulties in solving them. This method is applicable to examples in either one or two em 100 _»_,, 1 60 60" . " 40 20 - ^ 0 20 . \ iO - '\ 1 60 80 : \ ' 1 100 120 J40 lao N - lO c o 1 Fig. 15. Red Sea — range of M2 constituent. (After Defant, 1919.) .... observed xxx calculated dimensions. Moreover the computation of tides in a one-dimensional channel can be arranged in such a way as to make it possible to express the values of ^ in the grid points by the boundary values in a direct manner. This has been done by Defant. After this method he investigated a large number of elongated channels and seas, and especially he obtained remarkable agreement between observed and computed tides in areas of very small width. Fig. 15 gives the result for the Red Sea (Defant, 1919). In two-dimensional cases the amount of computing work increases with the number of grid points, and, in addition, it becomes complicated in cases of small values of the determinant of the above-mentioned system of linear equations. As already noted, the choice of the grid-size is determined by the demand that all essential features of the tides in the sea under consideration are taken into account. On the other hand there is an upper limit for the number of grid points caused by the amount of computa- tion work. Normally, iteration processes have been used for solving the equa- tions. This method has been applied to the tides in the North Sea, the Enghsh 784 HANSEN [chap. 23 Channel, the Irish Sea and the North Atlantic Ocean. Holsters (1959) used this method to obtain information in Belgian coastal areas. Rossiter (1958) con- tinued the work of Proudman and Doodson in connection with oceans bounded by parallels of latitude and longitude using methods of finite differences. In this special case he pointed out : "Thus the distribution of a tidal constituent is obtained without the use of observations from a knowledge of the tide generating forces only. The method may be extended to oceans with boundaries of irregular shape." In all other cases, apart from this more theoretical task treated by Rossiter, the magnitude of the M2 constituent is prescribed on the boundary, partly based on coastal observations and partly on estimated values on the artificial boundary. To obtain some idea of the accuracy of these methods applied to actual seas, it is necessary to compare theoretical values and ob- served data. This is possible to a certain degree in the North Sea (Hansen, 1952) and also in the English Channel and the Irish Sea (Doodson et al., 1954). In the North Atlantic Ocean there is only a small number of observations avail- able from oceanic islands. This comparison shows that there is some con- formity, and it seems to be useful to tackle certain tidal problems by means of this method. Fig. 16 gives as an example observed and computed |i and I2 values for the southern part of the North Sea. Boundaries are shown by open circles with crosses and interior points by full circles. The |i and ^2 values are drawn up as vector components. This method may be used to get information on the tides in the open sea, if these are known on the boundary. As already mentioned, this method is usually applied to the principal M2 constituent. In a similar way all other constituents may be ascertained, and with this the tide can be represented by a series of terms like equation (1), but it should be kept in mind that this is feasible only in the case of linear equations. This is true in the deep sea, but in shallow waters the hydrodynamical equations are essentially non-linear. This is the reason why the method discussed above is not applicable to tides in tidal estuaries or in shallow bights. Model trials can solve these problems of shallow- water tides and have been undertaken for both scientists and coastal engineers. Besides hydraulic and electric models often used by engineers, several methods have been developed to compute the tides in such areas as, for ex- ample, the Netherlands. Dronkers (1935) extended the values of tide and tidal currents into power series of the space variable x, and in this way he arrived at solutions of the non-linear equations. Fig. 17 shows the difference between observed and computed sea-level on the Nieuwe Waterweg. Schonfeld applies the characteristic method to tidal problems and compares his results with observations in a hydraulic model. In 1960 Pekeris gave a lecture concerning his computation of the principal constituents of the tides for the ocean as a whole. Starting with the hydro- dynamical equations in spherical co-ordinates, but without friction terms, he introduced a time factor and, after eliminating the components of velocity u and V, he arrived at an equation similar to (7) taking into account the boundary condition — normal flux equals zero — expressed in terms of derivatives of ^. He SECT. 5] TIDES 26— s. I. 786 [chap. 23 had to solve a system of more than 300 hnear equations. This was done by inverting the matrix of this system. The result seems to be in good agreement with tidal observations. Another computation was made by Gohin in 1960 for the North Atlantic. He found remarkable agreement between observations and computation. Fig. 17. Height of sea-level in the Nieuwe Waterweg at Kruiteland. (After Dronkers, 1935, Fig. 2.) Full curve after observations; broken curve after computations. 7. The Application of DifFerence Methods to Initial -Boundary Problems Recently finite difference methods have also been applied to non-linear hydrodynamic differential equations in one and two dimensions. By generaliza- tion of Defant's idea, all derivatives of space and time variables in the system (3) are replaced by finite differences. If for a certain time t — to the elevation ^, the components of current velocity u, V and the external forces are known, the equations (3) allow the time deriva- tives of I, u, V to be determined. This means it is possible to have at least approximate information about these values at a following time t + r, where t is sufficiently small. In this way the variations off, u, v are determined step by step for all time. It must be realized that this process is only a formal one, and it must be ascertained that this method becomes convergent ; in other words, the numerical computation must be stable. This problem differs from the task discussed above (the boundary-value problem) : 1. The system of differential equations is now of the hyperbolic type. In the former case, ^ was the solution of a differential equation of elliptic type. 2. No time factor is introduced; functions are not restricted to simple harmonic terms but an arbitrary time dependence is allowed : the same is valid for the boundary values. SECT. 5] TIOES 787 3. To start the computation not only boundary values are required but also initial values. If friction is taken into account, the final solutions — that means f, u, V after a sufficiently long time — become independent of these initial values. The process to set up a system of difference equations is the following. The sea under consideration, G, is covered by a grid with mesh-size I. All derivatives in space direction x,y are replaced by central differences according to (10). Derivatives in time direction are substituted by forward differences with time interval t, m f{t + r)-f{t) et ^ T Introducing these finite differences into the equations (3) for a grid point with co-ordinates x, y at the time t, it follows that u{t + T, X, y) = ( 1 — T • i?<^) )u{t, X, y) -fTv{t, x, y) -Yi[^\t,x + l,y)-^'{t,x-l,y)l v{t + T,x,y) = [l-T ■B] This difference converges to zero when r becomes small, the real component of a = ai + i-a2 as a friction term is positive and T<2ail{ai'^ + a2^). The result is that a suitable choice of t makes it possible to approximate the exact solution of the differential equation by means of difference methods. In the more general case of (11), the condition of approximation is that the matrix must be a normal one, and the absolute value of eigen values must be smaller than one. Kreiss has shown that it is possible to fulfil this condition in each properly posed problem taking the mean of values in neighbouring grid points. In some cases it is sufficient to take t< l|2^/{gh). 8. Numerical Solutions of Initial-Boundary- Value Problems of Tides in One and Two Dimensions It appeared appropriate at first to apply this difference method to one- dimensional problems. There have been some calculations made for tidal estuaries ; in these regions the equations are non-linear. At the mouth of the SECT. 5] 789 I 0+3+6 +9h +3 +6 +9 +12 +I5h +3 +6 +9 +12 +I5h -9 -6 -3 0 +3 +6h 9-6-3 0 +3 +6h 9-6-3 0 +3 +6h 9 -6 -3 0 +3 +6h Fig. 18. Tidal curves in rivers Ems, Hunte, Elbe and Eider. Full lines, observations at tide gauges; broken lines, values as computed by difference method. 790 [CHAP. 23 river the elevation of sea-level recorded at a gauge is needed. Up-stream the freshwater input must be known ; width and cross-sections must also be pre- scribed. The diiference method gives the distribution of sea-level and current in time and space. The results of a number of examples from rivers flowing into the south-eastern part of the North Sea are shown in Fig. 18 and compared with observations for the rivers Ems, Hunte, Elbe and Eider. The differences Profiles from which r Existing channel profile curves obove and J »„ chonnel development ''".°I,'ni*'" I 13m chonnel devebpment (5) (g) I ^ 0| \(R '60' 70' ■ ie'oT 1 CUXHAVEN OTTERNDORF BRUNSBUTTELKOOC 90 STADERSANO LUHEORT I SCHULAU I WITTENBERCEN ' llJO' 'I' ' l50' ' ' ' 1401 '|iJo' ' I' ' l50' ST, PAULI K HOOf E OVER ZOLLENSPIEKER II I ELBBRUCKEN CR. STACKDRT knllOS 140km CEESTHACHT 12 IS IS Fig. 19. Generalized River Elbe. Upper part : distribution of tidal currents as a function of time. Lower part : tidal depth as a function of time. Middle part : depth distribution in the river. between observations and computations lead to the assumption that this method is also applicable to non-linear problems, in spite of the fact that Kreiss only demonstrated convergence in cases of linear systems. In a more mathematical way Kreiss (1957) treated the non-linear problems in an elongated channel. Rose (1960) investigated tides and tidal currents as functions of length, width and other parameters in numerous artificial canals. Fig. 19 represents the influence of depth on the distribution of elevation and SECT. 5] TIDES 791 currents in an idealized channel with dimensions of the River Elbe. A general representation of methods appropriate for computing the tides in rivers is given by Hensen (1958). These results in the one-dimensional case suggested that an attempt should be made on the two-dimensional problem. A first start was made with North Sea tides using a grid with large mesh-size and a small number of grid points to avoid wearisome computations. Later on, the Swedish electronic computer became available for tide and storm-surge computations with small grid-size in the North Sea (Hansen, 1956; Fischer, 1959). These first attempts were not completely satisfactory although approximate solutions were obtained. Recently this work has been continued with electronic computers of large capacity and again tides and wind-caused motion in the North Sea are being considered. The North Sea is limited by a boundary in the north running from the Firth of Forth to a point north of Bergen on the Norwegian coast. In the south the boundary is situated in the Straits of Dover. On these boundaries the principal M2 constituent is prescribed in the form i = ii cos (yt + ^2 sin at. These values are derived from observations. On all other boundaries (i.e. along the coastline) the values are calculated and the normal component of velocity is zero. Skagerrak and Cattegat are included in the area under in- vestigation, and it is assumed that for all intents and purposes the entrance to the Baltic is closed. The hydrodynamical meaning of this assumption is that there are no currents through these straits and, as on coastal boundaries, the normal component of velocity is zero. In brief, the variations of sea-level and currents in the whole North Sea are uniquely determined by these para- meters ; no observations, other than the ^-values on the northern boundary and in the Straits of Dover, are needed. In the difference equation the Coriolis force and the variation of depth are taken into account. The friction term is determined as follows : The coefficient r has a constant dimensionless value r = 3 x 10~3. This number seems to be applicable to problems in estuaries and in open seas as well as in the ocean. The mesh-size was 1= 18.5 km and the time step t= 147 sec. Con- vective terms have been dropped out of the equation. Investigations in shallow water have shown that these terms are only important in areas with large variations of depth ; this is not the case in the open North Sea. From friction and divergence terms there arise non-linear influences effective especially in regions of small depth. In Fig. 20 the tide curves for spring and neap are drawn up according to observations in Heligoland, and the computed tide curve in the grid-point next to this island is reduced to a uniform scale. In order to obtain information as to the accuracy of computed tides and tidal currents, these have been compared with observed data. In Fig. 21, for each 792 HANSEN [CHAP. 23 Fig. 20. Tidal curves for Spring ( ), neap ( -.-.-.- line computed for M2 constituent. -) after observations in Heligoland. grid-point computed, |i has been plotted along the x-axis and observed ^1 along the y-axis. In Fig. 22 the same has been done with ^2- In both drawings a correlation of observations and computations may be seen. Larger differences are found in the Wash, a small bight on the English east coast where there is a complicated depth pattern. Figs. 23 and 24 show the distribution of error for ^1 and I2 respectively. The x-axis represents the difference between the ob- served and computed values ; the y-a,xis represents the number of occasions between integral numbers of cm/sec when these differences existed. In the case 2.5 20 •./' / 1.5 1.0 X''' 0.5 / 2 5 -20 -15 /' ./' ./ / -10 ./ -0.5 -0.5 -10 -1.5 -2.0 -2 5 Q5 10 15 20 25 Fig. 21. Correlation between observed (vertical axis) and computed (horizontal axis) elevation ^\ of the M2 constituent in the North Sea. SECT. 5] TIDES 703 of li, 89.05% of all differences are located in the interval ± 5 cm. For ^2 the corresponding figure is 88.62%. In both cases the total number of differences are restricted to the interval ± 26 cm with only one exception. The ^2 difference in the Wash reaches 68 cm. The difference method yields not only the variation of sea-level but also the components of currents. In the North Sea there is a relatively large number of records of tidal currents which render 2.0 1.5 1.0 0.5 / I- -1.0 0.5 1.0 1.5 2.0 m -1.0 -■ Fig. 22. Correlation between ob.served (vertical axis) and computed (horizontal axis) elevation ^o of the M2 constituent in the North Sea. possible the comparison between observed and comptited currents. Fig. 25 contains on the a::-axis computed maximum values, and on the ?/-axis observed maximum values of tidal currents. In Fig. 26 the x-axis represents the dif- ference between the observed and computed velocity in cm/sec, the ^/-axis represents the number of occasions on which, between integral numbers of cm/sec, these differences existed. 73.68% of all differences are in the interval ± 5 cm/sec and 91.02% are in the interval ±10 cm/sec. 794 [CHAP. 23 56 i,iiinii,ii 25 20 1 5 1 0 5 0 16.14 47.84 66.86 5 10 15 20 25 L 79. 54-1 L- 84.73—' — 89.05 ' 1 — 100.00% 30 Difference in cm Fig. 23. Distribution of errors for the elevation ^i of the M2 constituent. Horizontal axis: differences of observations minus computations, in cm. Vertical axis: number of differences for 0, ± 1, etc., in cm. -66 -30 -25 -20 -15 -^^ 0 I 21.261 5 10 15 20 25 30 35 -10 45 50 55 60 65 SI. so 7SIS- '-8I.74-' ^ 85.03 ' — 88.62 — ' 96.71 - 99.10 ■ 99.70 lOO.OO*/.- 70 Difference in cm Fig. 24. Distribution of errors for the elevation ^2 of the M2 constituent. Horizontal axis : differences of observations minus computations, in cm. Vertical axis: number of differences for 0, ±1, ± 2, etc., in cm. SKPT. "»] 795 no 100 / 90 y 80 ■ / 70 / ' 60 y i. Hydrog. Paris, 4, 269. VI. TURBULENCE K. F. BOWDEN 1. General Properties of Turbulence A . Reynolds Stresses and Turbulent Transports In the turbulent flow of a fluid, an irregular fluctuating motion is super- imposed, on the general pattern of movement. The velocity measured at a given point in the fluid varies rapidly with time and there are comparable variations from one point to another at a given time. The laws of hydro- dynamics are applicable to the instantaneous distribution of velocities at all points in the fluid and would, in principle, enable the subsequent motion to be determined. To render the problem tractable in practice, however, it is neces- sary to separate the complete flow into mean motion and turbulent motion. While it is possible in this way to deal with the details of the mean motion, the properties of the turbulent flow can only be treated statistically. Let rectangular axes OX, OY, OZ be taken, with OX, OY in a horizontal plane and OZ vertically upwards. Let u', v', w' be the corresponding com- ponents of the instantaneous velocity at a point {x, y, z). Then u' = U + u, v' = V + v, W = W + w, (1) where U, V, W are the components of the mean velocity and u, v, w are the components of turbulent motion. U, V, W may be defined by an averaging process which is carried out over a specified length of time at a given point or over a specified volume at a given instant. The method of averaging and the fundamental interval or volume will depend on the scale of the motion and the aspect of it which is being considered. It was shown by Reynolds (1894), in his pioneer work on turbulence, that the effect on the mean flow of the existence of the turbulent fluctuations of velocity was to introduce additional components of internal stress, given by Pxz = -pU^, Pyy = -PV^, Pzz = - pw"^, _ _ (2) Pxy = — pUV, Pxz = — pUW, etc., where Pxy denotes the stress per unit area on a surface perpendicular to OX in the OY direction, and the bar denotes a mean value taken over the fundamental interval. These stresses, known as the Reynolds stresses, are formally analogous to the viscous stresses arising from molecular viscosity which, in the case of the mean motion, are given by i>.. = 2/x— , p,, = ^(^— + —j, etc., (3) [MS received July, 1960 ] 802 SECT. 6] TURBULENCE 803 where fx is the coefficient of viscosity. For a fuller treatment, reference may be made to Lamb (1932) or Proudman (1953), Since the effect of turbulence is to produce shearing stresses analogous to, although usually much greater than, those due to molecular viscosity, it seems natural to attempt to allow for the Reynolds stresses by