s., ^^.^> , .^ Not* To c fRc00 - 8.000 - 7.000 - 6.000 - 5.000 - AOOO - 3.000 - ZOQO ATMOSPHERIC -SS'C 0.00 038 g cm"' ATMOSPHERK STRATOSPHERE TROPOSPHERE Fig. 1. Diagrammatic representation of the main boundary surfaces in tlie structure of the Earth and the density changes at each. The figures at the right are the heights or depths in metres above or below sea level. land and sea. The coastal limits of the continents projecting above the surface of the ocean are known almost everywhere with satisfactory accuracy. It is only in the polar regions where vast areas of land are buried under ice that it is difficult to determine accurately the limits between continent and sea. These uncertainties have recently been considerably reduced, however. Apart from this reservation, of the 510-01 million km^ of the Earth's surface not less than 361 -1 million km^ is ocean and only 148-9 milHon km^ is land (Kossinna, 1921). The ratio of land to sea is 1 : 2-43 or 29-20 relative to 70-80%. The uncertainty in these values is not more than a few hundredths. The Earth's surface is thus mostly oceanic. Similar relationships hold for the Northern and Southern Hemispheres taken separately: in the Northern Hemisphere 60-7% water, 39-3% land; in the Southern Hemisphere 80-9% water, 19-1% land. Water still pre- dominates in the Northern Hemisphere, while in the Southern Hemisphere land is very markedly in the minority. A great circle can be drawn dividing the surface of the The Ocean 3 Earth into a land and a water hemisphere, one containing the largest possible land area and the other containing the largest possible water area. The pole of the land hemi- sphere lies at 47-25° N., 2-5° W. near the mouth of the Loire, and this hemisphere contains 52-17°o sea and 47-3% land, corresponding to a ratio of 1 : 0-90; the water area is still shghtly greater than that of the land. The centre of the water hemisphere lies at 47-25° S., 177-5° E., south-east of New Zealand, and this hemisphere contains 90-5° 0 water and 9-5% land corresponding to a ratio of 1 : 0-11; shghtly less than 10° o is land. For many phenomena affecting the Earth as a whole this division into land and marine sides is of some importance. The distribution of land and water areas given in percentage is very irregular and apparently completely asymmetric. Table 1 gives the percentages of land and sea in zones of 5° of latitude. Table 1. Distribution of sea and land for zones of 5° of latitude (In per cent, according to E. Kossin-na, 1921) Latitude Northern Hemisphere Southern Hemisphere zone Water Land Water Land 90-85° 1000 00 00 1000 85-80° 85-2 12-8 00 100-0 80-75° 77-1 22-9 10-7 89-3 75-70° 65-5 34-5 38-6 61-4 70-65° 28-7 71-3 79-5 20-5 65-60^ 31-2 69-8 99-7 0-3 60-55° 450 550 99-9 01 55-50° 40-7 59-4 98-5 1-5 50-45° 43-8 56-2 97-5 2-5 45^0° 51-2 48-8 96-4 3-6 40-35= 56-8 43-2 93-4 6-6 35-30= 57-7 42-3 84-2 15-8 30-25° 59-6 • 40-4 78-4 21-6 25-20° 65-2 34-8 75-4 24-6 20-15° 70-8 29-2 76-4 23-6 15-10° 76-5 23-5 79-6 20-4 10- 5° 75-7 24-3 76-9 23-1 5- 0° 78-6 21-4 75-9 241 90- 0° 66-66 39-34 80-92 19-08 90° N.-90° S r total ocean 361-059 X 10'' km2, 70-80' \ total continents 148-892 x 10^ km^, 29-20% The thin dotted lines in Fig. 2 for 50% and 25% land show that land predominates only in two places, between 70° and 45° N. across the Eurasian and North American continents and at about 70° S. in the region of the Antarctic continent. In the Southern Hemisphere, with the exception of the polar area, the land is nowhere more than 25% of the total area. Between 55° and 65° S. the ocean forms a continuous belt around the Earth, a fact which is of fundamental importance for many oceanographic phe- nomena. 77?^ Ocean Fig. 2. Percentage distribution of water and land areas in five degree zones. The arrangement of the continents outlines the irregular distribution of the sea. The sea fills the depressions between the continents as far as its volume allows. On closer inspection a division into three major oceans can be recognized: the Atlantic, the Pacific and the Indian Ocean. They are all connected with each other, forming a continuous ocean belt in the higher latitudes of the Southern Hemisphere. This can be seen very clearly on Steinhauer's star projection centred on the south pole. Here the Atlantic and the Indian Oceans appear as very large and extended bays radiating out from the circumpolar Southern Ocean (Fig. 3). The main boundaries of the three oceans are fixed in the first place by the conti- nents. Conventional boundaries are necessary only to the south of Australia, South America and Africa where distinct morphological boundaries are missing. These have been fixed by international agreement (Intern. Hydrogr. Bureau, Monaco, 1937; WiJST, 1939). The three major oceans are subdivided by the continental coast lines which in some places are remarkably irregular. There is a particularly marked contrast between the open ocean and the seas enclosed between mainland and groups of islands. The sea areas which are separated from the ocean and project to a greater or lesser extent into the continents are denoted adjacent seas, and according to the degree of separation from the open ocean they may be either marginal or mediterranean seas. The demarca- tion from the ocean is usually topographical. The more important adjacent seas are listed in Table 4 (see p. 1 7), together with the area, the mean and maximum depths of the The Ocean Fig. 3. Steinhauer star projection to show the distribution of oceans and continents. three major oceans (with and without adjacent seas) as well as for the marginal and mediterranean seas (Kossinna, 1921; Landolt-Bornstein, 1952, article by Dietrich, p. 460). 3. Sea -level and its Variations. Chart Datum The surface of the ocean which forms the boundary between the ocean and the atmosphere is in a physical sense a free boundary that may assume different forms at different times under the influence of various internal and external forces. This bound- ary surface is called the "sea-level". If the Earth was covered entirely by a homogeneous ocean unaffected by atmospheric phenomena such as winds and atmospheric pressure or the tidal forces of the sun and the moon, then there would be only a single force acting on the sea : gravity. In the equilibrium state there can be no component of the force of gravity along the surface of the sea and the direction of the force of gravity must be perpendicular to the surface. This "ideal" sea-level is thus a geopotential surface or a gravitational equipotential surface. If minor variations in the force of gravity due to the irregular distribution of the mass of the outer crust of the Earth are disregarded, the ideal sea-level will coincide with the surface of a rotational ellipsoid. Even if the sea does not cover the entire Earth, the ideal sea-level will correspond to the surface of this rotational ellipsoid. When the small irregularities in gravitational force due to the irregular mass distribution of the Earth crust are taken into account. 6 TJie Ocean the sea-level as a geopotential surface will no longer have the same simple ellipsoidal form but will show little variations to either side. This irregularly shaped surface is called in the theory of the Earth figure the "geoid". The geoid can be regarded as dis- placed from the surface of the rotational ellipsoid by the distortions of the continental masses. The geoid rises on passing from the sea towards the continents and falls on passing towards the sea again. Figure 4 illustrates the undulations of the geoid around Ocean Continent Rototionol ellipsoid ^^llXlIIJilL-J 'n^^^^^ Fig. 4. Undulations of the geoid about the rotational ellipsoid. the rotational ellipsoid. The ideal sea-level (geoid) lies below the rotational elHpsoid in sea areas and above it in land areas. The magnitude of these deviations depends on the magnitude of the gravitational anomahes in the upper crust of the Earth. It was at first thought from theoretical considerations that the undulations of the geoid must be rather large. However, it was found that due to the almost perfect isostatic adjustment of the masses of the outer crust (hydrostatic equilibrium), these remain rather small and amount to not more than rhlOOm. The forces that cause periodic variations of the actual sea-level from the geoid were mentioned above. Amongst these are the forces due to the attraction of the sun and the moon which produce the tides in the ocean and the tangential force of the wind on the surface of the sea which causes ordinary sea waves. Both of these effects on the sea-level initiate waves that can be considered as oscillations to either side of a mean sea-level. It can be fixed at any coastal station by continuous observation of the water level, because the influence of the tides can be excluded if full-yearly observations are available while the effect of the ordinary wave motions disappears in a daily mean of observations. Other forces affecting the ideal sea-level may cause long lasting displacements of the actual sea-level from the geoid. If these forces are steady the corresponding displace- ments will also be steady and give a static equilibrium state. Also in the case of slowly changing forces the time will be sufficient for the sea-level to follow the changes. If, however, there are rapid changes in the intensity of the force the situation will be more complicated and an oscillation may develop depending on the size of the water masses involved. An important source of steady displacements of this kind from the ideal sea-level is the effect of barometric pressure. The ocean reacts to steady changes in the atmos- pheric pressure on the surface like an enormous water barometer: as the atmospheric pressure rises the sea-level will fall below the geoid, as the atmospheric pressure falls it will rise above it. When conditions are stationary there can be no pressure difference between two points at the same level within the ocean. The pressure at a depth h^ below ideal sea-level in a homogeneous sea of density pq will be The Ocean 7 where p^ is the air pressure at the surface, g is the gravitational acceleration and |^o is the deviation of the surface from ideal sea-level. At another place it will be P = Pi + gPi(fh + O- The pressure difference between the two places will then be ^P = -g(po - PiVh - gipo^o - Pi^i)- (I-O For a completely homogeneous sea (pq = pi) the relative deviation of the sea-level from the geoid will be J^=-A.zlp. (1.2) gp If the average density for sea-water is taken as 1 -028 then J I in dynamical cm = — 0-973/1/7; (Ap in mbar),^ }■ (1.3) J ^ in cm = —0-993Ap; (Ap in mbar). J The numerical factor in the last equation will be 1 -326 when Ap is expressed in mm Hg, because 1 millibar (mbar) corresponds to 0-75 mmHg. For a steady difference in air pressure the displacement of the sea-level from the geoid in cm will be 0-993 times the local variation in atmospheric pressure measured in mbar, in the opposite direction. From a knowledge of the steady pressure distribu- tion at sea-level the deviation from ideal sea-level can easily be found. In January the barometric pressure in the high-pressure cell near the Azores is about 1020 mbar, in the Icelandic low-pressure area it is about 990 mbar. It can therefore be expected that the sea-level in the area of the Irming Sea will be about 30 cm higher than at the Azores. Comparisons between changes in barometric pressure and changes in sea- level made at polar stations, where the ice covering allows them to be followed more easily, have shown satisfactory agreement between observed and calculated values of sea-level (Hessen, 1931; Wegener, 1924). Other effects due to the inhomogeneity of the water in the ocean and to the currents associated with it, and also to phenomena caused by the blocking of ocean currents at continental coasts (water level rise, Anstau) are harder to deal with. All these aperi- odic stationary deviations of the actual sea-level from the ideal are included in the concept of the physical sea-level. This physical sea-level is, under steady conditions, the true boundary between the ocean and the atmosphere. The methods used to fix the position of the physical sea-level relative to the surface of the geoid will be described later (Part II). The effect by itself of different distributions of density in different water masses within the ocean can be found using equation (I.l). Assuming the barometric pressure being the same at both stations (Ap = 0) it follows approximately P where h^ is the depth at which the pressure difference within the water mass vanishes. For a density difference of 10~^ and a water volume of 100 m vertical extent, the 8 The Ocean lighter of the water masses must be 10 cm higher than the heavier. If the density differ- ence changes with the depth the above equation will include the integral of Po — Pi dz taken from the surface to the depth /;. For practical purposes the mean water level is determined at coastal stations by a tide gauge. Calculation of a mean value will eliminate the periodic factors (tides and waves) but other factors will remain ; in the first place the aperiodic changes in mete- orological factors such as the wind, barometric pressure, precipitation and evaporation that can only be eliminated by taking a mean value over a number of years. However, even this mean value cannot be taken as invariable. It will reflect secular (long period) changes in meteorological factors and also slow deformations of the Earth and slow changes in the total water mass of the oceans. For comparison and inter-relation of mean sea-levels fixed at different places along a coast, precision levelling between these points is essential. This must be taken over land and be independent of the conditions in the sea in order to show whether the mean sea-levels are in one and the same or in different niveaus. On the subject of precision levelling along the Baltic coast (1896-8) see Westphal (1900), along the east coast of North America see Anvers (1927) and Bowie (1936), and on the interpretation of these see Dietrich (1937). Sea-level at almost all coastal stations shows clearly an annual period which is related principally to wind conditions along the adjacent sea coast; thus the sea-level at Aden is connected with the monsoon in the Arabian Sea (Krummel, 1907), while in Japanese waters the annual changes in barometric pressure and in density of the water are of greater influence (Nomitsu and Okamoto, 1927). On the annual variation in the sea-level along the Baltic coast see Hahn and Rietschel (1938), and Bergsten (1917). Along the coasts of those seas where there are strong tides the determination of mean sea-level is more complicated since the effect of the tides has first to be eliminated. This is best done by subtracting the mean tide level calculated by means of the har- monic tide constant from the actual change in water level as shown by the tide gauge. The remaining part is the aperiodic deviation in water level (in addition possibly free- oscillations of water masses) which must be related to meteorological factors (Marmer, 1927). If this ideal method is not possible the mid-point of each tide can be found by taking an average of hourly readings over a full tide period and it can be assumed that this value is reasonably free from any cosmic influence. An investigation of this type has been carried out for the German Bay (North Sea) by Leverkinck (1915). The changes in sea-level recorded on a tide gauge can also be simulated by a rise or a fall of the land on which the gauge stands. Movements of the coast line forming the boundary between land and sea may be compounded of two movements, those of the water and those of the land (Penck, 1934). As the ocean may be compared with a large vessel filled with water, changes in the water surface may arise through changes in the volume of water in the ocean or by alteration of either its size or the position of the water surface in the vessel. All changes in sea-level that affect the entire ocean surface in the same direction are termed, following Suess (1888), eustatic. This in- cludes two very slow changes: the nomic and \h& juvenile motion. The first is due to the slow erosion of the land that lifts the sea bottom, the second is due to the continuous The Ocean 9 addition of juvenile water from the interior of the Earth by volcanic and thermal activity. According to Penck, about 12 km^ of solid material are carried into the sea annually and this would raise the level of the sea by about 33 mm in a thousand years. This nomic movement will continue as long as there is land that can be eroded. When this final state of erosion has been reached the sea-level will have risen about 250 m higher than it is at the present time. The juvenile increase in the level of the sea amounts, according to Penck, to not more than about 2-8 mm in 1000 years or barely one- twelfth of the nomic. It will continue as long as volcanic activity on the Earth persists. A faster change than either of these eustatic movements is that due to the melting of glaciers. During the ice ages there was approximately 40 miUion km^ more ice covering the land than there is at the present time. This melted during a period of 10,000 to 20,000 years and raised the surface of the sea by 100 m or by 5-10 m in 1000 years (Ramsay, 1939; Penck, 1933). Melting of the present-day ice of glaciers covering the land (22-2 million km^) would raise the sea-level by 55 m. The level of the ocean varied during the ice ages over a maximum range of 155 m. The movements of the solid crust of the Earth may be of either tectonic or volcanic origin or they may be due to isostatic elevation or subsidence of single parts of the crust. The first may be accompanied by considerable local changes in a short time. Chart datum. Sea charts showing depths at different places give a picture of the topography of the sea bottom. These depths are not calculated from sea-level (as a reference level) but from a so-called chart datum. This has been done for purely practical reasons concerned with navigation. Chart datum on English and German charts is that of mean low-water springs; on French charts it is the level of the local I Nash Point I 0 Portishead 0 5 10 15 Fig. 5. Mean sea level and chart datum in the main shipping route in the Bristol Channel. Dungeness g^-^ ^^^ \ Le Colbart \ J, 0 0 5 10 15 Sm Fig. 6. Mean sea level and chart datum in the straits of Dover. 10 The Ocean lowest low water and on American charts it is the level of local mean low water. Only the sea charts of tideless mediterranean seas relate their depths to mean sea-level (e.g. the Baltic). Chart datum is nowhere the same as normal datum {NN) for carto- graphical surveys on land but is generally lower. Since the tidal range varies from one coastal station to another the chart datum forms an undulating surface which in general falls as it approaches a coast. This fall is greatest in funnel-shaped bays where the tidal range rapidly increases towards the inner end. On the open sea there are only small differences between chart datum and mean sea-level. Chart datum must be taken into consideration in more accurate hydrographic cal- culations. Raverstein (1886) first pointed out the importance of chart datum and prepared two charts of a part of the English Channel. One of these showed isobaths according to the sea chart (calculated from chart datum), and the other showed iso- baths calculated from the surface of the geoid. These charts demonstrate clearly the importance of considering a reference level. Figures 5 and 6 show two profiles of the differences between mean sea-level and chart datum for a longitudinal section along the Bristol Channel and for a cross-section of the Straits of Dover. For further informa- tion on the often very complex question of chart datum see especially Horn (1944). B. THE THREE-DIMENSIONAL STRUCTURE OF THE OCEAN 1. Methods of Recording Deep-sea Data The safety of shipping in coastal waters requires an accurate topographical survey to considerably greater depths than the 12 m draught of the biggest ships, usually down to 200 m. This is about the maximum depth at which soundings can be made with any accuracy using a hand lead line. Soundings taken in this way can also be used to measure the depth of water under a vessel anchored in shallow water and hence to 'feeeste" StationM «. * *. •x • * J • " ^ , • :, * K • • 1 ■ « »• 1 , • • ^K k m. • J » • K I 1 1 1 \ '- 1 : 1 1 ■ «« 13- 8 10 12 14 16 18 20 22 24 2 4 6 8 10 12 I7-2II-34 hr Fig. 7. Tides determined by sounding from an anchored ship (0 = 53° 55-9' N, A = 7= 52-2' E). determine the range of the tide at that point. This is the simplest method of determin- ing the tidal range at a distance from the coast in adjacent seas that are not too deep and along the continental shelves. The hemp lead line should have a piano wire trace at the upper and lower ends. Soundings of this type can, with some practice, be de- termined to within ±5 cm even for wave motion. Figure 7 gives an exam.ple of a tidal cycle measured in this way at a station in the southern North Sea. The Ocean 11 Soundings at depths greater than 200 m cannot easily be made with a hemp line by hand since the weight of the line and lead is too great and it is difficult to feel the contact with the bottom. The measurement of greater depths is extremely difficult and it took several decades of experimental work before deep-sea soundings could be made reliably at any point in the ocean. The measurement of the depth of the sea (Stahlberg, 1920) is the determination of the perpendicular distance between the surface and the sea bottom. At great depths this is difficult because: (1) the bottom contact is not easy to detect, and (2) hauling in the increased sounding weight is very laborious unless it is done by machine. Two conditions are necessary for a reliable deep-sea sounding: (1) the use of a thin steel wire in place of the hemp line used previously, and (2) the release of the sounding weight on contact with the sea bottom. The wire sounding method used at great depths will not be described in detail here since it is essentially a technical question. Further details can be found in technical handbooks [see especially Pratje (1952), and Oceanographic Instrumentation (Re- port of conference, Rancho Santa Fe, Cahfornia, 21-23 June 1952)]. The development of echo sounding has revolutionized the investigation of sea- bottom topography; wire soundings could never have been made in such large numbers nor have given such good results for the rapid and precise elucidation of conditions at the bottom of the ocean, and centuries would have been needed to get the results that can be obtained without difficulty in a few years by echo sounding. The basic principle of echo sounding is very simple; it measures the time required for a sound wave to travel from the bottom of a vessel (the sea surface) to the sea bed and back. The returning wave can be detected as an echo and amplified. To calculate the depth, knowing the speed of sound in sea water, it is only necessary to determine the time from emission of the sound until the echo is detected- — the echo time. If the time is /, the speed of sound in water v and the depth of the sea h, then Echo sounding makes it possible to sense the bottom of the sea accurately and to ascertain its actual topography. A vessel equipped with echo sounding can fix the depth of the sea without loss of time while moving at full speed. Scientifically, sonic sounding is of value only when: (1) it is combined with an accurate determination of the position of the vessel which in general should not be determined less accurately than ± 1 nautical mile and (2) when the mean velocity of the sound emitted by the echo sounding apparatus is known in addition to the echo distance. Only then is it possible to convert the value obtained to the true depth. The enormously increasing number of echo soundings requires the establishment of an international office to correct and unify the mass of data and to chart it after critical interpretation. This would give results of great utility both scientifically and for the improvement of the sea charts of all nations (Defant, 1938). Echo sounding has only one disadvantage compared with wire sounding; it cannot be combined with the collection of bottom samples which are necessary to ascertain the nature of the bottom sediments. If these are needed wire sounding is indispensable. However, it is possible with more modern types of echo sounding equipment to draw some conclusions about the nature and thickness of the bottom sediments from the 12 The Ocean appearance of the echo in the receiver. The structure and form of the returning wave is dependent on the nature of the reflecting surface. If the oscillatory form of the re- flected wave can be ascertained in the receiver it is possible to decide whether the bottom is rock, sand, mud, or other material. It is very frequently found that the echo is split into broader or narrower bands which are clearly connected with the different layers in the bottom sediment (mud or rock). The echo sounder thus gives a pre- liminary idea of the nature of the bottom and often the thickness of the soft upper sediment. This was first mentioned by Stocks (1935). For further details reference may be made to Evving, Crary and Rutherford (1917), Bullard (1938) and EwiNG and Vine (1938). Another method of studying the structure and thickness of the deep sea sediments has recently been developed by Weibull (1947). Very good results were obtained with this by the Swedish "Albatross" Expedition (Pettersson 1946). Indirect depth determination with an unprotected reversing thermometer. Ruppin (1906. 1912) first suggested the use of the difference between protected and unpro- tected reversing thermometers for the measurement of the depth at which the reversing frame or the water sampler on which the thermometers are mounted is reversed. The usefulness of the method has been shown by the investigations which he carried out at depths up to 100 m and by those of von Perlewitz at up to 1000 m. Brennecke (1921) on the "Deutschland" Expedition of 191 1-12 made valuable use of it, and it was used systematically for the first time on the "Meteor" Expedition of 1925-7 (WiJST, 1932). In both wire sounding and in oceanographic serial observations there is always a wire angle of greater or lesser magnitude and it is therefore extremely valuable to have a method available which allows a reduction of the temperature and salinity values to true depth or which ascertains a determination of depth independent of the wire angle. For the construction and function of the reversing thermometer, the corrections applied and the accuracy of the depths obtained [see particularly Oceanographic Instrumentation (Report of conference, Rancho Santa Fe, Cahfornia, 21-23 June 1952, p. 55)]. 2. The General Morphology of the Sea Bottom The topography of the bottom of an ocean or part of an ocean can be conveniently shown on a depth chart on which all available soundings are recorded after critical interpretation. The reliefs of the sea bottom can be shown by drawing lines of equal depth (isobaths) at fixed intervals. Constructing the isobaths between separate soundings isessentially a question of interpolation which is considerably facilitated if the soundings are distributed as evenly as possible over the whole area. This condition is unfortu- nately very rarely satisfied, even less so after the introduction of sonic sounding. Apart from the more sporadic distribution of earlier wire soundings there is now a greater concentration of soundings along isolated fines of echo soundings resulting in an extremely uneven distribution of depths and, while some parts are extremely well surveyed, there are very large areas with only single soundings. The task of preparing isobaths for an entire ocean has thus become more difl[icult than before the introduction of echo sounding. The construction of the isobaths for an ocean area depends on subjective considera- tions; the lines must of course be fitted to the soundings, but the available points I The Ocean 13 usually allow considerable elbow-room for the use of ideas and speculations on the bottom topography afforded by other knowledge (for example, geological). In par- ticular, the construction of the isobaths requires good use of oceanographic view- points. The distribution of temperature and salinity at the sea bottom and in the water immediately above it are dependent on the bottom topography and often allow greater accuracy than is possible from the records of depths alone, for example in the determination of depths on saddle points or the position of cross-ridges and others. Indicators such as these of the course of the isobaths are always valuable and deserve full attention. In this connection, see especially Stocks and Wust (1935) in the addenda to the chart of the Atlantic Ocean in the "Meteor" volumes. Good charts are not available at the present time for all the oceans and adjacent seas; it is to be expected that there will be considerable improvement here in the future. Apart from the older depth charts in the Sailing Directions for single oceans and charts produced by single expeditions the following may be noted : (1) The Carte Generale Bathymetriqiie des Oceans, scale 1 : 10 million, produced by the Hydrographic Bureau in Monaco; 16 sheets on Mercator projection: second edition, 1911-30, third edition from 1935. (2) The ocean chart published by Groll (1912) in which all depths available up to that time were interpreted in a uniform way and used for careful construction of the isobaths; equal-area projection on a scale of 1 : 40 million, (3) The chart of the total Atlantic Ocean on the records of the "Meteor"; a general chart, 1 : 20 milHon on the Lambert equal-area azimuthal projection with iso- baths at 500 m intervals (Stocks and Wust, 1935). In addition to this there is a basic chart of oceanic soundings on a scale of 1 : 5 million in 1 3 sheets (4 sheets published, Stocks, 1937) showing all the critically checked soundings in this ocean. (4) A more recent chart of the Indian and Pacific Oceans has been given by Schott (1935) on an equal-area projection, on a scale of 1 : 60 million, with the nature of the bottom topography of these oceans indicated with sufficient accuracy. (5) An excellent chart of the sea bottom topography of East-Indian Seas was con- structed by VAN RiEL (1934) and was published in the scientific results of the "W. Snellius" Expedition. For more recent charts of parts of the oceans and adjacent seas, see the sections on the special morphology of these areas. The charts accompanying this book(Plate 1) give a summary of what is known of the main features of bottom topography of the oceans. Much of the knowledge obtained by more recent expeditions by echo sounding has been taken into consideration here, in so far as the small scale will allow. In these charts the isobaths are drawn for every 1000 m and the 200 m isobath has been shown where the scale permits to show the limits of the continental shelf. The coloration of the depth-intervals gives a clear picture of the general bottom topography in spite of the confusion of fines at some points. In order to make the characteristic bottom configurations such as deep-sea basins, troughs and ridges and of the cross-ridges, deep-sea canyons and other forms which may occur, more visible, a somewhat schematic chart has been prepared and is reproduced in Plate 2 (Defant, 1947). All the important peculiarities of bottom topography of the ocean have been indicated by letters and numbers. 14 The Ocean The first scientific interpretation of the topographical chart of the ocean bottom taken in conjunction with a contour map of the land areas of the Earth was a general investigation of the relationships of heights and depths on the surface of the Earth crust. This was a purely statistical analysis of the variations of the surface of the solid crust about an average value, the mean crust level. If the whole of the solid crust of the Earth were levelled off" to give a single solid sphere, the mean level of the solid surface would be 2440 m below the present sea-level. The level of the sea itself would then be about 260 m above the present level, that is, the solid crust would be covered by a layer of water 2700 m thick (Kossinna, 1921). It would be expected that the fre- quence of occurrence of individual heights and depths was entirely random. The mean crust level (taking the present sea-level as zero: —2440 m) should occur most fre- quently, and the frequencies of individual heights and depths around this should form a probability curve. In these chance cavities the water would collect as oceans and the formation of the oceans would then offer no problems, since they would obviously form in the deepest depressions of the crust. The statistical distribution of the heights and depths of the Earth crust has, however, led to the striking result that the frequency in no way approaches a Gaussian-probabi- lity curve. On the contrary, there are two height-intervals which occur with high fre- quency while the other, less frequent, intervals group themselves around these two culmination points as two probability curves (Fig. 8, Table 2). 6000 4000 2000 2000 E 4000 6000 I K \ Sea level > - ^ Aver( ]ge crust level ^ - :> r""^ 1 1 1 ! 12 16 Frequency percentoge 20 24 Fig. 8. Frequency distribution of different height and depth intervals over the entire surface of the Earth. The two maximal frequencies lie at the height-interval of 0-1000 m and at a depth interval of 4000-5000 m; nearly 45% of the entire surface of the Earth falls within these two intervals, while only 10% falls on the other eleven steps. It is especially noticeable that the mean crust level of —2440 m (depth interval —2000 m to —3000 m) occurs infrequently, and is indeed very near the minimum between the two maxima. The Ocean 15 Table 2. Frequency and areas of individual height- and depth-intervals of the earth crust (According to Kossinna, 1921) Interval (m) Areas (10« km-) Per cent Interval (m) Areas (10' km^) Per cent >5000 0.5 01 >5000 0-5 01 4000-5000 2-5 0-5 >4000 3 0-6 3000-4000 3 0-6 >3000 6 1-2 2000-3000 10 20 >2000 16 3-2 1000-2000 24 4-7 >1000 40 7-9 1000-500 27-] 5-3^ 6-5 yii-i > 500 67 13-2 500-200 33 ;^108 48j > 200 100 19-7 200-0 9-4J > 0 148 28-1 0 200 fs.iy^ t^y^ >-200 176-5 33-7 - 200- -1000 >-1000 192 36-7 -1000- -2000 15 2-9 >-2000 207 39-6 -2000- -3000 24-5 4-8 >-3000 231-5 44-4 -3000- -4000 71 13-9 >^1000 301-5 58-3 -4000- -5000 119 23-3 >-5000 421-5 81-6 -5000- -6000 84 16-5 >-6000 505-5 98-1 >-6000 4-5 0-9 > -10,000 5100 1000 The position of the two maxima can be fixed more closely by investigation of denser intervals. It is apparent that one maximum falls within the interval 0-200 m and the other within the depth interval 4600-4800 m. The structure of the crust thus in- cludes two special areas: (1) a land area with a height of 100 m to 200 m, and (2) a sea area at a depth of about 4700 m. These two areas together include almost 65% of the entire surface of the Earth. These relationships can also be shown in another way in the "Hypsographic curve for the surface of the Earth" (Fig. 9) which depends on the areas in each separate height- and depth-interval over the surface of the Earth. 10 8 6 4 I 2 i 0 Q -2 -4 -6 -8 -10 Average level of the physicol earth surface J +_245m Continental block I I I I I I I I I ^Continental slope I ■ 1270m I I Average crust level".- 2.440 m I M I I I I I I Average ocean depth:— 3800 m 100 400 200 300 mill, qkm Fig. 9. Hypsographic cur\e for the surface of the Earth. 16 The Ocean This shows a stepwise form and is divided by four inflection points into five parts which may be regarded as natural regions of the land and of the sea: (1) Summits. All land above 1000 m (approx, 40 million km-, mean height 2040 m, maximum height: Mount Everest 8882 m) (2) Continental plateaus. Land below 1000 m and the continental shelf to —200 m (approx. 136 million km^, mean height 230 m) (3) Continental slope. From the edge of the shelf at —200 m to mean crust level —2440 m (approx. 39 million km^ mean depth 1270 m) (4) Deep-sea bottom. Sea bottom from —2440 to —5750 m (approx, 284 million km', mean depth 4420 m) (5) Deep-sea depressions and trenches. Below —5750m (approx. II milhon km^ mean depth 6100 m, greatest depth: "Emden" deep in the Philippines trench 10,800 m). This marked distribution into high and low areas divides the surface of the Earth into: (1) a high continental block which includes all land areas, the adjacent and parts of the marginal seas and the continental shelf and projects about 3100 m above the mean crust level, and (2) the deep sea which lies in basins in the Earth's crust whose bottom is about 2000 m below the mean crust level. The division of the Earth's crust between the continental block and the deep sea is shown in the summary in Table 3 and is illustrated schematically in Fig. 10. These show clearly the sharp division between the two parts: the continental block and the deep sea; the continental slope Table 3 Oceans per cent per cent of total Earth surface 3611xl0»km2 70-8% of the Earth surface Adjacent and ' Shelf 43-7 3-51 mediterranean seas: 400xl0«km2 Continental 31-8 2-5 1 7.9 7-9% of Earth ^ slope surface | I Deep sea 24-5 1-9J ' Shelf 2-7 1-7'] Oceans: 32Mx]0'km2 62-9% of 1 Continental slope 4-8 30 >62-9 Earth surface Deep sea 92-5 58-2] JO-8 Continents 148-9 X 10" km2 29-2% of the Earth surface 29-2 Total 1000 Total deep sea: 601%; Continental plateau (continents plus shelf): 34-4%; Continental slope: 5-5 % 1000 The Ocean 17 Fig. 10. Schematic representation of the Earth's crust by a continental block and a deep sea. Table 4. Area, volume and mean depth of oceans and seas (For Atlantic Ocean according to Stocks 1938, otherwise according to Kossinna 1921) Body Area Volume Mean depth Greatest depth (10« km2) (10« km^) (m) (m) Atlantic Ocean 106198 353-498 3331 85261 Indian Ocean 74-917 291-945 3897 7450! Pacific Ocean 179-679 723-699 4028 1 0,800 § Atlantic Ocean (excluding adjacent seas) 82-216 318-078 3868 8526 Arctic Mediterranean 14-057 21-453 1526 5180? American Mediterranean 4-311 9-373 2174 6269 Mediterranean Sea and Black Sea 2-969 4-318 1458 4404 Baltic Sea 0-422 0-023 55 463 Hudson Bay 1-232 0-158 128 229 North Sea 0-575 0-054 94 665 English Channel and Irish Sea 0178 0-010 58 263 Gulf of St Lawrence 0-238 0-030 127 549 Indian Ocean (excluding adjacent seas) 73-443 291030 3963 7450 Red Sea 0-438 0-215 491 2359 Persian Gulf 0-239 0-006 25 84 Andaman Sea 0-798 0-694 870 4177 Pacific Ocean (excluding adjacent seas) 165-246 707-555 4282 1 0,800 ii Asiatic Mediterranean 8-143 9-873 1212 6504 Bering Sea 2-268 3-259 1437 4273 Okhotsk Sea 1-528 1-279 838 3374 Japan Sea 1008 1-361 1350 3712 East China Sea 1-249 0-235 188 2377 Gulf of California 0162 0-123 813 2274 Bass Strait 0-075 0-005 70 — All oceans (including adjacent seas) 361-059 1370-323 3795 — t Puerto Rico trough north of Puerto Rico. % Java trench, south of Java. § Philippines trench north-east of Mindanao ("Emden" depth). li Mariana trench, about 11 ° N., 143° E. Gr. greatest depth 10,363 m (according to Cabruthers and Sawfori, 1952). II 77?^ Ocean includes not more than 6% of the surface of the Earth, and this percentage is being decreased rather than increased by the results of echo sounding. These figures empha- size that the deep-sea basins are not just chance depressions in the crust of the Earth. This division of the structure of the Earth is one of the most important of geophysical phenomena and requires a special explanation that must be very closely connected with the history of the Earth. Charting the sea bottom by means of isobaths and measurement of the areas of the different depth-intervals makes it possible to calculate the volume of each ocean and of the total ocean. The quotient of the volume and the surface area gives the mean depth. The volume of the ocean (including all the adjacent seas) amounts to 1370-6 million km^ and the mean depth is therefore around 3800 ± 100 m. The volume and mean depth can also be worked out for parts of the ocean and for the adjacent seas: the values for most areas according to Kossinna are given in Table 4. The Atlantic, Indian and Pacific Oceans have the mean depths 3930, 3960 and 4280 m respectively. These figures are not very different; the mean deviation is little more than 4^0. In addition to this general agreement, the figures for the depth-intervals in all three oceans, as shown in Table 5, demonstrate a very similar morphological structure of the Earth crust. This is further proof of a uniform structure in different parts and an indication that the existence of the two favoured levels of the Earth's crust repre- sented by the continents and the deep-sea bottom is a universal phenomenon prevailing over all parts of the Earth's crust. If the average density of sea water, taking the com- pressibility into account, is as 1-037, the total mass of the ocean will be 1-42 x 10^^ = 1-42 trillion tons which is only 1/4200 part of the mass of the Earth. Table 5. Morphological structure of the three oceans (exchiding mediterranean seas). Areas of the different depth-intervals given in percentage of the total Earth surface (Atlantic Ocean according to Stocks 1938; otherwise according to Kossinna 1921) Depth-interval in km 0-0-2 Atlantic Ocean Indian Ocean Pacific Ocean All oceans 5-8 3-2 1-7 3-1 0-2-1 1-2 2-3 3-4 4-5 3-8 3-7 7-5 21-3 33-9 2-7 3-1 7-4 24-4 38-9 2-2 3-4 5-0 19-1 37-7 2-8 3-4 6-2 20-1 36-6 5-6 23-3 19-9 28-8 26-2 6-7 0-7 0-4 1-8 1-2 0-3 0-3 0-1 Sum 100-0 100-0 100-0 100-0 3. Special Characteristics of Sea-bottom Topography The larger and smaller oceans and parts of the oceans are usually considered as more or less extended volumes sunk into the solid crust of the Earth. From this one is guided to assume that the sea bottom taken as a whole is concave inward. In reality this is so only in exceptional cases; in general the sea bottom arches upward and follows the surface of a sphere with a somewhat larger radius than that of the surface of the Earth. Expressed in another way the radius of curvature of the sea bottom points towards the centre of the Earth almost all the time and differs little from the radius of curva- ture of the Earth. If large areas are considered, really concave basins occur very in- frequently and are limited to the margins of the deep ocean trenches, to crater- shaped basins and especially to individual adjacent seas. Bathymetric charts of the The Ocean 19 ocean bottom, based on a few wire soundings, gave rise earlier to an impression of a certain smoothness and evenness of the sea bottom. Especially the bottom slope between two sounding points was ascertained and in most cases was found to be less than the smallest deviation from horizontal that the human eye can still detect, (a slope of 1 : 200 or a slope angle of 0° 17'). Actually, values found in this way showed very few vertical divisions over wide stretches of the ocean. This very smooth sea- bottom topography has, however, been shown by the much closer values given by echo sounding to be at least partly a misapprehension caused by the small number of wire soundings. Without doubt the sea bottom on the whole and especially away from areas where orogenetic and volcanic forces are active is on a small scale far more smooth and even than the surface of the land. The effects of the atmosphere, weather- ing and erosion by running water which all contribute to the variety of small forms which occur on land surfaces are of course all absent. However, echo-sounding pro- files at close intervals very often show considerable bottom irregularity. All echo- sounding profiles so far obtained are similar in this respect. The morphological inter- pretation must be made with the greatest caution since for greater clarity the results are usually shown with a strongly exaggerated vertical scale. Some vertical distortion is, however, essential when the profile extends over such great distances in order to show the details of the sea bottom clearly. Figures 1 1 and 12 show the "Meteor" profile Echolot of "meteor" on profile 5K 5000 Fig. 1 1 . Echo sounding profile across the South Atlantic obtained by the "Meteor" at 23 "^ S. (profile VII: 21-25-24° S.); with 180-fold enlargement of the vertical scale and disregarding the curvature of the Earth. Fig. 12. The same echo sounding profile as in Fig. 11 taking the curvature of the Earth into account. Upper curve: vertical enlargement 1:3; lower curve: 1:30 (according to Stocks). VII (21-25-24° S.) in two different forms (according to Stocks, 1936). The upper diagram shows the echo-sounding profile along a line from Rio de Janeiro to Whalefish Bay with a 180-fold magnification of the vertical scale and without taking the curva- ture of the Earth into consideration. In the lower profile, on the same horizontal scale, the curvature of the Earth at latitude 23° has been taken into account; the outer arc is the surface of the sea, and below this the upper curve shows the sea bottom with a vertical exaggeration of 3 : 1 while the lower curve shows the sea bottom with a vertical exaggeration of 1 : 30. The details of bottom topography and changes of slope are still easily recognizable on the curve with a 30-fold vertical exaggeration and 20 The Ocean are closer to reality than that in the upper diagram. A quantitative reading of differ- ences in height is, however, hardly possible here, and with a vertical magnification of only X 3 the thickness of the thinnest lines on the diagram is significant. The appear- ance of prominent features such as the Whalefish ridge can scarcely be seen and any qualitative differentiation into areas of greater or lesser irregularity is hardly possible. Magnification of the vertical scale is thus necessary from the topographical point of view, but must be used with appropriate caution. No accurate numerical evaluation of the echo-sounding profile, in order to fix the degree of bottom irregularity in different parts of the ocean, has yet been made. The superficial appearance of most of these profiles shows that the bottom relief varies from one area to another, and care is needed in making generalizations as these sound- ings give more and more detail. In most cases there is a relatively smooth bottom pro- file in the broad extended deep-sea basins and considerably greater irregularity over the central ridges and over the rises that separate the broad basins; considerable elevations above the mean surface of an area occur frequently in the vicinity of great depths and depressions so that extreme variations in depth are very often situated close together. Only certain especially characteristic forms of the commonly occurring typical bot- tom features will be discussed here. Stretching out to sea from the edge of the land there is first the beach which at high water is part of the sea bottom and, at low water, is part of the land. This amphibious part of the Earth's surface according to the estimate of Schott has an area of 1-6 million km^ or about 0-4% of the ocean area. Outside this the ledge-like rim appears, sometimes narrow, sometimes broad, but rarely completely absent, and is called the continental shelf. From the boundary between the land and the sea the sea bottom, except along coastal cliffs, slopes gently down at a slight angle, at the most 1-1-5°, This angle gradually increases and near the 200 m isobath it changes abruptly to the steeper gradient of the continental slope. The mean slope angle is about 3° here but in isolated cases it may be appreciably larger (6-10° or more). The edge of the shelf is normally at a depth of between 100 m and 200 m, but in some cases it appears only at a depth of 400-500 m. The continental shelf is seldom a uniform surface. It is very frequently broken by canyons, furrows and troughs, and shows clearly the effects of the more intense movements of the water because of the shallow depth (ocean and tidal currents). These effects of the action of the ocean are not found everywhere; in some places the sea bottom has clearly been formed during the ice ages by glacial action and has the character of a drumlin landscape as in the Irish Sea, for instance, between Ireland and Scotland. The continental shelf can usually be regarded as a part of the continental block which has been flooded by the sea, and its formation and topography are partly the product of the separation of the continents through accumulation and partly due to the erosion of the coast by wave action (Penck, 1934). Up to the present time no detailed investigation of the extent of the continental shelf has been made. Usually the 200 m isobath is taken as the outer limit of the shelf and the area of the shelf within this is usually designated as "bathymetric". In his statistics of the ocean depth Kossinna has listed these areas for each continent (Table 6). The bathymetric shelf extends over an area of 27-5 milhon km^ or 7-6% of the area of the ocean; Wagner has given the value 30-6 and Kegel has given 29-5 million km^. The mean depth of the The Ocean 21 shelf has been estimated by Kossinna as less than 100 m and is probably between 50 m and 70 m. Table 6. The shelf-areas (0-200 m) of the continents and oceans respectively (10" km^, according to Kossinna, 1921) Continents Areas (excluding mediterranean seas) Mediterranean seas Europa 311 Atlantic Ocean 4-59 of the Atlantic Ocean 9-52 Asia 9-38 Indian Ocean 2-37 of the Indian Ocean 0-80 Africa 1-28 Pacific Ocean 2-89 of the Pacific Ocean 7-32 Australia 2-70 North America 6-74 Sum 9-85 Sum 17-64 South America 2-42 Antarctic 0-36 Sum 25-99 + rather distant islands 1-50 Sum 27-49 x IC km% 7-6 % of the sea surface The shelf near the continental slope, often at a considerable distance from the coast, shows remarkable canyon-like troughs stretching over the bottom of the shelf and the adjacent continental slope. While previously only a few of these remarkable structures were known it has been shown recently, especially by the work of the United States Coast and Geodetic Survey, that they are of wide occurrence. Their topography can be rapidly and accurately determined by echo sounding. They were first thought to be drowned, sunken valleys, but it has been shown that they probably have a different origin. Two trough forms are found : submarine valleys in areas which have at some time been strongly glaciates (for instance around Iceland) and submarine canyons in regions which have remained unglaciated. The latter are usually found only at the edge of the shelf in the area of transition to the continental slope; these reach large depths (2000-3000 m) and often have little apparent connection with the topography of the neighbouring coastal area. Several series of these submarine canyons have been found and accurately charted: on the continental shelf and the edge of the shelf along the North American coast north of Cape Hatteras among which is the long-known submarine valley of the Hudson mouth (Fig. 1 3), along the coast of Cahfornia and along the coast of Washington and Vancouver Island (Smith, 1939). Individual submarine valleys are known along the east coast of Korea, along both coasts of Japan and on the eastern and southern coasts of Formosa. Submarine valleys frequently occur at the mouths of large rivers, such as the Ganges, the Indus (Fig. 14), the Congo (Fig. 15), the Ogowe and the Niger. They are also present in different parts of the European and American mediterranean seas. Some parts of the continental shelf are free from these canyons, for example the North American coast south of Cape Hatteras or the eastern coast of Asia south of the Yellow Sea. A summary of the distribution of can- yons in all oceans and the possible nature of their origin was recently given by Shepard (1948). The walls of these subm.arine canyons are usually very steep on both sides, often with a slope of 5-10° and sometimes 20-35° or even more. These canyon walls must be made of hard rock since thick layers of soft loose sediments could not be 22 The Ocean Fig. 13. Submarine valley off the mouth of the Hudson (according to Smith). expected to remain at such steep angles for any length of time without collapsing. Nor can it be supposed that they have been washed out of thick soft bottom sediments since they would then hardly be permanent. On the other hand, they appear definitely to be quite young formations that have been formed only in recent times; they appear, at least in part, to be connected with earthquakes, tectonic breaks and fissures. For a description of the morphology of these canyons see especially the work of Shepard and his collaborators (1933, 1938); concerning their probable origin see particularly Daly (1936) and Kuenen (1938); reference might also be made to the interesting work of Cooper and Vaux (1949), of Kullenberg (1954), Hecson, Ericson and Ewin (1954). They have been discussed from the purely geological standpoint by Jessen (1943), and a survey has been given by Kaehne (1941). Turning to the general form of the deep sea bottom it is immediately obvious that the rises and ridges that divide the ocean are features of such enormous size that they could scarcely occur on the land. The most prominent of these features is the Atlantic Ridge that extends from Iceland through the Azores, Ascension and Tristan da Cunha to Bouvet Island and resembles an enormous mountain range 20,000 km The Ocean 23 23°40' 23° 20' 67^20 67°40 . Interval between contours=50fathoms Fig. 14. Submarine valley off the mouth of the Indus. E 12° 20' Tnfervol between contours = 50 tbtfioms Fig. 15. Submarine valley of the Congo. 24 The Ocean long. It divides the Atlantic Ocean into two parts: the eastern and the western At- lantic troughs. These two elongated depressions are further divided into basins by transverse ridges. The peculiar relief features of the Atlantic Ridge which forms the axis of the Atlantic and runs roughly parallel to the continental coast on both sides is regarded by many as the beginning of a mountain fold, but it could also be the rump of an old one (Kossmat, 1931). The Indian Ocean shows a similar division. Here also there is an Indian Ocean Ridge dividing it into an eastern and a western half, though these two halves appear to be less subdivided. The Pacific Ocean, on the other hand, is largely a single basin (see p. 29). Amongst the most prominent features of the oceanic bottom topography are the narrow elongated arcs of marginal deeps that lie near the surrounding mountain chains (or island chains) of the Pacific basin and contain the greatest ocean depths. These remarkable depressions are confined exclusively to the margins of the Pacific Ocean; they can also be found in the Sunda arc in the eastern Indian Ocean, in the Caribbean, in the middle Atlantic Basin and in the south Sandwich marginal deep in the western part of the South Atlantic. They are usually termed "deep-sea trenches" or "troughs". This has reference only in a morphological sense and not to its origin. They are very closely connected with folding processes in the earth's crust, and to some extent are the counterpart of the mountain chains of the land, and have a related origin. As an example, the Mariana marginal deep is shown in Fig. 16 both on an isobathic chart and in a profile perpendicular to its longitudinal extension (Sigematsu, 1933). Its topographical form is typical of all well-developed marginal deeps. On the side towards the land, towards the submarine ridge which runs alongside the deep and is always of mountainous character, the slope is steep, on the ocean side of the deep the slope is more gentle. On the landward side the angle of the slope may be as much as 20° or more; according to Schott the mean value for a large number of Pacific deeps is 6-3°. They are always long and narrow. Table 7. The most important trenches (With reference to soundings up to 1954) Greatest Greatest depth (m) depth (m) North Pacific Ocean East Pacific Ocean Alaska-Aleutian Trench 7679 Chile-Peru J Atacama Trench Trough \ Arica Trench 7634 6867 West Pacific Ocean South Mexico /Acapulco Trench 5342 Japan Trench Trough \ Manzanillo Trench 5121 (Kurillen, Hokkaido, East-Hondo) 10,554 California Trench 4867 Bonin Trench Mariana Trench Ryu-Kyu Trench 9156 10,897 7507 East Indian Ocean Sunda Trough 7455 5664 5257 Philippines Mindanao Trench Yap Trench 10,497 7141 Andamana Trough Palau Trench 8138 Atlantic Ocean Bougainville-New Britain Trench 9140 Puerto Rico Trough 9219 New Hebrides Trench 7570 Cayman Trough 7200 Tonga-Kermandec Trench 10,633 South Sandwich Trench 8264 The Ocean 25 Fig. 16. The Mariana marginal trench; isobathic chart (lines of equal depth at 1000 m intervals) and cross-section taken normal to longitudinal axis of the trench. 26 The Ocean Table 7 gives a list of the marginal deeps and the greatest depths that have so far been measured in each. Without doubt these marginal deeps contain the deepest fissures in the Earth's crust, and in their neighbourhood are the greatest vertical differences in height that are to be found within a short horizontal distance on the Earth's crust. The marginal deeps are conspicuously associated with the volcanic belt which stretches along the landward side (on island chains or submarine ridges) parallel with the line of deep-sea trenches and with the earthquake belt which is also present in the immediate neighbourhood of the trenches, especially on the landward side. This connection with seismic and volcanic activity is always present and indicates a causa- tive connection between these phenomena. Another phenomenon closely associated with the marginal deeps is the strong negative gravitational anomaly occurring along a very narrow line. The investigations of Vening-Meinesz (1932, 1934) on the gravi- tational field in the East Indies and later the investigation of Hess (1938) in the West Indies have clarified this connection. The belt of abnormal gravity does not coincide exactly with the line of deep-sea trenches, but is displaced towards the adjacent moun- tain ridge. There exists in all cases a parallelism with the deep-sea trenches, but the relationship to the topography is more complicated than this. In the Philippine trench the line of negative anomaly lies directly underneath the trench (see Fig. 17) but it is 800 Fig. 17. Gravity profile over the Philippine Trench at Surigao (isostatic anomaly; observed values indicated by black dots; the bottom profile shown schematically with a vertical enlargement by 1:15) (according to Vening-Meinesz). weak, although the trench is particularly deep; in the Java trench the gravitation anomaly is very pronounced but lies at the side of the trench (Fig. 18). Since a line of negative gravitational anomaly is present wherever there is a deep-sea trench, there must undoubtedly be some connection between the two phenomena. This is also indi- cated by the relationship of seismic activity and the distribution of volcanoes mentioned above. For the explanation of this relationship, see especially Vening-Meinesz (1940). In addition to the deep-sea trenches there are also the differently shaped, nearly circular depressions. It cannot yet be decided whether these should be regarded as deformed marginal deeps but those between the Sunda Islands, the Moluccas, and the Philippines (Celebes, Sulu, Banda and other deeps) occur in close connection with the I The Ocean 11 East Indian negative gravitational anomaly. There are similar shaped deeps in the European Mediterranean, in the Gulf of Mexico and in other places, though not of the same depth or extent. Amongst these may be reckoned the comparatively small but very deep Romanche deep which divides the mid-Atlantic Ridge in two, at about 18-19° W. on the equator. The corresponding lowering of the mid- Atlantic ridge is as low as 4500-4800 m. The great significance of this deep connection between the eastern and the western troughs for the hydrographic structure of the water masses of -100 Isost Anomaly 600 Fig. 18. Gravity profile from Benkulen (Sumatra) towards the Indian Ocean (see Fig. 17). the eastern trough will be discussed later (see Chap. Ill, 5, b). The greatest depth measured in the Romanche deep is 7230 m. A bathymetric chart of the area has been given by Stocks and Wtisx (1935). While the slope of the deep-sea bottom is in general slight and only reaches larger values at the continental slope, occasionally very steep gradients occur near islands, submarine banks and reefs. As on land there has often been major volcanic activity on the sea bottom, partly in extended zones associated with the deep-sea trenches and partly more widely spread. The steepest slopes are always those of the purely oceanic islands which are all of volcanic origin; these slopes are of the same order of magnitude as those of land volcanoes. The slope of the island St Helena, for example, over short distances is as much as 38-40° and the Atlantic island St Paul has slopes of 62°. In numerous cases the volcanic forces have been insufficient to build an island cone up to the surface. They form submarine peaks, whose summits may still be some hundreds of metres below the surface and seldom come up to normal anchorage depths. These submarine volcanic cones were only occasionally found by wire sound- ing, which allows them to be quickly and accurately charted. In this connection there might be mentioned the surveys of the area of the Bogoslov volcano (Bering Sea) by the United States Coast and Geodetic Survey (Smith, 1937) and the survey of the "Altair" peak (Defant, 1939). 4. Arrangement of the General Bottom Topography of the Individual Oceans For an elucidation and abbreviation of the following discussion Plate 2 is presented, and it shows all the main characteristic features of sea-bottom topography in a clear manner. For each ocean there is a list of the principal features which have been desig- nated by letters and numbers on the plate. The capital letters show the deep-sea basins (troughs) in succession for each ocean, the small letters denote the ridges and rises that separate these basins, and the numbers indicate the deep-sea trenches. 28 3 Im Ocean c Ocean Deep-sea basins Ridges and rises A North America Basin a North and South Atlantic Ridge B Brazil Basin b Rio Grande Rise C Argentina Basin c Whalefish Ridge D Cape Verde Basin d Atlantic Indian Ridge E Sierra Leone Basin e Guinea Rise F Guinea Basin f Sierra Leone Rise G Angola Basin H Cape Basin Deep-sea trenches and troughs J Agulhas Basin 1 Cayman Trough K Atlantic-Antarctic Basin 2 Puerto Rico Trough L South Antilles Basin 3 South Sandwich Trench 4 Romanche Trench The topography of the Atlantic Ocean bottom is characterized by its division into East and West Atlantic Troughs by the Atlantic Ridge. This ridge begins at Iceland ; from the Iceland shelf it runs south-westward as the narrow Reykjanaes Ridge whose bottom form was fixed by the soundings of the "Meteor" (Bathymetric chart by Defant, 1930, 1931, 1936). At 5 1 ° N. the ridge broadens out somewhat towards the west (Telegraph Plateau), and then runs into the Azores Plateau which can be regarded as a great extension of the central ridge to the east and south-east. The ridge then narrows and remains at a depth of 2500-3500 m and apart from St Paul Island supports no islands as far as the equator. At 7-8° N. 36° W. there is a gap which reaches to a depth of 4400 m. The greatest gap is, however, on the equator near the Romanche Trench (see p. 27). South of this the ridge is broad and rounded and carries the islands Ascension (height 860 m), Tristan da Cunha (2329 m), Gough (1335 m) and Bouvet (935 m). St. Helena belongs to a minor ridge farther to the east. These extended minor ridges are peculiar to the section of the ridge between 0° and 20 "" S. The South Atlantic Ridge is connected west of Bouvet Island by the Atlantic Indian Ridge to the Crozet and Kerguelen Ridges of the Indian Ocean. The Atlantic Ridge extends over 20,300 km and is by far the longest underwater mountain system on the Earth. The Eastern and the Western Atlantic Basins are further divided by transverse ridges. An outline of the main division is shown in Plate 2 where the geographical arrangement of the basins is particularly clearly shown. A special characteristic of the Atlantic Ocean is that it is completely closed in the north towards the Arctic Sea and the Norwegian Sea below a depth of about 500 m. This has far-reaching oceano- graphic consequences. In contrast to this nearly complete blocking of the deeper layers to the north, the Atlantic in the south is completely open down to great depths to the Atlantic-Antarctic Basin. There are topographical differences between the eastern and the western troughs that have a considerable effect on the oceanographic structure. The transverse ridges are not as well developed in the western trough as in the eastern, and particularly the Rio Grande Ridge, which is somewhat better developed, has deep openings that per- mit continuous communication from the Atlantic-Antarctic Basin through the Ar- gentina Basin, the Brazil Basin and the Guiana Basin to the North America Basin below 4000 m. In the eastern trough, on the other hand, the Whalefish Ridge, which The Ocean 29 separates the Cape Basin from the Angola Basin, forms a continuous diagonal transverse barrier. It rises steeply from a depth of 5000-5500 m to only 964 m, forming an unbroken submarine wall connecting the mid-Atlantic Ridge between Tristan da Cunha and Gough Island with the broad shelf of the African mainland. All the other ridges in the east Atlantic trough have openings that reach below 4000 m. {b) Indian Ocean Deep-sea Basins Ridges and rises A Arabian Basin a Bengal Ridge B Somali Basin b Carlsberg Ridge C Madagascar Basin c Diego Garcia Bank D Agulhas Basin d Central Indian Ridge E South-westlndian Antarctic Basin e Mascarene Radge F South-east Indian Antarctic Basin f Atlantic-Indian transverse Ridpe G South Australian Basin g Crozet Ridge H India-Australia Basin Deep-sea trendies h i Kerguelen-Gaussberg Ridge Macquarie Ridge 1 Sunda Trench 2 Nicobar Trench It is only in more recent times that it has been found that the Indian Ocean is also divided into two large troughs by a central ridge. This central ridge runs north- westward from the Kerguelen-Gaussberg Ridge, gradually narrowing, then through the elevation around the volcanic islands of New Amsterdam and St Paul in the section between the 20° and 0°, where it reaches its highest elevation. Here it carries the shallow waters and banks of the Saya da Malha and the Nazareth Bank. Two outlying ridges run out from this point, one to the north-west to the Seychelles and the Amirantes, and the other to the south-west, here it carries the islands of Mauritius and Reunion. In this middle section the ridge stretches over more than 10° of latitude. From here it splits into two parts running towards the north. The eastern part carries the Chagos Islands and runs up through the Maldives and the Laccadives, gaining a connection to the south-west Indian shelf. The western part, which was first mapped by the Danish "Dana" Expedition (Carlsberg Ridge), is much narrower and not as high. This Indian Ridge is also of enormous length and runs from the South Arabian Sea to the edge of Antarctica at Kaiser Wilhelm Land (WusT, 1934). (c) Pacific Ocean Deep-sea basins Deep-sea trendies A Central Pacific Basin 1 Aleutian Trench B Philippines Basin 2 Kurile Trench C Caroline Basin 3 Japan Trench D Coral Basin 4 Bonin Trench E Fiji Basin 5 Mariana Trench F Tasman Basin 6 Japan Trench G South Pacific Basin 7 Philippines Trench H Berlinghausen Basin 8 Riukiu Trench J Peru-Chile Basin 9 Bougainville-New Britain Trench K Califomian Basin 10 New Hebrides Trench L Banda Sea 11 Tonga Trench M Celebes Sea 12 Kermadec Trench N North China Sea 13 Chile Trench 14 Peru (Atacama) Trench 15 Califomian Trench 30 The Ocean Ridges and rises a Bonin Ridge b Eastern Pacific longitudinal Ridge c South Pacific transverse Ridge d Macquarie Ridge e Fanning Ridge f Hawaii Ridge g Fiji Ridge h New Hebrides Ridge As has already been mentioned above (see p. 24), the deep-sea trenches that are a major characteristic of the Pacific Ocean are marginal, that is, they occur around the rim of the ocean, either near the coast or beside outlying island chains. The main part of the ocean forms a vast deep-sea basin that, judged by the rather sparse sound- ings available, is not as strongly subdivided as the Atlantic and the Indian Oceans. The western, and especially the north-western open Pacific Ocean, contains the greatest continuous extension of the sea bottom below 5000 m and wide areas have a depth even greater than 6000 m. The eastern and south-eastern parts are less deep. Sound- ings have confirmed the deep-sea division, apparent from the individual chains of islands, along a direction from north-west to south-east. In the central part of the ocean, especially to the south, there are groups of islands that are not associated with deep-sea trenches and that occur in clusters. It was earlier supposed that these were on top of plateaus or ridges at no great depths. More recent soundings have shown, however, that this is not the case; only islands that are very close have any submarine connection, and the others usually rise separately as volcanic cones from very great depths and form a very characteristic topographical feature of the South Pacific. {d) Mediterranean and Adjacent Seas The Atlantic Ocean is connected with the greatest number of mediterranean seas, which have also greatest extent. These are the Arctic Sea, which can also be regarded as a continuation of the open ocean across the Greenland-Iceland-Faroes Ridge, and the American and European mediterranean seas. The North Polar Sea, also known as the Arctic Mediterranean, includes: (1) the North Polar Basin surrounded by the seas of the flat shelf of Northern Europe and Northern Asia (Barents Sea, Karelian Sea, West Siberian Sea, Nordenskjold Sea, the East Siberian Sea and the Tjuktjen Sea) and of North America (Beaufort Sea and the large number of sea straits in the North American archipelago); (2) the European North Sea south of the Spitzbergen Ridge (depth 1750 m); and (3) the Baffin Sea. The total area amounts to 14-06 million km^. The European North Sea is divided by a ridge at a depth of about 2400 m, running from Iceland through Jan Mayen to the Bear island into two basins; the southern Norwegian deep and the northern Greenland deep, both with a depth of over 3000 m. For the bottom topography of the North Polar Basin see WiJST (1941). The American Mediterranean is divided by the coastal orography and by the bottom topography into three areas: the Mexico Basin (1-602 million km^), the Yucatan Basin (0-760 million km^), and the Caribbean Basin (1-948 million km^) with a total area of 4-310 million km^, A new bathymetric chart has been prepared by Stocks The Ocean 31 (1938) taking into account numerous recent soundings. The Caribbean Basin is itself further subdivided by two north-south ridges the Beata and the Aves Ridges into three parts : the Magdalena Basin in the west, the Venezuela Basin in the middle and the Aves Basin in the east. The general form of the bottom topography of the whole of the American medi- terranean basins shows considerable regional differences that can be explained by their different origins (see Dietrich, 1937, 1939). All three basins are to a large extent cut off from the Atlantic Ocean; this is of decisive importance for the question of renewal of the deep water of the individual basins. The Gulf of Mexico is connected with the free ocean only through the Florida Straits (sill depth 800 m) and with the Yucatan Basin through the Yucatan Channel (sill depth 1600 m). The Yucatan Basin and the Caribbean Sea are connected over the Jamaica Ridge with a sill depth of not more than 1400 m. The Yucatan Basin has a single connection with the Atlantic Ocean, the Windward Passage between Haiti and Cuba with a sill depth of about 1600 m. The Caribbean Sea is connected with the open ocean by several gaps between the West Indian Islands, the deepest of these are the Mona, the Jungfern and the Anegada Passages, which are the only ones concerned in the renewal of the deep water of this mediterranean sea. Their sill depths are 1600-1620 m and 1780-1800 m, respectively. The European Mediterranean Sea. This falls into two clearly separated main divi- sions, the Western Mediterranean from the Straits of Gibraltar (sill depth 320 m) to the Sicilian Ridge (sill depth 324 m), and the Eastern Mediterranean. To the latter are connected the Adriatic Sea and the Aegean Sea which in turn is connected through the Dardanelles (sill depth 57 m) with the Sea of Marmora and further, through the Bosphorus (sill depth 37 m) with the Black Sea, A modern bathymetric chart for the European Mediterranean has been given by Stocks (1938). The Western Mediterranean is separated by a ridge running from Tunis through Sardinia, Corsica and Elba to the Italian mainland into two basins: the Balearic Basin in the west and the Tyrrhenian Basin to the east (greatest depth 3731 m). The Eastern Mediterranean goes down to considerable depths (more than 4000 m) especially in the Ionian Basin ; the greatest depth is 4715 m south-west of Cape Matapan. Of the smaller mediterranean seas around the Atlantic, the Baltic and the Hudson Bay may be mentioned, but will not be described further since they have largely the character of shelf seas. The mediterranean seas of the other oceans are also of the same type except for the Red Sea which is an elongated canyon-like trough with depths of more than 2000 m and forming a real trench between the coastal strips of the Arabian and Egyptian plateaus. Its outlet in the south is the Strait of Bab el Mandeb with a sill depth of about 150 m. The Persian Gulf is a shelf sea with depth less than 100 m (Stocks. 1944). Chapter II The Sea- water and its Physical and Chemical Properties 1. Collecting Oceanographic Samples The ocean basins are filled with a liquid that is essentially the same as rain water formed by the condensation of water vapour. An accurate knowledge of the different contents of sea-water is indispensable in order to be able to learn something of the geophysical-chemical structure of the ocean. This knowledge of the structure must be derived from samples collected at oceanographic stations. It cannot be limited to the surface layers of the sea but must include all layers down to the sea bottom and must be based on a network of observation stations placed as systematically as possible. The precise determination of the spatial distribution of the oceanographic factors is a major achievement of modem oceanography and its observational technique. Collecting samples from the surface of the sea offers no real difficulties, or at the most only those that can be overcome by simple means. The collection of unob- jectionable and homogeneous material of definite origin from deep layers of the sea is, however, not easy and it has required the work of several decades to overcome the difficulties. The differences in the oceanographic factors (such as temperature and salinity) at deeper levels become continuously smaller both in horizontal and vertical direction; the accuracy of measurements at great depths must therefore be increased, and it has only been possible by the use of modern analytical techniques to do this with the degree of accuracy needed to follow small local variations. Almost all the properties of sea-water, apart from the temperature, can be deter- mined if genuine samples of water are available from each particular depth, because these properties show no appreciable alteration when the sample is brought from the deep sea to the surface. The temperature of the water must, however, be determined at the place and at the depth from which the water sample was taken {in situ). To collect oceanographic data at a station it is necessary to lower a thermometer in order to measure the temperature at different depths, and to bring back genuine samples of water from these depths in sampling bottles. The work at such an oceanographic station is done with a series-machine so-called because it is usually used for series observations, that is, the sampling bottles and thermometers are lowered at the same time to predetermined depths and a series of samples is collected and brought back together with temperature measurements. More recently, specially built machines have been used for this, but sounding winches or hydrographic winches were used previously. The oceanographic series machine and its operation on board ship will not be described here, but details are given "'Meteor'" Work, 4, No. 1 (WiisT, Bohnecke and Meyer, 1932, Berlin). 32 The Sea-water and its Physical and Chemical Properties 33 i ' 'Nil Before turning After turning Fig. 19. Water bottle used on the "Meteor" Expedition and method of operation. 34 The Sea-water ami its Physical and Chemical Properties Sampling bottles and thermometers are the most important of the instruments used at an oceanographic station. To be suitable for series observations the sampling bottle must be as light as possible; while still having sufficient capacity, it must allow free circulation of water and it must function and close reliably. There are many differen models of sampling bottles. They are all lowered open, allowing the water to pass through freely as the bottle sinks and are closed automatically for hauling to the sur- face. The most successful design is that of Nansen with two plug valves. The series water bottle used by the "Meteor" Expedition 1925-27 was constructed on the same principles but was a little larger and had a number of minor improvements. This water bottle and its function is illustrated in Fig. 19 (WiJST, 1932). It had a capacity of 1250 cm^, weighed 44 kg (with thermometer frame approx. 5 kg) and had an over- all length of 75 cm. Among the older designs may be mentioned that of Ekman (1905) with improvements by Knudsen (1923) and a special 4 1. water bottle {'"Meteor" Report, 4, No. 1, 1932). Small 100-200 cm^ bottles of ordinary green glass are suitable for storage of water samples (for chlorine titration and analysis) since they have been found by the investigations of Helland-Hansen and Nansen to have very slight solubility; they are fitted with a patent stopper with a porcelain head carrying the sample number. Before use the bottles must be boiled, cleaned with chromic acid-sulphuric mixture, rinsed with distilled water and very carefully dried. A definitive programme has been worked out for the work required at each oceano- graphic station and this has been found to be very successful as, for instance, during the "Meteor" Expedition 1925-27, and has been described in Vol. 4, No. 1 of the ''Meteor'' Report. It is worth mentioning particularly that a machine and an obser- vations schedule containing everything of importance in the working programme for the series should be kept for each oceanographic station. Very often the results of an oceanographic series depend on the careful compilation of the machine and observa- tions schedules. Apparently unimportant details may become important later during the interpretation of the observations and can contribute to the uniformity and homo- genity of the observations. 2. Temperature Determination for all Layers of the Ocean The determination of the temperature of the surface layer of the sea offers little difficulty. A sample taken from water collected in an ordinary bucket, lowered into the sea for a short lime while the vessel is under way, is put immediately in a shady place and its temperature is taken with a sensitive thermometer while at the same time it is kept stirred. The water sample must be drawn from as far forward as possible (on steam ships forward of the condenser exhaust). See Lumby (1927) on the measurement of surface temperatures and the collection of suitable water samples. New surface sampling bottles have been designed by Sund (1931) and improved by Schumacher (1938). The determination of the temperature of the deeper layers of the sea is considerably more difficult, and this also needed the work of almost a decade to reach an accuracy suitable for scientific requirements. In the upper layers temperatures correct to 0-1 °C are usually sufficient, but in the deep layers the variations both horizontally and ver- tically are usually so small that an accuracy of 0-01 °C is needed to get some idea of The Sea-water and its Physical and Chemical Properties 35 the spatial variations in temperature. This accuracy is also necessary for the calculation of densities accurate to the fifth decimal place. Deep-sea thermometers are thus extremely accurate and sensitive instruments which cannot be handled skilfully just by anyone. An ordinary thermometer suspended freely in the water will not show the correct temperature since the pressure of the water will compress the thermometer bulb and force the mercury to a higher level. It is therefore necessary to protect the thermometer against the water pressure by enclosing it in a thick-walled glass tube. The part of the tube surrounding the thermometer bulb is filled with mercury to improve the heat transfer between the water and the bulb. Since the temperature usually decreases with depth the instrument first used was a maximum and minimum thermometer con- structed by Six and adapted for deep-sea use, and this was the classical instrument used on the "Challenger" and the "Gazelle" Expeditions. Since 1874 the reversing thermometer, first produced commercially by the firm Negretti and Zambra, has been used instead, and with numerous modifications is still used at the present time as the standard instrument for oceanographic temperature recording. This is a thermo- meter with the capillary considerably constricted a little above the mercury bulb, so that the mercury thread will break at this point when the thermometer is turned through 180° and slide down to the other end of the capillary. The higher the tem- perature when the thermometer is reversed the longer the mercury thread that is broken off. This thread gives a direct reading of the temperature at that time when read against a scale running in the reverse direction with appropriate corrections. The accuracy of the thermometer is very dependent on the shape of the constriction. It must, of course, be made so that the mercury thread always breaks at the same point and it must be designed so that further mercury cannot follow the thread if the ther- mometer passes subsequently through a warmer layer of water. All the initial diffi- culties were overcome by the work of Richter (of the firm Richter & Wiese, BerHn) so that the reversing thermometer is now a true precision instrument. The shape of the constricted part of the capillary is shown in Fig. 20. Further details are given in the ''Meteor'' Report, 4, No I, by Bohnecke (1932), and in "Oceanographic Instrumenta- tion" (Rep. Conf. Rancho Santa Fe, Calif. 