introducing a kinematic "eddy viscosity" N, analogous to the kinematic molecular viscosity v^fx/p. In most cases, when dealing with the mean motion, the stresses due to molecular viscosity may be neglected compared with the Reynolds stresses. Thus, if the mean current U is in the OX direction and the shearing stresses are also in the OX direction, it follows that — AT ^^ Tyx = - pUV = pJSy — ^ (4) Tzx = — PUW = pJSlz -T—' cz where Tyx is taken to be the shearing stress on a surface perpendicular to OY in the OX direction, and the stresses due to molecular viscosity have been neglected. In general Ny will differ from Nz in magnitude and the coefficients of eddy viscosity are not physical constants like the molecular viscosity v, but depend on the type and scale of the motion and on the degree of stability. In a given pattern of flow they will vary, in general, from one point to another. The Reynolds stress component — puw represents the mean rate of turbulent transport of w-momentum by the w component of flow. Similar considerations may be applied to a property of the fluid other than a component of momentum. If 8' is the concentration per unit mass of fluid of any property, such as the heat content or salinity, at any instant, 8' may be separated into a mean value 8 and a fluctuation s, i.e. 8' = 8 + s. (5) Then across a plane perpendicular to OZ, for example, there will be a turbulent transport of the property 8', given by pws. By analogy with molecular diffusion, the turbulent transport may be regarded as a process of eddy diffusion. In the example just considered, a coefficient of eddy diffusion, Kz, may be defined by Tsz = pws = -pKz^-' (6) cz where Tsz represents the turbulent transport of 8 in the OZ direction. Co- efficients Kx and Ky may be defined similarly for turbulent diffusion in the OX and OY directions. The coefficients Kx, Ky and Kz, like the corresponding coefficients of eddy viscosity, will, in general, differ from one another, be functions of position in the fluid, and depend on the type and scale of motion and the stability. The existence of turbulence in a field of motion gives rise, therefore, to two types of effect : the production of shearing stresses and the process of eddy 804 BOWDEN [sect. 6 diffusion. Whereas the turbulent shearing stresses react on the mean motion and have an essentially dynamical effect, the turbulent diffusing processes affect the distribution of a particular property of the fluid without reacting directly on the flow, B. Mixing Length Theory There is an obvious analogy between the random motion of molecules con- sidered in the kinetic theory of gases and the irregular movements of elements of fluid which occur in turbulent flow. The introduction by Prandtl (1925) of the concept of a "mixing length" I, analogous to the mean free path in the kinetic theory of gases, led to a valuable line of development in the theory of turbulence. From considerations of similarity between the turbulent motion in various parts of the mean flow, von Karman (1930) was able to relate I to the properties of the mean motion. For the special case of flow near a solid boundary at 2 = 0, and in the region in which tzx does not vary appreciably with z, the theory of Prandtl and von Karman leads to the well-known logarithmic law for the velocity distribution : U ^ }- iZ^Y \n'-±^, (7) where to is the stress at the boundary, zq is a parameter known as the "roughness length" and ko is von Karman's constant, having a numerical value of approxi- mately 0.40. {toIpY'- is frequently denoted by U^ and termed the "friction velocity". The logarithmic law has been much used in all branches of fluid mechanics, including meteorology, and its validity in conditions of neutral stability appears to be well established. Equation (7) implies that, within the limits of z for which it is valid, I = ko{z + zo) (8) Nz = koU^{z + ZQ). If the velocity U is measured at several values of the distance z and U plotted against log 2, the shearing stress to and the roughness length 20 may be deter- mined from (7). C. Diffusion by Continuous Movements The analogy between turbulent motion and the kinetic theory of gases is imperfect, however, in that the interaction of an element of fluid with its sur- roundings takes place continuously and the element itself gradually loses its identity. An alternative approach was initiated by Taylor (1921) in his theory of diffusion by continuous movements. He defined the function E{r) as the coefficient of correlation between the turbulent velocity ut of a particle of fluid at time t and the velocity ut+r of the same particle at a time t later, i.e. E{t) - {utut+r)lu2. (9) SECT. 6] TURBULENCE 805 If I is the displacement of a particle of fluid at time t from its original position at time ^ = 0, it follows that |2 = 2^^ f r R{T)dTdt'. (10) Jo jo For small values of t, such that R{r) does not differ appreciably from 1 in the interval 0 to, then Ii{r) dr will reach a limiting value jo for t' > TO, and for large values of t, such that t^ro, where =f R{t) dr. (12) If the particles had been dispersed by a diffusion process according to Fick's law, with a constant diffusion coefficient K, the mean square displacement would have been F = 2Kt. (13) In the special case considered above, for large values of t, the dispersion corre- sponds to diffusion with a coefficient K given by K = u^I. (14) D. Richardson's " Neighbour Separation'' Theory Instead of considering the distance of a particle from a fixed point, Richard- son (1926) approached the diffusion problem by considering the variation with time of the distance separating two particles. He defined the instantaneous distance I between two particles as the "neighbour separation", and introduced a "neighbour concentration function" q{l) to represent the proportion of pairs of particles having separations between / and l + dl. The ordinary diffusion equation was then replaced by the equation where F{1) is the "neighbour diffusivity". From data based on observations in atmospheric diffusion, Richardson deduced that F{1) = eZ^/3, (16) where e is a constant. The application of this approach to conditions in the sea is considered in section 5 of this Chapter, page 818. 806 BOWDEN [sect. 0 E. Statistical Theory and Local Isotropy In the statistical theory of turbulence, various correlation functions are used to relate a velocity component at one point with a velocity component at another point. Thus if f{r) represents the correlation between the velocity component Ux at a point x and the component Ux+r at a point x + r, f{r) = {UxUx+r)luK (17) A space correlation /(r) in the OX direction may be expressed as a time correla- tion between the velocity at point x at time t and the velocity at the same point at time ^ + t by putting r = Ur. Then /(r) = {utut^.)iu'^. (18) The correlation function /(r) is related by a Fourier transformation to the spectrum function F{n), where F{n) dn is the proportion of the total turbulent energy due to fluctuations with frequencies between n and n + dn (Taylor, 1938), i.e. f[r) = F{n) cos 27mT dn •'» (19, /•CO F{n) = 4 /(t) cos 277nT dr. Jo The correlation functions which are derived from observations are usually of the type (17) and (18) and are Eulerian, whereas the function R{t) defined in (9), in connection with the theory of diffusion by continuous movement, is Lagrangian. The concept of isotropic turbulence, i.e. a state of motion in which the mean properties of the turbulence are independent of the axes of reference, was introduced by Taylor (1935). In this case the Reynolds shearing stresses are zero and the turbulence can neither gain energy from the mean motion nor react on it. For this reason, isotropic turbulence, although studied extensively both theoretically and in wind-tunnel experiments, found little application to meteorological and oceanographic problems. A further advance was the introduction of the theory of locally isotropic turbulence by KolmogoroflF (1941) and its development by von Weizsacker (1948) and Heisenberg (1948). This assumes the existence of a continuous spectrum of fluctuations which may be interpreted as a spectrum of eddy sizes. The largest eddies receive energy from the mean motion and are anisotropic. The theory of local isotropy applies particularly to eddies in an intermediate range, which receive their energy from larger eddies, the motion of which is considered as "relative mean motion" in this connection, and pass it on to smaller eddies by the action of turbulent stresses due to the latter. This transfer of energy may be expressed in terms of an effective eddy viscosity N, which is a function of the fundamental volume, of typical length L, used to separate SECT. 6] TURBULENCE 807 relative mean motion from turbulent motion for the particular scale of eddies considered. From similarity considerations it is found that N oc i4/3. (20) Thus the effective eddy viscosity increases with the scale of motion considered. The result (20) is similar to the relation for the neighbour diffusivity (16) found by Richardson. The smallest eddies lose all their energy by viscous dissipation. More recently, much attention has been given to the properties of the "large eddies", i.e. those of dimensions comparable with the scale of the mean flow and largely responsible for abstracting energy from it. These are markedly anisotropic and would appear to be a more ordered type of motion than the remainder of the turbulence. Some progress has been made by representing them as arrays of vortices of defined properties (e.g. Grant, 1958). A comprehensive account of recent work on turbulent shear flow was written by Townsend (1956). Frenkiel (1953) and Batchelor and Townsend (1956) have written reviews of work on turbulent diffusion and a general account of the problem of atmospheric diffusion has been given by Deacon (1959). A new approach to a theory of turbulent shear flow has been made by Malkus (1956). F. Influence of Stability The energy of turbulent motion is continuously being dissipated by viscosity and, in order to maintain a steady state, energy must be derived from the mean motion by the action of the Reynolds stresses. In the case of a horizontal shear flow, in which the mean current U is in the OX direction and varies in the OZ direction only, the rate of generation of turbulent energy per unit volume is r-r ^^ dz (21) —8U „ (dUY The rate of dissipation by viscosity is D = 110, where )■ (22) (h - ^(^'^V '?(^^Y 9(^^Y (^^ ^^Y (^^ ^^Y l^^ ^^\ W ^'^\d^) ^^l"^/ "^lai^a^j ^\d^'^~dl) ^yd^^W If there is a vertical gradient of mean density in the fluid, the density may be regarded as a property capable of turbulent transport. Let the instantaneous value of the density at a point be p + p', so that p' represents a density fluctua- tion. Then vertical turbulent mixing gives rise to an increase in potential energy at a rate per unit volume of P = g7^= -gKz^' (23) 808 BOWDEN [sect. 6 In a fluid of uniform density, the intensity of turbulence may be maintained at such a level that all the energy supplied from the mean motion is dissipated by viscosity, i.e. G = D. In the presence of a vertical gradient of density, a portion of the energy supplied is used in increasing potential energy, so that G = P + D', where D' is the rate of dissipation in this case. Hence, for a given G, D' < D and 0' <0. Since 0 depends on the turbulent velocity gradients, a decrease in 0 implies a decrease in the general level of intensity of the turbulence. It also follows that G> P for this case, and hence from (21) and (23) NzjKz > Ri, (24) where Ri is a parameter known as the Richardson number. Thus, the effect of a stable density gradient is to reduce the intensity of the turbulence and hence also the coefficients of eddy viscosity Nz and eddy diffusion Kz, but Kz will be decreased more than Nz. A parameter Rf, known as the "flux Richardson number", which has been introduced in meteorology, may be defined by i^/ = ^ = |iei. (26) The condition (24) may then be expressed as Rf < 1. (27) 2. Turbulence in the Sea In the sea, the vertical and horizontal components of turbulence usually differ so much in scale and in intensity that their effects are considered sep- arately. These differences arise, firstly, because the horizontal dimensions of the bodies of water concerned are much greater than the vertical dimension, and, secondly, because of the influence of stability. The presence of a stable vertical gradient of density causes a reduction in the intensity of the iv fluctuations but has no direct effect on the fluctuations in u and v. A striking example of the difference in the scale of horizontal and vertical diffusion was given by Revelle et al. (1955). Radioactive material released at a point below the thermocline spread horizontally, or more strictly over an iso-density surface, covering an area of 100 km^, while its vertical extent did not exceed 1 m. In general, the SECT. 6] TURBULENCE 800 eddy coefficients of viscosity and diffusion in the vertical direction, Nz and Kz, fall within the range 1 to lO^ cm-/sec while the corresponding coefficients in a horizontal direction, Nn and K},, are in the range 10^ to 10^ cm^/sec. The generation of vertical turbulence arises from the association of horizontal shearing stresses and a vertical gradient of velocity. The main generating processes are, therefore : ( 1 ) The action of wind stress on the surface layer ; (2) The effect of bottom friction on currents, particularly tidal currents ; (3) The presence of shear in currents due to horizontal pressure gradients. Horizontal turbulence may be generated by : ( 1 ) Horizontal variations in the stress of the wind on the surface ; (2) Lateral stresses at coastal boundaries ; (3) Horizontal shear in currents, or between adjacent currents. The very wide spectrum of horizontal motions existing in the oceans has been described by Stommel (1949). The main energy input is into the large-scale ocean circulations which are due directly to the major wind systems. Next in scale are the large eddies which develop in the main currents and may become detached from them. These in turn give rise to a cascade of eddies of diminishing size. It would appear at first sight that, between the largest and the smallest eddies, there would be a wide spectrum of eddies receiving their energy from those larger than themselves and passing it on to those smaller. Such eddies would be expected to fall in the "inertial sub-range" of the theory of locally isotropic turbulence, so that the results of this theory, such as the L^l^ law of eddy viscosity, could be applied to them. The situation is complicated, how- ever, as Stommel pointed out, by the possibility that energy may be injected directly into eddies of almost any intermediate scale, e.g. by local storms or squalls or by tidal currents. These eddies would not then be in the energy equilibrium postulated in the theoretical spectrum and could not be strictly isotropic. In dealing with horizontal motions, the choice of a suitable scale of averaging when separating them into mean motion and turbulence is highly significant. Eddies of, say, 50 km in diameter might be treated dynamically as individual entities, if observations spaced sufficiently closely in space and time were available ; or they might be treated statistically as turbulence aff"ecting the main circulation. In addition to the paper by Stommel (1949), the application of general ideas on turbulence to oceanographic problems has been discussed by von Karman (1948) and Defant (1954). The mixing of sea-water by turbulence was con- sidered by Proudman (1948) on the basis of Taylor's theory of diffusion by continuous movements (page 804). He showed that, except in special cases, the distribution of a property, such as salinity, could not be expressed in terms of coefficients Kx, Ky and Kz as usually defined. Eckart (1948) distinguished between "stirring" and "mixing" processes, designating by "stirring" the dis- tortion of elements of fluid by a shearing current. The large gradients in the 810 BOWDEN [sect. 6 concentration of a property produced in this way give rise to diffusion or "mixing" along the gradient directions. These two processes interact to produce the diffusion effects which are observed. Particular problems involving vertical or horizontal turbulence relate either to the dynamical effects, which have usually been treated in terms of eddy viscosity, or to the diffusion effects. In each case there are two possible methods of treatment : (1) The turbulent fluctuations and the corresponding turbulent transports may be considered directly ; (2) Distribution of mean properties of the water, e.g. velocity or salinity, may be interpreted in terms of turbulence parameters, usually eddy coefficients of viscosity or diffusion. These alternative approaches are considered in the following paragraphs. 3. Turbulent Fluctuations and Turbulent Transports This section deals with investigations which are based on the direct measure- ment of the turbulent fluctuations of velocity and, in some cases, also of temperature. Oscillations of an apparently periodic character, such as those associated with surface waves or tides, are excluded and the fluctuations under consideration are understood to be those of a more irregular character. The distinction, however, is not always clearly defined. Let x be the displacement of a particle from a fixed point and u — dxjdt be its velocity. Then Eckart (1955) distinguished between a "random oscillation", in which x^ remains finite as t increases indefinitely, and a "random drift", in which x^ increases indefinitely with time, although u'^ remains finite. The difference between the two oases can be recognized in the form of the velocity spectrum as the frequency ap- proaches zero. A general type of wave motion, as discussed in Chapter 15, Section 3, corresponds to a random oscillation. Turbulent motion, on the other hand, would appear to correspond to a random drift and it may be that this is a valid criterion for distinguishing turbulence from wave motion. A record of fluctuations over a given length of time may always be represented as a spectrum, in which a certain proportion of the total energy is associated with each of a continuous series of frequency intervals. If a given narrow band of frequencies contains an appreciable proportion of the energy, one may speak, rather loosely, of fluctuations of this frequency as "occurring" in the spectrum of turbulence. Measurements of velocity at a fixed point in the sea show the occurrence of fiuctuations of a very wide range of periods, from the order of 0.01 sec, as recorded by Patterson (1957) to a number of hours or even days, as reported by Ekman (1953) or Swallow and Hamon (1960). It seems probable, therefore, that the periods of fiuctuations found in a particular experiment will be limited at the lower end only by the response-time of the instruments and at the upper end only by the duration of the record. It may happen, however, that all the SECT. 6] TURBULENCE 811 fluctuations which contribute significantly to a particular effect, such as the shearing stress, lie within a limited range of periods. The experimental method may then be designed to cover this range adequately. Fluctuations in the measurements made by current meters have long been known but have usually been regarded as undesirable features to be eliminated by averaging. The first attempts to study the fluctuations themselves were probably those of Thorade (1931) in the River Elbe and later (1934) in the Kattegat, using the Rauschelbach current meter, giving records of the speed and direction of the current every 10 sec. The fluctuations which he attributed to turbulence had periods of several minutes. Mosby (1947, 1949) used twelve sets of revolving cups, mounted on a stand at different heights, from 9 cm to 249 cm above the bottom, to measure the current in the Alvaerstrommen, near Bergen. Fluctuations with periods in the range 2 to 15 min were found and usually corresponding fluctuations could be identified at all levels. Similar observations were made (Mosby, 1951) on the Viking Bank, in a depth of 100 m, at a distance of 100 km from the nearest coast. In order to record short-period turbulence, Doodson (1940) designed a current meter which would respond to fluctuations with periods down to about 1 sec. In the first experiments, with the meter suspended from a boat, fluctuations were recorded but there was some doubt as to whether they were influenced by the rolling of the boat. Clear evidence of the short-period fluctuations associated with the flow of a tidal current was obtained when the Doodson meter was used in a stand on the bottom in the River Mersey (Bowden and Proudman, 1949). The values of u'^, their relation to the mean current U at various heights and the auto-correlations of the u component were determined. In another series of observations, two current meters were used, mounted in the same stand with varying separations, so that the spatial correlations of u could also be investigated (Bowden and Fairbairn, 1952). The introduction of an electromagnetic flow-meter (Bowden and Fairbairn, 1956) enabled measurements of the vertical component w as well as the longitu- dinal component u to be made, and also extended the range to rather shorter periods. The measuring head of the electromagnetic flow meter produces a local magnetic fleld and the water flowing through the field has an e.m.f. induced in it. The potential gradient set up in the water is measured by two pairs of electrodes, so that two components of the velocity fluctuations can be recorded simultaneously. The over-all diameter of the head is 10 cm, while the frequency response of the equipment is flat for periods above 1 sec and falls to one half at 0.25 sec. A series of observations with this apparatus was made in tidal currents off" Anglesey, North Wales, within 2 m of the bottom in depths of water ranging from 12 m to 22 m. There was no measurable vertical gradient of temperature or salinity, so that the measurements corresponded to flow in conditions of neutral stability. For mean currents U in the range 25 to 50 cm/sec, average values of the root-mean-square fluctuations were ('m2)'/2/C/ = o.13, {w^y^^lU = 0.066. These values did not vary much with height in the range 50 to 175 cm. 812 [SKCT. 6 From a number of simultaneous records of u and w the stress - puw was determined, the coefficient of correlation between u and w averaging —0.38. In the case of the u fluctuations, the vertical scale appeared to be only about one-third of the scale in the direction of the mean flow. Fig. 1 shows the average spectra of u'^, iv^ and uw, computed from six u,w records at a height of 75 cm above the bottom. The u fluctuations are seen to contain considerably more energy at the lower wave numbers than do the w fluctuations, while most of the shearing stress appears to be due to wave numbers k< IO-2 cm-i, with the peak at A: = 6 x IQ-^ cm-i approximately. 0.3 0.2- 0.1 - 1 1 1 1 1 ^-^ 1 / \ / \ / V ^ ^^ \ / / / / y \ / / '"■■••.._ \ y ''""■"* -»» \ /* ^ y ,■' \^^ \ 1 III 1 ^ 0.1 0.2 0.5 k (IQ-^ cm-') 20 Fig, 1. An example of turbulence spectra in a tidal current : from six u,w records, 75 cm above the bottom, off Anglesey : /2 . 7y,2 . A paper by Francis el al. (1953) is noteworthy as representing the first attempt to measure directly the turbulent transports of both momentum and heat. The observations were made at various depths in the Kennebec Estuary, Maine, the horizontal velocity fluctuations being measured by a current meter designed by Von Arx (1950) and the vertical fluctuations obtained from the movements of a vane pivoted on a horizontal axis. The temperature fluctua- tions were measured with a thermistor. The shearing stresses derived were very variable and some values obtained near the surface were as large as those near the bottom. Since the stresses were determined from readings at 3-sec intervals over a record only 2 or 3 min long, it seems likely, in view of later experience, that the sampling variations would be large. Four current meters arranged in a frame were used by Hamada and Okubo (1952) to measure the u fluctuations in a tidal current at Suminoe Harbour. The time response of the meters permitted fluctuations of periods down to 6 sec to be recorded. Velocity measurements at a series of depths between the surface and the bottom, in water 10 m deep, were made by Nan'niti (1956) in SECT. 6 J TURBULENCK 813 a tidal current in Uraga Strait at the entrance to Tokyo Bay. Fluctuations attributable to turbulence were recorded with periods from a few seconds up to several minutes. An application of the hot-wire technique, much used in wind-tunnel and atmospheric work, to the study of turbulence in the sea has been described by Patterson (1957). He reported observations on the high-frequency turbulence in the flow behind obstructions in a tidal current. A development of the hot- wire method was used in a study of the high-frequency end of the turbulence spectrum by Grant, Moilliet and Stewart (1959). In place of the hot wire they employed a probe consisting of a platinum film of 4 x 10"^ cm thickness and maximum length 1 mm on the tip of a glass cone. The probe was mounted on the nose of a heavy body streamed from the stern of a ship and observations were made in Discovery Passage, British Columbia, in a depth of 60 m and tidal current velocity of 100 cm/sec, corresponding to a Reynolds number of 4 x 10'^. The response of the instrument extended to very high wave numbers but was limited at the lower end by movements of the towed body. The authors regard the spectrum of the u fluctuations, derived from a 30-min record, as reliable for wave numbers k from 0.02 to 1 cm~i. Within this range the energy is very nearly proportional to A;~5/3^ ag would be expected from the theory of local iso- tropy in the inertial sub-range. Most of the dissipation of energy appears to occur from fluctuations of A;> 1 cm~i with a peak at about k = ^ cm~i. It was estimated that the measured portion of the energy spectrum contributed only about a fifth of the total energy of w^ and that the fluctuations of A; > 0.02 cm~i could not contribute significantly to the Reynolds stress. As far as the condi- tions were comparable, this view is consistent with the results of Bowden and Fairbairn, described above, which apply to the range k< 0.01 cm~i. A turbulence meter for measuring fluctuations of velocity and temperature, as well as their mean values, has been developed by Kolesnikov et al. (1958). The horizontal and vertical components of the velocity fluctuations are measured by crossed hot wires, while a single hot wire measures the mean velocity. Thermistors are used for recording the mean temperature and its fluctuations. Observations with this equipment were made below the ice from the drifting station "North Pole 4" in 1956. The intensity of the turbulence was found to be high within 1 m of the lower surface of the drifting ice, but to fall off fairly rapidly at greater distances. Correlation functions and spectra were computed. An interesting example of the application of the turbulent flux method on a much larger scale was provided by Ichiye (1957), who computed the momentum transport across the Kuroshio from current observations made by GEK (towed electrodes). Taking the a;-axis parallel to the direction of maximum current in the Kuroshio and the ?/-axis across it, the components of mean flow U, V and turbulent flow, u, v were found from the series of observations at each GEK station. From these, the turbulent transport uv and the transport UV due to the mean flow were computed. The results showed that in most cases the turbulent transport of momentum was large compared with the mean 814 BOWDEN [sect. 6 transport. In the main current of the Kuroshio, where the momentum flux could rehably be related to the gradient of mean velocity, dU jdy, the corre- sponding eddy viscosity Ny was of the order 10^ to 10'^ cm^/sec. 4. Vertical Turbulence A . Eddy Coefficients and the Influence of Stability Turbulence is generated in the surface layer of the sea by the stress of the wind. The rate of generation depends on the shearing stress and the velocity gradient, while the latter is itself dependent on the effective eddy viscosity. In its simplest form, Ekman's theory of wind-driven currents is based on a con- stant eddy viscosity Nz within the surface layer, and it then follows from observational data (Sverdrup et al., 1942, p. 494) that Nz is related to the wind velocity W by the equation pNz = 4.3 If 2 for W > 6 m/sec, where W is measured in m/sec. In fact it is more probable that the intensity of wind-induced turbulence, and hence also the magnitude of Nz, will have a maximum value a small distance below the surface and will then decrease with increasing depth. Theories have been developed by Rossby and Montgomery (1935) and others which treat Nz as a function of z. However, it is still true, as stated by Sverdrup et al. (1942), that "no observations are as yet available by means of which the results of a refined theory can be tested". When the density of the water increases with depth, due, for example, to a downward flux of heat, the effect of stability is to reduce the values of Nz and Kz. In the wind-mixed layer, the diffusing effect of the turbulence is sufficient to keep the density gradient small, but at some depth the rate of generation of turbulent energy may no longer be great enough to overcome the stability effect and a transition region, known as the thermocline, will be formed. This region is characterized by a steep density gradient, with a low intensity of turbulence and, hence, low values of Nz and Kz. Munk and Anderson (1948) developed a theory of the thermocline in which they considered the shearing stress and heat flux as functions of depth, with the coefficients Nz and Kz as functions of the Richardson number Ri. In the present notation, they took Nz = Ao{l+^vRi)-y^, Kz = ^o(l +i8Ti^^)-'/^ where Nz = Kz = Ao for Ri = 0. The forms of these functions and the numerical values of the constants ^v and ^T were chosen to be consistent with the data of Jacobsen (1913) and Taylor (1931). The appropriate values of the constants were ^v= 10, ^r = 3.33. Thus a value Ri = 0.1 would correspond to iV^2 = 0.71^o, ir2 = 0.65^0 and a value Ri=l to iV2 = 0.30^o, Kz = 0.11Aq. In spite of the great simplification of the SECT. 6] TURBULENCE 815 problem which was necessary, the theoretical treatment yielded results on the depth and characteristics of the thermocline w hich were in general accordance with experience. The influence of stability on the vertical mixing processes due to turbulence has also been treated by Mamayev (1958), who considered that the appropriate forms for Nz and Kz were Nz = Aqe-^R^, Kz = Aoe~^^\ where n — m>0. From Jacobsen's data it was deduced that n = O.S, m — OA. As discussed in section 1 of this Chapter, the quantity Rf=KzRilNz, known as the flux Richardson number, represents the fraction of the turbulent energy being generated by the shearing stresses which is used to maintain turbulent mixing against the density gradient. Ellison (1957) has deduced on dimensional grounds that Rf should approach a critical value Rf cm., less than 1, as Ri increases indefinitely. Data from atmospheric turbulence indicate that RfcT\t. = 0.15 approximately. It may be noted that the forms of Nz and Kz used by Munk and Anderson correspond to Rf—^Rfcrit. as Ri-^cc, but the critical value is i?/crit. =0.52. Mamayev's forms of iV^ and Kz, on the other hand, correspond to Rf -^ Q as Ri ->cc, but Rf would reach a maximum value, Rfmax., at some intermediate value of Ri. He envisages the possibility of Rfm&x. exceeding 1, but this would no longer correspond to a steady state as the intensity of turbulence would be decreasing with time. Methods of determining the numerical values of the coefficients of eddy viscosity Nz and eddy diffusion Kz, from stationary distributions of current and temperature or salinity or from periodically varying conditions, have been given by Sverdrup et al. (1942) and by Proudman (1953). Among the more recent determinations of Kz in deep water are those of Wiist (1955), based on a reconsideration of the Meteor data. In the core of the Antarctic bottom current, in the West Atlantic Trough, he derived values of Kz ranging from 7 to 50 cm-j sec with an average of 29 cm^/sec over the latitude range 50°S to 10°N. Values of the same order of magnitude were obtained for the North Atlantic deep current. Koczy (1956) used the distribution of radium with depth to determine Kz, assuming that the radium content of the water was maintained by the production of radium from ionium in the bottom sediments and its transport upwards by eddy diffusion. Data from four stations, one in the Atlantic, one in the Indian Ocean and two in the Pacific, gave values of 4 to 30 cm^/sec near the bottom, decreasing with height to quite small values at 2000 to 3000 m above the bottom. B. Velocity Profile near the Sea-Bed One of the most firmly established results in the general study of turbulent flow is the logarithmic law for the velocity profile near a solid boundary (7) and its use as a means of determining the shearing stress. The first measurements of the velocity profile immediately above the sea-bed appear to be those of Revelle and Fleming at the entrance to San Diego Harbour (reported in 816 BOWDEN [sect. 6 Sverdrup et al., 1942, p. 480). From measurements at three levels, 21, 51 and 126 cm above the bottom, with velocities from 15 to 26 cm/sec, they deduced that the logarithmic law was valid and that the bottom was rough, with a roughness length 20 = 2 cm. Mosby (1947, 1949), using the cup-wheel current meter described on page 811, recorded the current simultaneously at 12 levels, from 9 to 249 cm above the bottom in the Alvaerstrommen. From 30- min averages, it appeared that the profile was approximately logarithmic, although there were considerable variations from one period to another. Lesser (1951) made observations at four levels, 20, 40, 80 and 160 cm above the bottom, using four Ekman current meters mounted on a tripod. The measurements were made in homogeneous water, in depths of about 45 m, on three types of bottom. Interpreting the data in terms of the logarithmic law, Lesser deduced that, in two cases, with bottoms of gravel-sand and mud- sand respectively, the flow was rough with 20 from 1.3 to 1.6 mm. In the third case, with a mud bottom, the flow appeared to be smooth, but in this case the velocities were very low, reaching 12.7 cm/sec at 160 cm. Observations were made in a tidal current off Anglesey by Charnock (1959), using an instrument consisting of five cup-wheels at heights from 30 to 200 cm above the bottom. Most of the velocity profiles recorded were approximately logarithmic and corresponded to hydrodynamically rough flow with zq from 1 to 3 mm. The frictional stresses estimated in this way were in reasonably good agreement with those determined by other methods in the same area (Bowden and Fairbairn, 1952a and 1956). Bowles et al. (1958) have described observations made with the same equipment at a f)osition in the English Channel, south of Arish Mell Gap. The velocity profile was found to follow the logarithmic law, with values of zq which were rather variable but averaged 2 mm. The apparatus designed by Charnock was also used by Bowden, Fairbairn and Hughes (1959), in conjunction with a Doodson current meter for current measurements at greater distances from the bottom, to determine the variation of shearing stress with height. This investigation yielded estimates of Nz at various depths which indicated that its value was somewhat greater near mid- depth than nearer the surface or bottom, reaching the order of 300 cm^/sec during the flood and 150 cm^/sec during the ebb. In the case of tidal currents in homogeneous water it might be expected on dimensional grounds that the maximum value of Nz would be proportional to Uh, where U is the amplitude of the tidal current and h the depth of water. From the Anglesey data, the average value of (NzUaxJUh is 2.5 x 10-3. C. Tidal Mixing in Coastal Waters It has been known for many years that in shallow areas with strong tidal currents the water remains homogeneous throughout the year, whereas in areas of similar depth with weaker tidal currents a thermocline develops in the summer months. In such cases there are two sources of vertical turbulence : that due to the wind being most intense near the surface and decreasing with SECT. 6] TURBULENCE 817 depth, while that generated by bottom friction in the tidal current decreases in intensity upwards from the bottom. In regions of complete vertical mixing, the intensity of turbulence is sufficiently great at all depths to ensure efficient mixing. A summer thermocline will be formed, however, if there is a level of minimum turbulence at some intermediate depth, where the vertical eddy diffusion is not adequate to transmit the downward flux of heat without an appreciable temperature gradient developing. The conditions governing the formation of a seasonal thermocline appear to be quite critical and the transi- tion from one regime to the other frequently occurs in a distance of a few miles. Dietrich (1950) j^ointed out that, in the English Channel, the areas with com- plete vertical mixing correspond closely to those where the amplitude of the tidal current exceeds 100 cm/sec. The influence of a tidal current on vertical mixing was considered quantita- tively by Hansen (1950), who introduced the Richardson number into the equations of motion and mixing and derived a relation between the density gradient and the coefficient of eddy diffusion. The problem was also considered by Dietrich (1954) in relation to the temperature stratification in different parts of the North Sea. It was assumed that if Ri < 0.5, turbulent mixing was effective but when Bi > 0.5 the vertical turbulence was suppressed sufficiently for a temperature gradient to persist. The critical density gradient dp/dz, corresponding to Ri = 0.5, was determined as a function of depth, taking the vertical variation of velocity to follow the power law V = vo{zlh)", where 1^0 = velocity at surface, 2 = depth below surface, A = total depth and the index a has a value between ^ and y. The critical gradient was compared with that which corresponded to the temperature gradient associated with the downward flux of heat under conditions of summer heating. The intensity of turbulence existing in a current also affects the concentration of sediment which it is able to maintain in suspension. This problem has received considerable attention, experimentally and theoretically, in hydraulics (e.g. Vanoni, 1946; Hunt, 1954). Joseph (1954) made a study of the effects of current velocity and stability on the concentration of suspended particles and the optical extinction coefficients at two stations off Texel, in the southern North Sea, A theoretical treatment was given, relating the eddy dififusivity to the particle concentration, and values of Kz were derived which were of the order of 100 cm^/sec near the bottom and 200 cm^/sec near the surface in homogeneous water. When a thermocline was present, the value of Kz near the bottom was reduced only slightly but its value decreased to the order of 10 cm2/sec in the discontinuity layer. 