21-23 June 1952, p. 55). In use the reversing thermometer is enclosed in a suitable holder (a brass tube) which is attached directly to a reversing sampling bottle or to a frame which can be reversed at the desired depth (reversing frame, propeller frame). The reversing thermometer does not show the true temperature {in situ) directly since it will have been brought back to the surface through layers of water at different temperatures. After removal from the sampling bottle on deck it is placed immediately in a water bath and allowed to adjust to the water temperature before it is read. To show the temperature of the water bath every reversing thermometer is fitted with a normal auxiliary thermometer. To correct the reading to the temperature in situ a small correction given by the formula {T -t){r+ Kq) 6100 J ^ {r~t){r+ V,) 6100 must be applied. In this equation T' is the uncorrected reading of the reversing ther- mometer, t the reading of the auxiliary thermometer (bath temperature), Vq is the 36 The Sea-water and its Physical and Chemical Properties Fig. 20. Reversing thermometer (with visible constriction). volume of the small bulb and the capillary of the main thermometer until 0°C and expressed in degree units on the capillary scale, 1/6100 = jS being the coefficient of expansion of mercury. The corrections given by the formula are listed in tables to allow quick accurate working (Schumacher, 1923, 1933; Hidaka, 1933; Geissler, 1934). Kalle (1953) has given a simple graphical method for the determination of the corrections (C). A calibration correction has to be added to the corrected reading of the thermometer. By very careful attention to all the factors involved (continual checking of the re- versing apparatus, accurate readings using a magnifying glass, checking the zero point, proper correction) the mean error in the temperature determination can be kept down to, on the average, ±0-01 °C. This method gives the temperature at single points in the ocean and is of considerable use in series observations at oceanographic stations. For a special purpose, however, it may be desirable to have a continuous record of the temperature at a fixed depth or to obtain quick successive readings of the temperature in a particular layer. A thermograph is usually used for the first purpose (at coastal stations or for continuous recording of the temperature at the surface of the sea from a moving vessel). For greater depths diff'erent types of electrical resistance thermometers have been designed but they have not yet proved very satisfactory in use. For a rapid survey of the upper 150 m of the sea or for a continuous registra- tion of the vertical temperature gradient of this upper layer to about 200 m, Spil- HAUS (1938, 1940) has developed and tested a bathythermograph. This has proved successful and offers considerable advantages where rapid changes of temperature can be expected. For greater depths Mosby (1940) has designed a "thermosounder" that has given useful results. 3. Salinity and its Determination One of the most important properties of water is its ability to dissolve a very large number of solids and gases without chemically reacting with them. As a consequence The Sea-water and its Physical and Chemical Properties 37 of this property all the water on the earth is more or less impure, that is it contains in addition to chemically Hnked hydrogen and oxygen (HgO) a number of other substances in varying amounts. If the salt content, the salinity, were defined as the weight of all the salts dissolved in a kg of sea-water this would provide to be the simplest numerical specification of the amount of dissolved salts in the water. Unfortunately it is rather difficult to measure this definite quantity since, when sea-water is evaporated to dry- ness and heated to red heat to remove the last traces of water, some hydrogen chloride, carbon dioxide and a small amount of hydrogen bromide are also lost. This loss is not easily compensated with sufficient accuracy by adding a corresponding correction. At the suggestion of Forch, Sorensen and Knudsen (1902) the salinity has been de- fined as the total amount of solid material in grammes contained in 1 kg of sea-water when all the bromine and iodine have been replaced by the equivalent amount of chlorine, all the carbonate converted to oxide and all organic matter has been com- pletely oxidized. The salinity defined in this way can be determined with great accuracy and can thus serve as a basis for the investigation of the relationship between any single component and the total salinity. Sea-water is a dilute solution of a mixture of salts; in such an aqueous solution salts, acids and bases are more or less completely electrolytically dissociated (Arrhenius and van't Hoff). The chemical compounds precipitated on evaporation of such solu- tion are in solution split into atoms or groups of atoms with an electric charge, either positive (cations) or negative (anions). The electrical charges balance exactly so that the solution remains electrically neutral. The constituents of this mixture of salts are therefore listed as their ions. Table 8 shows the composition of a typical sample of sea-water with a salinity of 34-40%o. Table 8. The principal constituents of sea-water (34-40 /oo salinity) Cations Sodium Potassium Magnesium Calcium Strontium g/kg 1 mmole/kg 10-47 0-38 1-28 0-41 0-013 455-0 9-7 52-5 10-2 0-15 percent- age of S 30-4 1-1 3-7 1-2 0-05 Anions Chloride Bromide Sulphate Bicarbonate Borate g/kg 18-97 0065 2-65 0-14 0027 mmole/kg 5351 0-81 27-6 2-35 0-44 percent- age of S 55-2 0-2 7-7 0-4 0-08 It was formerly customary to give the constituents of sea-water in terms of the com- pounds that were precipitated on evaporation. Dittmar (1884) has given the figures shown in Table 9 as the mean of seventy-seven very complete analyses of sea-water samples made by the "Challenger" Expedition; they have been calculated on the basis of a salinity of 35 g of salts in 1 kg of sea-water. In the open ocean the total concentration of salinity varies between moderate limits, usually between about 33 and 38%o depending in the first place on the climate (precipitation, evaporation and in polar regions ice melting). In coastal areas where there is a considerable inflow of fresh water from rivers and from ground water the salinity may have a considerably lower value. Especially in the almost closed adjacent seas of higher latitudes (such as the Baltic) with low evaporation, a considerable 38 The Sea-water and its Physical and Chemical Properties Table 9. The salts obtained from sea-water (Calculated as 35 g of salts per kg) Salt weight in g/kg sea-water Percentage of total salts Sodium chloride OJaCl) 27-213 77-758 Magnesium chloride (MgCl) 3-807 10-878 Magnesium sulphate (MgSO^) 1-658 4-737 Calcium sulphate (CaS04) 1-260 3-600 Potassium sulphate (K2SO4) 0-863 2-465 Calcium carbonatef (CaCOg) 0123 0-345 Magnesium bromide (MgBrg) 0076 0-217 Total 35000 100000 t Includes all the other salts present in trace amounts. inflow of fresh water and precipitation on the surface may have a low saHnity (8- 5%o) and at the inner ends mostly only brackish water with 1%^ or even lower. The highest salinities are to be found, on the other hand, in the subtropical adjacent seas with almost no inflow of fresh water, no precipitation and strong evaporation as, for instance, in the Red Sea and in the Persian Gulf which at the inner ends have maximum salinities of almost 40%o. While the salinity is always liable to show some variations the proportion of the different ions in sea-water is remarkably constant. This constancy which is of con- siderable oceanographic importance is only further confirmed by all carefully made analyses. Accurate chemical analysis of the samples collected by the "Challenger" Expedition from almost all parts, and depths of the ocean demonstrated this constant proportion between the individual constituents and more recent investigations as shown in Table 10 have led to the same results. Table 10. Analysis of the salt content of sea- water (percentages) (DiTTMAR, 1884; Makin 1898; Wheeler, 1910) No. of samples 77 22 5 CI 55-29 55-18 55-29 Br 0-19 0-13 — SO4 7-69 7-91 7-56 CO3 0-31 0-21 0-37 K Ml Ml M4 Na 30-59 30-26 30-76 Ca 1-20 1-24 1-22 Mg 3-72 3-90 3-70 The mean ratio Mg : CI is 0-0682 and for SO4 : CI the ratio is 0-1397. The most recent analyses by Matthews, Thompson, and others, have given a value of 0-6802 (with limits of 0-6785 and 0-6814) for the first ratio and 0-1395 (with limits of 0-1387 and 0-1403) for the second. Using very accurate analyses calcium and bicarbonate 77?^ Sea-water and its Physical and Chemical Properties 39 show sometimes smaller variations from the above-mentioned general propor- tionality (not more than 1%) which are due to biological processes (precipitation of calcium carbonate), to the solution of calcium carbonate from sea bottom and in coastal areas to the inflow of river water (containing calcium carbonate). The very constant proportions of the ions present in sea-water allow chlorine to be used as a measure of the salinity of a sample of sea-water. This was done many years ago by Forchhammer (1859, 1865) and later by Knudsen (1902), from a very careful examination between 2-69 and 40-18%o, derived the simple equation S = 0-030 + 1-8050 CI, which is now used generally for the calculation of the salinity (S) from the chlorine content. This salinity is that given in the definition above. It is a little smaller than the actual salt content (by about 0-14%o) but since it is the differences in salinity that are important this has very little significance. The most convenient method for the determination of salinity is that of Mohr (1956) in which the sample is titrated with silver nitrate with a calcium chromate solution as indicator; this is also suitable for use on board ship. This chemical method gives a relatively fast and accurate determination of the chlorine in sea water, and the salinity can be calculated from this value using the equation given above. This method is the usual method used at the present time in practical oceanography (see especially Meyer (1932) for the practical details of the titration and the necessary working rou- tine). The chlorine titration is only a relative determination, and to find the absolute value it is necessary to standardize the solution used for titration against the "Normal water" introduced by Knudsen (1903, 1925); this standardization very largely elimi- nates the effect of the subjective assessment of the colour of the indicator. Normal water is sea-water kept in sealed glass tubes of which the chlorine content has been very accurately determined, formerly by the central laboratory of the International Hydrographic Institute in Copenhagen, and at the present time by the Woods Hole Oceanographic Institution. The difference between the value obtained by titration of the normal water and that marked on the tube gives the total error in the titration. Knudsen (1901) has prepared hydrographic tables for the comparison of chlorine determinations of sea-water with different salinities with the chlorine determination made on normal water. If the average salinity of the ocean is taken as 35%o then calculation gives the total amount of salt in the ocean as 4-84 x lO^*' tons; this corresponds to a volume of 21-8 miUion km^ which, spread evenly over the sea (361 million km-), would be a layer of salt 60 m thick. In addition to the substances already mentioned, sea-water also contains traces of a large number of elements which are of little importance for oceanography, though they are probably important in the metabolism of marine organisms. The determination of the concentrations of these elements presents very great analytical difiiculties and the older determinations must be treated with great caution. Table 1 1 shows a more recent list of the elements present in the sea according to Kalle (1945), which is based on a similar one given earlier by Watterberg (1938). In many cases the figures given represent only the order of magnitude of the concentration of an element. Of 40 The Sea-water and ifs Physical and Chemical Properties the elements that are present in somewhat greater concentration may be mentioned iron, copper and gold. Iron is present in extremely small quantities and sea-water is probably one of the naturally occurring materials poorest in iron. The importance of copper can be seen from its occurrence in place of iron in the blood pigments of many marine animals (hccmocyanin). The occurrence of gold in sea-water at one time aroused particular interest since, according to older determinations, the isola- tion of gold from sea-water was technically promising. These older determinations have, however, been shown by the results of Haber (1928) and Jaenicke (1935) to be incorrect, and the gold found came largely from the reagents used, from the air and from the glass of the apparatus. The gold content of sea-water found by analysis of the samples collected on the "Meteor" Expedition was only 4 x lO"'* g/kg of sea- water, a concentration which would be of no technical use. Table J I. Concentrations of the trace elements present in sea- water in milligrams per cubic metre (According to Kalle, 1945) Fluorine 1400 Selenium 4 Silica 1000 Uranium 2 Nitrogen (NO", NO;, NH3) 1000 Caesium 2 Rubidium 200 Molybdenum 0-7 Aluminium 120 Cerium 0-4 Lithium 70 Thorium 0-4 Phosphorus 60 Vanadium 0-3 Barium 54 Yttrium 0-3 Iron 50(2) Lanthanum 0-3 Iodine 50 Silver 0-3 Arsenic 15 Nickel 01 Copper 5 Scandium 004 Manganese 5 Mercury 003 Zinc 5 Gold 0004 Radium 00000001 The radioactivity of sea-water has been accurately investigated in recent times, and detailed examinations have been made principally by Pettersson (1937, 1938), and Thompson and his collaborators (1932). According to these investigations the radium content of sea-water with a salinity of 35%o varied between 0-04 and 0-2 x IQ-^^ o/^ (or between 0-04 and 0-2 billionth parts of a gramme per litre); 0-07 x 10-^^ 0/^ radium can be taken as mean value. Deep water has a uranium content of 1-5-2 x 10"'' %o; surface water has a somewhat lower value. The thorium content is less than 0-5 x 10-« %«. Since the radium content of sea-water is 10-000 times less than that of rocks of the Earth crust, it is extremely small and corresponds to only 10% of the amount that would be in equilibrium with the uranium content. According to the view of Petters- son, this remarkable deficiency of radium in the sea can be attributed to the very rapid precipitation of the iron carried into the sea, almost entirely as ferric hydroxide. In the precipitation the thorium and its isotope ionium that immediately precedes radium in the disintegration series are co-precipitated. The ionium produced from the uranium in solution in the sea is thus steadily removed by precipitation of the iron. The Sea-water and its Physical and Chemical Properties 41 Only that part of the element remaining in solution disintegrates to give radium and its disintegration products in the sea (see also Hess, 1918). 4. The Density of Sea-water and its Dependence on Temperature, Salinity and Pressure The density p of a material is the mass of a unit volume [g cm"^]. Frequently the specific weight is given instead of the density; this is defined as the quotient of two densities p/p„., where p is the density of the substance in question and p,,, is the density of distilled water at a fixed temperature. The specific weight is thus a dimensionless quantity. In the CGS system the density and the specific gravity are numerically equal if distilled water at 4°C is taken as the comparison liquid. Due to its salt content sea-water is heavier (more dense) than pure water. The den- sity is always fairly close to 1 and varies depending on the salinity S, the temperature / and the pressure p between narrow limits ; for example, at the surface of the open ocean between 1-02750 and 1-02100. For oceanographic purposes it is necessary to know the density correctly to at least 5 decimal places. For simplicity instead of using p it is customary to use a density value o derived from the equation a = (p— 1) x 10^; for instance instead of p = 1-02754, a = 27-54 is used. Very often the reciprocal of the density 1/p = v, the specific volume [cm^ g-^] is used. This also is required cor- rect to the fifth decimal place and for simplicity and convenience only the last three figures are given according to the equation a = (i; — 0-97) x 10^. For example when V = 0-97320, a = 320. The dependence of the density and the specific volume on the temperature, the salinity and the pressure were first investigated at the beginning of this century (1899) by an international commission headed by K>ajDSEN (1902, 1903). The relationship of the density at 0°C and atmospheric pressure at sea-level to the chlorinity is given by ao = -0-069 + 1-4708 CI - 0-001570 Cl^ + 00000398 C\\ This equation is valid for chlorinities between 1-47362 and 22-2306. The dependence of the density of sea-water on the temperature requires a knowledge of the thermal expansion of sea-water. The thermal expansion coeflficient determined in the laboratory shows that the density has a pronounced dependence on the tem- perature; at atmospheric pressure (sea surface) is given by o-^ — CTq — Z). Z) is a very complicated function of a^ and of the temperature / and has been given to the fifth place in Knudsen's hydrographic tables (1901). Schumacher (1922) has also given graphical tables, and further tables for the determination of the density of sea-water under normal pressure have been given by Matthews (1932) and Thorade and Kalle (1940). These tables show that an increase of 0-01%o in the salinity gives an approxi- mate increase in the density (ct^) of 8 units in the third decimal place. The increase is about the same for all temperatures and salinities. For low and high temperatures the density change is very different and depends also somewhat on the salinity. Figure 21 (Helland-Hansen, 1911-12) shows the eff"ect of variations in temperature on the densities of distilled water and of sea-water with sahnity 35%o. From the re- lationship between temperature and density the temperature of maximum density can be determined for different salinities. This is also given with somewhat less ac- curacy by the equation /max = 3-95 - 0-266ao. 42 The Sea-water and its Physical and Chemical Properties -3 -2 ^ Seowater5 = 357oo ^ -^ ^ ^ ■^ 0 1 / ^ ^Pure water ^ 1 1 1 1 i -2 0 12 16 t, "C 20 24 28 Fig. 21. Effect of changes in temperature on the density of pure water and of sea water at 35^00 salinity. Thus for different salinities, where S in %„ = 0 10 20 cmax = 000 818 16-07 /max in °C = 3-947 1-860 -0-310 25 30 35 40 2010 24-15 28-22 32-32 -1-398 -2-473 -3-524 -45-410 Since water is compressible, though only slightly, the density depends on the pressure. In the deeper parts of the ocean the pressures are enormous and have a considerable effect on the density of the water. The change in unit volume of a material per pressure unit is termed its compressibility coefficient /x. If the pressure unit is taken as 1 bar (= 10*' dynes/cm^) then the compressibility coefficient of sea-water is of the order of magnitude of 450 x 10"'; it increases somewhat with increasing pressure, increasing salinity and increasing temperature and its extreme values lie somewhere between the limits 510 and 390 X 10~". Ekman (1908) derived a precise empirical formula for the effect of pressure on the density that takes into consideration the changes in the compressibility coefficient with salinity and temperature (see Landholt-Bornstein, 1952; Dietrich, p. 484). This gives the density of sea-water for a given salinity, given temperature and a fixed pressure and thus gives the density in situ a,^ ^ of a water sample directly from a,. Bjerknes and Sandstrom (1910) have presented complete tables to allow the spe- cific volume anomaly or the density to be quickly found from the basic values for a homogeneous sea at 0°C and with 35%o S for depths down to 10,000 m or pressures of 10,000 decibars. Hesselberg and Sverdrup (1915) have given a method by which the vertical variations in density can be calculated in a fairly simple way from the temperature and the salinity. This simplification is due largely to the elimination of part of the work by starting in the first place from the value for a,. If only the anomaly is required, the tables prepared by Sverdrup (1933), which are still further simplified and which give more accurate results, can be used. In general the relation a^ j^ = ^35, 0. 0 + S can be used where S is the specific volume anomaly. 5 is the sum of three terms: 5 = A,j + Sgj, + S,,p. As shown by the indices the first term depends on the temperature and the salinity, the others depend on the pressure and on one of the other two factors each. Since The Sea-water and its Physical and Chemical Properties 43 S, X 10-3 and then "35. 0. 0 = 0-97264, A^^^ = 0-02736 ^.,t= 1 + X 10- 1 + a, X 10-3 1 + a, X 10-3 The values of the three terms J j,,, 6,,,^ and S,,^, can be given in short tables from which the anomaly can be found correct to five decimal places. The same accuracy can be obtained by accurate graphical methods or with the ingenious slide rule of Sund (1929). The usual method for determining the density in oceanography is by calculation from the temperature, the salinity and the pressure. The physical methods of de- termining density such as the hydrostatic weighing and the pycnometer are unsuited for oceanographic purposes, but the hydrometer has however often been utilized in oceanography. Some very troublesome sources of error present with the ordinary stem hydrometer have been discussed in detail by Krummel (1900), Buchanan (1884) and Nansen (1900). They originate from insufficient attention to temperature differ- ences between the instrument and the water sample and within the water sample itself, the variable wetting of the instrument (traces of oil on the surface), the air content of the water sample and not least to the variable capillary rise of the water in the stem of the instrument which is often difficult to allow for. With proper use this instrument gives values for a^ correct to two units in the second decimal place. Nan- sen (1900) avoided the errors due to varying surface tension at the stem by using a "hydrometer of total immersion" in which the ffoat is balanced in the water sample by the addition of suitable weights. This method gives a^ correct to the third decimal place (SvERDRUP, 1929). Since work with small weights is inconvenient on board ship O. and H. Pettersson (1929), used a diff"erent method of loading a float hydro- meter which is very simple and requires no handling of the float. A fine chain is sus- pended from the float (chain hydrometer) so that the length of chain supported above the bottom is a measure of the density. Another method for the direct determination of the density which has been used in older investigations (Pulfrich refractometer) utilizes the difference in refractive index of the water sample from that of distilled water. This is measured either by the Hall- wach method or by interferometry. The first method was used by Krummel (1889) on the "Plankton" Expedition and later in 1892 by Drygalski on the Greenland Expedi- tion. The interference method is more sensitive, although it requires suitable labora- tory work to give the desired accuracy. (Askania Interferometer, Bein, Hirsekorn and Moller, 1933, 1935). This interference method has been developed to give greater precision and will give the density to the third decimal place in a^. As well as the optical refractivity it is also possible to use the electrical conductivity for the determination of densities. This method has several times been recommended but has seldom actually been used. A survey of these experiments has been given by Bein (1936). An instrument suitable for routine use was first developed by the Bureau of Standards in Washington (Thuras, 1918; Wenner, 1930). It was in continual use by vessels of the Ice Patrol in the North Atlantic Ocean from 1921 and was used by the 44 The Sea-water and its Physical and Chemical Properties oceanographic vessel "Carnegie". Experience with this "saline tester" was not very encouraging and the accuracy attained was, in spite of the greatest precautions, not entirely satisfactory. 5. Vapour Pressure, Freezing Point, Boiling Point and Osmotic Pressure of Sea -water Sea-water is a "dilute" solution and has the properties of such a solution. Due to the low concentration of the dissolved material these will in several respects approach those of the pure solvent, i.e. of pure water. It was shown quite early that the vapour pressure p of a dilute solution is always less than the vapour pressure p^ of the pure solvent and that the elevation of the boiling point is accompanied by a depression of freezing point. As shown by Raoult and van't Hoff the relative lowering in vapour pressure is independent of the nature of the material in solution and of the temperature of the solution, and is proportional to the amount of dissolved material in solution in the solvent. For a solution of « moles of a substance in Nq moles of a solvent: Po Po n No Figure 22 shows the different phase states for pure water and for sea-water; it illus- trates more clearly the relationship between the three well-known properties of dilute solutions mentioned above. The curve G'S' showing the lowering of vapour pressure is always lower than the vapour-pressure curve for pure water by the amount of the 760mm s r^/ /r / / \ ' / m Water phaseyp/ y i / z'' Ice / y phase y/ / Sub cooled W ^ ^y^ ^y^p Sea water j u :r^^ yo-p^^ ^L^'" ^-^ Vapour phase /*(?■ tc Temperature fs h' 0°C IOO°C Fig. 22. Phase states for pure water and for sea-water (schematic). depression p^ — /;. Since for a given concentration (po — pMPq is constant, this de- pression increases with increasing pressure and therefore also with increasing tem- perature. The Po-curve for pure water cuts the line for a pressure of 760 mm Hg at the point 5; this is the boiling point of pure water for which the corresponding tem- perature ts = 100°C. The vapour-pressure curve for sea-water cuts this isobar first at S' and this boiling point corresponds to a temperature z^- which is higher than /,. The elevation of boiling point /1/s of sea-water of a given concentration is given by At, = /v - r,. 77?^ Sea-water and its Physical and Chemical Properties 45 The depression of freezing point by a dissolved substance can also be inferred from this diagram. The intersection G of the solid and liquid phases (the triple point) corresponds to a temperature of 0-0075 °C. At 760 mm Hg the freezing point to of pure water is 0°C and is fixed by the position of the intersection of the melting-point curve Grwith the 760 mm isobar. It is the temperature at which the two phases (water and ice) have the same vapour pressure, and therefore are in equilibrium with each other. On the other hand, the freezing point of sea-water is at the intersection G' of the vapour pressure curve for sea-water and that for ice; at this point the vapour pressures over sea-water and over ice are the same. This corresponds at 760 mm Hg to the freezing point of sea-water to' which is lower than to- The freezing-point de- pression for sea-water is given by J/c = to' — to- From this diagram it can immediately be deduced that both quantities Ate and At^ are larger the larger the value oi p^— p of the relative lowering of vapour pressure ApJ p, that is the larger the concentration of the solution of the salinity. Quanti- tatively it has been shown experimentally and theoretically that for low concentrations the elevation of the boiling point and the depression of the boiling point are both proportional to the concentration. In dilute solutions of substances termed in physical chemistry "strong electrolytes", amongst which sea-water is included, it is found that the electrolytic dissociation of the molecules is equivalent to an apparently larger molecular concentration so that the simple proportionality no longer holds. The accurate determination of saturated vapour pressures and of boiling points is experi- mentally difficult and has been described in detail. The freezing point has been de- termined by Hansen on eleven samples of sea-water, and by Knudsen (1903), using determination of the constants, and the following empirical equation has been found to -0-0086 - 0-064633 a^ - 0000 1055 al. This gives freezing temperatures correct to ±0003°. Table 12. Freezing point and osmotic pressure of sea-water Salinity (%) 5 10 15 20 25 30 35 40 Freezing point (°C) -0-267 -0-534 -0-802 -1-074 -1-349 -1-627 -1-910 -2-196 Density (ctq) 3-96 8-00 12-02 16-07 20-10 24-14 28-21 32-27 Osmotic pressure (atmos.) 3-23 6-44 9-69 12-98 16-32 19-67 23-12 26-59 Table 12 shows related values of salinities, freezing point t„ the density of sea- water at this temperature and also the osmotic pressure (see later, p. 48). For the relative lowering of vapour pressure Witting (1908) has given the equation Aplp^ = 0-538 X 10-3 5. the elevation of boiling point can as a first approximation be obtained from At, = 0-01585. Table 13 gives related values for the elevation of boiling point and the lowering of vapour pressure at boiling point (at 760 mm Hg). 46 The Sea-water and its Physical and Chemical Properties Table 13. Elevation of boiling point and lowering of vapour pressure in sea-water Salinity, %„ 5 10 15 20 25 30 35 40 J/,(X) 005 016 0-23 0-31 0-39 0-47 0-56 0-64 Jp in mm Hg at 760 mm Hg 213 4-23 6-45 8-47 10-73 12-97 15-23 17-55 A comparison of the temperature of the freezing point / a, that of maximum density d and their corresponding densities at different sahnities is of some interest. Figure 23 shows the change in these temperatures with increasing sahnity. The temperature of 2 - \ \i> 0 24 695%<.J h \. -2 -4 -1 332-0 ->- br^ V .^ - N 1 1 1 10 30 40 20 S, %„ Fig. 23. Dependence of freezing temperature to on the salinity S. maximum density decreases with increasing salinity more rapidly than the temperature of the freezing point. At a salinity of 24-695%o (ctq = 19-839) both temperatures are the same and {) = fa = -1-332°C and a^ = a,^ = 19-852. Reference might be made here to an oceanographic use of this (Helland-Hansen, 1911-12). Suppose a surface layer of a sea area is homo-haline with a salinity less than 24-695%o and that its surface is subject to strong cooling in winter. This cooling will increase the density of the surface water, and as a consequence a vertical convection must occur and will continue until the whole homo-haline surface layer reaches the temperature of maximum density. It will then cease. The surface only will now be cooled further by radiation until it reaches the freezing point and ice begins to form. This will increase the salinity, and convection will again be set up and will be maintained by the double effect of the increase in salinity and the decrease of temperature. These conditions may occur, for example, in the Baltic. The homo-haline surface layer with 5 = 10%o during the winter cools and the vertical convection continues until the The Sea-water and its Physical and Chemical Properties 47 temperature reaches +1-86°. The whole layer then has the maximum density a^ = 8-18. If cooling proceeds further the temperature falls only at the surface until this reaches the freezing point tc — —0-53°, where ct,^ = 8-00, while the remainder of the water mass remains at 4-1-86° and a^^ = 8-18. On further loss of heat ice is formed and the density is raised by the liberation of salt until it reaches 8-18 when convection starts again and continues as long as ice continues to form. If, on the other hand, the surface layer has a salinity greater than 24-695% then the vertical convection continues until the whole layer reaches the temperature of the freezing point and proceeds further without interruption as long as fresh ice continues to form. The difference between the two densities cr^g, and ua^ is, however, not large. As shown in Fig. 24 these differences are largest at salinities of 6-7%o and very small between 20%o and 35%o. C-25 0-20 ^ 015 's' 010 005 0 / ''"^ \ / \ <>= fG 1 y 10 15 20 25 30 35 40 5, %o Fig. 24. Density at the freezing temperature and maximum density of sea water as a function of salinity. Two adjacent water masses of different salinity will not, as far as their salinity is concerned, be in equilibrium. In solutions of different concentrations in contact in this way the material dissolved in the water will move from the region of higher con- centration to that of lower concentration, that is, in the direction of the concentration gradient. Known as molecular diffusion, it follows the same laws as thermal conduct- ivity. If the salinity gradient is —{dSjdx) x 10~^, where S is given in %o, there will, by diffusion, pass in unit time (sec) through unit area at right angles to the direction of the gradient (1 cm-) an amount of salt Mg given by Mg = —K(dSjdx) x 10"^ where k is the molecular diffusion coefficient with the dimensions (g cm"^ sec~^). The change with time in a given distribution of salinity follows from the differential equation dSjct =^ k{c'^SIcx^) where /c is a constant independent of the time and the distance (Fickian-diffusion equation). The diffusion coefficient for sea-water is very small (0-0189 g cm"^ sec"^ at 35%o), molecular diffusion thus proceeds extremely slowly, and long periods are needed to eliminate larger differences in salinity by pure molecular diffusion. In this respect diffusion is quite analogous to thermal con- ductivity. Osmotic pressure is a phenomenon that is closely related to the properties of dilute solutions described above. It is of very considerable importance for the biology of living organisms in the sea. If a tank II (see Fig, 25) filled with sea-water of salinity 48 77?^ Sea-water and its Physical and Chemical Properties S%o is separated from a tank I containing distilled water by a semi-permeable mem- brane M which is permeable only for water and not for the substances in solution, water will pass from tank I through the membrane M into tank II which contains the the salt solution, and as a result the pressure in the tank II will rise. The sea-water could be said to draw the pure water through the membrane. This process will continue until the excess pressure in TI exceeds that in I by a fixed value P. This excess pressure at which the system is in equilibrium is termed the osmotic pressure. According to physi- cal chemistry it has been shown (see Nernst, Theoretische Chemie, 4th ed. 1903, Fig. 25. For explanation of the osmotic pressure. p. 157) that there is a relationship between the osmotic pressure and the depression of freezing point which for sea- water at 0° takes the form P = —M-AAta- Stenius (1904; see also Thompson, 1932) found the proportionality value 12-08 atm for the constant in this equation. For other temperatures Pq must be multiplied by (1 + 0-003670- Table 12 gives values for the osmotic pressure at 0° according to Stenius. The size of the osmotic pressure gives an idea of its biological importance. Or- ganisms that live in the water are usually covered by a skin that is partly permeable to water. They live in osmotic equilibrium with their environment. If one of these organisms is placed in water of lesser salinity, water will pass in through its skin into its body ; if the salinity is higher, water will be removed. Both processes, if they occur to any extent, are unfavourable to the life of the organism since thecapacity of adaptation is fixed within narrow limits. 6. Other Physical Properties of Sea-water Other properties of sea-water that are also of importance in oceanography and should be briefly mentioned are the heat capacity and the thermal conductivity, the surface tension and the internal viscosity. {a) The heat capacity of the specific heat of a body is the number of calories required to heat 1 g of the material through 1 °C. The specific heat of pure water is dependent on the temperature and shows a minimum of 0-947 at 34°C. It rises more rapidly to- wards lower than towards higher temperatures and at 18°C it is 0-999. A series of experimental determinations of the effect of the salinity was made by Thoulet and Chevallier (1899) and their results have been utilized by Kriimmel to prepare the figures shown in Table 14. The experimental value for the specific heat of sea-water c^ is less than would be expected from the amount of salt in solution. The Sea-water and its Physical and Chemical Properties Table 14. The specific heat of sea-water at 17-5^ 49 Salinity (°bo) c„ 0 1000 5 0-982 10 0-968 15 0-958 20 0-951 25 0-945 30 0-939 35 0-932 40 0-926 The dependence of Cp for sea-water on the temperature has not yet been closely investigated, but presumably it is of the same form as that for pure water. Figure 26 shows the effect of temperature on Cj, for pure water and for sea-water with 35%o S. The dependence of c^ on the pressure/? can be found using well known thermodynamic I-OI 100 "S \ ^ure wate r s ^ :a w Iter 3-99 0-95 0^4 0-93 10 20 30 40 50 r, "C Fig. 26. Specific heat for pure water and for sea water at 35o/(,p salinity. principles (Ekman, 1914). If the pressure/? is taken in decibars, and the density of the water is p, the absolute temperature T, the coefficient of thermal expansion /S, and J is the mechanical equivalent of heat (4-1863 x 10^ ergs/cal or dyn cm/cal), then dp pj \ 8t ^' Ekman has calculated the value of c^ for atmospheric pressure and for pressures from p = 2000 top = 10,000 decibars, corresponding to depths of about 2000 to 10,000 m (Table 15). At great depths c^ differs appreciably from 1 and this must be taken into account in accurate theoretical calculations. Table 15. Specific heat of sea-water at different pressures when ct = 28 (34-8%o) Temperature ' (^C) -2 0 5 10 15 20 Pressure in dbar 0 0-942 0-941 0-938 0-935 0-933 0-932 1000 0-933 0-933 0-930 0-929 0-928 0-927 2000 0-925 0-925 0-924 0-923 0-922 0-921 3000 0-910 0-912 0-913 0-913 0-913 — 6000 0-898 0-901 0-904 — — — 8000 — 0-892 0-896 — — — The relationship k = c^jc^ is also of interest. The specific heat/constant volume c,; is a little less than Cp. From thermodynamics the equation Cp = c„ + ^^ pixj 50 The Sea-water and its Physical and Chemical Properties can be derived, where jj. is the cubic compressibiHty. For sea-water where Oq = 28 (34-84%o S) at temperatures of 0" and 30°C respectively, /3 = 15 x 10^« and 334 x 10-« grad-i and ix = 46-59 x IQ-^^ and 42-07 x lO-^^ jyn-i cm^. From this it can be found that k -= 1-0004 and 1-0207 for 0°C and 30°C respectively. At greater depths ^J. is smaller and there k is larger than at the surface. (h) The thermal conductivity coefficient A is defined by the equation Q = -x(ddidx), where Q (cal/sec) is the amount of heat passing through 1 cm- at right angles to the flow and dd ( C) is the change in temperature along a distance d.x (cm) in the direction of flow. A thus has the dimensions (cal cm~^ sec~^ grad"^). For pure water A = 0-001325 + 4 X 10-«/. A has not been determined directly for sea-water; as a first approximation, according to Weber's rule, the ratio of the thermal conductivities of two substances is the same as that of the thermal capacities of equal volumes. This gives the values shown in Table 16 for the coefficient of thermal conductivity for different salinities. Table 16. Coefficient of thermal conductivity at different salinities Salinity (%„) 10 I 20 30 I 35 40 Thermal conductivity I ! coefficient (X 10-») 1-400 1-367 1-353 • 1-346 ! 1-341 1-337 For oceanic water (35%o S) the thermal conductivity coefficient is about 4-2% less than for pure water. The temperature conductivity coefficient is the quantity a = XJipCp) and has the dimensions (cm- sec~^). For sea-water pCj, is not very different from 1 and the numerical difference between A and a is slight. (c) In fluids with motion there is a shear stress between every layer in the direction of flow and the adjacent parallel layer, and this shearing stress is proportional to the velocity gradient perpendicular to the direction of flow, that is dv ^ dz The proportionality factor /x is a measure of viscosity or inner (molecular) friction (g cm~^ sec~^). For many flow phenomena there occurs the coefficient i- = /x/p, the kinematic viscosity (cm- sec "^). These frictional coefficients decrease rapidly with increasing temperature. For pure water, the values shown in Table 17 are ob- tained. According to the investigations of Krummel and Ruppin (1905) viscosity increases very little with salinity; at 0°C by 3-9 or 5-2% for 25%o S and 35%o S re- spectively and at 30X' by 6- 1 or 8-2"o. The effect of pressure appears to be negligible. Table 17. Viscosity coefficients for pure water (g cm"i sec"^) Temperature ( C) 0 10 j 20 30 40 /x 0-0179 0-0131 , 0-0100 1 00080 00065 The Sea-water and its Physical and Chemical Properties 51 The magnitude of the molecular viscosity was eariier attributed some importance in the biological and dynamic processes in the sea, but it has since been recognized that processes in oceanic currents are always turbulent and the coefficient of turbulent viscosity is considerably larger than the coefficient of molecular viscosity. This has very much reduced the importance of the latter. (d) Surface tension. Krummel (1907) investigated the dependence of the surface tension on the temperature and the salinity; it decreases with rising temperature and with decreasing salinity. Fleming and Revelle (1939) have taken more recent values to derive the equation surface tension in dyn/cm^ = 75-64 - 0-144/ + 0-0399 CI. Impurities in the water always lead to a considerable reduction and this must be taken into consideration for surface waters of the sea. 7. The Optical Properties of Sea-water (a) The Extinction of Incoming Radiation Parallel radiation entering a layer of sea-water is gradually weakened in three ways: (1) By absorption by the pure sea- water. (2) By scattering by the pure sea-water. (3) By scattering, diffraction and reflection by suspended particles in the water (impurity of sea- water). The last two factors do not change the form of the energy but divert a part of the radiation from its original direction. A beam of radiation of wavelength A passing through a distance dx in water is reduced in intensity by an amount dl which is pro- portional to the intensity and to the distance ^.v travelled through the water, so that dl = —Kidx. K the extinction coefficient (cm~^) is dependent on the wavelength A. If the intensity of the radiation is /q when x = 0, then for a distance ,v I = I,e-^\ The reduction in intensity of the radiation is often characterized in practice by the extinction E for a layer of thickness 1 m and is given as a percentage of the incident radiation £■= 100 ("l - ^ The transmission D may also be used, and gives the percentage of the incident radia- tion passing through a layer of fixed thickness i) = 100 - - 100 e--^^ Detailed measurements have been made of the extinction coefficient for water over the whole spectral region from 0-186/x in the ultraviolet to 8-5 ju. in the infra-red. The spread of 2-3% in the values obtained in different series of measurements are largely due to the difficulty of preparing "pure water". Dietrich (1939) has given a comparison of the older measurements of Aschkinas (1895) and more recent values by Kreusler (1901), Sawyer (1931) and Collins (1925, 1933) from which the values shown in Table 1 8 have been abstracted. 52 The Sea-water and its Physical and Chemical Properties Table 18. Absorption coefficient k (cm"^) for pure sea-water Wavelength Wavelength Wavelength Wavelength X in fj. \ K Ain/i K A in /x K Ain/x K 0-20 000899 0-70 00084 1-30 1-50 200 85 0-30 000151 0-80 00240 1-40 i 160 2-10 39 0-31 00084 0-90 f 00655 1-50 i 19-4 2-30 24 0-40 000072 100 0-397 1-60 8-0 2-40 42 0-50 000016 110 0-203 1-70 7-3 2-50 85 0-60 000125 1-20 1-232 1-90 73 2-60 100 0-2-0-3 /x according to Kreusler (1901), 0-31 -0-60 /x according to Sawyer (1931) and from 0-7 /n according to Collins (1933). Figure 27 shows the spectral range from 0- 1 86 /x to 2.65 /x. From about 0-48 fi towards the red end of the spectrum and beyond, the absorption coefficient increases strongly and continuously. According to the measurements of Aschkinas, weaker absorption bands follow stronger bands between 2-86 /x and 3-27 /x, and at 6-7 ij. where there is almost complete extinction of the radiation. The absorption depends slightly on the 0-01 100-0 60-0 400 J .-^ . r \ / 200 ■ /N K / v/ 10-0 e-0 40 J \ / ; visit le spectrum / vy 20 - / 1-0 0-6 0-4 /**-J ; t \ 0-2 - / \J 0-1 006 004 / ; J 002 • r 0-01 0006 0 004 f u 0 002 ^\ t 0-001 0-0006 00004 \ f : j 00002 -\.n.r,r,\ V / 0-5 1-0 I-! \ in /i 2-0 2-5 100 1000 10000 Fig. 27. Absorption coeflRcient for pure water (pure sea water for parallel radiation (wave- length range 0-186-2-65 /j.) (From 0-2 to 0-3 ^ according to Kreusler; from 0-31 to 0-60 /u. according to Sawyer; from 0-70 /x on . . . according to Collins). temperature and an effect of the salinity has been found but from the summary given by Dietrich it can be seen that the absorption in pure sea-water is almost the same as in pure fresh water. The extinction coefficient k takes account of the effects of both scattering and ab- sorption. The scattering of light in a turbid medium is caused by reflection and diffrac- tion of the incident light by the small particles suspended in the medium. If the size The Sea-water and its Physical and Chemical Properties 53 of these particles is very small compared with the wavelength of light and if the con- centration is not too large, the scattering is due to pure diffraction following Rayleigh's law; according to this the reduction in intensity of the incident light is inversely pro- portional to the fourth power of the wavelength. Amongst the phenomena due to scattering is included that known as the Tyndall ejfect, where a beam of light passing through a turbid medium produces a more or less intensive illumination of those por- tions in the medium affected by light. This is due to reflection and scattering of the light by the suspended particles. Since the shorter wavelengths are more strongly scattered, the Tyndall-light is bluish. The water molecules themselves can be regarded as scattering particles. Thereby one thought to explain also the blue colour of the scattered light in pure water. However, it has later been recognized that a direct scatter- ing by the water molecules can hardly occur since there are too many compressed into a small space and the distances between them are too small relative to their diameter. According to the theory of Smoluchowski irregular molecular movements give rise to an optical inhomogeneity (streaks; Schlieren) of very small dimensions and are therefore responsible for the scattering of light. Table 19. The energy distribution in the spectrum of sunlight after passing through water layers of different thickness Wave- Thickness of the water layer length 0^) 0 001 01 1 1 10 1 10 100 mm mm mm cm cm m m m 0-2-0-6 237 237 237 237 237 236 229 172 14 0-6-0-9 360 360 360 359 353 305 129 9 0-9-1 -2 179 179 178 1 172 123 8 1-2-1 -5 87 86 82 i 63 17 1-5-1-8 80 78 64 27 1-8-2-1 25 23 11 — — , 2-1-2-4 25 24 19 1 — 2-4-2-7 7 6 2 — 2-7-3-0 0-4 0-2 — — — — — — — Total 10000 993-7 952-1 859-4 730-2 549-3 358-1 181-5 13-9 The only natural parallel radiation occurring in the upper surface of the sea is direct sunlight. On passing through water the spectrum of sunlight undergoes great changes. Schmidt (1908), on the basis of the extinction values of Aschkinas and values according to Langley for the distribution of radiation energy from the sun on the surface of the sea, has calculated the spectrum of the sunlight at different depths and obtained the values given in Table 19 for water layers of difiTerent thickness; the total radiation from the sun incident on the surface of the sea is taken as 1000 (Fig. 28). The total extinction for different layer-thickness is given in Table 20. The reduction in intensity of sunlight after passing through very thin layers of water is quite consider- able. For a layer 1 cm thick, wavelengths >l-5^t are completely eliminated and the spectrum extends only to 0-9 /x. For layers 100 m thick the remaining energy has fallen to less than 1-5%. 54 The Sea-water and its Physical and Chemical Properties Wove length, // Fig. 28. Energy distribution in solar radiation after passing through water layers of different thickness (according to Schmidt). A-B, at the water surface; A-C, after passing through 1 cm of water; A-D, after passing through 1 m of water; A-E, after passing through 100 m of water. Table 20. Extinction values for sunlight passing through sea-water Down to a depth of j 00 1 mm Extinction in per cent' 0-6 01 mm ] 1 mm 4-8 i 14-1 1 cm 270 10 cm 45-1 1 m 64-1 10m 81-8 100 m 98-6 The extinction coefficients in Table 1 8 are valid only for pure sea-water. The water of the sea is, however, not optically pure, and always contains more or less large amounts of suspended organic and inorganic particles. The intensity of the light passing through the water is still further reduced by scattering on these particles as well as by the ordinary extinction. It may be so strong that the actual absorption, especially in the presence of very small particles Rayleigh's law applies, but for larger particles the scattering is almost independent on the wavelength. It depends primarily on that part of the total surface influenced by the sun radiation of all the individual particles present in a unit volume. Scattering by large particles is then no longer colour selective (Pernter, 1901). The reduction in the intensity of radiation in the sea under natural conditions has, for the first time, recently been subjected to more accurate investigation, because of its special biological interest (see especially Jerlov, 1951; Joseph, 1952). These measurements have been made principally with photo-electric cells which have a sensitivity extending over a considerable range of wavelengths, while the extinction coefficients mentioned above were measured by spectrobolometric methods. The re- sults are thus only comparable after appropriate corrections. The most detailed measurements have been made on lakes (Sauberer and Ruttner, 1 941) ; measurements in the sea which are of greater interest in the present connection are rather few in number. The extinction coefficient applies to the solar radiation and the diff'use sky radiation taken together. When radiation passes through water it undergoes a pro- gressive alteration both qualitatively and quantitatively. The long wave and short wave parts of the spectrum are filtered out almost at once so that the light soon takes on a bluish-green or blue colour. With a greater degree of optical impurity the effect of the scattering is less colour selective; the remaining light is more greenish, or with strong turbidity even yellowish green (Pettersson, 1936). At the same time the light The Sea-water and its Physical and Chemical Properties 55 undergoes a progressive change in direction since the most obUque light is diminished most while the diffuse light formed by scattering increases continuously. The first light measurements on the open sea were made by Poole and Aitkins (1924). Detailed measurements have been made more recently by Clarke (1933, 1936, 1938) and by Clarke and Oster (1935); (see also Utterback 1936). For an example Figs. 29 and 30 show the percentage reduction in intensity of light in different parts of the spectrum for the surface layers of the Sargasso Sea and of the Gulf of Maine. Percentage of surface light OOI 005 0-1 05 1-0 50 10 50 100 20 40 60 80 ^ iOO 120 140 160 180 Q 1 1 1 mi: 1 1 1 ,^ iin ff ^M - y^ y/ / " 309, ^ Red } Y ^/ / - ^y V / - Jl V - Green ji ■y ^ / r 31 ^ ^ / P^ ^ '^Violet i Bju^ / ■^ 3i2ldr "^310 Fig. 29. Decrease in the intensity of light in the Sargasso Sea for different spectral ranges as a percentage of the intensity at the surface (according to Oster and Clarke). OOI Percentoqe of surface light 005 0-1 05 10 5 10 50 100 _ 1 1 1 1 III - 1 1 I 1 111 ' "!"" ^A 7 \ ' .-^'"''xfd ^ E Red j^ X M E ^ J20 BlueV''^ / E 20 Bor^iolet / E 319^' 32lK -een Fig. 30. Decrease in the intensity of light in the Gulf of Maine for different spectral ranges, as a percentage of the surface intensity (according to Oster and Clarke). As a striking feature the extinction curve is almost linear with depth so that within the spectral region investigated the extinction coefficient is almost constant and is independent of the depth. The violet and the blue are most strongly affected by the turbidity, the red is least affected. The extinction coefficient for shelf and coastal water is considerably larger than for ocean water, approximately two to three times larger or even more. Its size represents only the order of magnitude of the coefficient since these types of water show large variations both in time and locality. Swedish light measurements, which have been 56 The Sea-water and its Physical and Chemical Properties made, principally by the Oceanographic Institute in Goteborg (Pettersson), since 1933 in fiords, in the Skagerrak, in the Kattegat, and in the Baltic have given similar results, but they also show a particularly strong dependence of the reduction in in- tensity of the light near to the thermocline (discontinuity in vertical density distribu- tion). This intensification of the extinction is undoubtedly due to an enrichment of suspended particles at such layers. This enrichment shows considerable local diff'erences and causes strong variations in the extinction coefficient. If the scattering and the absorption due to the suspended particles is removed by filtering the water samples there remains a selective absorption which must be due to strongly absorbing humic material dissolved in the water. This "yellow material" must be an organic metabolic product, either from the land or from the remains of decomposed plankton. The turbidity of the water can now be determined continuously from a moving ship by the self-recording transparency meter (Joseph, 1950, 1952) and the results can be used in suitable cases to determine the origin of a water mass since the extinction value pro- vides a persistent characteristic (Dietrich, 1953; Joseph, 1953; Jerlov, 1953; see also Wyrtki, 1950). The distribution of particles in suspension can be studied with the Tyndall-meter which measures the intensity of the scattered light produced from a parallel beam of light, by comparison with the known intensity of an illuminated glass filter using a Pulfrich photo-meter. This apparatus can also be used for the measurement of the scattering from suspended and dissolved material in especially transparent ocean water, corresponding measurements of this type have been made by Jerlov (1953) in the three oceans during the "Albatross" Expedition. {h) Refraction and Reflection of Radiation Parallel radiation incident on the surface of the water will be partly reflected and in part will enter the water. The angle of reflection will be the same as the angle of inci- dence but the ratio of the intensities of the incident and the reflected beam will be dependent on the angle of incidence of the original radiation itself. Radiation entering the reflecting medium undergoes a change of direction on passing through the surface, and the angle of this refracted beam is given by the equation sin / -^ — = n, sm r where / is the angle of ncidence, r is the angle of refraction and n is known as the refractive index. For air and pure water it is almost exactly 1-333338 or -^4/3. That is, in water which is optically denser the beam is refracted towards the perpendicular (Fig. 31). The refractive index for a ray passing from the water into air is Xjn ~ 0-75. If the angle of incidence of radiation passing from the water into air increases, the angle / will increase faster than the angle r until finally the value of / reaches 90°; the outgoing ray then passes along the surface of the water. This occurs when r = 48-5° = R (see Fig. 31). If/- increases still further, radiation cannot enter the air but is reflected entirely within the water; R is known as the critical angle for total reflection. SoRET and Sarasin (1889) have measured the refractive index of mediterranean water (approx. 37%o S) for various wavelengths and compared these values with those for pure water. Table 21 shows the results. The dependence on salinity is, however, suflUciently large for use in the optical determination of salinity (refractometer) ; The Sea-water and its Physical and Chemical Properties 57 wm'////myMMM'/M//m}/m'M- Fig. 31. Reflection and refraction of radiation at the interface between air and water. Table 21. Values of the refractive index for sea-water and for pure water (After SoRET and Sarasin, 1889) Frauenhofer A in /i PureH^O / = 20°C ■ Sea-water 37%o Sea -water- line 20 C 10 C 20 ^C c 1 ^ i F 1 h 0-6563 0-5896 0-4861 0-4102 1-33120 1-33305 1-33718 1 -34234 1-33816 1-34011 1-34437 1-34973 1-33906 1-33092 1-34518 1-35064 000696 000706 000719 0-00739 besides this it is much stronger dependent on the temperature. More recent investiga- tions on the dependence of the refractive index of sea-water on the temperature and saHnity have been carried out by Bein (1935) at the Physikalisch Technische Reichs- anstalt in Berlin. Table 22 shows the deviation of the refractive index of sea- water «s from the refractive index for pure water, /z„. = 1-333338 (at 15°C, A = 587, 6 m/ii) at different temperatures and sahnities. The dependence on the wavelengths of the light used is not as large and only has to be taken into consideration for more accurate treatment. Table 22. Variation of the refractive index {n^ — «„.) x 10® with temperature and salinity (According to Bein, 1935) . /^C \. 0 10 20 30 35 40 5%„\ 20 4001 3814 3697 3621 3594 3571 25 4989 4759 4617 4524 4491 4463 30 5977 5708 5538 5429 5390 5357 35 6966 6657 6463 6337 6292 6254 40 7956 7610 7391 7250 7199 7157 58 The Sea-water and its Physical ami Chemical Properties The relationship between the intensities of the incident and the reflected radiation is expressed by Fresnel's law. If 7 is the intensity of the incident radiation and R that of the reflected radiation, the relationship between them is given by R J sin^ (/ — /■) tg^ (i r) sin2 (/ + /-) tg^ (/ + /-) Ify = 100 and n == 1-333 this gives the values shown in Table 23. If the angle of inci- dence is 0°, only 2% of the radiation is reflected and almost the whole of the energy penetrates through the surface. Table 23. Reflected radiation R and refracted radiation Dfor different angles of incidence i of radiation on a water surface {J = 100, n = 1-333) / 0° 10° 20° 30° 40° 50° 60° 70° 80° 100^ r 0° 7° 29' 14° 52' 22° 02' 28° 50' 35° 05' 40° 31' 44° 49' 47° 38' 48° 35' R 20 2-1 21 2-1 2-5 3-4 60 13-4 34-8 1000 D 980 97-9 97-9 97-9 97-9 96-6 940 86-6 65-2 00 With increasing angle of incidence the reflected energy increases only slowly up to about / = 60^ and thereafter very rapidly. The larger the angle of incidence the more is reflected, at 70° more than 13%, at 80° more than 35%. This is shown in Fig. 32. The rays coming from the upper left incident on the surface are split into reflected and 10° f Fig. 32. Graphical representation of the proportions of reflected and transmitted radiation incident on the surface of water at different angles. For each ray incident from the upper left with an intensity of 100 there will be a reflected ray and a ray transmitted into the water. Both are represented by vectors which give the intensity and the direction of the ray. entrant rays; the incident rays have an intensity of 100 and the vectors marked on the diagram correspond in intensity and direction to the reflected and entrant rays. Larger values for the reflected energy only occur with obliquely incident light and especially in that range, where the entrant radiation falls to very low values. It can be seen that The Sea-water and its Physical and Chemical Properties 59 the direct incident radiation coming from a whole quadrant is concentrated into a fairly narrow beam range from 0° to 48-5°, while at angles of incidence more than 65^ the intensity of the entrant radiation is rather small. Schmidt (1908) showed by actinometric measurements at the surface of pure water that the same conditions apply for the total solar radiation as for the D line of sodium (n = 1-333). More recent measurements by Poole and Atkins (1926) and by Whitney (1938), as well as by Angstrom (1925) using the pyranometer, show that the theorectical values for re- flection are also obtained essentially in practice. However, the reflection is more or less strongly increased by waves on the surface of the water; it may be increased in this way by more than 50% (Lauscher, 1944). (c) The Behaviour of the Water Surface for Diffuse Incoming and Outgoing Radiation As well as the direct sunlight, which may be regarded as unilateral parallel radiation, there is also a general diffuse radiation for which conditions relative to the sea surface are rather different. The diffuse radiation on the surface of sea includes: (1) diffuse sky light (daylight) which is essentially short-wave radiation (between 0-38 ju and 0-75 /^i) and is only present in the day time; and (2) the long-wave radiation from the atmosphere which is long-wave (maxima at 7-5 /z and 12-5 /x), and is present both day and night. Each single beam of the diffuse radiation that is incident on the surface of the water at an angle / is partly reflected following Fresnel's law and is thus subject to a corre- sponding reflection loss as shown by the values given in Table 23. Since the diffuse radiation comes from all directions and the radiation with a greater angle of incidence is more strongly reflected, it is necessary to find the sum of the losses for each angle of incidence in order to determine the total loss by reflection. The calculation of this total from the values r(i) given in Table 23 gives the reflection losses (forn = 1-333) as 0-660, that is 6-6% of the diffuse radiation is reflected from the surface of the water. Considering the refractive index to be slightly different for different parts of the spectrum this value varies between 5% and 10%. Mention should also be made here of the properties of water as a source of radiation (Schmidt, 1915). Since the extinction coeflftcient of water for long-wave radiation is particularly large and the thermal radiation from the surface of the sea contains only longer wavelengths (around lO^u) it can be expected from Kirchhoff's law that as a source of radiation water would behave as a black body. Nevertheless, water radiates less than a surface of the same temperature since each beam coming from the interior of the water mass will suffer a reflection loss at the surface which will reduce the intensity of the total from the surface outgoing radiation (Fig. 33). In addition to this reflection loss the intensity of the radiation suffers a further de- crease since in passing through the surface to the air it must spread out into a larger space. The radiation from water within a space angle of 2 x 48° 35' = 97° 2' is spread out over a full 180°. If this is taken into account (Schmidt, 1916) it is found that for a temperature range of 0-20°C the outgoing radiation from a water surface is about 9-10% less than that from a black surface. Since the radiation from a black body according to the Stefan-Boltzmann law is given by £" = aT'^ where a = 1-374 x 10~^- cal cm"2 sec"^ grad"^ the radiation from a flat water surface will be given by ^4 = 0-904CTr''. Angstrom has found experimentally that for long-wave radiation the effici- ency of emission of sea-water is 96% of that of a black body. The constant in the above equation should therefore be not very different from 0-95 for the temperature range 60 The Sea-water and its Physical and Chemical Properties Fig. 33. Back radiation from the interior of the sea towards the water surface. concerned. Lauscher (1944) has obtained the same result in another way and found the value 0-9535 for the constant. Falkenberg (1928) has made similar calculations and has found the somewhat lower value 0-937 for this constant. {d) The Colour of Sea-water The colour of sea-water in the scientific sense is taken to include all those colour phenomena which arise because of the optical properties of sea-water and the sub- stances dissolved and suspended in it. The colour of the sea can vary widely and may assume any shade from a yellowish green to the deepest blue. To observe the colour of the sea undisturbed by external reflections it is best to look through a tube which is blackened inside, dipped in the water. The colour can be determined by comparison with standard colours or by spectrophotometry. Kalle (1938) has designed a special colour measurement tube in which the colour of the sea can be determined with a comparator. In practice, the colour is for preference determined with standard colours, using the Forel-Ule scale. Accurate colour determinations in the open sea are by no means frequent and have been made almost only by oceanographic expeditions. The largest part of the surface of the ocean is blue {Forel 1 and 2), particularly, the regions within the tropics and subtropics, while the green colour is prevalent in coastal areas and shallow seas, especially in adjacent seas and polar regions. In the Atlantic Ocean (Schott, 1942) there is a certain symmetry in the distribution of colour. From 15° to 35° N. and from 10° to 30° S. it is a deep blue. The purest and richest colour is in the central parts of these areas, roughly from the Bermudas to near Madeira and off the Brazilian coast till St. Helena. In the Benguela current, generally in areas of upwelling, for example off the West African coast in the north and off the south-west African coast in the south the sea-water has a more greenish colour. In the Southern Hemisphere a tongue of greenish blue water runs from this coast of South Africa far up to the north between 0° and 10° S. (up to St. Paul Island). The higher latitudes in both hemispheres are always discoloured. Greenish blue predominates north of 40° N. and gradually changes to green. The waters of the English Channel, the North Sea and the Baltic are of the same colour. In the Southern The Sea-water and its Physical and Chemical Properties 61 Hemisphere the colder water of the Falkland current and the oceans areas around Bouvet Island are mostly greenish blue to green. An explanation of the colour of pure sea-water must be sought, in the first place, in the optical properties of sea-water. The Bunsen theory ascribed the blue colour of the sea to the combined effects of the spectral absorption of pure sea-water and re- flection by the particles suspended in the water (absorption theory). The light entering the water (direct sunlight and diffuse radiation from the sky) will be weakened least in the blue by absorption. Down into the deeper layers the light becomes more and more blue. This relative concentration of blue is further increased in the light reflected from small particles and passing back to the surface, the light returning through the surface is thus blue. Against this absorption theory, Soret has set a diffraction theory according to which the explanation of the blue colour of the sea is analogous to that of the blue colour of the sky and is due to the scattering of light in the water. Ramana- THAN (1923) has attempted to prove by experiment and theoretical investigation that pure sea-water should show an indigo blue colour by molecular dispersion and by selective absorption, and that small amounts of suspended matter have little effect on the colour. According to the theoretical investigations of Gans (1924), the colour is due principally to diffraction of higher orders (see also Lauscher, 1947). A third possible explanation for the widely occurring greenish colour was advanced by WiTTSTEiN (1860) and later by Spring (1886, 1898) in the so-called "solution theory". In this, blue was regarded as the actual colour of the water and all variations were due to different substances dissolved in the water. This effect was ascribed prin- cipally to organic humus materials that in increasing concentration made the water first green, then yellowish green and finally, in extreme cases, brown. It was first pointed out by Kalle (1938, 1939) that the physiology of colour vision must play a large part in the explanation of the colour assumed by the sea and must be taken into consideration. According to the Young-Helmholtz theory of colour vision, the human eye has three groups of colour-sensitive elements (cones), each of which is sensitive to one of the three primary colours, red, blue and green. The stimulation of two or all three of these groups at the same time gives the impression of a mixed colour. Every different colour impression is produced by a definite ratio in the strength of the stimulation of the three different types of cones. A "colour triangle" (Fig. 34) can be used to represent diagrammatically all possible colour impressions. The three corners of the triangle represent the total (100%) stimulation of only one group of receptors — red, green or blue. At every point on the triangle the sum of the oblique co-ordinates of the point is always 100%, and these co-ordinates represent the per- centage composition of the mixed colour characterized by that point. The point W = white which, by definition, is composed of a mixture of 33J% of each of the three primary colours hes at the centre of gravity of the triangle. All tones of the same colour lie along a straight fine that runs radially from the white point; the nearer a point on such a line lies to one of the sides of the triangle the more saturated is the colour it represents. The position of the spectral colours within the triangle is shown by the curve marked on the diagram. Since the spectral colours are the most saturated colours possible in nature, all colours found in nature must lie on the area within the spectral curve and the line joining its two end-points. In the light of the consequences of this theory, Kalle has investigated the effects 62 The Sea-water and its Physical and Chemical Properties of selective absorption and selective scattering and also of the interaction of these two processes on the colour of the sea. These results are summarized in Fig. 35 which shows a part of the colour triangle and the spectral curve. The absorption colour of sea-water lies on a curve running from the white point and approaching concave Fig. 34. Colour triangle of the Young-Helmholtz colour theory and spectral curve. Fig. 35. Part of the colour triangle showing colour points for sea-water colour. The Sea-water and its Physical and Chemical Properties 63 downwards the spectral curve asymptotically. With layers of increasing thickness the increasing saturation of the colour gives a slow displacement towards the blue, while at the same time the brightness of the colour decreases rapidly so that only a relatively thin surface layer is concerned in the colour of the sea. According to Kalle, the result is a colour with a wavelength approaching 492 m/x, a somewhat greenish blue, corresponding to a light path of 38 m. This shows immediately that the deep blue colour of the Sargasso Sea cannot be explained in this way. If selective scattering of the different colours is taken into account the colour curve lies further towards shorter wavelengths. As far as the colour is concerned the most important point on this curve approaches that corresponding to a 50 m thick layer where the colour value is 485 m/Li. This value agrees fairly well with the colour of the Sargasso Sea, especially if the higher order scattering which would give a further slight displacement towards shorter wavelengths is taken into account. The absorption and the scattering of light are thus responsible for the blue colour of the tropical and subtropical areas of the ocean and they are reinforced by the greater brightness of the sunlight and of the diffuse light from the sky and by the almost completely pure sea-water of these areas. For water masses that are not so pure and contain large numbers of suspended particles (mostly plankton), as is usually the case in higher latitudes, the depth from which the selective scattering is reflected is less, and the colour gradually reverts to a value of 495 m/x. This would be more or less the longest wavelength for the colour of the sea if only absorption and scattering were involved during its formation. Other causes are, however, required to explain the greenish colours of longer wavelength than 495 m/i that are also of frequent occurrence in the open ocean. Investigation has shown that these are due to coloration caused by yellowish substances dissolved in the water. These substances appear to be related to humus and are apparently to be re- garded as products of phytoplankton metabolism. They displace the colour of the water towards the green especially in water masses such as in the English Channel and in the North Sea where values of 498 m/x to 505 m^u may occur. In coastal regions further humus material carried by fresh water flowing into the sea from rivers is added to the more oceanic yellow material and causes a further displacement towards yellow-brown colours. In addition to these yellow substances there may also be fluorescence phenomena in the seas as Ramanathan, and later Kalle, believed; these would give a further displacement towards the green but the extent to which such fac- tors are present is not yet certain. A qualitative survey of the contribution of each single factor to the colour of the sea has been given by Kalle in Fig. 36. In the clearest water and with a depth of visi- bility of 50-60 m, selective scattering plays to a very large extent the principal part. If cloudiness due to the presence of plankton occurs, the depth of visibility gradually decreases and the natural absorptive colour of water which tends towards a greenish shade begins to predominate. At the same time small amounts of yellowish substances may be formed as the colour tends more and more towards green. With the increasing turbidity the yellow material becomes more and more important until finally, at very small depths of visibility, the discoloration is due to the natural colour of the material causing the turbidity. Very close to the coast the natural colour of the bottom begins to show through the shallow water, and the colour of the water is clearly altered to- wards this. 64 The Sea-water and its Physical and Chemical Properties Turbidity discoloured 70ny/F60. ;-5l5m/^F25. |i-488ny/F5. 477ny/ FO. Magnitude of porticipotion of the individuol factors giving rise to ttie colour of the sea Fig. 36. Quantitative representation of the contribution of the individual factors giving rise to the colour of the sea (according to Kalle). 8. The Chemistry of the Sea In general, liquids have the property of absorbing gases with which they are in contact to give a solution of the gas in the liquid. The solubility of the gas in the liquid is not unlimited, but usually fairly soon reaches a limit; the liquid is then saturated. According to Henry's law the amount of gas dissolved in a saturated solution is pro- portional to the pressure of the gas in contact with the liquid. If the liquid is in contact with a mixture of gases then each separate gas is absorbed according to its partial pressure. When the liquid is completely at rest the process of solution depends on the process of diffusion, and thus requires time for the pressure of the gas in the liquid to come to the same pressure as the gas outside it. In nature, the wave motions, turbulent currents and convections can accelerate considerably the uptake of gas by the liquid. By the gas content of a sample of water is understood the amount of gas in the water expressed in volume units (ml/litre) at NTP (0°C and standard pressure of 760 mm Hg). The actual gas content may of course differ more or less from the amount present when saturated. Besides, by this absolute definition the gas content may also be characterized by the ratio of actual content to that by saturation. It is then specified by the relative gas content which is expressed in per cent of the amount required for saturation. The absorption coefficient is taken as that volume of gas which can be absorbed by unit volume of liquid at a given temperature and standard pressure. The Sea-water ami its Physical and Chemical Properties 65 If the solution process is limited to purely physical absorption the absorbed gas does not enter into chemical combination with the water; the situation is then fairly simple. It is, however, possible for the gas to combine chemically with the liquid. Both possibilities occur in the atmosphere-ocean system. Oxygen, nitrogen and the rare gases obey the pure physical absorption ; carbon dioxide, on the other hand, fol- lows the second possibility since it reacts both with the water itself and in part also with the salts dissolved in it. (For chemistry of sea waters see especially Harvey, 1955.) (a) Oxygen, Nitrogen and Hydrogen Sulphide Contents of Sea-water The composition of the air absorbed by pure water can be calculated from the absorption coefficients of the gases present in the atmosphere and is shown in Table 24 for 0° and 30°C. It is different from that of atmospheric air since the absorption coeffi- cient of the individual gases is very different. In atmospheric air the ratio of oxygen to nitrogen is 21 : 78 or about 1 : 4, but in the air dissolved in water at 0°C it is 35 : 62, and at 30°C 33 : 64 or about 1 : 2. The air dissolved in water is thus twice as rich in oxygen as atmospheric air, but it should not be forgotten that while a litre of air con- tains 210 ml of oxygen, a litre of water saturated with air contains only about 10 ml. Table 24. Distribution of atmospheric gases at saturation dissolved in sea-water Oxygen Nitrogen . i Carbon -r . i ^••g^" i dioxide 1 T^^^' ml/l.|3oec 10-31 5-60 1811 10-74 0-54 0-30 0-51 29-47 0-20 j 16-84 In «/ / ^°^ 35-0 33-2 61-5 1-8 1 1-7 100-0 63-8 i 1-8 1-2 1000 The solubility of gases in water is very strongly dependent on the temperature and falls off rapidly as the temperature rises. The sea-water as a dilute salt solution shows also a dependence on the salinity and the absorption coefficients fall with increasing salinity. Fox (1905, 1907, 1909) has carried out extensive research on this subject, and Rakestraw and Emmel (1937, 1938) have made further investigations. Table 25 shows saturation volumes at different temperatures and salinities for oxygen and nitrogen. The weights present in mg can be obtained by multiplying the figures for oxygen by 1-4292 and those for nitrogen by 1-2542. If oxygen-nitrogen ratios Oa/Ng for different temperatures and salinities are worked out, it can be seen that there is little variation; the dependence on salinity is small; with temperature it falls off slightly. For chemical methods of determining the oxygen and nitrogen contents of a sample of sea-water see Report of "'Meteor' Expedition, 3 or "Oceanographic Instrumenta- tion. Chemical Measurements" (Carrit, Nat. Acad. Sci. Nat. Res. Coun., no. 309, pp. 166-85, 1952). At the surface of the sea, in contact with the atmosphere, there is ample oxygen and nitrogen available and it would be expected that the upper layers of the ocean were saturated with both gases. This is generally the case, especially for nitrogen which is 66 The Sea-)vater ami its Physical and Chemical Properties Table 25. Saturation values for oxygen and for nitrogen in sea-water in mill Hit res per litre for a dry standard atmosphere Temp. Oxygen salinity (%o) Nitrogen salinity (%„) (. «-J 20 25 30 35 40 20 25 30 35 40 -2 9-50 916 8-82 8-47 812 0 901 8-68 8-36 803 7-71 1602 15-46 14-90 14-34 13-78 5 7-94 7-67 7-40 7-13 6-86 1408 13-62 1317 12-72 12-27 10 710 6-86 6-63 6-40 6-17 12-74 12-32 11-92 11-51 1110 15 6-43 6-23 604 5-84 5-64 11-57 11-20 10-84 10-48 1011 20 5-88 5-70 5-53 5-35 5-18 10-53 10-21 9-91 9-61 9-30 25 5-40 5-21 503 4-93 4-77 9-69 9-42 9-16 8-88 8-62 30 4-96 4-80 4-65 4-50 4-35 — — — — less reactive than oxygen and is biologically inert. Water samples from different depths show mostly only minor deviations in nitrogen content from the saturation values. This could be used to draw conclusions about the origin of deep layers and about the vertical and horizontal displacements that they have undergone since their last contact with the atmosphere, but since very few nitrogen determinations have been made in the open ocean the method has so far been of little use. In any case, care would be required in the interpretation of such results since super-saturation or in- complete saturation may be due to other causes: to subsequent heating and cooling, to the mixture of saturated water masses at diflTerent temperatures which always leads to small super-saturation, to variations in atmospheric pressure and to the production of nitrogen by bacteria that decomposes nitrite or nitrate. Since the equal- ization of existing differences in saturation always proceeds slowly these deviations will be conserved for a long time and can simulate water movements that would otherwise be quite impossible. The oxygen is also for the most part in equilibrium between the air and water at the surface of the sea, but the deviations are more frequent and more marked than for nitrogen. Besides the causes of more physical factors mentioned above (temperature and pressure alterations, mixing, etc.), there are also biological factors which cause variations. The respirations of plants and animals produces carbon dioxide and uses up oxygen. Animals, however, obtain their essential carbon compounds from in- gested organic material, while plants, on the other hand, obtain it by the assimilation of carbon dioxide. This is converted with the help of sunlight into organic substances and oxygen, which is set free, raising the oxygen content of the water. The oxygen in sea-water is consumed not only by the respiration of living organisms but also by the bacterial oxidation of dead organic matter and of organic compounds in solution. This oxygen consumption is proportional to the rate of oxidation, which is in the first place dependent on the temperature and also on the amount and nature of the organic material present. In the assimilation layer (the upper layer of the sea), usually down to the thermo- cline (rapid density change in the vertical) conditions are rather complicated due to the mutual interaction of the different factors. Oxygen is being steadily absorbed from The Sea-water and its Physical and Chemical Properties 67 the atmosphere and produced by photosynthesis. Usually this addition is not exceeded by removal of the respiration of the organisms present and by the oxidation of dead material. Super-saturation by oxygen is thus quite possible and is occasionally found. The surface layer is generally, however, the layer which is nearest to equilibrium with the air. In the deeper layers of the ocean, below the assimilation layer, the oxygen is provided almost exclusively by transport of the water from the surface by vertical and horizontal movements. On the trajectories which the water particles perform there is a continuous progressive consumption of oxygen so that the oxygen supply in deeper layers depends either on the distance covered since the water mass left the surface or on the speed with which it moved. A stationary state is only possible when the supply of oxygen by renewal of the water mass and the oxygen consumption are in equilibrium. Estimation of the oxygen distribution in the deeper layers of the ocean, especially of the vertical and horizontal differences in saturation, until very recently gave only the "age" of the water mass, i.e. the time since it left the surface layers. After that some clarification had been obtained of conditions for similar processes in lakes, the chemical-biological processes of oxygen depletion in the sea were further elucidated by Seiwell (1937, 1938), Sverdrup (1938) and Wattenberg (1938). The last one has discussed in detail the relevant chemical-biological factors in the ""Meteor'"' Report and has pointed out its great importance for a proper understanding of the distribution of oxygen in the ocean. This distribution within the ocean shows that the explanation given can account qualitatively for the oxygen producing and consuming factors mentioned above. The maximum oxygen content is always found in the surface layers; in this skin layer mix- ing by the wind and the waves and the turbulence due to ocean currents gives a more or less even distribution that normally differs little from equilibrium with the atmos- phere. The lower limit of this oxygen-rich layer, which coincides with the assimilation layer, follows essentially the thermocline in the general oceanic structure. At this transition layer, when it is strongly developed as is always the case in lower latitudes, the oxygen content falls to a minimum. According to the geographical position of the part of the ocean and the range of the annual convection at that point the depth of this minimum varies between 100 and 1500 m. This oxygen-poor intermediate layer is the most prominent feature of the oxygen distribution of the ocean in middle and low latitudes. Below this minimum layer there is always oxygen-rich water with up to 70-90% saturation. As is explained later, this oxygen content of the deep-sea circula- tion of the oceans originates from the major convection areas of the subpolar and polar regions of the ocean where the water masses in the surface layers can sink to great depths, and from there also fill the depths at middle and lower latitudes. In spite of the long path travelled by these water masses there is little depletion because of the low temperature and the small amount of organic material present, and the oxygen content shows only a slight decrease. Figure 37 shows as an example the vertical distri- bution of oxygen at about 10° S. in the South Atlantic; the vertical variation of density is also shown and the density transition can be clearly seen. The right-hand side of the figure shows the vertical changes in percentage oxygen saturation and in density a, at a station in the North Atlantic near Greenland in the area where, according to a view expressed by Nansen (1912), the North Atlantic deep water is formed and sinks during the late autumn and winter. The almost constant value of the density down to 68 The Sea-water and its Physical and Chemical Properties below 2000 m and the high oxygen content proves the possible presence of con- vection descending to great depths and the considerable ventilation it would give. The renewal of the deeper water layers has a major effect on the oxygen distribution in them. If renewal did not occur oxygen depletion processes would in time reduce the oxygen content until it would be finally zero. It is to be expected that enclosed, stag- nating water masses will always have a very low oxygen content when their thermo- haline structure prevents the thermal circulation from the surface reaching the bottom. If the surface layer density is so low that it does not become heavy enough when the temperature decreases in the autumn and winter in order to change places with the 6t 35 26 02% 20 I W 500 1000 1500 2000 2500 3000 Fig. 37. Left : Vertical distribution of oxygen and density at about 10° S. in the South Atlantic (according to the values of the "Meteor" Expedition). Right: the same for "Meteor" station 122 (Greenland, ^ = 55° 3' N., A = 44° 46' W). more saline deeper layers thus carrying oxygen to the layers beneath, the oxygen content of the deep stagnating layers may fall to zero, especially when a lateral addition of fresh water, due to the orographic conditions, is hindered or completely missing. In this case hydrogen sulphide will be formed either by the decomposition of proteins or by the reduction of sulphate by the carbon compounds of organic material under the action of certain bacteria. The classic example of these conditions is the Black Sea, where the water from about 200 m down to the greatest depths contains considerable amounts of free hydrogen sulphide and thus forms a "Kingdom of the Dead" from which all life has disappeared and where the organic world is represented only by the lowest forms of plant life (Schokalski, 1924; Nikitin, 1927; Neumann, 1942, 1944). The thermo-haline structure of the Black Sea is indicated in Fig. 38 which shows the vertical distribution of temperature, salinity, density, oxygen content and The Sea-water and its Physical and Chemical Properties 69 hydrogen sulphide content at three stations in July in different places in the eastern part of the Black Sea (Neumann, 1943). The station PM 298 lies in the southern part, the station PM 308 lies in the northern part of the central eastern basin near an area with little current, and the station PM 303 lies south-west of Sochum in the area of the strong current along the Caucasian coast. Hi^}° t'7 9 II 13 15 17 19 5%«I7 IB 19 20 21 22 23 a-fW 12 13 14 15 16 17 7 cm' 2rc Ks'Y -^^ H,st 5 6 7cmyi S%ol7 t7>l| II 13 15 17 19 21 "C 19 20 21 22 23 %<. 