5. Horizontal Turbulence The dynamical effect of horizontal turbulence has received most attention in relation to the major ocean currents of the wind-driven circulation. Thus Munk 27— s, I 818 BOWDEN [sect. 6 (1950) considered the energy acquired by the circulation from the wind to be dissipated by lateral eddy viscosity in the western boundary currents. The magnitude of the lateral shearing stresses required corresponded to a value of Nh of the order of 5 x 10'^ cm^/sec. (The notation Nn and Kh will be used to denote coefficients of horizontal eddy viscosity and diffusion when no distinc- tion is made between the x and y directions.) Hidaka and other workers have found that values of Nh from 10^ to 10^ cm^/sec are needed to account for the observed features of the currents. An alternative explanation of a western boundary current, such as the Gulf Stream, is to regard it as an inertial boundary layer (Charney, 1955) in which the pressure gradients and Corolis forces are balanced by field acceleration terms. Energy might then be dissipated in a frictional boundary layer between the inertial flow and the coast. Alternatively, energy might be abstracted from the main current by large eddies which become detached from it and are themselves dissipated later. In view of the uncertainty about the significance of lateral shearing stresses, methods of determining the Reynolds stress — puv directly are of special interest. The analysis by Ichiye (1957) of GEK observations in the Kuroshio, mentioned on page 813, gave values of —puv consistent with Ny of the order of 10^ to 10'^ cm^/sec. Stommel (1955) applied the same method to current measure- ments in the Straits of Florida by Pillsbury in 1885 and found stresses corre- sponding to Ny of the order of 10^ cm^/sec. A summary of data on horizontal eddy viscosity was given by Sverdrup et al. (1942, p. 483) and methods of determining Nx or Ny from the observations were given by Proudman (1953). The applicability of such methods to a particular current system usually depends on the assumption that lateral shearing stresses do, in fact, play a dominant part in its dynamics. Problems involving horizontal eddy diffusion are of two main types : (1) The maintenance of a steady distribution of a property, such as salinity, in the presence of advection and a source or sink of that property. Periodically varying distributions may also be included in this group. (2) The dispersion of a cluster of particles or the spreading of a patch of pollutant, initially concentrated within a small volume. Problems of the first type arise in many oceanographic investigations and are treated, almost invariably, in terms of effective eddy coefficients Kx or Ky, which are, in general, functions of position and also of the scale of the process under consideration. The methods of analysis involve special solutions of the diffusion equation : dS 8S dS d /' ^ dS\ 8 /^^ dS\ Typical examples of such investigations have been given by Sverdrup et al. (1942), Proudman (1953) and Dietrich (1957). The average rate of increase of the separation of pairs of particles, or marked elements of fluid, is the problem treated directly in Richardson's "neighbour SECT. 6] TURBULENCE 819 separation" theory (1926, referred to on page 805). A number of tests made in the sea with parsnip floats, paper markers and dye spots verified the relation F{1) = €^4/3 (29) between the "neighbour separation" I and the "neighbour diflfusivity" F{1) in the range 1 = 25 cm to Z=100 m (Richardson and Stommel, 1948; Stommel, 1949). Olson (1952) made use of drift-card observations to extend the tests to values of I up to 10 km. More recently Olson and Ichiye (1959), by including drift-bottle observations in Japanese waters (Ichiye, 1951), showed that the relation F{1) = 0.0246Z4/3 (30) fitted the whole series of data over the range Z=10to 10^ cm. The same authors have also shown that the Z^/s law may be derived from the general theory of the turbulent diffusion of two particles (Ichiye and Olson, 1960). Fig. 2 shows Fig. 2. Neighbour diffusivity F{1) as a function of neighbour separation I. (Taken from Ichiye and Olson, 1960, Fig. 1. By permission of the German Hydrographic Institute, Hamburg.) Sources of data : x Stommel (1949). V Olson (1952). n Platania (quoted by Olson, O Ichiye (1951), A, open sea; 1952). B, near shore. the data on F{1) plotted as a function of I. Experiments with paper floats have also been described by Ozmidov (1957), who found that the Z^/s law was valid provided I > IQh, where h is the depth of water. In another experiment Ozmidov (1958) used pairs of indicators of different diameters d, and found that their rate of dispersion was consistent with the postulate that only eddies of wave 820 BOWDEN [sect. 6 number k, within the range given hy d< HkS') x IQio dynes cm-2. (1) The measurements were all made on ice at — 20°C. Very little data are available about the other elastic parameters of sea-ice. Poisson's ratio has been measured for fresh ice using torsional oscillations of a bar, seismic methods and sonic methods. The results are in good agreement. Northwood (1947) obtained 0.33 and his value is considered the best. Peschansky (1957) quotes 0.29 for sea-ice. C. Visco-elastic Properties Pure ice ranges from being almost perfectly elastic for rapidly varying stresses (small enough in amplitude not to produce fractures) to being ex- tremely plastic for static loads. Numerous investigations of the visco-elastic nature of pure ice and snow have been carried out in recent years. Most of these have consisted of measurements of creep curves using static loading (see Glen, 1952; Glen and Perutz, 1954; Jellinek and Brill, 1956; and Griggs and Coles, 1954, for example), but Nakaya (1959) has also applied sonic methods to obtain both elastic and visco-elastic parameters. Comparable studies on sea-ice are rare, the main investigator being Tabata (1955, 1958) who measured creep curves for in situ ice beams and for statically loaded, small cylinders. He found that the visco-elastic behaviour could be described adequately by a rheological model consisting of a Maxwell unit and a Voigt unit in series. The temperature range ( — 1° to — 7°C) was too limited to obtain any information about the dependence of the parameters on tempera- ture, which one would expect to be large. The range in salinity was also small. 3. Thermal Properties Owing to the brine cells in sea-ice, any change of temperature will involve a phase change of some of the ice. The concepts of specific heat and latent heat are thus thoroughly interrelated. The specific heat of sea-ice of any appreciable salinity is much higher than that of pure ice, particularly near the freezing point, because of this melting or freezing of additional ice. The ordinary definition of latent heat of fusion is not applicable to sea-ice, both because there is no fixed melting point and because the phase change is not reversible. If sea-ice is melted and then cooled to its initial temperature, the resulting material may be in quite a different state since the formation of sea-ice is so dependent on the freezing rate. SECT. 7] THE PHYSICS OF SEA-ICE 833 These phenomena were investigated by Malmgren (1927) and very Httle additional work has been done. His results are summarized in The Oceans, pp. 73-75, where tables are given showing the relationship of specific heat to temperature and salinity, the variation of the coefficient of thermal expansion with the same variables, and the total heat required to melt one gram of sea-ice of given salinity and temperature. This last table gives an effective latent heat for melting. For the freezing sea-water, an equation of Malmgren frequently used is ^^1^4: (2) 2 4 6 8 10 15*. Fig. 3. Specific heat of sea-ice as a function of temperature and salinity. where L, Lp are the latent heats of fusion of sea-ice and pure ice, and Si, 8w are the salinities of the sea-ice and sea-water. Malmgren's results will not be repeated here except that some of the data on specific heat are plotted in Fig. 3. The thermal conductivity of sea-ice is, like most other properties of this odd material, dependent on its physical state. The air bubble content has a marked influence. Malmgren found values from 1.5-5.0 x 10"^ cal cm~2 sec"i, with the higher values applicable to the lower portions of an ice cover. The effect of gas content and pressure on the thermal conductivity of ice was investigated by Vlasov and Uspenskii (1931) and by Shuleikin et at. (1931). They showed that for pure ice an increase of pressure decreased the thermal conductivity, but that for porous ice containing carbon dioxide a pressure rise increased the conductivity. At constant pressure an increase in gas content reduced the con- ductivity. A number of observers have reported values of thermal conductivity, all within the range found by Malmgren. These values have usually been computed from measurements on growth rates of ice sheets and are thus not too 834 POUNDER [sect. 7 reliable because of the many uncertain factors involved in the calculations. A good average figure is about 3.5 x 10~3 c.g.s. units. 4. Electrical Properties Pure ice can be considered as a readily polarizable, poor conductor or as a dielectric with a large loss factor. Mantis (1951) and Dorsey (1940) summarize the extensive investigations of the electrical properties of ice. Sea-ice, with its inclusions of ionic salt solutions, would be expected to show a higher electrical conductivity and a markedly anisotropic conductivity because of the crystal structure discussed earlier. The only extensive investigation, that of Dichtel and Lundquist (1951), confirmed these conclusions. Pounder and Little (1959) showed that the resistivity in annual sea-ice increases in the later months of the winter owing to brine drainage. 5. Growth and Disintegration of an Ice Cover Here it is proposed to summarize the physical factors and give references to some of the many papers with empirical or theoretical growth and disintegra- tion equations. Two distinct problems exist for the cooling period of fall and winter : first, to predict the date of first ice formation and, second, to predict the rate of ice growth. The first problem depends on both oceanographic and meteorological conditions. Its solution is usually carried out using the "ice potential" method introduced by Zubov (1938). Data from oceanographic stations are used to find the depth of the thermocline and the salinity and temperature distribution above it. Models of convective mixing in the water and of heat transfer to the air are assumed. Climatological data permit a prediction of the rate at which freezing exposure will accumulate and hence a prediction of the date of first ice formation (Simpson, 1958). The freezing exposure, Et, sometimes called the degree-days of frost, is the product of time in days and the average difference between the air temperature and the freez- ing point of sea-water. Once a cover of sea-ice forms the processes of heat removal change. The latent heat of fusion of the sea-water is the principal heat source and contribu- tions from the bulk of the water are small. In the Arctic, with the very light snowfall, conduction through the ice cover is the limiting factor on growth. Elsewhere, the amount of snow cover may be very important ; Holtsmark (1955) measured growth rates for varying snow thicknesses. Stefan (1891) set up the problem of ice growth mathematically and for a particular case obtained a solution for the ice thickness h in the form, /i2 = {2klLd)Et, (3) where Et is the freezing exposure since first ice formation, k, L and d are the thermal conductivity, latent heat and density of the ice respectively. This solution ignores the snow cover, any radiation imbalance and heat transport SECT. 7] THE PHYSICS OF SEA-ICE 835 through the water. A number of attempts have been made to obtain more general solutions of the problem. Kolesnikov (1958) gives a discussion with references. Most of these solutions have been too involved to be of much use in practical calculations. Stefan's equation works sufficiently well that a number of equations based on it are frequently used; Assur (1956) may be cited as an example. In these, equation (3) or a slight modification of it is usually assumed, but with an empirical coefficient dependent on snow cover and type of ice. The decay of an ice cover is largely controlled by solar radiation and by the albedo of the surface. The ice stops growing and starts to decay a considerable time before the air temperature rises to the melting point of ice. The later stages of the decay of an annual ice cover in the Arctic are startlingly rapid. It takes place at a season with 24-h daylight and, until there are significant amounts of open water, under usually cloudless skies. The albedo of the snow cover changes within days or even hours from a value of 0.9, typical of clean snow, to as low as 0.45. The snow cover melts rapidly leaving wet ice whose albedo is almost as low. An ice cover 8 ft thick can melt completely within 6 weeks. The discussion above applies to annual sea-ice. The regime of the perennial pack-ice in the Arctic and Antarctic is more complex. In winter this ice grows by accretion of sea-ice at its bottom surface ; in summer the upper surface melts. Since the ice cover is broken into floes at this time, the relatively fresh meltwater drains to the underside of the ice and may refreeze there since its freezing point is higher than that of the saline water. These processes lead to an ice cover of fairly constant thickness of between 3 and 4 m. Solar radiation seems to be the dominant control. For more details see Yakovlev (1958) and Untersteiner and Badgley (1958). 6. Theory of Sea-Ice Structure and Properties Important theoretical developments were reported in Anderson and Weeks (1958), Anderson (1958) and Assur (1958). The primary purpose of these theories is to account for the extreme variation with temperature and salinity of the strength of sea-ice, from much less than that of pure ice at high tempera- tures to at least twice as great at low temperatures. Space permits only a brief and over-simplified summary of the arguments. It was pointed out earlier that a sea-ice crystal contains brine inclusions distributed regularly in parallel planes which are perpendicular to the c-axis. For stress applied parallel to the c-axis, which is the direction of minimum tensile strength of the ice, these liquid cells reduce the effective cross-sectional area. In addition they act as stress concentrators. Anderson explains the reduced strength of sea-ice as jointly caused by these two effects and writes the general equation a = crp{l-^e)lk, (4) where a and o-j, are the strengths of sea-ice and pure ice, /3e is the fraction of unit area of the failure plane occupied by brine, and k is the stress concentra- tion factor, which is equal to 3 for small, isolated, circular cylinders. Assur 836 POXJNDER [sect. 7 agrees with this equation but points out that measurements on natural fresh ice do not give dp since such ice always contains air bubbles and other flaws which also act as stress concentrators. He prefers to use an equation of the form (7 = ao(l-|8,), (5) where ctq is the "basic" strength of sea-ice obtained by supposing all the brine (and air) pockets filled with ice but with minute stress concentrators present. The next step is to relate ^e to the relative brine volume v of the ice, where v is the fraction of the total volume occupied by liquid. The salinity S obtained from a melted ice sample does not give v directly since at any temperature below — 8.2°C some of the salt will be in the solid form. From the phase diagram of sea-ice it is possible to calculate ;^ as a function of salinity and temperature. The calculation of ^e from v requires a model of the size, shape and distribution of the brine cells. Assur considers several models and concludes that the most likely equation is o- = aoil — a-y/v). (6) The constant a depends on the model chosen and can be determined empirically. As sea-water is cooled the various salts are precipitated from solution starting at various temperatures. Figures for these temperatures for the hydrates of sodium sulphate and sodium chloride have already been given ; these are the important ones in practice. The deposition of solid salts reduces V but also has a second and more important effect ; the salt reinforces the ice surrounding the reduced brine pockets and thus gives the much higher strengths at lower temperatures. This is particularly true below — 23°C since sodium chloride is the major constituent. The theory just outlined has had considerable success in quantitative pre- dictions of ice strength and has also been used to derive equations for thermal and electrical conductivities. Additional experimental work^ on phase relations in sea-ice and on its petrography will provide more accurate values of the parameters, but a most promising start has been made on a quantitative theory of sea-ice. References Anderson, D. L., 1958. A model for determining sea ice properties. Arctic Sea Ice, 148, NAS-NRC Pub. No. 598, Washington. Anderson, D. L., 1958a. Preliminary results and review of sea ice elasticity and related studies. Trans. Eng. Inst. Can., 2, 116. Anderson, D. L. and W. F. Weeks, 1958. A theoretical analysis of sea ice strength. Trans. Amer. Geophys. Un., 39, 632. Amol'd-Albiab'ev, V. I., 1939. Strength of the ice in the Barents and Kara Seas. Problemy Arktiki, No. 6, 21. Assur, A., 1956. Airfields on floating ice sheets, for routine and emergency operations. SIPRE (Snow, Ice and Permafrost Research Establishment, Corps of Engineers, U.S. Army, Wilmette, 111.), Tech. Rep. 36. 1 Assur (private communication) reports that much additional information was obtained during the last two years which will modify and refine the relations presented in Fig. 2. SECT. 7] THE PHYSICS OF SEA-ICE 837 Assur, A., 1958. Composition of sea ice and its tensile strength. Arctic Sea Ice, 106, NAS-NRC Pub. No. 598, Washington. BrowTi, J. H. and E. E. Howick, 1958. Physical measurements of sea ice. NEL (U.S. Navy Electronics Laboratory, San Diego) Res. and Develop. Hep. 825. Butkovich, T. R., 1956. Strength studies of sea ice. SIPRE Res. Rep. 20. Butkovich, T. R., 1958. Recommended standards for small-scale strength tests. SIPRE Tech. Rep. 57. Dichtel, W. J. and G. A. Lvmdquist, 1951. An investigation into the physical and electrical characteristics of sea ice. Bull. Nat. Res. Council. No. 122, Washington. Dorsey, N. E., 1940. Properties of Ordinary Water Substance. Reinhold, New York. Faraday, M., 1860. Note on regelation. Proc. Roy. Soc. London, 10, 440. Glen, J. W., 1952. Experiments on the deformation of ice. J. Glaciol., 2, 111. Glen, J. W. and M. F. Perutz, 1954. The growth and deformation of ice crystals. J. Glaciol., 2, 397. Griggs, D. T. and N. E. Coles, 1954. Creep of single crystals of ice. SIPRE Rep. 11. Holtsmark, B. E., 1955. Insulating effect of a snow cover on the growth of young sea ice. Arctic, 8, 60. Jellinek, H. H. G. and R. Brill, 1956. Visco-elastic properties of ice. J. Appl. Phys., 27, 1198. Kolesnikov, A. G., 1958. On the growth rate of sea ice. Arctic Sea Ice, 157, NAS-NRC Pub. No. 598, Washington. Kriimmel, O., 1907. Ice in the ocean. Handb. Oceanog., Stuttgart, 498-526. (Text in German.) Laktionov, A. F., 1931. The properties of sea ice. Trudy Inst. Izucheniiu Severa, 49, 71. Langleben, M. P., 1959. Some physical properties of sea ice. II. Can. J. Phys., 37, 1438. Malmgren, F., 1927. On the properties of sea ice. Norweg. N. Polar Exped. with the ''Maud" 1918-1925, Set. Res., 1, no. 5, 67 pp. Mantis, H. T., 1951. Review of the properties of snow and ice. SIPRE Res. Rep. 4. Xakaya, U., 1959. Visco-elastic properties of snow and ice in the Greenland ice cap. SIPRE Res. Rep. 46. Nelson, K. H., 1954. Deposition of salts from sea water by frigid concentration. Tech. Rep. 29, Dept. of Oceanography, Univ. Washington. Nelson, K. H. and T. G. Thompson, 1953. Desalting of sea water by freezing processes. Tech. Rep. 13, Dept. of Oceanography, Univ. Washington. Northwood, T. D., 1947. Sonic determination of the elastic properties of ice. Can. J. Res., A25, 88. Oliver, J., A. P. Crary and R. Cotell, 1954. Elastic waves in Arctic pack ice. Trans. Amer. Geophys. Un., 35, 282. Percy, F. G. J. and E. R. Pounder, 1958. Crystal orientation in ice sheets. Can. J. Phys., 36, 494. Peschansky, I. S., 1957. On certain problems of Arctic ice study. Problemy Arktiki, No. 2. Peschansky, I. S., 1958. Physical and mechanical properties of Arctic ice and methods of research. Arctic Sea Ice, 100, NAS-NRC Pub. No. 598, Washington. Petrov, I. G., 1955. Physical -mechanical properties and thickness of the ice cover. Vol. II, Sect. 6 of Observational Data of the Scientific -Research Drifting Station of 1950-51, Izd. Morskoi Transport. (English translation by American Meteorological Society, Boston.) Pomeroy, P., 1956. Seismic studies in Hopedale, Labrador. Prelim. Rep., U.S. Air Force Cambridge Res. Centre. Povmder, E. R. and E. M. Little, 1959. Some physical properties of sea ice. I. Can. J. Phys., 37, 443. Pounder, E. R. and P. Stalinski, 1960. General properties of Arctic sea ice. Intern. Assoc. Sci. Hydrol. Pub. 54, 25. 838 POUNDER [SECT. 7 Pounder, E. R. and P. Stalinski, 1960a. Elastic properties of Arctic sea ice. Intern. Assoc. Sci. Hydrol. Pnh. 54, 35. Ringer, W. F., 1928. Changes in the composition of sea-water salt during freezing. Rapp. Cons. Explor. Mer., 47, 226. Shuleikin, V. V., 1953. Physics of the Sea, 3rd Ed. Izdatel'stvo Akademii Nauk SSSR, Moscow. (Text in Russian.) Shuleikin, V. V. et al., 1931. The relation between the thermal conductivity of ice and its structure. Zhur. Geofiz., 1, No. 1-2, 179. Shumskii, P. A., 1955. A study of ice in the Arctic Ocean. Vestnik Akad. NaukS.S.S.R., 25, No. 2, 33. Simpson, L. S., 1958. Estimation of sea ice formation and growth. Arctic Sea Ice, 162, NAS-NRC Pub. No. 598, Washington. Stefan, J., 1891. The theory of ice formation especially in the Arctic. Ocean. Ann. Phys. Chem. N.F., 42, 269. Sverdrup, H. U., 1956. Arctic Sea Ice in the Dynamic North. Book I, part VI, Technical Assistant to Chief of Naval Operations for Polar Projects, U.S. Navy, Washington. Sverdrup, H. U., M. W. Johnson and R. H. Fleming, 1942. The Oceans, their physics chemistry and general biology. Prentice-Hall, New York. Tabata, T., 1955. A measurement of visco-elastic constants of sea ice. J. Oceanog. Sac. Japan, 11, 185. Tabata, T., 1958. Studies on visco-elastic properties of sea ice. Arctic Sea Ice, 139, NAS- NRC Pub. No. 598, Washington. Untersteiner, N. and F. I. Badgley, 1958. Preliminary results of thermal budget studies on Arctic pack ice during summer and autumn. Arctic Sea Ice, NAS-NRC Pub. No. 598, Washington. Vlasov, L. I. and P. N. Uspenskii, 1931. The variation of the thermal conductivity of pure and porous ice with pressure. Zhur. Geofiz., 1, No. 1-2, 187. Weeks, W. F., 1958. The structure of sea ice. Arctic Sea Ice, 96, NAS-NRC Pub. No. 598, Washington. Whitman, W. G., 1926. Elimination of salt from sea-water ice. Amer. J. Sci., 211, 126. Workman, E. J. and W. Drost-Hansen, 1954. The electrical conduction and dielectric properties of ice. Phys. Rev., 94, 770. Yakovlev, G. N., 1958. Solar radiation as the chief component of the heat balance of the Arctic Sea Ice, 181, NAS-NRC Pub. No. 598, Washington. Zubov, N. N., 1938. Marine Water and Ice. Gidrometeorologicheskoe Izdatel'stvo, Moscow. (Text in Russian.) Zubov, N. N., 1945. The icing of the Arctic. Izdatel'stvo Glavsermorputi, Moscow. (Text in Russian.) Zubov, N. N., 1947. Dynamic oceanography. Gidrometeorologicheskoe Izdatel'stvo, Moscow. (Text in Russian.) AUTHOR INDEX Authors' names of Chapters in this book and the page numbers at which these Chapters begin are printed in heavy type, page numbers of citations in the text are in ordinary type and those of bibliographical references listed at the ends of the chapters (including joint authors) are in italics. Where the same author is mentioned in more than one chapter it may be convenient to differentiate by subject between the successive ranges of page numbers by consulting the list of contents at the begirming of the book. Names associated with known apparatus, equations, laws, principles, etc. are not entered here but in the Subject Index. Abbot, R., 167, 287 Agamat, E.-H., 10, 28 Airy, G. B., 779, 799 Akima, T., 660, 663 Alaka, M. A., 173, 291 Aldrich, H. L., 556, 563 Alford, R. S., 554, 564 Anderson, D. L., 827, 831, 835, 836 Anderson, E. C, 316, 320 Anderson, E. R., 66, 80, 82, 84, 814, 824 Anderson, L. J., 84 Anderson, V. C, 513, 514, 537 Angstrom, A., 449, 456, 466 Arnold, J. R., 316, 320 Arnol'd-Albiab'ev, V. I., 831, 836 Arons, A. B., 20, 21, 28, 30, 68, 84, 89, 138, 293, 325, 343, 347, 355, 365, 367, 379, 380, 393, 395, 537 Arrhenius, G., 304 Arthur, R. S., 660, 662, 679, 680, 696, 747, 762 Arx, W. S. von, 694, 696, 812, 825 Assur, A., 830, 831, 835, 836, 837 Atkins, W. R. G., 410, 440, 449. 466 Austin, R. W., 447, 449 Avsec, D., 208, 211, 287 Azhazha, V. G., 541, 549, 563 Backus, R. H., 464, 466, 498, 504, 505, 507, 511, 512, 525, 534, 535, 537, 538, 540, 555, 560, 561, 563, 565 Badgley, F. I., 58, 85, 835, 838 Baer, L., 623, 644 Bagnold, R. A., 682, 696 Bagrov, N. A., 127, 128, 129, 134, 287 Baker, A. de C, 510, 537 Balay, M. A., 662 Ball, F. K., 71, 84 Balls, R., 499, 537 Banerji, S. K., 700, 701, 704, 717 Banks, R. E., 616, 645 Banos, A., 472, 475 Barber, N. F., 569, 578, 579, 580, 585, 588, 664, 676, 679, 684, 692, 693, 694, 695, 696 Barbour, G. B., 296, 302 Barham, E. G., 504, 509, 537 Barnes, H., 512, 537 Bartels, J., 764, 799 Batchelor, G. K., 84, 807, 822 Bath, M., 706, 716, 717 Batzler, W. E., 500, 537 Baur, F., 127, 287 Baylor, E. J., 466, 466 Bean, B. R., 167, 287 Beer, T., 462, 466 Bein, W., 3, 9, 28, 29 Benard, H., 208, 287 Benioff, H., 713, 714, 717 Beranek, L., 543, 563 Bergstrom, S., 497 Berliand, T. G., 75, 84 Bernard, P., 701, 715, 717 Beuttell, R. G., 438, 450 Beyer, L., 558, 563 Bien, G., 317, 318, 320 Bird, J., 320 Birkhoff, G., 571, 573, 588 Bjerknes, J., 115, 280, 281, 282, 283, 284, 287 Bjerknes, V., 10, ^9, 31, 41, 323, 393 Blackadar, A. K., 54, 56 Blackman, R. B., 586, 588 Blackwelder, E., 304 Blaik, M., 717 Blanchard, D. C, 305, 307, 308, 311, 312 Boden, B. P., 441, 443, 444, 450, 458, 466, 467, 500, 508, 534, 537, 539 Boenninghaus, G., 553, 563 Bolin B., 316, 317, 320 839 840 AUTHOR INDEX Bourret, R. C, 238, 293 Bowden, K. F., 598, 609, 623, 625, 643, 773, 775, 799 802, 811, 813, 822 Bowen, I. S., 105, 287 Bowles, P., 816, 821, 822 Bradley, W. H., 301, 302 Bramlette, M. N., 301, 302, 304 Brauer, A., 463, 466 Bray, J. R., 316, 320 Breder, C. M., Jr., 550, 559, 563, 565 Breslau, L. R., 458, 461, 466, 510, 537 Bretschneider, C. L., 671, 675, 696 Brewer, A. W., 135, 287, 438, 450 Bridgman, P. W., 514, 532 Brill, R., 832, 837 Brocks, K., 55, 56, 58, 59, 64, 66, 69, 70, 84 Broecker, W. S., 316, 317, 319, 320 Brooks, C. E. P., 595, 609 Brown, J. H., 831, 837 Brown, P. K., 463, 468 Brown, P. R., 65, 84 Brown, P. S., 463, 468 Brown, W. F., 295, 302 Bruch, H., 54, 69, 84 Bryan, K., 88, 124, 125, 287 Budyko, M. I., 49, 75, 84, 97, 99, 100, 102, 103, 104, 105, 110, 114, 115, table 116, table 117, 119, 120, 121, table 122, 124, 125, 126, 127, 128, 129, 133, 134, 135, 136, 137, 140, table 157, 167, 288 Bull, G. A., 297, 303 Bunker, A. F., 77, 84, 88, 95, 99, 100, 113, 147, 156, 188, 205, 207, 208, 209, 210, 214, 217, 268, 269, 271, 288 Burkenroad, M. D., 552, 559, 563 Burling, R. L., 533, 537 Burling, R. W., 317, 320, 723, 725, 729 Burns, R. B.., 822 Butkovich, T. R., 829, 831, 837 Byers, H. R., 201, 235, 288, 292 Callendar, G. S., 315, 319, 320 Carder, D. S., 706, 708, 710, 717 Carlisle, D. B., 463, 466 Carnevale, E. H., 497 Carpenter, J. H., 9, 29 Carrier, G. F., 348, 350, 394, 589, 657, 663 Carritt, D. E., 9, 29 Cartwright, D. E., 567, 573, 580, 582, 584, 588, 608, 609, 647, 669, 671, 683, 685, 690, 695, 696 Castaing, R., 298, 303 Catton, D., 576, 588, 685, 696 Cedarstraw, C, 450 Cepeda, H., 651, 663 Chaffee, M., 231, 235, 290 Chalmers, R., 450 Chandler, 606 Chang, S. S. L., 585, 588, 692, 696, 698 Chappius, P., 9, 29 Charney, J. G., 350, 356, 394, 818, 822 Charnock, H., 59, 76, 84, 103, 107, 181, 199, 288, 293, 816, 822 Chase, J., 688, 693, 696 Chevallier, A., 12, 13, 30 Chigrin, N. J., 3, 30 Chilowsky, C, 498 Christensen, R. J., 499, 55 S Chrystal, G., 767, 799 Clack, B. W., table 27 Clark, E., 550, 559, 563 Clarke, D., 135, 288 Clarke, G. L., 449, 456, 457, 458, 461, 462, 464, 466, 504, 534, 535, 537, 538 Clauser, F. H., 49, 50, 84 Clerici, E., 295, 303 Cochrane, J. D., 660, 662, 147, 763 Coe, J. R., table 27 Coles, N. E., 832, 837 Collatz, L., 767, 799 Colon, J., 88, 91, 104, 112, 148, 149, 151, table 152, 153, 154, 156, table 157, 158, 159, 160, 161, 163, 167, 169, 170, 288 Cooke, 461 Cooper, R. I. B., 703, 717 Corkan, R. H., 611, 627, 635, 638, 643, 800 Cornish, V., 671, 696 Cote, L. J., 577, 588, 696 Cotell, n., 837 Coulson, C. A., 567, 588 Courant, R., 782, 799 Cousteau, J. Y., 495, 496, 510, 538 Cox, C. S., 63, 84, 658, 695, 697, 720, 721 723, 724, 726, 728, 729, 752 Cox, D. C, 659, 663 Cox, R. A., 12, 29 Craig, H., 5, 29, 316, 320 Craig, R. A., 77, 84 Craig, R. E., 510, 538 Cramer, H. E., 46, 84, 572, 588 Crapper, G. D., 726, 729 Crary, A. P., 837 Crease, J., 394, 638, 643, 657, 662 Curcio, J. A., 433, 434, 450 Cushing, D. H., 464, 467, 499, 502, 519, 532, ■538 AUTHOR INDEX 841 , 59, 822 767, 800, Dalton, 49 Damste, B. R., 645 Dantzig, D. van, 633, 643, 644 Darbyshire, J., 60, 64, 85, 588, 635, 644, 673, 675, 696, 700, 707, 708, 71 1, 713, 716, 717 Darbyshire, M., 60, 64, 85, 635, 644, 675, 696, 717 Darwin, C, 298, 303 Darwin, G. H., 598, 609 Davidson, B., 181, 290 Davies, T. V., 570, 588 Davis, J. O., 588 Davis, L. C, 523, 538 Dawson, L. H., 433, 434, 450 Deacon, E. L., 43, 48, 49, 51, 54, 55, 58 64, 69, 72, 84, 106, 179, 180, 288 Deacon, G. E. R., 700, 701, 717 Deardorff, J. W., 58, 85 Defant, A., 738, 740, 747, 762, 764, 770, 778, 779, 781, 783, 799, 799, 809, 822 De Groot, S. R., 23, 28, 29 Deland, R. J., 45, 46, 86 Del Grosso, V. A., 477, 478, 496 Denton, E. J., 456, 461, 463, 467 Depperman, C. E., 235, 288 Devik, O., 323, 393 Devin, C, Jr., 517, 518, 538 Devoid, F., 499, 538 DeVries, H., 317, 520 Dichtel, W. J., 834, 837 Dietrich, G., 3, 29, 394, 817, 818, 822 Dietz, R. S., 461, 467, 486, 496, 500, 508, 538, 745, 762 Dijkgraaf, S., 553, 563 Dinger, J. E., 705, 717 Disney, L. P., 596, 609 Dittmar, W., 5, 29 Dobrin, M. B., 554, 562, 563 Donn, W. L., 639, 644, 657, 658, 660, 662, 663, 676, 696, 705, 714, 717 Doob, J. L., 572, 588 Doodson, A. T., 591, 593, 594, 597, 600, 609, 624, 644, 764, 775, 779, 780, 800, 801, 811, 822 Dorrestein, R., 688, 696 Dorsey, N. E., 9, 29, 834, 837 Dow, W., 496, 496 Doyle, D., 696 Draper, L., 685, 686, 687, 696 Dronkers, J. J., 784, 786, 800 Drost-Hansen, W., 826, 838 Drozdov, O. A., 114, 129, table 130, 131, 137, 169, 288 661, 608, 784, Dufosse, A., 541, 558, 563 Dunn, G. E., 221, 231, 288 Dunn, H. K., 492, 497, 524, 539 Duntley, S. Q., 448, 450, 452, 454, 455 Duvall, G. E., 499, 538 Ebeling, A., 518, 535, 539 Eckart, C, 5, 10, 11, 12, 29, 31, 33, 36, 40, 41, 642, 644, 752, 762, 809, 810, 822, Edgerton, H. E., 458, 466, 495, 496, 510, 537 Edrisi, 295 Egedal, J., 596, 609 Ehrenberg, C, 295, 298, 303 Ehrenfeld, S., 581, 555 Ekman, V. W., 9, 10, 29, 275, 323, 341, 394, 731, 762, 810, 822 Elder, J. W., 822, 823 Elliott, W. G., 438, 451 Ellis, L. G., 533, 538 Ellison, T. H., 53, 85, 815, 823 Emerson, R., 415, 450 Emery, K. O., 682, 698 Emling, J. W., 554, 564 Emmel, V. M., 314, 321 Eppley, R. A., 706, 710, 715, 717 Erickson, D. B., 302, 303 Eriksson, E., 311, 312, 316, 317, 320 Essapian, F. S., 559, 560, 565 Everest, A. F., 553, 556, 562, 563 Everest, Y. A., 564 Ewing, G. C., 824 Ewing, M., 302, 303, 316, 320, 481, 496, 496, 497, 637, 644, 657, 658, 659, 661, 662, 706, 707, 714, 717, 718, 719 Eyring, C. F., 499, 538 Fairbairn, H. W., 301, 303 Fairbairn, L. A., 779, 800, 811, 813, 816, 822 Fairbridge, R. W., 313, 321 Faller, A. J., 89, 138, 293, 355, 395 Faraday, M., 828, 837 Farmer, H. G., 688, 695, 697, 823 Fedorov, K. N., 633, 644 Feely, H. W., 297, 303 Felsenbaum, A. I., 323, 394 Fergtisson, G. J., 316, 317, 319, 320 Ferris, H. G., 40, 41 Feshbach, H., 513, 539 Fessenden, R. A., 498 Ficker, H. von, 288 Fischer, G., 639, 644, 673, 698, 791, 800 Fish, M. P., 550, 551, 554, 558, 562, 563, 564 842 AUTHOR INDEX Fisher, G. H., 705, 111 Fisher, L. R., 463, 4:61 Fjelstad, J. E., 731, 752, 753, 162, 799, SOO Fleagle, R. G., 58, 65, 69, 85, 675, 691 Fleming, R. H., 6, 50, 68, SI, 128, 29S, 323, 395, 749, 7(53, 801, 815, 824:, 838 Fletcher, A., 462, 461 Flint, R. F., 595, 609 FofonoflF, N. P., 3, 5, table 14, table 15, 17, 29, 323, 350, 352, 353, 356, 368, 394 Folsom, T. R., 647, 663, 821, 823, 824 Forbes, A., 745, 162 Forrel, 767 Forsch, C, 6, 29 Fraenkel, G. S., 464, 461 Francis, J. R. D., 59, 60, 62, 84, 85, 107, 181, 199, 288, 293, 675, 697, 823 Fraser, F. C, 553, 557, 564 Frassetto, R., 537, 538 Fredriksson, K., 298, 303 Freeman, J. C, 623, 624, 643, 644 Frenkiel, F. N., 807, 823 Friariksson, A., 499, 538 Fritz, S., 288 Froese, C, table 14, table 15, 16, 29 Fry, W. J., 491 Fuglister, F. C, 153, 288, 333, 392, 394 Fukuda, M., 430, 450 Fultz, D., 89, 93, 99, 138, 198, 288, 292 Futi, H., 296, 303 Gabites, J. F., 154, 165, 289 Gail, F. W., 440, 451 Gangopadhyaya, G., 91, 289 Gardiner, A. C., 464, 468 Garner, D. M., 317, 320 Garstang, M., 215, 224, 225, 227, 228, 229, 289 Gelci, R., 674, 691 Gentry, C, 221, 292 Gerard, H., 316, 320 Gershun, A., 401, 445, 450 Gerstner, F. J. von, 587 Gherzi, E., 700, 111 Gifford, M. M., 307, 312 Gilmore, M. H., 710, 118 Ginnings, D. C, 30 Gissler, 593 Glawion, H., 295, 303 Glen, J. W., 832, 831 Godfrey, T. B., table 27 Gohin, 786 Goldberg, E. D., 295, 299, 300, 304, 824 Goldie, E. H., 463, 461 Goldsbrough, G. R., 138, 289, 638, 644, 779, 800 Goode. G. B., 558, 564 Goodman, N. R., 584, 588 Goody, R. M., 201, 289 Gordon, A. R., table 27 Gossard, E., 658, 662 Grant, H. N., 807, 813, 823 Gray, W. S., 210, 215, 292 Greene, C. W., 541, 564 Greenspan, H. P., 642, 644, 657, 662 Greenspan, M., 477, 478, 491 Griffin, D. R., 300, 541, 564 Griggs, D. T., 832, 831 Green, P., 611, 621, 627, 628, 632, 634, 639, 644, 646, 691, 752, 162 Groves, G. W., 350, 394, 591, 609, 611, 614, 615, 616, 617, 644 Guggenheim, E. A., 4, 29 Gunn, D. L., 464, 461 Gutenberg, B., 596, 597, 609, 662, 700, 705, 713, 714, 118 Haeggblom, L. E., 68, 85 Hamada, T., 812, 523 Hamon, B. V., 28, 29, 333, 395, 810, 824 Hansen, W., 639, 644, 764, 773, 778, 784, 791, 800, 817, 823 Hanzawa, M., 820, 823 Harbeck, G. E., 73, 85 Harden Jones, F. R. : see Jones, F. R. Harden Hardy, A. C., 463, 461 Harris, D. L., 617, 633, 637,- 641, 644 Harris, J. E., 465, 461 Hart, S. R., 301, 303 Hashimoto, T., 545, 551, 564 Haskell, N. A., 706, 118 Hasselman, 586 Hastings, J. W., 460, 468 Hatherton, T., 716, 118 Haubrich, R., Jr., 606, 609 Haurwitz, B., 288, 636, 645, 1^1, 753, 754, 162, 799, 800 Hays, E. E., 478, 491, 537, 538 Haxo, F. T., 460, 468 Hebb, D. O., 556, 564 Hecht, F., 298, 303 Heezen, B. C, 302, 303, 316, 320 Heisenberg, W., 806, 823 Hela, 593 Helland-Hansen, B., 15, 29, 261, 289, 1^1, 162 AUTHOR INDEX 843 Hellman, G., 295, 303 Hellstrom, B., 60, 85 Hensen, W., 791, 800 Herdman, F. H. P., 502, 538 Hersey, J. B., 476, 494, 497, 498, 500, 504, 505, 507, 511, 522, 523, 524, 525, 531, 532, 533, 538, 539, 540, 543, 545, 547, 558, 564 Hesselberg, Th., 16, 29, 37, 41, 323, 393 Hewson, E. W., 260, 289 Hidaka, K., 181, 289, 332, 394, 818 Hilbert, D., 782, 799 Hildebrand, S. F., 559, 564 Hinde, B. J., 709, 714, 718 Hintenberger, 313 Hirona, T., 718 Hirsekorn, H.