13 14 6 16 17 ri 9 II 13 15 17 19 21°C 5%ol7 18 19 20 21 2 2 23 %o o-,ll 12 13 14 15 16 17 18 100 200 400 600 800 1000 1200 1400- 1500 P.M. 298 42''00'IM 38°00'E I6/I7-3ZII-1925 Is It !\ li j i ! i P.M. 303 42°23'N 40°33'E 20-2Ii925 i ! it il ij il il i I :i — li- w ^•^ 5%..' H,S\. P.M. 308 43''04-9N 38° 29-8'E 22-3ZIIi925 Fig. 38. Vertical structure of the water masses in the eastern part of the Black Sea. (Sept, 1925 stations: P.M. 298, 303, 308; temperature, salinity, density, oxygen content and sulphur content.) The vertical structure of the Black Sea is characterized by two layers. The upper layer shows a very rapid increase of density with depth and usually extends down to about 200 m. After a sharp bend in the a^-curve the density changes little with depth. The boundary between these two layers coincides approximately with the upper limit of the hydrogen sulphide; its depth varies from place to place depending on the dy- namics of the currents prevailing. The upper layer (the troposphere) is divided in summer at a depth of about 50-70 m by a definite temperature minimum at 6-5°- 7-5 °C. This surface zone has a constant salinity and shows a pronounced vertical thermal convection; it is well ventilated and has a rich oxygen supply from the at- mosphere and also from plant assimilatory activity. In the lower part of the surface layer the oxygen falls off rapidly with depth and finally disappears, and in places is replaced by hydrogen sulphide. The oxygen of these upper layers comes partly from above and partly from horizontal advection but the latter effect is limited to the immediate vicinity of the Bosphorus. The whole of the layer from below the oxygen zone down to the bottom at about 2000 m has an almost constant temperature, about 8-8-9-0°C; the slight increase from 300 m is largely an adiabatic effect. The principal characteristic of this lower water is the hydrogen sulphide content which increases down to the bottom (see Table 26). Similar conditions, though on a smaller scale, are shown by several Norwegian fiords where in most cases there is a considerable depth, a fresh-water influx at the 70 The Sea-water and its Physical and Chemical Properties Table 26. Average vertical distribution of t, S and Hydrogen Stdphide in the Black Sea Depth (m) 0 100 200 300 500 1000 1500 2000 Temp. {°C) Potential temp. (°C) Salinity (%«) 13-80 7-95 7-94 20-36 8-69 8-67 21-35 8-80 8-76 21-73 8-83 8-77 22-09 8-93 — 8-81 — 22-24 22-31 9-00 8-75 22-34 HjS content ml/1, (standard pressure and C) 0-0 00 0-45 1-42 3-45 5-55 6-09 6-24 inner end and access to the open ocean only over a bar or a very shallow sill. They have recently been reviewed in detail by Munster (1936). Of 30 fiords on the western and southern coasts, 16 showed hydrogen sulphide in the bottom layers; the other 14 had very low oxygen values varying between 0-22 and 5-47 ml/1. The ventila- tion of the deeper layers depends in the first place on the sill depth and the width of the passage to the free ocean. In some fiords changes in the hydrogen sulphide content were found which must be due to the addition of ocean water. The formation of hydrogen sulphide is only possible in closed or very poorly ven- tilated deep basins. The Baltic which has a much lesser depth than the Black Sea shows very similar hydrographic conditions, although the deep water in the Baltic is renewed occasionally by the spasmodic entry of masses of North Sea water (Watten- BERG, 1941) so that it is only in stagnant periods that the oxygen content is depleted by the respiration of animals and by the oxidation of organic material in the water and on the bottom. Table 27 shows typical conditions at a summer station in the middle of the Baltic. Table 27. Gotland Basin (57^24' N., 19°52' E.); ''Skagerrack'' Station Alf. 96, 31 July, 1922 Depth (m) Temp. (°C) 5%o Density Oxygen Free CO2 Ot ' ml/I. j %o satur. ml/1. 0 15-52 6-64 4-19 6-78 101 0-23 20 10-60 7-27 4-37 7-64 103 0-28 30 415 7-36 5-91 — — — 40 2-95 7-63 614 8-74 99 0 64 60 2-20 7-92 6-35 7-86 87 0-83 80 405 10-42 8-34 3-72 44 2-9 100 4-35 10-90 8-70 , 3-62 43 2-7 150 4-55 12-59 10-04 3-72 45 2-4 209 4-60 12-81 10-21 2-45 30 3-3 The temperature minimum caused by winter cooling is at 60 m depth. The oxygen content falls from the well-ventilated upper layer with 7-8 ml/1, to less than 2 ml/1, a little above the bottom where it reaches only 30% of saturation. The carbon dioxide content here is 3-3 ml/1., which is eight times the normal concentration (Schulz, 1924), The Sea-water and its Physical and Chemical Properties 71 (b) The Carbon Dioxide Cycle in the Ocean and its Relationship to the Atmosphere Unlike nitrogen and oxygen, the carbon dioxide in the sea is present not only in solution but also in considerably larger amounts chemically combined as salts. Conditions are thus much more complicated, and the situation has only been clarified in recent times by Buch and McClendon using modern dissociation theory. Funda- mentally one realized by this that the free and the combined carbon dioxide in solu- tion are not independent of each other, but according to the law of mass action are in chemical equilibrium with each other. The combinations occurring can be repre- sented by the following equations : CO2 (in the air) ^ CO2 (in solution )+ HgO ^ H2CO3 (carbonic acid). The carbonic acid splits partially into its ions according to: H2CO3 ^ H^ which can dissociate further by: HCO3 (bicarbonation), HCO3 ^ H+ + COg^ (carbonation). All these forms derived from carbon dioxide are present in sea-water principally as carbonate and bicarbonate ions, and only to a lesser extent in the free state. Equili- brium exists between these forms, the carbonate and the bicarbonate ions, free carbon dioxide and the hydrogen ion, and it will be discussed later. The reasons are now understood for the long time needed in oceanographic research to obtain suitable accuracy in the determination of the carbon dioxide in solution in sea-water. Free carbon dioxide and carbon dioxide pressure. The solubility of carbon dioxide in sea-water is relatively large, almost thirty times that of nitrogen. Fox investigated its dependence on the temperature and on the chlorine content of NaCl solutions, and corresponding measurements have been made by Krogh for sea-water. On this basis of these investigations Buch and collaborators (1932) Wattenberg, (1936) prepared tables showing the dependence of the solubility of carbon dioxide in sodium chloride (NaCl) solutions on the temperature and the salinity. Table 28 shows a condensed extract from these tables. The solubility of carbon dioxide decreases considerably with increasing temperature and salinity. One litre of sea-water at 0° and 35T9%o S, when in equilibrium with the atmosphere (partial pressure of carbon dioxide 0-0003 Table 28. Solubility of carbon dioxide in sodium chloride (NaCl) solution in millilitres per litre at a carbon dioxide pressure of 1 atm. 10 15 20 25 30 0 1890 1713 1424 1194 1019 878 759 665 20 1708 1554 1291 1088 932 808 702 617 30 1622 1479 1228 1038 890 774 674 594 32 1605 1464 1215 1028 882 768 669 590 34 1588 1449 1203 1018 874 761 663 585 36 1572 1 1435 1191 1009 866 755 658 581 40 1540 1406 1167 990 850 742 648 572 72 The Sea-water and its Physical and Chemical Properties atmosphere) contains 0-42 ml/1, of free carbon dioxide, which is very little. In water in the uppermost layer of the open ocean the carbon dioxide content is usually not far from the equilibrium value with the atmosphere. According to Krogh's measurements in the North Atlantic the value for the carbon dioxide pressure varies between 1-55 x 10^*and2-9 x 10~^. According to Brennecke's values in the Weddell Sea ("Deutschland" Expedition) the carbon dioxide pressure was higher than that in the atmosphere. Carbon dioxide in solution comes only slowly to equilibrium with the atmosphere. Detailed investigations along this line have been made by Buch (1917), in the waters around Finland, by Schulz (1923) in the Baltic, by Wattenberg (1933) on the "Meteor" Expedition (1925-7 principally between Africa and South America), by Deacon (1934) especially in the Arctic and Antarctic regions and finally by Buch (1939, 1939^) in the North Atlantic and on a cruise in the Arctic. All these measurements of the carbon dioxide pressure show variations around the equilibrium position, sometimes the pressure in the water is higher than in the atmosphere and at other times it is lower. These variations,, however, are small as in the course of the long time which has been available, sea and atmosphere have come into a mutual adjustment. Wattenberg (1936), from the ob- servations available, arrived at the following conclusions (Fig. 39): (1) There are limited areas of the sea where the carbon dioxide pressure of the water is definitely greater than that of the air; these are principally places where rising water currents bring water rich in carbon dioxide to the surface from intermediate layers rich in carbon dioxide (west coasts of North and South Africa and of North and South America). (2) In other places there are, however, large areas where the carbon dioxide pressure is somewhat less than the normal partial pressure of the atmosphere. These occur especially in temperate and cold zones during the spring and summer, when rich plant plankton is actively assimilating. There may be pronounced annual changes here in the carbon dioxide pressure at the surface of the sea: a strong reduction in spring at the beginning of diatom development and a gradual rise in autumn when dead organisms start to decompose. See p. 77 for the distribution of carbon dioxide in deep water and at the sea bottom. Total carbonic acid. If the sea was neutral it would contain little carbon dioxide. Sea-water is in fact alkaline and has a total carbon dioxide content that is much greater than would be concluded from the carbon dioxide pressure. By far the largest part is chemically combined in the sea salt. The total amount of carbon dioxide present depends on the one hand on the car- bon dioxide pressure and on the other on the amount of base available for combina- tion with the carbon dioxide which is termed the alkalinity. Since the carbon dioxide pressure is small, there is an almost linear relationship between the total amount of carbon dioxide present and the alkalinity, and thus also the salinity since the alkalinity is dependent very largely on this. Thus Buch (1914) for the Pojowick under average conditions found the relationship A = 0-07 + 1-00 CO, and COg = 0-32 - 0-1735' where CO, is expressed in millimoles/litre and A in milliequivalents. Similar relation- ships were also derived for the Gulf of Finland and the Gulf of Bothnia. The Seo-water and its Physical and Chemical Properties T 73 Fig. 39. Distribution of carbon dioxide pressure (given in 10-* atm.) at the surface of the South Atlantic (according to Wattenberg). In the open ocean the average value for total free carbon dioxide is usually between 45 and 55 ml/1. Ruppin has found for the middle North Sea 45-9 for the Beltsea 36-7 and for the southern part of the Baltic 31-9, while Brennecke (1909) found values be- tween 46 and 55 in the Atlantic and in the Indian Ocean and between 45 and 59 in the Antarctic Ocean. In the North Sea Knudsen (1899, "Ingolf" Expedition) found lower values, between 34-1 and 46-6 ml/1. Alkalinity. In sea-water the sum of the cations of bases (Na+, K+, Mg2+, Ca2+), is always a little greater than the sum of the anions of strong acids (SO^-, CI", Br-). This excess of base is known as the "alkaline reserve"; it gives sea-water an alkaline 74 The Sea-water and its Physical and Chemical Properties reaction and is very largely present in the form of carbonates and bicarbonates. Since Tornoe, it is also known as the "alkalinity", a term which is also used for the hydrogen- ion concentration. To avoid confusion the sum of the carbonate and bicarbonate ions is termed in oceanography (following Buch) the "titration alkalinity". This is ex- pressed in the equation A = 2[C02-] + [HCO3], and can be found by simple titration with hydrochloric acid (Wattenberg, 1933; see also 1930). The alkaline reserve in sea-water is largely combined with carbonic acid, but a smaller part is also combined with other acids the most important of which is boric acid. Sea-water of 35%o S contains 4-7 mg/1. of boric acid (Buch, 1933). The last anomalies in the carbon dioxide system of sea-water have only been eliminated by taking this acid into consideration since it and its ions are definitely concerned in the equilibrium despite their small concentration. Since the individual constituents of the salt in sea-water are in almost constant ratio to one another, it would be expected that the amount of base available for the formation of carbonate and bicarbonate, that is the titration alkalinity, would be directly dependent on the salinity. This is the case. The dependence between the two was first shown by Hamberg (1885) and the investigation by Brennecke of the surface samples collected on the "Deutschland" Expedition gave the relationship between them as A = 0-06119S (according to Schulz, 1921). Later investigations have shown that for the open ocean the dependence of alkalinity on the salinity is given with suflftcient accuracy by the relationship A = 0-068S%o = 0-123 CI (in milliequivalents). This simple proportionality does not apply to the sea-water of the marginal and ad- jacent seas as has been shown by Ruppin and Buch; these variations appear to be due to the inflow of fresh water from the land. The North Sea and the Baltic, especially in coastal areas, show alkalinity values that are higher than would correspond to the salinity (addition of carbonate in river water). Similar conditions are found in the Gulf of Bothnia, the Gulf of Finland and in the Adriatic. Carbonate at the sea bottom passing into solution has the same effect as the addi- tion of carbonate from the land. The investigations of the "Challenger" Expedition clearly indicated that the water immediately above the sea bottom was more alkaline than that at the surface or in the middle layers (Dittmar, 1884; Brennecke, 1921). The more accurate alkalinity determinations of the "Meteor" Expedition 1925-7 showed definitely that the specific alkalinity (the ratio of alkalinity to chlorinity, A : CI) almost always increased near the sea bottom. This increase can only be ex- plained by calcium carbonate from the bottom sediments going into solution (see p. 85). Hydrogen-ion concentration. Pure water dissociates according to the equation HoO ^ H+ + OH-. The Sea-water and its Physical and Chemical Properties 75 H+ is the positively ciiarged hydrogen ion and OH~ is the negatively charged hydro xyl ion. The law of mass action gives the equation [H+] • [OH] _ [H,0] '''"' where the square brackets indicate concentrations in mols per litre. The concentration of pure water [H2O] is approximately the same for all dilute aqueous solutions such as sea-water. Since [HgO] is constant for a given temperature it can be included with the constant K^^ so that [H+] • [OH-] = K,,. At 18^ 25° and 50°C K,, has the values 0-61 x 10-", 1-0 x 10-^^ and 5-4 x 10-^* respectively. The concentration of either of the ions can be calculated if that of the other is known. Solutions where [H+] > [OH"] are acid and where [H+] < [OH-] are alkaline; in neutral solutions the two concentrations are equal. The character of the solution is thus specified completely by [H+]. In pure neutral water at 25°C [H+] = [OH"] = VK^ = 10"'^. The hydrogen ion concentration of a solution is usually not given as [H+] but as the quantity —log [H+] = pH. For pure water at 25 °C the pH is thus 7-0. Carbon dioxide system in sea-water. There is an equilibrium between the different chemical species derived from carbon dioxide that are present in sea-water and this must follow the law of mass action. As for every electrolyte there is a reciprocal re- lationship between the concentrations of the undissociated substance and those of its ions. For the first and second dissociations of carbonic acid [H^l • [HCO;l ^ i^y^^^K,. [H2CO3) ' [HCO;l To these equations can be added the equation for the titration alkalinity 2[C02-] + [HCO3] = A. Since the dissociation constants Ky and Ko are known, these three equations contain four unknown quantities [H+]; [HCO;]; [CO^-] and [H^COg]. If one of these can be determined, for instance the pH = (—log [H+]) then the other three can be calculated. The dissociation constants for carbonic acid in pure water (18°C) are Ky = 3-06 x 10"' and K^ = 5 x 10"". In sea-water the values of these dissociation constants are different because of the effect of the considerable amounts of other ions present in sea-water. The ions of the neutral salts such as Na+, K+, Mg2+, S0^~ also affect the carbon dioxide equilibrium but not 76 77?^ Sea-water and its Physical and Chemical Properties in proportion to the total amount present: according to the theory of interionic forces developed by Milner, Bjerrum, Debye and Huckel, amongst others, only a small frac- tion is involved. This fraction of the total concentration is termed the "activity"; the equilibrium thus involves not the total concentrations of the different ions, for instance [H+] but the activities, in this case/JH+J, where/is the "activity coefficient" and the above equations are replaced by others where the factors on the left-hand side are multiplied by the activity coefficients /i, /a, /g and/4. The constants Ki and K2 remain unchanged; they are termed "activity constants". However, instead of taking the effect of the neutral salts directly into consideration it can be allowed for by its effect on the dissociation constant; the apparent dissociation constants K[ and Kl are termed the "concentration constants". At the suggestion of the International Council for Oceanography Research they have been determined by Buch and co-workers (1932) Wattenberg, (1936). Table 29 gives numerical values for —log K[ and —log K2 for different temperatures and salinities (see also Buch, 1951). The calculation of the concentration of the individual forms of carbon dioxide in sea-water (free carbon dioxide, carbon dioxide pressure, carbonate and bicarbonate Table 29. Values of the first and second dissociation constants of carbonic acid in sea- water at different temperatures and salinities -\ogK[ _ log a:; S%o 0°C 10°C 20°C 30°C 0°C 10°C 20°C 30 °C 0 6-66 6-57 6-49 6-43 10-56 10-56 10-45 10-34 10 6-32 6-23 616 611 9-59 9-46 9-35 9-24 15 6-29 620 612 607 9-47 9-34 9-23 912 25 6-23 614 606 600 9-32 9-20 909 8-98 35 619 610 602 5-95 9-24 912 9 00 8-80 ions and total carbon dioxide) is now a simple calculation if the hydrogen-ion con- centration, the pH, is measured directly and the titration alkalinity is found from the salinity using the relationship given on page 74. This calculation can be shortened considerably if the carbon dioxide pressure and the total carbon dioxide are tabulated or plotted graphically for the most frequently occurring values of salinity, temperature and pH. The relationship between pH and the concentrations of free carbon dioxide, carbonate and bicarbonate can be shown clearly in a diagram (Fig. 40), where the percentage of each form is given as a function of the hydrogen-ion concentration. The S-shaped curves separate these factors in such a way that for any value of the pH the composition of the total carbon dioxide present is given along the ordinate. The curves for sea-water are drawn with full lines, the curves for pure water with dashed lines; the first is displaced towards lower pH-values. The presence of neutral salts in sea-water displaces the equilibrium towards the acid side because the apparent dissocia- tion constant increases. It can be seen that at very low pH-values there is almost only free carbon dioxide present, as the pH rises the concentration of bicarbonate increases and reaches a maximum at pH = 7-5; the carbonate ion becomes important only at higher pH-values, The two vertical lines in Fig. 40 show the normal range of the pH The Sea-water and its Physical and Chemical Properties 11 in the open ocean. It comes within the range where all three factors: HCO3, COg" and free CO2 are present in measurable amounts, although bicarbonate predominates considerably. The above values for the apparent dissociation constants are for water at a pressure of one atmosphere. If the pressure is increased the constant also increases since the pressure strengthens the dissociation both of the carbon dioxide and of the neutral Fig. 40. Percentage distribution of the three forms of carbon dioxide (free carbon dioxide, bicarbonate, carbonate) in pure water and in sea water as a function of pH (according to Buch). salts. This dependence implies, as shown in Table 30, that water displaced from the surface downwards to great depths will be more acidic, and inversely that of a sample brought from a definite depth with a collecting bottle will as a consequence of the decrease of pressure show a higher pH (be more alkaline). Table 30. Dependence of the concentration constants for carbon dioxide CO2 on the hydrostatic pressure Depth in m 0 2000 4000 6000 8000 10,000 10 10 1-26 110 1-58 1-20 200 1-32 2-45 1-41 i 3-02x^nat ' 1-55 X K'.y'^ Wattenberg gives the example shown in Table 3 1 of this effect. This pressure effect has a practical significance in processes involving the hydrogen-ion concentration such as the life of deep-sea organisms and the solubility of calcium carbonate at the sea bottom. Carbon dioxide in the deep layers of the ocean. The work of the "Meteor" Expedition 1925-7 gave the first reasonably good information on the distribution of carbon di- oxide in the deep layers of the sea. The essential results have been summarized by Wattenberg (1936). For the most part there is an approximate equilibrium at the surface of the sea between the partial pressures of carbon dioxide in the sea and in the 78 The Sea-w'oter ami its Physical and Chemical Properties Table 31. Variations of pH with depth at constant carbon dioxide content due to the change in pressure (After Wattenberg, 1936 ) Depth (m) 2000 4000 6000 8000 10,000 "1 7-80 7-75 7-70 7-65 7-60 7-55 pH >■ 800 7-95 7-91 7-87 7-82 7-78 J 8-20 816 812 808 804 8 00 atmosphere (see p. 72). Down to 50 m there is a slight reduction in the carbon dioxide content due to the assimilatory activities of the phytoplankton. Then follows a thin layer where the effects of assimilation and respiration are in balance. Beneath this layer the carbon dioxide pressure rises until it reaches a pronounced maximum at a depth of 500-1500 m (intermediate layer) depending on the latitude; it then falls off again, at first steeply and finally in the deeper layers approaches the values found at the surface. This carbon dioxide inversion (see Fig. 41) is accompanied by a change in the pH which is almost the exact mirror image. Figure 42, which shows the carbon di- oxide pressures along a cross-section through the subtropical South Atlantic, illus- trates how clearly marked these changes are. According to Wattenberg, these pro- nounced variations in the carbon dioxide distribution are due principally to the follow- ing factors : ( 1 ) The strong renewal of the deep water of the oceanic stratosphere by water masses of polar and subpolar origin which sink during the late autumn and winter in pH 0 7-6 78 80 8-2 / ,-' y ^ P: " -■ - — ^ 1 P H'\ f Ix / cc t- ^.y 1000 \ , - 1 \ X i ' /' 2000 \ i 1 1 1 1 i 5000 W :o. 1 i i i \/ A jpH 4 6 8 CO,- pressure 10 20 rc Fig. 41. Vertical distribution of the carbon dioxide pressure ^002 CO"* atm), the hydrogen- ion concentration pH and the temperature in middle latitudes of the Atlantic (according to Wattenberg). The Sea-water and its Physical and Chemical Properties 79 lOOO 2000 300 4000 5000 600O Fig. 42. Carbon dioxide pressure cross-section through the subtropical part of the South Atlantic (8-5 -13' S., profile VIII from the "Meteor" Expedition; given in 10"* atrri). higher latitudes and reduce the carbon dioxide content of the water at middle depths (2000^000 m). (2) The decomposition of dead organisms that takes place principally in the upper layers beneath the transition layer. In shallow seas dead organisms reach the bottom before decomposition is complete and the carbon dioxide pressure thus increases down to this depth. In the deeper layers of the major oceans decomposition occurs largely in the upper layers and the carbon dioxide pressure then decreases with further increase in depth. (3) The respiration and oxidation processes that produce carbon dioxide proceed more rapidly at the higher temperatures in shallow depths than at greater depths where the temperature is lower. All three factors combine to bring about the observed distribution, although a sta- tionary state can naturally only occur when the addition and the consumption of carbon dioxide are in equilibrium. However, for quantitative considerations of this type there is as yet no numerical estimate of the effect of the different processes. In the last hundred metres immediately above the sea bottom there is a more or less large increase in the carbon dioxide content above the almost constant value of the 80 The Sea-water and its Physical and Chemical Properties deeper layers (see Fig. 41). This apparently almost universal phenomenon may be due partly to the slower renewal of the water in the layer next to the bottom and partly to the gradual decomposition of material, not easily oxidizable, which with the shells and skeletal parts of organisms forms the sediments of the bottom and makes possible the formation of carbon dioxide in the bottom layer. This bottom layer with a definite increase is particularly well developed and sharply separated from the upper layers in the western half of the South Atlantic in the area of Antarctic bottom water (see Fig. 43). The carbon dioxide system between the ocean and the atmosphere (BuCH, 1942). The state of equilibrium at the surface of the sea between the ocean and the atmosphere 9(r 80" 70" 6(r 50" • , Release from rocks; • ooooo>. , Forest and prairie fires ; /^^ , Respiration; > tension differences in the sea; ► ••••»- , Deposition in sediments and minerals; • ••••»> , Combustion of coal and oil ; , Assimilation; »- , Flux following CO2 f = value smaller than 0-5 rel units (Lettau, 1954). between the two media that will cease only when equilibrium is established. Schlosing, in laboratory investigations, has clarified these exchange phenomena and shown that the sea always has a levelling effect on pressure differences that occur between the atmosphere and the sea. Since the sea has a carbon dioxide content several times greater than that of the atmosphere it suppresses fluctuations in the atmospheric carbon dioxide content and it tends to hold the atmospheric carbon dioxide at a constant value. In this respect the sea acts as a "regulator" of the carbon dioxide content, opposing changes in the content of this gas in the atmosphere. Nevertheless, recent investigations have shown that the variations in carbon dioxide content in both the ocean and the atmosphere are of the same order of magnitude. The sea in acting as a damper thus undergoes the same variations as the atmosphere, and under these conditions it is not easy to decide which is the "regulator" and which is the passive part. When changes occur and a new equilibrium is established, the sea of course takes up a much larger amount of carbon dioxide than the atmosphere. According to Table 32 the annual production of carbon dioxide amounts to about 0-0008 g/cm^ of the Earth's surface. The amount of carbon dioxide already present in the atmosphere amounts to about 0-4 g/cm^. If the whole of the carbon dioxide produced remained The Sea-water and its Physical and Chemical Properties 83 in the atmosphere the present carbon dioxide content would be doubled in 500 years. In actual fact if there is a pressure difference between the ocean and the atmosphere the sea takes up carbon dioxide until this difference vanishes. Buch (1939) has cal- culated that if the ocean and the atmosphere are always in equilibrium then five-sixths of the carbon dioxide produced is absorbed by the sea while only one-sixth remains finally in the atmosphere. Thus, if the sea absorbs the industrial carbon dioxide so rapidly that equilibrium is always maintained, then its present content would double at first in 3000 years. However, some time is needed to reach a new equilibrium and this is probably not reached as quickly as is customarily assumed. The cause could lie in the very slow vertical circulation within the ocean. In a short time only a very thin contact layer can interchange with the atmosphere. The equihbrium time for the whole volume of the ocean should certainly be more than several thousand years, and it must also be remembered that the initial pressure differences are very small and at first rise only slowly. According to the investigations of Buch in the North Atlantic in summer 1935 and in the sub-arctic regions in summer 1936, this part of the ocean and of course the corresponding region in the Southern Hemisphere appear to be the only areas where over long periods carbon dioxide is absorbed from the air in water masses which, by convective sinking in the autumn and winter, convey it to the rest of the ocean. Only in these layers is a rapid renewal of the surface water to be expected and these are thus the principal sites of equilibration in the carbon dioxide interchange between the ocean and the atmosphere (see also Buch, 1948). At the present time insight into the dynamics of these processes is rather inade- quate due to the scarcity of carbon dioxide pressure determinations. Extensive syste- matically collected series observations are needed for a better understanding of these phenomena. A more accurate investigation of the distribution of carbon dioxide in an adjacent sea (the Baltic) has been described by Buch (1945). (c) Calcium Carbonate in the Sea The solubility of calcium carbonate in water increases with the carbon dioxide content. This can be explained chemically as follows: calcium carbonate in solution is almost completely dissociated into Ca^^ and C0|" ions according to the equation CaCOg ^ [Ca2+] + [CO^-]. Since the concentration of undissociated calcium carbonate is very small and, if the sea-water is saturated, must be constant, the solubility product is given in a first ap- proximation by [Ca2+] . [CO^-] = Ki In the carbon dioxide equilibrium shown on p. 75 most of the hydrogen ions present combine with the carbonate ions to form bicarbonate ions since the bicarbonate ion, HCO^, is dissociated only to a small extent. This alters the calcium carbonate equili- brium, and calcium carbonate will thus go into solution until [Ca^+] increases suffi- ciently to satisfy the equilibrium equation. The equilibrium thus depends on the con- centrations of all the ions, H+, HCOg", CO^- and Ca^^ involved (Wattenberg, 1933, 1936). 84 The Sea-water and its Physical and Chemical Properties As well as the concentration of free carbon dioxide there are other factors also that affect the solubility and, while they are not so important, they must still be taken into consideration. The first of these is the concentration of Ca2+ derived not from the dissolved calcium carbonate but from the calcium sulphate and calcium chloride, that is, the excess of calcium ions above that corresponding to the combined carbon dioxide. These calcium ions, by the law of mass action, reduce the solubility of the calcium carbonate. An additional factor affecting the situation is the increase in the solubility product due to the presence of neutral salts in the same way as for carbon dioxide. Table 33 shows that the constant K^ is a hundred times greater in sea-water than in pure water. The solubility constant depends not only on the salinity but also on the temperature and, unlike most salts, decreases with increasing temperature. The pressure (at constant carbon dioxide pressure) also has a considerable effect on the solubility of calcium carbonate, but it is not yet certain how large this effect is. The factors affecting the solubihty of calcium carbonate in sea- water thus fall into two groups: (1) those that increase the solubility such as increasing carbon dioxide concentration, salinity and hydrostatic pressure ; (2) those that decrease the solubility such as increasing tempera- ture and calcium concentration. The values for solubility given in Table 34 show that in sea-water these factors more or less compensate each other so that there are no major differences from the solubility in pure water. Table 33. Dependence of the solubility product, K'^,for calcium carbonate on the salinity and temperature (After Wattenberg, 1936) 5 in %„ (at 20^C) '\ Temp, in °C (at 35 %„ S) 0 10 25 ! 35 ' 0 i 10 20 30 ^3 1 0-5 22 j 48 62 X 10-** j 8.3 7-4 6-2 1 4-4 X 10-' The calcium carbonate content can be found by determination of the alkahnity which varies in direct proportion to the variations (in milliequivalents/litre) in calcium car- bonate. Since the variations in calcium content are not very large it is necessary to determine the alkalinity very carefully (Wattenberg, 1930). Table 34. Solubility of calcium carbonate (CaCOg) in milligrams per litre in sea-water (355'%o)/or different tempera- tures and carbon dioxide pressures (in 10-4 atm) (After Wattenberg, 1936) Pco2 X 10* t°C 1 2 3 5 10 0 60 75 80 100 135 10 45 60 65 78 105 20 35 45 50 60 83 30 25 35 35 45 60 The Sea-)\'ater and its Physical and Chemical Properties 85 The mean vertical distribution of calcium carbonate in the Atlantic between 20° N. and 20" S. is shown in Fig. 44 from the results of 236 determinations made during the "Meteor" Expedition. The variations in calcium carbonate content can be divided into two groups: (1) Those due to differences in the total salinity — the alka- linity and therefore the calcium carbonate both increase with increasing salinity; CaCojmg/l. CoCojg/kg Solt 3-32 3-36 3-40 3 44 348 352 1000 2000 ^ 3000 4000 5000 Fig. 44. Mean vertical distribution of salinity (S^/q^, calcium carbonate (mg per litre of water) and calcium hydroxide (in g per kg of salts). The last curve gives a measure of the deviation of the calcium hydroxide content from proportionality with the salt respective to chlorine contents. (2) Those caused by chemical and biological changes. To show the last more clearly the calcium content is given not in terms of unit volume of water but in unit weight of salt; these are therefore expressed in g CaCOg per kg of salt or as per mill (%o). This then shows the variations in the calcium carbonate content of sea- water from propor- tionality with the salinity or the chlorinity. These are particularly important; they are furthest from normal at two places: (1) in the surface layers close to the atmosphere; (2) in the layer immediately above the sea bottom. The relatively small calcium car- bonate content of the very surface layer must be attributed to consumption by plank- ton, while the sharp increase at the sea bottom must be due to calcium carbonate dissolving from the sediments at the bottom. The normal calcium content of the water in the open ocean can be taken as about 3-40 g CaCOg per kg of salt. The depletion of calcium carbonate in the surface layer is about 2% and the enrichment at about 50 m above the bottom is about 4%. Special measurements would be needed to determine whether there is ii further in- crease nearer the bottom. The maximum value in the bottom water is not the same everywhere. It appears to be larger the greater the depth of the bottom, as is shown in Table 35 (Wattenberg, 1931). At depths of 3000 m this increase begins at about 300 m above the bottom, at 4000 m depths at 600 m and at 5000 m depths at about 1000 m. There are two fac- tors involved in producing this apparently stationary state: (1) the continuous upward flux of the calcium carbonate content as specified above ; and (2) the advective transport 86 The Sea-water and its Physical and Chemical Properties Table 35 Bottom depth in metres 2000 3000 4000 5000 CaCOg-content in g/kg salt of the deep water (at 50 m above the bottom) 3-398 3-448 3-500 3-525 of more or less calciferous water by the bottom currents. Approximate calculations made by Wattenberg (1935) have given an intensity for bottom currents which agrees well with that deduced from oceanographic factors. The surface and bottom waters show regional differences, while the intermediate water masses of the ocean show practically the same calcium carbonate content 90" 80° 70° 60° 50° 40° 30° 20° 10° 0° 10° 20° 30° 40° 50° Fig. 45. Chart of the calcium percentage saturation of the surface water in the Atlantic Ocean (CaCOa) (according to Wattenberg). The Sea-water and its Physical and Chemical Properties 87 throughout. In the polar and subpolar regions the surface minimum is largely absent because of the winter convection which eliminates the minor depletion of calcium carbonate by the few calcium-carbonate-consuming organisms during the brief summer. In low latitudes, on the other hand, the depletion of calcium carbonate is particularly pronounced due to the isolation of the upper layer by the thermocline in the tropics. The regional differences in the bottom layers are shown principally by the degree of saturation with calcium carbonate. The degree of saturation of sea-water by calcium carbonate in solution is found by comparison of the actual content with the solubility of the water in situ. Calculation of the degree of saturation shows that the surface water in equilibrium with the atmos- phere is supersaturated with calcium carbonate at all temperatures found. In the tropics and the subtropics the supersaturation is very large and the water may contain up to three times more calcium carbonate than that corresponding to the equilibrium value; Fig. 45 shows the percentage saturation with calcium carbonate in the surface water of the Atlantic. The large supersaturation in low latitudes shows very clearly; only the presence of this supersaturation allows an equilibrium between the addition of calcium in river water and its consumption by various organisms and sometimes by spontaneous inorganic precipitation at the bottom. It can be readily understood that the production of calcium by various organisms is facilitated and favoured by this supersaturation. Beneath the thermocline in low latitudes the saturation value falls rapidly with in- creasing carbon dioxide pressure to below 100% and reaches a minimum in the inter- mediate layers; in places the saturation may fall to less than 92%. While there are no large differences in the deep water below 1500 m (degree of saturation 98-100%) the bottom water in the Atlantic Ocean differs somewhat in saturation just as it also differs in carbon dioxide content. Calcium is also involved in a closed cycle. In the upper layers of the sea there is a strong withdrawal of calcium carbonate, partly by biological processes associated with calcium using animals and plants and partly by inorganic precipitation. Compensation is performed at the sea surface by the supply of calcium due to river water and at the sea bottom by solution from the bottom sediments. Without an accurate quantitative estimation of the individual components in the cycle it is im- possible to state whether supply or consumption predominates, or whether the present condition of considerable supersaturation at the surface of the sea is a stationary state. Chapter III Temperature in the Ocean ^ the Three- dimensional Temperature Distribution and its Variation in Time 1. Heat Sources, Heat Exchange and Heat Budget in the Ocean All changes of state in the liquid and gaseous envelopes of the Earth are due basically to energy changes. Energy is very largely supplied from outside the Earth, principally from the sun which provides an inexhaustible source of radiant energy for the Earth. There is a constant inflow of energy from the sun and a constant outgoing radiation from the Earth into space. The Earth does not retain the energy supplied to it but returns all except a vanishingly small part to outer space in the same form (radiation) in which it received it. The possibility of life on the Earth and all changes of state on the Earth depend not so much on the inflow of solar energy as on the enormous supply of entropy involved in the conversion of the high-temperature radiation from the sun to the low-temperature radiation from the Earth. These considerations lead to the concept of a stationary state as far as the heat energy of the Earth, taken as a whole, is concerned. This constancy in heat energy can be confirmed for the solid part of the Earth and for the atmosphere, and it can be expected that it holds as a close approximation for the energy budget of the oceans. There are, of course, small variations with time in the temperature of the ocean, but these can be taken as variations around a mean value which remains essentially un- changed . Heat budget of the ocean. Tn this quasi-stationary state all the supply in energy is balanced by equally large losses of energy. The most important factors are the radia- tion, the interchange of sensible heat with the atmosphere above the sea and evapora- tion from the surface of the sea or the condensation of atmospheric water vapour. Other minor sources of heat that may be mentioned besides the above stated ones are listed in Table 36. The order of magnitude of the heat amounts involved in each of these processes varies considerably. The largest is certainly the heat absorbed from solar and sky radiation which is the principal factor in the heat budget of the very upper layers of the sea. At its upper limit the Earth atmosphere obtains per cm^ by normal incidence an energy of l-94g cal/min (solar constant). The entire surface of the Earth receives per cm^ on the average 0-485 g cal/min or during the entire day 700 g cal/cm^. This incoming radiation from the sun is largely short wave. Its intensity is decreased on passing through the atmosphere so that only 43%, that is 0-21 g cal cm - min"^ 88 The Three-dimensional Temperature Distribution and its Variation in Time 89 Table 36. Heat sources Heat losses 1. Absorption of solar and sky radiation 1. Radiation from the sea surface Qb Qs 2. Convection of sensible heat from atmos- 2. Convection of sensible heat from sea to phere to sea atmosphere Q/, 3. Conduction of heat through the sea bot- 3. Evaporation from the sea surface Q^. tom from the interior of the earth 4. Conversion of kinetic energy into heat 5. Heat produced by chemical and bio- logical processes 6. Condensation of water vapour on the sea surface 7. Radioactive disintegration in the sea- water reaches the surface of the sea. Of this, 27% is direct solar radiation and 16'^o is diffuse radiation from the sky (sky light). From the other sources listed in Table 36 those with a comparatively smaller effectiveness can be neglected. The sources listed under item 2 for heat gain and loss can be added giving one source. The same applies to item 6 (heat gain) and item 3 (heat loss). The heat obtained from the interior of the Earth is about 50-80 g cal/cm^ per year or on the average about 10 x 10"^ g cal/cm^, min. The heat supplied from the in- terior of the Earth has recently been measured directly for the deep-sea basins of the Pacific Ocean by Revelle and Maxwell (1952) and for the Atlantic Ocean by BuLLARD (1954). These measurements gave in agreement the value 6-2 x 10~' g cal/cm- min which corresponds to the value for the continents. Compared with the heat from solar radiation this is unimportant; it probably causes only small local variations in the thermal structure of deeper, enclosed stagnating water (see Chap. III. 4d). The kinetic energy which the sea obtains by the tangential action of the wind on the sea surface and by the dissipation of tidal energy by turbulent friction and which will be transformed into heat gives only a very small heat contribution. The energy im- parted by the winds amounts scarcely to a ten-thousandth part of the solar and sky radiation energy and can therefore be neglected. Also the tidal energy dissipated by turbulence is only of any appreciable influence in shallow waters. For example, Taylor found the value of 1050 g cal/cm^ per year = 0-002 g cal/cm^ min for the Irish Sea (see Vol, U. Chap. XV, 3). If this heat could accumulate in the Irish Sea for a whole year the temperature rise would be only 0-2 °C. The item 5 in Table 36 has no significance in the general budget of the sea and only requires to be taken into consideration where there are local concentrations of plant life. The disintegration of radioactive material in sea-water will afford barely 4 x lO'^gcal/cm^ min. Under these conditions the heat budget of the ocean requires the following equation Qs - Qb- Qh ~ Qe = 0. 90 The Three-dimensional Temperature Distribution and its Variation in Time For particular parts of the sea and for short intervals of time it may also be necessary to take into consideration the heat carried by ocean currents, or by mixing processes into or out of the oceanic region under consideration and also the heat which causes over short periods of time changes in water temperature. The above equation is, however, sufficient for the ocean as a whole. The individual terms will now be dis- cussed in some detail. (a) Direct Solar Radiation Of the solar constant /^ is 1-94 g-cal cm~- min"^ one horizontal cm^ at the sur- face of the Earth obtains for a zenith distance z of the sun (altitude /; = 90^ — r) and due to the angle of incidence and the reduction in intensity due to the atmos- pheric absorption only the intensity / = /q e~^" sec z cos r. Sec z is the relative thickness of the air through which the radiation passes (equals 1 for an atmosphere pressure of 760 mm Hg with the sun at zenith; equal to 2 when the sun is at an altitude of 30°). e-^« = ^ is the transmission coefficient and q has under normal conditions a value between 0-6 and 0-7, a = 0-1 28-0-054 log sec z and ris the "turbidity factor". If z and Tare known then the direct solar radiation inci- dent per cm^ on a horizontal surface can be calculated directly for any altitude of the sun. Part of the solar energy reaching the sea surface will be reflected there. This part depends on the angle of incidence, that means from the zenith distance of the sun. Schmidt (1915) has calculated that due to the compensatory effects of solar radiation, which decreases with increasing zenith distance and of the simultaneously increasing reflection the intensity of the reflected radiation for approximate calculations can be put as ^ = O-OlO-O-013/o. By using the known total amounts of heat received by 1 cm^ of the Earth's surface by direct solar radiation with an average value of the transmission coefficient, and by knowing the reflection loss at the sea surface, it is possible to calculate the amount of energy obtained by 1 cm^ of the sea surface in one day. Table 37 gives the mean daily total sum for a year for q = 0-6-0-7. The figures show that even assuming a continuously clear sky the equator receives barely one-half and the pole only a fifth of the solar radiation incident on the upper atmos- phere. When the transmission coefficient is 0-6 the entire surface of the Earth receives only 44% of the theoretical amount of heat. This value will be still further reduced by the presence of clouds. If the cloudiness is w (as a fraction of the visible sky) then the radiation actually reaching the surface of the sea is only ^2 = (I " u') S,, where S^ is the value given in Table 37. Table 37. Mean total daily sums of direct solar radiation on g cal/cm^ day^). (J^ = 1 -94 g cal/cm^ mir a free water surface Latitude 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° «7 = 0-6 i 402 392 365 q = 0-7 493 481 452 322 402 270 211 341 274 155 206 105 146 74 109 1 60 94 The Three-dimensional Temperature Distribution and its Variation in Time 91 Since the radiation on the surface of the ocean is difficult to measure and only few determinations have been made, Mosby (1936) has given an empirical equation for the mean monthly and annual values of the radiation incident per cm^ on a horizontal surface for given values of the mean altitude of the sun and of the mean cloudiness Qs == kh{\ - 0-07 liv); g cal cm-^ min-^. The bars indicate mean values and k is a factor which depends on the turbidity of the atmosphere; at the equator it is 0-023, at 40^" latitude 0-024 and at 70° latitude 0-027. {h) Diffuse Sky Radiation During the day the surface of the Earth also receives general scattered short-wave radiation from the atmosphere and also direct solar radiation reflected from clouds. Estimates based on the direct measurement of total radiation (direct + diffuse radia- tion) show that in general the average value of the diffuse sky radiation for the whole Earth and for a cloudless sky amounts to about 56% of the total radiation on the upper limit of the atmosphere. If we take this value as an average for all latitudes, for a cloudiness u-, the direct radiation S2 will be increased by diffuse radiation amounting to 0-56vr • Si. At the surface of the water this more or less generally scattered radiation will suffer a reflection loss of 6-6%. The fraction of diffuse radiation from the sky entering the water is thus given by D = 0-52h' • S^. (c) Long-wave Radiation of the Atmosphere The effective back-radiation R^ is the difference between the radiation according to the Stefan-Bo I tzmatm law (E = err*) and the long-wave radiation of the atmosphere and depends, for a cloudless sky, on the absolute temperature T of the lowest layer of the atmosphere and on the water-vapour pressure in this layer (e in mm Hg) (Ang- strom, 1936). The effect of clouds is shown in a reduction of the effective back-radia- tion and can be calculated if the cloudiness is given. With this equation it is possible to calculate numerically the longwave radiation of the atmosphere for a given tem- perature, water-vapour pressure and mean cloudiness. The effective back-radiation can be measured directly, but such measurements have only seldom been made over the sea. Angstrom has derived an empirical formula which has been given by Moller in the following form ^eff = oT^[l - (0-210 + 0-174 X 10-»-»55eo)(l - 0-675vv)], where a is the Stefan-Boltzmann radiation constant, T is the absolute temperature, Re^ is the vapour pressure above the surface of the sea and w — as before — is the mean cloudiness. For the surface of the sea it can be rearranged to give Q^ = 0-954ar4 - (7r[(0-210 + 0-174 x lO-oo^^e^^^i _ o-765m-)]. Since, as shown by Lauscher (1944), the radiation from a plane water surface is decreased by 6-6% by back-reflection (see p. 60). The effective radiation is the first loss in the heat balance (see Table 36, item 1 (heat loss)). 92 The Three-dimensional Temperature Distribution and its Variation in Time (d) Evaporation (see Chapter VII). A further debit item is the heat lost by evaporation. The amount of heat involved can be easily found from the mean zonal values for evaporation (WiJST, 1922), since for the evaporation of 1 mm of water from 1 cm^ of a water surface 60-65 g cal are needed. (h If a layer of water of e mm thickness evaporates from the top of such a column of water with 5'%o salinity then the increase in salinity when evenly distributed over the column of water is given with sufficient accuracy by AS, = ^ €. If at the top of a similar column an ice layer of e cm thickness is formed with a salt The Three-dimensional Temperature Distribution and its Variation in Time 97 content 5^, smaller than S, then, as a first approximation the corresponding increase in salinity is given by 0-9g {S - S,) If the ice contains no salt (Sg = 0) then In these quantities AQ,AS, and ASe (heat loss, salinity increase by evaporation and by ice formation) lies the primary cause of every thermo-haline convection. In lower lati- tudes where there are only small variations in the temperature the heat loss is out- weighed by the effect of evaporation ; in temperate latitudes the heat loss by radiation is the decisive factor, while in polar regions, in addition to these processes, the increase in salinity due to the formation of ice is also effective. Only very small changes in the specific volume are needed to initiate convection in the uppermost surface layer since the resistance to be overcome is not large, a hun- dredth %o salinity or a hundredth degree centigrade is sufficient. The range of effectiveness of convection depends entirely on the vertical density distribution in the water mass in which it occurs. For a given surface disturbance it can only extend down to that depth at which the displaced quantum of surface water reaches, water having the same specific volume. If there is a rapid decrease in the spe- cific volume, then the convection will cease in the upper layers ; this is liable to occur particularly at the density transition layer (thermocline) which acts as a barrier layer and confines the thermo-haline convection to the top layer of the sea (thin homo- geneous layer of uniform density). On the other hand, a randomly initiated disturbance of any size at the surface leads to convection which extends in a homogeneous water mass down to the bottom. The range of effectiveness of convection is a maximum only when the density disturbance of the sinking water quantum is retained while it sinks. If, as is to be expected, it mixes with the surrounding water the disturbance will be rapidly decreased and the depth of influence of convection will be correspondingly less. The larger the density difference between the sinking water and its surroundings the more rapidly the difference between them will be diminished and the greater the reduction in the depth of the convection layer. When the sea has a normal stable structure (tropics, subtropics and temperate latitudes) the nocturnal convection before sunrise will extend to a depth of 10 or 20 m. The seasonal convection processes, caused by prolonged cooling during the autumn and the winter, will extend to greater depths, normally to about 300 m. The con- vection is developed to its greatest extent in polar and subpolar latitudes, where it is assisted by a very uniform temperature and salinity distribution. The question for the primary cause initiating these major convection processes, which are of decisive importance for the deep-sea circulation of the ocean, has been the subject of a con- troversy that is still not without interest. The initiation and maintenance of the vertical convection in higher latitudes could be due to the cooling of the upper layers by radia- tion, or it could be due principally to contact with melting ice. Pettersson (1904) supported the strong cooling effect of the ice that is so plentiful in these latitudes, while Nansen (1912) favoured the direct cooling of the surface layer by outgoing 98 The Three-dimensional Temperature Distribution and its Variation in Time radiation. This controversy was settled by important and interesting experiments in the sense of Nansen's reasoning. He suggested that the winter convection in parts of the Norwegian Sea and of the North Atlantic (south and south-east of Greenland and in the Irminger Sea) could reach very great depths because of the almost uniform den- sity structure of the sea, so that the autumn and winter cooling thus continued almost to the bottom. This should therefore be the place where the uniform North Atlantic Deep Water was formed. The observations of the winter cruises of the "Meteor" in the Iceland-Greenland waters during 1929-35 have shown that these views of Nansen were correct. The cause of this convection is certainly the radiation of the surface layer during the late autumn and early winter. In the North Polar Basin conditions are somewhat different. The very large rivers of Asia and North America bring large amounts of fresh water into this basin and these overlay the saline water that flows into the deeper layers from the Atlantic Ocean. Any deep-reaching convection is scarcely possible here, cooling is limited to the sur- face layer and is correspondingly stronger. The melting of ice in the spring and summer sets up a barrier against the denser water masses in the deeper layers so that the sum- mer heating does not penetrate far. Characteristic examples of a convection that extends to great depths, and can be attributed primarily to an increase in the salinity of the surface layer caused by strong evaporation, are found in the Mediterranean Sea and in the Red Sea. The low precipi- tation, the small amount of river water flowing in, and the high rate of evaporation raise the salinity of the surface layers especially in the summer, though at this time only a limited convection occurs, since the increase in density is largely offset by the effect of the summer heating. However, in the autumn and winter a well-developed con- vection is set up due to the lowering of the temperature of the surface water and reaches to great depths because of the uniformity of the vertical structure of the deeper layers. The accurate mathematical treatment of thermo-haline convection processes is not easy. It can be attempted in the following way (Defant, 1949). To begin one considers two thin layers of thickness h-^ and h^, temperature ^i and d'z, salinity ^i and S^ and density pi and p^. A disturbance introduced in the entire upper layer so that p^ = p^ will cause mixing of the two layers /zj and h^ by convection and the final result will be the layer h^ + h^ of density p^. If the disturbance in the upper layer is assumed to be due entirely to a reduction in the temperature of the upper layer by '&-y — Ad'i then the final temperature at the end of convection results to (^, - A{^,)h, + ^Jh h, — h^^. — ^^--^:;^^- when the mean temperature that would be obtained by simple mixing of the initial water masses is given by ^'•'^ h, + h, ' The final salinity after ceasing of convection is given by Syh^ + S^h^ ^'''~ h, + h, and corresponds to the salinity obtained on simple mixing. The Three-dimensional Temperature Distribution and its Variation in Time 99 If the disturbance in the upper layer is due to an increase in the salinity by AS^, then the final temperature and salinity are h ^1,2 and 5i,2+i-^ ^S^- A disturbance in the second layer can in the same way be passed on to a third and from this to a fourth and so on while at the same time its intensity decreases continually. If the disturbance in the layer /?! + //g is due to a temperature decrease of J )?i,2 then progression of the convection to the third layer in an analogous way gives the tem- perature and saUnity at the end of the convection process as /7iJl^l + //l, 2^^1,2 ^1.2, ^1,2,J and iSi, 2, 3' If the disturbance is due to an increase in the salinity ofASx,^ then the temperature and salinity are t^i.2,3 and ^1,2, 3 H r . "1,2.3 Thesimplestwayofcalculatingzl'!^andJ5'is to use a [r5]-diagram (see Chap. VI). In Fig. 46 the thin Unes are lines of equal density (isopycnals). The point A shows the values 20° 347o< 35%< 36%, 377o< 15' 10' 25 25'.5 ^ :^ ^ 26 X 26-5 /a /; ^y ^"y^ A\, X y / 2 7 /b / / 27.'5 / / / 28 / / / / 28.'^ 29 / / ^ Fig. 46. [r5]-diagram for the determination of degree of disturbance during the initiation of convection processes in the sea. 1 00 The Three-dimensional Temperature Distribution and its Variation in Time of 19' and S of the upper layer hy. The density p^ of the second layer //o corresponds to the isopycnal that passes through the points B and C. Since the density of the upper layer must be equal to the density of the lower layer, as must be the case at the end of the mixing by convection, then either the temperature must decrease hy AB = —A'd'i if the salinity is constant, or the salinity must increase by AC — ASi if the temperature is constant. A convenient connection of the point A with a point D on the ispycnal p2, between B and C, gives the value of the disturbance for the tem- perature and the salinity if both are present at the same time. The determination of magnitude of the disturbances from the [r5]-diagram in this way offers little difficulty, A simple schematic diagram gives a convenient representation of the results of convective mixing. Fiaure 47 shows the normal vertical distributions of d and S and of %^^<=^ At AS Fig. 47. Change in the thermo-haline structure of the sea produced by convection processes. the specific volume a; they represent the conditions before the mixing of the upper layers by a convection extending only to a depth h. If the convectional disturbance is entirely due to a reduction in & (by radiation) then the state of the upper layer at the end of the convection process is characterized by the broken straight line; if, on the other hand, the convection disturbance is entirely due to an increase in salinity the dotted straight line shows the final state. It can be seen that the convection levels out any differences in the vertical gradient for the different factors. (d) Dynamic Convection (forced vertical mixing) While thermo-haline convection is produced by external sources of disturbance and continues as long as these disturbances remain, dynamic convection depends on the forced mixing of superimposed layers of water embedded in a turbulent current. The disordered eddying flow of larger quanta of water within such a current causes a con- tinuous mixing of the water mass in both vertical and horizontal directions. This mixing The Three-dimensional Temperature Distribution and its Variation in Time 101 process affects not only the vertical distribution of velocity within the current, but also plays a considerable role for the distribution of the properties of the water mass. The importance of such a mixing process, due to turbulent flow in a water mass, was realized much earlier in oceanography than in meteorology. Gehrke (1909, 1912) was the first to show that the mixing of the water masses in an ocean current must give rise to a vertical transfer of heat. He found that this vertical heat transfer is pro- portional to the product of the specific heat and the vertical temperature gradient, so that it corresponds to the ordinary equation for the molecular thermal conductivity, but with a coefficient which is dependent on the intensity of mixing and is consider- ably larger than the coefficient for molecular thermal conductivity. Gehrke termed this a "coefficient of turbulent mixing"; it has the dimensions [cm^ sec-^]. Following Gehrke, Jacobsen (1913, 1915, 1918), in particular, has dealt in detail with the "apparent" thermal conductivity and with the "apparent" diffusion which are con- nected with turbulent processes. He pointed out that for all the processes initiated by the mixing of the properties of the water (temperature, salinity and the content in sea-water of other dissolved and suspended materials and of organisms) the tur- bulent mixing coefficient should be the same and should be dependent only on the intensity of the turbulence in the current. Through the turbulence also the flow mo- mentum (impulse of the current) is affected by the "mixing" process, i.e. a vertical equalization that manifests itself in the turbulent (apparent) viscosity. Already Jacobsen has put forward the view that in the transfer of the small quanta of water from layer to layer within the turbulent flow produces an immediate and complete equalization of the momentum; however, complete equalization of the properties of the water does not necessarily follow. This would imply that the "intensity of mixing" of the momentum (turbulent viscosity coeflricient) must always be larger than that of, for example, the temperature or the salinity (apparent thermal conductivity coefficient, apparent diffusion coefficient). These views of Jacobsen appear to be confirmed by the quantitative determination of these coefficients. Following these investigations which gave a deep insight into the nature and efficiency of turbulent ffow, Schmidt (1917, 1917^, 1925) and Taylor (1915, 1918, 1922), at about the same time, carried out extensive work on turbulent flow and on the phenomena connected with it, which has had a wide utility for the explanation of several oceanographic phenomena. These started from the basic approach that due to the random movement of individual small quanta of water in a turbulent flow there is not only an equalization of the momentum in the direction of the largest velocity gradient, but that every property can be transferred to an adjacent mass in the direc- tion of its largest gradient. The simplest derivation of the most important and funda- mental equation for the interchange of properties within a turbulent flow has been given by Schmidt. Consider a horizontal unit area (1 cm^) in such a horizontal flow, whereby the vertical direction z is counted positive upwards and negative downwards of it (the zero point (z = 0) lies in the surface itself). Due to the turbulence of the flow there will pass through this unit area a mass of water m^ upwards and a mass ma downwards. Since, however, there is on the average a displacement of the water only in a horizontal direction it follows that over a long period of time Hmy, = lima. Every small quantum of water will, however, carry its properties with it during its turbulent displacement. If one of these properties is designated by s (for instance the 102 The Three-dimensional Temperature Distribution and its Variation in Time salinity) and j is a function of z only, then at the unit surface as a first approximation ds s = s, + ^z, where Sf is the value in the surface where z = 0. Every small particle of water passing through the surface from below will take with it an amount w„ s^, while those from above will carry an amount m^ s^. The final exchange flux S through the unit surface upwards can be expressed as the difference S = I^m^Su — ^tn^Sai whereby the summation has to be taken for all the small particles moving upwards and downwards through the surface. Now ds J ds Su = -^z + 3^ ^w and s^ = Sf + —Za, where the values of Zy are all negative and the values of z^ are all positive. This gives 8s S = {Sm^z^ — i:maZa) ^^ . Considering the different signs of z, the quantity in brackets gives a negative sum —Em I z I , where every small mass m of water moving through the surface is now multiplied by the initial absolute distance | z | from the unit surface. This sum de- pends only on the state of turbulence of the flow. Schmidt has called it the "Austausch (exchange) coefficient" t]. It has the dimensions g cm~^ sec~^. The basic equation for the exchange is thus The most important exchange quantities involved in oceanographic turbulent trans- fer processes are: heat-temperature, salt-salinity, gas amount-gas content, number of organisms-organism content. The flow momentum-flow speed also follows this law (see later). It appears that the assumption that every small quantum of water starts from its initial position with a property s corresponding to the mean vertical distribution at that point does not entirely accord with the actual conditions. Only for the pair flow momentum-velocity does there appear to be a complete and immediate equaliza- tion of the velocity diff'erences. For all other properties a correction must be applied to the above basic equation. Ertel (1942) has attempted to take these circumstances into account, and obtained the equation ds ds S = -^(X - 2n) -= - A j^. The Three-dimensional Temperature Distribution and its Variation in Time 103 Thereby, it was assumed that a small particle of water passing through the unit sur- face is not immediately mixed completely with the surrounding water but is mixed in the proportion 1 : n. For the velocity in a turbulent flow, n would be equal to zero and the exchange coefficient A for the property s (eddy conductivity and eddy diff"us- ivity) would then be less than the eddy viscosity coefficient -q. Determinations of A and T] also verify this. Table 41 gives list of such determinations measured in currents in different parts of the oceans. It can be seen that -q is of the order of 100-200 or more while A is of the order of 5^0, on the average about 20 g cm^^ sec^^ The ratio -qjA is of the order 5-20. Taking an average value of about 10, then Afrj = l-2n, n = 0-45, that means that the small quanta of water in random movement are mixed with the surrounding water only to the extent of about 45% of their mass and accordingly the temperature and salinity, for example, tend towards the values of their surroundings at this rate. This value is not unreasonable considering the difficulty of mixing water of different densi- ties and the constant tendency for water masses of different densities to separate again. The exchange equation applied to the pair heat-temperature has the same form as that for the molecular thermal conductivity (p. 50), except that the thermal conduc- tivity coefficient a = {^lcj,p) is replaced by the quantity CpA (specific heat x exchange coefficient). The exchange coefficient A is of course not constant and will vary from layer to layer. Taking a mean value of about 20 g cm-^ sec~^, then since Cj, is approximately equal to 1, Cj,A will be about 15,000 times greater than a. The molecular thermal conductivity is thus of no importance compared with the eddy conductivity (dynamical convection). The thermal conductivity equation for turbulent heat trans- port is therefore d^_A 8^& Tt~'^ a?' where A is assumed to be independent of the depth. If this is not the case the equation is 8^ _l 8 / 8^ Tt^'p 8z y-^ Temperature changes at the surface will be transmitted much more rapidly by turbu- lent thermal conductivity down to the deep-ocean layers. For the process of molecular heat conductivity surface disturbances were shown to require a half-value time of some miUions of years (see Table 40), however, for conductivity it would take only some hundreds of years according to Table 42. Indeed, in the upper layers surface changes will penetrate downwards by turbulent action remarkably rapidly; only a few days are required to spread completely through the layer down to 50 m. Periodic changes will of course reach deeper. For values for A of 20 and 100 gcm~^ sec~^ the amplitude of a diurnal variation will decrease to 1/100 of its value at the surface in 34 and 75 m, respectively. For the annual variation the corresponding values are 644, 1440 m, respectively. This corresponds better with the values given by tem- perature observations. 104 The Three-dimensional Temperature Distribution and its Variation in Time {e) Horizontal Convection and Lateral Mixing The dynamic convection discussed in the preceding section applies only to mixing of water masses in a vertical direction moving in a horizontal turbulent flow. In addition to this vertical mixing process there will also be a mixing process, largely Table 41. Coefficients of eddy conductivity, eddy diffusivity and eddy viscosity Coefficient Current or oceanic region Depth of layer (m) Magnitude (g cm~^ sec~^) Reference Eddy conductivity Philippine Trench 5000-9788 20-3-2 Schmidt, 1917 and diffusivity Algerian Coast 0- 20 35^0 Schmidt, 1917 from temperature Mediterranean 0- 28 42 Schmidt, 1917 and salinity Cahfornia Current 0- 200 30-40 McEwen, 1919 measurements, A Caspian Sea 0- 100 1- 3 Stockman, 1936 Barents Sea — 4-14 Subov, 1938 Bay of Biscay 0- 100 2-16 Fjeldstad, 1933 Equatorial Atlantic Ocean 0- 50 320 Defant, 1932 Randesfjord 0- 15 01-0-4 Jacobsen, 1913 Schultz Grund 0- 25 004-0-74 Jacobsen, 1913 Kuroshio 0- 200 30-80 Sverdrup-Staff, 1942 Kuroshio 0- 400 7-90 Suda, 1936 Southern Atlantic Ocean 400-1400 5-10 Defant, 1936 Arctic Ocean 200- 400 20-50 Sverdrup, 1933 Carribean Sea 500- 700 2-8 SeiweU, 1938 South Atlantic Ocean 3000-Bottom 4 Defant, 1936 South Atlantic Ocean Near Bottom 4 Wattenberg, 1935 Eddy viscosity -q Wind currents Surface layer l-OIw^w < 6 m/sec) Thorade 1913/1914, 1914 Wind currents Surface layer 4-3 (^^(m' > 6 m/sec) Ekman, 1905 North Siberian Shelf 0-60 (tide) 75-260 Sverdrup, 1926 North Siberian Shelf 0-60 (tide) 10-400 Fjeldstad, 1936 North Siberian Shelf 0-22 H^m' Fjeldstad, 1929 Schultz Grund 0-15 1 •9-3-8 Jacobsen, 1913 Caspian Sea 0-100 0-224 Stockman, 1936 Kuroshio 0-200 680-7500 Suda, 1936 Japan Sea 0-200 150-1460 Suda, 1936 Table 42. Advance of a sudden temperature change penetrating into the sea by thermal turbulent conductivity {half value time of surface disturbance) Depth (m) 1 10 50 100 500 1000 3000 6000 Time when Alp = 20 cm^/sec Time when Ajp = lOOcm^/sec 9 min 1 5 h 16 days 64 days 1-8 min 3h | 3 days 13 days 4-4 years 320 days 17-4 years 3 J years 185 years 37 years 624 years 125 years The Three-dimensional Temperature Distribution and its Variation in Time 1 05 effective in the horizontal direction caused by currents moving side by side carrying small masses of water at greater or lesser velocity and by eddies of varying size with vertical axes, that is, by the lateral turbulence in the flow. In the horizontal direction the disturbances are of greater dimensions than in the vertical direction, particularly those due to atmospheric effects (wind, squalls and rapid changes of pressure), which affect the surface layer of the sea and to some extent the deepei layers also. Disturbances due to coastal and bottom topography are also able to produce turbulence in an horizontal direction with turbulence elements which must obviously develop on a much larger scale than the vertical turbulence. The corresponding exchange coefficient will be much larger than for vertical mixing. In a certain sense there is an analogy with the large-scale lateral turbulence in the atmos- phere which is also quasi-horizontal (isentropic). In this case the coefficient is on the average of the order of 10^ g cm~^ sec~^ as compared with an average value of 50- 100 of ordinary vertical turbulence. That lateral large-scale turbulence is also im- portant in oceanic phenomena was first pointed out by Defant (1926), who determined the order of magnitude of this exchange coefficient as about 5 x 10^. Later, Witting (1933) discussed both vertical mixing and lateral mixing, and has attempted the de- termination of the exchange coefficient by large-scale coloration experiments. Rossby and co-workers (1936) have clearly shown that there occurs in the ocean, as in the atmosphere, a lateral mixing of this type along the isotropic surfaces, which is essen- tially in the ocean the same as along the or^-surface. Parr (1938) has shown the large effect of this lateral mixing on the distribution of temperature and salinity in the water masses around Newfoundland; Sverdrup and Fleming (1941) have found the same effect in the coastal water off California and Stommel (1950) has determined the lateral mixing coefficient Ajp in the Gulf Stream to be 2-3 x 10^ cm^/sec. For a given horizontal gradient in any of the properties of a mass of water the horizontal convection will play a large part in the long-period equalization of this gradient. This presupposes a transport of the property along the direction of the gra- dient. Furthermore, if a small mass of water has a property s (for instance, temperature) present in amount S (for instance, heat), then the horizontal transport of S across the horizontal turbulent flow in the direction n is as before, Sn = —An(Ssldn). An is now the horizontal exchange coefficient. Its order of magnitude is several times larger than that of the coefficient for vertical mixing A^. Since, in general, the vertical gradient of a water property (such as temperature, sahnity) dsjdz is considerably larger than that in the horizontal direction dsjdn, the horizontal transport Sn may still be of the same order as the vertical transport S^, since in the above equation the product of the two quan- tities is essential. This appears to be the case in reality so that lateral mixing is no less important than the vertical. Consider a volume element dx, dy, dz through which there is a turbulent flow with velocity components u, v, w; the exchange coefficients in the three directions A a;, Ay, Ay. Then, for the individual change with time in the property s the following equation will apply ds 8s 8s 8s 8s \ f 8 / 8s\ 8 / 8s\ 8 / 8s] dI = 8t-^''8x-^'8y+''8-z--p[8x [^^8xj^8y l^^a^j + ^:^ l^^FrJ If the .Y-axis is taken as the direction of the turbulent flow (positive in the flow 106 The Three-dimensional Temperature Distribution and its Variation in Time direction) {v — w = 0), then stationary conditions in the distribution of the property s in the volume element {^dsfdt = 0) are only possible if the equation d'^s dh d^s 8s is satisfied, where A;^, Ay and A^ are taken as constants. From this general equation can be derived more special cases : d^s ^•y « / X ^* aF^ - ^" a^ ^ ^ ^^^ if there is vertical mixing only (A^. = Ay = 0); 8^s ^s ^ • /,x if there is transverse mixing (in a horizontal direction normal to the flow) (Ax = A:i = 0); 8^5 8^s ^'-^ _ n if there is mixing in all directions but no aveiage water transport in the ^r-direction (u = 0). Cases (a) and (b) are mathematically identical but solely the vertical and horizontal directions are interchanged. A solution for the equations (a) and (b) has been given by Defant (1929) 77 TT^ Az s = Sa + m e""* cos -^, z and a. —-r^ — . For case (b), the co-ordinate z is replaced by the co-ordinate y. The distribution of the property s along the homogeneous turbulent flow has been found to be tongue-shaped if the 5-content initially has a maximum value at the centre of the flow (at x = 0, s = Sq -{- m cos (7r/2/)z). This is also the case when the velocity is the same over the whole transverse cross- section. Figure 48 gives an example of the course of the i--lines for Aj p = 4 cm^sec, M = 10 cm/sec and / = 2 x 10'* cm. The further the cross-section is taken from the initial section {x = 0) the lesser are the horizontal and vertical differences in s. By the extension of the above solution to different initial conditions for x = 0 (Thorade, 1931) it became evident that neither the tongue-form of the distribution of the proper- ty s nor the distribution of the velocity u in the cross-section is considerably affected. In addition, the initial distribution of 5 at x = 0 has equally little effect. The tongue- form of the j'-curves is always re-established in a short time and is very largely a consequence of the turbulent mixing. This is shown particularly well in Fig. 48a, which shows the distribution of the property s in the case where the velocity is constant across the transverse section, and initially for a: = 0 the property s is constant within the distance 2/ {s = 100), while outside of this range there is no content of ^ in the water {s = 0). In the flow a tongue-shaped distribution of s is produced immediately. This The Three-dimensional Temperature Distribution and its Variation in Time 107 0 1000 2000 3000 WOO 5000 km SOOOlm ' Fig. 48. Formation of a tongue-shaped distribution in a property of sea water by advection and mixing (turbulence). zooo 3000 iWGO 5000 km 5000 hm Fig. 48a. Tongue form produced by turbulent mixing at constant flow velocity shown in a cross-section (tongue of i'-content for a steady current, which attains a constant ^-content in its total cross-section when it enters into a second water type.) case corresponds to the conditions present in the spreading of a current of water of high saHnity penetrating into a body of water of lower salinity. In the horizontal and vertical distribution of the temperature and saHnity over a large space in the ocean there are often found cases wheie the isolines have a tongue- form. This distribution allows the numerical determination of the relationship be- tween the exchange and the velocity of the flow, that is, of the quantity Ajpu provided that this is imposed by exchange processes. Such calculations are fairly numerous: they have been made, for instance, by Defant (1936) for the subantarctic intermediate current and for the Antarctic bottom current in the South Atlantic (AJpu =-- 1 — 10 which for u = 1-5 cm/sec gives A^ as about 0-5-10 gcm-\sec-^); by Montgomery (1939) for the equatorial counter current in the Atlantic (maximum value for A^ 0-4 g cm-Vsec-\ for Ay 4 x 10^ g cm-Vsec-^ by Sverdrup and Fleming (1941) for the coastal water off California at 200 and 400 m depths {AJp = 2x10*' cm^/sec) and by Seiwell for the distribution of temperature and salinity in the Caribbean Sea {AJp larger than 1C« cm^/sec). Recently Defant (1955) in an investigation of the ] 08 The Three-dimensional Temperature Distribution and its Variation in Time spreading of the Mediterranean water into the North Atlantic found a horizontal exchange coefficient of 5-5 x 10^ cm^/sec. There is no doubt that the exchange coefficients for lateral mixing A^ and Ay are about a million times larger than that for vertical mixing. The lateral mixing has thus, despite the low values of the horizontal gradients for the different properties of sea- water, at least the same importance as the vertical exchange. It can, however, be stressed that the nature and inner mechanism of these two exchange processes are different; the vertical mixing is small-scale, the lateral operates over a large space. It may be expected that they are related to different ranges in the total turbulence spectrum (see Chap. XIII, 3). The third special case is for mixing operating in all directions but without any dis- placement of water in a particular direction ; it shows therefore the effect of mixing alone unaffected by advection. In the two-dimensional case (.v- and r-directions) the solution takes the form (Sverdrup, 1940) s = Sq + m cosh [a{h — z)] cosh [ah] sin 27-v, whereby 4/2 For z = 0, that is at the surface of the sea, the distribution of a property s is s = Sq -{- m sm {ttI21)x. Selecting, for example, h = 4 km, Sq = 0, m = 5 and AJA^ = 6 x 10^, then a = 0-384 and Fig. 49 gives the distribution of s in an ocean of a horizontal extent Fig. 49. Distribution of a property "5" in the total ocean due to mixing alone (according to Sverdrup). 