-G., 29 Hiser, H. W., 238, 293 Hochstrasser, U., 707, 718 Hodgson, W. C, 499, 538 Hofsommer, D. J., 633, 634, 644 Holmes, R. W., 416, 451 Holsters, 784, 800 Holtsmark, B. E., 834, 837 Horton, J. W., 477, 480, 488, 489, 490, 497, 541, 564 Hosokawa, H., 553, 564 Hough, S. S., 138, 289, 779, 800 Houghton, H. G., 289 Houghton, J. T., 97, 102, 114, 126, 127, 128, 135, 287 Houtz, R. W., 662 Howick, E. E., 831, 837 Hubbard, C. J., 441, 443, 444, 450, 457, 458, 466, 504, 538 Hudswell, F., 822 Hughes, P., 816, 822 Hulburt, E. O., 433, 434, 450 Hunt, I. A., 623, 645 Hunt, F. v., 498, 538, 639 Hunt, J. N., 817, 823 Hurley, P. M., 301, 303 Hurt, D. A., 231, 289 Ichiye, T., 323, 365, 370, 394, 570, 589, 818, 819, 823, 824 Ide, J. M., 483, 497 Iglesias, H. W., 647, 663 Imbert, B., 715, 718 Inman, D. L., 651, 655, 662 Inouye, W., 705, 718 Isaacs, J., 510 Isaacs, J. D., 679, 696, 824 Iselin, C. O'D., 88, 282, 284, 289 Ishiguro, S., 660, 662 Ivanov, A., 415, 431, 450, 466, 467 Ivanov-Frantzkevich, G. N., 16, 29 Ivanov, V. N., 823 Iyer, H. M., 709, 711, 712, 713, 714, 718 Jacobs, L., 297, 303 Jacobs, W. C., 43, 49, 66, 85, 93, 99, 100, 105, 106, 114, table 116, table 117, 119, 129, table 130, table 157, 169, 262, 263, 289 Jacobsen, J. P., 814, 823 Jahn, T. L., 561, 564 James, D. G., 297, 303 James, H. R., 456, 466 James, R. W., 675, 698 Jander, R., 465, 467 Jardetzky, W. S., 481, 496 Jaumann, G., 31, 33, 41 Jeffreys, H., 62, 211, 289, 604, 607, 609, 669, 697 Jellinek, H. H. G., 832, 837 Jerlov (Johnson), N. G., 430, 435, 436, 438, 439, 447, 450, 466, 467 Jessen, V., table 27 Johnson, H. R., 497, 504, 507, 511, 523, 532, 538 Johnson (Jerlov), N. G.: see Jerlov (John- son), N. G. Johnson, J. W., 679, 697 Johnson, M. W., 30, 87, 127, 293, 323, 395, 500, 508, 512, 538, 553, 556, 562, 563, 564, 749, 763, 801, 824, 838 Johnstone, N. B., 314, 320 Johnson, N. K., 58, 64, 68, 69, 72, 85 Jones, D., 433, 451 Jones, F. R. Harden, 501, 539, 552, 564 Jones, H. Spencer, 605, 609 Jones, O. A., 715, 718 Jones, W. M., 715, 718 Jordan, C. L., 231, 289 Joseph, J., 3, 29, 447, 450, 817, 820, 821, 823 Jung, G. H., 623, 644 Junge, C. E., 297, 303, 311, 312 Kajiura, K., 617, 620, 645, 657 Kampa, E. M., 441, 443, 450, 458, 466, 467, 534, 539 Kampe de Feriet, J., 572, 589 Kanwisher, J., 319, 320, 507, 518, 535, 539 Kaplan, S., 588 Karman, T. von, 81, 107, 202, 289, 804, 809, 823 Katsuragi, Y., 297, 303 844 AUTHOR INDEX Katzin, M., 721, 729 Keeling, C. D., 319, 320 Kellogg, W. N., 560, 564 Kelvin, 764 Kennedy, D., 463, 467 Kerr, D. E., 84 Kesteven, G. L., 541, 564 Ketchum, D. D., 695, 697 Keulegan, G. H., 60, 85, 726, 729 Khintchine, A., 577, 589 Kientzler, C. F., 21, 28, 68, 84, 306, 312 Kierstead, H. A., 587, 589 Kimball, H. H., 121, 127, 289 Kivisild, H. R., 639, 645 Kleinenberg, S. E., 553, 564 Knudsen, M., 9, 12, 21, 29 Knudsen, V. O., 554, 564 Koczy, F. F., 815, 823 Koenig, W., 492, 497, 525, 539 Kohler, M. A., 85 Koizumi, M., table 27 Kolbe, R. W., 298, 299, 303 Kolesnikov, A. G., 813, 823, 835, 837 Kolmogoroff, A. N., 806, 823 Kolthoff, I. M., table 27 Komplanetz, M. V., 716, 719 Konig, 313 Korringa, P., 465, 467 Korvin-Kroukovsky, B. W., 683, 690, 697 Kotik, J., 571, 573, 588 Kozlyaninov, 434, 437, 447, 450 Kraus, E. B., 158, 170, 199, 201, 218, 289 Kreiss, H. O., 787, 788, 790, 800 Krilov, J. {or U.) M., 670, 671, 697 Kristjonsson, H., 499, 538 Kritzler, H., 556, 564 Kriimmel, O., 3, 26, 29, 837 Kuenen, P. H., 301, 303 Kuhn, 464 Kuiper, J., 560 Kullenberg, B., 557, 564, 733, 762 Kuwahara, S., 16, 30, All, 478, 497 Lacy, L. Y., 492, 49 7, 524, 539 LaFond, E. C, 3. 30, 658, 682, 698, 730, 732, 733, 734, 735, 738, 742, 745, 747, 757, 762, 763 Laitenen, H. A., table 27 Laktionov, A. F., 829, 837 Lamb, H., 32, 41, 567, 568, 570, 573, 589, 598, 609, 654, 657, 662, 665, 666, 682, 686, 697, 720, 725, 729, 742, 744, 763, 764, 801, 803, 823 Lambeth, D., 709, 718 Langleben, M. P., 831, 837 Laplace, P. S., 764, 801 Larsson, W., 295, 303 LaSeur, N. E., 292 Laurila, E., 60, 86 Lauwerier, H. A., 633, 634, 639, 645 Lawford, A. L., 729 Lawrence, B., 553, 556, 560, 561, 565 Leavitt, B., 514 Lebel, F., 674, 697 Lee, A. J., 501, 502, 538, 539 Lee, A. W., 707, 718 Lee, O. S., 735, 738, 747, 763 Leet, L. D., 707, 718 LeNoble, J., 397, 450, 457, 467 Leppik, E., 645 Lesser, R. M., 816, 823 Lettau, H., 181, 290 Levine, J., 90, 175, 216, 290 Liapounoff, 572 Libby, W. F., 297, 303 Liebermann, L. N., 469, 480, 497 Liljequist, G., 416, 450 Lindberg, R. G., 560, 564 Lineikin, P. S., 324, 365, 369, 394 Lisitzin, E., 593, 595, 609 List, R. J., 151, 290 Litovitz, T. A., 497 Little, E. M., 834, 837 Lock, R. C, 728, 729 Lohman, K. W., 298, 303 Lohr, E., 3fl, 4i London, J., 97, 102, 104, 114, 127, 128, 134, 165, 166, 167, 290 Longard, J. R., 616, 645 Longuet-Higgins, M. S., 571, 573, 574, 575, 576, 577, 578, 581, 582, 587, 588, 589, 669, 682, 695, 697, 701, 702, 703, 704, 706, 707, 715, 717, 718 Lorenz, E. N., 138, 290 Love, A. E. H., 752, 763 Lowenstein, O., 561, 564 Lowrey, C. A., 231, 289 Loye, D. P., 562, 564 Ludlam, F. H., 92, 216, 290, 293 Lundquist, G. A., 834, 837 Lyman, J., 6, 30 Lysenko, L. N., 716, 719 McBride, A. F., 556, 557, 560, 564 MacDonald, G. A., 659, 663 Macdonald, G. J. F., 607, 609 Machlup, S., 513, 522, 523, 524, 531, 532, 539 AUTHOR INDEX 845 Macelwane, J. B., 710, 718 MacGinitie, G. E., 560, 564 MacGinitie, N., 560. 564 McGough, R. J., 588 McGuiness, W. T., 678, 696 Mcllroy, I. C, 46, 59, 64, 85 MacKenzie, K. V., 478, 497 McVehil, G. E., 54, 86 Malkus, J. S., 88, 90, 91, 95, 104, 112, 133, table 134, 147, 148, 161, 162, 164, 166, 170, 173, 181, 195, 199, 201, 211, 214, 216, 217, 221, 223, 232, 243, 247, 249, 251, 252, 288, 292, 293 Malkus, W. V. R., 108, 138, 174, 175, 177, 188, 202, 210, 215, 219, 235, 290, 823 Malmgren, F., 829, 833, 837 Mamayev, O. I., 815, 823 Manabe, S., 252, 256, 257, 258, 259, 262, 263, table 264, 265, 266, 290, Mandelbaum, H., 729 Maniwa, Y., 545, 551, 564 Mantis, H. T., 267, 291, 834, 837 Mapper, D., 298, 304 Marciano, J. J., 75, 84, 85 Markowitz, 606 Marks, W., 579, 588, 589, 696, 697 Marr, J. C, 499, 538 Marshall, N. B., 519, 533, 535, 539, 562, 564 Marshall, P., 296, 303 Marti, M., 497 Mason, B. J., 312 Massa, 544 Matthews, D. J., 3, 15, 30 Matthews, L. H., 478, 553, 564 Mehr, E., 588, 696 Meinardus, W., 129, 137, 291, 295, 303 Meixner, J., 31, 33, 41 Mellis, O., 301, 303 Meredith, 69, 72 Miche, M., 701, 718 Michell, J. H., 665, 697 Miles, J. W., 178, 291, 658, 662, 670, 697, 728, 729, 730 Milkman, R. D., 463, 467 Miller, A. R., 611, 618, 639, 645, 657, Miller, B. I., 288 Miller, C. S., table 27 Miller, D. J., 231, 267, 291, 659, 663 Miller, G. R., 650, 663 Millis, B. G., 576, 588 Mintz, Y., 133, 291, 303 Mintzer, D., 488, 497 Miyake, Y., 21, table 27, 30, 297, 303 Miyazaki, M., 263, 266, 291 Moilliet, A., 813, 823 Moller, L., 29 Monin, A. S., 52, 78, 85 Montgomery, R. B., 3, 27, 28, 30, 51, 67, 69, 71, 73, 77, 80, 81, 84, 85, 86, 99, 129, 106, 109, 181, 291, 293, 814, 824 163, Moore, H. B., 464, 465, 467, 500, 534, 538, 200, 539 231, Morgan, G. W., 363, 394 290, Morrison, T. J., 314, 320 Morse, P. M., 513, 522, 539 176, Mortimer, C. H., 741, 763 249, Mosby, H., 100, 104, 291, 811, 816, 823, 824 Motais, R., 463, 467 Moulton, J. M., 551, 552, 553, 555, 556, 558, 559, 561, 562, 565 261, Mowbray, W. H., 562, 564 291 Munk, J., 648 Munk, W. H., 62, 64, 65, 84, 86, 87, 90, 178, 218, 291, 323, 342, 344, 346, 348, 350, 394, 568, 578, 589, 595, 596, 605, 606, 607, 609, 620, 643, 645, 647, 649, 650, 651, 652, 657, 658, 660, 662, 663, 665, 671, 672, 675, 676, 679, 680, 682, 685, 695, 696, 697, 698, 721, 723, 724, 726, 729, 730, 749, 753, 754, 762, 763, 814, 552, 817, 821, 824 Munz, F. W., 463, 467 Murai, G., 718 Murie, J., 553, 565 Murphy, T., 462, 467 Murray, C. A., 607, 609 Murray, J., 298, 303 Nakano, M., 659, 663 Nakaya, U., 832, 837 Namias, J., 273, 274, 275, 277, 284, 291 Nanney, C. A., 716, 718 Nan'niti, T., 812, 821, 824 Nansen, F., 273, 289, 14:1, 762 Neeb, G. A., 301, 303 671, Nelson, K. H., 828, 837 Neumann, G., 64, 86, 218, 291, 672, 673, 675, 697, 698, 724, 730 663 Newcomb, S., 606, 609 Newton, C. W., 267, 291 Nichols, J. T., 559, 565 Nicol, A. J. C., 459, 460, 462, 467 Nishimura, M., 545, 551, 564 Norris, K. S., 554 Northwood, T. D., 832, 837 846 AUTHOR INDEX O'Brien, P. A., 695, 697 Obukhov, A. M., 51, 52, 78, 85, 86 Officer, C. B., 482, 497 Ogawa, T., 560, 565 Oholm, L. W., table 27 Okami, N., 451 Okubo, K., 812, 821, 823, 824 Oliver, J., 831, 837 Olson, E. A., 316, 317, 319, 320 Olson, F. C. W., 819, 823, 824 Onishi, G., 294 Onsager, L., 35, 40, 41 Osborne, N. S., 14, 50 Oshiba, G., 451 Oswatitsch, K., 31, 41 Owen, D. M., 504, 505, 507, 511, 538 Ozmidov, R. V., 323, 394, 819, 824 Palmen, E., 60, 86, 91, 95, 133, table 134, 170, 173, 181, 239, 243, 267, 291 Palmer, C. E., 217, 223, 291 Panofsky, H. A., 45, 54, 86 Panteleyev, N. A., 823 Parson, D., 823 Pasquill, F., 79, 86, 110, 291 Patterson, A. M., 810, 813, 524 PattuUo, J., 153, 154, 157, 165, 263, 291, 593, 599, 608, 609 Patzaic, R., 298, 303 Pekeris, C. L., 483, 484, 497, 784 Peres, J. M., 512, 539 Perey, F. G. J., 827, 837 Perutz, M. F., 832, 837 Peschansky, I. S., 831, 832, 837 Petrov, V. P., 823, 831, 837 Pettersson, H., 298, 303, 416, 449, 450, 466 Petty, C. C, 433, 434, 450 Phillips, H., 90, 127, 138, 178, 287 Phillips, N. A., 291 Phillips, O. M., 292, 586, 589, 670, 671, 673, 698, 722, 728, 730 Pierson, W. J., 569, 571, 573, 578, 583, 588, 589, 673, 675, 692, 694, 696, 697, 698 Piest, J., 699 Pillsbury, 818 Pinson, W. H., 301, 303 Pirenne, M. H., 461, 467 Platzman, G. W., 637, 645 Poincare, H., 781, 801 Polli, 593 Pomeroy, P., 831, 837 Poole, H. H., 449, 466 Posner. G. S., 292 Post,.R. F., 497 Pounder, E. R., 826, 827, 829, 831, 832, 834, 837, 838 Prandtl, L., 107, 202, 292, 804, 824 Preisendorfer, R. W., 397, 399, 407, 411, 412, 413, 433, 434, 445, 450, 451, 454, 455 Press, F., 481, 496, 644, 657, 658, 659, 661, 662, 706, 707, 714, 717, 718, 719 Priestley, C. H. B., 46, 47, 53, 76, 77, 78, 86, 106, 157, 181, 271, 292 Prins, J. E., 660, 663 Pritchard, B. S., 438, 451 Proskuryakova, T. A... 719 Prosvirnin, V. M., 716, 719 Proudfoot, D. A., 562, 564 Proudman, J., 386, 394, 598, 599, 600, 609, 610, 625, 633, 637, 645, 764, 773, 775, 780, 784, 801, 803, 809, 811, 815, 818, 822, 824 Pumphrey, R. J., 462, 467 PurveSj 553 Putz, R. R., 573, 589 Pyrkin, Yu. G., 823 Radczewski, O. E., 295, 299, 303 Rafter, T. A., 316, 319, 320 Raitt, R. W., 499, 507, 538, 539 Rakestraw, N. W., 317, 318, 320, 321 Ramirez, J. E., 710, 719 Rand, C., 541, 565 Rao, P., 732, 747, 763 Rattay, M., 799, 801 Rayleigh, Lord, 31, 41, 211, 292, 497, 513, 539, 706 Record, F. A., 46, 84 Redfield, A. C., 618, 645, 657, 663 Reickel, A., 558, 565 Reid, J. L., 747, 763 Reid, R. O., 617, 625, 633, 642, 645, 646, 657, 663 Reik, H. G., 31, 33, 41 Renard, A. F., 303 Revelle, R., 304, 313, 319, 321, 596, 605, 606, 609, 749, 763, 808, 815, 824 Rex, R. W., 295, 299, 304 Reynolds, O.. 49, 802, 824 Rice, S. O., 571, 580, 581, 584, 587, 588 Richards, A. F., 491, 497, 545, 565 Richards, F. A., 3, 30, 314, 321 Richardson, I. D., 501, 502, 519, 532, 538, 539 Richardson, L. F., 498, 805, 819, 820, 824 Richardson, W. H., 451 Richardson, W. S., 416, 441, 443, 444, 450, 733. 763 AUTHOR INDEX 847 Rider, N. E., 48, 49, 51, 79, S6, 110, 292 Riedel, L., 27 Riehl, H., 88, 90, 91, 93, 95, 99, 112, 129, 133, table 134, 138, 148, 156, 159, 161, 163, 164, 166, 169, 170, 173, 174, 175, 176, 177, 181, 191, 194, 201, 204, 210, 215, 217, 218, 221, 224, 232, 233, 239, 243, 245, 247, 249, 289, 290, 291, 292 Rikitake, T., 660, 663 Ringer, W- F., 828, 838 Robinson, A., 365, 370, 372, 374, 375, 377, 394 Robinson, R. A., 20, 21, 30 Rodewald, M., 279, 280, 292 Roll, H. U., 54, 55, 64, 65, 86, 218, 292, 673, 698, 727, 728, 730 Ronne, C, 95, 104, 176, 210, 215, 235, 290, 292, 293 Ronne, F.C., 696 Ropek, J. F., 588 Rose, D., 790, 801 Rosenblatt, H., 572.. 589 Rossby, C. G., 51, 86, 99, 106, 109, 165, 292, 293, 324, 325, 349, 380, 385, 386, 394, 814, 824 Rossiter, J. R., 590, 591, 596, 609, 610, 611, 638, 646, 780, 781, 784, 797, 800, 801 Rouch, J., 3, 30 Rudnick, P., 747, 763 Ruppel, A. E., 492, 497, 525, 539 Russell, F. S., 464, 465, 467, 468 Russell, S., 726, 730 Rykunov, L. N., 716, 719 Saint-Guily, B., 323, 329, 394 Saito, Y., 633, 646 Salmi, M., 295, 304 Sandstrom, J. W., 10, 29, 323, 393 Sargent, R. E., 304 Sarkisian, A. S., 323, 395 Sasaki, T., 430, 451 Savarensky, E. F., 716, 719 Saville, T., Jr.., 639, 646 Schalkwijk, W. F., 611, 623, 636, 638, 639, 646 Scheidig, A., 297, 304 Schevill, W. E., 540, 544, 553, 556. 560, 561, 565 Scholte, J. G., 707, 719 Schonfeld, J. C, 624, 625, 646, 784, 801 Schooley, A. H., 723, 727, 729, 730 Schott, 137 Schroeder, W. C, 559, 564 Scorer, R. S., 92, 215, 293 Seiwell, H. R., 698, 739, 763 Sekerzh-Zenkovich, Ya. I., 726, 730 Seligman, H., 821, 824 Sendner, H., 820, 821, 823 Senn, H. V., 238, 293 Shand, J. A., 731, 763 Shaw, J. J., 710, 719 Shaw, N., 608, 610 Sheehy, M. J., 486, 496 Shelford, V. E., 440, 451 Shepard, F. P., 659, 663, 682, 698 Sheppard, P. A., 54, 58, 59, 64, 70, 72, 76, 81, 82, 83, 84, 85, 86, 106, 107, 179, 180, 181, 197, 199, 288, 293 Shida, T., 560, 565 Shimozuru, E., 660, 663 Shishkova, E. V., 541, 551, 565 Shtokman, W. B., 323, 395 Shvileikin, V. V., 670, 671, 698, 833, 838 Shumskii, P. A., 827, 838 Silvester, R., 683, 698 Simpson, G. C, 127, 293 Simpson, L. S., 834, 838 Slocum, G., 316, 321 Smales, A. A., 298, 304 Smith. G. D., 297, 304 Smith, H. M., 541, 565 Smith, N. D., 12, 29, 695, 697 Smith, P. F., 514, 519, 539 Smith, R. C, 195, 293 Snodgrass, F. E., 589, 647, 648, 649, 650, 652, 657, 658, 663, 676, 685, 687, 697, 698 Snodgrass, J. M., 441, 443, 450, 491, 497, 545, 565 Sommerfeld, A. O., 472, 475 Sorensen, S. P. L., 29 Sorensen, W. E., 541, 559, 565 Southern, R., 464, 468 Spencer Jones, H.: see Jones, H. Spencer Spiegel, E. A., 201, 293 Stalinski, P., 829, 831, 832, 837, 838 Stampehl, H., 558, 565 Stefan, J., 834, 835, 838 Steinberg, M., 519 Stephenson, G., 588 Stern, M. E., 90, 138, 194, 195, 283, 290, 293 Sterneck, R., 767, 801 Stevens, R., 588 Stevenson, G., 696 Stevenson, T., 671 Stewart, R. W., 60, 86, 813, 823 Stimson, H. F., 30 848 AUTHOK INDEX Stoker, J. J., 567, 589, 665, 681, 682, 698 Stokes, G. G., 587, 657 Stokes, R. H., table 27 Stommel, H., 88, 89, 90, 92, 138, 155, 211, 215, 283, 288, 293, 294, 323, 324, 325, 343, 346, 355, 363, 364, 365, 367, 369, 370, 372, 374, 375, 377, 379, 380, 381, 382, 386, 393, 393, 394, 395, 646, 753, 754, 762, 809, 818, 819, 820, 823, 824 Stoneley, R., 707, 718, 719 Strong, E., 609 Stuhlman, O., 312 Subov, N. N., 3, 30 Suess, H. E., 313, 317, 318, 319, 320, 321 Sugiura, Y., 297, 303 Sund, O., 499, 539 Suthons, C. T., 671, table 675, 698 Sutton, O. G., 76, 87 Svansson, A., 639, 646 Sverdrup, H. U., 3, 16, 21 30, 37, 64, 66, 68, 69, 72, 80, 81, 82, 83, 87, 104, 114, 121, table 122, 124, 125, 128, 149, 151, 155, 275, 293, 323, 335, 395, 665, 671, 672, table 675, 698, 749, 763, 764, 801, 814, 815, 816, 818, 824, 829, 838 Swallow, J. C, 333, 347, 395, 495, 497, 772, 810, 824 Sweeney, B. M., 460, 468 Swinbank. W. C, 46, 47, 48, 51, 77, 85, 86, 87, 157 Swindells, J. F., table 27 Symons, G. J., 659, 663 Tabata, T., 832, 838 Takahashi, T., 58, 64, 69, 87 Talwani, M., 496, 497 Tanunann, G., table 27 Tannehill, I. R.. 231, 293 Tavolga, M. C, 560, 565 Tavolga, W. M., 552, 554, 559, 562, 565 Taylor, G. I., 73, 381, 395, 607, 610, 766, 801, 804, 806, 814, 822, 825 Taylor, J. H., 454, 455 Taylor, J. K., 9, 30 Taylor, R. J., 87 Tchernia, P., 499, 508, 539 Thayer, M. C, 88 Thiesen, M., 9, 30 Thomas, B. D., 28, 30 Thompson, T. G., 3, 8, 30, 828, 837 Thomson, W., 15 Thorade, H., 764, 773, 775, 801, 811, 825 Thorarinsson, S., 295, 304 Thoulet, J., 12, 13, 30 Tick, L. J., 572, 586, 587, 588, 651, 663, 698 Tilton, L. W., 9, 30 Timofeev, M. P., 110, 293 Tokarev, A. K., 550, 551, 565 Tolstoy, I., 661, 662 Tomaschek, R., 599, 610 Tomczak, G., 635, 646 Tomilin, A. G., 556, 565 Tomoda, Y., 715, 719 Tower, R. W., 541, 565 Townsend, A. A., 329, 395, 807, 822, 825 Traylor, M. A., 679, 680, 697 Trout, G. C, 501, 539 Tschiegg, C, 477, 478, 497 Tsuchiya, M., 332, 394 Tucek, C. S., 316, 320 Tucker, G. B., 95, 294 Tucker, G. H., 500, 501, 508, 509, 539 Tucker, M. J., 584, 585, 587, 588, 589, 649, 650, 652, 657, 658, 663, 664, 682, 685, 689, 692, 696, 697, 698, 713, 714, 719 Tukey, J. W., 584, 585, 586, 588, 589, 693, 699 Turner, H. H., 598, 609 Twomey, S., 307, 312 Tyler, J. E., 397, 410, 415, 416, 429, 434, 435, 438, 439, 441, 445, 446, 451, 457, 468 Ufford, C. W., 738, 747, 757, 763 Unna, P. J. H., 699 Unoki, S., 659, 663 Untersteiner, N., 835, 838 Upton, P. S., 715, 719 Ursell, F., 569, 588, 657, 663, 669, 676, 679, 696, 699, 701 U.S. Navy, 294 U.S. Weather Bureau, 294 Uspensku, P. N., 833, 838 Utterback, C. L., 30, 449, 466 Valembois, J., 684-, 687, 699 Van Dorn, W., 57, 60, 65, 87 Van Dorn, W. G., 647, 658, 660, 661, 663, 726, 730 Vanoni, A. A., 817, 825 Vantroys, L., 608, 610 Veley, V. F. C, 729, 729 Veltkamp, G. W., 627, 633, 646 Veronis, G., 249, 251, 252, 283, 290, 294, 365, 369, 381, 382, 393, 395, 646 Verploegh, G., 639, 646 Vetter, R. C, 588, 696 Vigoureux, P., 476 AUTHOR INDEX 849 Villain, C, 779, 801 Vine, A. C, 821, 823 Vines, R. G., 65, 87 Vinogradova, O. N., 59, 64, 81 Vlasov, L. I., 833, 838 Volkmann, G., 507, 539 von Arx, W. S.: see Arx, W. S. von von Karman, T.: see Karman, T. von Vries, H. de: see De Vries, H. Vuorela, L. A., 95, 133, table 134, 173, 291 Wald, G., 463, 468 Walden, H., 699 Walden, R. G., 696, 697 Waldram, J. M., 434, 451 Walker, T. J., 560 Wallace, D. H., 562, 566 Walther, J., 295, 304 Warburg, H. D., 600, 609 Warren, F. J., 461, 463, 467 Watanabe, ^.,451 W^aterman, J. H., 415, 431, 450 Waterman, T. H., 465, 466, 467 Watt, J., 693 Wattenberg, H., 447, 450 Watters, J. K. A., 582, 589 Webb, E. K., 43, 54, 58, 64, 66, 72, 75, 78, 85, 87, 106, 179, 180, 288 Webster, J., table 116, 124, 125, 287 Wedderburn, E. M., 749, 763 Weeks, W. F., 827, 835, 836, 838 Weenink, M. P. H., 623, 625, 626, 627, 628, 632, 634, 636, 639, 646 Weizsacker, C. F. von, 806, 825 Welander, P., 323, 370, 371, 372, 395, 623, 639, 646 Wernelsfelder, P. J., 685, 699 Wertheim, G. K., 333, 395, 504, 538 Wesley, J., 472, 475 Westenberg, J., 541, 550, 566 Westerfield, E. C., 500, 537 Weston, D. E., 493, 497, 502, 537, 539 Whipple, R. T. P., 822 Whitman, W. G., 828, 838 Whitney, C. G., 697 Wiechert, E., 700, 719 Wiegel, R. L., 687 Wilcox, R. E., 297, 304 Wilkins, E. M., 295, 304 Willis, B., 296, 304 Wills, M. S., 433, 451 Wilson, B. W., 658, 663 Wilson, K. G., 20, 30 Wilson, W., 477, 478, 497 Wilton, J. R., 665, 699, 726, 730 Winston, J., 267, 294 Winterhalter, A. C, 533, 538 Wirth, H. E., 8, 30 Witt, G., 90, 211, 216, 290 Witting, R., 68, 594, 610 Wolfe, U. K., 465, 467 Wood, A. J., 298, 304 Wood, F. G., Jr., 550, 556, 559, 560, 566 Woodcock, A. H., 203, 204, 208, 215, 219, 294, 305, 307, 308, 311, 312, 745, 763 Woodward, B., 2.15, 294 Wooster, W., 279, 294 Workman, E. J., 826, 838 Worthington, L. V., 347, 395 Worzel, J. L., 302, 304, 496, 497 Wulff, V. J., 561, 564 Wtist, G., 69, 71, 73, 87, 91, 100, 129, table 157, 169, 294, 825 Wyman, J., 95, 147, 203, 204, 205, 206, 208, 215, 219, 294 Wyman, T., 763 Wyrtki, K., 595, 610 Yablokov, A. V., 553, 564 Yakovlev, G. N., 835, 838 Yamamoto, G., 294 Yeh, T. C., 292 Young, A., 462, 467, 595, 605, 607, 610 Young. R. W., 553, 556, 562, 563, 564 Zeilon, N., 732, 763 Zerbe, W. B., 659, 663 Zetler, B. D., 663 Zeuner, F. E., 595, 610 Zobell, C. E., 307, 312 