2/ = 20,000 km. In this case it would reach from pole to pole. The abscissa in Fig. 49 is therefore divided into meridional degrees from 90° N. to 90° S. For a per- sistent maximum accumulation of the property s at the surface of the sea in equa- torial regions, the effect of mixing alone would in the stationary state force a distribu- tion of j: in ocean space shown by curves of equal s in this representation. For a per- sistent temperature difference at the sea surface along a meridian, essentially the same as that produced by the combined effect of the solar and back-radiation, the effect of a mixing process acting alone ovei the entire ocean would give a vertical temperature distribution such as that shown by the isotherms in Fig. 49. The temperature decreases The Three-dimensional Temperature Distribution and its Variation in Time 109 everywhere with increasing depth, most rapidly at the equator and least at the poles. This case will be considered later in connection with the actual temperature distribu- tion in the deep ocean (see p. 123). Another solution for the third special case is ^• = ^11 m e' cos z with jS = V 11 ^ 2/ Obviously the solution of this distribution of 5 is identical with that of the first case on p. 106 if j8 is put equal to a, that is if This means that in a vertical cross-section a tongue-shaped distribution of s can be equally well regarded as the effect of a horizontal advection with velocity u in the direction of the tongue and as a vertical turbulence with an exchange coefficient A'^, or as the sole effect of pure mixing in horizontal and vertical directions without any advection. See Vol. I, Part II, Chap. XIII, 3, for a theoretical discussion of turbu- lent mixing in ocean currents. 3. Diurnal and Annual Variation of the Temperature in the Ocean The daily variations in temperature at the surface of large bodies of water (lakes and seas) are confined within narrow limits as was mentioned previously. In lakes, away from the shore, there may be diurnal variations exceeding 2 °C. They decrease rapidly with depth so that at 4-6 m they may be not more than 0-1 °C (see particu- larly the investigations by Homen (1913) in Lake Logo (Finland). Some idea of the diurnal temperature variation (of the air and the water) is afforded by the in- vestigation of Merz (1911) in the Gulf of Trieste (an enlcosed basin, relatively close to the land). The amplitude of the water temperature was 0-87 °C, for the air it was 3-l°C, which is considerably more. For a discussion of the diurnal and annual varia- tions of the surface temperature in a shallow water especially in the North Sea and in the Bahic, see Dietrich (1953). (a) The Diurnal Temperature Variation in the Open Sea The diurnal variations of temperature in the open sea are even smaller than in lakes; usually smaller than 0-4 °C and can rise at the most to about 1 °C in calm and fair weather. The most accurate measurements of the daily temperature variation in the open sea are obtained at anchor stations (fixed location). Four equatorial stations between 12-5° N. and 4° S. of the "Meteor" Expedition in the Atlantic Ocean (De- FANT, 1932) gave the following values (Table 43). Table 43. Mean daily temperature variation from four "'Meteor'' anchor stations Local Ampli- time 0 2 4 6 8 10 12 14 16 18 20 22 tude (hours) 1 dt (°C) -7 -10 -12* -10 -8 -1 + 11 + 19t + 15 +7 -1 -5 1 31 Minimum; j Maximum 1 1 10 The Three-dimensional Temperature Distribution and its Variation in Time Latitude 12i°N. 4° N.^° S. 8°-14° S. 211° S. Mean Diurnal variation At the surface (°C) At 50 m depth (°C) 019 0034 0-40 0056 0-23 <005 016 (009) 0-25 (004) The diurnal temperature variations always decrease towards higher latitudes; the maximum occurs at 14.00 h and the minimum at 04.00-05.00 h. The diurnal course corresponds almost exactly to a pure sine curve. KuHLBRorx (1938) obtained the same results from a study of the daily temperature records of the "Meteor" Expedition by the elimination of the effect of changes in position of the vessel. The average daily amplitude for all areas of the South Atlantic amounts to only 0-26 °C. This value, which was obtained by averaging all days without any selection, is somewhat smaller than the value obtained by Wegemann (1920) from the "Challenger" observations and by Meinardus (1929) from the "Gauss" observations. The heating of the sea surface begins soon after sunrise due to the absorption of solar radiation in the uppermost layer of the water, but the largest part of the added heat is used for the evaporation of water (about two-thirds) and only a small part remains for a temperature rise. The temperature thus rises only slowly to the maximum at 14.00 h. After sunset the temperature fall continues due to outgoing radiation. There are very few measurements of the depth to which the diurnal temperature variation penetrates. The only information for 50 m depth is given by the hourly observations at the anchor stations. However, for these depths near the thermocline the influence of tides through the associated vertical currents (internal tide waves) cannot be entirely excluded. Table 43 contains some values for the diurnal temperature variation at 50 m depth showing that for these depths the amplitude is less than 0-05 °C. Aime has made measurements of the diurnal temperature variation at different depths off the Algerian coast. It is, however, not entirely certain that all the observed changes can be attributed to the diurnal cycle; however, if this assumption is made it is found that the nocturnal cooling at 14 m is one-fifth of the surface amplitude and that the heating during the day, which is three to four times stronger than the nocturnal cooling, falls to a tenth at 28 m. Schmidt (1925) calculated from this decrease of temperature the vertical exchange coefficient as 35-40 g cm"^ sec~^. The observations of Knott on the "Pola" Expedition in the eastern Mediterranean show a decrease in the amplitude to a tenth at 29 m, which corresponds to an exchange coefficient of 42 g cm~^ sec~^. Since the ocean covers more than two-thirds of the surface of the Earth it can be said that over much the largest part of the surface the diurnal tem- perature variations remains less than half a degree. Therefore, the considerably greater diurnal temperature variations of the continents play only a minor part in the total heat budget for the Earth. {b) The Annual Temperature Variation Changes in temperature over longer periods can be investigated in two different ways. They can be recorded as "individual" temperature changes in a water mass which is followed in its course in the ocean; they are then described by reference to "oceanographic" co-ordinates. On the other hand, they can be followed at fixed The Three-dimensional Temperature Distribution and its Variation in Time 1 1 1 points and are then referred to "geographic" co-ordinates. These last changes are more complicated, since they are a combination of thermal changes within an indi- vidual water mass and of changes caused by the displacement of different water bodies (ocean currents). t For most parts of the ocean the annual displacements of the currents are known and most of the major annual changes in these areas can be ascribed to these. The seasonal displacements of the Gulf Stream system and the Labrador Current in the region of the Grand Banks of Newfoundland are well known. The large annual temperature variations in this region of the sea are associated with these displacements Similar conditions are found off the Norwegian coast where the seasonal displacements of the coastal current and the Atlantic current cause pronounced seasonal variations in temperature and salinity. For smaller areas the annual temperature variation at the sea surface can be derived only from a statistical evaluation of ship's observations and for the deeper layers from series observations made by oceanographic expeditions. Averaging the values that fall for different parts of the year into one, two or more degree squares gives mean tem- peratures for these subsections of the year with sufficient accuracy, provided there is a reasonable number of observations available. This of course gives only values related to "geographic" co-ordinates. Such a rough statistical method can only be used with some reliability for the sea surface. All the available data on surface temperature in the Atlantic Ocean have been collected and studied by Bohnecke (1936) and presented in a comprehensive form. For the Indian and Pacific Oceans a less complete presentation has been given by SCHOTT (1942). These show that there is an absolute minimum in the annual variation of surface temperature of all oceans in the tropics where over extended areas, especially in the Indian and Pacific Oceans, this variation is less than 1 °C. There is also a second- ary minimum in the Southern Hemisphere everywhere in the water encircling the Antarctic continent which also shows values less than 1 °C. In the Northern Hemi- sphere there is a decrease in the annual temperature variation in the Norwegian Sea and the variation becomes gradually smaller towards the north; this is true also for the North Pacific, but the northward decrease is slower. The maximum annual tempera- ture variation always occurs in the subtropical high-pressure belt where, near to the Bermudas and near the Azores, the maximum value is greater than 8°C. This region is connected with that showing the absolute maximum surface temperature variation t The individual change in temperature d?^ldt in a given unit mass is caused by the addition or abstraction of a given quantity of heat Q. This quantity Q is due to the absorption of radiation, to back-radiation, to thermal conductivity, to evaporation and to mixing and others. If the local distri- bution and that with time of these properties is given along the path followed by the unit mass of water then the "individual" variation in temperature d^ldt can be found. If the "local" temperature change d^jdt is required for a fixed point occupied successively by different masses of water, then for a given flow (velocity u) in the direction n the following equation is valid: db db db 1 dt ct dn Cp The advection term u(8bl8x), which includes the effects of the transport and the displacement of different masses of water at different temperatures in the direction n, thus plays an important role for the assessment of the local temperature change d^ldt. 1 1 2 The Three-dimensional Temperature Distribution and its Variation in Time (larger than 15 °C) off the coast of North America and in the region of the Newfound- land Banks. There, as already mentioned, the annual variation in temperature is caused by the fluctuating seasonal movements of ocean currents. Similar conditions occur in the North Pacific; the absolute maxima of more than 20 °C in the Yellow Sea, and in the Sea of Japan, are associated with a zone of maximum annual ampli- tude (greater than 9°C) extending from Japan eastward towards the east coast of North America. In the Southern Hemisphere the subtropical maxima of temperature variations are of a smaller extension. The annual temperature range is also large (8-10°C) in the areas of cold water upwelling (off" western Africa in the Northern and Southern Hemispheres and off California) in accordance with the seasonal variations in these phenomena. The geographical distribution of the annual temperature variation at the sea surface is not difficult to explain. In the tropics the small amplitude is due in the first place to the constant high altitude of the sun throughout the whole year and also to the relatively high cloudiness, so that there are only small annual variations in the in- coming radiation. In the subtropics the absorption of solar radiation has a much greater influence on the development of a marked annual temperature variation be- cause of the already larger seasonal changes in the zenith distance of the sun, and also because of the stronger effect of back-radiation due to the low cloudiness prevalent in these areas. With increasing latitude the incoming radiation becomes less effective and the autumn and winter convection, which is able to penetrate down to greater depths here, still further reduces the annual amplitude of the temperature variation until it reaches a minimum in the polar regions. Table 44 shows mean annual variations in temperature for equatorial, temperate and high latitudes, by the use of the mean temperatures for zones of 10° latitude given by Bohnecke (1938). In the equatorial zone there are two maxima at the time of the equinoxes. In the subtropics the maximum occurs in September and March, respectively, and in the extra-tropical regions in August and February, respectively. The minimum values in the first area occur in March and August respectively, and in the latter, in February and September, re- spectively. Table 44. Annual variation in the water temperature at the sea surface in the Atlantic (Deviation from annual mean 0-1 °C) i _ c a 0 Latitude Jan. Feb. Mar. Apr. May June July Aug. Sept. ! Oct. Nov. Dec. 50 '^-70" N. -16 -21* -13 -17 -11 +2 + 15 +26t + 13 + 6 -5 - 6 4-7° 10°-50° N. -23 -29 -30* -24 -12 +9 +28 +38t +34 + 19 0 -14 6-8° 20=N.-20°S. + 1-5 +2-5 + 7 +8-5t + 6 0 - 5 - 8* - 6 - 2 -0-5 + 1 1-7° 20°-50° S. + 19 +27t +23 + 15 + 4 -8 -15 -20 -23* -18 -8 + 7 50" 70^N.-60 S. - 1 - 1 0 - 3 - 5* -1 : 0 +4 + 2 + 2 -1 + 1 0-9° * Minimum; f Maximum In general the surface temperature minimum is retarded about two to three months after the sun reaches its lowest height; the maximum is retarded also by about the The Three-dimensional Temperature Distribution and its Variation in Time 1 1 3 same. A comparison of the oceanic and continental annual temperature variations is given in Table 45. Table 45. Annual temperature variations (°C) Latitude Equator 10° 20° 30 = 40° 50^ Oceans Continents 2-3 (1-3) 2-4 3-3 3-6 7-2 5-9 10-2 7-5 140 5-6 24-4* * Only Northern Hemisphere Figure 50 shows isopleths for the annual surface temperature variations in the At- lantic. It can be seen at once, that there is a narrow zone just north of the equator, where there is a six-monthly temperature variation so that over the whole of the tropics the amplitude of the annual temperature variation remains very small and that the middle latitudes between 30° and 50° show a maximum which decreases towards the pole, especially in the Southern Hemisphere. The annual temperature variation is transmitted to the deeper layers beneath the surface by the effect of convection and turbulence, with a corresponding reduction in amplitude and a retardation of the extremes, until it finally disappears. However, the annual displacements of water masses can also simulate an annual temperature variation, which is then not due to the total production and expenditure rates of heat at that point, but to others at more distant parts of the sea. Our present knowledge of these phenomena is still very poor. To obtain the exact annual temperature varia- tion at deeper layers it is necessary, because of the small number of observations Jan. Feb. Mor. April Moy June July Aug Sept. Oct. Nov. Dec. Joa Fig. 50. Isopleths of surface temperature in the Atlantic Ocean. 114 The Three-dimensional Temperature Distribution and its Variation in Time available, to eliminate aperiodic changes. This elimination is done by the "tempera- ture anomaly" method given by Helland-Hansen (1930). According to the mean [rS'l-diagram (see Chap. VI, 3) there is normally, for every value of the salinity in the water mass under consideration, a definite mean temperature {>. If an observation (^1, S) is obtained in this area, then the difference ?^i — i^ is termed the "temperature anomaly" of this observation. Experience shows that the temperature anomalies in a given set of data is of a considerably smaller scatter than the original values and that the aperiodic change of it has very largely been eliminated. The annual temperature variation in particular is shown much better than by the original values. By these methods Helland-Hansen has worked out the annual temperature variation for the water layer down to 200 m depth for three ocean areas in the North Atlantic. Figure 51 gives the results for the Bay of Biscay (area B) for the surface, as well as for I n m nz 2 M 211 vnr IX X XI xn '^- v\ B // \, / \ f \ / f \ ^ / / 1 l\ . ' / ,-- ~ V\ / ^ C'. --.r~- ,rf5< ■z^„-^ ' — ~~ ■ — 1 N- Fig. 51. Annual temperature variation in the water layer down to 100 m depth in the Bay of Biscay (area B) (according to Helland-Hansen). the depths of 25, 50 and 100 m. Table 46 presents the time of occurrence of the maxima and minima in this area and in the area between Portugal, Morocco and Madeira and also gives the amplitude at different depths. The amplitude at 25 m is still quite considerable and not very much smaller than the surface amplitude. However, lower down it decreases more rapidly and at 200 m the annual variation is more or less insignificant. The shape of the curve is almost the same in both areas and quite characteristic. In late autumn and in winter the surface water cools rapidly and the resulting convection also involves the deeper layers in this coohng. Thus, the ver- tical temperature gradient decreases continuously and becomes almost zero in spring. Heating now raises rather rapidly the temperature of the uppermost 25 m layer. Table 46. Annual temperature variation in the Bay of Biscay (B) and in the area between Portugal, Morocco and Madeira (C) AreaB AreaC Depth Min. Max. Variation Min. Max. Variation (m) (°C) (°C) 0 Jan. Aug. 7-7 Feb. Sept. 5-3 25 Feb. Aug.-Sept. 6-8 Feb. Sept. 4-7 50 Mar. Sept.-Oct. 2-4 Mar. Oct. 1-4 100 Mar. Dec. 0-7 Mar.-Apr. Nov. 0-9 200 — — 0-3 — — 0-25 The Three-dimensional Temperature Distribution and its Variation in Time 115 This effect may still be noted down to 50 or 100 m, but the heating process is inter- rupted in June and only reappears later at 50 m. The reason for this remarkable phenomena can be seen in the circumstance that turbulence due to the wind affects the greater depths in spring, while the water mass processes an indifferent or weakly stable stratification, so that surface heat can penetrate still to depths below 50 m. The rapid temperature rise of the surface layers soon builds up such a strong tempera- ture gradient thai turbulence is unable to prove a match for the created strong vertical stability of the water masses and the turbulent transport of heat therefore ceases. The upper layer is heated further by continued incoming radiation, and because of mixing becomes almost isothermal while the lower layers remain cold. The temperature in these layers rises again. Only when the density gradient is destroyed in the autumn can the effect of mixing and convection again extend to deeper layers. Only then can further heat be carried to the layer beneath the thermocline (Defant, 1936fl). In places where the ocean currents are subjected to considerable displacements, in both direction and strength during the year, the annual temperature variation can be considerably affected down to great depths by these current displacements. A typical example for this is the annual temperature variation in Monterey Bay, Cali- fornia (Skogsberg, 1936). Here there are three different periods in the annual variation: the period of the Davidson Current from the middle of November to the middle of February, when the temperature varies only slightly with depth down to almost 100 m; then follows a period of upwelling water from the middle of February to the end of July with low temperatures and stronger stratification ; while from the middle of July to the middle of November the Californian Current prevails and the temperature variation shows normal oceanic conditions. On the other hand, the temperature variation in the Kuroshio south of Japan (KoENUMA, 1939) shows almost exactly the same conditions as in the Bay of Biscay which was mentioned above. Fjeldstad (1933) has attempted to use the observations of Helland-Hansen in area B to calculate the eddy conductivity coefficient from the changes in the annual temperatuie variation with depth. He developed the annual temperature variation in the individual depths into harmonic series, and obtained in that way the values c„ and a„ as the amplitude and phase of the «th term of the series. Fjeldstad then showed that A na dz. p K (^««/^^) P" where a = l-n-jT, T is the annual period and h is the depth at which the ampUtude vanishes. A better representation of the observations can only be achieved by assum- ing a seasonal variation of the eddy conductivity coefficient. The mean value at the surface is 16gcm-isec-\ at 25 m it is 3gcm-^sec-^ and at 100 m the annual mean is only 3, in summer 0-5 and in winter 5-5. The same method has been applied by Sverdrup (1940) to values for the Kuroshio which in this case appears to be permissible, since the advective effects are outweighed by radiation and the eddy conductivity. In the Kuroshio area, where the strength of the current is large and the turbulence correspondingly high, the annual temperature 1 1 6 The Three-dimensional Temperature Distribution and its Variation in Time variation penetrates almost to 300 m depth with an eddy coefficient of about 70 g cm~^ sec"^ at the surface, 30 and 27 at 50 m, and 200 m, respectively. For a selection of 1 and 2 degree squares in the area between the Faroes and the Bay of Biscay, Neumann (1940, unpublished manuscript) using simple statistical methods has derived the annual temperature variation for 20, 40 and 100 m and has obtained rather similar results. The effect of stratification on heat transport in the deeper layers of a water mass also appears in lakes in the same way as described above and the annual temperature variation can be explained only under consideration of these processes. In shallow seas (Shelf seas) it is possible to study more accurately the penetration of heat and especially the eff'ect of turbulence generated by the wind, and also to investigate the eddy viscosity caused by strong tidal currents at the sea bottom. Recently Dietrich (1950, 1953, 1954) has given an instructive example of the various possibilities by the use of isopleths of temperature in vertical cross-sections in diff'erent shelf waters which illustrates the conditions present in the best possible way. Figure 5\a shows the annual variation of temperature and salinity from the sur- face to the bottom. 31-1 35-0 35-1 Fig. 5\a. Example of annual temperature and salinity variations from the surface down to the bottom (according to Dietrich), a, Irish Sea, North-Channel; b. Central North Sea; c, Baltic, Bomholm deep. {a) In the Irish Sea, north channel, with strong tidal currents where even in summer a thermocline cannot form and the strong turbulence evens out the annual temperature variation in the whole mass of water from the surface to the bottom (extremely strong heat transport from the surface downwards due to turbulence). {b) In the middle of the North Sea where the weak tidal current has little eff'ect. The formation of a thermocline prevents the development of a more pronounced annual temperature variation beneath it. (c) In the Bornholm deep in the Baltic, no noticeable tidal current and a strong increase in salinity with depth (strong density stratification during the whole year) so that the lower layer is isolated and shows no annual temperature variation. See p. 115 and Munk and Anderson (1948 on the theory of the thermocline). The annual heat budget of a limited water mass can also be calculated without diffi- culty if sufficient observational data are available. A number of calculations of this type have been made for lakes and similar calculations have been carried out for more or less enclosed seas. According to O. Pettersson ( 1 896) the Baltic gives off" 1 37-500 kg cal/m- from August to November and a further 385-500 up to March, in total about 523-000 The Three-dimensional Temperature Distribution and its Variation in Time 117 kg cal/m^. The annual heat budget for the Ionian Sea has been calculated by Hann (1906, 1908) as about 371,000 kg cal/m^; for the sea south of Cyprus he found 426,000, for the Bay of Naples 432,000 and for the Black Sea 482,000 kg cal/m^. In the polar regions the annual heat budget is much smaller. Malmgren (1927) has made corre- sponding calculations for the North Polar Basin; he estimated that the atmosphere received 68,000 kg cal/m^ from the sea annually. This mean annual value was obtained from the difference between the loss of 76,700 kg cal/m^ from September to April and a gain of 8,700 kgcal/m^ from June to August. See Deitrich (1950) for a dis- cussion of the annual variation of heat content in the English Channel. 4. The Vertical Distribution of Temperature in the Ocean Figure 52 shows the vertical temperature distribution for a series of oceanographic stations along a meridional cross-section through the middle and central parts of the Atlantic. The general and common characteristics of the vertical temperature distribu- Temperature, °C 0 4 8 12 Fig. 52. Vertical temperature distribution at a series of stations along a meridian in the Atlantic Ocean : 1. '•Will. Scoreby' 554 63° 20' S. 17° 23' W. 1 5143 m 5. ii. 1931 2. "Meteor" 58 48° 30' S. 30° O'W. 4989 m 7/8. X. 1925 3. "Meteor" 83 32° 9'S. 25° 4'W. 4506 m 29. xi. 1925 4. "Meteor" 170 22° 39' S. 27° 55'W. 5454 m 9. vii. 1926 5. "Meteor" 191 9° 7'S. 2° 2'W. 1 4533 m 9/10. ix. 26 6. "Meteor" 212 0° 36' N. 29° 12' W. 3773 m 19. X. 1926 7. "Meteor" 283 17°53'N. 39° 19' W. 5748 m 22/23. iii. 1927 8. "Dana" 1376 33°42'N. 36° 16' W. 1 10. vi. 1922 9. "Armauer Hansen" 17 58° O'N. 11° O'W. 1860 m 29. vii. 1913 10. "Fram" 29 78° I'N. 9 10' E. , 1075 m 22. vii. 1910 tion thereby stand out clearly. Conditions in the other oceans are also essentially the same. The typical curve for the vertical temperature distribution in the open ocean is ana- thermic, that is it shows a decrease of temperature with increasing depth, though this decrease is not uniform. In latitudes between about 45° S. and 45° N. the thermal 118 The Three-dimensional Temperature Distribution and its Variation in Time stratification of the sea is characterized by two principal layers. The upper layer ex- tends from the surface down to about 600-1000 m and is termed the oceanic tropo- sphere; its uppermost part down to about 100 m is subject to the direct influence of the atmosphere. This is the layer of diurnal and annual convections originating at the surface and it shows the strongest mixing due to the effects of the wind and waves ; it can be designated as the layer of surface disturbances. The troposphere shows the strongest temperature decrease with depth and in low and middle latitudes forms an upper warm layer of water overlying the cold water masses underneath and separated from them by a more or less sharply marked thermocline. Table 47. Mean vertical temperature (°C) distribution in the three oceans between 40° N. and A0° S,. Atlantic Ocean Indian Ocean Pacific Ocean Mean Depth (m) ^" 100 m ^° A^°l 100 m ^° ^^7 100 m ^° A^°l 100 m 0 200 22-2 21-8 21-3 100 17-8 2-2 18-9 3-3 18-7 3-1 18-5 2-8 200 13-4 4-4t 14-3 4-7t 14-3 4-4t 140 4-5t 400 9-9 1-8 110 1-6 9 0 2-6 100 20 600 70 1-5 8-7 1-2 6-4 1-2 7-4 1-3 800 5-6 0-7 6-9 0-9 5-1 0-65 5-9 0-75 1000 4-9 0-35 5-5 0-7 4-3 0-4 4-9 0-5 1200 4-5 0-20 4-7 0-4 3-5 0-4 4-2 0-35 1600 3-9 015 3-4 0-3 2-6 0-2 3-3 0-22 2000 3-4 012 2-8 015 2-15 01 2-8 012 3000 2-6 008 1-9 009 1-7 005 21 007 4000 ■ •8 008 1-6 003 1-45 003 1-6 005 t Maximum The lower part of the thermal stratification is the oceanic stratosphere which extends from the bottom of the troposphere (thermocline) down to the sea bottom; to it belong the major water masses of the deep sea which are characterized by the very small changes in temperature both in horizontal and vertical direction. Table 47 presents the mean vertical temperature distribution in the three oceans for latitudes between 40° N. and 40° S. and also the vertical temperature gradient at each depth in degrees per 100 m. The approximate limits between the zone of disturbance, troposphere and stratosphere are indicated in Fig. 53. This twofold subdivision in the thermal structure of the ocean is limited to the tropical and subtropical parts of the ocean. As is shown in Fig. 52 the troposphere becomes less well developed towards higher latitudes and the stratosphere comes closer to the sea's surface. In the subarctic and subantarctic regions (polewards of the oceanic polar front, see Chap. XIX) the troposphere disappears and the cold-water masses of the stratosphere extend generally to the surface. The water masses of the troposphere lie on top of and are embedded in the cold-water mass of the stratosphere in tropical and subtropical areas, but thin out and disappear in higher northern and southern latitudes. Because of the decrease of salinity with depth it can be expected, just for reasons of stability, that the temperature must also decrease with depth. Solar radiation is con- verted into heat in the upper layers and from here the heat spreads rapidly downwards. The Three-dimensional Temperature Distribution and its Variation in Time 119 0 2 0 2 4 6 8 10 12 14 16 18 20 22 0 2 4 6 8 10 12 14 16 18 20 22 Fig. 53. Mean vertical temperature distribution in the three oceans. The temperature distribution of the ocean must be regarded as quasi-stationary and this leads to the deduction that the vertical temperature distribution is a phenomenon closely connected with the oceanic circulation. Assuming that there was no motion in the very deep ocean this vertical temperature distribution could not be understood. Humboldt (1816) emphasized at an early date that the low temperature at great depths in the tropical ocean can only be explained by assuming an equatorward flux of cold-water masses originating in high latitudes. {a) The Oceanic Troposphere In general, the troposphere shows a well-developed subdivision into three parts. In the top layer the vertical differences in temperature and salinity are very small — so frequently that this top layer can be regarded as homogeneous. Its thickness is seldom greater than 100 m. In the Atlantic an isothermal surface layer (tempera- ture gradient <0-015°/m) is present only in the region between about 35° S. and 25° N. polewards from these limits the isothermal stratification is slowly destroyed and the effect of the seasons begins to predominate (disturbance zone). Table 48 presents mean Table 48. The quasi-isothermal top layer in the Atlantic Ocean Total no. of stations 6 3 3 6 6 3 Mean geographical position 24 °S. 16° W. 15° S. 15° W. 9°S. 17° W. 0°S. 22° W. 8°N. 23° W. 18° N. 36° W. Depth (m) 0 25 50 20-36 20-32 20-38 20-37 20-30 24-10 24-44 24-45 23-46 Temp 24-40 24-36 24-28 23-79 .(°C) 26-50 26-43 26-28 25-80 25-82 25-43 24-55 22-78 22-86 22-91 75 22-77 17-02 13-42 22-65 100 20-65 17-10 20-32 14-60 19-77 12-98 22-50 150 17-72 20-22 1 20 The Three-dimensional Temperature Distribution and its Variation in Time conditions at several stations in the central part of the Atlantic. In the subtropics (30°-20°S. and 20°-25°N.) the isothermal layer extends down to about 100 m, but is more shallow in the tropics and in regions close to the equator (in the west about 75 m and in mid-latitudes 50 m or less). Off the African coast, especially in the Gulf of Guinea, the thickness decreases to 25 m or less, and in the regions with cold water upwelling it is entirely absent. Underneath the top layer there is a strong tem- perature decrease that continues, gradually weakening, down to the lower limit of the troposphere. The maximum of vertical temperature gradient (thermocline) is generally found between 100 and 200 m, with a mean value of nearly 5°C per 100 m. The meridional variation of the depth of thermocline is shown in Table 49 (Fig. 54). Table 49. Meridional variation of the depth of thermocline in the Atlantic Ocean (Mean values for the entire ocean) Latitude Depth (m) 20° S. 141t 15° 121 10° 108 5°S. 77 0° 69* 2-5 °N. 83t 5° 81 10° 53* 15° 89 20° 160 25° N. 195* * Minimum; f Maximum 50 S r f' \ /] \ 100 - /j \ / ^ \ 150 / V - ^ 200 - 1 1 \ 20° S 10° 0° 10° 20° N Fig. 54. Meridional distribution of the depth of thermocline in the Atlantic. The thermocline rises steadily from a depth of 150 m in the subtropics to minimum values in the equatorial regions. Approaching the equator from the Southern Hemi- sphere a minimum of about 70 m is reached directly at the equator; however, coming from the north the minimum (about 55 m) already shows in 10° N. Between these two highest locations the thermocline drops about 1 5-20 m to a deeper level (approx. 80 m) at 2-5° N. These changes in level are rather characteristic for the entire width of the ocean and due to dynamical reasons are associated with the zonal oceanic circulation of the equatorial water masses (see Chap. XVIII and XIX). The intensity of the thermo- cline is greatest in the equatorial areas, where it has a mean value greater than 0-4 °C/m. An actual transition layer (temperature gradient >0-rC/m) properly speaking only occurs between 15° N. and 15° S.; on either side of this belt the gradient falls rapidly toO-05° C/m or lower and the transition layer shows only as an intensification of the vertical temperature gradient. The Three-dimensional Temperature Distribution and its Variation in Time 121 Table 50. Heat transport downwards assuming a temperature gradient of 1 °C/ 1 00 m Vertical exchange coefficient (/Ij g cm~^ sec"^) 20 10 5 2-5 1 Heat amount (g cal cm-2 day-i) 172 86 43 21-5 8-6 Beneath the thermocline from about 200-300 m the water masses of the sub- troposphere are remarkably constant in their nature and geographical distribution. The vertical temperature gradient in these waters rapidly decreases with depth and gradually changes its magnitude into that of the stratosphere. Considerable amounts of heat are transported by dynamic convection through the layer immediately be- neath the almost isothermal top layer to the layer below. Table 50 gives an idea of the quantities of heat involved; it assumes a mean temperature gradient of 1°C/100 m. These amounts of heat are surprisingly high. Even for small values of A^, the down- ward heat flux amounts to 10-40 gcal cm "May ""^. Since there is always a tem- perature gradient, this raises the very natural question of where all this heat goes to. In the lower layers of the troposphere the temperature gradient is again smaller and therefore the downward heat flux becomes smaller again in the middle layers of the troposphere ; the accumulation of heat in these layers should soon destroy the vertical temperature gradient and thus also the thermocline. It must therefore be true that the vertical temperature gradient in the troposphere can only be maintained if the lateral influx of colder water compensates the flow of heat from above and indeed the heat from above and the horizontal advection must compensate each other exactly. The vertical temperature distribution in the troposphere is thus maintained in a stationary state by the oceanic circulation (Defant, 1930). The cause for formation of the thermocline below an almost isothermal top layer in the tropics and the subtropics is therefore as follows: The top layer is certainly more or less in thermal equilibrium with the atmosphere above. The lower tempera- tures of the lower subtroposphere and of the stratosphere are essentially of polar origin; as they flow towards the equator these water masses mix with warmer water and thereby gain heat, but are continually renewed and are thus kept at a relatively low temperature. It would be expected that the diff"erence between the high temperature at the top and the low temperature of the deeper layers would give rise to a roughly linear vertical temperature gradient in the middle layer; instead a homogeneous top layer is formed and the transition to the lower temperatures of the subtroposphere takes place abruptly in a well-developed transition layer (thermocline). The explanation of this thermal stratification in the tropics and the subtropics lies in the same circumstances that give rise to the summer transition layer in lakes as well as in the ocean. The turbulence induced by the wind and the waves will slowly trans- port the heat from the upper layers downwards and the temperature diff"erences thus formed will work their way down into deeper and deeper layers. However, further rise in temperature in the top layers will also increase the vertical density gradient. The downward transfer of heat from above by turbulence will cease when the increase in vertical stability diminishes the intensity of the turbulence. If the vertical density gradient is very strong the turbulence of the flow cannot overcome the great stability of the stratification and a transfer of heat to a deeper level through the thermocline can no longer occur. In the top layer the turbulence leads finally to a complete 1 22 The Three-dimensional Temperature Distribution and its Variation in Time equalization of temperature thus forming an upper isothermal top layer. Beneath this, at a definite constant level, lies the thermocline, which acts as a barrier for all turbulent processes. The important point in the explanation of the formation of tropical and subtropical thermoclines is the exclusion of turbulence and their consequences in a fixed depth due to the increase of the vertical density gradient above a critical value. The condition for the reduction and final elimination of the turbulence in a non- laminar flow is that the dimensionless quantity (Richardson number): {glp){hpidz) Ab {dujdzY -^ At' In this relation u is the basic velocity of the turbulent flow, /Ib is the exchange coefficient for the flow momentum (apparent viscosity) and At is the exchange coefficient for density differences (temperature and salinity). In the ocean the ratio between these two quantities is between about 5 and 20 (see p. 103). In the thermocline of the equa- torial region of the Atlantic the quantity Spjdz is of the order of 5 X 10"* for a vertical interval of 20 m. In drift currents dujdz can be taken as about 10 cm/sec for every 20 m. The left-hand side of the above inequality is thus 100, which is considerably more than the value of the right-hand side. With such a stratification the turbulence in a current cannot be maintained (Defant, 1936), The basis of the theory for the formation of the thermocline has been given by MuNK and Anderson (1948). They have shown that the sharp transition between the top layer with mixing and the thermocline can be explained theoretically on the as- sumption that the eddy coefficients are a function of the vertical stability and of the wind shear. This theory gives a value for the depth of the thermocline that is some- what too sm.all but it is of the correct order of magnitude. This depth depends on the wind velocity, on the latitude, on the heat flux and on the [r^Sl-relation in that order. This theory undoubtedly penetrates deeply into the important processes that control this phenomenon but it does not yet completely satisfy all points. Experimental investigation and systematically planned observations would very probably improve the basis of the theory. (b) The Oceanic Stratosphere The vertical temperature differences in the very deep layer of the oceanic strato- sphere are small. Here also the distribution is almost everywhere anothermic; however, the temperature gradient at depths below 1000 m falls rapidly to values less than 0-4°C per 100 m, at 2000 m it is at the most O-TC and at 3000 m and below it is barely 0-05 °C/ 100 m. Departures from this anothermic distribution are found only in the Western Atlantic (Brazilian and Argentinian basins) and in the south-western Indian Ocean where at a depth of 1300-1600 m there is a very weakly marked tem- perature inversion, a phenomenon of particular importance for the oceanic circulation of these oceanic spaces. Table 51 shows particularly well-developed inversions at some "Meteor" stations. Inversions such as these occur only rarely in the eastern half of the South Atlantic and are very weak. They appear to be due to long-term changes asso- ciated with aperiodic variations in intensity of the deep-sea circulation (Merz, 1922; WiJST, 1936, 1948). The Three-dimensional Temperature Distribution and its Variation in Time 123 Table 51. Temperature inversions in the western Atlantic (°C) "Meteor" station Depth (m) 800 900 1000 1200 1400 1600 1800 2000 170: 22-6° S., 27-9° W. 158: 15-9° S., 300° W. 201: 9-5° S., 300° W. 4-55 403 4-125 3-91 3-70 3-91 3-53 3-605 3-79 305 3-79t 3-97 302 3-78 4-13t 3-465t 3-55 3-86 3-45 3-34 3-54 3-30 3-14 3-31 t Maximum Also in a horizontal direction the temperature differences in the stratosphere are small. The temperature distribution here must certainly be due to the stratospheric circulation which starts from the locations where the stratosphere extends up to the surface, that is in the polar and subpolar regions where it is in direct contact with the atmosphere. The water masses that sink in these places, where the major convection processes (see p. 97) originate, spread out very largely in a quasi-horizontal direction towards the equator to fill up the greater part of the space underneath the troposphere of the tropics and subtropics, and are thereby subjected to considerable lateral mixing at the same time. SvERDRUP (1938) has pointed out that the stratospheric temperature distribution can be mainly explained on the assumption that there is extensive lateral and vertical mixing of the water masses. This mixing takes place along the isopycnic surfaces that rise towards the surface in the polar and subpolar parts of the oceans. Figure 55 shows that the temperature distribution in a meridional cross-section through the Atlantic below 1000 m can be interpreted roughly as due to the effects of this lateral and vertical mixing; the theoretical isotherms calculated from the equation on p. 108 taking Ax : Ay as 6 x 10^ follow a similar course than the observed isotherms. The temperature distribution in the Atlantic asymmetric to the equator is partly due to the effects of an inflow of warm water from the Mediterranean and partly due to the strong cooling effect of the Antarctic. It cannot be doubted that mixing along the isopycnic surfaces in the oceanic stratosphere is of very considerable importance in the distribution of the oceanographic elements. (c) Adiabatic Temperature Changes and Potential Temperature Since sea-water is compressible, although only slightly, the pressure changes undergone by a small mass of water in the ocean must be accompanied by adia- batic changes in temperature which can be significant for oceanographic problems. Nansen (1900, 1902) first drew attention to the thermal effects of the compressibility of sea- water. If a mass of water is raised from a given depth to a shallower one, will be subjected to less pressure and will expand, performing work against the external pressure, and the water will be cooled by a definite amount. Analogous conditions will apply for a water mass which sinks ; its temperature will increase. Since the compressi- bility of water is not large these temperature changes will remain only in hmits ; how- ever, since the vertical temperature gradient in the deeper layers is extremely small, these adiabatic ejfects must be taken into account. The adiabatic temperature change Si^ for a displacement from a depth /z^ to a depth 1 24 The Three-dimensional Temperature Distribution and its Variation in Time 00091- OOOt^l 000 a 0009 J -) -9, a O u • — V =5 "3 .i2 i2 o "O ^.2 o o y t (\J c — o ILI o ■- c O r- o 2 y o ^ S 0 05 i^eq H \ C.2 y 3 ^ O X) 0 — O-C c O to o ~1 3 b < « (M •5 cu 'o 8 o o o CM ai O O o ro • 'mdSQ o o o o o 2 ?r 3 c ^ 2^ o ^ o 3 "" .-3 60 -J o 'J IT) w The Three-dimensional Temperature Distribution and its Variation in Time 125 h^ can be calculated using a formula derived from the energy principle by Lord Kelvin j v^'hich in c.g.s. -units take the form: ''" Ta*g 8& r dz, hi ^v' where T is the absolute temperature of the water, a* is its coefficient of thermal ex- pansion, Cp is the specific heat at constant pressure, g is the gravitational acceleration and J is the mechanical equivalent of heat (4-1863 x 10' erg/cal). The adiabatic temperature change hd for a displacement from a depth h^ to a depth //o is thus dependent on the coefficient of thermal expansion and on the specific heat of sea-water, which are both effectively dependent on the temperature, the salinity and the pressure (see p. 49). After solving the above equation, Ekman (1914) has presented numerical values which allow an easy determination of the adiabatic effects for sea-water. Helland- Hansen (1930) later prepared from these values tables giving directly the adiabatic heating and cooling in sea-water of o^ = 28-0 (corresponding to a salinity of 34'85%o) when raised from a given depth to the surface with a given temperature; a further table gives the adiabatic temperature change for the upper 100 m for salinities be- tween 30-0%o and 38-0%o. With these tables or the corresponding diagrams, any adia- batic change can be determined without difficulty. Table 52 is extracted from these tables. Example: at a depth of 9788 m (Philippine Trench) a temperature of 2-60° C was measured and a density o- = 28. What would be the temperature of the water for an adiabatic ascent to the surface? Table 52 gives, by interpolation, a T-change at 2-60° of — M37°C for 9000 m; for 10,000 m the change would be —1-319° and this for 9788 m —1 -280-0. If the water at 9788 m rises to the surface there will be an adia- batic temperature change from 2-60°C to 1-32°C. The temperature of a water mass after being moved adiabatically to the surface is known as the potential temperature. It is given hy d = d -\- 8§. If the vertical stratification of the sea were such that the salinity were constant, so that the density would only depend on the temperature, then the equilibrium state ofthe sea could be shown by the vertical distribution of the potential temperature in the same way as in the atmosphere. Complete mixing of the water masses in vertical direction would eliminate t This above equation can be derived without difficulty from the first and second laws of thermo- dynamics. If the state of a body is defined as a function of the temperature T and the pressure p, then T da dQ = c,dT-j^dp. Taking the definition of the coefficient of thermal expansion (see p. 48) as 1 da -^ = «* a ct and the static equation as dp = gp dz then for an adiabatic process (dQ = 0) with pa = \ and /& = JT 8 »= f dz or for the interval from h^ — h^ the above formula is derived. 1 26 The Three-dimensional Temperature Distribution and its Variation in Time Table 52. (A) Adiabatic cooling {in 0-01 °C) resulting from an ascent of a water particle of temperature d'm up to the sea surface (a = 28-0 ; S = 34-85%o) i>™(°C) Depth (m) -2 0 2 4 6 8 10 1000 2-6 4-4 6-2 7-8 9-5 no 12-4 2000 7-2 10-7 14-1 17-2 20-4 23-3 26-2 3000 13-6 18-7 23-6 28-2 32-7 37-1 41-2 4000 21-7 28-4 34-7 40-6 46-3 51-9 57-2 6000 42-8 52-2 61-1 69-4 — — — 8000 — 81-5 92-5 102-7 — — — 10,000 — 115-7 128-3 140-2 — — — (B) Adiabatic temperature change (in 0-01 **€) for the upper 1000 m at different salinities »(°C) S%o 0 4 8 12 16 20 30 3-5 7-0 10-3 13-2 16-1 18-9 32 3-9 7-3 10-6 13-5 16-4 19-1 34 4-3 7-7 10-9 13-8 16-6 19-3 36 4-7 8-1 11-2 14-1 16-9 19-6 38 5-1 8-4 11-6 14-4 17-2 19-8 all temperature differences except those due to adiabatic effects. The temperature of each depth would be fixed by purely adiabatic displacements of water from the surface or from the bottom to the given depth and in that way the vertical distribution of temperature would remain invariable. In this case a mass of water from the surface would be subjected neither to a force upwards nor to a force downwards, but would always be in equilibrium with its surroundings (indifferent equilibrium). In a vertical direction the potential temperature within it would be constant. The vertical distribu- tion of temperature in such a case for some initial values at the sea surface is shown in Table 53. In neutral equilibrium there is thus a slight increase of temperature with depth which does not reach a temperature of 1-5 °C in the 10 km depth. Table 53. Vertical temperature distribution of indifferent equilibrium Potential temperature Depths (km) CC) 0 1 2 3 4 5 7 9 0 5 10 0-00 5-00 10-00 0-045 5-087 10125 0-109 5-191 10-265 0-192 5-312 10-419 0-293 5-448 10-587 0-412 0-698 1-044 If the vertical temperature gradient is greater than the adiabatic, i.e. if the potential temperature calculated from the temperature in situ increases with increasing depth, The Three-dimensional Temperature Distribution and its Variation in Time 1 27 then the equilibrium state is unstable in the vertical. If a small water mass in such a thermal stratification is displaced downwards, it will remain colder than its surround- ings in spite of adiabatic heating, and it will be forced down further and further from its initial position. If it is displaced upwards then it will remain warmer than the surroundings and will therefore continue to rise. If, on the other hand, the vertical temperature gradient is less than the adiabatic, particularly if the temperature de- creases with depth, then the potential temperature will also decrease with depth and the stratification is stable. Table 54. Vertical distribution of potential temperature (°C) below 3000 m for several stations in the western and eastern troughs of the Atlantic Ocean Western trough ] Eastern trough North Depth Argentina Brazil Basin America Antarctic Cape Basin Cape Verde (m) Basin Basin Basin Basin Met. 56 Met. 249 Dana 1356 Met. 129 Met. 77 Met. 264 48-4° S., 50° S., 300° N., 58-9° S., 34-0° S., 10-2° N., 42-6° W. 26-4° W. 59-6° W. 4-9° E. 30° E. 26-6° W. 3000 + 1-40 +2-44 +2-66 -0-60 +207 +2-46 3500 +0-96 +2-28 +2-32 -0-73 + 1-73 +2-22 4000 +0-33 + 1-66 +2-03 -0-82 +0-94 +2-06 4500 -005 +0-48 + 1-85 -0-85 +0-68 + 1-92 5000 -0-27 (+0-25) + 1-66 -0-86 +0-60 + 1-84 5500 -0-27 — + 1-61 -0-88 — + 1-77 The vertical temperature distribution present in the ocean is such that the stratifica- tion, in so far as it depends on the temperature, is stable. In the oceanic troposphere the temperature decrease is so large that, in spite of the vertical decrease in salinity, the equilibrium state remains quite stable. In the upper layers of the stratosphere the stratification is still stable, however, it becomes continuously less stable with in- creasing depth. Table 54 shows the vertical distribution of the potential temperature below 3000 m for several stations in the eastern and western troughs of the Atlantic Ocean which show these conditions rather clearly. The same is usually also found in the open sea of the Indian and Pacific Oceans. At very great depths, below about 4500 m, especially in the more or less extended deep-sea basins, the vertical temperature distribution approaches the adiabatic and may even exceed it a little, so that there is an indiff'erent stratification at great depths or sometimes it may even be slightly unstable. It is principally in the deep- sea trenches of the Pacific and Indian Oceans that this occurs. In these there is nearly always temperature increase, but it seldom exceeds the adiabatic gradient and if it does then only by very little. Such a condition of indiff'erent stability is formed only when there is an almost complete separation of the water mass from the surrounding waters. More or less fully enclosed deep inland seas such as the individual basins of the European Mediterranean and the North Polar Basin show this phenomenon to a marked degree in their deeper parts. The classic example of conditions in a deep-sea trench is the vertical temperature distribution in the Philippine Trench (Schott, 1914; SCHULZ, 1917; Wust, 1937; van Riel, 1934; Schubert, 1931). According to 128 The Three-dimensional Temperature Distribution and its Variation in Time more recent calculations from the observations made in this trench there is no in- stability in the deepest layers as was previously supposed. Table 55 presents the data obtained in this case. Table 55. Vertical temperature distribution in the Philippine Trench: ""Will. Snellius" Exp., Stat. 262 (9° 40-5' N., 126° 50-5' E.) Depth (m) 1455 1970 2470 2970 3470 3970 4450 5450 6450 , 7450 8450 10035 Temperature (^C) 3-205 2-27 1-825 1-66 1-585* 1-595 1-64 1-78 1-925 2-075 2-23 2-475 Potential temp. (°C) 3 095 2-13 1-65 1-44 1-31 1-26 1-25 l-26t 1-25 1-24 1-22 116 Change over 500 m 1 (orc) _9.4 _4.8 -2-1 -1-3 -0-5 -01 +0-05 -005 -005 -005 -0-1 Salinity (%„) 34-58 0-605 0-64 0-66 0-67 0-67 0-67 0-67 0-67 0-685 0-695 0-67 * Minimum; j Maximum Between 3500 and 10,035 m with almost constant salinity there is an increase in temperature from 1-58 to 2-47°C, an increase of 0-89°C; this increase is, however, less than the adiabatic one ; the stratification is thus still stable but very close to the in- different equilibrium. At a level of 5500 m there exists a small anomaly because a thin layer of water, with a warmer potential temperature (1-26°C), is situated under- neath another layer with a colder potential temperature (1-25°C). The difference is, however, only small. The stratification here is thus very close to a vertically unstable state. However, if the salinity would decrease only a little more with depth the weakly stable temperature stratification could be changed by the salinity into an indifferent or even into a slightly unstable one. It was at first supposed that the almost adiabatic or slightly superadiabatic tempera- ture gradient, in the deep-sea trenches and the deep troughs of the major oceans, was due to a heat gain from the solid Earth. The heat transferred from the interior of the Earth to the lowermost water layer per second is Q = -2-1 X 10-«gcal/cm2 (see p. 88). This heat amount would accumulate in the layer very close to the sea bottom, until such a temperature gradient is formed that the incoming heat per unit time would equalize the heat transfer to the layers above. If the water mass were to be completely motionless, then according to the calculations of Schmidt (1925), the stationary temperature gradient would be determined by the heat entering the layer from the Earth and by the coefficient of thermal conductivity, so that in this case there would be a temperature decrease away from the bottom of 1-5°C in 10 m. dO 2-1 X 10-« , ^ ,^3 _, -, = y-. :r,r-„ == 1-5 X 10-=^ °C/cm. dz 1-4 X 10-^ Thus, in a deep-sea trench below 5000 m the temperature should rise linearly due to the heat transferred from the Earth to the water, and at the bottom (10,000 m) would be over 700°C. Since this does not occur it must be concluded that even the deepest The Three-dimensional Temperature Distribution and its Variation in Time 129 layers are not at complete rest, and that due to the exchange produced by turbulent motions in these layers the heat is more rapidly dissipated than by normal conductivity. An exchange of about 4 g cm-^ sec~^ would be sufficient to account for the observed slightly superadiabatic temperature gradient. This appears, however, not entirely conclusive. Even when the water masses in these more or less enclosed deep-sea troughs and trenches do not participate in the general horizontal circula- tion of the deep sea and can therefore be regarded as motionless in horizontal direc- tion, there may still occur vertical convection currents produced by the continuous influx of heat from the Earth which will carry this heat to the layers above. Such a convection will be effective if there is the smallest vertical instability. Once such instability exists in the bottom layer there will be a steady interchange of small water quanta rising and sinking, and this convectional circulation will be maintained by the steady inflow of heat through the sea bottom. In the water masses above an adia- batic temperature gradient will be established; a gradient greater than the adiabatic can, however, form only in the very bottom layer, though even here it will be scarcely possible to detect it by physical measurement. It is required here in order to maintain the vertical circulation against the internal viscosity. This might be the cause that the water masses of the deep-sea trenches and the deep basins in the ocean show a vertical stratification approximating closely indifferent equilibrium state. {d) The Vertical Temperature Distribution in Adjacent Seas While a steady decrease in temperature with increasing depth is characteristic for the open oceans, in adjacent seas connected with the open ocean over shallow sills the temperature below a certain depth is almost constant no matter how deep they may be. The adjacent seas can be divided into two groups according to their tempera- ture stratification : the first includes all those adjacent seas where the surface water in winter cools to a temperature which is lower than that of the open ocean at the greatest depth at which they are in communication (sill depths). Provided there is an almost homo-haline structure in these adjacent seas, the autumn and winter convection causes the cooled surface water to sink to the bottom, and the deeps in these adjacent seas are thus filled with water masses at approximately the lowest surface temperature occurring during the coldest month of the year. The deep layers in this show roughly the winter temperature of the region concerned, provided the convection is not pre- vented from reaching the greatest depths by irregularities in the thermo-haline struc- ture of the surface layers, for instance, by a layer of low salinity. Examples of this type of adjacent sea are the Red Sea and the European Mediter- ranean. In the first case, in the Straits of Bab-el-Mandeb (north of Perim island), the sill depth is 1 50 m; in the second, in the Straits of Gibraltar, about 350 m. In the Mediterranean during the summer there is a pronounced anothermal stratification in the upper layers, while depths below about 300-400 m are essentially homo- thermal. Towards the end of the winter this homo-thermal state extends upwards to the surface. The temperature of this deep layer is thus about 12-9-1 3-2 °C in the Balearic Basin and in the Tyrrhenian Basin, and about 1 3-6-1 3-9°C in the Ionian Basin and in the eastern basin near the Syrian coast. The northern Adriatic Sea shows values near to 12°C. These temperatures are all in good agreement with the winter temperatures in these regions (Table 56). 130 The Three-dimensional Temperature Distribution and its Variation in Time Table 56. Vertical distribution of temperature and salinity in the European Mediterranean Tyrrhenian Sea Ionian Sea "Dana" 4119 (30.V. 1930) "Thor" 144 (23.vii.1910) Depth (m) 40° 13' N., 12° 6' E., 3400 m 34°31'N., 18°40'E.; 3340 m rrc) 5 (%„) r(°C) S (%o) 0 20-0 37-72 26-05 38-49 25 17-36 37-80 22-50 38-13 50 14-79 1 38-00 17-28 38-26 100 13-81 38-30 15-40 38-35 150 14-09 38-50 14-66 38-64 200 14-12 38-60 14-41 38-78 400 14-13 38-69 13-96 38-77 600 13-76 38-61 13-76 38-72 1000 13-19 38-49 13-58 38-66 1500 13-06 38-46 13-55 38-64 2000 13-04 38-44 13-56 1 38-64 3000 13-21 38-41 Bottom Bottom (3200 m) 13-30 38-42 (3000 m) 13-69 38-64 During the autumn and winter tlie deep water forms at the surface and is carried by the convection to the deep basins. This is not influenced through the Straits of Gibraltar, since the bottom current through the straits carries water out from the Mediterranean and the influence of the upper current on salinity and temperature does not reach very far to the east. Conditions in the Red Sea are similar (see Table 57, Riel, 1932). Table 57. Vertical distribution of temperature and salinity in the Red Sea and in the Gulf of Aden. ''Will. Snellius" Exp., April 1929 Depth (m) Red Sea St. 18 15° 52' N., 44° 43' E. Straits of Bab-el-Mandeb St. 19 13° 27' N., 42° 51' E. Gulf of Aden St. 20 12° 55' N., 45° 48' E. r(°c) S(%o) T{°0 S (%o) r(°c) S{%o) 0 25 50 100 150 250 500 600 700 900 1000 26-70 26-10 25-99 22-51 21-94 21-66 21-59 21-58 21-60 21-63 21-66 37-07 37-11 37-42 40-27 40-46 40-57 40-61 40-57 40-60 40-60 40-60 Bottom 1030 m 27-60 27-41 27-28 24-53 36-12 36-33 36-36 38-57 28-80 125 m 22-80 39-98 Bottom 135 m 36-25 25-36 36-05 22-70 35-87 18-12 35-52 14-76 35-45 14-89 35-36 13-50 35-24 12-51 35-22 10-26 35-88 The Three-dimensional Temperature Distribution and its Variation in Time 131 Below about 300 m down to the greatest depths it is filled with a water mass at a temperature between 21-5°C and 21-6°C. The deep water has its origin at the surface in the northern half of this sea, where in March and April the v/ater temperature is 21-5°C combined with salinity values of 40-5-40-7%o increased by evaporation. The currents present definitely exclude any influence from conditions outside the open straits in the south. With this group can be included the temperature distribution in the deeper layers in the Norwegian Sea (from 1000 to 3500 m approximately homo-thermal, —0-8 to — 1-3°C and 34-9%o). Presumably this water mass must be formed at the surface to the north of Jan Mayen. The second group of adjacent seas belongs exclusively to the warmer zones, where the surface temperature during the whole year is so high that the temperature at the sill depth is the determining factor for the thermal structure of the sea below the sill depths. Only oceanic water has access in this case to the deeper layers below sill depth. The sinking of oceanic water into the enclosed space produces a potential temperature extending to the bottom, that is determined by the potential temperature of the open ocean at the level of the sill. This phenomenon is in many cases so marked that inversely the sill depth can often be deduced from the vertical temperature distri- bution in the adjacent sea, A characteristic example of this second group is the quasi-homo-thermal structure of the water masses in the Australian-Asiatic deep-sea basins beneath the depths of the sills over which they are connected with the Pacific Ocean or with the neighbouring basins. An accurate and detailed investigation of these conditions based on the ob- servations made by the "Willebrod SneUius" Expedition has been made by Riel (1934), Table 58. Table 58. Vertical distribution of temperature and salinity in the Australian-Asiatic Basins {""Will. Snellius" Exp.) Sulu Sea Celebes Sea Banda Sea 7°N., 120° E., 3°N., 121° E., 7°S., 128° E., Depth Sept. 1929 Sept. 1929 Apr. 1930 (m) TCO S(%o) r(°C) SCYoo) TCO S(%o) 0 27-8 33-46 28-4 34-22 28-4 33-48 50 27-75 33-59 27-33 34-33 27-07 34-20 100 24-26 34-32 24-41 34-68 21-42 34.52 150 18-66 34-40 20-44 34-81 17-46 34-60 200 15-25 34-48 17-26 34-70 13-71 34-56 400 11-50 34-50 8-99 34-42 8-83 34-57 600 10-53 34-47 6-90 34-52 6-62 34-55 800 1015 34-45 5-54 34-52 5-71 34-59 1000 10-08 34-46 4-49 34-55 4-70 34-59 1500 10-09 34-47 3-78 34-58 3-71 34-59 2000 10-14 34-47 3-61 34-57 3-24 34-61 3000 10-28 34-46 3-60 34-58 3-06 34-61 4000 — — 3-72 34-59 — — Bottom 10-42 34-45 3-77 34-59 310 34-61 Bottom 3950 m Bottom 4773 m Bottom 3308 m 132 The Three-dimensional Temperature Distribution and its Variation in Time The Sulu Sea between Borneo and the Philippines is connected in the north with the Pacific through a sill with a maximum depth of about 400 m. Below this depth the vertical temperature gradient becomes very small and down to the greatest depth at approximately 5580 m the temperature remains almost constant (minimum 10-07°C at 1225 m, rising to 10-42°C at the bottom). The deep basin of the Celebes (greatest depth 6220 m) has an almost constant temperature below 1400 m (sill depth at 1400 m in the Kawio Strait). The broad Banda Basin has a sill depth of 3130 m and in the northern part shows a temperature minimum of 3-04 °C at 2990 m, in the southern part 3-06 °C at 2720 m. Similar conditions are also present in the American Mediterranean. The main morphological structure consists of three major basins: the Gulf of Mexico, the Yucatan Basin with the Cayman Trench and the Caribbean Basin (Parr, 1932, 1937, 1938; see also, Dietrich, 1937, 1939). Table 59 shows the vertical distribution of temperature and salinity at three stations in the three major basins of this adjacent sea. Figure 56 shows several characteristic vertical temperature distributions for four adjacent seas. Table 59. Vertical distribution of temperature and salinity in the American Mediterranean Gulf of Mexico Cayman Trench Caribbean Sea "Mabel Taylor" 1104 "Atlantis" 1570 "Atlant is" 1509 Depth 25-8° N., 92-5° W., 19-3° N., 77-5° W., 140° N., 68-6° W., (m) 17 Apr. 1932 24 Apr. 1933 23 Mar. 1933 r(°C) s(7oo) r(°C) 5(%o) r(°c) 5(%o) 0 22-94 36-16 27-32 35-99 26-08 36-38 50 21-90 3612 27-07 36-02 26-01 36-28 100 19-30 36-31 2508 36-04 24-97 36-68 150 16-00 3618 22-86 36-66 21-86 36-80 200 13-405 35-70, 20-35 36-67 1815 36-35 400 7-99 34-96 15-25 36-06 10-90 35-25 600 5-77 34-87 10-61 35-31 7-71 34-82 800 4-94 34-93 704 34-94 610 34-75 1000 4-54 34-93 5-14 34-90 5-18 34-84 1500 4-16 34-97 4-26 34-97 4-20 34-96 2000 4-16 34-97 4-14 34-99 408 34-96 3000 4-23 34-966 4-09 34-99 4-13 34-96 4000 — 4-20 34-97 4-25 34-96 5000 — — 4-34 34-97 — — Bottom depth > 3000 m 5373 m 489 2m The question of the origin and renewal of the deep water in individual basins from different sides has been discussed on the basis of the modern oceanographic data collected by the "Atlantis" Expedition in the spring of 1933 and 1934 in the Carib- bean, and by the "Mabel Taylor" Expedition in 1932 in the Gulf of Mexico. There are only two passages through the Antilles that are important for the conditions in the deep layers of the American Mediterranean: the Windward Passage between Cuba and Haiti (sill depth at 1600 m), and the Anegada-Virgin Passage (sill depths at The Three-dimensional Temperature Distribution and its Variation in Time 133 14° 16° 18° 20°C 22° 24° 26°C 500 1000 1500 2000 2500 3000 3500 4000 ^J'^r— — I — — ■" vT ■^ 1 .--->>-- ; .'-' 1 / / - .^fe / - / \z - / j -/ i -;' 4 '3 I L _ I 1 ! _ ! 1 1 •- I ! fe X. i i : ! 1 \ 1 IT^I 1 1 1 1 ! 1 8° 10° 12° 14° 16° 18° 20° °C Fig. 56. Vertical distributions of temperature for four adjacent seas. 1780-1800 m and at 1600-1620 m) between the Virgin Islands and the northern Lesser Antilles. The Caribbean and Yucatan Basins show similar and almost constant values for the temperature and salinity below sill depth, and it is not easy using these values to determine the sources of the water in each basin. This was even more diffi- cult using the older observations. However, an unequivocal solution was reached only on the basis of the vertical oxygen distribution. Having the same potential temperature (Yucatan Basin 3-79-3-8rC, Caribbean Basin 3-81-3-83°C) the water in the Windward Passage contains more oxygen than that of the Anegada Passage. Since the mean oxygen content at 2500 m (ml/1.) in the Caribbean Basin is about 5-0, in the Yucatan Basin about 5-5-6-0 and in the Gulf of Mexico about 5-0, it follows that the renewal by transport through the Windward Passage and that in the Caribbean Sea is determined by that of the Anegada-Virgin Passage. The depth of the two sills can be deduced very reliably, as shown by Dietrich, from the potential temperatures. Earlier determinations based on the observed temperatures recorded in situ resulted in much too large a depth. The potential temperature along a cross-section through the Anegada-Virgin Passage is shown in Fig. 56a. The renewal of the deep water in the Gulf of Mexico is more simple to decide. Since the transport through the Florida Straits with a rather shallow sill depth of about 600 m is not likely to be of great influence, the renewal must come from the Yucatan Basin through the Yucatan Strait (sill depth 1 600 m). {e) Vertical Temperature Distribution in Adjacent Seas at Higher Latitudes and in the Polar Regions; Autumn and Winter Convection and Ice Formation The basic condition for the formation of a quasi-homo-thermal state in adjacent seas is the presence of an approximately constant sahnity at all depths below the sill 1 34 The Three-dimensional Temperature Distribution and its Variation in Time depth. If this condition is not satisfied the convection processes in the autumn and winter will not be able to extend to the bottom. The consequence of this limitation of the convection to a surface layer of greater or lesser thickness is a dichothermal temperature stratification during the warmer period of the year. There is a colder intermediate layer situated between a warmer upper and a warmer lower layer, which can be interpreted as the remainder of the convectional flux extending to this depth during the cold period of the year. 1000 100 Nautical miies Fig. 56a. Vertical distribution of the potential temperature beneath 1000 m over a vertical section through the North American Basin, the Anegada-Virgin Passage and in to the Caribbean Basin (according to Dietrich). Vertical enlargement by 1 : 1500. This cold intermediate layer is typical of the whole of the open Baltic Sea during the summer. The approximately homo-haline top layer heated by solar radiation extends down to about 30-50 m depth; underneath a depth of 50-80 m there is a core of relatively cold water with a temperature of 2-3°C, while still further down to the bottom the temperature gradually rises to 4-5 °C. This cold intermediate layer re- sults from cooling of the surface water during winter. The temperature distribution of the top layer during this time shows an almost isothermal state due to mixing by turbu- lence and convection, whereby at the same time the temperature at the surface may fall to near or sfightly below the freezing point (see Fig. 51^, c; p. 116). Similar conditions can be found in the Black Sea. For further detail see Skorzow and NiKiTiN (1927) and especially a monograph on conditions in the Black Sea by Neumann (1944). During the summer the water masses in the polar waters may also show a similar temperature distribution in the upper 100-150 m. Conditions in this layer at the end The Three-dimensional Temperature Distribution and its Variation in Time 135 of the winter can be represented by the curve shown in Fig. 57. The winter cooling reaches down to a depth hy,. The heating during spring and summer initially affects only the uppermost layer and penetrates very slowly downwards to lower layers. During the summer it reaches to a depth /?, and the vertical distribution can then be *- Temperature Fig. 57. Development of the vertical temperature distribution in the polar seas. represented by the broken curve. The formation of a cold intermediate zone is clearly shown ; it is tiot in a stationary state, but is gradually weakened by continuous heating from above and by mixing with the warmer water masses above and below, and may even disappear towards the end of summer to be reformed the following winter. Table 60. The cold intermediate layer in the polar waters Depth (ra) Barents Sea "Poseidon" 15 2 Aug. 1927; 214 m 75-2° N., 260° E. TCO S(%o) Cape Farewell "Utekor" 43 9 Aug. 1930; 173 m 59-6° N., 44-0° W. T{°C) 5(%o) Baffin Bay "Godthaab" 50 13 July 1928; 215 m 69-7° N., 57-4° W. rrc) siXo) Labrador Current "Marion" 1251 11 July 1931; >200m 54-6° N., 53-5" W. TCO 5(%o) 0 10 25 50 75 100 150 175 200 +2-49 + M9 000 -0-79 -0-79 -007 +0-26 +0-47 +0-56 30-30 3200 34-16 34-74 34-83 34-88 34-94 34-96 34-96 +0-49 +0-63 +0-98 -0-79 -0-81 + 1-12 +2-82 32-35 32-69 .32-90 33-08 33-31 33-71 34-14 +4-10 +3-60 +0-64 -1-60 -1-56 -0-91 +0-65 + 1-20 165 m +2-02 34-16 213 m +0-61 34-96 33-35 33-37 33-40 33-68 33-75 33-86 34-13 34-29 + 3-85 +0-01 -1-19 -0-72 -0-24 +0-51 + 1-36 32-26 3305 33-27 33-69 3400 34-21 34-47 1 36 The Three-dimensional Temperature Distribution and its Variation in Time The cold intermediate layer is particularly pronounced and lasts longest at the edge of pack ice and polar ice. Table 60 presents several examples. Figure 58 shows the temperature along a longitudinal cross-section through the northern Barents Sea 74°-77° N., ]9°-38° E.) along the pack ice Hmit in August 1927 according to series observations made by the "Poseideon" (Schulz and Wulf, 1929). From west to east exists a layer of increasing thickness of cold winter water at a depth between 20- 1 00 m, while above this there is a layer heated by solar radiation, partly also melt water. With distance from the ice limit this cold intermediate layer weakens and is gradually eliminated by mixing. This cold intermediate layer forms the core of the cold ice carrying currents around Greenland, in Baffin Bay and in the Labrador cur- rent (Defant, 1936). 200 240 St90 74°0'N I9°0'E St 15 St8283 St52 53 St50 49 75°I3'NI 76°I5'N 76°32'N 77°I6'N 26°0'E 30°0'E 33°30'E 38°0' E Fig. 58. Longitudinal temperature section in the northern Barents Sea 19°-38' E.) along the drift-ice limit (August 1927). (74°-77° N. The thermal structure of the Polar Sea in the layer beneath the top layer, in con- trast to the cold intermediate layer, is determined by the deep circulation of the polar water. In the European North Polar Basin between 250 m and 750 m underneath the cold top layer, a relatively warm intermediate layer of water of Atlantic origin is introduced with a temperature of about 0-5 °C (maximum of 2-0°C). Its salinity, 34-94-34-96%o, shows its Atlantic origin clearly. Underneath this layer spreads cold deep and bottom water that reaches its lowest temperature of — 0-83°C to — 0-87°C between 2000 m and 3000 m (WiJST, 1941, 1942). In high latitudes of the Southern Hemisphere there is generally a similar vertical temperature distribution in all the oceans as shown in Table 61. Some numerical values were given previously for the annual heat exchange in ad- jacent seas and in more or less enclosed parts of the ocean (see p. 116). The method used for this can also be applied, as mentioned on p. 98, to the special case of the The Three-dimensional Temperature Distribution and its Variation in Time 137 Table 61. Vertical distribution of temperature and salinity in high latitudes of the Southern Hemisphere Atlantic Ocean Indian Ocean Pacific Ocean "WiU. Scoresby" 554 "Gauss" "Discovery" II Depth (m) 5 Feb. 1931; 5143 m 26 March 1903; 13 Jan. 1931; 3098 m 3397 m 63-3° S., 17-4° W. 65-3° S., 80-5° E. 66-2° S., 71-8° W. T(°C) s (%„) r(°c) SiVoo) TCO s(7oo) 0 -0-20 33-96 -1-82 33-69 + 1-21 33-71 50 -1-75 34-46 -1-5 33-69 -1-54 3406 100 -1-80 34-42 -1-6 34-35 -0-90 34-25 150 -0-35 1 34-51 -1-6 34-33 +0-30 34-43 200 +0-22 34-60 -1-6 34-30 1-43 34-58 400 0-37t 34-63 +0-05 34-48 l-65t 34-70 600 0-37 34-68 1-05 34-61 1-53 34-72 800 0-29 34-68 0-90 34-62 1-42 34-72 1000 0-20 34-67 0-75 34-63 1-25 34-72 1500 000 34-65 0-15 34-60 0-88 34-72 2000 -014 34-66 0-0 34-58 0-62 34-71 3000 n-37 '\A-f\'\ ■ 0-38 34-70 4000 -0-46 34-64 3397 m -0-25 34-58 t Maximum heat exchange at single stations. For polar stations it affords some idea not only of the heat amounts involved in such a winter convection, but also of readiness for ice formation at the surface of the sea which finally occurs after the temperature has been reduced to the freezing point due to convection. These conditions can be illustrated by an example recorded by station 888 of the "Andrey Perwoswanny" ("Murman" Expedition) on 6 August 1903 at 71° 5' N. and 49° 0' E. in the south-eastern part of the Barents Sea (Breitfuss, 1906). Table 62 gives the oceanographic conditions down to a depth of 120 m, with mean values of the temperature and the density in each layer. Layer 1 is in direct contact with the atmosphere and is exposed to all the disturbances proceeding from it. Table 62. "Andrey Perwoswanny" St. 888; 6 Aug. 1903 (7M° N., 49-0° E.; 126 m) Thick- Depth r(°Q 5(%„) Layer ness TCC) S(%o) Specific (m) (m) (m) volume 0 2-84 33-96 _ 5 2-78 34-04 0-5 5 2-71 34-00 359 10 4-55 34-33 5-10 5 3-665 34-185 352 15 4-64 34-33 10-15 5 4-595 34-33 352 20 3-85 34-33 15-20 5 4-245 34-33 347 30 0-07 34-45 20-30 10 1-96 34-39 322 40 -M2 34-56 30-40 10 -0-525 34-5O5 300 50 -1-35 34-63 40-50 10 -1-235 34-595 291 75 -0-65 34-72 50-75 25 -1-00 34-675 285 100 -0-41 34-74 75-100 25 -0-53 34-73 282 120 -M3 34-81 100-120 20 -0-77 34-775 277 138 The Three-dimensional Temperature Distribution and its Variation in Time At the beginning of the winter convection the temperature in this layer falls while the salinity remains constant. When the specific volume of the first layer becomes the same as that of the second there will be complete mixing of the two layers by con- vection; the resultant layer will have the mean specific volume of the second layer, given in Table 62 as 352, while the salinity will be the mean of the original salinities, that is 3409%o. This specific volume and salinity correspond on the [r^l-diagram to Table 63. Heat available from convection and the readiness for ice formation at St. 888 "Andrey Perwoswanny". Thick- ness of Before nixing After mixing 25° 18 28 66 14 126 35 > 20° 41 38 97 16 191 53 Table 66 shows total areas with mean annual surface temperatures above 25 and 20°C; the warm parts of the oceans aie really of enormous horizontal extent. More than half of the entire ocean surface is warmer than 20 °C and of this 50% more than two-thirds has a mean annual temperature above 25 °C. The oceans over much of their surface are decidedly warm. The coldest parts of the ocean are at — 1-7°C (close to freezing point of salt water) in the North Polar Basin and in the circum- polar Antarctic waters. Referring to the general distribution of the isotherms at the sea surface the following points may be mentioned: The Three-dimensional Temperature Distribution and its Variation in Time 143 (1) The isotherms tend to be arranged zonally, especially in higher southern latitudes in all three oceans, where they almost parallel the latitude circules. This is due to the homogeneous climatic conditions over this almost exclusively oceanic area. (2) The major equatorial ocean currents to a large extent run from east to west. At east coasts of the continents they diverge and the isotherms do the same. The western sides of the oceans are thus appreciably warmer than the eastern sides. These differences are particularly pronounced in the Atlantic; here in temperate and higher latitudes this difference between east and west is actually reversed, and from about 35° N. the east is appreciably warmer than the west. However, this phenomenon does not occur in the Southern Hemisphere. Again, the major current system at the sea surface can be considered to be the cause of different behaviour of both hemi- spheres. The horizontal advection of water with a different temperature produces almost stationary contrasts in temperature between the eastern and western side of the ocean. In addition the distribution of land and sea and in some regions local oceanographic-meteorological phenomena, such as upwelling water, and piling up ("Anstau"), influence the temperature distribution. (3) There is another phenomenon apparent on the chart which is not clearly shown in the Southern Hemisphere because of the sparsity of the observations, although it has long been recognized in the Northern Hemisphere. This is the uneven, stepwise change in temperature towards higher latitudes. Already Fig. 50 (see p. 1 13)shows clearly this phenomenon, as it appears in the Atlantic. In both the Northern and the Southern Hemisphere there is an increase in the meridional temperature gradient in the zone between 40° and 50° which, during the year, is displaced towards and away from the poles following the movements of the sun. The concentration of the isotherms into a narrow belt between the Gulf Stream and the Labrador Current and between the Atlantic water and the Greenland Current is quite obvious. This boundary is called, in analogy with the atmospheric polar front, the "oceanic polar front" which indicates the position of the Arctic convergence where the two different types of water are brought into close contact. Its southern continuation along the east coast of North America has long been known as the "cold wall". This discontinuity appears in the chart of mean values because the aperiodic displacements of the ocean currents are confirmed within narrow limits. Accurate information about this sharp discon- tinuity has only been obtained from numerous thermographic recordings made by ship- ping across the whole system of currents off the east coast of North America (Church, 1937; Spillhaus, 1940). Figure 60 shows the most important of the results obtained by analysis of these recordings. The coastal water with a slowly increasing temperature eastwards borders the warm belt of water in the Gulf Stream which is barely 50 km wide. Towards the east the Gulf Stream is separated almost as sharply by a rapid fall of temperature from the water of the Sargasso Sea, where the temperature rises again slowly towards the east and south-east. The "band" character of the Gulf Stream does not show very clearly in the hori- zontal temperature charts, since the temperature is recorded at one or two degree squares which completely blurs this phenomenon, and the strong aperiodic dis- placements of the discontinuity along the right-hand side of the band (looking down- stream) contribute to this blurring when mean values are taken. The observations are also not strictly synoptic but are only obtained with differing time. 144 The Three-dimensional Temperature Distribution and its Variation in Time 32 24 u ° 20 OJ ^ 16 o i> Q. E _^^^- — ~J n Gulf She Co streonr f woter istol wc ' / Sargasso- Sec / 1 1 I 2° 0 200 400 600 800 1000 1200 Sea miles towards SE Fig. 60. Surface temperature distribution in the western North Atlantic (in the area of the Gulf Stream) from repeated temperature recordings made along shipping routes (according to Church). In the western part of the North Pacific there is also a similar phenomenon at the boundary between the warm Kuroshio and the cold Oyashio where arctic water and subtropical water come advectively in close contact. Due to the lack of data it was for a long time impossible to determine the position of this discontinuity in the circumpolar water in the Southern Hemisphere. Meinardus (1923) first showed its presence from observations made in the southern Indian Ocean. Its position in the Atlantic was deduced later from the current charts and it was recognized as the line of covergence between the oceanic west wind drift and the Ant- arctic water (Defant, 1928). It runs from about 48° W. to well out into the Indian Ocean (80° E.) between latitudes of 50° and 48° S. and then gradually turns south- wards to about 62° S. at Drake's Passage. (4) A second temperature discontinuity which is sometimes more sharply marked, though it can still only be detected on continuous recordings, lies where the sub- tropical water meets the subarctic water of the oceanic west wind drift {subtropical convergence). The frontal discontinuity in the region of the subtropical convergence shows large local meridional displacements and is therefore completely smoothed in mean temperature charts. Figure 61 shows two thermograph recordings given by Deacon (1938) that were taken on passing through the subtropical convergence and the Antarctic convergence {oceanic polar front). They show clearly the character of frontal discontinuity of this dynamically important phenomenon. (5) A useful aid in comparing temperature conditions in the oceans, especially in a zonal direction, are charts with lines of equal deviation from the normal value charac- teristic for each latitude. Such isoanomalic charts show which parts of the ocean are cold and which are warm relative to a normal latitude. In the Atlantic the heat surplus from the Gulf of Mexico across the North Atlantic to the Norwegian Sea as far as The Three-dimensional Temperature Distribution and its Variation in Time 145 Spitzbergen is particularly noticeable. This warm zone is associated with the Gulf Stream. There are negative anomalies showing the advection of polar water in the east Greenland Sea and the Labrador Sea down to Newfoundland. The Moroccan and the south-west African areas of upwelling water also show negative anomalies, and the eastern side of the Atlantic south of 35° N. is colder than the west side. A similar phenomenon also appears in the South Atlantic. The Pacific generally shows a similar subidi vision, with the western half decidedly warmer and the eastern half too cold. 12 IS 20 24 4 12 16 20 24" 4 20 24" 4. 8 12" 16 20 24 4 6 12 \ \ v \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\\\ rl^^fc^pff^^^ri^^^ Fig. 61. Thermograph recordings made passing through the subtropical and antarctic con- vergences (according to Deacon). {b) Horizontal Temperature Distribution at Different Depths and Vertical Temperature Sections The horizontal temperature distribution remains similar to that at the surface down to a depth of at the most 50-75 m and then changes rapidly. It was already shown in Table 48 that the thickness of the top layer {disturbance layer) is least in the equa- torial areas and this is where the cold water masses of the subtroposphere come closest to the surface. It is thus to be expected that horizontal temperature charts, even for shallow depths, will show a band of cold water embedded between the warm- water masses of the subtropics which becomes greater in width with increasing depths. This can be seen on horizontal temperature charts at 200 m intervals both for the Atlantic and also on charts for the other oceans. The subtropical warm-water areas of both hemispheres are thus separated by a cooler equatorial zone almost 30° wide and are limited polewards by two cold-water areas in higher latitudes. In the layers between 400 and 800 m the highest temperatures are always found on the western side of the oceans, particularly in the Atlantic. This is a dynamic consequence of the stationary distribution of the currents at these depths. The chart for a depth of 800 m shows already the asymmetry typical for the tem- perature distribution in the deep layer of the Atlantic, which is due to the cold sub- antarctic intermediate current in the south and to the influence of the Gulf Stream and the inflow of Mediterranean warm water in the north. This asymmetry domi- nates the temperature distribution down to depths of more than 3000 m. The influence 1 46 The Three-dimensional Temperature Distribution and its Variation in Time The Three-dimensional Temperature Distribution and its Variation in Time 147 ji 3 O Ilk _>i . 60 en >^ '5 "o c "-^ O <4_ w o .3 •a 3 a o 148 The Three-dimensional Temperature Distribution and its Variation in Time of the Gulf Stream extends down to about 3000 m and that of the Mediterranean water down to 2000 m, so that there is always a considerable heat surplus even at these great depths in the North Atlantic. Below 3000 m the cold Antarctic bottom water first appears and at deeper levels spreads northward with slowly increasing temperature. There are insufficient systematic data available for the Indian and Pacific Oceans below 2000 m to allow any reasonably accurate description of the horizontal temperature distribution in the deeper layers. The importance of horizontal charts of the distribution of temperature and other oceanographic factors, as a geographic aid to the comprehension of the distribution of these factors throughout the ocean, has in the past been somewhat overestimated. Oceanic processes never, or very rarely, occur along horizontal planes or are quasi- horizontally arranged. Because the three-dimensional field of oceanic elements is arbitrarily intersected by horizontal planes, connected phenomena will therefore be cut by such planes. They are thus, for example, quite insufficient for following water movements in the depths of the oceans. The same is equally true for the study of atmospheric phenomena. Before these were deduced in other ways it was difficult to interpret the arrangement of the isotherms in horizontal sections. In all cases vertical cross-sections must also be used to clarify the three-dimensional field of any oceano- graphic element. Vertical temperature sections can be taken in any direction and thus can give a far better idea of the thermal stratification of a water mass than a horizontal chart. It is, of course, best and most convenient to take the vertical section either along the axis of major spreading of the water mass in the ocean concerned or across it. At the present time there are several such longitudinal or transverse vertical sections (relative to the direction of flow) for all three oceans, showing temperature, salinity and in part also the oxygen content. Those for the Indian Ocean (Moller, 1929; Clowes and Deacon, 1935) and for the Pacific Ocean (Wust, 1929; Sverdrup, 1942, 1945; ScHOTT, 1942) are less accurate because of the smaller number of stations than those for the Atlantic Ocean (WiJST, Defant, 1936). It is neither possible nor appro- priate to describe and interpret these vertical sections individually. An interpretation can only suitably be given in conjunction with the phenomena of the oceanic circula- tion in the deeper layers. Figure 62 shows, as an example, a longitudinal section along the western side of the Atlantic giving temperatures and salinities (after WiJST, 1928). This runs from 75° S. near the area of formation of the Antarctic bottom water, through the Weddell Sea and the South Antilles Sea, along the western side of the West Atlantic Trough to the Newfoundland Banks through the Labrador Basin to the Davis Ridge. There is a vertical distortion of the section by a factor 1 : 1300. This section is quite typical of all sections through the Atlantic Ocean and shows the im- portant characteristics of the meridional vertical temperature distribution: the two large warm-water accumulations in the subtropical troposphere of both hemispheres, the approach of the cold-water mass in the equatorial subtroposphere towards the surface, the concentration of the isotherms at the polar limits of the troposphere between 40° and 50° S. and 45°-55° N., and the oceanic polar fronts. This western section also shows at about 1000 m an intrusion of colder water from 55° S. towards the north as a tongue-shaped bulge on the isotherms which is visible even across the equator. In a central section this is only weakly developed, in an eastern section it is The Three-dimensional Temperature Distribution and its Variation in Time 149 not visible at all. It is caused by the intrusion of subantarctic intermediate water and represents the same phenomenon as the isothermal layer or actual inversion in the vertical distribution which was mentioned previously (see p. 123). South of 55° S. the oceanic space all around the Antarctic is filled down to the greatest depths with cold Antarctic water. The isotherms here steeply descend from the surface to 2500- 3000 m, clearly showing the extension of this cold-water type northward along the deep basins that open to the south. This, like all other longitudinal sections, shows the considerable asymmetry in the temperature distribution of the oceans. As previously mentioned this asymmetry is caused by topographic conditions of the Atlantic, which allow only a spreading of the cold heavy Antarctic bottom water towards the north. This is, of course, also the case in the Indian Ocean but not entirely so in the Pacific where, although only to a small extent, there is an Arctic component from the Okhotsk Sea to be taken into account. The meridional temperature contrast between high-southern and high- northern latitudes, which is especially well shown in the Atlantic and can also be seen in the Pacific Ocean, is the main cause of the deep-sea circulation of these oceans and also gives rise to their asymmetry relative to the equator. (c) Bottom Temperatures in the Three Oceans The question of the origin and the spreading of the lowermost layer of bottom water in the oceans was raised at a very early stage in the development of oceano- graphy— much earlier than the problems dealing with the oceanic circulation of the middle layers. This was due to the existence of a greater amount of data for the bottom layer than for the middle and deep layers, since bottom temperatures were measured from cable-laying ships as well as from research vessels. The low tempera- tures found in the bottom layers clearly indicated at an early stage a polar origin of the bottom water and formed the main basis for the assumption of a deep-sea circula- tion. An historical account of the exploration of the nature of the bottom water has been made by WiJST (1936), who has also given a description and comparison of the movements of the bottom water spreading out into the three oceans based on a critical inspection of all the available data (Wust, 1938). Plate 4 gives a chart of bottom temperatures on the deep-sea basins. The course of the isotherms is much more cer- tain in the Atlantic than in the other incompletely explored oceans. The temperatures given are potential temperatures in order to give a clear picture of the spreading of bottom water influenced by the relatively large irregularities of the bottom topography. Table 67 gives mean values for 10° latitude zones in the three oceans and for the total ocean. In general, there is a continuous rise in the bottom temperature to be seen from high southern latitudes across the equator as far as to temperate northern latitudes. The maximum temperature that can be taken as the boundary between Arctic and Antarctic influences at the bottom is situated rather asymmetrically at 40° N. in the Atlantic and at 30° N. in the Pacific. In almost all latitudes the coldest bottom water is found in the Indian Ocean. The coldest water is in the deepest depressions in the Atlantic South Polar Basin; the cold pole with — 0-92°C lies at the western edge of the Weddell Sea, where according to Brennecke (1921) and Deacon (1937) that thermo-haline stratification in the autumn and early winter exists, which per- mits the ice-cold shelf water to sink by convection along the continental slope down 1 50 The Three-dimensional Temperature Distribution and its Variation in Time to the ocean bottom. From here this cold heavy water spreads out in general towards the east within the Antarctic circumpolar Ocean to form the source of the meridional northward outflow along the deep-sea troughs of the Ocean. It is still uncertain whether there are other regions of bottom-water formation in the Antarctic, but that in the Weddell Sea is in any case the most important and the most intense one. In each ocean the Antarctic bottom water spreads out both in zonal and meridional direction according to the bottom topography. There are seven cold streams of bottom water spreading out along the seven major longitudinal troughs of the oceans towards the north. These are listed in Table 68. Table 67. Mean zonal distribution of bottom potential temperature (°C) in the deep sea (> 4000 m); mean for each latitude circle, (After WiJST 1938) Latitude Atlantic Indian Pacific All oceans Ocean Ocean Ocean S. 70" -0-71 -015* -0-43* 60 -0-87* "0-54* 006 -0-42 50 -0-33 0-25 0-49 012 40 017 0-36 0-67 0-44 30 100 0-53 0-84 0-76 20 104 0-61 103 0-90 S. 10" 119 0-86 103 101 0 1-32 0-93 106 107 N. 10 1-66 M6t 108 1-20 20 1-89 — . 108 1-32 30 1-83 — llOf l-33t 40 l-95t — 100 1-32 N. 50° 1-81 — 106 1-22 Strongest meridional difference 2-82 1-70 125 1-76 * Minimum; f Maximum Table 68. Initial temperatures and northward extent of the cold Antarctic bottom water Initial temp. (X) at 55° S Northern extent of cold water Deep-sea Trough tongue (potential temp. 1 •0°C) To lat. To cross ridge 1 . West Atlantic -0-8 8°N. Para Rise 2. East Atlantic -0-7 22° S. Whalefish Ridge 3. West Indian -0-6 10° N. Carlsberg Ridge 4. East Indian -0-3 5°N. — 5. Western Pacific (Tasman Basin) 0-2 24' S. Coral Rise 6. Central Pacific 0-4 25° N. Hawaii Rise (?) 7. Eastern Pacific (South Polar Basin) 00 37 S. Eastern Rise (?) The Three-dimensional Temperature Distribution and its Variation in Time 151 The distance to which each of these streams extend in each meridionally-oriented trough is very largely dependent : (1) on the morphological form of the trough, on whether there are deep passages through cross-ridges or whether the stream can flow over any rises, and (2) on the kind of water mass spreading above the cold bottom water towards the equator. It combines and interchanges with this and shows much stronger conserva- tism in its character of Antarctic water the lesser the influence of the water above. The most extended is the central Pacific cold stream which, due to the favourable topography and partly also because of the absence of deep warm currents in the North Pacific, reaches as far as 25° N. Also, in the Indian Ocean, the cold-water currents on both sides of the central ridge extend almost to the northern limit of the ocean. The most impressive one of these streams is, however, the west Atlantic cold water spreading where the Antarctic water penetrates through gaps from deep-sea basin to deep-sea basin as far as the Para Rise at 8° N., and finally warms up by mixing with the relatively warm North Atlantic deep water and flows into the North American Basin. In the East Atlantic Trough the Whalefish Ridge completely prevents further extension north and there is therefore a large difference in the temperature of the bot- tom water on the north and south sides of this cross-barrier. The bottom layers of the Atlantic Eastern Trough north of the Whalefish Ridge are formed by colder West Atlantic bottom water flowing in through deep gaps in the central parts of the Middle Atlantic Ridge at 0° latitude (Romanche Deep) and at 10° N. There are cross-rises also in the eastern and western deep-sea troughs of the Pacific that prevent the north- ward extension of Antarctic water beyond 22° and 37° S., respectively. There is very little bottom water of Arctic origin. The most productive source is probably the outflow from the Okhotsk Sea which extends southwards as a cold stream, with an initial temperature of less than 0-6 °C about 15° N. In the Atlantic deep-sea troughs there are indications of bottom water at less than 1-8° between 53° and 45° N. which is probably of subarctic origin. A detailed investigation of the horizontal spreading of the Antarctic bottom water in the Atlantic has been made by WiJST (1936). Figure 63 shows the potential tempera- ture along a quasi-meridional section through the Western and Eastern Troughs below 3000 m. In the western section the bottom water is separated from the water mass above by a marked discontinuity in the vertical temperature (and salinity) distribution. It descends from south to north with a gradient of about 20 m in 100 km and follows the bottom topography closely. Such influences on the temperature (and salinity) are recognized as far north as 40° N., 16,500 km away from the origin of the stream at the rim of the South Polar Basin. The eastern quasi-meridional section is rather different. The barrier due to the Whalefish Ridge shows even more prominently here and the eff"ect of the local inflow of Antarctic- West Atlantic water through the Romanche Trench is also clearly visible. From here and from the saddle at about 10° N. the bottom water spreads north and south in the eastern Atlantic Basin. The increase in temperature and salinity along the core of spreading of the relatively shallow bottom water is due to mixing processes with the warmer North Atlantic deep water above, comparatively of larger vertical extent. The distribution of temperature and salinity in the bottom water can be re- garded as stationary and this can only happen when advection and mixing are in 1 52 The Three-dimensional Temperature Distribution and its Variation in Time l-K- \:M. .i C « (3 I g § s ^ Xi C bO Si o B ■" ^W o :3 W) ^ .S .2 3 2 o » § >^ ^o 3 S CM ^ O .2 O S o 13 o C XI ' ■5 '-> 3 '-n ■>-• o o c ; J « o £ o X MH 60 i: o c ^ *- t? w •^ I ^ '" 6--3 SsS LU 'mdSQ 'Mtdea The Three-dimensional Temperature Distribution and its Variation in Time 153 balance. From the distribution of these factors the ratio of the vertical exchange ^2 to the velocity u of the spreading can be calculated (Defant, 1936). The value of A ^lu is between 2 and 3 over the transverse rises and between 5 and 6 in the troughs, with a maximum value of 10. Because this ratio as a first approximation is proportional to the Prandtl mixing length (see Chap. XII I) and this length is more suited for the charac- terization of a turbulent flow than A 2 the above result therefore means that the mixing length is greater in the troughs than over the rises. In the core of this flow for a narrowing of the gap and corresponding increase in the velocity the mixing is somewhat reduced (more laminar flow), while in basins, on the other hand, the contrary occurs (velocity-decrease, stronger mixing). 6. Mean Vertically Integrated Temperature for Individual Oceans in Zonal Rings Calculations of mean temperatures of parts of the sea, or of particular zones of latitude or for the total ocean, are of course only of statistical value. Krummel (1907) determined the values of some of these mean temperatures on the basis of the hori- zontal charts then available; Table 69. The mean temperature of the total ocean of 3-8 °C appears very low especially compared with the surface value of 17-4°C. The decisive factor is the very large water masses of the oceanic stratosphere and the com- paratively shallow oceanic troposphere. The mean values for 10° latitude zones show again the marked decrease of about 5°C between the equator and higher latitudes, but the differences between 40° N, and 30° S. remain, in general, small. This is also true for differences in the values for the three oceans. Table 69. Mean vertical integrated temperatures °C for different oceans and the total ocean (According to Krummel 1907) Northern Hemisphere Southern Hemispheie Zone of latitude Atlantic Indian Pacific All Atlantic Indian Pacific All Ocean Ocean Ocean oceans Ocean Ocean Ocean oceans 0-10° 50 5-8 4-5 4.9 4.4 5-2 4-6 4-7 10-20° 5-1 7-4 41 4-8 4-2 4-8 4-7 4-7 20-30° 5-8 10-3 3-8 4-7 4-7 4-8 4-5 4-6 30-40° 61 — • 31 4-5 3-7 4-2 4-1 40 40-50° 51 — 2-4 3-2 2-1 2-6 30 2-8 50-60° 3-8 — 2-3 •2-8 0-6 0-8 1-4 10 60-70° 4.4 — — 2-2 -0-2 -0-2 0-4 00 70-80° — — — (-0-6) -0-2 -0-2 0-3 01 80-90° — — — (-0-9) — — — — 0-90° (resp. 80°) 5-3s 6-5, 3-6e 4-3, 7.0 3-4, 3-72 3-4, 90° N.- 80° S. 40, 3-82 3-73 — — — — — On the whole, the mean temperature of 3-8 °C for the entire ocean makes a rather cold environment for the living organisms in it, however, they are mainly concen- trated in the upper warmer layers. Chapter IV The Salinity of the Ocean, its Variation in Oceanic Space and in Time 1. Periodic and Aperiodic Variations of Salinity * If tidal effects are disregarded the most obvious periodic changes in salinity to be taken into account are the diurnal and annual variations. There is little data on daily variations. The diurnal variation of evaporation must give rise to a similar change in the salinity but it can have only little signification. Apart from the small diurnal variation in evaporation, the variations in salinity will be further diminished by the vertical convection set up immediately in the homogeneous top layer by increased salinity at the surface. The effect of an increase in salinity by a high evaporation rate will thus spread very rapidly over a large water mass and will scarcely be detectable. The true salinity variation uninfluenced by other factors can only be shown by ob- servations made at an oceanographic anchor station, and in this case also all stations that showed any appreciable vertical salinity gradient should be left out of account. At such stations the vertical displacements of water by the tides cause variations in salinity with a tidal period which are usually several times greater than the normal diurnal variations. A small diurnal variation can only be clearly shown in an almost completely homo-haline top layer. Five "Meteor" anchor stations between 21° S. and 4° N. gave the mean diurnal variation shown in Table 70. The second column of the table shows the diurnal salinity variation as hourly values taken over three days at the "Altair" anchor station (44-5° N., 34° W.); see Fig. 64. The range is very small and amounts to less than half of 1/100 part %o; there is a broad flat minimum during night time until sunrise after which the salinity rises, slowly at first and then rapidly, to a pronounced maximum in the late afternoon and falls off just as rapidly to the night values. Physically the process can be regarded as the effect of a positive transient source of salt at the surface, the surface amplitude of the effect being somewhat modified by vertical exchange with the layers under- neath. The variation proceeds so regularly that despite its small amplitude it deserves more attention than it has hitherto received. Visser (1928) deduced a value for the mean diurnal variation of the surface salinity by analysis of the observations of the "William Snellius" Expedition; this is similar to that found in the Atlantic: minimum at 04.00 h, maximum at about 17.00 h; but the amplitude was almost twice as large probably due to climatic conditions in the area. Knowledge of the annual salinity variation is also rather meagre. Bohnecke (1936) has prepared charts showing surface salinities for each month in the North Atlantic and seasonal means of salinity for the total Atlantic which allow the annual 154 Salinity of the Ocean, its Variation in Oceanic Space and in Time 155 salinity variations to be found; these are supplemented by a chart showing mean annual amplitudes. Over the major part of the open ocean surface away from coastal areas the annual range in salinity in middle latitudes is less than 0-5%o, usually less than 0-25%o. A zone with more than 0-5%o and a core with more than \%o and oc- casionally over l-5%o extends right across the Atlantic from South America to Africa between 5° and 15° N. and includes the area of the equatorial counter current. There is a further region with values greater than 0-5%o and several cores about l%o in the Gulf Stream region until the south-east of the Newfoundland banks. Otherv^dse the maxima of annual variation are found in coastal areas especially off the mouths of the larger rivers (Amazon, La Plata, and the inner part of the Gulf of Guinea) with large seasonal variations in fresh-water inflow or in polar areas with seasonal melting of the Table 70. Diurnal salinity variation Five "Meteor" "Altair" anchor- Time stations station (hours) 2-l = S.-4°N. 44-5"N.-34-0=W.; 3 days 1 35-468 35-800* 3 35-466 35-866 5 35-464* 35-887 7 35-464 35-876 9 35-466 35-882 11 35-470 35-889 13 35-480 35-885 15 35-490 35-893 17 35-540t 35-913t 19 35-486 35-900 21 35-474 35-883 23 35-466 35-879 Range 0-042 0-040 Minimum; f Maximum Fig. 64. Diurnal salinity variation. Above: the mean of five "Meteor" anchor stations between 21 ° S. and 4° N. in the Atlantic Ocean. Below: the mean for three days at the Altair anchor station (44-5° N., 34" W.). 156 Salinity of the Ocean, its Variation in Oceanic Space and in Time ice (especially around Greenland, Tierra del Fuego and similar places). A special investigation of the annual salinity variation in the open North Atlantic has been made by Smed (1943). Neumann (1938) has made a detailed investigation of the annual temperature and salinity variations over twelve five-degree squares for part of the Gulf Stream region between Newfoundland and about 25° W. (north and north-west of the Azores). These variations are presented graphically in detail in Fig. 65. It shows a rapid decrease Fig. 65. Annual salinity variations in the North Atlantic between the Newfoundland Banks and the Azores (according to Neumann). in the annual amplitude and a displacement of the maximum on moving from the west to the east and south-east away from the Newfoundland Banks, where the large annual change in salinity is due in the first place to seasonal changes in the inflow "of salt with the Labrador Current. This area is the starting point of an annual disturb- ance that spreads out to the east and south-east and gradually diminishes in intensity due to mixing. This phenomenon can be treated theoretically! and comparison with ob- t The differential equation governing the process requires that the local change dsjdt of salinity with time and the change by horizontal salinity advection u(8sldx) should be exactly balanced by the change in salinity due to mixing {Aylp)(8Hldy^) so that Ss , 8s A^ 8^s ^r + « — = — ^ — s- 8t 8x p 8y^- The boundary condition for a linear increase in salinity from y = —m to y = +m on which is superimposed a periodic disturbance at j: = 0 with a maximum amphtude at the zero point and vanishing at >> = ±'n may be formulated as S-^o = ^ + ^y + C cos -^ cos — . 2m T Then a general solution can be given in the form s = M + Ny + Cexp \~^'LA^] L 4i>rpiii This solution gives a salinity distribution that varies with time in the region from +m to —m as a function of distance and time. The intensity of the disturbance decreases in the direction of flow according to a power of e-function. nV COS — COS 2m ?-H'-l)l Salinity of the Ocean, its Variation in Oceanic Space and in Time 157 served values leads to a maximum lateral exchange coefficient of 4-9 x 10^ gcm-^ sec^^ which in view of the intense mixing in the Gulf Stream is of an order of magnitude in good agreement with this coefficient (see p. 105). From the extensive data available for the Australian-Asiatic Mediterranean (largely from the "William Snellius" Expedition) Visser (1928) has determined the annual temperature and salinity variations and has discussed them in detail. The rather large annual variations here (more than 2-5%o) are also mainly produced by advection. Table 70a gives, as an example, some values for the eastern Java Sea. While the tem- perature shows the equatorial double wave with maxima in April and December and minima in January and August, the salinity shows only a single main maximum in September and single minimum in May. These phenomena are due to the monsoon change and the associated changes in advection. During the east monsoon cold sahne water flows in from the east (May to August) and the salinity rises ; it remains almost constant during the monsoon change (September to November) and falls from De- cember to February, while the west monsoon carries water of lower sahnity in from western Java Sea. Table 70a. Annual temperature and salinity variations in the eastern Java Sea Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Annual range Temp. (°C) 26° plus: Salinity(%„) 31 plus: 1-88* 0-71 1-96 0-84 2-25 1-38 2-66t (0-88) 2-41 0-39* 1-79 (1-24) 105 210 0-74* 2-73 0-95 2-98t 1-99 2-57 2-44 2-56 2-47 1 (1-64) 1-92 2-59 * Minimum; t Maxi mum; ()Ap proxima te valu es In the interval between the monsoons from March to May the changes are only small. It is obvious that here also the advection of water masses of different sahnities is the principal factor involved. In the Polar regions the annual salinity and temperature variations may be due not only to the effects of advection but also to ice formation and melting thereby producing large amplitudes. The annual salinity variation may be increased to as much as 25%o or more, but this occurs only in a very thin top layer; the layers underneath show only a small annual variation with a maximum in winter and a minimum in summer. This small annual variation can be regarded as a consequence of ice formation. Table 71 shows, as an example, conditions in the homogeneous top layer of the east Siberian Sea from November 1922 to October 1923. SvERDRUP (1929) pointed out that between February and the end of May there was an increase of 0-47%o in the salinity of the layer below the top layer. If this is assumed to be due to ice formation, and the ice formed is assumed to have a salinity of 5%o, then the increase observed corresponds to an ice layer 67 cm thick which is in agree- ment with actual measurement of ice thickness. The salinity decrease between May and August is about 0-55%o, corresponding to the melting of 87 cm of ice which is also in agreement with the observed values. Footnote continued from opposite page Knowing the amplitude of the variation in the region of the flow the quantity Ayj pu can be calcu- lated and knowing p and u a numerical value of the lateral exchange coefficient Ay can be found (see p. 106. et seq.). 158 Salinity of the Ocean, its Variation in Oceanic Space and in Time Table 71. Monthly mean values for T and S in the homogeneous top layer in the East Sebirian Sea, Nov. 1922-Oct. 1923 Depth (m) 1922 Nov. Dec. 1923 Jan. Feb. Mar. Apr. Temp. °C Salinity (%„) 0 10-30 0 10-30 -1-63* -I-6I2 29-45 29-50 -1-61 -1-62, 29-56 29-50 -1-60 -1-593 28-99 29-23 -1-57 -l-59o 29-21 29-20 -1-58 -I-6O0 29-28 29-36 -1-57 -I-6O5 29-49 29-46 Depth (m) 1923 May June July Aug. Sept. Oct. range Temp. °C Salinity (%„) 0 10-30 0 10-30 -1-58 -l-62o* 29-67t 29-67 -0-98 -1-587 29-25 29-71t 0-80t -1-552 24-70 29-61 0-47 -l-48et 23-58* 29-14 -0-21 -1-498 24-79 28-74 -0-35 -1-524 27-57 28-56 2-43 0-134 6-09 1-15 * Minimum; j Maximum The annual variation in the surface salinity in an adjacent sea depends very largely on whether it has a humid climate with a large inflow of fresh water from rivers and from precipitation, or whether it is in an arid climate with little fresh-water gain but with a high evaporation rate. The latter type of adjacent seas with high salinities show only a small annual variation, since the evaporation has very little effect on the surface salinity; in the first type, on the other hand, the annual range may reach relatively large values. The annual surface salinity variation at the Adlergrund hght-ship in the south- western part of the Baltic is presented as an example (Neumann, 1938) (Mean monthly values for the period 1926-35) in Table 72. Table 72. Annual surface salinity variation at the Adlergrund light-ship (Baltic Sea) and total fresh-water inflow into the Baltic 1 Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Mean Range Salinity (%„) 7-51 Variation Inflow (km'/ 231 month) 7-52t 26-8 7-50 34-4 7-45 59-8 7-39 84-2t 7-30 71-0 7-31* 480 7-31 41-2 7-32 30-2 7-36 26-4 7-41 23-5 7-48 22-8* 7-41 40-09 0-21 61-4 • Minimum; t Maximum These values follow almost exactly a pure sine curve S = 7-41 + 0-103 sin | ^ t + 66-8 | + 0-006 l-^ t + 66-8 j + 0-006 sin I - / + 15-4' with a maximum in January-February and a minimum in July-August. The most important factor affecting the amount of salinity in the Baltic is the inflow of fresh Salinity of the Ocean, its Variation in Oceanic Space and in Time 159 water from rivers and from precipitation. There is therefore a very close correlation between the two phenomena if one applies a readily explicable phase difference of about two months. Gehrke (1910) has already pointed out that this phase difference becomes smaller and smaller approaching the coast from the open sea. Similar con- ditions are found in the eastern part of the Baltic (Granquist, 1938). Of the occasional disturbances in the surface salinity occurring only for very short time due to the influence of external agencies, probably the most interesting is that caused by precipitation. It is to be expected that heavy precipitation of long duration will reduce the salinity. However, this reduction, first affecting the surface, will extend when precipitation continues down to deeper and deeper layers beneath the surface due to turbulent mixing. After the cessation of the precipitation there is a continued equalization of the sahnity change by these turbulent processes that gradually ehmi- nates the disturbances. Some data are found in the literature (Krummel 1907) on the relationship between precipitation and simultaneous and subsequent decreases in salinity, but these observations have been made from moving vessels and therefore do not permit an unequivocal quantitative determination of the effect of dilution by precipitation. Neumann (1940) has given some more recent results on the determina- tion of salinity before, during, and after rain. Figure 66 shows three sets of observation made by the research vessel "Carimare" of the surface salinity given as deviations (l/100%o) of the value at the time of the rain. In total agreement with each other all three cases show minimum salinity at the time and at the end of the rain. After the rain the salinity rose, at first rapidly and then more and more slowly to the value -8 -7 -6 -5 -h -3 -2 -1 0 +1 +2 +3 *h +5 +6 +7 +8 +9 +10 %(?>oo ' r- ■ 1 1 ' 1 — 1 — I ' 130 ■^^,^^^ - 120 no 2 . \ 100 90 * - 1 1 1 1 - 80 1 1 70 - 1 1 - 60 50 1 - 1 1 1 1 1 - kO 30 20 10 • . — "^x 1 1 1 1 1 1 1 1 ^ - 1 1.. 1 1 1 f "■ T 1 3 I n " "• Fig. 66. Changes in surface salinity due to precipitation (according to the observations of the research vessel "Carimare"). The zero point on the abscissa corresponds to: in case 1 (6-7 June, 1938 at 1 7.00 h) ; in case 2 (8 June, 1938 at 04.00 h) ; in case 3(16 June, 1938 at 02.00 h). 160 Salinity of the Ocean, its Variation in Oceanic Space and in Time present before the disturbance produced by rainfall, so that 2-3 h after the rain had ceased the salinity values did not differ from the value before the rain by more than 0-05-0-10%o. A quantitative treatment of these processes has been given by Defant and Ertel (1939). The rain v/ater falling on the surface can be regarded physically as a salinity sink at the surface (z = 0) that consumes a quantity of salt —S per unit time and unit area; this corresponds to an intensity in the salinity flux —A^ids/dz) immediately at the sea surface z = 0 (^4^ is the vertical exchange coefficient, s is the amount of salt in unit mass, z is counted positive downwards). This reduction in salinity extends downwards into deeper layers by mixing during the precipitation period according to the exchange equation ds A 8^s 'dt^ ~p 8z^' At the start of precipitation (/ = 0) the salinity should be uniform {s = ^o)- ^ will be dependent on the intensity of precipitation and on the time t and therefore for a dura- tion T of precipitation 2J(t) > 0 for 0 ^ t ^ T while at the end of precipitation 2:(t) = 0 for t ^ T. At large depths the disturbance will vanish so that for z = oo and for any time s =s*. Solution of the problem for the given boundary conditions will give a complete answer for the entire process not only for the sea surface but also for all the layers underneath the surface. The simplest case is that where for the total duration T of precipitation 27 is constant for 0 S t ^ T, while after the rainfall 27 = 0 for / ^ T. In this case the solution for the total precipitation time T is ' = '*- (vSx)) ^' and when precipitation has ceased The maximum salinity disturbance q will reach by the end of the precipitation a value ^ ~ VipTrA) • The salinity disturbance at the sea surface during the precipitation will follow the equation s* - s = q hr for (0 ^ r ^ T). At the end of rain {t = T) q reaches a maximum value and then the disturbance decreases according to the formula _\l T \]\t for (/ Z T). I Salinity of the Ocean, its Variation in Oceanic Space and in Time 161 The change in time of a salinity disturbance at the surface of the sea caused by precipi- tation has the form shown in Fig. 67. Case 2 on Fig. 66 corresponds completely to the theoretical solution as far as the observations allow a comparison. For the time / required to reduce a salinity disturbance produced in a time T to a fraction q by turbulence alone the above equation gives for g = J / = 0-56 T 4 10' 3-52 T 24-95 T. Thus a salinity disturbance produced by precipitation lasting one hour would fall to one-tenth of its maximum value in about a day. Therefore, heavy rain can have an appreciable effect on surface salinity and in a discussion of frequent rainfall this circumstance deserves considerable attention. Fig. 67. Change in time in salinity due to precipitation at the surface according to the theory. In addition to the precipitation, the melting of icebergs which have drifted into warm water can appreciably reduce the salinity in the remote and in the close surround- ing waters. This process operates much more slowly than the precipitation but no data for investigation are as yet available. The physical process should not be so very different except that the limited extent of an iceberg will confine it to a smaller space and it will thus have to be considered in three dimensions. 2. The Horizontal Distribution of Surface Salinity The most detailed charts for the Atlantic Ocean are those prepared by Bohnecke (1936) based on all the available data. More recent charts for all the oceans have been given by Schott (1928, and in improved form 1934); corresponding charts are also given in his geography of the Indian and Pacific Oceans (1935). Plate 5 shows such a chart on an equal area projection. The salinity of the open ocean varies between less than 33%o in the north-eastern Pacific and a little more than 37%o in the horse latitudes of the North Atlantic. The range of variations is little more than 5%o. All three oceans have zones of maximum salinity in the subtropics with maxima of more than 37-25%o in the North Atlantic and the South Atlantic. In the open northern Indian Ocean the Arabian Gulf has maximum salinity values of more than 36-5%o in sharp contrast with the low sahnity of the Bay of Bengal. In the southern Indian Ocean towards Australia there is a subtropical oval region with a maximum salinity of more than 36-0%o. 162 Salinity of the Ocean, its Variation in Oceanic Space and in Time In the Pacific the zonally oriented cores of maximum sahnity lie between 30° and 20° N. with somewhat more than 35-6%o and between 15° and 25° S. with about 35-6%o. Between the areas of the subtropical salinity maxima there is a belt of low salinity for all three oceans located in correspondence with the region of the equatorial counter currents. On the polar side of the subtropical maxima in salinity there is a rapid decrease in salinity which is particularly pronounced in the Southern Hemisphere in all three oceans as far as the southern oceanic Polar Front (45°-50° S.). On the polar side of this the salinity remains everywhere a little less than 34%^ especially in the area of the Antarctic pack ice and drift ice. In the North Atlantic, due to the effect of the Gulf Stream and the Atlantic Current, there is a sharp difference between the eastern and western sides. In the eastern part there is only a slow decrease in salinity towards the north; in the western part shows a belt of low salinity (less than 32%o) associated with the Greenland and the Labrador Current which borders with a strong salinity gradient the warm, more saline Atlantic water. Table 73. Factors increasing or decreasing the surface salinity. Increasing Decreasing E = evaporation P = precipitation If = ice formation /„ = ice melting C+ = surface circulation (advection C- = surface circulation (advection of more saline water) of less saline water) M+ = mixing with more saline deep M^ = mixing with less saline deep water (turbulence and convec- water (turbulence and convec- tion) tion) L = Solution of salt deposits (Gulf R = Inflow of fresh water from the of Suez, Suez Canal, Gulf of land (rivers, glaciers, icebergs) Akaba) (run off). Table 73 shows the factors listed by Wust (1936), which increase or decrease the salinity at the surface of the ocean. For a stationary distribution of salinity the effect of all these factors at any point must balance. An analysis of the horizontal distribu- tion of salinity in this way is not yet possible at the present time. However, if only the mean meridional distribution in the space between 40° N. and 50° S. is considered the above factors are considerably reduced so that a good correlation equation of the form S =f(E - P, C, M) could be expected. At first attempts have been made to determine the dependence of the salinity on the quantity (E — P) from the available dita. Recent calculations of this type have been mad.' by WiJST (1930, 1936). Table 74 gives values of S, T and CT< separately for the three oceans and for the total ocean. Values for E — P are given later on in Table 87 (see Chap. VII, 3). Figu-e 68 shows the close relationship between the distributions of S and {E — P). It has been found by accurate analysis that the correlation equation 5" — f{E — P) for the entire ocean is linear: 70°N.-10°N.: S = 34-47 + 0-0150 {E - P) ± 0-1 l%o, 60° S.-10° N.: S = 34-92 + 0-0137 (E - P) ± 0-09%o. I Salinity of the Ocean, its Variation in Oceanic Space and in Time 163 80 36-5 / \ 60 1 f \ •v 360 1 •-, \ \ 40 h %\ 35-5 IT 1 // ft // \ 20 § 1 U 350 * . 0 > / 1 k 34-5 1 W V lii i \ -20 f \ 340 I x^ -40 \ i \ 335 ) 'j \ -60 ,/ \ 330 V -80 J V, 32 5 60° N 40° 20° 20° 40° 60° S Fig. 68, Mean meridional distribution of evaporation-precipitation {E-P) and surface salinity for the entire ocean (according to Wiisr, 1954). Table 74. Mean values of salinity, temperature and density for 5° latitude zones for each ocean and for the total ocean including adjacent seas (WusT, 1954) Zone Atlantic Ocean Indian Ocean Pacific Ocean Mean for all oceans °lat. 5(%o) r(X) "( 5(%o) WO "t S(°D r(°c) <'t 5(%„) WO "t N. 70-65 (33-5) 2-l)» (26-79)t . (30-0)* (-0-6)* (24-12) (33-4) (2-1)* (267 l)t 65-60 (32-45)» (4-4) (25-73) — — — (32-0) (0-8) (25-67) (32-35) (3-7) (25-73)* 60-55 32-90 6-6 25-83 — — — 32-37 3-6 25-76t 32-66 5-2 25-84 55-50 34-56 8-8 26-82t — — — 32-63 5-8 25-74 33-41 70 26-19t 50-45 34-80 11-4 26-53 — — — 32-98 7-7 25-74 33-69 9-2 26-08 45-40 34-90 14-9 25-94 — — — 33-53 11-8 25-51 34-14 13-2 25-70 40-35 36-47 19-3 26-08 — — — 33-98 16-2 24-93 35-11 17-6 25-48 35-30 36-9It 21-5 25-89 — — — 34-49 19-8 24-45 35-50 20-5 25-03 30-25 36-75 23-5 25-13 (39-57)t 25-6* 26-63t 34-95t 220 24-20 35-76t 22-7 24-62 25-20 36-74 24-8 24-74 36-92 26-2 24-44 34-90 24-4 23-47 35-C6 24-6 23-98 20-15 36-22 25-7 24-04 35-27 26-8 23-00 34-61 26-0 22-76 35-14 26-0 23-16 15-10 35-90 26-2 23-67 35-13 27-5 22-67 34-20 27-0 22-14 34-76 26-9 22-59 10-5 35-18 26-7t 22-98 35-12 27-6 22-63 34-04* 27-5t 21-85* 34-43* 27-4t 22-18* 5-0 3501 • 26-6 22-87* 35-07* 27-8t 22-49* 34-54 27-4 22-27 34-73 27-2 22-47 N. 70-Ot 3545 18-87 2511 3538 2718 22 90 3417 21-46 23 51 3471 2106 24-01 S. 0-5 35-65 25-5t 23-69* 3501 27-6t 22-55 34-91 27-Ot 22-67* 35-07 26-9t 22-85* 5-10 36-04 25-0 24-14 34-83 27-3 22-51* 35-20 26-6 23-01 35-25 26-5 23-09 10-15 36-65 23-9 24-94 34-62* 26-7 22-58 35-45 26-0 23-39 35-42 25-8 23-42 15-20 36-66t 22-7 25-26 34-93 25-3 23-22 35-65 24-9 23-88 35-62 24-6 23-96 20-25 36-34 21-7 25-34 35-34 23-4 24-09 35-70t 23-3 24-39 35-74t 23-0 24-51 25-30 35-98 20-6 25-37 35-69 21-2 24-98 35-53 21-2 24-86 35-68 21-1 24-99 30-35 35-53 18-4 25-59 35-81t 18-4 25-81 35-17 18-7 25-24 35-46 18-5 25-52 35-40 34-97 15-4 25-65 35-43 15-4 26-24 34-73 15-8 25-60 3504 15-6 25-89 40-45 34-42 11-0 26-34 34-66 11-2 26-49 34-51 12-8 26-07 34-54 11-8 26-29 45-50 34-07 6-6 26-76 34-07 6-3 26-80 34-24 9-6 26-48 34-14 7-7 26-58 50-55 33-87 3-0 27-01 33-85 2-9 27-00 34-12 6-6 26-80 33-96 4-4 26-94 55-60 (33-88)* (0-5)* (27-19)t 33-88 (0-7)* 27-18t (34-02)* (3-2)* (27-1 l)t (33-94) (1-7)* (27-18)t S. 0-60t 35 31 16 07 25 63 34 84 16 08 25 08 35 03 19 64 24 65 35 03 17 99 24 99 * Minimum; t Maximum; J Without polar zones; () Approximate values. 164 Salinity of the Ocean, its Variation in Oceanic Space and in Time For the individual oceans the deviations from a Hnear form are larger and Wiist was able to show that these were due in the first place to mixing of the surface layers with the layers underneath. This linear dependence although unequivocal cannot be taken as a casual physical relationship. This is readily seen, since if the surface salinity in an area was dependent only on the difference evaporation-precipitation the constant excess of evaporation (for always positive E — P) would cause it to rise continuously and a linear correla- tion could not be maintained. The simple linear dependence is only a part of the gener- ally applicable equation S = f{E — P, C, M) and to this equation adds the varying effect of advection and mixing (Defant, 1931). This influence enters into the above equation partly in the coefficient of the {E — P) term and partly in the first term which represents primarily the effect of vertical mixing. If surface water of salinity S is mixed with water of constant salinity 5*0 then the change of salinity due to mixing will be proportional to Sq — S. The change of salinity due to processes of evaporation and precipitation will be proportional to E — P. Under stationary conditions the local change in surface salinity will be zero. Thus ^4 = ^ ' y ' ' )f ^ ' \ 400 800 T" ""^ / - ) V '' "^ \ ^ ^ /^ / / \ I k \ / \ 1200 1600 2000 2400 2800 3200 3600 4000 - 1 ' \\^ ^. Ws \ 10 - \ ,^^ K^ \ ) / 1 1 - \ \ ] / - \ - - - - L ; I I'M 1 1 1 1 1 1 1 34 2 4 6 8 34 7 8 . 1 50 i i 1 5 1 1 1 Fig. 69. Vertical salinity curves for a series of oceanographic stations along a meridional section through the Atlantic (corresponding vertical temperature curves are shown in Fig. 52). Footnote from opposite page t This is demonstrated by the low salinity of the adjacent seas with a strong freshwater inflow (such as the Black Sea and the Baltic a.o.) and can also be seen in coastal areas where there is a large fresh-water inflow. The effect of these frequently turbid river v/aters is often found surprisingly far out at sea. Charts of the mouths of the major rivers (Amazon, Congo, Tajo, La Plata) usually con- tain a limited area in which the lighter water shows at the surface on top of the heavier sea-water; but this is usually only the case in a thin layer and already in the wake of a ship the sea-water of much more blue colour may be brought to the surface. An investigation of the mixing of the lighter river water and the heavier sea-water at the mouth of a large river would be of some interest. The Suez Canal shows the great effect on the salinity of solution of a salt deposit, in this case at the bottom of the great Bitter Lake which is connected by the canal with the Mediterranean and with the Gulf of Suez. Water of lower salinity flows in from both sides and causes a progressive dissolution of the salt deposit and maintains in that way the high salinity of the water above at 50%o at the surface and 56%o at the bottom (about 10 m depth). Since the canal was first built (1869) when the water depth was 7-56 m dissolution of the salty canal bottom has increased the depth here linearly to give a depth in 1921 of 11-7 m. At the same time the salinity of 68%o in 1872 had fallen to about 52%o by 1924. The available and, in parts, sparse data on the distribution of salinity in the Suez Canal and on the currents caused by it have been dealt with by WiJST (1934, 1935) in two interesting papers. 1 66 Salinity of the Ocean, its Variation in Oceanic Space and in Time decrease in temperature in these layers is thus associated with a strong decrease in the salinity, This extends down to about 800 m where the salinity reaches a minimum of 34-3-34-9%o. There is then a second increase to about 34-8-34-9%o at about 1600- 2000 m and then a further slow decrease is generally observed down to the bottom. The inversion in salinity at 800-1000 m becomes weaker and weaker towards higher northern and southern latitudes, and from the polar fronts of both hemispheres towards ^he poles it is entirely missing; the vertical differences are then small with usually a slight increase in salinity if fresh water has not been added to the surface layers by the melting of ice, but this becomes weaker and weaker towards the poles. In contrast to this vertical distribution generally found, the North Atlantic shows a pronounced peculiarity in middle latitudes which can be seen at some of the stations in Fig. 69. The intermediate salinity minimum at about 800 m is missing here, and from the core of upper layer of high salinity situated in middle latitudes the salinity decreases almost uniformly down to the bottom. There is thus a marked asymmetry between North and South Atlantic vertical distributions of salinity. ib) The Salinity of the Oceanic Troposphere The vertical distribution of salinity in the troposphere layers of the subtropics and the tropics is worth a somewhat more detailed description. It has, of course, been investigated more closely in the Atlantic (Defant, 1936). Almost all stations in the tropics and subtropics show a nearly homo-haline top layer. Its thickness is not the same as that of the thermal top layer but is usually somewhat smaller. In many cases just below the quasi-isothermal top layer, however, still in the upper part of the ther- mocline, there is a more or less well-developed salinity maximum. This maximum is one of the most characteristic phenomena of the vertical salinity distribution of the upper troposphere. Figure 70 shows an example of this. The "Meteor" 256 station shows the maximum particularly well developed ; in a thin layer from about 50 m the salinity rises from about 36-1 to 37-0%o and then falls again to the previous value. It is worth noting that the salinity maximum appears there where the first drop in tempera- ture occurs beneath the isothermal surface layer and not at about the maximum temperature gradient of the thermocline (see Fig. 71). The sahnity maximum thus extends just above the thermocline, but does not fully coincide with the density transi- tion layer, the position of which is in turn fixed by the high salinity value. Careful investigation of this sahnity maximum in the tropical and subtropical regions of the Atlantic has shown that it is almost always present. Starting from the extensive sub- tropical accumulation of very saline water (at about 25° S. and at about 30° N.), where in a top layer down to the thermocline a homo-haline structure is found, a thin layer of maximum salinity spreads out northward in the Southern Hemisphere and southwards in the Northern immediately above or directly inside the thermocline. This spreading occurs below the upper part of the top layer, in which salinity decreases in both hemispheres towards the equator. From this it can be concluded that the layer of the salinity maximum is formed from the lowermost parts of the subtropical high salinity water by currents flowing towards the equator. It thus represents the intrusion of highly saline water under the surface layers of lower salinity of the equatorial regions and forms a part of the upper tropo- spheric circulation. Salinity of the Ocean, its Variation in Oceanic Space and in Time 167 200 300 400 500 34-5 350 355 360 i" /OO 365 370 24 25 26 27 28 Fig 70 Vertical temperature, salinity and density curves for the troposphere at "Meteor" Stn. 256 (0 = 2-4° S., A = 39-3° W.). Increasing values of 5 and / Fig. 71, Position of the tropospheric salinity maximum relative to that of the thermocline (schematic). 168 Salinity of the Ocean, its Variation in Oceanic Space and in Time While the entire area between the behs of subtropical highly saline water in the Northern and the Southern Hemisphere show these salinity maxima, just above or inside the thermocline two belts without maxima stand out sharply; one between 10° and 15° N, and extending from 45° W. eastwards to the African continent and a second, but more narrow belt, between 2° S. and 3° S. and extending from 30° to 10°, which is particularly well developed in the central part of the Atlantic Ocean (see Fig. 72). These two belts without salinity maxima more or less mark the southern Fig. 72. Distribution of salinity in the tropospheric salinity maximum in the subtropic and tropics of the Atlantic. and northern limit, respectively, of the subtropical water masses spreading towards the equator. Between these two belts from about 7° N. to the equator the maximum appears again and may be very pronounced. This is the region of the Equatorial Countercurrent which is fed at a depth of 80-100 m from regions west of 35°-40° W., which are situated outside the area with no maximum. The salinity maxima of the tropics and the subtropics are thus very closely connected with the tropospheric circulation in these areas. The best illustration of the formation, extent and intensity of this very pronounced thin layer of high salinity lying between low salinity layers (above and below) is given by a vertical cross-section along the core of the Equatorial Countercurrent and the Guinea Current in the Atlantic Ocean. This section is shown in Fig. 73. It starts in the central Atlantic at about 18° N., 37° W., proceeds south- wards to 10° N., 38° W. and then along the core of the Equatorial Countercurrent, finally reaching the inner Gulf of Guinea. The layer of maximum salinity spreads southward from the homo-haline top layer of the subtropical North Atlantic below the low salinity surface layer towards equatorial latitudes. If the 35-5%o isohaline is Salinity of the Ocean, its Variation in Oceanic Space and in Time 169 c < C 4) •S 3 e «> 2 in i o h « X (53 C Is (D o (U U &o ^ J3 4) ^- §2 a" „ 60 c.S o c o p on 1 70 Salinity of the Ocean, its Variation in Oceanic Space and in Time taken as the upper and lower limit of the layer with maximum salinity it has an average vertical thickness of only 50 m ; it stays about the same thickness over its long course to well within the Gulf of Guinea, and the salinity of the core layer changes very little after it has lost its tongue-like form along the first half of its route. A comparison of the salinity section with a corresponding density section shows that the position of the salinity maximum along the greater part of the cross-section coincides with the strongest vertical concentration in the density field. The very saline water thus extends in a thin layer along the thermocUne itself. The spreading in this layer is caused by advection and turbulence but the latter factor must be of very little effectiveness, be- cause of the almost unchanged character in this remarkably thin layer over such large distances. It must be supposed that above and below the thermocline the transport of water with maximum salinity is accompanied by strong mixing with the water above and below, but that in the thermocline itself the stability strongly suppresses turbulence, so that the almost horizontal spreading takes the character of a laminar flow. This has been confirmed by calculations of the vertical exchange coefficient in the area of the Equatorial Countercurrent by Montgomery (1939), who found /i^ = 0-4 g cm~^ sec~^ along the axis of the Countercurrent. Since lateral mixing was neglected in these calculations the value found will be a maximum value; the true one must approach rather closely the molecular diff'usion coefficient for salt in water (0-011). As men- tioned above, the spreading must therefore be of laminar character. However, in horizontal direction lateral mixing is very eff'ective and the lateral exchange coeffi- cient Ay reaches the value of 4 x 10^ g cm~^ sec"S generally found. From the deep-reaching accumulations of warm and saline water in the subtropics there is not only a flow of this water towards the equator but also towards the poles in somewhat deeper layers. Thus at depths only a little below the upper layer, and the almost homo-haline top layer which shows decreasing salinity towards the pole, there is a secondary maximum in the vertical distribution of the salinity. In the Southern Hemisphere this poleward flow of highly saline water occurs first at a depth of 100 m, but descending to a depth of 150 m or more, and continues on over a very broad front across the entire ocean; however, the energy of this outflow is soon dissipated and the maxima disappear due to mixing. In the Northern Hemisphere this maximum is associated with the Gulf Stream and its continuation (the Atlantic Current) and it can be followed across the entire Atlantic Ocean into the Norwegian Sea and further polewards. Figure 74 shows a longitudinal salinity section given by Schott (1942) through the Atlantic Current from the Wyville-Thomson Ridge past the Shetland Islands as far as Spitzbergen. The Atlantic water soon descends underneath the cold and low saline polar water of the surface layer. Although the salinity maximum is decreased by mixing it can still be traced in the North Polar Basin and into the Barents Sea. Its occurrence here was discussed in connection with the description of the vertical temperature distribution in the North Polar Basin (see p. 133). An interesting and, from the point of view of the method used, important study of this spreading of At- lantic water {§ = 10-2°, S = 35-45%o) in the northern part of the North Sea, in the Norwegian Sea and in the Barents Sea and its mixing with the surrounding water {d = 2-5°, S = 34'90%o) made by Jacobsen (1943) should specially be mentioned here. From our knowledge of the tropospheric salinity maxima of the Pacific and the Salinity of the Ocean, its Variation in Oceanic Space and in Time 171 172 Salinity of the Ocean, its Variation in Oceanic Space and in Time Indian Oceans, we know their formation and spreading are still very pure. The much stronger intensity of this phenomenon in the Pacific Ocean has been shown by several recent oceanographic stations but detailed information about their extent is still lacking. (c) 772^ Salinity of the Stratosphere The vertical salinity distribution in the stratosphere of the three oceans can best be discussed by means of longitudinal sections through the Atlantic, the Indian Ocean and the Pacific. The longitudinal section through the Atlantic Ocean is that given by WusT (1936) through the Western Trough from the Weddell Sea to Davis Strait (see Fig, 62). In the Indian Ocean a central section (Fig. 75) from the Antarctic to the south- ern tip of India has been selected (Moller, 1929); the Pacific Ocean is characterized by a vertical section through its eastern half (Fig. 76). In the northern part this section 34-0 35-5 35-0. 35-5 35-0. BOCO ^000 ■""^60* S 50° 40° 30° 20° 10° 0° N 10° Fig. 75. Longitudinal salinity section through the central part of the Indian Ocean. 60° N 1000 2000 £ 3000 4000 5000 I Fig. 76. Longitudinal salinity section through the central part of the Pacific Ocean, Salinity of the Ocean, its Variation in Oceanic Space and in Time 173 is based on the "Carnegie" observations (Sverdrup) and south of 40° S. on the "Discovery" observations (Deacon, 1937). Longitudinal sections through the western and central parts of the Pacific Ocean have been given also by Wust (1929). (d) Subpolar Intermediate Water At 800-1000 m there is a characteristic lovv^ salinity zone extending across almost the entire ocean though not always equally well developed. In the south it begins always just south of the oceanic polar front where this special water mass sinks rapidly from the surface to a depth of 800 m and spreads out from here with decreasing vertical thickness and decreasing salinity in its core into the Atlantic across the equator to about 20° N. It can still be traced north of here until it joins the deep and saline water accumulations of the subtropics. There is little to be seen from an Arctic counter- part to this subantarctic intermediate water. Only in the western section weak indica- tions of such arctic intermediate water may be found as far as the Newfoundland rise. Also in the Indian Ocean this intermediate water is found everywhere underneath the high saline water mass south of the subtropics as an intrusion of low saline water with its core somewhat deeper than in the Atlantic (approx. 1000-1200 m). In the Pacific tongues of low saline polar water spread out below the high saline tropo- sphere almost to the equator, from both north and south. The Antarctic branch of low saline water forms just south of the oceanic polar front at 50°-60° S.; the arctic branch formed in the area of the Okhotsk Sea is weaker; in the western and central parts of the Pacific Ocean it can be followed to about 10° N. It is completely absent in the whole of the eastern part of the Pacific and there is thus an asymmetry in the salinity distribution similar to that in the Atlantic Ocean. The vertical thickness of the subantarctic intermediate water is about the same in all the three oceans (about 600 m) and it is separated from the troposphere above by a sharp salinity (and density) transition layer. It is of particular interest that the inter- mediate water is found with the same characteristics and thickness across the entire transverse section of the ocean, especially in the Atlantic. Evidence for this is given in Fig. 77 which gives a cross-section of salinity through the Atlantic at about 22° S. This uniformity of this water across the total cross-section can be regarded as a conse- quence of strong lateral mixing which leads to an equalization of all existing major horizontal salinity differences. A detailed investigation of conditions in the subantarctic intermediate water and its meridional spreading in the Atlantic has been given by Defant (1936). The vertical salinity distribution in successive cross-sections normal to the main direction of spreading is best characterized by the dimensionless quantity {sq — s)I{sq — s^, where ^o (=34-85%o) is the salinity which the subantarctic intermediate water takes on by continuous mixing with the surrounding water and s,n (=34-19%o) is the salinity of the subantarctic intermediate water in its region of origin before spreading out towards the north. The quantity {sq — s,n) corresponds to a potential difference present between the two oppositely moving types of water which is finally eliminated by mix- ing. Determination of this quantity in cross-sections, 500 km apart from each other, for the core layer (salinity minimum) and for several layers above and below this core allows of construction of lines of equal values of the quantity (5'o — 5)/(^o~ ■5' m) expressed in percentage of intermediate water. These lines then illustrate the mixing process 174 Salinity of the Ocean, its Variation in Oceanic Space and in Time _ _ 20° i(r 0° 10' Fig. 77. Cross-section of salinity through the Atlantic Ocean at about 23° S (profile VII at 24°-21-25° S., of the "Meteor" Expedition). 400 300 200 100 0 -100 -200 -300 -400 -500 7 95^-^89 83 ^79 76 737070 100 97 94 ) 87 83 80 79 82/72 95 /88 82 /7§ 74^7Cr^^67 68 68 65 /58 57 51 -50 47 ^67 66 §u6C>57---..^6l &\^0 53^49 43 45 4L ^59 57 53-50'46'^>^53 50^— '49 43^.-^S8 34 36 ■ 33 u4034- 43 ^17 30— 5000 6000 7000 8000 9000 10000 000 12 000 Fig. 78. Percentage of subantarctic intermediate water in the core layer of this water type the western side of the Atlantic. (Distribution of the quantity (^o — ^)/(*o — -yJ in per cent.) Salinity of the Ocean, its Variation in Oceanic Space and in Time 175 along the entire spreading area for the entire western section through the Atlantic Ocean (Fig. 78). This distribution has a clear similarity to that presented in Fig. 48 which shows the radial and turbulent spread of a particular water mass into surround- ing waters. This distribution also corresponds to the processes of spreading in a so- called "jet" (Freistrahl) (Prandtl, 1926; Tolmein, 1926; Ruden, 1933). Figure 79 100 X 80 *600 +800 60 40 80 1 ■ i f ^ 1 k u 1 1 1 - kf ■^ o - - /a > K - / \ - "; t 1 1 1 \ \ 1 ^J 0^ T" Fig. 79. Distribution of salinity relative to the minimum in the core layer of the subantarctic intermediate water along the western side of the Atlantic for different vertical cross sections. shows for each cross-section the distribution of salinity relative to the minimum in the core and shows that the general distribution is the same for all cross-sections and that the processes involved mu: t be essentially the same, geometrically and mechani- cally, as in a "jet". An accurate knowledge of the salinity values throughout the entire region of the subantarctic intermediate water allows the vertical distribution of the quantity Ajpu to be calculated from the equation on p. 106 for all cross-sections. A rough calculation shows at once that vertical mixing and advection are able to maintain the tongue-formed salinity distribution stationary in the subantarctic inter- mediate current. The vertical salinity distribution at a distance of 8000 km from the zeio point of the western section (about 13° S.) is as follows -300 + 200 +100 100 -200 -300 -400 (m) salinity in %o vertical gradient (per 100 m) 34-71 0-54 0-42 O^S" 0-44 0-52 0-60 0-69 -017 -012 -004 +006 +008 +008 +009 Considering a vertical water column of 1 cm^ base between +250 and —250 m the inflow of salt into the column from above and below is shown in Fig. 80, taking A = 4gcm-isec-^ The salt gain in the entire volume (5 x 10* cm^) thus amounts to 1-00 X 10"^ g/sec or 8-64 mg/day. Without an advective outflux this continuous gain of salt would soon eliminate the salinity minimum of the subantarctic intermediate water. Through the left-hand (southern) boundary of the water column (with an area of 5 X 10* cm2) there enters an amount of salt of 5 x 10* X m • .s X 10"^ g, where u 176 Salinity of the Ocean, its Variation in Oceanic Space and in Time is the velocity of the horizontal advection. At the right-hand (northern) boundary there is at the same time an outflow of 5 X 10* X m-^-i x 10"-^ g of salt from the entire volume so that the loss of salt in g/sec will be 50u{si — s). For a stationary salinity distribution this loss must be compensated by a gain due to mixing, that is by 1-0 X 10"' g/sec. Taking u equal to 5 cm/sec, this is only possible for a horizontal salinity gradient of(si — s) = 4 x 10^^° %o/cm in the current. The salinity at 7000 km 450 m = 0-12 MO" (vertical gradient of salt at 450m) 950 m I cm 068 X 10 g Inc rease of salt content 1-00 X lO'^g pro sec salinity S velocity U salt flux Fus« 10" I Fus, « 10 — ^ = 0-08 1 10 Az (vertical gradient of salt at 950m) 0-32 « 10 g salt loss Fu X I0''(s,-s) Fig. 80. Salinity exchange and advection. For w = 5 cm/sec results salinity gain = salinity loss: 5 X 10* X 5 X 10-^ (s^ - s) = 2-41 x 10"^ or {s^ - s) = 0-97 x 10-» "/oo per centi- metre. distance in the core of subantarctic intermediate water is 34'34%o, at 9000 km, however, 34-42%o, so that according to the observed values there is a salinity gradient of 0-08%o for the 2000 km = 2 x 10^ cm. This gives exactly the value derived above of 4 X 10"^"%o/cm. The vertical and horizontal salinity distribution in the subantarctic intermediate current at this point can thus remain stationary with values of 4 g cm~^ sec~^ for A and 5 cm/sec for u. The ratio Aj pu = t — 0'8 cm/sec satisfies therefore the condition of a stationary state of the phenomenon in time. It is fairly easy to see that the above calculation gives only the quantity Ajpu and not the absolute value of the individual quantities. Salinity of the Ocean, its Variation in Oceanic Space and in Time 111 Using the above mentioned relationship the quantity Ajpu can be determined numerically for every point more accurately than in the rough calculation made here by deliberately selecting a large water column. Table 75. Mean values for Ajpu for transverse sections {normal to the direction of spreading) through the subantarctic intermediate water in the Atlantic Position of Vertical distance from the core (m) Mean depth sections +300 +200 + 100 0 -100 -200 -300 (m) 37°-30°S. 27°-21°S. 25°-12° S. 3°S.-2°N. 4°-8°N. 3-5 2-8 2-4 (2-2) (2-3) 20 1-7 1-4 1-4 1-7 1-3 10 0-8 11 1-7 10 0-7 0-4 0-9 10 1-2 1-6 1-3 1-2 1-6 1-7 30 3-2 2-2 2-4 2-4 3 0 3-9 3-2 3-4 850 800 725 700 625 Mean 2-64 1-64 112 0-82 1-38 2-50 318 740 Table 75 shows that the vertical distribution of A I pu scarcely changes along the entire region of spreading: Al pu is least at the core of the spreading water (on the average 0-82) and rises steadily both above and below with the distance from the core to large values (about 3). If the distance from the core axis is denoted by z, then Alpu=^f(z). When / is the Prandtl mixing length and therefore Az) = Pcu u dz Integrating this equation for constant / from the core (2 below the core then, since dujdz is always negative, f* u /(zyz=-/Mn-, 0), to a distance b above and where Mq and u are the values of u in the core and at a distance b from the core. The value of the left-hand side can be found from Table 75 and this gives, knowing /, the ratio of m/mq as a function of the vertical distribution and ihus also of the quantity AJAq, where A^ and A are the exchange coefficients in the core itself and at a distance b from it. The constancy of / can be tested on the plausible assumption that the velocity of the current within a distance of ±300 m from the core falls to a low value; over a wide region / is almost constant with a value of about 150 cm. The results of the calculation are shown in Fig. 81. The relative distribution of velocity over a trans- verse section has a striking similarity to that distribution found experimentally in turbulent spreading processes in a "jet". This justifies the conclusion that the spread of the subantarctic intermediate water in the Atlantic northwards, along the boundary between the oceanic stratosphere and the troposphere, is very probably a process that 178 Salinity of the Ocean, its Variation in Oceanic Space and in Time is largely equivalent to the phenomena in a "jet" CFreistrahl) at some distance from the nozzle (Diise). The distribution of^ AjA^ in transverse section is also quite charac- teristic. The maximum appears in the lower part of the spreading layer (150 m be- neath the core) ; below this the ratio falls rapidly, but above only slowly. This striking distribution of the exchange coefficients can be readily explained by the different stability conditions above and below the core. +WOm 1.2 o,e OA 0,0 +200 -200 -^00 1 1 A ^, — ,i \ // . 1 1 *s^^ Fig. 81. Relative distribution of the exchange coefficients, AlA^ and the current velocity, w/«o along a cross-section through the subantarctic intermediate water along the western side of the Atlantic. (a) Salinity of the deep water below 1500 m. In the deep layers of the Atlantic the salinity increases slowly from the Antarctic regions across the equator as far as the deep-reach 'ng, warm and high saline water of the northern subtropics (20°-40° N.); from here towards the north it decreases slowly in the upper layers. However, in the deeper layers the increase to about 20°-40° N. is much less. The asymmetry of the salinity distribution shown so strongly in the subantarctic intermediate water is also present in the deeper layers but not to the same extent. This contrast is due in the first place to the strong accumulations of saline water in the subtropics, but in these layers it is also reinforced by the inflow of highly saline water from the Mediter- ranean through the Straits of Gibraltar. Everywhere in this area there exists a well- defined maximum in the vertical sahnity distribution at 1300 m (at about 20° N.) lowering to 2500 m (at 35° S.) that must be attributed to the spreading of the Mediterranean water. This effect of inflow from the European Mediterranean can be seen particularly on the salinity chart for 1000 m depth. The spread of this type of water will be discussed in greater detail later on (Vol. I, part 2, Chap. XVI, 3). The nature of the water beneath the upper part of the stratosphere in the Atlantic indicates an area of formation in higher northern latitudes (north of 50° N.) in the Western Trough. Here it is formed at the surface during the late autumn and early winter, sinks by thermo-haline convection to great depths and spreads out more or less horizontally below 2000-2500 m to fill the lower part of the stratosphere. The high oxygen content which characterizes this water type will be discussed later in connection with the oceanic circulation (see Vol. I, part 2, Chap. XX, 7. A similar contrast between the higher latitudes of both hemispheres is also present in oceanic stratosphere of the Indian Ocean. Here it is due in the first place to the inflow of highly saline water from the Red Sea. Coming from the Straits of Bab-el-Mandeb (seep, 182 and Fig. 84), it sinks to about 10(X)m, mixes with less saline water in the Gulf of Salinity of the Ocean, its Variation in Oceanic Space and in Time 179 Aden and from here extends southwards beneath the Antarctic intermediate water at a depth of 1500-2000 m as a tongue of highly saline water. This salinity maxi- mum shows very clearly throughout the western and central parts of the Indian Ocean. In the Pacific the few observations that have been made below 1 500 m show a remarkably uniform vertical and horizontal salinity distribution at all latitudes. Its average value is about 34-65-34-68%o, but it is nowhere connected with the equally high values in salinity of the surface layers. There is no tropical or subtropical ad- jacent sea acting as a source for saline water for the Pacific stratosphere like the Mediterranean does for the Atlantic one or the Red Sea for the Indian Ocean strato- sphere. It must therefore be supposed as pointed out by Sverdrup (1931), that the Pacific deep water below about 1 500 m depth for which there is no area of formation in the Pacific itself must be formed in the Indian Ocean or even in the Atlantic. Water masses from these two oceans must be carried to the east by the Antarctic ciicumpolar ocean current and then spread northward in form of current branches to fill the deep basins of the Pacific. (8) The salinity of the bottom layers. The salinity of the deepest layers shows also the same characteristic distribution already known from the bottom temperatures. In the Atlantic Ocean (Wust, 1936) it varies between 34-62 and 34-92%o in the most northern parts; this is explicable from conditions of formation of the bottom water. The deepest parts of the Antarctic regions are filled with Antarctic bottom water with a salinity of 34-67-34-69%o, formed at the continental slope of the Weddell Sea (see p. 14?). Above this the Antarctic deep water is found at 5000-4000 m with 34-62- 34-66%o that feeds the Antarctic bottom currents of the Eastern and Western Troughs. The isohalines of meridional sections demonstrate a clear conformity with the bottom profile and show the penetration of the water across the Equator in the Western Trough and the Eastern Trough as far as the Whalefish ridge. Figure 82 gives meri- dional salinity sections through the Western and Eastern Troughs of the Atlantic which show how the spreading of the bottom water is reflected in the distribution of the salinity in the same way as in the distribution of potential temperature (see p. 152) deduced previously. A typical Arctic bottom water cannot be recognized from the salinity distribution though traces of it can be detected in the Labrador Basin north of the Newfoundland Rise (WiJST, 1943). Our knowledge of the salinity of the bottom water of the other two oceans is still pure due to a lack of systematic salinity data. 4. The Horizontal Distribution of Salinity at Particular Depths Horizontal charts of salinity distribution are so far available only for the Atlantic: they are given for instance in the ''Meteor'" Report for depths of 200-800 m at 200m intervals, for depths of 1000-2000 m at 250 m intervals and for depths of 2000-4000m at 500 m intervals. Plate 6 shows charts for 400 m and 1000 m depths. It is clear that these charts do not give other information than the longitudinal and transverse sec- tions. The charts down to 800 m, of which the 400 m chart is given as an example, all show essentially the surface salinity distribution; only the horizontal differences become smaller with increasing depth. Of the two extensive regions with salinity maxima in the subtropics the northern is the larger. The highest values appear, 180 Salinity of the Ocean, its Variation in Oceanic Space and in Time 'mdaQ Salinity of the Ocean, its Variation in Oceanic Space and in Time 181 however, not in the central part of the Sargasso sea but are displaced'in the peri- pheral parts towards the west, partly on the right hand (north) side of the North Equatorial Current (especially at 200 m) and partly on the right-hand side of the Gulf Stream (especially at 400 m, but still visible at 1000 m). This distribution is a dynamic effect of the currents which cause an enormous water transport. Below 600 m the influence of the high salinity inflow from the European Medi- terranean begins to appear and extends already at 800 m to 40° W. It remains the principal phenomenon in all charts down to almost 2000 m and the remarkable asymmetry between the North and the South Atlantic shows particularly clearly here. Below 2500 m the horizontal salinity differences already become very small though there is still a noticeable salinity gradient from north to south. South of 40° S. more pronounced differences in salinity reappear which indicate the increasing influence of the Antarctic deep and bottom water. 5. Salinity in Adjacent Seas and Sea Straits In discussing the temperature distribution in adjacent seas (see p. 1 29) it was already emphasized that beneath the sill depth in all the adjacent seas theie is an almost constant salinity; in the adjacent seas without winter convection it is identical with the salinity of the open ocean at the sill depth off the passage ; in the adjacent seas with a winter convection, on the other hand, it is identical with the surface salinity at the time of the thermo-haline mixing (see Tables 56-66). When there are relatively large differences between the water masses of the free ocean and those of the adjacent sea, the equilibration movements in the more or less narrow sea straits connecting them show rather striking conditions which deserve particular attention. The interchange of water between the European Mediterranean and the Atlantic is a consequence of currents through the Straits of Gibraltar, which carry water at the surface and in the uppermost layers into the Mediterranean to- wards the east, but in the deeper layers beneath towards west. Corresponding con- ditions are also found in the Straits of Bab-el-Mandeb, but in other sea straits the thermo-haline structure imposes reversed flow conditions. In the Dardanelles and the Bosporus, Aegean water flows into the Black Sea in the lower layers, while the flow into Mediterranean occurs in the upper layers. Similar conditions also prevail in the connecting straits between the North Sea and the Baltic, where North Sea water enters through the Oresund and the Great and Little Belts along the bottom, while contrary the surface water flows out of the Baltic. All these water transports are associated with considerable changes in temperature and salinity. It could hardly be expected that these processes should be stationary ones. In fact they are turbulent and occur in pushes and therefore cause extremely large variations in both factors that they can only be investigated and understood with the aid of synoptic surveys. The available summarizing descriptions of the distribution of the different oceanic factors in such straits should thus be interpreted with some caution. Figure 83 shows the distribution of temperature and sahnity according to Schott (1928) through the Straits of Gibraltar for the transitional period from spring to summer when average conditions prevail in the currents. The isohalines of the longi- tudinal section show clearly that the highly saline Mediterranean water, for which. 182 Salinity of the Ocean, its Variation in Oceanic Space and in Time 8°7' 7°54' 7°20' Albocoral66Mbwe2 Thor9l MSars28 •aS-Z? 211 BllO 210 7°0'W 7°30' M.S(ys22Thor92aim.LoboXX[ 210 SHO 223 6°0' S'SO' Thor97 Almi.obaXXX2IIThor98fllmi.obi 3fflO 2123 --511021123 1500 Fig. 83. Temperature and salinity distribution through the Straits of Gibraltar at the transi- tion from spring to summer (mean conditions, according to Schott). due to mixing, a slowly westwards decreasing salinity with the surrounding water is characteristic, sinks beneath the weakly saline Atlantic water below about 300 to 400 m. The temperature distribution shows identical conditions. This water continues to sink to about 1000-1200 m off the Spanish Bay, and from here it spreads out into the Atlantic as a more or less horizontal layer of highly saline water. The di'.tribution within the strait shows strong seasonal variation : at the end of the winter the contrasts are reinforced, at the end of the summer they are weakened, but there is always a continuous outflow of water with a high salinity from the Mediterranean into the Atlantic and the submarine ridge never forms a barrier to the Mediterranean water as BuEN attempted to show (1927). Conditions in the Straits of Bab-el-Mandeb are rather similar ("Schott, 1929). The highly saline deep water of the Red Sea (S 37%o) flows over the sill at 150 m depth north of the strait of Perim into the Gulf of Aden (Fig. 84). It sinks here to 500-1000 m and then spreads out horizontally at such a depth, in which the density of the sinking water becomes equal to that of the surrounding water. Also the transition from the higher salinity of the North Sea (about 32%o) to the lower salinity of the Baltic (about 7%o) is not at all continuous, as one might easily be misled by studying mean charts only, but usually occurs rapidly, mostly in two steps (Wattenberg, 1941). The first rapid change occurs near the boundary be- tween Skagerrak and Kattegatt and changes its position very little in time; the second much sharper change has a more variable position between the southern edge of the Kattegatt through the Great and Little Belts to the rises leading to the actual Baltic (Darsser and Drogen Rises). These jumps in salinity have all the properties of true hydrographic fronts. They separate three water types: North Sea, Kattegatt and Baltic water. Figure 85 shows the distribution of the surface salinity from the Skagerrak to the Baltic in three diff'erent cases, and illustrates clearly the typical distribution at these fronts. The latter are not, however, stationary in location but move around continually I Salinity of the Ocean, its Variation in Oceanic Space and in Time 183 'mdaa 184 Salinity of the Ocean, its Variation in Oceanic Space and in Time Kattegat Baltic 30 20 10 30 20 10 Sk. L Jr. AKn. L.Gr. Or. Chr.O. 1 \ 1 1 1 M.G. \ \ \ \ \ > — \ \ \ \ 6.V. \|6E \2.BZ: 1938 "= --^ — ^^''Nt- \ \ \ \ \^ -V, \ >62. \l6.I2 N2.12 \ \ ^\ \ 1938 1 V III 11 ~ 1 Sk. L.R. O.FI. Sch. Gr Ky. Hfi. K.N. RB. G.R. Kattegat Large ondF Belt Baltfc Fig. 85. Changes in surface salinity betv.'een the North Sea and the Baltic (from the Skagerrak into the Baltic) in three cases (according to the individual values recorded on 2 and 16 April and 16 May 1938, according to Wattenberg). often at considerable speed in one or the opposite direction, and these displacements are ttien associated with jump-like changes in T and S at any given point. In the sea straits so far discussed the equalization currents are superimposed (one above the other) and the water movements occur along a boundary surface sloping in the direction of the strait. This superposition of the two types of water appears to be causally associated with the narrow width of these straits. If this surpasses a cer- tain value then the interchange of the different waters no longer takes place through currents flowing one above the other, but rather side by side in the strait, whereby the boundary surface now slopes transverse or normal to the main longitudinal axis of the strait. This type of water interchange is apparently present in the straits between the White Sea and the Barents Sea (Timonoff 1925), see Vol. I, Chap. XVI, p. 1-3 for a discussion of the dynamics of this process. Chapter V The Density of Water Masses in the Ocean ^ Vertical and Horizontal Density Distribution and its Stability 1. Diurnal and Annual Variations at the Surface The diurnal and annual variations are uniquely determined by that of the tempera- ture and salinity. Since the diurnal temperature variation is essentially parallel with that of salinity, the effects of both factors on the density partly cancel each other out, and apart from the fact that they are both small anyway, the diurnal surface-density variation is thus a rather insignificant phenomenon. In general, the aperiodic changes in density during the day are so large that they completely mask the regular diurnal variation. At anchor stations the average diurnal variation in density, taken as the average over several days, is of the order of 0-05-0-1 in a^ (Table 76), Table 76. Diurnal density (of) variation at the ocean surface (Atlantic Ocean) Anchor Hours Diurnal stations variation 1 3 5 7 9 11 13 15 17 19 21 23 "Meteor" 5° S.