Zubenok, L. I., 75, 84, 133, 137, 294 Zubov, N. N., 834, 838 SUBJECT INDEX Titles of Sections and Chapters and the page numbers at which they begin are printed in heavy type. Names of ships and of genera and species are printed in italics. Where a single index entry (such as a geographical name) is followed by a string of page numbers the reader should consult the list of contents at the beginning of the book to differentiate them by subject. Absorption coefficient in underwater acoustics, 540 light under water, 452 sound in water, 477, 479, 483 Acapulco, 653, 660 Acoustic: see also Sound impedance, 512 observations, 536 pressure, 476 telemetering, 496 Adelie Land, 715 Adiabatic lapse rate of temperature, 15 Adriatic Sea, 778 Aerodynamically rough flow, 81 smooth flow, 80 Aerosol marine particles, 307 Ageostrophic flow, 93, 172 Agulhas Current, 123 Air: see Atmosphere Airborne sea-salt, 308 ff Air bubbles: see Bubbles Aircraft, 90 direct flux measurements, 113 profile through trade cimiulus, 211 Woods Hole data, 207 Air-sea interchanges, 43 ff flux determination, 100 {see also Aircraft) comparisons, table 112 energy budget method, 101 transfer formulas, 106 flux divergence, 104 Alaska, 267, 297, 659 Albedo, 102, 835 Aleutian Islands, 349, 659 Alpheidae, snapping shrimps, 553, 555, 560 Alutera punctata, 558 Alvaerstrommen, 816 Amplifiers, 545 Analogue computer, 692 Anchoviella choerostoma, anchovy, 551 Anglesey, 811, 812, 816 Animal life. Light and, 456 (see also Marine animals) Antarctic bottom water, 368, 815 circumpolar cxirrent, 332 microseisms, 715 pack-ice, 835 Anticyclones gyres and western boundary jets, 178 mid-latitudes, 253 Antitrade winds, 149 Apparent optical properties, 400, 411 spectral contrast, 453 Arabian Sea, 139 Arctic ice cover, 834 ice crystals, 827 pack-ice, 835 Argentinian shelf, 651 Argon, 313, 314 Armauer Hansen, 772 Astronomical: see under Tides Atacama, 662 Atherina hepsetus, 550, 551, Atlantic Ocean (see also North and South), 104 bathyphytometer measurements, 458 equatorial zone, 224 ff geostrophic transport, 346 shearing stress maps, 181 storms distantly recorded, 716 tidal constituents, table 771 Atlantic City, 617, 620, 643 Atlas echo -ranger, 549 Atmosphere dust, 295 ff heat balance, annual, 134 near-surface layer, 67 poleward heat-energy transports, table 134 pressure variations, table 602 sea-salt particles, 308 stability, 64 total volume of gases, 313 transfer coefficients, 44 850 SUBJECT INDEX 851 Attenuation coefficient, 453 diffuse, 440 ff light, 431 ff sound, 479 spectral radiance, 452 Aiu-ora, 474 Autobaratropic fluid, 31 Backward scattering light coefficients, 406 radar from sea surface, 473 sovind under water, 487 Bairdiella, 558 Baleen whales, 553 Batistes, 552 Baltic, 8, 58, 594, 778 Band -pass long wave filter, 647 sound filter, 492 Bandwidth light measurement, 414 sound recording, 545 waves on sea surface, 568 Barbados, 677, 678 Baroclinic inertial flow, 354 ff, 362 internal flow, 343 transport, 339, 341, 342 velocity, 337 zonal transport, 342 Barotropic condition, 107 transport, 339 ff waves, 382 Basdic echo -ranger, 549 Bathygohius soporator, 552, 554, 559, 562 Bathypelagic animals, 508, 533 swim-bladders, 519, 535 Bathyphytometer, 458 Bathyscaphe, 461 Beckoning of crabs, 559 Bengal Bay, 742 Benguela Current, 118, 119, 123, 142 Bering Sea, 607 Bermuda, 268 ff, 280, 707, 753 Beroe, ctenophore, 462 Bikini, 739 Bimini, 562 Biological formations fikns on sea surface, 65 sediments, 298 ff "Biological" sounds, 550 Bioluminescence, 458 ff animal photography, 510 organisms scattering sound, 504 Blenny fish, 552, 559 Bony fishes, soniferous, 541 Bore, 682 Bothnia, Gulf of, 60, 61, 593, 595 Bottle-nose porpoise, 561 Bottom: see also Sea-bed reverberation, 498 Boundaries layer, 345 ocean, 344 ff salinity, 827 tide problems, 781 ff, 786 ff, 788 ff zonal, 348 ff "Bound wave" of sound, 482 Bowen ratio, 49, 83, 105, 110 comparative table, 117 extreme storms, 249, 251 Japan Sea, 262 Brazilian Current, 118 Breakers, 681 Brine, 828 Brownson Deep, 462 Brunt-Vaisala frequency, 37 Bubbles, 305 sound scatter, 487, 515 sound spread, 480 Budgets of heat and water of ocean- atmosphere system, 114, 126 Japan Sea, table 264 Bulk aerodynamic method, 48 Bulk evaporation coefficient, 75 Bulk relationships theoretical approaches, 80 Bulk stability parameter, 67 Burrfish, 559 Buzzard's Bay, 57 California current, 118, 119, 123 earthquake slumping, 662 gulf, 749 microseisms, 705 Mission Beach: see this name sea-level, 614, 616 swell from distant storms, 676 Camp Pendleton, 653 Canal Zone Region, 149 Canary Current, 118, 119, 123 Cantherines pulles, 558 Cape Cod, 556 Capellinhos volcano, 491 852 SUBJECT INDEX Capillary ripples, 65 Caranx hippos, 555 Carbon dioxide, 313, 315 ff, 319 exchange reservoirs, 316 radiocarbon, 318 Caribbean Sea air-sea energy joint budget, 148 ff, 152 easterly waves, 223 evaporation, annual, table 157 heat and water budgets, 158 trade cumuli, 205 volcanic dust, 295 Wyman- Woodcock expedition, 111 Caspian Sea, 59 Castle Harbor, 756 Cathode-ray oscillograph, 492 Cattegat, 594, 791 Cauchy-Poisson form of Fourier Integral, 570 Cauchy problem, 787 Cavitation, 489 Cephalopods, 533,, 551 Cetaceans, 541, 551, 553, 556 Challenger expedition, 282 Chandlerian nutation, 606 Chasmodes bosquianus, 552, 554, 559 Chemical potential difference, 17 ff, 32 Chesapeake Bay, 563 Chile, 506 Chilomycterus spinosus, 559 Chlorinity, 6 vapour-pressure relation, 68 Click sounds, 556, 557 Climatology of energy exchange, 114 ff mean annual energy transactions, 131 ff momentum exchange, 181 ff Clouds, cloudiness, 102 (see also cumulus) effects, 141 surveys by photography, 177 transports, 211 "Coast effect" on precipitation over sea, 131 Coastal waters surges, 640 tidal mixing, 816 ff transparency, 457, 458 wave approach, 678, 704 Coefficients: see after the quantities measured Cold currents, 118, 123 Communicative functions of sounds, 560 (see also Sexual) Compass reaction of animals, 464 Compression of sea- water, 10 Compression wave, 704 Computers, 692 Conduction current, 470 Conductivity electrical, of sea-water, 28 thermometric, of air, 44 Conservation of matter, equation of, 33 Conservation of salinity, equation of, 33 Continental components of sediments, 299 Convection circulation, 364 ff definitions, 39 ff forced, free and boundary, 77 Convective circulation, 364 ff Convective turbulence, 194 Copepods, 512 Coriolis force, 193, 284, 325 ff, 380, 621, 633, 635, 766, 791 Cornwall, 677 Crawford research vessel, 224 ff Croakers, 490, 554, 562 Crustaceans, 541, 552 Cuba, 151 Cumulonimbus clouds, 96 Cumulus clouds, 94, 215 (see also Hot towers) trade, 205 Currents: see also under their names dynamics of, 323 ff geostrophic, 335 ff heat transporters, 123 warm, 141 Cyclones East Asia, 267 exchange measurements, 268 mid-latitudes, 253 Cyclonic inertial circulation, 363 Cynoscion, 558 Dactylopterus volitans, 559 "Daisy" hurricane, 234 ff Dalton number, 49 Daphnia magna, 465 Decay height of heat source, 196 Deep-sea animals, 460 apparent optical properties, 413 eyes of fish, 461 scattering layers, 464, 499, 502, 505, 534, 536 sound velocity, 486 wave recording, 687 Delphinapterus, 557 Delphinus delphis, 548, 557 SUBJECT INDEX 853 Density, air, table 44 Desert dust, 295 Diaphus iheta, 509 Diffusion, diffusivity estimation, 27 Fick's law and coefficient, 34 heats of, 32 molecular, 43 salt in water, 26 thermal, of water, table 27 water vapour in air, 44 Digital computer, 692 Dinofiagellates, 458, 460 Discovery Passage, 813 "Dishpan" studies, 89, 93, 99 Displacement current, 470 Distribution functions of light, 409 Diurnal illumination under water, 534 vertical migration, 463, 499, 505 Dogfish, 541 Dolphin, 557 Doppler effect scattered radiation shift, 473 ships encountering waves, 684 Dorsal light reaction, 464 Dover Straits, 608, 627 Downwelling irradiance, 407 Drag coefficient, 49, 109 atmospheric stability, dependence on, 64 determined from eddy correlations, 59 determined from surface-tilt observa- tions, 60 determined from wind-profiles, 57 fetch, effect of 60, 63 rainfall, effect of, 66 slicks, effects of, 65 wind speed, dependence on, 61 Drogue, 687 Dust transport, 295 ff Dye marker, 742, 743 Dynamics of ocean currents, 323 ff Dynamic viscosity, table 27 Earphones, 546 Earth eccentricity effect on sea -bed, 606 rotation force, 193 Earthquakes causing tsunamis, 659 frequencies overlapping sound, 476, 491 submarine slumping, 662 Echo, 485, 487 iceberg-detection, 498 Echo-sounding and ranging, 495, 549, 685 Ecological importance of light, 456 Eddies air-sea interactions, 204, 210 diffusion problems, 818 drag coefficient. 59 evaporation, 66 transfer coefficient, 107 turbulent transfer, 45 viscosity, 49 Edge waves, 641, 656 Eels, 463 Ekman friction layer, 193, 372, 377 theory of wind drift and gradient cur- rents, 323 ff transport components, 340, 342 Elbe river, 811 Electrical charges of air, 307 conductivity of sea-water, 28, 307 Electromagnetic flow meter, 811 sound detectors, 491 tape recorder, 545 Electromagnetic radiation other than light, 469 Maxwell's equations, 469 noise in sea, 474 propagation, 470 ff sea-surface effects, 471 ff sea-water properties, 469 transmission window, 470 "El Nino", 278 Elsasser radiation chart, 259 Ems, Hunte, Elbe, Eider river tides, 790 Energy distribution deep scattering layers, 536 waves on sea surface, 567 Energy transmission within the sea, 397 {see also Light) English Channel, 593, 594, 769, 772, 783, 816 Enthalpy, 7, 23, 25 Entropy, 7, 12 ff, 25 Eolian materials, marine sediments, 298 ff Epinephelus striatus, 562 Equation of state for sea-water, 4 Equations of motion of sea -water, 31 field, 40 ff transformations of, 35 ff vertically integrated, 337 ff Equator (ial) Atlantic exchange fluctuations, 225 ff barotropic flow across, 347 854 SUBJECT INDEX Equator{ial) — continued moisture transport across, 133 trough zone, 224 energy transformations, 164 Equilibrium thermodynamic state, 4 ff Erie, Lake, 61 Esbjerg, 594, 603 Eubalaena, 556 Euler equation, 33 Euphausids, 508 oil globule, 514, 532 Eustatic changes, 605, 607 Evaporation, 94 bulk coefficient, 75 heat flux distribution, 118 heat transfer distribution, 115 laminar layer effect at sea surface, 80 sea siirface, 66 spray, 71 Exchange fluctuations in mid-latitudes, 253 formulas, 101 mechanisms and fluctuations, 202 trade-wind region, 203 Explosion, 485, 486 charges, 493 shock- wave, 522, 531 Extinction coefficient in underwater optics, 540 Extinction rates, 456, 457 Extra-terrestrial components of sediments, 295 Eye of hurricane, 232 ff Eyes of marine animals, 461 ff "Facts" are rarely useful in isolation, 286 Feeding sounds, 550 Feldspars, 300 Feruioscandia, 596, 597 Fermat's principle, 659 Fetch, 60, 63 Fick's law and coefficient of diffusion, 34 Fiddler crab, 559 Field equations, 40 ff Filefishes, 558 Filters for waves, 647 Finland: see Bothnia and Fennoscandia "Firebox" model of climate, 139, 163 Fish: see also Light and animal life eyes, 461 ff noise, 490 sound back-scatter, 487 Flashing of marine animals, 458 Florida Straits, 155, 363 Flow: see Currents Fluids: see also Hydrodynamics idealized, 31 streakiness, 176 Fluxes: see Air-sea interchanges Flying gurnard, 559 Forcing function, 195 Forerunner surge of hurricane, 618 Forward scattering coefficients of light, 406 Fossil fuel combustion products, 315, 317 Fourier covariance transform, 577 heat conduction law, 34 integral theorem, 569 ff Frequency migration of fish, 526 sea-surface waves, 568 sound scattering dependence, 525 Frictional drag, 197 Froude number, 330, 359 Fundamentals, 3 Gabriel's cycle, 608 Gas bubbles: see Bubbles Gases, 313 rare, 314 ff solubiUty, 313 volume, total, in atmosphere, 313 GEK (towed electrodes), 813, 818 Geneva, Lake of, 767 Geomagnetism, natural noise spectrum, 474 Geometric spread explosion waves, 493 sound, 479 Geophysics (see Fluids, Sea-ice physics. Tides light measurement applications), 446 Geopotential, 335 ff Georgia Strait, 731 Geostrophic flow, 93, 194, 274, 335 ff German Bight, 709 Gershun tube, 398 Gibbs-Duhem equation. 5, 7 Gibbs function, 7 Gibraltar Strait, 537 Glacial eustasy, 595 Glitter photographs, 63 Global heat and water budgets, 114 ff Glohicephala, 554, 558 Goby fish, 559, 562 Gonyaulax polyhedra, 460 Gravitation, acceleration potential, 335 Grazing ray of sound, 485 "Greenhouse effect", 101 Green's law, 651 SUBJECT INDEX 855 Ground swell, 665, 676, 682 Groynes, 682 Guadalupe, 651, table 653 Guantanamo, 151 Gulf Stream, 97, 118, 119, 123, 142, 254, 333, 346, 350, 818 dynamical response, 324 southward shift hypothesis, 284 streamlines, 281 Hankel functions, 572 Harbour resonance, 660 Harmattan dust haze, 299 Harmonic constants, 768 Hatteras, Cape, 363 Hawaii, 63, 148, 661 Hearing by fish, 561 Heat balance annual distribution of components, 119 ff, table 121, 128 seasonal march, 140 budget, Japan Sea, 266 conduction coefficient, 26 Fourier's law, 34 energy budget inflow layer, 245 ocean-atmosphere system, 126 exchange by radiation, 72 flux distribution latitudinal, 166 sensible, 118 latent and sensible exchange distribution, 114 of diffusion, 32 of fusion of ice, 19 of vaporization of sea-water, 19 source, decay height, 196 Heat and moisture fluxes, lateral, 169 Heat and water exchange, 144 Heat-energy conservation law, 27 Heat of transfer, 26 Heat source, extra-oceanic, 235 ff Heat source, oceanic, for hurricane, table 245 Heat transfer between sea and air, 66 ff annual heat and water budgets, 126 global budgets, 114 ff mean annual distribution, 114 seasonal march, 139 Hefner, Lake, 75 HeUgoland, 768, 772, 792 Heliimi, 313, 314 Hippocampiis, 558 Holocentrus ascensionis, 562 Homogeneous layer, 209 Hot towers, 175, 201, 214 giant, 215 hurricane, 234 Humidity over sea, 73 measurements, 75 Humidity -profile coefficient, 68 Hump-back whale, 556 Hurricanes, 148, 216 Atlantic City, 617, 643 "Carrie", 243 "Daisy", 234 ff, 243, 244 dynamic properties, table 241 eye, 232 ff exchange requirements, 246, tables 246 ff heat-engine model, 237 mature, 231 ff momentum budget, 239 surge, 618 tracking from microseisms, 709 ff Huyghens's principle, 708 Hybrid optical properties, 412 Hydrodynamics equations of motion, 31, 33 ff smoothness and roughness, 109 tidal problems, 765 ff Hydrophone, 496, 542, 543 Hydrosol, 413 Ice: see Sea-ice physics Icebergs, detection by echo-ranging, 498 Iceland, 283 atmospheric pressures, 603 volcanic ash, 295 Idealized fluids, 31 Illite, 300 Image recording equipment, 448 transmission under water, 452 Impedance, sound, 479, 480 Indian Ocean, 8, 119, 149, 181 Indonesia, 301 Inertial flow baroclinic, 354 ff cyclonic, 363 steady circulation, 349 ff two -layer ocean, 362 Inherent optical properties, 400, 401 ff, 411 spectral contrast, 454 spectral radiance, 453 Insolubles, 295 dust-transport meteorology, 295 856 SUBJECT INDEX Insolubles^ — continued eolian materials, 298 extra-terrestrial components, 298 Interchange of properties between sea and air, 43 small-scale interactions, 43 ff Interfering waves, 703 Interior transport equations, 340 ff Internal waves, 731 basins, 749 ff currents, 735 coherence reduced by turbulence, 761 coherence related to beam width, 759 ff differential equations, 752 ff direction, 738 dye markers, 742, 743 echo-sounding of scatterers, 537 isotherms, 734, 757 lakes, 749 ff measurements, 732 ff period, 737 shallow water, 735 slicks, 745 ff sound transmission, 740 ff spectrum, 753 speed, 738 temperature, 739, 753 thermistor beads, 733 tides, 747, 799 turbulence, 757, 761 Vaisala frequency, 752, 755, 758 International Geophysical Year, 590, 595, 715, 716 Inversion conditions over land, 79 Irish Sea, 773, 784 Irminger Sea, 821 Irradiance, 399, 439 ff if -functions, 453 scalar (spherical), 445 Irreversible processes, 33 ff Isaacs-Kidd mid-water trawl, 507, 509 Isohyetal distribution, 129, 131 Isopleth patterns, 123 Isostatic compensation, 596 Issik Kul lake, 716 Jack fish, 550 Japan Sea, winter monsoon, 254 ff Jet streams, 97 Kagoshima Bay, 58 Kamchatka, 659 Karman constant, 49 Kattegat, 733, 811 Kay Electric Co. instruments, 524, 547 -K-constituents of tidal harmonics, 764, 780, 820, 821 Kelvin edge waves, 657 tide gauge, 647 waves, 390 ff, 638, 720 X-functions of irradiance, 410, 452 ff Kinematic viscosity air, table 44 water, table 27 KlinotaxiS; 464 Krakatoa, 659 Krypton, 313, 315 Kuroshio Current, 97, 118, 119, 123, 143, 254, 267, 333, 346, 350, 813. 816 dynamical response, 324 Labrador Current, 118 Lakes: see names thereof internal waves, 749 ff Lamont Observatory, 705 Lampanyctus leucopsariis, 509 Langoustes, 552, 556, 559 Lantern fish, 509 Laplace equilibrium theory, 598 tide theory extension, 779 Lapse conditions, 76 Large-scale interactions, 88 momentum relations, 178 Latitudinal heat fluxes, 166 Leiostomus, 558 Light, 397 {see also Underwater visibility) absorption coefficient measurement, 445 attenuation, 431 ff, 440 ff colour, 457 data, table 446 distribution functions, 409 ecological importance, 456 extinction rates, 456 instrumentation, 414 ff irradiance, 399 ff, 439 X-functions, 410 lunar intensity changes, 465 optical properties, 401 ff, 407 ff, 411 photometer, 429 physical constructs, 397 ff polarization, 431 radiance, 397, 416, tables 417 ff reflectance functions, 409, 440 ff scattering, 434 ff seasonal changes, 465 sighting range, 454 4 SUBJECT INDEX 857 Light — continued spectrum, 457 transparency, 457 turbidity distribution, 821 volume absorption, 406 volume attenuation, 401 ff volume scattering, 403 ff Light and animal life, 456 Little Ice Age, 595 Long ocean waves, 647 coherence, 655 ff frequency of shelf waves, 653 instruments, 647 peak sharpness, 654 shelf waves, 649, 651 ff spectrum, 648 ff surf beat, 649 tsunamis 658 ff Long swell, 682 Long-term variations in sea-level, 590 {see under Sea-level) Loudness, 542 Lough Derg, 464 Lough Neagh, 60 Love waves, 706, 713 Luminescence: see also Bioluminescence flashing at night, 458 Lunar light intensity changes, 465 tide constituents, 764 Maenid fish, 550 Magnetic: see Electromagnetic Magnetostrictive sound detectors, 491 Manchurian loess, 297 Mar del Plata, 651, table 653 Marine animals and other organisms light and animal life, 456 sound production by marine animals, 540 sound scattering by marine organisms, 498 Marine sediments, 298 ff Maud expedition, 829 Maui, 653 Maxwell's electromagnetic equations, 469 Maxwell unit in rheological model, 832 Mean square oscillation, 567 "Mechanical" sounds of marine animals, 550 Mediterranean Sea bathyphytometer measurements, 458 deep scattering layers, 507 sound velocity profiles, 478 tides, 778 Megaptera, 556 Melichthys, 552 Menticirrhus, 558 Mersey River, 811 Meteor expedition, 75, 147, 770, table 772, 775, 815 Meteorology dust transport, 295 paleo-, 295 sjmoptic scale, 164 tropical, 144, 177 Mexico, Gulf of, 768 Mica, 300 Micropogon undulatus, 554, 558, 562 Microseisms, 700 approach direction, 709 ff barriers, 708 direction estimating, 711 ff instruments, 713 ff mobile stations, 715 nature of, 706 ff refraction, 707 sea-wave relations, 700 ff spectra, 705 storm tracking, 709 ff Mid-latitudes, exchange fluctuations, 253 ff Migrations of marine animals, light- stimulated, 463 Mission Beach, 736, 737, 738, 748, 758 "Mode" of sound propagation, 482 Moist layer, 214 Moisture transport across equator, 133 Molecular atmospheric transfer, 44 Momentum budget along trade -wind trajectory, 191 ff exchange, 179, 181 ff flow steadiness in trades, 1 90 ff flux formulation, 107 global distribution of exchange, 190 ff large-scale relations, 178 ff transfer and wind profile, 49 ff Monin-Obukhov series, 78 Monocanthus hispidus, 552, 559 Monsoons, 139 Asiatic, 133 Japan Sea winter, 254 Monterey Bay, 504 Moon, secular acceleration, 607 Motion steady and time-dependent, 326 tinie -dependent. 