-5° N. 22 + 0-75 0-75 0-76t 0-76 0-74 0-71 0-67 0-65* 0-69 0-71 0-73 0-74 Oil "Altair" 44-5° N.. 34° W. 26 + 019 019 019 0-20 0-21t 0-21t 018 017 016* 016 016 017 0 06 * Minimum; t Maximum The maximum occurs in the morning or in the forenoon; the density then falls, probably due to the rising temperature — and in spite of the increasing sahnity — to a minimum in the afternoon ; the amplitude is everywhere very small. The annual density variation is much larger and its amplitude usually is of the order of 1 -00 and 2-00 in Of depending on whether the annual variation in the temperature is parallel or inverse to the corresponding salinity variation. The annual density variation can be conveniently presented by plotting the monthly values on a [rS'J-diagram. This has the advantage of providing a visual impression of the variations in temperature and salinity, and also in density. For annual variations in Tand S, following pure sine curves, the annual variation in density will be shown on such a diagram as a straight line if the annual variations of the two factors run either parallel or inverse. If the amphtudes are normalized (choosing scales of equal length for T and S in the diagram) then the straight line will be at an angle of 45° with the temperature axis, but for inverse 185 1 86 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution variations of T and S (phase difference of 6 months) it will be at an angle of 135°. For a phase difference of three months the density values will lie either clockwise or anticlockwise around a circle. This method has been used by Neumann (1940) for a close investigation of the annual density variation in the area of the Gulf Stream north of the Azores. Figure 86 shows such annual density variations for some five- Sfoo 340 332 •-^■•^/ /////// Ijjjjl / /vi / / / / y/n / /Vs/ / / -y/A V////Z V77?/T West of New Fbundlond 50°-45''N 8 10 45°-50°W 35-6 2 13 T 12 14 16 .18 50°- 491^ 20°-25''W Between New Fcxjndland and Azores 27-0 26-0 T' 14 16 45''-40°N 20 22 20 40°-45''W 14 16 18 20 45''-40°N 25° 30°W Fig. 86. Annual density variation at the surface of the sea in the area of the Gulf Stream north-west of the Azores (according to Neumann). degree squares according to the above method. The amplitude is largest {Aa^ = 2-09) at the boundaries of the Gulf Stream and the Labrador Current, then decreases to the east and south-east to only 1-5-1 in o-^. The maximum occurs in late winter (February- March) and the minimum without exception in August. In the western squares the densities lie almost on a straight line inclined at an angle of 135° to the temperature axis. The more or less sinusoidal annual variations in T and S show therefore a phase difference of about six months. Similar investigations for other oceanic regions are entirely missing. Bohnecke (1936) has given a chart showing the annual variations in surface density over the entire Atlantic. As may be seen from this chart in the large areas of the North and the South Equatorial Currents the annual variation in ct< is generally less than I-O. It rises locally above 1-5 only at the boundary between the North Equatorial Current and the Equatorial Counter Current (about 10° N.). In the tropics and the subtropics the annual variation is on the whole large only in those areas, where there exists a large annual variation in salinity (mouth of the Amazon, Gulf of Guinea, region with upwelling water east of Cape Verde Island). In higher latitudes the annual density variation remains, in general, also between 1-0 and 1-5, only falling below 1-0 north of Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 187 50° N.; however, in regions close to the coasts seasonal displacements of different types of water also cause large annual density variations (>20). 2. Density Distribution at the Surface of the Ocean It is very characteristic of the density distribution at the surface of the ocean that in spite of the extended strong salinity maximum in middle latitudes there is a rather regular increase of density from the equatorial regions towards the poles in all oceans. This already points towards a decisive influence of the temperature. Figure 86 shows the distribution of density at the surface of the Atlantic Ocean according to Bohnecke (1936). This picture illustrates the meridional increase from about 23-0 at 7°-8°N. to a value somewhat larger than 27-0 in higher latitudes mentioned above. Table 77 gives mean values for successive latitude zones of 5 degrees width. The increase is not entirely uniform in all these zones ; the regions of subtropical convergence stand out as zones with a smaller density gradient and this gradient becomes larger again only near the oceanic polar fronts. Beyond the extensive areas of maximum density in subpolar and polar regions of maximum density the surface density seems again some- what to decrease. Table 77. Mean meridional density distribution in the Atlantic (o-^) Latitude 0° 10° 20" 30° 40° 50° 60° 70° Northern Hemisphere Southern Hemisphere 23-50 23-50* 23-28* 24-53 24-48 25-31 25-44 25-42 25-90 26-06 26-69 26-75 27-25t 27-15t 26-61 26-93 * Minimum; f Maximum For the Indian and the Pacific Oceans the surface density charts of Schott (1935) give only summer conditions for each hemisphere. These charts show essentially the same basic features as in the Atlantic. In the northern Indian Ocean only, conditions are somewhat complicated due to the large annual variations in salinity. The large differences in density between the Bay of Bengal with values of 22-0-1 8-0 and the Arabian Sea with an increase to 23-0 or even to 24-0 should particularly be mentioned. 3. Vertical Density Distribution and Horizontal Charts for Different Depths The density is equally expressed by the quantity a, for the deeper layers. In this quantity the effect of pressure acting on the water mass is not taken into consideration and it refers therefore to zero sea pressure. As a rough approximation, ct< can be taken as the density which would occur in a water mass after displacement ofthe mass with its in situ temperature and salinity from the depth to the surface (potential density); thereby only the adiabatic temperature effect remains out of consideration. For a study ofthe vertical density stratification ofthe ocean it is necessary to go back to the values of the density or the specific volume in situ. Table 78 contains values for a standard sea at 0°C and 35%o salinity, the vertical distribution ofthe density 0-^,^,5, and of the specific volume a^.^^p, and the corrections which must be applied to these as,<,^ to obtain the distribution at 35%o for 10° and 20°C, respectively, or at 0°C for 32-5 and 37'5%o, respectively. 188 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution Table 78. Density and specific volume for dijferent s, t, p (ct^, j, j, and a^, <, p) Depth (m) Pressure Density o„ 1, , (OX, 35°/oo) Specific vol. "it 1. V (0 C, 35»/„o) 357.0 ( 0°C ! dbar 10°C 20°C 32-5»/oo 37-5»/oe 2813 28-23 28-36 28-61 29-08 29-56 3003 30-50 32-85 37-52 41-09 46-40 50-72 0-97264 253 242 219 174 129 084 040 0-96819 388 0-95970 566 173 0 + 109 + 0 X 10-' + 318 + 0 X 10-' 1 -191 - 0 X 10-' + 190 + 0 X 10-* 25 50 100 200 300 400 500 1000 2000 3000 4000 5000 + 0 + 1 + 2 + 4 + 7 + 9 + 11 + 21 + 41 + 60 + 77 + 1 + 2 + 3 + 7 + 11 + 14 + 17 + 34 + 66 - 0 - 0 - 0 - 1 - 1 - 1 - 2 - 4 - 7 -11 -14 -17 + 0 + 0 + 0 + 1 + 1 + 1 + 2 + 4 + 7 + 11 + 14 + 17 This type of presentation was chosen in order to allow differences from the values for standard ocean to stand out. The correction terms enclosed by rectangles refer to the quantities already considered during the determination of o-^ and a^. It is obvious that these are the main correction terms. It is, however, generally customary to judge the vertical density stratification from the a^-values. This will also be done here and the more correct cr^^f^j, and as,t,p will be considered again later. From stability considerations it is to be expected that the values of cr^ will increase with depth. Apart from the surface layer down to about 50-100 m, this is always the case. In the tropics and subtropics the increase is characterized by a transition layer which begins just beneath the top layer, rising to a maximum gradient, then slowly changing towards the deeper layers to a much smaller gradient. Towards higher latitudes the intensity of the transition layer decreases more and more and beyond 35° N. and S. it becomes of no significance. In the Atlantic, for example, it can then scarcely be regarded as a transition layer. In these regions the vertical density gradient decreases steadily from the surface value downwards. In polar and subpolar regions the density gradient from the surface layer down to the sea bottom becomes minimal. Figure 87 shows the vertical distribution of o-j for some stations for which the T- and ^-distributions were given already in Figs. 52 and 69. In the uppermost layer a small increase in the a^-values with depth can occasionally be noted (see the first three stations of Fig. 87). This does not, however, necessarily mean that the vertical density stratification of these water masses is unstable. Because reduction of the a^-values to the more correct Og^f^p may remove these small differences as happens in the three cases in Fig. 87. There still remains, however, a large number of stations where there is undoubtedly a state of weak instability (see Chap. V, 6). A better insight into the nature of the vertical cr^-distribution through the entire ocean is given by constructing longitudinal sections. Wiist has prepared sections of this type for the Atlantic, indeed he chooses the same sections as for T and S (see Fig. 62 p. 146 and 147). Figure 88 presents the o-rsection along the Western Trough of the Atlantic ; the others show in principle the same picture. Although at the surface there is a general slow increase of density, from the equatorial zone towards high southern and northern latitudes, already at 100 m depth and below a diflferent distribution is Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 189 T 25 26 27 28 of 23 24 25 26 27 28 27 28 500 1000 1500 2000 2500 3000 - 3500 4000 / . ' 1 ,- \ i. ' 1 \:~-. \ \ 1 \ \ 1 \ \ \ \ \ \ \ \ 2j 3| 41 - - 1 1 1 26 27 28 Fig. 87. Vertical distribution of the density cr^ at some oceanographic stations in the Atlantic: 1. "Meteor" 254 2° 27' S. 34° 57' W. 2. "Meteor" 170 22° 39' S. 27° 55' W. 3. "Meteor" 8 41° 39'S. 30° 06' W. 4. "Meteor" Greenland 122 55° 03' N. 44° 46' W. found that resembles more closely that of the temperature at these depths. In the subtropics of each hemisphere the lighter water extends down to great depths while in the equatorial zone the heavier waters of the deeper layers extend higher upwards to just below the strongly developed density transition layer. This gives rise to a horizontal density gradient from the equatorial zone towards the two subtropical regions, that is opposite to the surface gradient. This gradient remains unchanged in direction, though becoming weaker and weaker down to about 2000 m below which the meridional density differences are usually rather small. In all the vertical sections there is, however, a weak density gradient from high northern latitudes across the equator to as far as 40-50° S. which is connected with the oceanic circulation of the deeper layers. It is readily understood that horizontal charts of a^-values show in principle the same picture. A comparison of such charts with charts of the relative topography of the isobaric surfaces (Helland-Hansen and Nansen, 1926) demonstrate that the course of the isopycnals on the horizontal charts is in essential agreement with that of the dynamic isobaths. The horizontal circulation of the water masses can thus be deduced approximately from the horizontal distribution of the o-^-values. In that way stream lines for the relative water flow are obtained (i.e., with reference to the lower layers). Arrows showing the direction of flow are thus often inserted on the isohnes on isopycnic charts of the upper layers to indicate the currents. These are subject to the 190 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 000/1 00091 O C tn O u ':3 '-' -B •2 S ^ & >% "^ g-a ^ g .s 3 '5b c o H-1 uj 'mdao Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 1 9 1 0° 70° 60° 50° 40° 30° 2(f W° 0° m° 20° 30° 40° 50° J20° ^110° ^ 100°^ 90[ Fig. 89. Density of sea water cr, at the surface of the Atlantic (according to Bohnecke). 1 92 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution rule that in the Northern Hemisphere the higher density values are found to the left of the direction of flow, while in the Southern Hemisphere they are found to the right (see Fig. 89). Horizontal charts of the density for 400 and 1 500 m in the Atlantic Ocean have been given in Plate 7 to supplement the above brief remarks. The first chart shows the Gulf Stream system very clearly by the strong concentration of the isopycnals into a narrow belt running from the Gulf of Mexico through the Florida Straits to the New- foundland Banks and beyond to the north-east. Compared with this very large hori- zontal density gradient, that connected with the equatorial currents is only very small. In the Gulf Stream region the 400 m chart indicates another phenomenon that is characteristic of stronger gradient currents and is apparently missing in pictures of the surface current. On the right-hand side of the Gulf Stream some isopycnals deviate outward and turn into a south or south-west direction, opposite to the direction of the narrow band surrounding a strong longish density maximum at the right-hand side of the current core. These backward-turning isopycnals indicate the presence of a countercurrent to the right (to the east) of the Gulf Stream which is of considerable importance for the dynamics of this ocean current near the American coast. In the Southern Hemisphere the isopycnals are strongly concentrated in the regions of the Agulhas Current, the Brazil Current and the Falkland Current. In addition, a steady rise of density exists in the Southern Hemisphere extending around the entire southern ocean which is associated with the broad circumpolar West Wind Drift of the higher southern latitudes. All density charts down to 800 m show very much the same picture, though the density gradient becomes gradually smaller and the density maxima of the subtropics are thereby somewhat displaced towards the poles. At first, a different distribution begins to appear below 1000 m, which dominates in the 1 500 m chart. This is the density gradient from high northern latitudes to the mini- mum zone between 35° and 40° S. This north to south density gradient becomes less and less pronounced with increasing depth and below 4000 m the horizontal density differences become already very small. 4. Potential Density and Isentropic Analysis In earlier times potential density was considered a significant property on which to form an opinion about the state of vertical equilibrium of oceanic stratification. As already stated (see p. 1 88) potential density is calculated from the in situ salinity and the potential temperature. Since the latter differs only at great depths from the in situ temperature and then by only a few tenths of a degree centigrade, the difference between o-^ and a^ remains very small and is almost insignificant as shown in Table 79. It thus makes little difference whether the vertical density distribution is judged by means of the customary a^ or of the more correct oq. The potential density has recently become of greater interest due to the introduction of the method of isentropic analysis. In meteorology, the investigation of the distribution of individual meteorological elements on surfaces of equal entropy has been modernized and this has led to ap- preciable success. Parr (1938, 1938^) has studied the spreading of oceanic water types in a similar way by following the changes in salinity and temperature on surfaces of equal density ct<. Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 193 Table 79. Density a^ and potential density oq at ''Meteor'' station 310 (19-3° N., 25-0° W.) Depth Temp. Salinity Of f^e CT0 — O, ("0 CC) (%o) (density) (potential density) (difference) 0 21-45 36-69 25^68 25-68 000 25 21-35 -65 ■68 -68 -00 50 20-27 •77 26-05 26-05 -00 75 20-01 ■795 •14 -14 •00 100 19-53 ■75 -24 -25 •01 200 15-83 -11 •65 -66 •01 300 13-63 35-74 -84 -85 •01 400 11-69 -44 2700 27-01 •01 500 10-58 •36 -16 -17 •01 600 903 -14 -24 -25 •01 800 7 00 34-94 -39 •40 •01 1000 604 -96 -53 -54 •01 1500 4-50 35-03 -77 -79 •02 2000 3-55 34-965 -82 •84 •02 3000 2-83 -93 -87 •89 •02 4000 2-43 •885 -87 •90 •03 Atmospheric isentropic analysis requires an investigation of conditions on a sur- face of constant entropy. In the atmosphere, provided there is no condensation, these surfaces are identical with surfaces of constant potential temperature and also with surfaces of constant potential density. For oceanic water the relationships between entropy, potential temperature and potential density are not so straightforward as for atmospheric air and in particular, under normal conditions the surfaces of con- stant entropy, constant potential temperature and constant potential density in the sea are not identical sets of surfaces. It can easily be understood that especially the surfaces of equal potential temperature are not identical with surfaces of equal po- tential density by considering the complete dependence of the latter on the locally varying salinity which plays only a minor role in the calculation of the potential temperature. Thus for an investigation of the spreading of the water masses neither one of these surfaces can be favoured, since each satisfies certain conditions which seem to be necessary for such considerations, but are not sufficient to give any of the two methods a special preference. It is thus equally incorrect to denote the method of using surfaces of equal potential density as reference surfaces as "isentropic" method because they have nothing to do with entropy which for sea-water is difficult to define. Since there is, as previously pointed out, very little'difference between the potential density ae and the density Cf (down to a pressure of 1000 decibars or a depth of 1000 m), instead of strictly "isentropic analysis" simply the distribution of the oceano- graphic factors on surfaces of constant a^ has been studied. The method is thus quite simple in practice, but its usefulness is rather limited if one considers strictly its proper limits of applicability, and it offers little advantage over the "core layer" method and other similar methods which will be discussed later. The displacement of water masses 194 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 60° 50° 40° 30° 20° 10° Fig. 90. Salinity distribution on the 25-5 o^-surface in the north-east Atlantic between 0° and about 30° N. (according to Montgomery) (only decimals have been entered as salinity values). within such an isopycnic surface must by definition proceed without changes in the potential density and thus without changes in the potential temperature and the salinity (or in the oxygen content also). If the distribution of the temperature and the salinity (or of the oxygen content) pictured on such a surface show signs of change, these must be due to mixing, and it is therefore possible to investigate these more closely and to follow the main direction of flow and the spreading of different water types by means of isolines. Thereby it was assumed that the mixing in such "isentropic" surfaces occurs pre- dominantly in horizontal direction (that is in the direction of the surface) and to a much smaller extent in vertical direction (normal to the surface). This assumption is not entirely justifiable and may be satisfied only in cases where the Cj-surface runs just within the density tran ition layer, since here the exchange coefficient in vertical direction is strongly reduced due to the great stability of the vertical stratification, and lateral exchange is thus very much favoured. Outside the density transition layer, however, there is no reason to assume that the effect of vertical mixing is less important than that of lateral mixing, especially as the reduced magnitude of the vertical ex- change coefficient is compensated for by rather pronounced vertical gradients of the oceanographic factors, as was seen earlier. Montgomery (1938) has applied this method to determine the oceanic circulation of the upper layers of the southern North Atlantic. The results of this investigation will be discussed later in connection with the dynamics of ocean currents; here only the method for the use of the o-^-chart will be presented. Figure 90 gives an example of such Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 195 an "isentropic" chart for the salinity distribution at the 25-5 c7<-surface in the North Atlantic between 0° and 30° N. This surface intersects the sea surface at the dotted line and south of this it lies mostly at a depth of between 75 and 125 m. The arrows show the main direction of spreading of the highly saline {S) and low saline (F) water according to Montgomery. The arrows pointing in the west-east-direction show the Equatorial Countercurrent and correspond to actual flow. Only the east-west-arrows in the low-salinity tongue between the Equatorial Current and the southern branch of the north Equatorial Current and those directed from south to north off the West African coast may have little relation to actual currents; the first low-salinity tongue represents the salinity minimum between the intrusions of highly saline water to the north and the south, the latter minima are due to upwelling water off the West African coast. 5. The Vertical Equilibrium in the Ocean and Stability The use of the potential temperature 6, or the potential density a^, as criteria for the equilibrium conditions in the sea is only correct if the salinity is constant everywhere. Under these conditions the equilibrium is stable, indifferent (neutral) or unstable according to whether daejdz = 0. Correct equilibrium conditions can be derived in the following way: a small mass of water displaced from a level r by a vertical distance A^ towards the surface comes to a density p, while the surrounding water at this point has a density p'. This displaced water quantum will then be subject to a vertical accelera- tion proportional to p — p'. If the difference is positive then the displaced water mass will be subject to a downward force tending to move it back to its previous position ; the equilibirium is then said to be stable; if the difference is negative then it is subject to an upward force tending to displace it further and further away from its new position — the equilibrium is then unstable. If, after a displacement, it always has the same density as the surrounding water then the equilibrium is indifferent (neutral). The difference p — p' per unit length is thus a measure of the state of equilibrium. Hesselberg (1918) therefore denoted the expression E = Spjdz as "stabihty", where Spjdz is the individual change in density (in contrast to dp/dz which gives the geo- metric change in p with height). For positive values of £ the stratification is stable and is not altered by vertical displacement of individual small water quanta. For negative values of E the stratification is unstable and the slightest disturbance is sufficient to cause a new adjustment in stratification (Ekman, 1920). Between layers with positive and negative stability there is always a surface with E — 0. A small mass of water on displacement to the side where E is positive is always driven back to the surface, but a displacement to the side where E is negative removes it more and more from that sui face. Hesselberg and Sverdrup (1914, see also, Schulz, 1917) have given a simple method for the calculation of the quantity E. If a small water quantum at a depth z at point a (Fig. 91) is subject to a pressure p and has a salinity s and a temperature ^, at a depth z + dz, the corresponding values are p -\- dp, s ~\- ds and {}• + d^. If the water quantum is displaced near to point a, it will be subject to the pressure p and it will retain a salinity s + ds, but its temperature will change due to adiabatic expansion 196 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution B >a • b P s+ds ^+d4-dt z-¥dz p + dp s+ds 4+di Fig. 91. Calculation of stability. by dr so that its temperature becomes & + d& points b and a will thus be dr. The density difference between dp dp Pp,s+ds,&+d»-dT — Pp,s,9 ~ ^ ^^ ^ 'M. ^^^ ~ ^'^^■ The stability E is then given by the expression : dp ds dp id'd' ds dz dd' \dz E = dr Jz The geometrical changes in salinity and temperature ds\dz and ddjdz for the depth z at a give station can thus be determined from the given values of T and S, and the temperature gradient drjdz as well as dpjds and dpjdd' can be found from hydro- graphic tables. If the salinity is constant in vertical direction (dsjdz = 0) then d^ dr d doe 1z This is in agreement with the previously given equilibrium condition for the potential temperature. For a given vertical change in salinity its effect on E is so large that it cannot be ignored. "Meteor" St. 310 (see Table 79) has been selected as an example for the vertical stability distribution; the E distribution is given in Table 80. In the top layer down to 25 m there is a very weak negative stability and just below the top layer E lises to very large values. This is the density transition layer where the stratification of the water is extremely stable. Underneath the stability decreases some- what to assume a value of about 100 at the boundary between the oceanic troposphere and the stratosphere. Tt then decreases steadily approaching neutral equilibrium in the greatest ocean depths. All tropical and subtropical stations show similar conditions. Towards polar latitudes the large positive values of £" in the upper layers disappear and are replaced by a more uniform, however, not espec'a!ly la ge stability; only the sur- face layer can be disturbed to any extent by changes from season to season. The vertical stability at great depths in the deep-sea trenches is of particular interest. Since in these the salinity is very largely constant the vertical stability conditions can be estimated fairly accurately from the potential temperature (see p. 127). According Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 197 Table 80. Stability in the Atlantic (10^ X E) Depth "Meteor" All Depth "Meteor" All im) St. 310 "Meteor" stations (m) St. 310 "Meteor" stations 0-25 -6 900-1000 75 63 25-50 1541 — 1000-1200 57 59 50-75 360 — 1200-1400 63 48 75-100 377 — 1400-1600 43 35 100-150 343 — 1600-1800 17 24 1800-2000 16 181 150-200 514 — 200-300 219 202 2000-2250 13 12-6 300-400 168 151 2250-2500 9-5 104 400-500 152 120 2500-3000 8-6 8-2 3000-3500 7-3 7-9 500-600 113 102 3500-4000 3-6 84 600-700 108 75 700-800 74 70 4000-4500 — 8-6 800-900 92 65 4500-5000 — 3-3 to the observations made by the "Snellius" Expedition (Schubert, 1931) the Philippine Trench shows the values of £■ X 10^ given in Table 81. PoLLAK (1954) has given a different definition for the stability which has some ad- vantages in many cases. It gives somewhat different values for E, but differences re- main in the limits. Table 81. Vertical stability (10^ x E) in the Philippine Trench according to the observations of the ''Snellius''' Expedition Depth interval 3500- 4C03 4000- 4500 4500- 5500 5500- 6500 6500- 7500 7500- 8500 8500- 10,030 108 X E + 1-2 +0-8 -0-3 +0-5 +0-7 +0-2 +0-8 It may also be of interest to deal with another equation for the vertical stability which shows clearly the difference of the £-values from the vertical density gradient datjdz which has often been used previously as a measure of stability. The density Ps,i^,j, in situ is calculated from hydrographic tables by applying three correction terms to the value The first of these e^, depends only on the pressure p, the second e.,,,, depends on the salinity and the pressure and the third e^.j, depends on temperature and pressure, • Then Ps,9,p = 1 + [o'l? + e-j, -f €5, J, -f 6^, J,] and Ps^ds, &^d&-dT, p — 1 + dp ^d+dd- -f ^j) + ^s+ds,p + ^9+dS, p ^ dr 198 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution From these equations one obtains hp = da, + -^ds-\- -^^ ^^ - a^ ^^ and thus dag. dcg^ J, ds de&^ ^ d'& dp dr ^ ^ ~dz '^ 8s dz "^ ~dif dz ~ ~8& dz' In this expression for E the first term is usually the main one and the others are only correction terms; the second term shows the effect of changes in salinity, the third shows the effect of changes in temperature on the compressibility while the fourth allows for the adiabatic temperature effect. Estimation of the order of magnitude of these terms shows that they cannot be neglected; the effect of the temperature differ- ence on the compressibility must already be taken into consideration for depths be- low 100 m; in deeper layers also the adiabatic effect is of the same order of magni- tude as the first term. In general only the effect of changes in salinity is mostly small. The quantity do^jdz for itself thus cannot give a very precise measure of the stability. 6. The Distribution of Stability in the Atlantic Ocean Schubert (1935) has carried out a detailed examination of stability conditions in the Atlantic Ocean — in particular of regional stability differences in vertical sections and on horizontal charts. Table 80 also gives mean values of E for the entire ocean calcu- lated as means of all "Meteor" stations; the surface layer down to 200 m, i.e. the zone of disturbance, has been omitted. Of the many irregularities in the vertical distri- bution at individual stations, only two remain in the mean values, the most important being that at 1000 m. This is a definite intermediate stabiHty maximum. From the location of this rather strong interruption, or sometimes even reversal, of the normal decrease of stability downwards, the decrease in stability is considerably larger than before. This irregularity is present at about the same depth throughout the total ocean in temperate and tropical latitudes, and is connected with the subantarctic intermediate water. Its basic cause is the reversal in the salinity gradient. There is another secondary maximum imposed on the regular decrease of the E- values at a depth of 2000-4000 m. In contrast to the more sudden change at 1000 m a weak and more gradual increase in stability is characteristic. In the regional variability of the stability in particular, a strong decrease towards higher latitudes stands out. The higher values of E disappear already beyond 50° latitude; the greater uniformity and lower values indicate that only in higher latitudes do favourable conditions for vertical displacements of water exist. Solely by this, higher latitudes become the principal regions of origin for the deep-sea circulation of the oceanic stratosphere. Characteristic stability conditions are found in the top layer down to 100 m or occasionally to 200 m where frequently negative values occur. Apart from cases in the upper 25 m, where they are very frequent, these negative stabilities were formerly regarded as due to observational errors (especially in the salinity). However, variations of 0-01%o are in fact quite sufficient to explain them (Helland-Hansen, 1910). Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 199 Observations of more recent expeditions have shown that negative stabilities extend- ing down at the most to about 250 m are of such a frequent occurrence, that they are difficult to account for by observational errors alone. For example, in ninety-five cases with E greater than —100 the observational errors must be 0-04%o in S or 1°C in temperature. There is, however, further confirmation of the reality of this pheno- menon. This comes from the occurrence of negative values throughout the entire layer, and the fact that mostly a pronounced regional distribution of stations with negative values of E is found which would scarcely be possible if random observa- tional errors would have been made. In the Atlantic, for example, there is an extended area with negative values of E in the entire open ocean from 50° S. to 20° N. The highest negative values (< —200) fall within a latitudinal zone between 15° and 20° S. and there is probably a corresponding zone also in the North Atlantic approximately between 20° and 30° N. This instability in the top layer in tropical and subtropical areas must be due to the eff"ectiveness of evaporation. The increase in salinity and the decrease of the temper- ature at the surface leads to an increase in density and to a reduction in stability. Solely incoming radiation during day time works in the opposite direction, which compensates the density increase by a corresponding rise in temperature, but during night time when incoming radiation is missing and evaporation continues, the density increase will predominate and negative stability values can persist for a considerable time as long as the intensity of evaporation is sufficient. It is, however, a rather pe- culiar phenomenon that a vertically unstable stratification can be maintained for a longer time over such an extended area in the top layer in spite of convection and mixing. Fig. 92. Circulation in a convection cell according to Benard. Perhaps a possible explanation lies in the "convection cells", first observed and investigated experimentally by Benard (1901). He was able to show that when a rela- tively thin layer of a liquid with volatile components was cooled by evaporation, the entire mass of the liquid divided into a number of cells. In each of these the liquid rises in the centre, diverges in the upper part of the cell and descends again in the outer parts as shown schematically in Fig. 92. The diameter of the cells corresponds to about three or four times that of the thickness of the liquid layer. Instability in the 200 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution stratification is associated with such convection cells and is maintained by the circula- tion. Rayleigh (1916) and Jeffreys (1928) investigated sucha Benard cell theoretically and showed that there could be an equilibrium state with an upper layer of greater density on top of a lower one with smaller density if the vertical density difference between the upper and the lower layer was less than a certain limiting value given by the inequality < Agli" ' where k is the molecular thermal conductivity coefficient, v is the kinetic viscosity co- efficient and h is the thickness of the liquid layer. The unstable density difference is largest in the upper part of the layer; as long as the loss of heat by evaporation tends to maintain the unstable stratification the circulation will continue. It will, however, cease immediately as soon as the evaporation ceases. If there is a steady current in any direction in such a liquid the convection cells resolve into long bands with a corres- ponding transverse circulation. It is not impossible that the existence and maintenance of density instability in the top layer of the ocean has something to do with such phenomena. However, in order to simulate conditions actually found in the ocean, the influence of radiation and evaporation and especially that of the eddy conductivity and eddy viscosity must be taken into account in the above inequality, instead of the molecular thermal con- ductivity and the molecular viscosity. For a layer 25-50 m thick resting on top of a transition layer with a stable stratification, the above inequality will give a value for (p' — p) of the order of magnitude of the observed negative stabilities. By the effect of the circulation a mechanical instability is thus changed into a dynamic stability. In more recent times the theory of convection cells has been considerably advanced and has been discussed in detail in a symposium on the problems of boundary layers and convection cells in the Section of Oceanography and Meteorology of the New York Academy of Sciences, 1942. Stommel (1947) has presented a summary of the theory of convection cells which should especially be mentioned. Neumann (1948) has paid special attention to cell convection in the sea and has shown that indifferent (neutral) stratification occurs only when ^0 A"- F = - ^ Pg h'' where A^'is a. dimensionless quantity of the order of M X 10^ in the ocean, A is the vertical exchange coefficient and h is the thickness of the layer. This equation follows directly from that given by Rayleigh if the above-mentioned change from molecular into turbulent conditions is introduced. The greater the thickness of the layer h and the smaller the exchange coefficient A, the smaller is the decrease in density with depth that is still compatible with static equilibrium. Convection starts only when denser water is situated on top of lighter and when A in the above equation exceeds the critical value 1 100. At the "Meteor" anchor station 385 (16° 48-3' N., 46° 17-1' W.; second continua- tion of the German North Atlantic Expedition, February 1938) it was found, as a Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 201 mean of sixty series of observations, that the water at the sea surface was always appre- ciably denser (heavier) than that at 6 m depth and even at 1 5 m depth the water was still specifically lighter than at the surface. Taking ^ = 100 g cm-^ sec-^ and h = 500 cm, then E = — 16 x 10~^ ; this means that convection is initiated in this layer at this value and not at £" = 0. If the turbulence becomes stronger the critical value of E increases rapidly and strong density gradients are required for any start of convectional motion. The long lines of foam often observed on the surface of the sea can be regarded as "convection rolls" formed by a combination of a strong current in a single direction, and circulations in convection cells in the above sense. Their frequent occurrence is an indication that regular formations of Benard convection cells occur in the sea. Chapter VI The [TS] -relationship and its Connection with Mixing Processes and Large Water Masses 1. Temperature as a Function of Salinity and Large Water Masses Temperature and salinity vary with the depth h or the pressure p, and an investiga- tion of the vertical distribution of these factors is based mainly on a graphical repre- sentation of the variation of these quantities with depth h. In this way it is almost unconsciously assumed that these factors (temperature and salinity) are independent of each other. This is, however, not the case. Assuming salinity as a function of tem- perature or plotting it against temperature in a system of co-ordinates (tempera- ture as ordinate, the salinity as abscissa) the points for each depth are not distributed at random over the diagram but fall on a definite, more or less smooth curve. It is found that for oceanic regions with uniform oceanographic and special climatic, as well as undisturbed flow conditions, the [r^J-relationship is quite characteristic. A given temperature corresponds to a given salinity regardless of the depth. The prac- tical significance of this [r^J-relationship was first pointed out by Helland-Hansen (1918) and since then it has become increasingly important. Any given water type, a water mass, formed continuously in a particular oceanic area for any kind of condi- tions is characterized by a definite temperature and a definite salinity. If this water mass is homogeneous then the oceanographic factors in it are constant and it can be represented on a [r5]-diagiam by a single point. If this water mass is moved in any direction without altering its physical-chemical structure the point does not change its position on the diagram. However, under influence of certain processes, for instance mixing, radiation or evaporation, the water mass loses its homogeneity and the position of the point in the co-ordinate system is changed. Such changes occur espe- cially in the top layer (down to 200 m), where climatic conditions are able to pro- duce continuous "disturbances" in the normal state. Beneath the top layer with dis- turbances, however, conditions in the ocean are qudL^i-stationary and thus every station has its characteristic [r5']-curve which for that special station remains largely in- variable. This constancy is, however, not only true for each individual station but applies also in a somewhat wider sense to more or less larger oceanic spaces. Standard curves can thus be constructed for diff"erent regions and conclusions can be drawn about the origin and spreading of a water mass from the deviations of the values at a particular station from those of the standard curves. 202 [TS]-relationship and Connection with Mixing Processes and Large Water Masses 203 Figure 93 shows an example of such a [rS']-curvefor "Meteor" station 171 in the cen- tral part of the South Atlantic. Its shape is characteristic for the entire South Atlantic from 40° S. to beyond 10° N. Its constancy over such a large area expresses well the strong conservatism of vertical stratification which is of course necessary under sta- tionary conditions. If, in addition, lines of equal density Cf (isopycnals) are also included in the same diagram, as was done in Fig. 93, a rather instructive although not com- pletely correct representation of the stability of vertical stratification is obtained. If 34-2 350 360 370 Fig. 93. [75] -curve for "Meteor" St. 171 (22° 1-5' S. 23° 470'W.) in the central part of the South Atlantic (the thin dashed curves are the isopycnals Of). the [TS]-cmyQ of a certain layer runs approximately parallel to the isopycnals the stability in the layer is only small but if the [rSJ-curve cuts the isopycnals at a wide angle the stability is larger. For greater accuracy the [J'5']-curve must be constructed by using potential temperatures, but the differences in most cases remain small. As with temperature, so can any other property of sea-water be combined with the salinity in exactly the same way. Such a combination was made in particular with the oxygen content in order to see how changes in the oxygen content affect the temperature and salinity conditions, which determine the water mass. 2. Practical Significance of the [T^S"] -curve The [rS'l-curve offers advantages in the scientific preparation of oceanographic data and is used to detect errors and to make it homogeneous. If the value for a particular depth at an oceanographic station does not fall on the simple, regular and usually smooth [rSJ-curve it can be confidently assumed that there is an observational error or a fault in calculation (for examples see Merz, 1925). The [r5']-curve is thus a reliable criterion of the accuracy and homogeneity of a set of data. Since curves for neighbouring stations are similar all values can be checked immediately, but a faulty ©> 204 [TS]-relationship and Connection with Mixing Processes and Large Water Masses observation can also thereby be replaced by an approximate, rather more correct value. Only in this way is it possible to perform an objective and satisfactory "inter- polation" of oceanographic values in order to fill gaps (missing data) in the observa- tional material. 3. The [r5'] -curve and the Mixing of Water Masses If two homogeneous water masses are mixed in any given proportion, the mixture will have a definite [rSJ-curve. Each of the two homogeneous water masses is characterized by the two points, 1 {s^, §i) and 2 {s2, i dz), in the co-ordinate system, proceeds in the ordinary way; if two masses are mixed in the ratio nti : Wg then the mixing final temperature and salinity of the mixture will be given by -& = mi + /«2 s = m^Si + AWa^a m^ nu An example is presented in Fig. 94 where a homogeneous water mass U (10°, 35%o) from 100 m to 500 m depth is situated above a second mass Z (5°, 34-5%o) which extends down to a depth of 900 m (Defant and WiJST, 1930). These two homogeneous water masses are represented in the [7'5']-diagram by the two points U and Z. The boundary surface at 500 m depth, which is initially a sharp physical discontinuity surface, gradually disappears due to mixing. Different stages of this destruction of the dis- continuity is shown on the left-hand side of Fig. 94 (Defant, 1929). It is obvious that, whatever the ratio of mixing of the two water masses may be, the mixture will be represented on the diagram only by points lying between U and Z. However, the graphical construction shows that all points of the mixture must be situated on the straight line from U to Z and that only the depth changes on this line according to the intensity of mixing. This is readily shown theoretically (Defant, 1935). It can also be demonstrated that the distance of any point along the straight line from the two end-points (representing the two original water types) is inversely proportional to the ratio of mixing, the result of which is the mixed water type at the point in question. It is thus simple to determine from the position of a point relative to the end-points U and Z in Fig. 94 to what degree (in percentage) the final mixed water mass under consideration is composed of each of the original water types. The case where three water types are mixed is illustrated in Fig. 95. The three types are: Water mass U Z T Layer thickness (m) 100-500 500-1 COO 1000-1500 Temperature (°C) Salinity (%„) 10 350 5 34-5 5 350 The thermal boundary surface at 500 m disappears in the same way as in the pre- vious case. The salinity boundary surface does the same up to the time when the inter- mediate water mass Z becomes involved in its total height in the mixing process and in that way is slowly destroyed at its core. An advanced stage of this is shown in the [TS]-relationship and Connection with Mixit^g Processes and Large Water Masses 205 0 Mixing of 2 homogeneous woter bodies - U U - 200 - 10° y* 400 - z u z .^ u 8° 6° >4 600 r:^'- - ^ - z/ 600 - 4- S- f{t) - z z . 1000 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2° 1 1 1 1 1 1 1 1 Q 4° 6° 8° 10° 34.4 346 348 35 0 S, 7oo 34 4 34 6 34 8 35 0 5, %o - u u - 200 - : , 10° 400 - z J] u Z ■ u 8° 6° A, - z4 600 1 / ( ■■'■■■' 800 - ; [■ 4° S--f[t) - z z - 1000 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 2° 1 1 1 1 1 1 1 ' 4° 6° 8° 10° 34 4 34 6 34 8 35 0 34 4 34 6 34 8 35 0 /, "C 5, %o S. 7oo Fig. 94. Mixing of two homogeneous water masses and the resulting [JSJ-relationship. Mixing of 3 homogeneous water bodies 500- 2 1000 1500 - u U - - A-::-^^ u y-'j:. -^) u ^ v^'" z "^ ~ ~ z r z 'S^i^. - r 1 1 1 1 1 1 1 1 1 ! 1 1 1 T 1 4 6 8 10 /, "C 34 4 34 6 34 8 35 0 S, 7o, 34 4 346 348 350 S--f\t) Fig. 95. Mixing of three homogeneous water masses. 206 [TS]-relationship and Connection with Mixing Processes and Large Water Masses ^'-distribution by the dotted line. In the [J^J-diagram the mixing of the three water types is represented by the broken hne UTZ, but the point Z remains on this [r^]- curve only as long as the core of the intermediate water is not involved in mixing with the water masses U and T. When this happens the Z-point moves into the acute angle and the [TS]-c\xr\c no longer has a peak point at Z but becomes rounded there. The reversal point and the concentration of the depth marks around it shows the core of the intermediate water already affected by mixing, but even at this advanced stage the mixing of the three water masses can be represented by a curve VZT made up of two straight lines. Analysis of actual [r5']-curves of the oceans show essentially these main theoretical characteristics; they are remarkably constant over large oceanic regions, they have characteristic reversal points associated with the cores of the individual water types and large parts of these curves often show a surprising approach to a straight line. In these cases the [r5']-curves allow a precise determination of the depth, temperature and salinity of the water masses, which finally combine and form individual water types in the deep layers and they also allow the percentage of the individual compo- nents to be found at all intermediate stages. In the foregoing discussion it has so far been assumed only that mixing proceeds according to the usual mixing rules ; the magnitude of the exchange coefficients is not involved. The percentages of the original component-waters before mixing give no information on this point. For obtaining a connection with the [r^j-curve the basic equation given on p. 106 d^s 8s is required. This implies that in order to secure a stationary state the vertical exchange and the horizontal advection must completely balance. By choosing for the origin of the A"-coordinate (pointing positively in the direction of movement of the water mass under consideration in a longitudinal section) that point where the water mass 1 is still pure, then the salinity at a distance x will be s^ and at a distance x -{- dx will be Sx+Ax and one obtains with sufficient accuracy: 8s Az 8^s — Ax = s^-\ ^-^ 8x pu 8z' ^x-^-Ax — ^x \ a^^-^ ^x y' p_2 ^•^• If Sj. is formed from s^ and S2 in the proportion m^ and mg of water masses 1 and 2 and Sj._^^x in the proportion m^ — Am and Wg + ^w, then using the mixing rule the above equation transforms to (52 — Si)Am Ag 8^s . nil + f^h P" ^^ If now m^ + Wg is replaced by the distance D of points 1 and 2 on the [r5']-curve and Am by the distance AD of the point (s -{- As; >& -\- Ad) from the point (5, d), then A 2 S2 — Si AD 1 pu D Ax (8^sl8z^) [TS]-relationship and Connection with Mixing Processes and Large Water Masses 207 This formula allows the value at any point along the line of spreading of a water type to be calculated from the [^l-curve if the vertical distribution of salinity (or of temperature) is known. In the special case of a tongue-like spreading s is given with sufficient accuracy by the simple form (see p. 106, et seq.) •s = -^0 + Rx) cos ^^ z. Then and for the core layer (z = 0) 8'^s _ 772 dh 7r2 Because /(I) = 5o — 5i and /(2) = ^o — Sz, and therefore S2-s^=f{l)-f(2) we obtain ±,_^ /(I) -/(2) ^J> pu TT^D fix) Ax ' The application of this equation to the core layer of the subantarctic intermediate water along the western section in the Atlantic gives values for between 0-6 and M which is in rather good agreement with those determined by other methods. However, this method, using the [75]- relationship, also gives only the ratio between vertical exchange and velocity. An interesting method that also uses the [r^l-relationship and allows a deeper in- sight into the process of mixing has been given by Jacobsen (1927). Consider a vertical column of water with cross-section of 1 cm^. From this column we consider two cubes (volume 1 cm^) ai A [z = 0) and also at a point J5 at a distance z beneath A. In the course of a mixing process, which should follow the laws valid for diffusion and occurs within the total column which we assume at rest, there will be an exchange of q cm^ of water in the time of / sec between the two cubes. If the displacement of the water quanta during the mixing process follows a Maxwellian distribution then q = ke-°-'^\ Since there is no increase in mass in the entire water column the integral of qdz from — 00 to +00 must be equal to 1 , This gives a^ = nk^. The amount of salt in cube B is ps X 10~^, where the salinity s is given in per thousand and the increase in salt amount in a small time dt according to the exchange equation is Corresponding relationships with Sq and Asq applies to cube A. The sahnity (sq + Asq) in the cube after a time t is the sum of the salt amount originally present and the 208 [TS]-relationship and Connection with Mixing Processes and Large Water Masses salt increase due to the exchange of water quanta by mixing and is thus I+cc ps X 10"^ qds. Putting with sufficient accuracy (ds\ 1 [d^s\ ^dzj Q 2 ydz^/Q and using the above expression for q gives for the point A (z = 0) Then k is expressed in terms of the exchange coefficient A. The [r^j-relationship for the water column under consideration is presented in Fig. 96, At the reversal point /i,z = Oandthedepthmarks +1, +2, +3, . . ,,and — 1, — 2, — 3, . . ., respectively, correspond to the centre points of water cubes of 1 cm^; the cube at A is thus the zero cube. The circle with a radius R (AO = R) approximates closely to the [r5']-curve at the point A. For the part of the curve under consideration the depth marks are so situated that the arc between each pair of depth marks always corresponds to the same angle a in point O. It is necessary to find the co-ordinates (temperature and salinity) after / sec of the zero cube initially at A . Two points M and A^ at vertical distances z and z -\- dz cut out a volume element of dz cm^ of water. According to the previous discussion a quantity of water gdz is transferred from this element to the zero cube in t seconds. The same quantity of water gdz is also transferred from the symmetrically situated volume element M'N' in the same time. These two quantities of water mix, and according to the mixing rule the mixture 2qdz corresponds on the [rS'j-diagram to the small interval BC which is situated on the radius of curvature AO. It is determined by the distance AB ^ r = R - Rcos (za) = ^Rah^ and BC = dr = la'^z'^dR. The water masses entering the zero cube during time / are not only transferred from the two cubes MN and M'N', but also from all other cubes above and below, and it is easily understood that the T and S values for all these water masses must lie on the radius AO. Mixing of all these differential quantities gives the co-ordinates of the zero cube after / sec. Its position on the [rS'j-diagram will be fixsd by the distance Z along AO. According to the mixing rule this must be given by 2q dr dz. 0 One therefore obtains Z = - RaH. [TS]-relationship and Connection with Mixing Processes and Large Water Af asses 209 On the other hand, the chord drawn through Z perpendicular io AO intercepts an arc on the [r5]-curve with a centre angle ha (the depth marks at the end-points of the chord are -\-\h and —\h) and Z = R- RCOS ajla) + i RaW. Comparison of the two values for Z finally leads to an exchange coefficient This equation can be used for the numerical determination of A if the [r5^-curve for a water column has been found by observation for successive times. In Fig. 97 I Fig. 96. Calculation of exchange coefficients by the method of Jacobsen. denotes the initial distribution which is followed by distribution II after / seconds. It shows the changes that have taken place in the water column during time t. The points are depth marks for the determination of h. The tangent at A cuts the [r^]- curve I at the depth marks h^ and //g; the size of ^ is thus /?! — h^. The equation then allows calculation of the exchange coefficient yi if Ms known. The Jacobsen method appUes almost only to oceanic regions which are practically motionless and in which the gradual disappearance of a disturbance in the vertical structure due to vertical mixing can be determined by successive measurements. An application to stationary water displacements is possible using the principle that phenomena occurring one after the other in time can be replaced by others occurring side by side in space. Then the [r^J-diagrams I and II in Fig. 97 represent two successive stations at a distance L in the direction of water displacement. If u is the velocity of this displacement then L = ut and from the above relation one obtains Ajpu =h ^jSL. It can be seen that this method again gives only the ratio Aju. 210 \TS]-relationship and Connection with Mixing Processes and Large Water Masses T^ Fig. 97. [rSJ-relationship in a water column at successive times. 4. Further Examples of the [J'S'] -Relationship Extensive use has been made of the [rS'] -relationship in oceanographic investiga- tions of different oceanic regions. A detailed discussion of these investigations belongs to the individual sections on special oceanography and would be out of place here. The attention of the reader will therefore at present be directed more to the method used rather than to the phenomena characteristic for different parts of the ocean. A most intensive analysis of the [TlSj-curves for a single ocean was first made by Jacobsen (1929) on the data collected by the "Dana" Expedition. He divided the North Atlantic into twenty-four areas with approximately uniform conditions, and he derived mean characteristic [r^SJ-curves for these areas, using then these curves to give an interpretation of the formation of the stratification by mixing of the five principal water types. A homogeneous set of data for the preparation of [T^J-curves for the South Atlantic as far as 10° N. has been provided by the "Meteor" Expedition. Figure 98 presents [rSj-curves for the West Atlantic Trough as an example for meri- dional changes. In this region extending over more than 44° of latitude (almost 5000 km) the thermo-haline structure follows the same law almost without exceptions. It is in principle fixed by five water masses U, Z, T, Bn and Bg and corresponding mixing curves. Basic values are given in Table 82. Five points on the diagram characterize each of these water masses together with straight lines joining them, on which the mixed water masses must lie. The variations of the actual [r^j-curves from these ideal curves of pure mixing are surprisingly small, especially when there is a sufficient mass of water in the cores. This is usually the case, though for the subantarctic intermediate water as it progresses from south to north the [7'6']-curve moves farther and farther into the angle between VZ and ZT, as is required by theory, showing that in this comparatively thin layer of water the core is also involved in the mixing process. This case can be used to calculate the ratio Ajpu for the spreading of the sub-antarctic intermediate water by applying the Jacobsen method (Defant, 1954). Figure 99 shows [r-SJ-curves at four successive oceanographic stations from south to north in the Western Trough of the South [TS]-relationship and Connection with Mixing Processes and Large Water Masses 211 Table 82. Water masses of the South Atlantic between 33° S. and 11° N. Temp. Salinity (%o) Antarctic components Subantarctic intermediate water Z 3-25 3415 Antarctic bottom water Bs 0-4 34-67 North Atlantic components < North Atlantic deep water T 40 3500 North Atlantic bottom water By 2-5 34-90 Beneath the disturbed top layerJ approx. 100-200 m \ Subtropical lower water u 180 35-93 Atlantic but solely for 400-1400 m depth. The values for L and h in the Jacobsen equa- tion on p. 210 can be obtained immediately from the curves and in that way Table 82fl is obtained. The mean value of Ajpu is 0-74 and for ti = 10 cm/sec the quantity A is 7-4 cm^/sec which is in good agreement with values determined by use of other methods. A 2°° : West Atlantic Irouqh ^ y S=f(t) J.^ >^ ^'^ 340 «» 3&0 340 14 5 3S0 340 M5 350 34-0 M-5 350 °f .'^ ™®^' J 'C '"• '"'^v -^ i-<'''''5''' /'^!c i[^' /ihU's^'^T i°t ■ Bs^ Bsa*- Bsv' "^ 3>< /€'^ : - / - "*46'E. V. 1. 1935 •5 8 I . .III 0- 35rOO 35-50 %o Fig. 103. Calculation of the equivalent thickness of Atlantic water typeaccording to Jacobsen for the "J. Hjorf'-Station 1 May 1935 (63° 0' N., 3° 08' E.). 50-100 m follows the straight line AP rather well and therefore shows that the water masses of this layer are composed principally of these two components. For different depths the participation of the Atlantic water in the vertical stratification of the ocean at this station can directly be read from the diagram. The following values are ob- tained : Depth in m 50 150 200 250 300 400 410 Participation of component A 0-73 0-67 0-60 0-57 0-43 004 001 The equivalent thickness of Atlantic Ocean water at this station is then computed in the following way : KO-73 X 50 + 0-70 X 100 + 0-64 x 50 + 0-58 x 50 + 0-50 X 50 + 0-24 x 100 + 0-02 X 10) = 108 m The resulting thickness is a measure of the amount of Atlantic water which partici- pated in the formation of the water column at this station. A geographical distribution of the equivalent thicknesses over the entire spreading region of Atlantic water in the Norwegian Sea and in the Barents Sea is a rather good representation of the effect of the Atlantic current and of the heat carried by this current towards the north, and also allows a quantitative evaluation. 5. The Water Masses of the Oceans An accurate analysis of the [rS'J-relation in different parts of the oceans leads to a closer classification of the water types of which the ocean is made up. By a somewhat [TS]-relationship and Connection with Mixing Processes and Large Water Masses 217 schematic treatment it is possible to derive from the complicated forms of the [TS]- relations graphical representations of the characteristic water types of each ocean that give a good insight into the thermal and haline structure of the sea. This also assists in a clarification of the formation, spreading and mixing of the individual w^ater types and thus facilitates a quantitative description of the oceanic circulation of the deep and bottom layers. Table 83 summarizes the characteristic water types of the three oceans and indicates the temperature and salinity ranges in them. These limiting values for temperature and salinity must naturally only be looked upon as a rather crude measure to demonstrate in what extreme limits oceanographic factors may vary. Table 83. Water masses of the Atlantic Ocean Salinity Salinity North Atlantic Temp. (°C) (%o) South Atlantic Temp. (°C) (%o) 1 . North Polar water -1 to +2 34-9 1. South Atlantic cen- 2. Suba ctic water + 3 to +5 34-7-34-9 tral water +5 to +16 34-3-35-6 3. North Atlantic cen- 2. Antarctic inter- tral water +4 to +17 35-1-36-2 mediate water + 3 to +5 341-34-6 4. North Atlantic deep 3. Subantarctic water + 3 to +9 33-8-34-5 water + 3 to +4 34-9-350 4. Antarctic circum- 5. North Atlantic polar water +0-5to+2-5 34-7-34-8 bottom water + 1 to +3 34.8-34-9 5. South Atlantic deep 6. Mediterranean and bottom water 0 to +2 34-5-34-9 water + 6 to +10 35-3-36-4 6. Antarctic bottom water -0-4 34-66 Water masses of the Indian Ocean Temp. CO Salinity (%„) 1. Equatorial water 4-16 34-8-35-2 2. Indian central water 6-15 34-5-35-4 3. Antarctic intermediate water 2-6 34-4-34-7 4. Subantarctic water 2-8 34-1-34-6 5. Indian Ocean deep and antarctic circumpolar water 0-5-2 34-7-34-75 6. Red Sea water 9 35-5 Water masses of the Pacific Ocean Temp. Salinity Temp. Salinity North Pacific (°C) (%o) South Pacific (°C) (%o) 1 . Subarctic water 2-10 33-5-34-4 1 . Eastern South 2. Pacific equatorial Pacific water 9-16 34-3-35-1 water 6-16 34-5-35-2 2. Western South 3. Eastern North Pacific Pacific water 7-16 34-5-35-5 water 10-16 34-0-34-6 3. Antarctic Interme- 4. Western North diate water 4-7 34-3-34-5 Pacific water 7-16 34-1-34-6 4. Subantarctic water 3-7 34-1-34-6 5. Arctic Intermediate 5. Pacific deep water water 6-10 34-0-34-1 and A.ntarctic cir- 6. Pacific deep water cumpolar water C-l)-3 34-6-34-7 and Arctic cir- cumpolar water (-l)-3 34-6-34-7 218 [TS]-relationship and Connection with Mixing Processes and Large Water Masses The central water at 500-800 m in each of the three oceans forms the principal mass which always has a structure with an almost linear [r^SJ-relation and thus manifests its normal mixing in both horizontal and vertical directions. Underneath, and sepa- rated from it by the Antarctic inteiTnediate water, the deep and bottom waters are found in the Southern Hemisphere which have a remarkably similar structure in all three oceans. In the Northern Hemisphere the Atlantic is blocked oif from the Arctic Sea and has little or no Arctic intermediate water, but the Pacific Ocean, on the other hand, definitely shows this water and thus this ocean is of a more symmetrical structure. The North Atlantic and the Indian Ocean show a strongly increased salinity in the layers between 800 and 2000 m due to the inflow of warm saline water from the Mediterranean and the Red Sea. These eff'ects are quite strong and are evidenced even in the southern parts of these oceans. There are no corresponding disturbances in the Pacific Ocean. On careful examination of Table 83 one cannot fail to regard the striking similarity of the thermo-haline structure of the oceans with astonishment. There can be no doubt that this is a consequence of an analogous oceanic circulation driven and maintained by the same forces. Chapter VII Evaporation from the Surface of the Sea and the Water Budget of the Earth 1. Introduction One of the most important problems with which both meteorology and oceano- graphy is concerned is the water budget of the Earth. It can be assumed with a very considerable degree of probability that the cycle through which the water passes is closed. This follows, if a sufficiently long period is taken into consideration, from the constancy of the amount of water on the earth and from the absence of processes which could alter, and especially decrease, the total amount of water present. A stationary water cycle requires that the amount of water passing through any par- ticular part of this cycle (as either liquid, solid or vapour) should not vary with time, and particularly that the amount of water entering the cycle by evaporation from the ocean is returned to it. In that way there is never any permanent gain or loss of water from any point of the cycle. For a quantitative assessment of the water cycle on the Earth it is necessary to make a numerical estimate of the amount of water circulating through it. This can be done either at the place where water reaches the surface of the Earth from the atmosphere (precipitation), or where water leaves the Earth's surface in form of water vapour (evaporation). In both cases the numerical basis necessary for an estimate must be obtained from observations. On the continents the amount of water vapour precipi- tated from the atmosphere can be determined with sufficient accuracy by direct mea- surement of the precipitation, and this quantity can be determined more accurately the denser the network of rainfall measuring stations. The determination of the mean precipitation amount over the sea is, on the other hand, very difficult and is never precise because of uncertainties in the measurement of precipitation on boardship. On the other hand, the accurate determination of the amount of mean evapora- tion on the continents is accompanied by considerable difficulty while the direct determination of the evaporation from the oceans seems possible and can be made more easily because of the more uniform conditions at this surface, however, in practice critical examination is still needed. These circumstances give particular importance to the question of the magnitude of evaporation amount from different regions of the oceans. 219 220 Evaporation from the Surface of the Sea and the Water Budget of the Earth 2. Direct Measurement of the Evaporation on Board Ship and Methods for Obtaining Corresponding Values for the Sea Surface The evaporation from the surface of the ocean can only be measured from ships under way and this involves — probably even more than in direct measurements ashore — a number of sources of error that require special attention. Evaporation differs from other meteorological factors such as barometric pressure, temperature, wind, and cloudiness, in that the apparatus used to measure it is able to exert a very con- siderable influence on the observed values. This greatly increases the difficulty of getting reliable and useful results, because the values obtained are always only relative vahies which give correct absolute values only after the application of suitable correc- tions. Measurements on board a moving ship are made by using a vessel filled with sea- water and hung in a cardan suspension. Mohn (1883) used a volumetric method of determining the amount of water evaporated at a given moment. The loss in weight due to evaporation of the cylindrical evaporation vessel was replaced by refiUing it with fresh water to bring it back to a zero mark ; the evaporation height could thus be determined. A more accurate and reliable method is the determination of evaporation by observing the change in salinity which occurs in the evaporating water as a conse- quence of evaporation. Following a suggestion of Penck, a cylindrical glass vessel is used which has a cross-section of 288 cm^ and a volume of 2400 cm^; it is filled with sea water and placed within a white-painted or nickel-plated mantle that protects the vessel against direct radiation. Chlorine titration before and after the evaporation period gives the increase in salinity and allows a very accurate determination of the evaporation. The mean error in a single measurement is seldom more than 3%; it derives from the uncertainties of the salinity determination and in refilling the vessel, while the diminution in volume during the observation can be disregarded. Denoting with g^ and gg the weights of the sea-water at the beginning and at the end of the evaporation time, withg^ and^j,, those of evaporated water and of pure water and finally with gs that of the salt at the beginning of the evaporation, then gi = gw + gs and gz = (gu, — ge) + gs- The salinity at the beginning and at the end of the measurement is then jj = 103 X — ^ — and s^ = 10^ x gw i g s \gw ge) ~r g s If p is the specific weight (density) of sea-water at the beginning of the measurement and J is the volume of the evaporated water amount from the vessel, it follows that, from the above relations, Si S<2, ^1 g2 = gi- and ge= pJ — ^ — . If p is the specific weight (density) of distilled water at the mean temperature t^ at the time of the measurement and o is the area of the evaporating surface, then the evaporation height becomes J p S2 — Sj^ «<. = -- op s^ Evaporation from the Surface of the Sea and the Water Budget of the Earth 221 If 30%o < s < 40%o and — 2°C < t^ < 30°C, then with sufficient accuracy pIP = 1 -027 and with the above values for o and / So — s-i h, - 88-3 . For mean conditions the accuracy of //^ is 3-4%, which is quite sufficient. If systematic errors in the measurement are avoided (such as spray from the sea, water drips, and inflow of ash, etc.), the difficulty at the present day is not in getting comparable mea- surements of evaporation but in the correction of the values in order to obtain sea surface values. The value required is not the evaporation height in a vessel on board but the considerably different values at the surface of the sea. For this it is necessary to know: (1) the factors on which in the most general case the magnitude of the evaporation depends; and also (2) the differences between these factors in the vessel on board the ship and at the free sea surface. For an answer to these questions it is thus essential, during long-term series of observations on board ship, to perform special additional measurements; for instance, of the temperature of the surface of the water in the evaporation vessel, etc., in addition to the self-evident meteorological observations on board. Measurements of the evaporation in this way have not been made very often. They were first carried out by WiJST (1920, see also Schmidt, 1921) in a fundamental investigation, and these values after critical interpretation were used to derive more correct average zonal values of the evaporation at the surface of the ocean. From all the formulae which have been used many times to calculate the effect of the meteorological factors, the best is the expanded Dalton evaporation formula in the form : he = cf{u){\ + at){0-9Se, - e„). e^ is the height of water evaporated in 12 or 24 h, c is a constant and f{u) takes into account the eff'ect of wind velocity. The last two expressions in brackets, which were termed the "evaporation potential p" by Marvin (1909), take into account the effect of the air temperature / and of the difference between the saturation pressure of water vapour at the temperature of the evaporating water eg and the water- vapour pressure in the air e^. The factor 0-98 takes into account the effect of salinity which hinders evaporation. As it is known if 30%o < s < 50%o at the sea surface then e^ can be put equal to 0-98^5 in the atmosphere whereby the evaporation is almost indepen- dent of the salinity. From reliable measurements on board a moving ship single values of the quotient ejp can be computed and can be related with the motion of the air at the time of measurement, which is identical to the actual wind measured on board the moving ship. A conversion of the evaporation measured on a moving ship into that which was measured at deck height with an evaporation vessel at rest and at the true wind speed over the sea can be done with sufficient accuracy. For correction of the true evaporation obtained from the instrument on board ship to that at the surface of the sea, i.e. of the free ocean, Wiist used the gradient of the meteorological factors between the evaporation vessel and the sea surface. A basis for estimating the gradients of air temperature, humidity and wind speed immediately above the sea surface was obtained from observations in the Baltic Sea (September 222 Evaporation from the Surface of the Sea and the Water Budget of the Earth 1919). The mean values of the gradients used by Wiist are, however, obtained from relatively few observations but appear, as confirmed by later observations, to be of the correct order of magnitude (Shouleikin, 1928, Montgomery, 1936-7/8, Bruch 1940). The total reduction factor for a conversion of this kind amounts to k = 0-48 ± 008. It is seen that the actual evaporation at the surface of the sea is on the average somewhat less than half of the true evaporation measured by an evaporation vessel on board ship. In this way Wiist obtained for the North Atlantic, for example, the following mean evaporation heights (Table 84) for average meteorological conditions. Table 84. Evaporation in the Atlantic Mean evaporation at Mean Mean vessel the sea surface wind evaporation Climatic regions Latitude speed according to Wiist According According (km/h) to Wust to Liitgens mm/day cm/year cm/year cm/year Variable winds 50°^0°N. 30 40 146 66 95 Subtropical region 40°-30° N. 24 5-8 212 95 160 North-east Trade 30°-8° N. 24 7-8 285 128 240 Doldrums 8°-3°N. 10 5-5 201 91 115 South-east Trades 3"N.-20=S. 22 7-3 267 120 225 Subtropical region 20°-40° S. 20 5-8 212 95 175 Variable winds 40'^-55°S. 28 2-8 102 47 100 This table also gives some idea of the values measured by an evaporation vessel in different climatic zones and of the meridional distribution of the evaporation amounts over the Atlantic Ocean. The last column on the right gives values obtained by LuTGENS (1911) from his excellent measurements of evaporation; due to unsuitable correction, however, the latitudinal differences are overestimated, especially the evapor- ation amount in the trade regions, relative to that in the doldrums. The total procedure of a direct redaction of the observed evaporation on board a moving ship suggested by Wiist was later again controlled by Cherubim (1931), and he found, after applica- tion of some refined but not very important corrections, a reduction factor of 0-54 which, however, he multiplied by 1 -08 in order to account for the influence of the motion of the sea giving the final value 0-583. This latter increase in the size of the correction factor by about 8% for the motion of the sea, for which there was no ob- servational evidence, was regarded by WiJST (1936) as unsuitable since there were other factors, some acting in an opposite direction which had not been taken into account and of which the magnitude was equally unknown. The uncertainties of the direct correction are certainly rather large but if the value obtained by Cherubim is taken as a maximum and that obtained by Wiist as a minimum then a mean of 053 can be taken at the present time as the most probable correction factor. Evaporation from the Surface of the Sea and the Water Budget of the Earth 223 3. Meridional Distribution of Evaporation over the Whole Ocean and its Determination from Energy Considerations The mean values of the true evaporation for different parts of the ocean which can be regarded as the direct result of observations have been used by Wiist to give values for latitude zones of 10° width in the Atlantic and for the total ocean. These depend on interpolation and in part on extrapolation and can thus be considered only as a first approximation. The values recalculated with a correction factor k = 0-53 are given in Table 85. The zonal variations in evaporation, with pronounced maxima in the trade wind regions and a low value in the doldrums, are less pronounced in the figures for the total ocean than in those for the Atlantic alone. Due to the relatively large proportion of the Polar Sea with a low evaporation the mean value for the Atlantic is less than that for the total ocean. The mean evaporation for the total ocean found in this way is 93 cm/year or 2-54 mm/day. The limits of error for this mean value and for the zonal values are about ±12%. Table 85. Zonal distribution of evaporation in the Atlantic and for the total ocean (According to Wiist. (Correction factor k = 0-53.)) Total ocean Zone Atlantic (mean over all oceans) (cm/year) (cm/year) 80°-70° N. 8 8 70°-60^ N. 12 13 60°-50° N. 44 44 50°-40° N. 78 78 40°-30° N. 107 107 30°-20° N. 138 130 20^-10° N. 146 133 10°-O° N. 107 112 0°-10°S. 141 125 10°-20°S. 138 133 20°-30° S. 125 125 30°-40° S. 99 99 40°-50° S. 65 65 50^-60' S. 26 26 60^-70° S. 8 8 Mean 91 93 The mean evaporation of the total ocean can also be determined by another method, suggested by Schmidt (1915). It has already been shown in Chapter III/ 1 (see p. 88), in discussing the heat budget, that evaporation is one of the most important items (loss) in the heat budget of the sea. From a comparison of the amounts of heat in- volved in the heat budget for the world ocean the maximum heat amount available for evaporation can be estimated. Denoting the mean annual energy gain of the total ocean surface due to sun and sky radiation by Qs, the energy loss due to outgoing radiation from the ocean to the 224 Evaporation from the Surface of the Sea and the Water Budget of the Earth atmosphere with Qi,, the loss by evaporation with Q^, and the loss by convection (turbulent heat conduction) with Q^, then for a stationary state Qs ^ Qb-\- Qe+ Qh- Introducing R = QJQe and E = QJL, where L is 585 cal/g, the latent heat of evaporation of water, into the basic equation for the heat budget of the ocean (see p. 89) then L(l + R) ' If the radiation terms Qs — Qb and R are known it is possible to calculate the evapora- tion. Schmidt carried out this calculation using, however, R' == QeliQs — Qb) instead of R, and determined R' from general considerations as about 0-70. This gives a mean correction factor k for evaporation measurements on board ship, and he found k = 0-43 as the most probable value. For the extreme case ^^ = 0 (disregarding all con- vectional processes) Kleinschmidt (1921) found an upper value for k of 0-61. The good agreement with the value of Wiist of 0-48 is remarkable. Angstrom (1920) showed that Schmidt's estimate gave too large a value for R. From measurements and energy considerations he concluded that the value of R is only 0-1, which means that of the total gain in energy Q^ — Q^, only 10% will be given off to the atmosphere by convection and approximately 90% used for evaporation. The method of Schmidt has been carried further by Mosby (1936), who attempted in particular to remove the uncertainty in the determination of the incoming radiation Qs by the use of an empirical formula (see p. 91). The values for Q^, thus obtained, are given in Table 86. Table 86. Heat budget for the ocean. (According to Mosby (g cal cm"^ min~^)) Areas of the Latitude Qs-Q, zones in million km^ 70°-60° N. 0040 5-3 60°-50° N. 65 110 50^-40° N. 93 150 40 "-SO"^ N. 125 20-8 30°-20" N. 150 25-1 20°-10°N. 167 31-5 10°-O^N. 171 340 0°-10°S. 175 33-6 10°-20° S. 171 33-3 20°-30° S. 150 30-9 30'-40° S. 129 32-2 40°-50° S. 097 30-5 50°-60" S. 067 25-4 60-70° S. 0041 171 The mean value between 70° N. and 70° S., taking into consideration the ocean area of the separate zones, is estimated to 0-132 g cal cm~2 min~^ Since on average for the entire ocean advective processes are assumed to be of no importance, this is the average amount of heat available for evaporation and convection. However, Mosby Evaporation from the Surface of the Sea and the Water Budget of the Earth 225 could give only an estimated value for the heat contribution to be ascribed to convective processes, v^hich was based principally on Angstrom's investigation. This quantity was finally assumed to be about one-tenth of the heat available for evaporation, so that a heat amount of about 0-119 g cal cm~^ min"^ must be available for evaporation. The estimation of the convectional flow discussed on p. 92 led to a value of about 20gcalcm-2day"^, i.e., about 0014 g cal cm-^min-^ The agreement with the value assumed by Mosby is rather good, but this estimate applies only to temperate latitudes and the value should be increased for warmer climates to 0-030 g cal cm ~-min~^ Choosing a mean value of about 0-022 g cal cm"2 min-^ then the amount of heat available for evaporation will be 0-1 II g cal cm~2 min~^. Since the evaporation of 1 cm^ of water requires approximately 590 g/cal this latter value gives a mean evaporation of 97 cm a year, while Mosby's value is 106 cm a year. The accuracy here is also scarcely more than 10%. These values are in good agreement and within the limits of uncertainty of the value derived by Wiist. Another possibility for determining the value of R was pointed out by Bowtn (1926). For identical eddy coefficients for the diffusion of water vapour and the turbu- lent conductivity of heat, the upward flux of the latent energy of water vapour and heat are given by 0-621 de ^ d§ Q, = -L -y- A -j-_ and Qn = -c^ A ^- (see p. 92 concerning the latter equation). From these equations it follows that O, 0-62 IL dejdz ' Putting p = 1000 mb and L = 585 and replacing the differentials by corresponding finite differences the Bowen ratio is obtained: R = 0-64 -^ es - ea where t?, and '&a denote the temperatures of water and air and e^ is the maximum vapour pressure of water at temperature 'Og and e„ is the actual vapour pressure in the air. Jacobs (1942, 1943) has determined the dependence of the Bowen ratio on latitude in the North Atlantic and the North Pacific and found that R decreases with latitude. The following values were found as the mean for both oceans: Latitude (" N.) 70-60 60-50 50-40 40-30 30-20 20-10 1(M) R 0-45 0.31 0-21 015 Oil 010 0 10 The northward increase is an effect of the continents from which the cold air flows out over the warm sea in the winter. In the Southern Hemisphere this effect is missing so that R may increase only to about 0-25 at 70° S. By making proper use of all observations and methods which were more or less independent on each other, WiJST (1954) has evaluated a mean meridional distribution 226 Evaporation from the Surface of the Sea and the Water Budget of the Earth of evaporation. These mean annual evaporation amounts together with mean annual values of precipitations are contained in Table 87. Table 87. Mean values of precipitation, evaporation and the difference between them (E — P)for the entire ocean {including adjacent seas) (According to Wust, 1954) Evaporation- Zone in Precipitation Evaporation Precipitation degrees cm/year cm /year cm/year 70-65 N. 34 12 -22 65-60 N. 65 20 -45 60-55 N. 77 34 -43 55-50 N. 105 55 -50: 50-45 N. 112t 66 -46 45^0 N. 102 84 -18 40-35 N. 86 108 22 35-30 N. 74 125 51 30-25 N. 63 132 69 25-20 N. 57: 137t 80t 20-15 N. 70 135 65 15-10 N. 103 132 29 10-5 N. 187t 126 -6I: 5-0 146 113+ -33 70-0 N.