380 ff turbulent, 329 858 SUBJECT INDEX Myctophids, 509 Myctophum punctatum, 460 Myoxocephalus, 559 Mysticetes, 553, 560 Nassau grouper, 562 Nautiltis cephalopod, 533 Neolatus tripes, 511 Neon, 313, 314 Nephelometer recording polar, 438 spectro -hydro, 437 Neutral wind-profile, 49 Neuwerk shoals, 58 New England, 534 Newlyn, 594 Nitrogen, 314 Noise electromagnetic in sea, 474 fish, 490, 550 natural geomagnetic spectrum, 474 sea spectrum, 489, 490 Non-equilibrium state, 22 Non-neutral wind-profile, 20 Nordcaper whale, 556 North Atlantic Ocean, 8, 118 currents, 142 deep scattering layers, 506, 534 recent oceanic warming, 115 secular changes, 280 tides, 784 Northern Hemisphere ocean basins, 124 "North Pole 4" station, 813 North Sea, 8, 58, 593, 594, 613. 621, 636, 773, 774, 783, 784, 785, 789, 792 ff, 798, 816, 821 Norwegian Sea, 549 Nova Scotia, 526 Nuclear explosions, atmospheric dust, 297 Oceanography descriptive 446 levelling, 607 ff Oceans: see also under their names atmospheric heat and water budgets, 126ff boundaries, 344 ff currents as heat transporters, 123 currents, magnitudes of forces, 329 storage term, 165 stratification, 37 surface stress determinations, table 189 turbulence, 35 volume, total, 313 water balance- table 137 Oceans — continued waves: see under Long ocean waves, Waves, Wind waves Odontocetes, 553, 554, 556, 560 Okeechobee Lake, 639 Opsanus tau, 554, 558, 559, 562 Optical properties: see under Light "Order" of sound propagation, 482 Osmotic pressure, 19 Ostariophysan fishes, 561 Oxygen, 314 Pacific Ocean, 8 cumulonimbus clouds 96 desert dust, 295 geostrophic transport, 346 northwest currents, 143 quartz sediments, 299 shearing stress maps, 181 sound scatterers, 507 southwest current; 123 trade trajectory momentum budget, 192 weather stations. 148 Pack-ice, 835 Pago Pago, 659 Paleometeorology, 295 Palinuru^ vulgaris, 552 Panulirus argus, 552, 556, 559 Panulirus interruptus, 560 Path function of radiance, 405 Periodogram for wave analysis, 692 Persian Gulf, 778 Peruvian Current, 118, 123 Phosphorescence: see Bioluminescence Phosphorescent Bay, 458 Photography sea bottom, 495 soiind scatterers, 510 ff Photokinesis, 456, 464 Photometers. 456 Photoperiodism, 456 Photosynthesis, 448, 456 Phototaxis, 456. 464 Phototropism, 456, 464 Physeter, 557 Physical properties of sea -water, 3 Physics of sea-ice : see Sea-ice Physoclistous swim-bladders, 535 Piezoelectric sovmd detectors, 491 Pinnipeds, 551 Pistol prawn, 542 Pitching of ships, 683 Planetary waves, 380 Plankters, 512 SUBJECT INDEX 859 Planktonic animals deep scattering, 508 swim-bladders, 502 Plunging breaker, 681 Pogonias cromis, 558 Poincare waves, 638 Point transmitter, 483 Polarized light animal responses, 465 instrument, 431 Pole tide: see Tides Poleward heat-energy transports, table 134 Poly tropic fluid, 31 Porpoises, 541, 548, 550, 557, 560, 561 Port Phillip Bay, 59, 73, 74 Port waves: see Seiching Postglacial uplift of land, 596 Potential energy anomaly, 338 Power spectrum, 492 Prandtl mixing-length method, 51 Prandtl number, water, table 27 Pre -amplifier, 543 Precipitation: see Rainfall heating, 132 Pressure gradient, 26 Prevailing surface winds, 145. 146 Prionotus carolmvs, 552, 555, 558, 559, 562 Profile methods heat and water vapour transfer measure- ments, 66 turbulent transfer measurements, 46 Progressive waves, 702 ff Propagation electromagnetic, 470 ff sound in water, 476 ff Propellers: see Ships Puerto Rico, 188, 206 Pulfrich photometer, 435 Pulse generators, 492 Purring gourami, 558 Quartz grains, 299 piezoelectric effect, 491 Radar, reflection from sea surface, 473 Radiative transfer air-sea interactions, 88 ff balance, 94, 101, 127 heat exchange, 72 mean annual distribution 119 underwater light, 397 ff irradiance, 399, 439 ff radiance, 397 ff, 452 scattering, 402 ff Radiocarbon, 318 Rainfall distribution over the oceans, 129 drag coefficient, effect on, 66 hurricane areas, 232 ff, 237 impact effects, 305 "land effect" correction over oceans, 129 salinity, 311 trade winds, 199 Rare gases, 314 ff Rayleigh scattering, 513 waves, 582, 706, 713 Rayspan sound filter, 548 Red Sea, 778, 783 Reflectance functions of light, 409, 440 ff Reflection of sound, 480 Reflectivity of electromagnetic waves from sea surface, 472 Research vessels, 113 Resonant bubbles, 515, 518 Resurgence from hurricane, 618, 620, 643 Reverberation, 487, 520 frequency dependence, 523 Reynolds number, 330 stress, 328, 802, 803, 806 Richardson neighbour separation theory, 805 ff number, 50, 66, 68, 77. 808, 814, 815 Ridges, subtropical, 132 Right whale, 556 Ringkobing Fiord, 61 Ripples, 720 contamination effects, 725 definition, 720 growth, 727 mean square slope, 721 phase velocity, 720 profile, 726 shape of water surface, 726 slicks, 724, 725 ff spectrum, 721 velocity, 726 wind speeds, 721 Rips, 682 River tides, 789 ff Rochelle salt, 491 Rolling of ships, 683 Rossby-Montgomery formula, 51, 179 Rossby number, 93, 330 waves, 380, 387 ff Ross Sea, 716 860 SUBJECT INDEX Routing of ships, 683 Salinity, salt airborne particles, 308 conservation equation, 33 diffusion coefficient, 26 freezing point depression, 826 gradient, 26 rain, 311 sea-level effects, 592 sound effects, 484 thermodynamic effects, 4 Samoa, 659 San Clemente Island, 650 San Diego, 534, 615, 815 San Francisco, 614 Sargasso Sea. 283 Savin o-Angstrom formula, 103 Scalar irradiance, 399, 445 Scandinavia, storms distantly recorded, 716 Scattering, light, 402 measuring instrument, 436 Sciaenidae, 558 sound deep layers, 464, 499 groups, 500, 501 marine organisms, 498 Scripps Institution, 187, 188, 317 Sculpins, 559 Sea and air feed-back mechanism, 278 long -period interaction variations, 272 ff radiative heat exchange, 72 Sea-bed, 480, 482 photography, 495 sound back-scatter, 487 velocity profile, 815 "Sea canary", 557 Sea-drum, 558 Sea-horses, 558 Sea-ice physics, 826 brine, 828 crystals, 827 elastic parameters, 831 ff electrical properties, 834 glacial eustasy, 595 growth and disintegration, 834 ff mechanical properties, 829 pack-ice, 835 structural theory and properties, 835 ff thermal properties, 832 ff visco -elastic properties, 832 Sea-level analysis of observations, 601 ff Sea-level — continued climatological factors of variation, 593 day-to-day variation, 614 determination of mean, 590 ff eustatic changes, 605 ff, 607 geophysical implications of long-term variation, 605 ff ice coverage effects, 595 International Geophysical Year, 590 long-term variations in sea-level, 590 oceanographic factors of variation, 592 oceanographic levelling, 607 ff Permanent Service for Mean, 591 pole tide effects, 606 secular variations, 601 ff, table 602 steric, 592 sunspot cycle, 608 tidal bench-mark distance from earth's centre, 596 tidal charts, 773 ff tidal constituents in variation, table 591 variation causes, 592 ff Sea-robins, 558, 559, 562 Sea-water adiabatic pressure change, 17 clarity meter, 447 compression, 10 cooling, 17, 826 cool surface skin, 71 electrical conductivity, 28, 470 electromagnetic properties, 470 equation of state, 8 ff gases content, table 313 global budget, 144 ff heat and salt fluxes, 26 heat of mixing, 24 interchange of properties with air, 43 ff salinity, 17, 826 specific heat, 14 steric changes, 592 thermal expansion coefficient, table 12 Secchi disk, 455 Seiching, 60, 658 Seismographs, 714 Sergestes, prawn, 463 Seriola quinqueradiata, 551 Sexual significance of sounds, 558 Shallow water ripples, 735 sound scattering layer, 503 tides, 767 Sharks, 541, 550 Shearing stress, 107, 108 Shelf waves, 651 ff SUBJECT INDEX 861 Ships internal waves initiated, 731 propellers' cavitation noise, 490, 492 wake interrupts sonar beam, 480 wave effects on motion, 683 ff Shock wave, 522, 531 Shore approach waves, 678 ff Sighting range under water, 454 Silverside fish, 550 Simrad echo-ranger, 549 Single scatterers, 513 ff Siphonophores, 533 SIPRE (Snow, Ice and Permafrost Re- search Establishment), 831 Skagerrak, 791 Skin depth, 470 Slicks, 65, 724, 725 ff, 731 internal wave relation, 731, 745 ff Small-scale sea-air interactions, 43 Smaris sfnaris, 550 Smoke-trail photographs, 203 Snapping shrimp, 490, 542, 553, 560 Snell's law, 484 Snodgrass wave recorder, 648 ff Sofar stations, U.S. Navy, 486 Solubles, 305 Sonagraph, 524, 528, 547 Soniferous marine animals, 490, 541 Sonobuoy surveys, 551 Sound in the sea, 476 (see also Echo) absorption, 477, 479, 483 amplifiers, 545 detectors, 491, 492 fluctuation, 488 frequency analysis, 492 internal waves, 740 ff listening instruments, 542 propagation, 476, 492 shallow water, 482 scattering, 487, 498 ff freqviency dependence, 525 ff large groups, 519 ff observations, 524 water boundaries, 537 shadows, 485 transducers, 495 transmitters, 492 Sound production by marine organisms, 498 analysis, 546 eliciting, 561 identification, 548, 549 instrumentation, 542 monitors, 545 orientation function, 560 Sound production by marine organisms — continued purposeful and adventitious, 550 recorders, 545 spectrum analysis, 547, 554 suppressing, 561 Sound scattering by marine animals, 540 identification, 507 ff South Atlantic, 119, 770 Southern Hemisphere ocean basins, 124 Southwest Pacific Current, 123 Specific enthalpy, free energy, thermo- dynamic potential, 7 Specific gravity anomaly, 8 Specific heat of sea- water, 12, 14 Specific humidity profiles, 69 Specific volume of sea-water, 8 Spectra: see under Sound and Underwater light and Waves Spectral radiance, 452 Spectro-hydro nephelometer, 437 Spectrophotometer, 439 Sperm whale, 556 Spherical irradiance, 399, 408, 445 Spheroides maculatus, 550 Spheroides nephulus, 559 Spilling breaker, 681 Spirula, 533 Spray, evaporation, 71 Squall lines, 220 Squalus acaiUhias, 550 Square-law rectifier, 492 Squeals, 553, 556, 558 Squeteague, 558 Squids, 533 Squirrel-fish, 562 Stability length, 51 Stability parameter, 50, 52 Stabilizers for ships, 683 Stanton number, 49 Stationary waves, 701 ff Station-keeping by marine animals, 560 Steadiness of lower trades, 197 Steady inertial circulation, 349 ff Steady motion, 326 ff Steady-state theory, 324 Stokes law, 300 Stokes-Navier laws, 33 Storm surges, 231, 594, 595, 796 ff tracking from microseisms, 709 ff Stratification, 37 thermal 50, 66 862 SUBJECT INDEX Streakiness of geophysical fluids, 176 Stress distributions, 185 ff Stridulatory sounds, 554, 555 Submersible monochromator, 444 Subtropical ridge lines, 97, 132 Suitcase amplifier, 545 Sunspot cycle, 608 Surface reverberation, 498 Surface-tilt observations, 60 sun's glitter photographs, 63 Surf beat, 650 ff Surf zone waves, 681 ff Surges, 611 air pressure effect, 628, 637 boundary conditions, 626 classification, 611 definition, 611 description, 613 dynamics and forecasting, 621 external, 638, 798 forecasting, 621, 638 North Sea, 797, 798 open coast, 640 qugisi-equilibrium theory, 628 transient, 617 wind effect, 629, 634 Swedish Deep-Sea Expedition, 299 Sweetwater Lake, 750 Swell, 665, 682, 731 Swell-fish, 550, 559 Swim-bladders, 502, 518, 532, 550, 552, 554, 555 deep scattering layers function, 535 Synalpheus, 555 Talking fishes, 541 Target strength, 514 Taylor characteristic diagram, 73 Telemetering hydrophone, 496 Television of sound scatterers, 512 Telotaxis, 464 Temperature gradient adiabatic, 15 depth profile, 534 Temperature inversions, 16 Temperature profile, 69 lapse conditions over sea, 76 Texel, 816 Thermal conductivity, table 27 Thermal expansion coefficient of sea-water, table 12 Thermal stratification, 66, 76 Thermals within a cloud body, 215 Thermistor beads, 733, 734, 813 Thermocline, 486, 827 models, 372 Thermodynamics, 32 ff equilibrium state, 4 First Law, 195, 250 irreversible processes, 22, 25 non-equilibrium state, 22 Second Law, 33, 35 Thermohaline transport, 343 Thermometric conductivity of air, 44 Thorium, 315 Tides, 764 Admiralty Manual, 600 astronomical, 597, table 598, 611 boundary -value problems, 781 ff, 786 ff breaking and bore, 682 charts, 773 ff classical theory, 779 ff difference methods, 786 ff equilibrium, 598 ff friction, 607 gauges, 590, 596, 647 harmonic constants, publication, 768 harmonic terms, 764 hydrodynamic equations, 765 ff internal, 799 internal wave relations, 747 ff loading from yielding earth, 599 nodal, 599, table 602 numerical methods, 781 ff, 788 ff observation, 768 ff pole, 606 ff rivers in North Germany, 789 Time -dependent motion, 326, 380 ff definition, 327 equations, simplified, 381 ff Toad-fish, 554, 558, 559, 562 Toothed whales, 553, 556 Towers: see Hot towers Trachurus trachurus, 550, 551 Trade winds, 95, 132 atmospheric conditions, 162 cumulus clouds, 94, 205 adequate as fuel pumps, 214 origin, 210 diurnal cycle, 220 ff easterly wave, 221, 223 energy transactions, 144 exchange mechanisms, 203 inversion, 143, 146 low level, 147 oceanic heat source, dual role, 193 steadiness, 197 momentum production and flow stead- iness, 190 SUBJECT INDEX 863 Trade winds — continued rainfall, 199 synoptic disturbances, 218, 220 ff Transfer equation for radiance, 406, 452 Transfer of heat, sea to air, 43 ff formulas, 101, 106 ff turbulent, 45 Transparency: see under Light and Under- water visibility Transport across latitude circles, table 98 ash beds correlation, 301 baroclinic and barotropic, 339, 342 Ekman components, 340 interior, equations, 340 ff mixed, 343 phenomena, table 27 thermohaline, 343 two-layer ocean, 356 Trichopsis vittatus, 558 Trigger fishes, 552 Trinidad, 141 Tropics distvirbances, 221 exchange fluctuations, 216 ff meteorology, 144, 177 trade- wind regions, 132 Tropopause, 158 Tropotaxis, 464 Trough position and movement, 165 ff Tsunamis, 658 ff Turbidity, distribution of optical, 821 Turbulence, 35, 89, 107, 329, 678, 802 air-sea boundary stress, 178 atmospheric transfer coefficients, 47 coastal water mixing, 816 ff convective, 194 diffusion by continuous movements, 804 ff exchange, 107 fluctuations and transports, 810 ff fluid, 202 general properties, 802 ff horizontal, 817 ff internal wave relations, 757 ff local isotropy, 806 ff meter, 813 mixing length theory, 804 neighbour separation theory, 819 parameters off Bermuda, 269 Reynolds stresses, 802 ff sea, 808 ff sea-bed velocity profile, 815 sound back-scatter, 487 stability infiuence, 807 ff, 814 ff Turbulence — continued statistical theory, 806 ff transfer measurements, 45 transports, 802 ff, 810 ff vertical, 814 ff Tursiops truncatus, 550, 554, 556, 559, 560, 561 Turtles, 551 Two-layer ocean, 354, 362 convective circulation, 364 Uca pugilator, 559 Underwater sound: see after Sound Underwater visibility, 452 (see also Light) image transmission, 452 ff inherent contrast, 454 sighting range, 454 transparency, 457 Unified European Levelling Network, 608 Unstable (lapse) conditions, 76 Upwelling irradiance, 407 Uranium, 314, 315 U.S. Navy Underwater Sound Reference Laboratory, 544 Vaisala frequency, 37, 752, 755, 758 Vancouver, 731 Vapour pressure: see Water vapour Vertical energy transport, 174 extinction coefficient, 457 migration of animals: see Diurnal Vibration generators, 494 Vibralyzer, 524, 547 Vineyard Sound, 533 Viscosity air, table '44 coefficient, 329 drag, 50 eddy, 49, 329 lawS; 33 Voigt unit in Theological model, 832 Volcanoes dust and sediments, 295, 299 vibrational energy, 491 Volume absorption coefficient of light, 408 attenuation coefficient, 401 back -scattering measurement, 531 reverberation of sound, 498 scattering function of fight, 403 ff total scattering coefficient, 408 Vorticity, 357 Wake Island, 620, 661 Warm currents, 118, 123 864 SUBJECT INDEX Wash (bight), 792 ff Water: see Sea -water Water vapour conservation equation, 158 diffusivity in air, 44 flux convergence, 132 fractional lowering of pressure, 1 9 pressure at sea surface, 68 pressure profiles, 69 sea-air transfer, 66 ff, 212 Waves (sea surface), 567 {see also Internal waves and Wind waves) analysis and statistics, 567, 690 ff autocovariance function, 577 baroclinic, 382 barotropic, 387 central limit theorem, 572 deep-sea recording, 687 ff detectors and surveys, 578, 580 ff, 588 directional energy spectrum, 571, 573 ff, 576 ff, 693 ff distant storms, 676 ff edge, 641, 656 energy per unit surface, 573 energy spectrum, 567 ff, 576ff equations, 568 ff filtering, 585 ff Kelvin, 390 long-crestedness, 575 long ocean waves, 647 long-shore coherence, 655 measurement of spectrum, 584 ff microseism relations, 700 ff moments spectrum, 574 offshore coherence, 655 periodic solutions, 569 ff principal direction, 574 quadrature spectrum, 582 random phases, 571, 572 Rayleigh distribution, 582 recorders, 647 Rossby, 380, 387 ff second-order approximations, 586 ff shelf waves, 651 ff ships motion, 683 ff shore approaching, 678 significant height and period, 691 slope, 576 slope detector buoy, 580 sound back-scatter, 487 stationary, 701 statistical formulation, 570 ff surf, 650 ff, 681 ff types, table 665 zonal barotropic, 387 Weather ships, 113 West Africa, 149 Whales, 541, 548, 553, 557, 560 "White caps", "White horses", 489, 665 Wind {see also Trade winds and under Waves and Wind waves) coefficient, 75 drag coefficient determinations, 57 drift theory of Ekman, 323 effect on drag coefficient, 61 effect on surges, 629, 634 generalized treatment, 51 geostrophic, 194 horizontal stress, 615 observational considerations, 54 profiles, 49, 77, 180 roses, 186 Wind waves, 664 {see also under Waves) approaching shore, 678 directional power spectrum, 684, 693 ff distant storms, 676 ff energy, 667 generation, 669 group behaviour, 666 kinematics, 664 ff observation and analysis, 684 prediction, 671 ff, 675 recording, 685 ff, 687 ff refraction, 679 ripples, 721 spectrum, 667 ff, 676 surf zone, 681 types, table 665 Woods Hole expeditions, 147, 161, 187, 193, 203, 205, 211, 214, 268, 695 Crawford research vessel, 224 ff Wiistenqviarz, 299 Wyman- Woodcock expedition, 187, 215, 510 Xenon, 313, 314 Yamamoto radiation chart, 259 Yellowtail fish, 551 Yelping sounds, 559 Yielding earth, 599, 606 Yucatan Channel, 154 Zeiss turbidity meter, 435 Zeroth approximation, 36 ff Zonal baroclinic transport, 342 Zonal boundaries, 348 ff Zonal index, 275 Zuiderzee Staatscommissie, 625 INSTITUTION ARCHIVES W.H.O.l. DATA LIBRARY WOODS HOLE. MA. 02543 Vs/OODSHOLLf'A..