§ 1010 110-6 9-6 0-5 S. 105t 125 20 5-10 S. 109t 137 28 10-15 S. 94 139t 45 15-20 S. 76 137 61 20-25 S. 68 133 65t 25-30 S. 65: 123 58 30-35 S. 70 110 40 35^0 S. 90 96 6 40-45 S. 110 78 -32 45-50 S. 117t 56 -61 50-55 S. 109 39 -70 55-60 S. 84 12: -72: 0-60 S.§ 91-45 102-1 10-7 t Maxima; : Minima; § Excluding polar zones 4. Geophysical Aspects of Evaporation Problem Evaporation is a physical process that takes place at the boundary surface between water and the air above it and depends on the conditions both in the water and in the air in the immediate vicinity of the surface. The formula showing the dependence of the evaporation height occurring in a certain time on the meteorological factors is usually given in the form hn =f(p) X fiT) X f,(u) X (e, - Ca), where each term represents the effect of one of the meteorological elements (p the pressure, T the absolute temperature, u the wind speed); e, is the maximum vapour Evaporation from the Surface of the Sea and the Water Budget of the Earth 227 pressure corresponding to temperature and salinity of water, Ca is the vapour pressure in the air. Different expressions have been chosen for the functions /i, /g and/3 ^nd a formula of this type is given on p. 220 which shows the dependence of observed evaporation on the prevailing meteorological conditions and with a suitable choice of constants gives satisfactory values. However, it can hardly be assumed that such an evaporation formula which is a product of different functions could give a correct and causative description of the actual physical process of evaporation ; it is rather to be expected that such a formula would be of the form fh = fiP, T, u) {e, — e^, where the function/is probably a complicated function of the meteorological factors. According to the results of research in turbulence, the transport of the water vapour continuously formed at the sea surface into the air immediately above it proceeds by turbulent exchange; the magnitude of this exchange depends on the roughness of the evaporating surface which in turn also depends on the velocity of the air over the water. SvERDRUP (1936, 1937-8, 1951) was the first to attempt to clarify the problem as to how the evaporation process operates at the surface of the sea with a well-defined roughness under the influence of the turbulent exchange. His ideas are based on two circumstances which aie essential for a solution of this problem: (1) Immediately above the water surface a thin boundary layer exists in which the water vapour transport proceeds only by ordinary (molecular) diffusion. (2) Above this boundary layer the water vapour transport proceeds through the turbulent exchange A in form of random movements of the air particles (turbulence). The exchange A (according to laboratory experiments) is a linear function of the height above the water surface and depends on the roughness of the water surface. The latter is described by the roughness parameter Zq, and according to the results of Rossby about the increase of wind velocity with height Zq is considered constant im- mediately above the sea surface (zq = 0-6 cm). This is valid for weak to moderately strong winds. Correspondingly, A = pk^iz — Zo) J- , where r is the tangential force (stress) of the wind, p is the density of the air and kg is the Karman constant with a value of 0-38-0-40 (see Vol. I, Pt. 2). The thickness of the boundary layer immediately above the water surface depends on the wind velocity. The layer itself can hardly be regarded as invariably composed of the same air particles. Since the turbulent eddies will sometimes penetrate down to and into the boundary layer, it must clearly be understood that this layer occasionally disappears completely; however, after some time it will always be re-formed so that a mean thickness of this layer can be introduced. In addition to the theoretical case built up on the basis of these ideas Sverdrup also discussed a second possibility where the water surface was assumed to be "smooth" and the transport of water vapour away from the sea, due to turbulence, starts from the sea surface itself. Observations seem to favour the first case with a diffusion layer and turbulent transport above, and therefore only this case will now be dealt with. For the exchange coefficient A we may write A = pko(z — Zo) u^. 228 Evaporation from the Surface of the Sea and the Water Budget of the Earth if the so-called 'friction velocity'' is introduced according to Karman: The values for Zq and u^ follow from measurements of the wind over the surface and depend on the character of this surface. In the turbulent layer the water vapour transport E (expressed in g cra"^ sec~^) directed upwards is due to the turbulent exchange process and is given by dz where q is the specific humidity which decreases upwards, q may be replaced in this formula by the vapour pressure e according to the well-known formula 0-623 P and one obtains with sufficient accuracy 0-623 de E = A -T . p dz The process of evaporation must be regarded as stationary (E = constant), so that with the above value for ^, if c is a constant, de c dz (z + Zo) * Denoting the value of e at the lower boundary of the turbulent layer or at the upper limit of the diffusion layer (thickness d, z = d) with e^, then integration gives 1 - + -0 ''-'^-'^''d^rj-^' On the other hand, the quantity E was found to be 0-623 . , . ^ dz 0-623 Je E= -— pkou^iz -}- Zo)~ = --^7- pkou^c. The transport of water vapour through the diffusion layer is given by the equation es — ea E' d where S is the diffusion coefficient of water vapour in the boundary layer with reference to vapour pressure. At the boundary of the two layers the water vapour transport is steady so that the necessary condition z = d, E = E' must be satisfied. Considering, in addition, 0-623 b = K p. Evaporation from the Surface of the Sea and the Water Budget of the Earth 229 where k is the diffusion coefficient in cm^/sec then the thickness of the diffusion layer is given by 'e. and for a roughness parameter Zq In d + •] Vp A-o 0-165 u^. = Uz, hl{(z + Zo)/Zo} "^ log{(z + zo)/zo} where «, is the wind velocity at a height z. Finally, the evaporation E is thus obtained from the above formula E = 8u, (es - e^). If the thickness of the diffusion layer is known then the evaporation E can be calcu- lated, if we observe: (1) the wind velocity at a height above the surface of the water, by means of which u^ is found; (2) the temperature and the relative humidity at this height, wherewith e^ is known; (3) the salinity, from which e, can be determined. Only observations can give information on the thickness of the layer d. For this Sverdrup used the values determined by Montgomery (1940) on board the research vessel "Atlantis", wheieby Zq = 0-6 cm was assumed. Table 88 contains this calculation. Table 88. Values of the friction velocity u^, the evaporation E and the thickness of the diffusion layer d for a rough water surface (zq = 0-6 cm) (According to observations of the research vessel "Atlantis") Observation «* 10«£ 10«£ d group (cm/sec) (g cm"- sec"^) ^u* ■" ^«cm (cm) *i 13-2 106 0-30 0-28 «i 14-3 1-34 0-34 0-22 Cl 170 1-98 0-30 0-33 d 18-7 2-86 0-33 0-31 g 24-8 5-15 0-39 0-29 f 25-3 4-64 0-45 0-23 03 25-6 5-82 0-53 016 Cz 29-2 8-49 0-74 009 e 29-2 5-68 0-70 010 h 36-3 6-98 0-80 010 The value of d decreases with increasing wind velocity, and Fig. 103a shows that as a rough approximation d increases linearly with 1/w^. With suitable weighting of each group Sverdrup obtained d = 4-12/z/^. Unfortunately, there are no simultaneous measurements of evaporation available to allow a close test of the theory. Sverdrup with these values of ^) + div S = +m. ot Local changes in the condensate k in a unit volume in unit time can occur in two ways : (1) By the evaporation of a definite amount of condensate or by condensation of a definite amount of water vapour respectively. If more condenses than evaporates, then according to the above argument this change is -\-m; however, in the opposite case, —m. (2) The water content in a unit volume (liquid or solid) can also change if, for instance, part of it is removed as precipitation or is advected by air currents to other levels. For each point in space this movement of condensate can be considered a condensate flow which can be described by a vector 51, The absolute value |Sl| is the amount of condensate which passes in unit time through a unit area of a surface perpendicular to the direction of movement of the condensate. At the point where there is no condensate or if the condensate shows no movement then 1^| =0. The flux of condensate through a unit surface along the normal /m is 5l„ = 51^, and in par- ticular, for z = 0 gives the precipitation amount per unit area and unit time at the surface of the Earth (z = 0). The change of k due to such processes of condensate movement is then given simply by the convergence —div £ of the condensate flux. The condensate continuity equation is then — = —div ^ — m. dt Adding the two continuity equations for water vapour and condensate gives the continuity equation for the total water content finally in the form ^'''""' + "^ + div (p„tt, + S + S) = 0. Ot For a stationary, average state in the atmosphere this equation reduces to div (p^lt) + S + SI) = 0. Imagine now a vertical surface of control B, which parallels the coasts of a (not neces- sarily continuous) continent and reaches upwards to the upper limit of the atmos- phere. Considering a surface element dB with a horizontal normal n directed towards the interior of the continent (landwards). Then, integrating the above equation over the total volume between the surface of control B, the surface of the Earth and the 234 Evaporation from the Surface of the Sea and the Water Budget of the Earth upper limit of the atmosphere, it follows, according to the Gaussian integral law,* that for the total column of air over the land (p^rt) + Q)dB -\\ (B,dL- I ^,dL = 0, where dL is an element of the land surface L ; the two terms of the first integral vanish for the land surface and the upper limit of the atmosphere since at these extreme limiting surfaces either p„ or it) will be equal to 0 or .4 will be 0; for the two other integrals the amounts passing through the control surface B disappear. Now, the mean precipitation amount per unit time on the continent is ^,dL P dL, L and the mean evaporation amount over the continent is Finally, the water vapour flux through the surface B towards the land is S,dL. JL W.- Wr (Pu,^ + S)„ dB. B The condition of a stationary state thus gives one of the basic Bruckner equations as Ec~Pc + {W^ - W,) - 0. If, on the other hand, the integration is taken over the total ocean, one obtains in the same way [the inwards (oceanwards) directed horizontal normal of dB is now —n] the second basic Bruckner equation P, + E, + {W^ - W,) = 0. Integration of the continuity equation over the entire atmosphere above the surface (C + O) gives Pe i Pq ^^ Ee Eg, which can, of course, also be obtained by subtraction of the first two equations. The basic equations for the water budget of the Earth involve five quantities; a knowledge of three is sufficient to evaluate the others numerically. In general, it does not matter which of them we presume as known and which we want to obtain. How- ever, the accuracy with which the different quantities can be determined from the available observations is not the same for each. The precipitation over the sea can be estimated only with difficulty. For that purpose in wide regions of the oceans only the rain density (the mean precipitation amount for a single rain day) and the rain fre- quency (the average number of days with precipitation) are available from ships' * The Gaussian integral law states that the volume integral of a volume Kwith a surface A taken over div a is equal to the negative surface integral of r„ taken over the entire surface A, where // is the normal to A directed towards the interior, so that III 1.va , Dew deposit; of water vapour; 6 e = values smaller than 0-5 rel. units. 4 o o o o ot> , Run off; Removal from and addition to horizontal advection 6. Energy Budget between Ocean and Atmosphere for DiflFerent Oceans and Oceanic Regions The heat turnover between the total ocean and the total atmosphere has already been discussed in previous chapters. It is also of considerable interest to know the energy budget between the ocean and the atmosphere for the individual oceans and for differ- ent parts of the ocean, since on this depend the effects of the sea on the atmosphere above it or, in turn, the influences of the atmosphere on the sea. Such investigations, in spite of their importance, have only recently been made and indeed have been car- ried out almost exclusively by Jacobs (1942, 1943, 1951^, h) and Albrecht (1949, 1951). These investigations are based on the calculation of the evaporation from the formula on p. 230 using the differences ^^ — 'da and e^ — e^ derived from climatologi- cal charts of the oceans. Doubts about these latter values have been expressed by Dietrich (1950), but it appears that any errors that may have been introduced in this way are not systematic but may vary from one region of the sea to another and should, at least in part, cancel out. Calculations of this type have been made especially for the North Atlantic and the North Pacific, for which the climatic charts are more reliable. Such calculations of course give only a rough estimate but they serve, however, to give an approximately quantitative idea of the interplay between ocean and atmos- phere. At first the most important is the pure heat gain by the radiation turnover Qs — Qb, whereby Qg is the absorption of solar and sky radiation and Qi, is the radia- tion loss from the sea surface. Figure 106 shows the geographical distribution according to Sverdrup (1943) of the annual surplus of radiation penetrating the water surface. Over the whole year the oceans have everywhere a gain of heat from radiation, but north of 25° N. this gain decreases rapidly with latitude, therefore from 10° to 45° N. it is smaller on the eastern Evaporation from the Surface of the Sea and the Water Budget of the Earth 237 238 Evaporation from the Surface of the Sea and the Water Budget of the Earth Evaporation from the Surface of the Sea and the Water Budget of the Earth 239 Oi 240 Evaporation from the Surface of the Sea and the Water Budget of the Earth Evaporation from the Surface of the Sea and the Water Budget of the Earth 241 30^ 40^ N Lot Fig. 1 10. Energy interchange between the sea surface and the atmosphere at different seasons of the year (1) between 35° and 40° N. in the Pacific and Atlantic Oceans; left side: North Pacif.c Ocean, rght side: North At'antic Ocean. (2) along the western (left side) and eastern (right side) sides of the North Pacific Ocean; (3) along the western (left side) and eastern (right side) sides of the North Atlantic. Dec. Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. 242 Evaporation from the Surface of the Sea and the Water Budget of the Earth side of continents than on the western side. This is mainly due to differences in cloudiness. The annual heat loss by evaporation, Q^, according to Jacobs (1951) is given in Fig. 107. Evaporation is particularly large in the v^estern parts of the two oceans, where the currents carry warm water northward (Gulf Stream and Kuroshio). It is, however, less in the eastern parts where there are cold currents flowing southward. The extreme seasons show considerable quantitative differences in evaporation. In middle and higher latitudes the evaporation is large in winter and small in summer, but conditions may be rather complicated on the western sides of the oceans where in winter cold air is advected out from the continents over the warmer sea. The heat loss Q^ of sensible heat by convection is shown in Fig. 108 for the same oceans. Also one notices here a distinct increase on the western sides of the oceans which is of the same type as in the distribution of Q^. In the over-all distribution of energy given off from the sea as heat, the values for the evaporation predominate and set the basic pattern. With respect to seasonal changes also the behaviour of both loss items is rather similar. The sum —Qa = {Qe+ Qn) gives a final value for the total heat turnover as far as it applies to a current-free ocean. If currents are present then the equation Qv= Qr— Qa must apply, where Q^ is the energy surplus which is obtained by each cm^ of the surface under influence of a complete heat exchange with the atmosphere. This energy surplus, when positive, is carried away from the water mass unber consideration by currents and mixing processes and represents that part of the radiational gain Qr which is stored in the water. A negative surplus implies that energy is supplied to the water mass by currents and mixing processes which then is dissi- pated by the excess in radiation into the atmosphere (Sverdrup, 1945). Figure 109 shows the total energy surplus of the oceanic water (g cal cm^^ day"^), and shows that in the water of larger ocean surfaces, especially in middle and lower latitudes along and near to the western coasts of continents, some energy is stored in the water while enormous amounts of energy are dissipated (lost by the ocean) in the Gulf Stream and Kuroshio systems. Thus, to a very noticeable extent, the areas in which large amounts of energy are available to the atmosphere are localized in definitive oceanic regions. Comparison of Figs. 109 and 107 clearly shows that the pattern of total energy ex- change corresponds to that of evaporation. In order to recognize the seasonal varia- tions in the energy turnover, it is of advantage to compare these quantities along definite latitudes or meridians along the eastern and the western sides of the oceans respectively. This can be seen from Fig. 110. The first diagram shows a marked con- trast between western and the eastern sides of the oceans. A narrow band representing the energy loss appears along with the Gulf Stream in the North Atlantic, while a corresponding and more broad band is connected with the Kurishio. This is under- standable from the direction of the two currents in the zone between 35° N and 40° N. The contrast of the two sides of the ocean in a meridional direction is shown in the other two diagrams. Along the western sides at all times of the year, except in summer, the largest amounts of energy are given off between 25° N. and 40° N to 50° N. Along the eastern sides there is a winter minimum in these latitudes. These energy transports arc undoubtedly of decisive importance for the climatic conditions in the effected regions and form the basis of the study of the inter-relation between ocean and atmos- phere. Chapter VIII Ice in the Sea Extensive icefields cover the polar seas. The outer boundaries where the ice borders upon the warmer surrounding waters of lower latitudes are subjected to a constant change due to the freezing and melting process. They may take a wide variety of forms depending on the given external conditions. A plastic and lively description of the magic of the polar ice world has been given by Weyprecht (1879). Besides the so- called sea ice, formed by the freezing of sea-water, other floating ice is introduced to the sea from the neighbouring land by the great rivers (river ice), and in addition icebergs from the glaciers reach the sea. Floating river ice is comparatively unim- portant, except in coastal Siberian and North American waters, therefore; sea ice and icebergs dominate ice conditions in the Arctic and the Antarctic, and may be carried by ocean currents to warmer oceanic regions. This ice drift prolongs the existence of the winter ice barrier in the polar regions into spring and summer, and is thus of considerable importance for navigation. 1. Formation and Terminology' of Sea Ice Ice crystals are formed in the water either on crystallization nuclei, which are the smallest possible particles of organic or inorganic origin that are always present, or at an aggregation of several molecules which meet each other grouped more or less by chance giving a configuration favourable for crystal formation (Nernst, 1909). It appears that the triplex molecules are decisively engaged in the first phase of ice formation. Besides the crystallization nuclei, supercooling of the water is also necessary. The greater the purity of the water and the less disturbed it is, the more supercooling is needed. In natural waters there are always sufiicient crystalhzation nuclei present, and the water is usually in movement so that a very small degree of supercooling of only some hundredths of a degree Celsius is required to initiate ice formation. How- ever, supercooling has to be continuous for the formation of ice crystals. Since the formation of ice releases a latent heat of 80 g cal/g, for a change of water into ice heat must be continually removed by an amount greater than the latent heat. The more intensive the cooling and the less disturbed the water, the smaller are the ice crystals so formed, which show a needle-like structure. If the water is in movement then the forming ice particles lose this needle-like character, looking then like flat plates with irregular rounded edges, about 2-A cm long, 0-5-1 cm wide and 0-1-1 mm thick. They usually accumulate and form muddy clumps. The dependence of the freezing point on the salinity is discussed on p. 45. Only pure water is involved in the actual freezing process. Part of the salt content of the water is separated during the formation of the ice and, as a more or less concentrated salt solution, fills the small separating layers between the ice crystals which themselves 243 244 Ice in the Sea consist of pure water. As the ice crystals grow they withdraw pure water from this enclosed salt solution, which thus becomes more concentrated and of more specific weight; it gradually percolates out between the ice crystals and increases the salinity of the surrounding water. This diffusion process beneath a forming ice layer is pre- sumably the reason why the crystal plates in sea-water are always oriented perpen- dicular to the freezing surface, while in fresh water they are parallel to it. The arrange- ment of the crystal plates is in similar groups and they are oriented approximately parallel, relative to each other, so that the structure of simple sea ice is fibrous; therefore the fracture surfaces of the ice lumps appear perpendicular to the surface of the ice layer. The classification and terminology of ice formation and ice forms can be made according to diff"erent viewpoints ; unfortunately there is still no uniform terminology. Drygalski (1930) has given a completely general classification of ice forms based on genetic relationships. The two main forms of ice are shelf ice and sea ice. Shelf ice represents a transitional stage between the forms of ice occurring on the land and those found at sea. It Ues along the coast over the continental shelf and is for the most part a mixture of sea ice and land ice (coastal snow ice). Shelf ice reaches its greatest thick- ness and extent around the Antarctic; a typical example of this type is that found along the northern coast of Grant Land which is known as palaeocrystalline ice (Ureis). Other forms of Arctic shelf ice are found along the east coast of Greenland (Wegener, 1902, "floating land ice"). In sea ice there occurs a gradual change of the ice crystals to pap ice (ice mud, ice slush); in calm weather and at low temperatures it freezes together to a hard layer of ice up to 5 cm thick and forms, especially at the surface, a weakly saline top layer. In a rough sea and at still lower temperatures small sheets of ice are formed which grow rapidly and assume a plate-shaped form with upwards bulging edges (pancake ice). The individual plates have a diameter of 0-5-1 m, with a maximum of about 3 m. In calm weather pancake ice and ice slush freezes together to form a solid layer of young ice with a thickness of between 5 and 20 cm, having a greenish blue colour; the surface is wet and still rather plastic. Further growth gives sheet ice, often forming large lumps which are broken and piled up by pressure forming pack ice. A detailed terminology of ice forms has been given by Maurstad (1935; see also ZuKRiEGEL, 1935). Sea ice is divided according to age into two groups: winter ice (including young ice) and polar ice. The first is not more than one year old, still rela- tively soft and plastic, and usually occurs in the form of ice lumps. Polar ice, on the other hand, is mostly two or more years old, contains little salt and is therefore hard. Due to ice pressure it soon takes the form of pack ice. With reference to its position and movement Maurstad distinguishes between solid ice and drift ice. The first is found for the most part in bays, fiords and above shallow waters. Also winter ice, as long as undisturbed, may remain stationary during the entire winter; however, it is usually broken up by long open cracks and drifts away. Drift ice can take all forms and reaches its greatest extent in the drifting ice fields of polar ice in the Arctic. In spring and summer, under influence of the increasing solar and sky radiation and the warm winds, the winter ice begins to melt. The volume of the salt solution enclosed in the ice increases and the inner structure of the pure ice crystals is weakened. Ice in the Sea 245 The ice melts in this way from the interior outwards and becomes "putrid". The sur- face takes on the appearance of a honeycomb (cells), and the entire mass of ice soaks through down to a considerable depth. In contrast to the ice formed from pure water, sea ice has no definite melting temperature, but begins to melt as soon as the tempera- ture starts to rise. Putrid ice breaks up easily, exposing a much larger surface to the effects of solar radiation and to warmer sea-water in which it is floating. Most of the winter ice melts in summer, but a large part still remains, especially along the edge of the Siberian Shelf, that survives the summer and then becomes polar ice and in consequence is explosed to a strong annual melting cycle. 2. Physical and Chemical Properties of Sea Ice {a) The Salinity of Sea Ice The salinity of sea ice is defined as that quantity of sohd matter (in g) remaining after evaporation of 1000 g of melted sea ice. The limitation that was found essential in the definition of the saUnity of sea-water (see p. 36) thus also applies here. The essential difference between the salinity of sea-water and that of sea ice is that the first is a rather conservative property of sea-water; while the second, in strict contrast, is a very rapid changing quantity for each single piece of ice. Nevertheless, the sahnity of a sample of ice shows only minor variations. This has been shown by the numerous analyses made by the "Maud" Expedition, 1918-25 (Malmgren, 1927). As has been noticed by all polar expeditions the surface of young ice is covered by a surface salt solution, which remains liquid even for low temperatures and keeps the surface of the ice continuously wet. For very low temperatures only this layer also freezes, giving a mixture of ice and salt crystals which isolate themselves in form of snow-white clusters. Beneath the surface a part of the salt solution remains enclosed between the ice crystals and determines the salinity of the sea ice. Its amount depends on the processes going on during the ice formation, specifically on three factors: (1) on the salinity of sea-water from which the sea ice was formed; (2) on the rapidity of ice formation; and (3) on the age of the ice. Referring to the first, the salinity of sea ice is less than that of sec-water, since the part of the salt solution between the ice crystals is always Table 91 Air temperature ("C) -16 -28 -30 -40 Salinity of young ice (%o) 5-64 8-01 8-77 1016 Table 92 Ice thickness below the ice surface (cm) 0 1 6 13 25 45 82 95 Salinity (%«) 6-74 5-28 5-31 (3-84) 4-37 3-48 3-17 able to escape. In the analyses of young ice samples made during the "Maud" Expedi- tion the salinity of sea ice reached a maximum value of 14-59%o, but usually the salinity of sea ice was between 3 and 8%o. Referring now to the rapidity of ice formation it 246 Ice in the Sea shows that the faster the ice is formed (at lower temperatures) the less salt solution can escape and the higher therefore the salinity of sea ice (Tabic: 91). Since ice is formed more slowly in the deeper layers than at the surface some dependence on depth can also be expected. For a young ice layer that began to freeze in November 1924, Malmgren found in April 1925 the values shown in Table 92. Referring finally to the age of the ice, the older the ice the smaller its salinity. The salt solution leaks through continuously and this process is accelerated by changes in temperature. Blocks of ice lifted by the pressure of the ice become almost completely salt-free in the summer by this process of deconcentration, and can be used after melting for drinking water. The changes in salinity in winter ice occurring during the course ola year have been summarized by Malmgren in a diagram given in Fig. 111. The ice formed in October gradually increases in thickness, and initially the salinity decreases from the surface downwards. Corresponding to their age the middle layers have the lowest salinity. 0 12345678 9 Fig. 111. Salinity changes in winter ice during the course of the year (schematic, according to Malmgren). but at the lower surface of the ice layer the salinity again increase:* since the water freezes here from below. This is due to melt water sinking below the ice layer and freezing again immediately due to temperatures below freezing point (about — 1-6°C). In a dilute aqueous solution freezing proceeds with the formation of pure ice only until the eutectic point is reached, the concentration of the solution increasing at the same time. This critical point depends on the salts dissolved in the watoi . When sea- water freezes the separation of the salts dissolved in the water begins only at — 8-2°C. For sea-water the situation is simplified only in so far as: (1) all types of water have the same salt composition; and (2) ice is always the first substance to freeze out. In conse- quence, no matter how great the salinity, the freezing process always proceeds in the same way. For a given temperature the concentration and the composition of the salt solution is to a close approximation the same for all types of sea-water, regardless of their original salinity (Malmgren). If Tj is the freezing temperature of 1 g of sea- water of salinity S, then for a temperature t between t^ and --8-2°C only pure ice will separate out according to the above discussion, and at this temperature there will be a-r g of pure ice and (1 + a^) g of salt solution. If the sali ity of the salt solu- tion is S-T, then necessarily (1 -f a;)S. = S. Ice in the Sea 247 For sea-water of salinity 5" there will be a similar relationship (1 + a.')S.' = S'. Since Sr = St' it follows S S' -I = ~ = const., \ — a-r I — Or' that means, the amount of salt solution per gramme is proportional to the salinity of the sea-water from which the ice has been formed. The first substance which begins to separate at temperature below — 8-2°C is sodium sulphate (Na2S04). However, the chlorine is retained since its separation begins only at — 23°C. This selective separation process during freezing changes the composition of the salts in the sea-water (Ringer, 1906; see also O. Pettersson, 1883). Thus in the polar seas sulphate is expected to be steadily withdrawn by the freezing process from the sea-water which thus becomes enriched in chloride. On the other hand, in areas where the ice carried away by the ocean currents melts sodium sulphate goes again into solution and the sea-water should show a surplus in SO3. Malmgren and Sverdrup (1929) have found that deviations of this type from normal behaviour are only very slight, and thus there occurs no selective process on a large extent during ice formation in nature. On the other hand, the investigations of Liakionoff, according to Wiese (1938), have shown that in the Barents Sea both in sea ice, as well as in melt water, there is a deficit of chloride and a surplus of sulphate (SO3). Further investigation is required to settle this point. (b) Density and Porosity of Sea Ice The density of pure ice at 0°C is 0-91676, while the density of water at the same temperature is 0-999867. The density of sea ice which is free of air bubbles in- creases with its salinity. If it increases at the same rate as the density of sea-water increases with salinity, the density of sea ice is expected to increase by about 0-0008 for every l%o in sahnity. The density of sea ice free of air bubbles and with a sahnity of 15%o would thus be about 0-9296. The first precise determinations of the density of sea ice were made by Makaroff (1901) by extensive measurements of the mean height It and the mean depth d (above and below the sea surface) of freely floating ice floes. If a^ is the density of the sea-water then the density of the sea ice is given by This gives a mean value for the entire floe. Makaroff"'s measurements apply only to summer floes of drift ice and give reliable values only for regular floes without any snow cover. These observations gave results between 0-96 and 0-85. These large variations are due to the considerable amounts of air and water which may be present in sea ice. The greatest eff"ect is that due to air bubbles enclosed in the ice, which can be of a twofold origin. One part originates already during the ice formation, due to a separation of gases dissolved in the sea-water which cannot always escape from the cells between the ice crystals. Thus the gas bubbles will be more numerous and larger the faster the rate of freezing of the ice. The upper parts of freshly frozen ice thus usually contain more air than the lower parts. 248 Ice in the Sea A second source of air-bubble formation is the penetration of air during the meUing process (Hamberg, 1895). In the upper part of a mass of ice which begins to melt from the inside the rise in temperature first widens the small intermediate spaces con- taining the salt solution. As the ice particles melt their volume decreases and empty spaces are formed into which air is pressed in due to the atmospheric pressure. These spaces finally become so enlarged that the melt water, together with the salt solution, can flow out and finally they are replaced entirely by air. The originally pure and clear ice thus becomes a porous mass penetrated by a number of air channels. On top the drift ice floes in the summer thus always appear white simulating a snow cover. The lower parts below the water surface are still cold and hard (solid). They do not melt from the inside and show, at first, only very little porosity. However, when the ice disintegrates more and more and still drifts in sea-water, the temperature of which is above freezing point, the already existing empty spaces become filled with water and the ice density increases rapidly. The air enclosed in sea ice, according to Hamberg, has an oxygen content greater than that of atmospheric air but less than that of the air mixture absorbed by sea-water (24-26% as compared with 20-95 for atmospheric air and 34-6 for sea-water at 0°C and 35%o salinity). The most accurate determinations of the density of sea ice in situ have been made by Malmgren (1927) on the "Maud" Expedition. These were made by determination of the loss of weight of a piece of ice on immersion in petroleum of specific weight Pf If the weight of the ice sample in air is G and in petroleum g grammes then the density is given by G Pi = Pf Table 93. Density of sea ice (According to Malmgren ("Maud" Expedition)) Depth of Sample No. Time K°C) Salinity sample Density (%o) (cm) (g/cm^) 1-71 Jan. -Mar. -26-4 91 0-919 -290 14-6 132 Max. 0-924 -220 3-6 2 Min. 0-914 8 and Sa^ Feb. -240 00 2 0-921 93 Feb. -290 1-9 2 0-918 10* Mar. -22-4 4-7 5 0-911 11^ Mar. -270 00 8 0-857 12" May -6-2 — 2 0-885 13 and 14« May -6-2 — 65 0-892 ^ Young ice partly broken open. * Young ice from a freshwater pool on a thick old ice floe. ^ Young ice frozen in autumn from low salinity water. * Thick broken young ice some time exposed to the sun. ^ Top peak of ice exposed to sun ("gesommert"). * Sample of old ice at the place of temperature measurement. Ice in the Sea 249 Petroleum is particularly suitable as an immersion liquid because it cannot penetrate into the small air-filled channels of the ice pieces. The results of these determinations are summarized in Table 93, The conspicuous result is the very small variation in the density of young ice, in spite of the strongly varying salinity of the samples and of the equally variable depth from which they were taken, as well as of the changing thickness of the floes. The smallest values (0-914 and 0-916) were given by two thin and highly saline young ice floes which had been formed at very low temperatures. The rapid freezing must of course have trapped a large number of air bubbles, probably more than normal. The uniformity of the values between autumn and winter disappears gradually in spring as melting becomes more and more eff"ective. There is a progressive fall in density in late spring, and this decrease becomes stronger as the disintegration of the ice proceeds during the summer. Values less than 0-90 show by the large number of enclosed air bubbles that the ice must have been exposed to the sun ("gesommert"). The lowest value in density was found at the top peak of a large floe. During the pre- ceding summer the salinity in this ice had been completely removed, and in winter the melting water of it could be used for drinking water. (c) Thermal Properties of Sea Ice and the Temperature in the Interior of Ice Flow It is characteristic of sea ice that its thermal properties such as specific heat, latent heat of melting and thermal expansion behave quite abnormally. During investiga- tions of the heat expansion of sea ice Pettersson (1883) found that highly saline sea ice expanded with decreasing temperature down to — 20°C, though for ice of lower salinity this temperature was considerably higher. Malmgren showed by investiga- tions during the "Maud" Expedition that Kriimmels' assumption, that this was due to the salt solution enclosed in the ice, was correct. This abnormal behaviour rela- tive to the specific heat, latent heat of melting and thermal expansion is thus also a consequence of the formation and melting of pure ice occurring in the interior of sea ice. At a temperature r, 1 g of sea ice of salinity l%o will contain a-r g of pure ice and (1 — At) g of salt solution. If the specific heat of sea ice at the temperature t is Ct then this quantity of ice for a temperature change dr will require a quantity of heat Crdr. It is made up essentially of: (1) the rise in temperature Orcdr of pure ice (with specific heat c); (2) the rise in temperature of the salt solution (1 — a-^Kdr (with specific heat k); and (3) the heat Kda^ required to melt da-r g of ice (with latent heat of melting A^). This gives the equation Cr = a-rC + (1 — a-)K -f Xr -^ . dr Since the second term is small and as a first approximation a^ = 1 then Ct = c + A, — -. dr For sea ice of salinity 5'%o the variable amount of ice is Sda-r and therefore one ob- tains for it the relation: c.^c^-Sx/--^. dr 250 Ice in the Sea According to p. 247 if Sr is the salinity of the salt solution (1 - ar)S, = 1, so that da, 1 dSr and Cr = C + Xr S2 dr ' S dSr 5? dt ' According to the investigations of Pettersson (1878) A^ = 80 + 0-5t, The factor of Xj can be calculated from investigations made by Ringer, so it is therefore possible to evaluate the above equation for different temperatures and salinities (Table 94). Table 94. The specific heat of sea ice (According to Malmgren) Temp.(°C) -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 r 2 4 2-57 100 0-73 0-63 0-57 0-55 0-54 0-53 0-53 0-52 0-52 4-63 1-50 0-96 0-76 0-64 0-59 0-57 0-57 0-56 0-55 0-54 5'%o « 6 6-70 1-99 1-20 0-88 0-71 0-64 0-61 0-60 0-58 0-57 0-56 8 8-76 2-49 1-43 101 0-78 0-68 0-64 0-64 0-61 0-60 0-58 10 10-83 2-99 1-66 M4 0-85 0-73 0-68 0-67 0-64 0-62 0-60 ll5 1601 4-24 2-24 1-46 102 0-85 0-77 0-76 0-71 0-68 0-65 Malmgren has also determined the specific heat of ice samples experimentally, and has obtained values in excellent agreement with the theoretical. At higher tem- peratures the heat capacity of sea ice is quite high, at — 2°C and 15%o salinity it reaches 16-0 g cal. These high values can be explained either by melting or freezing of large amounts of pure ice in the salt cells of the ice at temperatures close to freezing point and fjr temperature changes of about 1 °C, which is accompanied by release or uptake of large amounts of heat from the latent heat of melting. For sea ice the specific heat and the latent heat of melting are properties closely related to each other. The dependence of the latent heat of melting on temperature and salinity can also be calculated theoretically from Sr the salinity of the ice, and r^, the freezing tempera- ture of sea-water of salinity S. If t is close to zero, the latent heat of melting for pure ice will be constant between r and r^ and will be 80 g cal. The amount of heat required to melt 1 g of sea ice will be made up of: (1) the heat = 80[1 — ^(1 — a-r)] required to melt pure ice; and (2) the heat required to raise the temperature of the pure ice and the salt solution from r to Tj,. Since the specific heat of pure water is 0-5 this quantity of heat will be approximately 0-5 (xg — T)aT. The latent heat of melting of sea ice will thus be given by U = ^°('-|) + 0-5(t. - t) Ice in the Sea Table 95. Latent heat of melting of sea ice 251 Salinity (%o) 0 2 4 6 8 10 15 „ r-io°c 80 81 72 77 63 72 55 68 46 63 37 59 16 19 Table 95 shows values for different salinities and for temperatures equal to 1° and -2°C. The coefficient of thermal expansion can be calculated in a similar way; it is made up of the coefficient of thermal expansion of pure water (a = 1-7 x 10"'*) and a term which depends on the amount of ice forming or melting due to the change in temperature in 1 cm^ of sea ice. Since the freezing of 1 g of water at t° is accom- panied by an increase in volume of yr = 0-091, the coefficient of thermal expansion of sea ice will, according to the above discussion, be given by dttr S dSr Table 96. Coefficient of thermal expansion of sea ice (Ur X 10'*) (According to Malmgren) Temp. (°C) . . -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 r2 4 6 8 10 .'5 -22-10 -45-89 -69-67 -93-46 -117-25 -176-72 -4-12 -9-92 -15-73 -21-53 -27-34 -41-85 -1-06 -3-81 -6-55 -9-30 -12-05 -18-92 016 0-83 1-13 0-56 0-00 1-23 0-78 0-33 1-27 0-85 0-43 0-02 1-33 0-96 0-60 0-23 1-38 1-07 0-76 0-45 014 1-44 Salinity %„ -1-37 -2-90 -4-43 -5-95 -9-78 -0-02 -0-88 -1-73 -2-59 -4-73 1-18 0-93 -U-D7 -M3 -2-54 -0-13 -0-59 -1-72 0-67 -U-40 -1-45 -0-13 -1-03 0-42 -0-63 -0-22 From this equation Malmgren has calculated the values given in Table 96, and experi- mental determinations of Ur, on samples of natural ice have fully confirmed the theor- etical values. There is an essential difference between Uj for sea ice and freshwater ice. Pure ice always expands with increasing temperature; sea ice expands only to a lesser extent, and then only at very low temperatures and low salinities. Thus the second term in the above equation becomes unimportant. At higher temperatures and salinities the second term predominates; this means that the ice volume increases with decreasing temperature and at very low temperatures and high salinities this increase may be considerable. Extensive series of temperature recordings at different depths in sea ice (ice floes) have been made by the "Fram" Expedition 1893-6 and the "Maud" Expedition 1918-25. The latter were obtained by using electrical resistance thermometers and are much more reliable. Table 97 gives monthly means for five depths down to 2 m for every month during which the snow cover at the place of measurement was left undisturbed. The annual temperature variation at all depths can be approximated closely by a simple sine curve of the form Af + a sin 12 / + a 252 Ice in the Sea and the result of this analysis, given in the last lines of Table 97, shows how regular is the annual temperature wave, with a decrease in amplitude and a phase shift in the Table 97. Annual temperature variation at different depths in sea ice (According to the values of the "Maud" Expedition, North Siberian Shelf) Depth (m) 000 0-25 0-75 1-25 200 Jan. -280 -24-1 -18-9 -140 -6-5 Feb. -30-9 -26-9 -21-3 -16-3 -8-5 Mar. -291 -26-0 -21-0 -16-5 -9-6 Apr. -21-6 -20-1 -17-3 -14-4 -9-4 May -7-4 -8-6 -9-3 -9-2 -7-4 June -1-5 -30 -4-1 -4-5 -3-8 July -00 -01 -1-3 -1-7 -1-8 Aug. -00 -00 -0-8 -11 -1-2 Sept. -4-7 -1-3 -0-9 -11 -1-3 Oct. -12-3 -7-6 -3-3 -1-6 -1-4 Nov. -23 0 -17-8 -11-9 -7-1 -2-4 Dec. -29-9 -24-4 -17-7 -12-2 -4-6 Mean M -15-70 -13-32 -10-65 -8-31 -4-82 aCC) 16-82 14-60 11-17 8-36 4-40 a (degrees) 259-6 250-9 240-9 230-7 210-4 extremes, penetrating into the ice (Fig. 112). In both series of recordings there is good agreement in the upper layers of ice down to about 1 -5 m, but this is not true at greater depths. The "Fram" values are too low, probably due to the observational method using bar-thermometers. The decrease in the annual amplitude with depth shows the same. According to the "Maud" values the annual variation disappears at a depth of 2-9 m. At a depth of 2-8 m the temperature of sea-water underneath the ice floe reaches — 1-6°C and remains constant throughout the whole year. At the side under- neath an ice floe, the thickness of which varies on the average as seen from Table 97, the amplitude of the annual temperature variation thus falls to zero. Fundamental investigations on the thermal conductivity of ice have also been made by Malmgren. Stefan (1890) found a thermal conductivity coefficient k = 4-3 x 10~^ from theoretical investigation of the process of ice formation, but this value can only apply for freshly formed pure ice. Later Mohn (1 900) attempted to compute the thermal conductivity coefficient from the decrease in the annual temperature variation and from the retardation of the extremes with depth in ice ffoes. However, these methods cannot give reliable values since the theory is valid only for infinite thickness, while the thickness of sea ice is small and the lower side of a floe remains almost always at a temperature of — I-6°C. Correct values of A' can be determined, according to Malm- gren, from the temperature gradient and its change with time at diff'erent depths. Assuming a cylinder with a vertical axis through an ice floe, then definite amounts of heat will enter the cylinder through its upper surface in / sec. If the ice floe is of suflH- cient horizontal extent no heat will pass through the vertical wall of the cylinder and the heat flux will only occur normal to the surface of the ice floe. If the heat content of the cylinder for a given time remains constant then kiGi = k^G^, where Ati, k^ and Ice in the Sea 253 0 00. ^025 2 00 1 n E Ez: Y 3a ^zasnux x xixn Fig. 112. Annual temperature variation at different depths in sea ice. Gi, Gz are the thermal conductivity coefficients and the temperature gradients at the upper and the lower surface of the cylinder. However, if the mean temperature changes from Tj to T2, then the following relation holds: (ArjCi — koG^t = hco{r^ — Tg), where h is the height of the cylinder, c the mean specific heat and a the mean density. From observations of temperature in ice it is possible to find cases where the mean temperature of a layer is constant for a certain time, and cases where it undergoes large rapid changes. The above equation can then be used to calculate A^ and k^. Table 98 shows numerical values for k determined for the winter periods 1922-3 and 1923-4. They are of the same order of magnitude as the mean values obtained by Stefan but have a marked dependence on the depth (Fig. 113). Table 98. Thermal conductivity of sea ice at dijferent depths (According to Malmgren) Depth (m) 0 25 60 and 75 resp. 125 „,. , / 1922-3 2-4 Winter |j923^ 1-7 3-6 40 3-3 4-5 4-2 X 10-3 50 X 10-3 There is a rapid decrease in the thermal conductivity in the top layers of sea ice which must be due to the numerous air bubbles in these layers (density about 0-88). 254 Ice in the Sea Deeper in the ice the thermal conductivity approaches a limiting value of 5-0 x 10~^ which corresponds to the value obtained for clear freshwater ice without air bubbles. Malmgren's determination of the physical constants of sea ice are of considerable importance in questions of the heat balance in polar regions, since they allow the de- termination of the amount of heat gained by the surface of the ice in polar regions and % o 0 I 2 Fig. 113. Changes in thermal conductivity in sea ice with depth. thus also by the atmosphere immediately above it from the water below. For the greater part of the year the water underneath is warmer than the ice cover and the air above it, and therefore there is a continuous flux of heat upwards. Such a calculation can be made with the temperature observations of the "Maud" over a period of a year. The total amount of heat passing through the different depth-levels in a year amounts on the average to 6800 g cal/cm^ and should be the same for all levels. This amount of heat is released to the atmosphere above the ice year after year. In the cold season of the year when the temperature gradient is several times larger this flux of heat is greater; in summer it may even be reversed but is then never very large. Taking a depth of 0-75 m as representative for the entire layer of ice, the amount of heat, W^o-75 passing through this level per cm^ and month can be calculated, knowing the temperature gradient for each month during the colder season of the year. Part of this heat serves to raise the temperature of the 0-75 m thick surface layer. If the tempera- ture difference between the beginning and the end of the month is Jr then the heat gained by the atmosphere during that month is W, = f^o-75 - IScaAr = fFo.75 - 34-4 J T. The values calculated by Malmgren using this equation for the months from Septem- ber 1923 to April 1924 show that during the cold season of the year the atmosphere receives the very large amount of 76,700 kg cal/cm^, which is sufiicient to melt 96 cm of ice. However, large as this may appear, it is only a ninth part of the heat that the European Mediterranean, for example, provides to the atmosphere (676,000 kg cal/m^). However, in the polar regions its eff'ect is none the less still important. During the cold part of the year there is a thin layer of cold air over the Polar Sea, extending to a height of about 150 m (Sverdrup, 1926). This layer of air has such a stable stratification that it mixes only to a very small extent with the air above. The heat from below is thus imparted almost entirely to this layer and prevents a decrease of the Ice in the Sea 255 temperature to very low values. The increase in temperature per day due to the flow of heat Wa from below can be found from the mean height of this cold atmospheric surface layer. As Malmgren showed, this heat is quite large and it is obviously this source of heat that prevents an intensive cooling of the atmosphere above the North Polar basin. The temperature can thus never reach the low values found in central Siberia or central Greenland, where this heat source is not available. {d) The Mechanical Properties of Sea Ice The continuous formation of ice by freezing is counter-balanced by very effective processes that reform and destroy the ice fields. The mechanical properties of ice (elasticity, plasticity and resistance against deformation, bending and compression) are of the greatest importance in the interplay between these processes. Large ice surfaces seldom remain unchanged for longer periods. They are broken up rapidly from the edges, by the combined action of the wind, waves and periodic tidal currents, and in a short time become separate ice floes. With the aid of strong winds they are piled up by the large horizontal pressures and pushed one above the other. The resultant mass, when finally covered with snow, cemented together and built up into several layers, is pack ice. Pressure and tensions are common in the polar regions (especially in the Arctic). Gaps and open spaces may exist for a short time but are rapidly covered over by young ice which again re-unites the whole mass. These pressures are not due to the effect of the wind alone, because often the wind only influences far-off regions, thereby subsequently causing pressures in the Arctic (distant effect) ; they are often due to rapid temperature changes at the surface of the ice. Since the under-side of an ice floe is always at the temperature of the water (near freezing point) there will be tensions and stresses in the floe. Figure 1 14 shows schematically the cracks and fissures formed when the stresses due to thermal expansion at the surface exceed the elastic limit. In the same way thermal contraction at the surface forms in an Fig. 114. Changes in an ice floe due to thermally induced expansion. analogous manner cracks at the lower side. The cracks on the upper surface of the floe soon fill with snow and melt water and those in the bottom surface fill with ice due to the rapid freezing of sea water in contact with the cold ice. There is thus a continuous formation of ice. The ice-covered regions in the Antarctic are not basins surrounded by land, and therefore ice pressures occur less often and are considerably weaker. The humps, hummocks and ridges of piled-up floes, known by the Siberian name toross, which are sometimes up to 5 m or more in height are much less common in the Antarctic pack ice; instead the action of pressure often forms folds and flexures. The mechanical properties of ice, like its other properties, depend on the temperature and salinity, but due to the multiplicity of ice forms and conditions these determine only the order of magnitude, and there may be considerable variations caused by the 256 Ice in the Sea special structure of an ice floe and its past history. The most important of the mechani- cal properties is the elasticity, which is characterized by Young's modulus E and the modulus of rigidity /x. Ice is of course composed of ice crystals and its elasticity is not the same in all stress directions. An ice crystal can be regarded as built up of a large number of thin platelets at right angles to the crystal axis. Deformation at right angles to this axis meets a much smaller resistance than one in the direction of the axis. The different values for Young's modulus shown by different natural samples are probably due to this. Few direct determinations have been made of the elasticity constants for sea ice, but they have been determined more often for fresh water ice by a variety of different methods. The more reliable values for the modulus of elasticity E are those of Reusch, which give 23, 632 kg/cm^. Its variability with the position of the crystal axis relative to the axis of force has also been determined, giving between 18, 200 and 38, 300 kg/cm^. E increases with decreasing temperature. A more accurate determination of these constants can probably be made indirectly by measurement of the velocity of elastic waves in the ice, and a large number of de- terminations of this type have been made. Ewing, Gray and Thorne (1934) measured this velocity in thin ice rods and found the following values for the elasticity constants: Young's modulus E Rigidity modulus fi. Poisson constant a 9-17 X IQio dyn/cm^ 3.36x lO^odyn/cm^ 0-365 Seismic measurements of the thickness of the ice on alpine glaciers and in Greenland (Brockamp and Mothes, 1930) have given E = 6-82 X lO^o dyn/cm2; ^i = 2-51 X lO^" dyn/cm^; a = 0-361. Considering the difference between experimental and natural conditions these values agree quite well. The elastic limit in ice is not large; for river ice Weinberg found 0-57 kg/cm^; for granular glacier ice Hess found 0-09 kg/cm^. The plastic limit is, of course, much higher. The strength of ice of different origins provides a more useful comparison than the above numerical values and has been used by Makaroff. His measurements show clearly that freshwater ice is of much greater strength than sea ice and that an in- creasing salinity in the water in which it is formed and a higher temperature, makes the sea ice less resistant. Weinberg (1907) investigated the strength of a large number of sea-ice samples and found that the values obtained usually increased with decreas- ing temperature; compared with the values at — 3°C there were increases of 20%, 35% and 45% at -10°, -20° and -30°C respectively. Investigations of the deformation of ice under the effect of continuous pressure have been made by Andrews, and especially by Royen (1922). From their results, it is worth mentioning that the plastic deformation of ice under the influence of continuous pressure can be expressed by the equation pi^T 1 - T where p is the pressure (load) in kg/cm^, T is the duration of this pressure in hours, t is the mean temperature of the ice and k is a constant characteristic for each sample and Ice in the Sea 257 varies within the hmits 6 x 10-' and 9 X 10"*. These investigations showed the con- siderable effect of the temperature on the hardness of the ice. The strength of ice is very important in calculating the loads that can be put upon it. The following empirical data may be given based on experience : freshwater ice 4 cm thick will carry a man, from 10-12 cm thick a galloping horse, from 15 cm thick a heavy-loaded truck, and over 45 cm thick a railway train. This question is also of importance for aircraft landing on ice. Moskatov (see "Die Naturverhaltnisse des Sibirischen Seeweges" ("Conditions along the Siberian Sea route"), Oberkom. Kriegsmarine, BerUn 1949, p. 84) has given the following table for the minimum safety thickness of freshwater ice for aircraft landings : Aircraft weight (tons) Minimum thickness (cm) 2 15 5 24 10 32 15 20 39 45 The strength of sea ice, and that of salt-free ice formed from sea ice due to a decaying process of several years is considerably less than that of freshwater ice. To carry the same load the ice in the centre of the Arctic basin must be two to three times thicker. 3. Ice Conditions and their Seasonal and Aperiodic Variations in Arctic and Ant- arctic Regions (a) Ice Conditions of both Polar Caps In the Northern Hemisphere sea ice is largely confined to the Arctic Mediterranean, the central basin of which is always covered by it. Figure 115 shows the general out- lines of mean ice coverage in summer and winter (Budel, 1943, 1950). September is the time of minimum extension in ice cover, and the ice is limited to the inner part of the North Polar Basin, which at that time is most remote from the warm land masses. This ice lasts throughout the summer and then extends again enormously during the winter. Except in the area of Gulf Stream water it reaches everywhere to the northern coasts of the continents and extends as long tongues of pack ice along the eastern coasts of Greenland and Labrador. To this winter ice then adds the one-year-old winter ice of the adjacent seas. In winter, of the total area of the North Polar Basin (11-6 milhon km^) on an average 8-7 miUion km^, (or 75%) are covered by ice. If the pole were surrounded by land with a circular area of 2-9 million km^ then the above mentioned ice-coverage would extend southward everywhere to the 72-7° parallel (thus everywhere 17-3° lat. distance from the pole). In the Southern Hemisphere, where the Antarctic land mass surrounds the South- pole with a total area of 14-8 million km^, the ice-coverage is 29-0 million km^ and for an even distribution would then reach northward to the 55-8° parallel. These figures show the strong contrast in ice conditions between the two polar regions. The ice covers 3-35% of the total Northern Hemisphere, but 11-30% of the total Southern Hemisphere. In the Southern Hemisphere (see Fig, 1 1 6) the ice extends uniformly around the central Antarctic continent, enclosing it on all sides, and the symmetric circumpolar arrangement of the ocean surface and the ocean currents fix zonal drift ice limits 258 Ice in the Sea Fig. 115. Average extent of sea ice (mean drift ice limit) in the Northern Hemisphere for summer and winter: mm^ AAA m///m °o°o° AVERAGE DISTRIBUTION OF Polar ice coverage closed in summer (about beginning of September) Brocken polar ice coverage in summer (about beginning of September) Southernmost iceberg limit in summer (May to September) Closed polar ice coverage in winter (March to April) Brocken polar ice coverage in winter (March to April) Southernmost iceberg limit in winter (October to March) Closed ice on inland seas and lakes in winter (February to March) Brocken ice on inland seas and lakes in winter (February to March) without any large meridional irregularities. In the Northern Hemisphere, on the other hand, the continents and the eccentrical position of the large polar icelands confine the ice field on all sides, and allow warm ocean currents to enter at only one gate, between Iceland and Scandinavia where the warm Atlantic current pushes the limits of drift ice back to the northern coast of Spitzbergen and into the inner parts of the Barents Sea. Ice in the Sea 259 Referring to the special regional distribution of the three different types of ice (polar ice, pack ice and solid ice) the central area of the North Polar ice consists always of pure polar ice (Smith, 1931); it is 3-3-5 m thick at the end of the winter and 2-2-5 ra thick at the end of the summer. It covers about 70% of the entire Polar Basin, i.e. 5-2 million km^. It is usually a continuous layer, but especially towards the edges it is split up by ice pressure into large ice fields and ice floes. This large polar ice cap is closely confined to the 1000-800 m isobath and has a more or less elliptical shape lying much nearer to the continental coast and coastal islands on the Greenland- North American side than towards the coast between Spitzbergen and Alaska where the broad Siberian Shelf lies between. The centre of the polar cap is often called the "pole of inacessibihty" and is situated about 400 nautical miles north of Alaska. The maintenance of this polar ice cap represents a state of equilibrium with the total annual growth. The total gain consists at first of an addition of ice from the surround- ing pack ice zone due to freezing at the bottom layers of ice floes, secondly of snow falls on the ice surface and the re-freezing of open spaces. The ice loss is caused by Northern limit of drifting fiores • *' 'V- 8 1820 1640 I860 1880 Years 1900 1920 Fig. 124. Relative sunspot numbers and smoothed values for the amount of ice in Davis Strait. (The latter is displaced two years to the left relative to the sunspot curve (9 full periods) (full line : sunspot number, dashed line : amount of ice.) +0-86 between the number of icebergs south of Newfoundland (48° N.) and the pack ice off Newfoundland valid from February until May (47 years, Smith, 1926-7). For the Barents Sea particularly good ice statistics are available for areas in which ice-measurements have been made by the Danish Institute during the years 1896-1916 (Nautik-Meteorol. Aarbog 1916). Wiese (1924) has used these in a study of the rela- tionship between the occurrence of ice and variations in the atmospheric circulation. He was able to show that the ice intensity in this sea from May to June depends largely on the distribution of atmospheric pressure over the Norwegian Sea during the period from January to the end of April and that a larger (smaller) atmospheric pressure gradient directed from south-east to north-west between the Norwegian coast and the axis of the low-pressure trough over the Norwegian Sea causes a decrease (increase) in the ice coverage of the Barents Sea. By calculations from the regression equations with the factors affecting the ice coverage, it is possible to obtain reliable ice prognoses for this area. A very strong aperiodic change in the Arctic has been in progress since 1918. Since the summer of that year there has been a general retreat of the ice limit, and at the same time a warming up of the entire Arctic (Weickmann, 1942). This can be seen best from the mean position of the ice limit from April to August in the two periods 1898-1922 (25-year mean) and 1929-38 (10-year mean) (Fig. 125). The especially favourable conditions during the second period are very noticeable when compared with those for the 25-year mean which can be regarded as normal. Bear Island, for example, is normally still surrounded by ice in April and partly also in May. During this second period it was ice-free during all months, and although the northern part of Novaya Zembla is almost never ice-free the ice limit receded during the second period almost to the northern tip in July and during August was only a little south of Franz Josef Land and Wiese Island. Ice in the Sea 271 20° 40° 60° 80° 80° 80° 70° 65° 75° 75° 70° 70° 65° 65' Fig. 125. Mean position of the ice limit from April to August in the Barents Sea for the period 1898-1922 (25-year mean, normal period) and for the period 1929-38 (10-year mean, warm period). 4. Land Ice in the Sea (a) Glaciation in Polar Areas In the polar regions the climatic snow-line Ues so low that under the prevailing orographic conditions the glacial endings of the ice streams reach the sea and spread into the ocean. The coverage of polar regions by glaciers was given by Hess and is shown in Table 104. Table 104. Glaciation in the polar regions Area in 1000 kjn^ Northern Hemisphere Greenland including islands 1896 Spitzberger 58 Franz Josef Land 17 Novaya Zembla 15 Severnoja Zembla 45 North American islands 100 Total 2131 Southern Hemisphere Antarctica 13,000 In the Northern Hemisphere the overwhelming part of the total glaciation is on Greenland where only 0-325 million km^ of its total area of 2-16 million km^ is ice- free. Glaciers flow out from all sides from the inland ice and a large number of them 272 Ice in the Sea reach the sea in a broad front. The part played by the other Arctic islands in the production of icebergs is quite insignificant ; only very few of the ice streams of the other islands reach the sea as calving glaciers, and even these produce only small icebergs. In Greenland the inland ice reaches the sea through more or less narrow fiords which act as funnels collecting the converging streams of inland ice, but in the Southern Hemisphere the inland ice reaches the sea in an open front. At the edge of the Ant- arctic the snow-line is everywhere in or below the sea-level. Here the ice takes the form of an ice barrier which in the Ross Sea, for example, is about 750 km long and has a mean height of 36 m, but sometimes exceeds 50 m. Enormous icebergs break away continually from the edge of the inland ice cover, and though at first often trapped in the shelf ice, they are carried away with it later on or melt completely in their place. {b) The Productivity of Glaciers Calving into the Sea in the Arctic Statistics of iceberg production by glaciers calving into individual oceanic regions are rather poor; reasonably reliable figures can only be given for very few ice streams. Smith (1931) has attempted to give such a preliminary survey for the Arctic. In the Eurasian Arctic there are only a small number of glaciers producing icebergs. In Spitz- bergen, probably the Negri glacier in the Storfjord; the east coast of North-east Land has some calving glaciers as has the completely glaciated Franz Josef Land. But the number of icebergs produced, which are seldom large, is not known and is presumably small. The productivity of Novaya Zembla and Sevemaya Zembla is equally not known, but is probably also very small. The few icebergs which are formed at the islands of the Siberian Shelf move mostly to the west and increase somewhat the number from the East Greenland icebergs. Smith estimated the number of icebergs produced annually in the north-east sector of the polar Atlantic ocean as about 600, which is only about 4% of the annual supply of icebergs from Greenland. Smith believed that the productivity of the eastern Greenland glaciers was some- what less than that of the west coast (7500 icebergs per year). There is, however, the important difference that in the east most of the icebergs are retained in narrow fiords and are prevented by the solid ice-barrier of the East Greenland Current from drifting southwards. Their importance to the Atlantic is therefore slight; about twenty to thirty a year reach Cape Farewell and then drift northwards with the West Greenland Current. They reach Davis Strait in a collapsing state. The iceberg survey of the "Marion" Expedition during the summer of 1928 found only seventeen icebergs off the south-west coast of Greenland; a very small number compared with the enormous amount that were found in Disko Bay to the north. On the western side of Baffin Bay only Ellesmere Land with two large ice caps shows any extended inland ice. About sixty glaciers reach the sea as calving glaciers, but according to Smith the productivity is not very large (about 1500 icebergs a year). The major source of icebergs is in the great glaciers of West Greenland from Cape Alexander to Disko Bay. The main part, from the North-east Bay as far as Disko Bay has more than 100 calving glaciers, the twelve largest and most productive ones alone producing more than 5400 icebergs a year. The most important of this group, the Jacobshavener Glacier calves about 1350 icebergs a year into Disko Bay. Not all of these reach the open sea immediately — on the contrary most are trapped in the fiords for longer periods. In the summer of Ice in the Sea 273 1928 the "Marion" Expedition found that all the icebergs produced during the pre- vious 3-4 years (about 4000 to 6000) were accumulated in the Eisfjord. They were all released from their ice chains during favourable weather at once. Then they arrived all together in Baffin Bay and drifted slowly to the south. Table 105. Production of icebergs in different regions on the Arctic Annual contribution North-eastern sector of Atlantic Islands on the European-Asiatic side 600 East Greenland 7500 From those arriving at Cape Farewell and passing in the East Greenland Current 20-30 North-western sector of Atlantic Eastern North America 150 North Greenland 150 Cape Alexander to Cape York 300 Cape York to Svarten Huk 1500 North-east Bay to Disko Bay 5400 Total 7500 Table 105 is a summary given by Smith of iceberg production in the Arctic. This estimate gives a total annual contribution from Baffin Bay of 7500 icebergs, of which 70% come from North-east and Disko Bays. The table gives only a rough idea of the ice amount available in Baffin Bay. Direct estimates of the ice outflow from the Green- land Inland ice by measurement of the speeds of the different glaciers along the western side of the island still differ widely. De Quervain and Mercanton (1925) estimated this ice flow as being between 10 km^ and 100 km'' a year. Assuming an average size of a large iceberg to be about 1-5 miUion m^ and assuming, further, that on calving about one-third of the ice forms icebergs and two-thirds gives debris and smaller pieces, then the total mass of ice released on calving is about 4-5 million m^. About 7500 such calvings per year gives approximately 35 km^ of ice. This value lies within the above Umits. Helland (1876) found values of 5-8 km^ and 2-3 km^ for the annual ice supply from the Jakobshaven and the Torsukatak Glaciers. Drygalski (1897) found 13-5 km^ for the large Karajak Glacier. According to these figures about 2-6 milhon m^ of ice are broken off in an average calving of a medium sized glacier ; from this about one-third is used for production of icebergs. (c) Calving, Size, Shape and Destruction of an Iceberg Careful observation of iceberg calving at the eastern Greenland glaciers led Dry- galski to distinguish according to the size of the icebergs formed between three types of calving. The third one proceeds almost continually over several days ; small blocks of ice break away from the face of the glacier and fall into the sea, often in such large amounts that the surface is covered with these broken pieces far out into the fiord. In the second type, large masses are suddenly released in the water from the lower 274 Ice in the Sea part of the glacier and rise as icebergs to the surface ; this leaves the edge of the glacier unchanged. The largest icebergs are produced by the first type: under the influence of the further continuous supply in ice mass the glacier pushes out into the sea for 200-300 m depending on the morphology of the fiord bottom ("fore part of glacier"). The fiord water slowly penetrates into the projecting ice mass and, due to buoyant forces, the forehead of the glacier gets lifted until it finally breaks off. Calving usually occurs exactly there where the depth of the fiord has increased to such a rate that the forward pushing ice-tongue loses contact with the sea bottom and starts floating. In addition to the increasing buoyancy, lifting due to the tides may also upset the equi- librium in the glacier tongue. Presumably the formation of icebergs proceeds in the same way in the Antarctic; however, the process there is of much larger dimension and produces enormous flat-topped icebergs. The direct production of icebergs proceeds at about the same rate throughout the year, but the number of icebergs reaching the open sea depends on the nature of the fiord and more especially on the season of the year. In winter the fiords are frozen and the icebergs are trapped. They are released with the coming of summer, all within a short time and mostly all at once, and they then drift away. This gives rise to the so-called ""iceberg swarms'" which often occur in Baffin Bay and Davis Strait, The shape of icebergs is remarkably variable: the pure-chance forms after calving are remodelled by the action of sea waves and by melting above and below the water; classification of these diff"erent forms is thus rather pointless. The height of icebergs varies widely, but the largest are of course found in the area where they are formed. Measurements made by Drygalski on eighty-seven icebergs frozen into the sea ice in the East Greenland fiords gave the results shown in Table 106. Table 106 Height (m) 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100 100 Number 7 6 12 10 12 10 4 4 1 The height decreases rapidly after their formation. The highest iceberg measured by the International Ice Patrol Service south of Newfoundland was 80 m high; it was flat-topped and 517 m long. Its volume was estimated as about 25-5 milUon m^. According to Smith, the icebergs in the Davis Strait have an average volume of 1-5 milUon m^; those of the Newfoundland Banks between 0-1 and 0-15 million m^; they are about 30 m high. The ''depth of immersion'' of an iceberg depends on the specific weight of glacier ice. Since icebergs contain a large percentage of air and numerous cracks and holes this depth does not correspond to that calculated solely from the specific weight. For mean densities of 0-8997 for the ice and 1-02690 for polar water, flat-topped icebergs will have one-eighth of their volume above the surface of the sea and seven-eighths will lie below the surface, but the shape of an iceberg has a considerable effect on the depth to it which immerses. Smith has made a summary of direct measurements and has found that for the most peculiarly-shaped icebergs of of the north-western Atlantic the ratio is 1 : 3, The flat-topped Antarctic icebergs immerse to greater depths. Ice in the Sea 275 The destruction of icebergs proceeds by calving, melting and erosion. Icebergs are often rapidly decreased in size by the breaking away of large and smaller pieces of ice. This may change the equilibrium of an iceberg so that it capsizes or rolls over. In cold water the melting process goes on mainly at the water line of icebergs (by the formation of holes). Melting increases greatly when they drift into warm water (e.g. south of the Newfoundland Banks in mixed water or in the warm Gulf Stream). Destruction from above is due to melt water running down the sides of the iceberg, by erosion and the action of the waves and rain. According to measurements made by Drygalski in North-east Bay, an iceberg in the summer months may lose from 3 to 4 m in 7 days. Between Greenland and Newfoundland the ice mass may decrease to an eighth, corresponding to a daily loss of 1-8 m a day. In the same time the height decreases by a half. {d) Iceberg Drift in the Arctic and Antarctic Icebergs in the open sea are subject to the eroding action of winds and currents. These effects are dependent: (1) on the ratio of the masses of ice above and below the water; (2) on the strength and duration of the wind; and (3) on the velocity and direc- tion of the currents. Mecking (1906) has emphasized the great importance of the wind and currents for iceberg drift in Baffin Bay. The coastal current plays the decisive part and the wind determines the course of the icebergs only when this current is weak. The continuous off-shore wind along the coast of western Greenland in the summer thereby determines the number of icebergs reaching the Labrador current and thus the number of icebergs off Newfoundland in the following spring. The International Ice Patrol Service, in order to determine the influence of the factors mentioned above on the course of the icebergs, has followed the drift of a large number in the area of the Newfoundland Banks and has recorded the meteoro- logical and oceanographic conditions at the same time and Smith (1931) has discussed this data in detail. The effect of the wind was made up of two parts: (1) the direct force of the wind exerted on the exposed surface of the iceberg above the water; and (2) the movement of the floating iceberg with the wind drift set up in the top layer of the water. For the latter influence it must be kept in mind that for a steady state the wind drift at the surface of the sea is deflected by 45° to the right of the wind direction (Northern Hemisphere). This deflection increases with depth and a mean deflection of 72° can be assumed for the upper 50 m. For the two cases of (a) deep- immersing larger icebergs and (b) smaller icebergs with immersion ratios of 1 : 1 and 1 : 2 average conditions of the effects of these two forces are given in Table 107 (Fig. 126). The drift speed of larger, deeper-immersing icebergs with a deflection of 40° to the right of the wind is less than that of smaller icebergs of lesser depth of immersion for which the wind force and the force due to wind drift act more closely together. In this case the deflection from the direction of the wind is only 20°. For more ac- curate information on the distribution of icebergs in different parts of the sea it is necessary to make a survey of the existing iceberg accumulations. The International Ice Patrol Service carried out a systematic investigation of this type with the patrol boat "Marion" and at the same time the research ship "Godthaab" (Riis Carstensen, 276 Ice in the Sea Table 107. Direct wind force and force due to wind drift on icebergs (According to Smith.) fl, deep-immersing large icebergs; b, smaller icebergs). Direct wind force in the wind direction (km/day) Wind drift, deflection 70° to the right of wind (km/day) Resultant icebergs drift Wind velocity Beaufort Speed (km/day) Direction, to the right of wind direction 2-6 40 8-8 13-7 3-2 4-8 40 60 4-5 6-9 10-8 16-4 40° 40° 18° 2X' Fig. 126. Diagram of the forces affecting the drift of icebergs (according to Smith), (a) Effect of wind on large icebergs; (b) Effect of wind on small icebergs. 1929, 1936) made an oceanographic survey in Baffin Bay. Figure 127 shows the distribu- tion of icebergs in the Davis Strait and the Labrador Sea during the summer of that year. The few icebergs along the south-west coast of Greenland are from the East Greenland Current. Most of the icebergs are carried southwards by the cold Labrador Current which runs close to the coast. The central parts of Davis Strait and the Labra- dor Sea are almost completely free of icebergs. The Labrador Current along the coast thus forms the channel along which the icebergs pass towards Newfoundland. The track of the icebergs, especially to the east and south of Newfoundland, has been Ice in the Sea 277 accurately fixed by tracking numerous icebergs with the patrol ships. The main ice- berg track as shown by these detailed surveys is shown in Fig. 128. An increased fre- quency is to be expected along the eastern slope of the Newfoundland Banks where the Labrador Current turns towards the west and its cold and weakly sahne water mixes along the southern side of the current in large eddies with the warm and highly saline water of the Gulf Stream. A careful study of these eddies by the Ice Patrol vessels has N 60 Fig. 127. Extent and distribution of icebergs in Davis Strait and the Labrador-Sea in the summer of 1928 according to the "Marion" Survey. been made and thereby an explanation was found for the continuing presence of ice- bergs in this part of the sea, since the eddies keep the icebergs quasi-stationary Occasionally individual icebergs withstand the destructive effects of the warm At- lantic water and reach much further south than usual. The most southerly position so far recorded was 30° 20' N. and 62° 32' W. near the Bermudas for an iceberg about 9 m long, 5 m broad and 1 m above the water, which was sighted by the "Baxter- gate " on 5 June, 1926. Knowledge of iceberg drift in the polar seas of the Southern Hemisphere is very scanty. The approximate northern limit of drifting icebergs is shown in Fig. 122. It is, of course, far north of the northern limit of drifting ice floes since the compact mass of a large iceberg can better withstand the destructive action of warm water and air. It must be assumed that here also winds and currents must be the factors that determine the drift of an iceberg. In some individual cases a relationship to the course of low-pressure areas has been demonstrated, but in view of the irregularities of the latter a strict relationship is hardly to be expected (Mecking, 1932). Icebergs are especially important in the Falklands area where they are sometimes carried, accompanied by drift ice, far to the north in large numbers. They have been sighted as far north as 42° S. and in 1906 even reached as far as 37° S. (59° W.). The aperiodic variations in the occurrence of ice appear to be particularly large here. In 278 Ice in the Sea Fig. 128. Main iceberg tracks off Newfoundland and the Grand Banks. the years 1891, 1892, 1893 and 1906 a remarkable accumulation of icebergs appeared in the area south of Cape Horn and northward of the Falkland Islands as far as 40° S. They occurred mainly along the edge of the shelf; farther to the west they were com- pletely absent. They are trapped and melt rapidly inside the numerous eddies along the boundary between the Falklands Current and the Brazil Current in a similar way to those south of the Newfoundland Banks. (e) Seasonal and Aperiodic Variations in Iceberg Frequency off Newfoundland Surveillance of the distribution of icebergs in the area of the Newfoundland Banks since 1900 has given the mean annual iceberg frequency and its variation from month to month shown by the data in Table 108 for Newfoundland (south of 48° N.) and for the area south of the Grand Banks. Table 108. Mean annual variation in iceberg frequency (a) off Newfoundland south of 48° N, and (b) south of the Grand Banks (For the period from 1900-26) Month Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Total (a) ib) 3 0 10 1 36 4 83 9 130 18 68 13 25 3 13 2 9 1 4 0 3 0 2 0 386 51 Ice in the Sea 279 The iceberg season usually lasts from 15 March to 15 July, but the number of ice- bergs decreases rapidly after the middle of June (see Fig. 117). From the middle of July to the following spring the area south of the Grand Banks is almost free of ice- bergs. The variations in iceberg frequency from year to year are very large. South of 48° N. there were in 1929 a total of 1351 icebergs, in 1924 only eleven. An accurate monthly record of these values is available starting from 1900. Together with the previous data compiled by Mecking there is now a complete series of records available, covering a period of 50 years for the iceberg frequency off Newfoundland. This is shown graphically in Fig. 129. With these more or less homogeneous data it is possible - i 1 I \ k 1 - ■ \aM^ A. A rl'vik il \A l\ ^t 1 H< p \A '\\ / '' i ' l/*i '/ \ 4 1 M \ r^ } 1880 1890 1900 1910 Years 1920 1930 Fig. 129. Character of the iceberg-years from 1880 to 1930 off Newfoundland (ten step- scale, according to Smith). to investigate with some hope for success the causes of the aperiodic variations in numbers of icebergs. In the first place there appears to be a correlation between the atmospheric pressure gradient from Iceland to Greenland-North America a few months previously and the iceberg maximum off Newfoundland. Determination of the air pressure anomaly for the North Atlantic for the months December to March during 6 years of low iceberg frequency, in this case with a total of 275 icebergs, and for the same months in 5 years with a high iceberg frequency with a total of 774 showed completely opposite conditions (Fig. 130). A weak Icelandic low pressure area during the spring and the autumn with a weak pressure gradient over Baffin Bay and Davis Strait seems to be followed by the low-frequency iceberg years. During ice- berg-rich years, on the other hand, the Icelandic low-pressure area is intensified and the strong pressure gradient to the west is accompanied by strong air movements and stronger wind drift in the iceberg area along the North American coast. Smith has also tried a quantitative determination of this relationship using the correlation method, and has obtained a prognostically valuable formula. An increase in the ice- berg frequency in the north-western Atlantic is thus accompanied by an intensification of the atmospheric circulation in the polar areas which corresponds to an increase in the outflow of polar air and of the arctic water towards the south. These relation- ships of course take into account only the meteorological effects and not the possible fluctuations in the production of icebergs by the western Greenland glaciers. At the present time it is not possible to make an estimate of these. 5. EflFect of Polar-Ice Conditions on the Atmospheric and Oceanic Circulation The total annual ice outflow along the whole of the west coast of Greenland has been estimated by Smith as between 42 and 63 km^ (see p. 272) ; the American coast 280 Ice in the Sea 80° 60° 40° 20° 0° 20° 40° 60° 80° 100° 80° 60° 40° 20° 0° 20° 40° 60° 100° Fig. 130. Iceberg frequency off Newfoundland and atmospheric pressure anomaly over the North Atlantic. Ice in the Sea 281 adds only about 1-9 km^. This is the amount of land ice that exists in Baffin Bay on a yearly average and drifts southward to melt in Davis Strait, along the Labrador coast and in the Newfoundland area. The amount of sea ice melting during one year can be calculated from the average area covered by pack ice and drift ice. Smith has made an estimate of this kind based on reliable data collected by the Ice Patrol cruises. The bases of this are contained in Fig. 131 which also shows the areas which stand in question; the most important are the shelf areas where the ice-covered area is about 1-6 million km^. Taking the mean thickness of drift and pack ice as about 1-8 m, the total amount of sea ice will be about 3000 km^. In contrast to this, the land ice amounts to only 44-65 km^, so that of the average annual amount of ice melting in the north-west Atlantic only between a hundredth and a two-hundredth part comes from icebergs. This is vanishingly small (see Fig. 131). This comparison shows that the amount of pack ice and drift ice is the decisive factor. If for any oceanographic or meteorological problem a consideration of the effects of ice destruction in the north-west Atlantic — which vary considerably from year to year — is needed, it is thus not justifiable to compare it with variations in the ice frequency, as has often erroneously been done. In dealing previously with convection processes (see p. 97) two possibilities were discussed for the initiation of such a process, which are of the greatest importance to the thermal structure of the middle and bottom layers of the oceans. It was assumed by Pettersson that the necessary heat loss of the upper water layers was mainly due to the melting of ice in polar and subpolar oceanic regions. However, laboratory experiments by Nansen showed that this hypothesis was untenable. For the special case of the conditions in the north-west Atlantic it is possible, using the values given by Smith to determine directly the amount of heat which is required for the observed yearly melting of pack ice and drift ice and therefore is not available for heating the ocean and the atmosphere. This can be compared, as has been done by Smith, with the amount of heat suppHed during the summer by solar and sky radiation which is required for the increase in temperature of the upper 150 m layer of water (the average depth to which the increase reaches downwards into the sea). From the num- bers given in Fig. 131 it can be seen that the mean summer increase in the tempera- ture of the water masses in this area (down to 150 m) is about 1-2°C. It can also be calculated that the annual melting of pack ice and drift ice in the same area is sufficient to decrease the temperature of the layer down to 150 m depth by 0-6 °C. Thus in the north-western part of the North Atlantic the water is cooled by the melting of the ice by only about half of the amount of the summer increase in temperature due to the absorption of solar and sky radiation. Dynamic treatment of the oceanographic data of the "Marion" and "Godthaab" Expeditions permits the calculation of the amount of the heat deficit at the Newfoundland Banks due to the continuous supply of cold polar water by the Labrador Current. Comparison of this heat reduction with that due to ice melting shows that the latter accounts for only 10% of the cooling effect of the Labrador Current. The dominant factor in the cooling of the water masses of the northern part of the North Atlantic is thus neither the mehing of icebergs nor of the pack ice and drift ice, but much more the continuous advective supply of polar water which the Labrador Current carries southwards towards the warm water of the Gulf Stream. The "Meteor" cruise in Icelandic and Greenland waters have given the same 282 Ice in the Sea Fig. 131. a quantitative representation of a number of comparisons between ice-melting effects and related phenomena. The shaded area bounded by the full line in the normal pack- ice area. The dotted line marks the mixing zone. The entire melting area, with a uniform thickness of 150 metres is divided into six parts; in summer the southernmost is heated an average of 5°F, and the northernmost only 0-5''F. The spot "M" off Cape Farewell repre- sents the annual crop of glacial ice expressed in the same scale as the pack ice and as one large berg. The shaded area 'W represents the total annual discharge of glacial ice into BaflSn Bay, expressed on the same scale and in terms of pack ice 6 ft thick. Ice in the Sea 283 results (Defant, 1933). The formation of the East Greenland Current and the main- tenance of its polar character as far as Cape Farewell is not due to melting processes ; its Arctic nature is mainly acquired from its direct connection with the North Polar Basin causing a continuous supply of polar water and from the climatic conditions maintained over Greenland by the inland ice. This advective supply of Arctic water from areas where the effect of solar radiation is very small is the determining factor, and sea ice and icebergs are only accessory phenomena. A marked effect of the ice masses of the polar seas on the atmospheric circulation has been assumed by many prominent meteorologists. Hildebrandsson (1914) especially has attempted to show that the cause of the secular variations in meteoro- logical factors is to be seen in the aperiodic variations in the amount of polar ice. More recent data from later investigations has lent support to this hypothesis, but a definite proof is difficult. Both phenomena are not independent of each other, so that it is reasonable to assume, of course, a mutual interaction between the ice conditions and the atmospheric circulation; it is not easy to separate cause and effect (Wiese, 1924). Conditions are probably such that variations in the atmospheric circulation change the equilibrium conditions in the polar reservoirs of cold air. Years with weaker circulation favour an increase in the thickness of the cold air masses in the polar regions. This increases the atmospheric pressure in the polar region and corre- spondingly winds and currents become stronger, which causes a greater extension of the polar ice towards the south. The increased ice surface in turn increases the air pressure; the atmospheric pressure anomaly thus acquires a certain permanence, and due to this mutual reinforcement the effect may last a long time. The atmospheric pressure in the polar areas is thus a very sensitive indicator of the general condition of the atmosphere. Since, however, the atmospheric pressure conditions in these re- gions is reflected in the ice conditions, the distribution of ice in the polar seas can be taken as a measure of the variations in the general atmospheric circulation, provided sufficiently accurate information is available. The major variations in the atmospheric circulation usually extend throughout the entire atmosphere over the whole Earth, both in the Northern and Southern Table 109. Parallelism between changes in ice conditions of the north and south polar regions Shown by the relation between ice conditions from March to May at the South Orkney Isles (years with close or open ice) and corresponding deviations of the ice coverage in May to August from an average value of the period 1896-1916 in the Barents Sea South-Orkney-Isles Character of ice conditions for March to May Close ice { Open ice . . . .■! Barents Sea deviations of the ice coverage (in 1000 km^) for May to August from an average value of the period 1896-1916 1903 1909 1910 1911 1912 +88 +102 +17 +97 +176 (above average) 1904 1905 1906 1907 1908 -158 -165 -61 -130 -121 (below average) 284 Ice in the Sea Hemisphere. Thus, for example, it has been shown with sufficient certainty that there is a high positive correlation between the atmospheric pressure pulsations in the North Polar regions and those in the South Polar regions. If this connection is real, certain parallelism would be expected between the variations in ice conditions in the Arctic and in the Antarctic. To test this assumption Wiese has compared 10-year records of ice conditions at the South Orkney Islands from 1903 to 1912 (Mossman, 1923) between March and May, with the area of ice in the Barents Sea between May and August in the same years. The results are given in Table 109 and show the existence of a positive correlation. 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Temperaturschichtung und Vertikalzirkulation im Siidatl. Ozean nach den "Chal- lenger" und "Gazelle" Beobachtungen. Z.d. Ges. Erkimde. Berlin. Merz, A. (1925). Die Deutsche Atl. Expedition auf dem Forschungs- und Vermessungsschiff "Meteor". /. Ber. Sitzbcr. preuss. Akad. Wiss. 31. Beriin. Meyer, H. H. F. (1932). '' Meteor" Wcrk 4, no. 1, 262. Berlin. Die Chlortitirierung. MoLLER, L. (1929). Die Zirkulation des Indischen Ozeans auf Grund von Temperatur- und Salz- gehaltstiefenmessungen und Oberflachenstrombeobachtungen. Verdff. Inst. Meeresk., Univ., Bert., N.F. Reihe A. Heft. 21. MoHN H. (1883). Den Norske nordhavs Exp. 1876-78. Vol. II. Meteorologi, S. 135 ff. Christiania. MoHN, H. (1900). The Norw. North Polar Exp. 1893-6. Sci. Res. Christiania. Montgomery, R. B. (1938). Circulation in upper layers of southern North Atlantic deduced with use of isentropic analysis. Pap. Phys. Oceanogr and Meteor. Mass. Inst. Tech. and Woods Hole Oceanogr. Inst. 6, no. 2, Cambridge, Mass. Montgomery, R. B. (1939). Ein Versuch, den vertikalen und seitlichen Austausch in der Tiefe der Sprungschichte im aquatorialen Atl. Ozean zu bestimmen. Ann. Hydr. Mar. Met. p. 242 Montgomery, R. H. (1940). Observations of vertical humidity distribution above the ocean surface and their relation in evaporation. Pap. Phys. Oceanogr. Meteor. Mass. Inst. Tech. and Woods Hole Oceanogr. Inst. 7, no. 4, pp. 30. MosBY, H. (1936). Verdunstung und Strahlung auf dem Meere. Ann. Hydr. Mar. Met. Bd. 64. MosBY, H. (1940). An oceanographic thermo-sounder. Un. Geod. Geophys. Intern. Ass. Oceano- graphy Phys. Proc. Verb. no. 3, 190. Mossmann, R. (1923). On Indian monsoon rainfall in relation to South American Weather 1875- 1914. Mem. Ind. Met. Dep. 23. Calcutta. MiJNSTER Strom, K. (1936). Land-locked waters. Hydrography and bottom deposits in badly- ventilated Norwegian Fjords with remarks upon sedimentation under anaerobic conditions. Skr. Norske Vidensk. Akad. Oslo, Math.-Natiirv. Kl. no. 7. MuNK, W. H. (1947). A critical wind speed for air-sea boundary processes/. Mar. Res. 6, no. 3; or Contr. Scripps Instn Oceanogr., N.S. no. 349. MuNK, W. H. and Anderson, E. R. (1948). Notes on a theory of the thermocline. /. Mar. Res. 7. Nansen, F. (1900). On hydrometers and the surface tension of liquids. The Norw. North Polar Exp. 1893-96. Sci. Res. 3, 10. Nansen, F. (1900). Some oceanogr. results of the Exp. with "Michael Sars" 1900. Nyt Mag. Naturvid. Kristiania. Nansen, F. (1902). Oceanography of the north polar basin. The Norw. North Polar Exp. 1893-96. Sci. Res. 3. Christiania. Nansen, F. (1912). Das Bodenwasser und die Abkuhlung des Meeres. Int. Re:, ges. Hydrobiol. Hydrogr. 5, Suppl. Nernst, E. (1909). Theoretische Chemie. 6. Aufl. p. 247. Stuttgart. Neumann, G. (1938). Zur Frage des jahrl. Ganges des Oberflachensalzgehaltes bei Adlergrund- Feuerschiff. 6. Bait. Hydrol. Konf. Lubeck, Aug. 1938. Berlin. Neumann, G. (1940). Die ozeanogr. Verhaltnisse an der Meeresoberflache im Golfstromsektor nordlich und nordwestlich der Azoren. Aus den wiss. Ergbn. der intern. Golfstromuntemehmung 1938. Ann. Hydr. Mar. Met. Beiheft. Neumann, G. (1943). liber den Aufbau und die Frage der Tiefenzirkulation des Schwarzen Meeres. Ann. Hydr. Mar. Met. pp. 1, 164. Neumann, G. (1942). Die absolute Topographic des phys. Meeresniveaus des Schwarzen Meeres. Ann. Hydr. Alar. Met. p. 265. Neumann, G. (1944). Das Schwarze Meer. Z. Ges. Erdkunde. Berlin. NiKmN, W. N. (1927). L'Oceanographie de la Mer Noire d'apres les explorations hydrogr. russes. Ann. Geogr. no. 203. Paris. Nomitsu, T. and Okamoto, M. (1927). The causes of the annual variation of the mean sea-level along Japan's Coast. Mem. Coll. Sci. Kyoto, Imp. Univ. 10, no. 3. Kyoto. Nusser, F. (1952). Das Vorkommen von Meereis um den antarktischen Kontinent zur Zeit des siidlichen Hochsommers. Auf der Eisubersichtskarte no. 19. Dtsch. Hydrogr. Inst. Hamburg. Parr, A. E. (1935). Report on hydrogr. Observations in the Gulf of Mexico and adjacent straits. "Mabel Taylor"-Exp. 1932. Bull. Bingham Oceanogr. Coll. 5, New Haven. 292 Bibliography Parr, A. E. (1937, 1938). A contribution to the hydrography of the Carribean and Cayman seas. Bull. Bingham Oceanogr. Coll. 4, 1937; and 5, 4, 1938. New Haven. Parr, A. E. (1938) Isopicnic analysis of current flow by means of identifying properties /. Mar. Res. 1, no. 4. Parr, A. E. (1938a). Further observations on the hydrography of the eastern Carribbean and adjacent Atlantic water. Bull. Bingham Oceanogr. Coll. 6. Penck, a. (1933). Eustatische Bewegungen des Meeresspiegels wahrend der Eiszeit. Geogr. Z., Heft. 6. Penck, a. (1934). Theorie der Bewegung der Strandlinie. S.B. preuss. Akad. Wiss. 19. Berlin. Pernter, J. M. (1901). tJber die Polarstation des Lichtes in triiben Medien und des Himmelslichtes mit Riicksicht auf die Erklarung der blauen Farbe des Himmels. Den/. (IX.4) Since the gravitational acceleration g changes with depth according to (IX. 2) the difference between two dynamic depths in the ocean is given by the relation 1 g dh. (IX.5) hi 10 As a first approximation (IX. 3) thus gives D = 0-98/2 and /; = 1-02Z). (IX.6) The numerical difference between a dynamic metre and a geometrical metre is thus about 2%. Tables for converting one unit into the other according to more accurate formulae have been given by Bjerknes and co-workers (1912, 1913). 3. The Field of Mass The mass field is given by the distribution of the density p or its reciprocal, the spe- cific volume a. In the sea it can be represented in a suitable way by surfaces of equal density {isopycnic surfaces) or by surfaces of equal specific volume (isosteric surfaces). The latter are used preferably in oceanography. The field of specific volume a^^, „ can be regarded as made up of two separate fields. The first of these agg, o, p represents the mass field of a homogeneous sea at 0°C and 35%o S (standard ocean) ; it is in this way completely defined and invariable. The second is the field of the specific volume anomaly S and this set of surfaces of equal anomaly 6 is quite sufficient for the charac- terization of the mass field in the total oceanic space. In a vertical section of the mass distribution the isosteres and the isopycnals appear as curved or wave-form lines deviating only slightly from the horizontal. A large 304 The Geophysical Structure of the Sea exaggeration of the vertical scale is required to show the slope of the lines in a better way. The geometrical depth, the dynamic depth or the pressure can all be used as the vertical co-ordinate. Such graphic representations are termed dynamical vertical cross- sections, in short dynamic sections. 4. The Pressure Field and its Relationship to the Mass Field. Solenoids The internal stress in a liquid such as the ocean is characterized by the pressure per unit area. In a liquid in equilibrium, due to the absence of any resistance to deforma- tion, this pressure acts perpendicular to any arbitrarily oriented surface through the liquid and is equal for any point and in all directions. This state is denoted as hydro- static stress state. The water masses in an ocean at rest is subject to the influence of gravity and the static pressure p at a depth h is defined as that force produced by the weight of a water column of unit cross-section extending from this depth to the surface of the sea. This does not take into account the atmospheric pressure at the surface of the water so that p is defined solely as the water pressure. Thus P = pmgh, where Pm is the mean density of the water column /;. The dimensions of p is [g cm^^sec"^]. According to (IX. 3) the dynamic depth D can be substituted in place of the geometric depth /; so that P = PmD. (IX.7) The pressure of a column of pure water (p„j = 1) of a height of 1 dyn. m is defined as 1 decibar. This is a tenth part of a bar which is defined as 10^ dyn/cm^ and is the pressure of a column of pure water of lOdyn.m. The practical pressure unit "one atmosphere", is only about 1% greater than one bar. Fractions of the bar in addition to the decibar are the centibar and the millibar. The latter corresponds to a water pressure of one dynamical cm of pure water and is equivalent to a pressure of 0-75 mm of mercury. For an ocean of pure incompressible water the following rule applies: The numerical value of "sea pressure" expressed in decibars is the same as that of the depth in dy- namic metres at which this pressure is exerted. Since p^ in the sea is not very diff'erent from 1 this rule also applies in very close approximation for sea-water. From equation (IX. 7) is obtained D = a^p, (IX.8) where a,„ is the mean specific volume of the water column. If p or a vary, equations (IX. 7 and 8) will be replaced by the integral forms \ pdD and Z) = a p=\pdD and D=\adp, (IX.9) where the integrals must be extended over the whole water column h. For numerical calculations the integral is split up into sums for the thinnest possible layers with approximately constant density or specific volume (see later). The relationships between pressure, geometrical and dynamic depth and the vertical distribution of specific volume and of density are shown in Table 110 for a homo- geneous sea at 0°C and 35%o salinity (standard ocean). The Geophysical Structure of the Sea Table 110, Vertical stratification of a homogeneous ocean at 0°C and 35%o salinity (standard ocean) 305 Pressure Geom. Dynamic Spec. Density Dynamic Pressure depth depth volume depth (dbar) (m) (dyn.m) (lO^a) ( + ^s + ft + ^s, t + ^s. D + ((,D and in addition to the basic values for the homogeneous oceans (Table 1 1 0) six tables are also given for the numerical determination of the six terms on the right-hand side. One term, e^, ;, />, is usually so small that it does not have to be taken into consideration. Hesselberg and Sverdrup (1914-15) have simplified the calculation of the density in situ by introducing the value of ot, which is known from Ps,uo=l -}- 10-=^ a,. Putting P35, 0,D= P35. 0. 0 + ^ D gives Ps./.o= 1 + 10-^c7,.fl, (IX. 14) where (^t,D 10-^ = (Tt \0-^ -f ej) -i- €,,D + et,D. If Gt is known then only three tables are required instead of six for the calculation of the density in situ. The calculation of the specific volume and especially the specific volume anomaly can be simplified in the same way (Sverdrup, 19336): ««, /. p = a35. 0. p + S. where S = 8, + 8^ + 8,, < + 8,, p + 8^, ^ Putting S. + 8, + 8„, =A,,t gives 8 = J„, + S„p+ 8,.^. (IX.15) The first term can be readily found from ct^ and then a. X 10-3 A..,= \ I + Gt X 10- 310 The Geophysical Structure of the Sea The numerical value of 035,0,0, is 0-97264 so that J„ , = 0-02736 - -, "!' ^ ^^in . . *' * 1 + (^< X 10-3 The specific volume anomaly S can then be determined quickly and easily using three small tables. The pressure /? at a dynamic depth D is by definition CD /? = Ps,t,D dD. Table 111. Example of the dynamic evaluation of oceanographic observations of a single station ("Meteor" St. 267, 18.11.27; (p = 13-7° N., A = 19-8° W., 4206 m) Depth Pressure Temp. Salinity Density 10^/1 ,,« lO^S,,, 10^8*.. lO^S AD AD (m) (dbar) (°C) (%o) {od (dyn.m) 0 0 21-20 35-236 24-64 331-3 _ 1 331 00 1 0-0827 25 21-12 35-2I5 24-645 330-8 — 1-0 332 0-0827 00651 50 16-23 35-57 2615 187-6 01 1-6 189 0-0434 0-1478 75 16-25 35-425 26-48 156-3 01 20 158 01912 00350 100 13-52 35-355 26-58 146-8 0-1 2-7 150 0-0725 0-2262 150 12-58 35-255 26-69 136-4 0-1 3-8 140 00650 0-2987 200 11-78 35-16 l&ll 128-8 0-1 4-8 134 0-1270 0-3637 300 10-62 3506 26-93 113-6 0-0 6-8 120 0-1160 0-4907 400 10-22 35-14 27-04 103-2 0-1 8-7 112 0-1080 0-6067 500 906 35 02 27-14 93-7 0-0 9-9 104 0-1000 0-7147 600 8-32 34-98 27-23 85-2 -0-1 11-2 96 0-0920 0-8147 700 7-23 34-89 27-32 76-6 -01 11-5 88 00850 0-9067 800 6-68 34-88 27-39 700 -0-1 12-2 82 0-9917 0-0795 900 5-92 34-83 27-45 64-3 -0-2 12-7 77 0-0740 10712 1000 5-51 34-85 27-52 57-7 -0-2 13-0 71 0-1320 1-1452 1200 4.99 34-92 27-63 47-3 -0-1 14-0 61 0-1160 0-1050 00960 0-0900 1-2772 1400 4-68 34-975 27-71 39-7 -0-1 15-5 55 1-3932 1600 4-14 34-975 11-11 340 -0-1 15-8 50 1-4982 1800 3-74 34-97 27-81 30-2 -01 16-2 46 1-5942 2000 3-44 34-965 27-84 27-4 -01 16-6 44 1-6842 0-1100 2250 3-21 34-955 27-85 26-4 -01 17-4 44 01100 1-7942 2500 3 02 34-945 27-86 25-5 -0-2 18-2 44 0-2175 1-9042 3000 2-73 34-93 27-875 24-1 -0-4 19-4 43 0-2175 2-1217 3500 2-51 34-90 27-87 24-6 -0-5 20-4 44 0-2250 2-3392 4000 2-37 34-89 27-87 24-6 -0-6 21-6 46 2-5642 The Geophysical Structure of the Sea 3 1 1 Replacing Ps, t, o by the relation (IX. 14) gives p = D -\- 10-3 f ^^^ dD (IX.16) Here only the last term requires numerical integration and has to be summed only to the depth at which the pressure is required. The anomaly in dynamic depth AD for given pressures is also obtained in the same way. Since D = Dss. 0. V + ^ A AD 8 dp. (IX. 17) If S is known it can also be found by numerical integration. Using the tables given by SvERDRUP (1933Z)) the complete dynamic calculation of the values for an oceanographic station down to 5000 m can with a little practice be done in less than half an hour since the numbers in the tables are always small. The absolute values for the specific volume and of the dynamic depth can be ob- tained by adding the anomalies to the standard values for the standard ocean at 0°C and 35%o ; they are given in Table 1 10. If o-^ is known accurately to the second decimal place, then the table will give the density in situ crs,t,D correct to the second decimal place, and the pressure for a given dynamic depth correct to the third decimal place. The specific volume anomaly and that of the dynamic depth at given pressures can be found accurately to the fifth and fourth decimal places, respectively, but the last two places in the anomaly of the dynamic depth have only computational significance. Table 111 shows as an example the complete dynamic evaluation for the "Meteor" station 267 (18.11.1927; cf^ = 13-7° N., A = 19-8° W., 4206 m), and also the calculation of the specific volume anomaly and that of the dynamic depth at given pressures in decibars according to the simpUfied method of Sverdrup. Chapter X Forces and their Relationship to the Structure of the Ocean 1. External, Internal and Secondary Forces (a) Of the external forces that give rise to or maintain the ocean currents, the most important are the air currents, the changes in atmospheric pressure at the surface of the sea, and the periodic tide-generating astronomic forces. These forces can also initiate water movements in a homogeneous sea. The changes in atmospheric pressure are transmitted through the entire mass of water down to the sea bottom and thus give rise to horizontal pressure differences and the formation of gradient currents. The effect of air currents is twofold: First, the tangential force of the wind on the surface of the sea (wind stress) produces a surface current which is transmitted by the effect of viscosity (turbulence) to the water layers beneath the surface. Secondly, the wind produces waves at the surface of the sea and the pressure exerted by the wind on the windward side of these waves also initiates water movements in the direction of the wind (wind drift). These currents produced by the wind and by the changes in at- mospheric pressure are considerably modified by the deflecting force of Earth rotation and by the boundary surfaces of the sea (coasts, continental slopes and sea bottom). The piling up of the water by coasts (Anstau) is by far the most important effect of the external forces and is responsible for the formation and the maintenance of an oceanic circulation in the deeper layers of the ocean. In a sea of homogeneous structure the external forces can produce no change in the physical stratification of the water mass. In a non-homogeneous sea, however, the water movements displace different types of water relative to each other, and thus either directly or due to the boundary conditions produce changes in the thermo- haline structure of the ocean. This upsets the system of internal pressures forces and give rise to ocean currents. (b) The internal forces arise from the vertical and horizontal distribution of mass within the ocean. These differences in the mass distribution (in horizontal and vertical direction) are the consequence of changes in the heat content (temperature) and in the salinity. If at first the water masses are in an internal equilibrium state, this equilibrium can be disturbed by changes of this type, thereby initiating ocean currents which in turn tend to restore the system to a new equilibrium. The principal sources for dis- turbances in the mass distribution can be found at the surface of the sea, where solar and atmospheric radiation and outgoing radiation first influence the ocean, and where evaporation also takes place. At the other boundary surface of the sea (the sea bottom) the intensity of the disturbances is small and usually of no importance in changing 312 Forces and their Relationship to the Structure of the Ocean 3 1 3 the distribution of mass. Within the sea, the turbulence in the moving water masses may presumably also produce changes in the physical-chemical structure of multi- stratified water bodies. All these disturbances are, however, small compared with the changes in mass distribution due to atmospheric influences effective at the sea surface. The only internal force dependent on the mass distribution is the gradient force. This force per unit volume is given by the pressure gradient G (see equation (IX. 10)). The pressure force per unit mass can be obtained by multiplication with the specific volume so that r ^P ^ "^P (X n dn p dn The pressure field also determines the field of force per unit mass, since the normal to the isobaric surface gives the direction, and the thickness of the isobaric unit layers gives the intensity of the pressure gradient at any point in oceanic space. Bjerknes (1900) by analogy with the pressure gradient introduced a "mobility vector" B which gives the variations in specific volume in the direction n of increasing specific volume perpendicular to the isosteric surface. 5 = ^. (X. 2) dn The degree of concentration of the dynamic isobaths on an isobaric surface is of course also a measure of the gradient force, and is at the same time also a measure of the potential energy stored in the mass distribution. Figure 133 presents a section through an ocean and two oceanographic stations are indicated by A and B (L km apart). As a first approximation the pressure surface can be regarded as horizontal and coincident with the surfaces of equal geometric depth. The surfaces of equal geopotential (equal dynamic depth) are inclined relative to these so that the same pressure Pn at dynamic depth Da at station A is found at the greater dynamic depth D^ at station B. Di, — Da is then the difference in potential energy between A' and B'. This potential difference can be regarded as a force along L which, if present alone, would set the water masses in motion. The force per unit mass resulting from the internal pressure difference is then K = ^' ~ ^°. (X. 3) According to (IX. 12), D^ — Da is the number of solenoids enclosed within the cross- section between the two stations A and B from sea-level to the depth in question. This number per unit length is thus a measure of the internal force resulting from the mass distribution. (c) Among the secondary forces are included all those apparent forces that in them- selves do not give rise to a current but which, when motion is present, are of decisive importance in determining the final form of the water displacement. These include the deflecting force arising from the rotation of the Earth (the Coriolis force) which affects solely the direction of the water movement, the viscosity (boundary fric- tion and turbulence) which affects more the velocity of a current, and finally the centrifugal force, which for motion along a curved path (velocity V, radius of 314 Forces and their Relationship to the Structure of the Ocean ture R) gives a force V^IR away from the centre of curvature; since for an angular velocity Q,V =^ QR the centrifugal force for unit mass will be Q^R. Fig. 133. Cross-section through an ocean. A and B are two oceanographic stations. Full lines: isobaric surfaces (p = const.); Dashed lines: surfaces of equal dynamic depth (D = const.); The pressure surface p„ appears in station A at the dynamic depth Da, in station B at the dynamic depth D^ ; L denotes the horizontal distance of both stations (schematically). (a) All observations on the rotating Earth are usually made with reference to a co- ordinate system rigidly connected with the Earth and therefore rotating with the Earth, although in the classical mechanical sense this is not a permissible reference system. Such a system should not follow the rotation of the Earth, but would for example have to be assumed at rest relative to the fixstars (absolute system). If the basic principles of Galileo-Newton mechanics are used and applied to the rotating Earth, deviations will appear which are due solely to the movement of the reference system imposed by the rotating Earth — a fact which we simply have to accept. These deviations have the character of two apparent forces which are additional to those forces present in the absolute system. One of these forces depends only on the geographical location ; this is the ordinary centrifugal force due to the rotation of the Earth ojV (oj is the angular velocity of rotation of the Earth — one total revolution per one sidereal day = (27r)/(86,164 sec) = 7-29 X 10~^ sec~^, r is the distance from the axis of rotation of the particle under consideration). Since this additional force acts both on a stationary or on a moving mass particle it can be combined with the gravitational force and becomes in this way part of the force of gravity. The second force, however, depends both on the geographical location and on the velocity of the mass particle set in motion on the Earth. This is denoted to CorioUs force and as a vector acting on unit mass has the form g = 2[b tu]. (X.4) Its absolute value is C = le| == 2Kcosin(t) to) and it is directed at right angles to the direction of the velocity vector b and to the angular vector of the Earth's rotation to, which is in the direction of the Earth's axis, so that I to| = to. It therefore acts at right angles to the tangent to the path of movement Forces and their Relationship to the Structure of the Ocean 3 1 5 as well as in the direction of the equatorial plane. A complete derivation for the Coriolis force, unobjectionable in every respect, has been given by Bjerknes and co-workers (1933). If at any point on the surface of the Earth a co-ordinate system is chosen with the (A:_y)-plane coinciding with the tangential plane to the Earth (x- positive to the east, j^-positive to the north, z-positive towards the Earth's interior), the components of the Coriolis force acting on a material particle on the Earth moving in any direction with a total velocity V (ii, v, w) can readily be calculated using equa- tion (X. 4). This gives the components in the three co-ordinate-directions Cx — 2ctjr sin — 2ww cos (f), Cy = —2wu sin 0, C^ — —Icou cos ^. (X. 5) From these it can be shown that every movement in the tangential plane to the surface of the Earth will be deflected by the Coriolis force to the right in the Northern Hemis- phere and to the left in the Southern Hemisphere. The terms cum sole and contra solem suggested by Ekman can be used respectively for rotation in the direction of the azimuthal movement of the sun, i.e., to the right in the Northern Hemisphere and to the left in the Southern Hemisphere {cum sole), and for rotations in the opposite direction to the azimuthal movement of the sun, i.e. to the left in the Northern Hemisphere and to the right in the Southern Hemisphere {contra solem).'\ Thus every movement in a horizontal direction is deflected cum sole by the Coriolis force. It also follows from equation (X. 5) that there is a vertical component of the deflecting force only for zonal movements (.v-component u) and not for meridional movements. The importance of the vertical component for the dynamics of moving masses is quite small since it acts in the same direction as gravity relative to which it is vanishingly small. ^ The horizontal component is very important, however; at the poles (<^ = 90°) it amounts to 1-46 x 10~^ cm sec"^ for m = 1 cm sec"^ and is thus of the same magnitude as other forces acting in the same direction (gradient forces, tidal forces); it is zero at the equator and reaches the above maximum at the poles. Since it acts at right angles tothedirectionofmovementitisunabletoproducechangesin velocity and is incapable of doing work; it can only produce changes in the ^//-ecZ/o/iofmovement, but these changes are of decisive importance for the finally established patterns of motion. Due to the effect of the Coriolis force a mass particle moving freely in a horizontal plane with a velocity V will follow a curved track. Since the deflecting force acts at right angles to the velocity (apart from the effect of change in latitude) and its absolute value is constant, this path will describe a circle which is known as the circle of inertia. The t Another terminology uniform for both hemispheres is that customary in meteorology: cyclonic = contra solem and anticyclonic = cum sole. X The usual statement, that the vertical component of the Coriolis force need not be taken into consideration, since it is small by comparison with the gravitational acceleration is not entirely correct. In the static equilibrium state of the sea, gravity and the vertical pressure gradient neutralize each other. However, under quasi-static conditions the difference between gravity and vertical pressure gradient is so small that it may be of the same order of magnitude as the vertical component of the Coriolis force. Nonetheless, it is customary to neglect the latter in calculations ; it can be regarded as an increase or decrease of gravity so that the acceleration towards the centre of the Earth is now g + Imi cos (p. It can also be regarded as causing a small change in density in the ratio {g + Iwu cos (p) : g. At the equator, when m = 30 cm sec-^ it may amount to 5 units in the sixth decimal place in p or 5 units in the third decimal place in at, which can be disregarded. 316 Forces and their Relationship to the Structure of the Ocean radius of this circle follows from the condition that the forces acting on the moving mass particle must balance each other (centrifugal force = Coriolis force). so that J/2 'r R 2a>Ksin ^, V (X.6) 2ca sin (j) The term "circle of inertia" has been chosen to indicate that relative to the movement of the rotating Earth this circular movement in a certain sense replaces the linear inertia of absolute motion. The radius of the circle of inertia is a function of the latitude and the velocity. Table 112 gives values for this functional relationship. Table 1 1 2. Radius of inertia circle as a function of V and (V in mm sec-^, cm sec-^, m sec-^: R in m, 10m, km units) 0 . . 5° 10° 20° 30° 40° 50° 60° 70° 80° Poles V= 1 79 40 20 14 10 9 8 7 7 7 2 157 79 40 27 21 18 16 15 14 14 3 236 118 60 41 32 27 24 22 21 21 4 315 158 80 55 43 36 32 29 28 27 5 393 198 100 69 53 45 40 36 35 34 6 472 237 120 82 64 54 48 44 42 41 7 551 277 140 96 74 63 55 51 49 48 8 629 316 160 110 85 72 63 58 56 55 9 708 356 180 124 96 81 71 66 62 62 10 786 395 201 137 106 90 79 73 70 69 The time required for one complete rotation on the circle of inertia {inertia period) is given by IttR it 1 2 sidereal hours T = CO sin (f) sin If the period of rotation of the plane of oscillation of a Foucault pendulum is a pen- dulum day = {2tt)1{o) sin of latitude and cos (^ by 1. Inertia movements super- imposed on horizontal and zonal currents play a large part in the dynamics of ocean currents especially the occurrence of long waves and in vortical disturbances. (See in this connection Defant (1956) and Vol. I, Pt. II, Chap. XIII, 6.) X The origin of viscosity can be sought in the continuous equalization of velocity between super- imposed layers of water gliding over each other in a moving water mass. This equalization is due to the interchange of individual molecules and the consequent transfer of velocity from one layer to the next. This viewpoint is, however not entirely correct since the molecules in a hquid are so closely packed that usually they can only oscillate within the small intermolecular spaces present and therefore only 318 Forces and their Relationship to the Structure of the Ocean For this assumption concerning the inner friction, the effect of the solid, stationary sea bed appears as a corresponding boundary condition. If n is the direction of the normal to the sea bottom (z = 0) then (1) for completely frictionless movement of the water over the sea bed (r = 0): dVldn = 0; (2) if the water is stationary at the bottom (z = 0): K = 0; (3) for part-time gliding at the sea bottom, that is for a discontinuity of the velocity at z = 0: dVjdt = f(V), where /(K) is a certain function of V, for example, kpV^. In a volume element 8x 8y 8z (see Fig. 134) in a current in which the velocity V in a direction perpendicular to the vertical direction z is very much stronger, there will be a shearing stress rSxSy on the lower surface 8x8y and a corresponding - (X. 11) Fig. 134. Computation of the frictional force from the shearing stresses. ^T j^ {8Tl8z)8z}8x8y on the upper surface at a distance Sz from the lower. On the entire volume element there acts thus a frictional force (8Tldz)8x8y8z so that accord- ing to (X. 10) the frictional force per unit mass in the direction of the x-co-ordinate will be given by fi 8^V 8z^ ' R:r. — (XJ2) Where fx can be regarded as a constant. From the general theory of friction in hquids it follows that for an incompressible fluid (and thus also with sufficient accuracy for sea-water) the components of the fric- tional force per unit mass in a viscous liquid are given by the three expressions : u ix M , ;?^ = - An, Ry = ^ Av, i?, = - Aw, PR P (X.13) Footnote continued from p. 317 seldom change position. These occasional changes in position are facilitated by the action of a tan- gential shearing stress especially in the direction of the stress itself and this alone permits the individual layers to glide over each other. The more frequent the changes in position of the molecules, the lower is the internal friction (viscosity) characteristic of the liquid. Forces and their Relationship to the Structure of the Ocean 319 where ii, v, w are the velocity components in the direction of the three co-ordinate axes and A is the Laplace operator 8^l8x^ + 8^l8y^ + 8^l8z^. The quantity v = ixjp is called the kinematic viscosity coefficient and has the dimensions [cm^ sec"^]. For numerical values of ju and v for pure water and for sea-water see Vol. I, Pt, I, p. 104. The actual movement of water masses in the oceans does not correspond to a simple ordered gliding of the individual superimposed layers relative to each other, but is rather a random disorganized movement that takes place in vortices and rolls similar to those which can be seen in a smoke plume. The first type of motion is called layered or laminar and the second turbulent. In turbulent flow there is a transfer of the flow momentum from one layer to another, not by the interchange of molecules as in physical internal friction but by the exchange of large elements of water (eddies) which move rather irregularly back and forth between the diff'erent layers and thus bring about a reduction in the velocity in the direction of the basic current; this is then referred to as virtual internal viscosity or eddy viscosity, which in an analogous way to the molecular viscosity can be characterized by a special eddy viscosity coeffi- cient. It is easily seen that the eddy viscosity, by its nature, will be more eff'ective than the molecular viscosity and is also understood by the numerically much larger viscosity coefficients. However, apart from this, the turbulent coefficient is no longer an in- variable quantity like the molecular viscosity at constant temperature, but depends on the nature and the intensity of the turbulence itself. Further, the components of the frictional force of turbulent viscosity can be expressed in exactly the same way as those in equations (X. 12 and 13) if ju, is replaced by the turbulent viscosity coefficient rj. If this is not constant then, for example, equation (X. 12) is replaced by the expression 1 a p 8z ('£) X.14 and the same applies for the other expressions in (X. 13). To a very large extent ocean currents are movements along quasi-horizontal planes so that the turbulent viscosity for small oceanic spaces is limited to that appearing in connection with layered gUding motion of the water masses. Within the moving water mass turbulence creates a definite vertical velocity profile and tends to maintain it. If there is no viscosity this profile must be linear (see Fig. 135, 1). The velocity of the Fig. 135. Main types of vertical velocity distributions: (I) in the case of no friction; (II) in the case when friction retards the mean current filament; (III) in the case when friction accelerates the mean current filament; (IV) for a constant frictional force. filament (a) in the middle of the current is then the mean of the velocities of the ad- jacent upper and lower masses. The accelerating influence of the upper layer will be exactly compensated by the retardation at one of the lower. In case II, where the 320 Forces and their Relationship to the Structure of the Ocean velocity of this middle layer is greater, the adjacent layers will exert, due to the transfer of their flow momenta, a retardation on the current maximum in the middle and will eventually eliminate it. The middle layer in case III will be accelerated by the equalization of velocity in the turbulent flow. Equation (X. 12) also shows that for a constant internal friction the vertical profile must take the form of a parabola. 2. The Basic Hydrodynamic Equations For a complete description of the water movement in ocean currents, it is necessary to know on the one hand the path of each small element of water in it, and on the other hand the position of such a small element along this path at any time; i.e. it is neces- sary to know the co-ordinates of a small element of water as a function of time. The basic hydrodynamic equations of motion in their most general form are the mathe- matical-physical tool for dealing with and for a theoretical understanding of the different successive states of a water mass. The motion can be looked at from two different view-points. The different mass elements may be followed as they pass a. fixed point in space and particular attention may be paid to the changes in the state of motion of the water mass which occur at this point. Alternatively, the changes of state of individual small elements moving along their track may be followed, and thereby a description of the conditions in the current in the course of their displacement can be obtained. The first approach gives the Eulerian basic hydrodynamic equations of motion (Euler, 1755) and the second leads to the equations of motion of Lagrange (Lagrange, 1781); both of these con- cepts are applied in oceanography according to the type of problem to be solved. For a small element of water in the point .v, y, z the components of the velocity are denoted u, v, w in the directions of the co-ordinate-axes of a left-hand system (xy- plane horizontal, x-axis positive to the east, >'-axis positive to the north, z-axis posi- tive towards the centre of the Earth), They will be functions of x, y, z and of the time /. First of all the basic Newtonian relationship of mechanics is applied: Mass X acceleration = sum of all forces. The individual accelerations dujdt, dvjdt and dwjdt are made up of two parts. The first part arises from changes in the state of motion at the point; it is given by dujdt, dvjdt and dwjdt (local change). The second arises since after a small time dt the water elements under consideration are no longer found at the initial point (.v, y, z) but are displaced by udt, vdt and wdt, respectively (advective change). Thus to the local part must be added an advective part, so that the total individual acceleration in the x- direction of the small elements of the liquid under consideration will be du du du du du Similar equations apply for dvjdt and dwjdt. It may be emphasized here that the partial derivative djdt always represents the change in the quantity under consideration at a fixed point, while the total derivative djdt represents the individual change in a quantity for one and the same element (which changes its position with time). Taking the mass of unit volume as p, and considering that since the only external Forces and their Relationship to the Structure of the Ocean 321 conservative force is gravity acting in the positive direction of the z-axis (downward), the pressure gradient forces will be given by \ dp I dp I dp p dx* P dy* p dz ' respectively, and introducing the CorioHs force according to (X. 5) and the frictional forces according to (X. 13), then the basic hydrodynamic equations of motion will take the complete form (X.16) du dt~ du du du du . I Sp fi ^ w: + u ^ + V ^+w -^ = + 2wv sin 0 ^ + -Au, dt dx dy 8z ^ p dx p dv dt~ dv dv dv dw ■ . ^ ^P H- . 8t dx oy dz p dy p dw dl^ dw dw dw dw \ dp a -77+ u -^ + V -^ + w w- = g — 2wu cos (^ — + - Z dt dx dy dz ^ p dz p Aw The third equation in the 2-direction can be considerably simpUfied, which shall be done at once. Since the movements of the water in the ocean occur very largely in a horizontal plane and w, dw/dt and the frictional term in this direction can always be assumed to be small, and further, since the vertical component of the Coriolis force can be neglected, the third equation in (X. 1 6) reduces to 1 dp (X.17) 0=^ dz which corresponds to the basic hydrostatic equation (see p. 337). For problems involving the whole or an extended part of the rotating Earth it is convenient to use polar co-ordinates. The reference surface selected is the free sea sur- face in a state of equilibrium (usually it is sufficiently accurate to take a spherical surface with the mean radius R of the Earth) and as co-ordinates can be taken the pole distance {^ = 90 — (j), the longitude A and the distance z from this surface (along the radius of sphere R, positive outwards). The velocities relative to the Earth along the three axes are then u = (R + z) d& dt V = Rs'm >& dX It and w dz Jt (X. 18) If the external forces have a potential Q-f and if the frictional terms are omitted, the equations of motion take the following form du ^ - ^ d /p \ dt 2ojv cos §■ — — R + z dd'\p dv dt dw df + 2 T' —,■ 3nd --, ^, ~, db db db dc dc dc respectively, and finally can be added. If the forces have a potential Q, the Lagrange form of the equa- tions of motion is obtained Idhc \ 8x id^ \ a V IdH ^Y^ ,^ ^P_^ \dl''~ ^) aa + U/2~ ^ ) da + \dl^~^)da^~p 8'a~ ^' \dt^ J db^\dt^ I db ^ W/2 ^ I 8b^ p db Id^x \ dx id^y \ dv (d^z \ dz I 8p b/2 - ^} e-c + \d^ - ^ } Tc^ \dt^~ ^ } 8c^ 0, p cc The hydrodynamic equations of motion form a very complex set of equations. They have to be solved in order to obtain a complete description of the state of motion but only in very rare and in the most simple cases it is possible to arrive at a final and definite solution. In most cases it is considered sufficient to determine, if possible, the state of motion at each place and at each time without paying any attention to the further history of the individual water elements. There is a considerable simplification possible when dealing with so-called stationary currents. These are currents in which the state of motion at each point does not change with time and is thus completely fixed by specifying its direction and velocity. The condition for a steady state in the current is thus 8u dv dw 8", = a, = a? = 0- <'^-2«> Forces and their Relationship to the Structure of the Ocean 323 Some kinematic properties of the motion should perhaps be referred to here. The path of a small water element is obtained from the three simultaneous equations : dx = udt, dy = vdt, dz = wdt. (X.21) The integration constants for f = 0 are then the three initial co-ordinates a, b, c of the water element under consideration. The instantaneous state of motion in a water mass is given by the stream lines (see Chap. Xll) which everywhere indicate the direction of a current by the tangent at the point under consideration. Their differential equations are dx dy dz — - — = —. (X. 22) U V w Since the state of motion in a steady current does not change with time it is under- standable that the stream lines in this case coincide with the trajectories of the water elements. Steady currents are not without accelerations since only the local part of the acceleration disappears; the advective part, for example, u(duldx) + vidujdy) + w{8ul8z) requires that the moving water element reaches any point with a velocity equal to that prescribed for that point, 3. The Continuity Equation and the Boundary-surface Conditions To the equations of motion must be added, as a special condition, the continuity equation which is based on the law of the conservation of mass. This states that in any volume element specified in the interior of a liquid the mass entering it at a given time must be equal to that leaving it at the same time. Any excess in one or the other direc- tion must appear as a corresponding change in the density if the liquid will permit such a change. Taking a volume element SxSySz, investigation of the extent by which, as a consequence of flow through the boundaries the amount of liquid enclosed in it varies, shows that for a conservation of mass the continuity condition is given by the equation dp dpu dpi) dpw Using the relationship equation (X. 1 5) this can be given the following form \ dp ] da 8u 8v 8w p dt a dt 8x cy 8z In an incompressible liquid dpldt = 0 the continuity equation reduces to 8u 8v 8h' ^ + ^ + — = 0 - (X.25) ex oy oz ^ -^ This does not assume that the liquid has the same density everywhere (homogeneous medium). The expression cujdx + cvjcy + bwj8z indicates the volume increase in unit time per unit volume of the element and is usually termed the three-dimensional or total divergence of the vector (m, r, vv). The continuity equation for an incom- pressible medium is then div (//, r, h) =-- 0. (X.26> 324 Forces and their Relationship to the Structure of the Ocean Since the rotation of the Earth does not affect the conservation of the mass, the con- tinuity equation does not contain the angular velocity of the Earth's rotation when a polar co-ordinate system is used for the rotating Earth (co-ordinates: pole distance '& — 90° — (f), longitude A and r along the Earth's radius R). However, there are changes in the cross-section of a current for meridional motion due to the convergence of the meridians and for vertical displacements of mass due to the divergence of the Earth's radii. Thus in the continuity equation for polar co-ordinates, in addition to the previous terms derived from flow through the volume element, there will be two further terms considering these further circumstances in this special co-ordinate system. These give the following equation : dp 1 87 '^ R sin ^ dpu sin '& 8pv' dpw 2pw + ^ + -^ = 0. (X.27) The effect of the convergence of the meridians is expressed in the term (puIR) cot g'& which is obtained by differentiation of the first expression in the brackets and the effect of the divergence of the Earth's radii is contained in the term 2pwjR. Since for vertical displacements of mass in the sea, which is shallow relative to the Earth's radius, the vertical velocities appearing are very small, this last term is not too important and can safely be neglected. For small oceanic spaces the convergence of the meridians can also be disregarded in first approximation, though not for large-scale ocean currents (see Chap. XXI).t If the liquid has boundary surfaces either at a solid body (the sea bottom) or when it is surrounded by differently stratified liquids (other water bodies) the continuity equation will take special forms and must be replaced or supplemented by special boundary conditions. At a solid boundary, in order to secure a reasonable state of motion with no empty spaces, the component of the velocity perpendicular to the surface must be zero. If /, m, n are the direction-cosines of the normal to the surface then a necessary condition is lu -{- mv + nv — 0. (X.28) t The continuity equation which corresponds to the Lagrange equations of motion is more difficult to derive and reference should be made to text-books of hydrodynamics. Taking the functional determinant Bx 8y 8z 8a da 8a 8x 8y 8z 8b 8b 8b 8x 8y 8z 8c 8c 8c 8(x, y, z) 8{a, b, c) ' the condition of constancy of mass in a volume element 8a 8b 8c will be 8(x, y, z) d{a, b, c) where Po 's the initial density at the point {a, b, c). For incompressible liquids p = Po the continuity equation takes the form 8{x, y, z) 8(a, b, c) = 1. Forces and their Relationship to the Structure of the Ocean 325 At all inner boundary surfaces, on the other hand, the velocity component perpendicu- lar to the boundary surface must be the same on both sides of the surface. If the values for the quantities on both sides of the boundary are specified by separate indices, then this kinematic boundary condition can be represented as a special case of equation (X. 28) /("i - «2) + rn{vi - ^'2) + n{yv\ - w^) = 0. (X.29) From the point of view of continuity it is allowed to make a free choice about the velocity component parallel to the inner boundary surface and solid surface, respec- tively. If the liquid has Sifree upper surface this will be subject to the condition that all the small fluid elements which belong to it will always remain in the liquid. If/Cv, y, r, /) = 0 is the equation for the free upper surface the foregoing condition requires that In addition to the kinematic, there is also a dynamic boundary-surface condition that must be satisfied at inner boundary surfaces as well as at a free surface. This requires that at the discontinuity surface where the individual quantities are subject to sudden changes, the pressure must be the same on both sides of the boundary. If /(x, y, z, /) = 0 is the equation for the discontinuity surface, which may be either moving or stationary, and if/7i and/72 give the pressures in the medium on both sides of the surface as functions of .Vi, y^, z^ and x^, J2, z^, respectively, then the dynamic boundary condition will require that values of x, y, 2 and t, in order to satisfy f{x, y, z, t) = 0, must also satisfy the equation PiiXi, >i, Ti, 0 — p^ix^, J2, Z2, /) = 0. (X.31) 4. Potential Flow, the Bernoulli Equation, Impulse and the Impulse Form of the Hydrodynamic Equations In very many cases the velocity components u, v, w can be expressed by a function 9, so that 80 S(D do This function then is called the velocity potential, and movement for which a function of ' this type is valid has been termed o. potential flow. By this kind of definition it is shown that if such a potential is present: (1) The stream lines will be everywhere perpendicular to the surfaces 9 = const, (equi-potential surfaces of velocity). This follows from (X. 22) when combined with (X. 32). (2) The following combinationary relationships : du 8v dv 8w 8w 8u 8y 8x ' 8z 8y ' 8x 8z will apply ; these state that the current in the presence of a velocity potential is irrotational (free of vorticity). 326 Forces and their Relationship to the Structure of the Ocean (3) The continuity equation for an incompressible medium will take the form S^cp c^cp 3^9 Neglecting the Coriolis force and the frictional forces, the three Eulerian equations of motion equation (X. 16), on multiplication by dx, dy and dz, respectively, and taking further into account the identity du du I d ^ ^ rf^ = «7 + 2 8Tx("^ + '-^ + "'^) ('^■33) and by subsequent addition, can be compressed into the single equation where F(t) is an arbitrary function of t alone and Q is the potential of the external forces. For a steady current (8u 8v 8w \ di^ 8i ^ 8t ^^) in which the stream lines coincide with the trajectories of the fluid elements U^ + V^ + H'^ p ^ +~+^=C, (X.35) where the quantity C is constant along each stream line but changes on passing from one stream line to another. The equation (X. 35) is called the Bernoulli theorem (equation). It shows that for steady motions the pressure at points along a stream line is greatest where the velocity is smallest and vice versa. Considering that a fluid particle on transfer from higher to lower pressure is subject to an acceleration (increase in velocity) the above statement is readily understood. This is another way of expressing the conservation of energy, since for unit mass the first term is the kinetic energy of motion, the second is the work done against pressure and the third is the potential energy; in a steady flow the sum of these energies along a stream line must be constant. If the water movement is solely influenced by the gravity force, then since Q = gz, the Bernoulli pressure equation will have the form ^ + - + ?z = const., with m2 m f2 ^ ^^,2 - c\ (X.36) 2 p For a two-dimensional potential flow it is convenient to introduce a stream function ifj defined by the relations «=-^, v=+^^ (X.37) and therefore from (X. 32) Sep Si/» ^9 8ijj 8x^ 8)'' 8y^ ~ 8x' Forces and their Relationship to the Structure of the Ocean 327 In addition, the differential equation J0 = 0 must also be satisfied by i/-. Since the curves ^ — const, are perpendicular to the curves 9 = const, \dx 8y '^ 8y dx "' They represent stream lines (hence, the name stream function). It can easily be shown that every analytical function of the complex variable r = .V + iy satisfies the continuity equation Jcp = 0, i.e. represents a solution for the equations of motion. If this function is given by F(z) = F(x + iy), then its real part is the velocity potential (p and the imaginary part is the stream func- tion ifj or vice versa. This important consequence allows simpler current systems to be completely worked out kinematically. Use will be made of this later (see Chap. XII, 3). In a few important cases the use of the impulse theorems for steady currents in a water mass has considerable advantages. The product of mass and velocity is termed the impulse or momentum; as a vector, like velocity, it has three components. The impulse theorem states that for any arbitrarily limited water mass (the outer boundary sur- faces all together are usually termed "control surface") the change with time of the impulse in it is equal to the sum of the external forces acting on the mass. The internal forces in the system balance each other according to the principle of action and reaction. The change in momentum can be divided into two parts. The first gives the change with time of the impulse in the volume under consideration enclosed by the control surface; for a steady current this term vanishes. The second is the momentum entering or leaving it in unit time through all the boundaries (total control surface). For a steady current the vector sum of all pressures acting on the control surface must be equal to the transport of impulse through it. As an example, the following two cases will be considered here. Fig. 136a shows a straight current tube formed by stream lines ; we consider the part between 1 and 2. At the cross-section 1 (surface F^) the current enters with a velocity V^. The water Fig. 136a mass transported in unit time is pV-^F^, the impulse transport (momentum flux) through Fj into the volume under consideration is p Fj^Fi ; similarly, at cross-section 2 (surface F^ an impulse amount pV^Fz leaves the enclosed space; as a "counter action" it has to be taken with a negative sign. The impulse amount remaining in the space is thus p(Ki^Fj — V^F^. In a steady current, in order to secure an equilibrium state, 328 Forces and their Relationship to the Structure of the Ocean it has to be balanced by the vectorial sum of all the surface pressures, that is, by Fj/?! — F^Pz- This gives for the current tube the equilibrium equation Ki2 + a-i P2. K22+-IF2 which corresponds to the BemouUi pressure equation. If the current tube is curved (Fig. 1 36^) the forces at both places 1 and 2 will have different directions and the resultant R of the two forces (indicated at point A) shows the effect of the pressure exerted by the curved flow on the adjacent water masses. (b) Fig. 1366 By the introduction of the contmuity equation, the equations of motion can be put in a form which expresses changes in impulse more clearly {impulse form of the equation of motion). Multiplying the continuity equation (X. 23) by m, v, w and adding these expressions respectively to the first, second and third of the equations of motion (with- out Coriolis force and friction terms, X, Y, Z are the external forces), then dpu dpuu 8puv dpuw dt + 8x + dy + 8z pX dp dx' dp 8pv 8pvu 8pvv 8pvw '8i '^ ~8x '^ "ajT "^ ~aF~ ^ ^ 8/ > (X.38) 8pw 8pwu 8pwv 8pww "aT "^ "ax" "^ ~e^ "^ 8z pZ 8p 8z' These show that the changes in the momentum within a volume element can be re- garded either as the result of forces acting on the mass contained within the volume element, or as the result of the mass flux passing through the boundary surfaces carrying its own momentum with it. The impulse-form of the equations of motion (X. 38) can be used with advantage in considerations concerning the internal structure of turbulent currents (Reynolds, 1 894). At any point of a turbulent flow there will be more or less strong variations in the flow velocity. These variations will, however, balance each other completely if on the average the current is steady, and if a sufficiently long period is considered. The velocity components at a given point can then be represented by M = W + «', V = V -\- V', W = W + H'', (X.40) Forces and their Relationship to the Structure of the Ocean 329 where m, d, w are the mean values of these components and u', v', w' are the compo- nents of the superimposed turbulent motion for which by definition u' = 0, V = 0, w' = 0. (X.40) The bar over these symbols indicates mean values considered over a sufficiently long time. It should further be noted that the mean values of the squares and products of «', v', w' of course must not necessarily vanish. If the impulse equations (X. 36) are apphed to such a turbulent flow it is not suffi- cient to consider the equations for the mean steady flow alone, since also the turbulent parts of the velocity changes are involved in the relationship between the mean steady current and the forces acting on the masses. This can be derived directly from the impulse theorem. Considering, for example, a part of the "control surface" that is at one time vertical to the x-axis and at another time vertical to the jv-axis, then in the first case a mass pu will pass through a unit area in unit time; the impulse transport due to the x-component u of the velocity is then pun and its mean value over a longer period puu. Now uu — {it -\- u'Y + «^ + 2wm' + u'^. In deriving the mean value uu it should be noted that u is already a mean value of u and w' = 0, so that puu — pu^ = pu'^. To the impulse of the steady mean current a turbulence contribution is added in form of the square of the turbulent variation in velocity, which when inserted in equation (X. 38) has the effect on the mean motion of an additional pressure. Similarly, a mass pv will pass through unit area of the control surface perpendicular to the >^-axis in unit time. The x-component of the impulse transferred through the surface is thus, in this case, puv and taking an average gives puv per unit time. With uv — uv ■}- u'v -\- uv' + u'v', puv = puv + pu'v'. In addition to the impulse of the steady mean current puv must be added a turbulence contribution which in general does not vanish; because positive values of m' are mostly correlated with positive values of v' and vice versa, so that the products are preferably positive. In the opposite case the products are mostly negative. If this turbulent contribution of the impulse transport is transferred to the right- hand side of equations (X. 36) it can be taken as a force acting along the .v-axis, which in all cases will be perpendicular to the >'-axis. It can therefore also be considered an apparent shearing stress r = -pTv' (X.41) arising from the turbulence of the current and was previously regarded (see pp. 3 1 7-3 1 9) as an apparent internal frictioH. Equation (X. 41) mediates between this viewpoint and the equation (X. 10) which defines the turbulent viscosity coefficient r]. 5. Circulation and Vorticity The Bjerknes theorem concerning the formation of vortices and circulation accelera- tion (1898, 1900, 1901) has been found very useful in the theoretical treatment of 330 Forces and their Relationship to the Structure of the Ocean problems arising with oceanic currents. This applies to the dynamics of moving ""non-homogeneous'' media in which the effects of friction are considered unimportant. This method of treating problems of oceanic movements has the particular advantage that it takes into account the total ejfect of the mass field on the water movements including all their smaller details. It can only be used in its simpler form by neglecting friction; in general, however, at a distance from the boundary surfaces the friction does not change to any large extent the nature of the currents set up by the internal forces. {a) Circulation for an Earth at Rest and for a Rotating Earth In the presence of (/?, a) solenoids, motions are always initiated the nature of which is that of a circulation, i.e., motions following in the most simple case a closed path. In a moving fluid a continuous chain of material elements may lie in a closed curve s. The velocity component of one of these small elements tangential to the curve s shall be F<. The sum of all these components along the curve s is defined as the circulation C of the curve s C = & Vt ds, (X.42) where ds is a linear element of the curve s. An expression for the change of C in time is easily obtained from the equations of motion (X. 1 6) (stationary Earth, frictionless motion). (X.43) Since normally the external forces (gravity) have a potential, the first integral vanishes and the equation becomes — = -^ ^ y.dp = N, (X. 44) where A'^ is the number of isobaric-isosteric unit solenoids, enclosed by the curve s (see p. 307 equation (IX. 1 1)). Assuming that the curve s lies in a plane, then: (1) The circulation is constant with time (dCldt = 0) if a is constant over the whole of the space under consideration (homogeneous sea) or if it is a function of pressure. The isobaric and the isosteric surfaces then coincide and the mass distribution is barotropic. (2) A circulation acceleration will be present if the specific volume is dependent not only on the pressure but also on other properties of the water (temperature, salinity). The mass field is then baroclinic. Form equation (IX. 12) for a curve 5 in a dynamic section formed by two vertical lines, the physical sea-level (/? = 0) and an isobaric line at depth p^, the number of solenoids enclosed will be given by the diff'erence in dynamic depths of the isobar p^ at the two stations. This gives dC -^ = N= Da- D,. (X.45) Forces and their Relationship to the Structure of the Ocean In the two-dimensional case {x, z) and from Stokes's law it follows that a dp = dp dx + a ^r- dz cz r , { C /8a 8p da 8p\ f^'^ = ]\ [8xTz-Tz8-xj dxdz 331 (X.46) (X.47) If now e and /3 are the angles of the ascendent of the pressure 8plcn and the ascendent of the specific volume 8al8n, respectively, with the .r-axis, then da da da da ^ = ^ cos ^, ^ = ^ sin ^, 8x 8n 8p 8p ^ = ^ cos €, 8x dn 8z dz dn ^dp . ^T Sin e en and from equation (X. 47) and adp =^ da dp dn on sin (e — P) dx dz dC ~di da dp -^ ■?- sm (e on dn iS) dx dz. (X.48) (X.49) The two possible cases are shown graphically in Fig. 136c. If e > j8 then the circulation acceleration dCldt < 0 and produces an anticyclonic circulation. If, on the other hand, '1 a'2 a'J p p*l p+2 p*3 p*4 p*5 p+6 Fig. 136c. Cyclonic and anticyclonic circulation movements for different pressure gradients and specific volumes. e < /8 then dCjdt > 0 and the resultant movement is cyclonic. In the two cases (see Fig. 136c) the circulation proceeds from the ascendent of pressure to the ascendent of specific volume. In oceanography, as a first approximation, the isobaric surfaces are horizontal, i.e. e = 90°, and thus dC 'dt g da a dn cos ^ dz (X.50) A cyclonic circulation is present when ;8 > 90° and thus the isosteres decline towards 332 Forces and their Relationship to the Structure of the Ocean the left relative to the isobars and an anticyclonic circulation will be present when i8 < 90° and so the isosteres decline to the right. The circulation theorem gives the change in absolute circulation C, i.e. the circula- tion referred to a co-ordinate system at rest. For oceanographic problems, however, it is the change in the circulation relative to the Earth which is of interest. The abso- lute velocity Va referred to a fictitious Earth at rest can always be represented as the sum of the relative velocity Vr relative to the rotating Earth and the velocity V^ of Earth rotation. Thus in the direction of the tangent / to the curve s Va,t = Vr,t + n., and thus C, = Cr-\- Ce. (X.51) The circulation Ce can be calculated. If the curve s lies in the equatorial plane then the velocity Vg for each point on the curve will be cor where r is its distance from the Earth centre. The component of it coinciding with the direction of the tangent / to the curve s will be given by Ve, t= rw cos P, where )S is the angle between the tangents to the circle r and to the curve s. Thus Ce, t = 60 r cos ^ ds =^ 2co \ r cos P ds = 2io F, (X.52) where Fis the area enclosed by the curve s. If the curve s does not lie in the equatorial plane it can be resolved into its projections on the equatorial plane and on the meri- dional plane. Since the velocity V^ is perpendicular to the meridional plane it will have no component in the direction of the tangent to the projection of the curve on the meridional plane and its contribution to Cg.t will therefore be zero. The contribution of the projection of the curve on the equatorial plane is identical with equation (X. 52); F is now the area within the projection of the curve s on the equatorial plane. Thus for the relative circulation acceleration is obtained dCr ^ dF .,, ^^^ -^ = N - 2aj -J-. (X.53) dt dt ^ ' As a first approximation, if the area is not too large, the latitude ^ is assumed constant and Fcan be put equal to F^ sin ,'\ where F^ is the area within the projection of the curve s on the sea surface. Thus ^ = TV - 2co sin 0 ^. (X.54) The acceleration is made up of two terms; the first is the number A^ of solenoids en- closed by the curve and acts always in the direction from the ascendent dajdn to the t More exactly dF dF„ . , ^ dtp dFm . , ^ V -^- = -^- sm (f + Fm cos f ^1= ^i sin (p + Fm cos (p ^. Here v is the south-north velocity; in middle and higher latitudes the second term is insignificant but towards the equator the first vanishes and the second becomes important. Forces and their Relationship to the Structure of the Ocean 333 pressure gradient dpjdn (Fig. 136c); the second represents the product of the CorioUs parameter with the change in time of the projection on the sea surface of the area en- closed by the curve. This term gives rise to a cyclonic circulation for a decrease in the area. If the vertical stratification of the sea is autobarotropic (see p. 308) then N = 0 and a change of the circulation with time can only be caused by the effect of the Earth's rotation. If a small horizontal layer of water (area F) moves polewards, its projection on the equatorial plane F^ will increase. If N — 0 there will be an acceleration in anti- cyclonic circulation according to equation (X. 54). If, on the other hand, it moves to- wards the equator it will be subject to a cyclonic circulation acceleration. The Bjerknes circulation theorem shows clearly the importance of the baroclinic stratification of the sea for the dynamics of ocean currents. For application see Chap. XV, 5. (b) Vorticity for an Earth at Rest and for a Rotating Earth A further important quantity in the dynamics of ocean currents is the vortichy. The horizontal area F enclosed by the curve s can be divided by two arbitrary sets of curves into a large number of very small surface elements 8F. It can readily be seen that the sum of all the circulations SC, in the same direction along the boundaries of these surface elements 8F, is equal to the circulation along the outer boundary s around the entire area F. Thus c = £ac. The limiting value of the ratio SC/SF is termed the vorticity and is denoted by ^. It is thus given by C = Hm 1^. (X.55) The vorticity is thus the circulation around a horizontal surface unit and therefore C = (h {u dx -}- V dy) = idxdy = \ t 8F. (X.56) F The circulation around a closed curve s is equal to the integral of the vorticity over the surface F enclosed by the curve s (Stokes's law). This is the two-dimensional case and C is thus only the vertical component of the total three-dimensional vorticity vector (curl V). For a horizontal surface element 8j.8y (see Fig. \36d), along the boundary (in a positive sense of rotation) of a horizontal surface element 8C — u dx -i- \v -\- and from (X. 55) 8v dx \ / Su \ (8v du\ 8,j - |„ + _ Syj - „ S^ = (- - -j 8.V Sy (X.57) ' 8v du\ 334 Forces and their Relationship to the Structure of the Ocean In the three-dimensional case analogously dw 8v £u dw .. 8v du i = 8y C Ox cy (X.59) If the velocity has a potential (see p. 325) the vorticity will vanish and the movement is irrotational (vorticity-free potential current). y+by y - dF Xi-dX Fig. \36d. Rectangular surface element for the derivation of vorticity. The vorticity for polar co-ordinates can be derived in a similar way and it can be assumed that the Earth and the co-ordinate system which is rigidly connected with it rotate with constant angular velocity a>. The vorticity is then made up of the vorticity of the rotating Earth and the relative vorticity of the water moving relative to the Earth. To derive the vertical component Ca of the absolute vorticity it is necessary to consider further a surface element 8F formed by the intersection of two latitude circles and two meridians. If the latitudinal difference is d(f> and the longitudinal ^A, then the total area SF is 8F = R^ cos cf> 8 SA. The zonal velocity along a latitude circle 4> is u = RQ cos ^, where i3 = tu + dXjdt. However, along a meridian A the meridional velocity is v = R(8l8t) and some simple calculations give for the vertical component of the absolute vorticity S/^ 1 S2JL 1 ^ L = 8C 1 8^ 1 8F cos ]. (X.57a) For a small water column at rest relative to the Earth 8Xj8t = 8 For small oceanic areas in which the latitude can be regarded as approximately constant, equation (X. 59a) reduces to '8v ta-f + 8u\ 8x cy' (X.60) Forces and their Relationship to the Structure of the Ocean 335 The vertical component of the absolute vorticity is thus always equal to the sum of the relative vorticity (vertical component) and the Coriolis parameter, (c) Vorticity and the Equations of Motion; Potential Vorticity Starting from the horizontal equations of motion (without frictional effects), equation (X. 16) gives du ~dt -fv 1 dp p dx' 8v \ cp ot -^ p cy (X.6]) Taking as a first approximation that p is independent of x and y or assuming baro- tropic conditions so that p — p(p) (a function of pressure only) then, by cross-wise differentiation of these equations and subtraction and simple calculation considering dfjct = 0 gives ^^^^ + a+ f) divH r = 0; ia=i-\-f (X.62) This is the relative vorticity theorem of Rossby (1939); it is used for the analysis of stream fields in steady currents and for the analysis of moving oceanic waves. The total change in the Coriolis parameter with time is d4> d -,- = Zoj COS 6 —r and smce y = — -. . dt ^ dt R dt The theorem of relative vorticity then takes the form df dt 2(jo cos (f) 2a) cos (f) V = pv with /3 = (X.63) (X. 64) R " ''^ '' ' R If the horizontal current (m, v) is non-divergent then equation (X. 62) reduces to i-^"- (X.65) The quantity /3 = cfjcy is called the '' Rossby parameter'' and represents the meridional change in the Coriolis parameter (change with latitude). It is positive in both hemis- pheres so that the relative vorticity always increases when small elements move southward and decreases when they move northward. The value of /3 at different latitudes is shown in the following Table 113. Table 113. 10" ^ [cm-i sec-i] = 90° 00 75° ; 60' 45° [ 30° I i 0-593 1 1145 1-619 1-983 15° 0° 2-212 2-290 In theoretical practice ^ is usually taken as a constant, that is, as independent of j\ This approximation is more or less justified near the equator where /S is a maximum 336 Forces and their Relationship to the Structure of the Ocean and its change with latitude amounts to only a few per cent. In higher latitudes, how- ever, taking ^ as constant is only a rough assumption since between 45° and 60° the increase in ^ is about 29%. If the current is non-divergent then from equation (X, 62) it follows that |(^+/) = 0, ^« = ^+/= const. (X.66) In a non-divergent, barotropic current the vertical component of the absolute vorticity is constant and the change in the relative vorticity must be compensated by a corre- sponding displacement in latitude. To use the vorticity equation for a water mass of thickness h which is variable with both time and space, it is necessary to take the continuity equation for the water layer // into account in addition to (X. 62). For a horizontal current (w, v) it is easy to show that the continuity equation must have the form dh dhu dhv ^ dh , ^. ^ ,^^ ^_. ^+-^-f-p=0 or -w;-h divn v = 0. (X.67) dt ex dy ot Combined with (X. 62) this gives It is obvious that the relative vorticity now is variable not only with latitude, but also with the thickness of the water layer under consideration. The value it, + /)/// is thus invariable for a given water mass; it is termed tht potential vorticity. Chapter XI The Ocean at Rest (Statics of the Ocean) 1. The Basic Static Equation and the Conditions for Static Equilibrium If a water mass in the sea is at rest relative to the Earth, the only external force acting on it will be the conservative force of gravity. In the stationary state its effect is balanced exactly by the resistance of the masses underneath. The elastic force of the substratum is thus opposed by the weight of the water masses and any vertical dis- placement is extinguished, when both effects are equal (i.e. when the weight of the water masses above any surface is equal to the pressure exerted upwards by the water masses underneath this surface). The condition for internal equiUbrium thus requires that no resultant of the gravity and the pressure force should act in the direction of the gravitational level surfaces. A horizontal cross-section through a water column enclosed between two vertical walls will carry a greater weight of water the deeper it is placed. At a depth z it shall be p^^. At a small distance dz below this there will be a pressure dp A = A + 7- ^-. The increase in pressure p.-^^ — p^ will be identical with the weight of the water masses per unit area between the two surfaces : P2— Pi= pg dz. From these two equations the "basic static equation" is obtained 1 dp Since the negative derivative of the potential

i Ipo = . P2 Pi approx. — ia(Si — So) (XI.5) if the densities are replaced by corresponding anomaUes of specific volume. This equation permits the relative inclination of the isobaric surfaces to be readily deter- mined from the distribution of the specific volume anomaly in a dynamic section. It also allows a determination of how closely isobaric and isosteric profiles fit together in dynamic profiles that have been obtained and plotted from oceanographic data. 3. Disturbances and Re-establishment of Static Equilibrium According to the principle of Archimedes, a stationary water mass will remain floating and at rest within a more extended water mass if its weight is equal to the weight of the displaced water. If it is heavier than the surrounding water it will sink 340 77?^ Ocean at Rest {Statics of the Ocean) under influence of a downward force. If it is lighter the corresponding upward force will cause it to rise. The forces initiating vertical displacements can be easily found from the third equation of motion in equation (X. 16). Neglecting Coriolis forces and friction they are given by dw dp 'dt^^~'"'8z' If the surrounding water masses are in hydrostatic equilibrium and have a specific volume a' then From these two equations it follows that the enclosed water mass will be subject to an acceleration given by dw a' — a , ^ The upward or downward forces (buoyance force of Archimedes) is thus proportional to the difference between the specific volumes of the surrounding and the enclosed water masses ; for water masses of either the same sahnity and with a temperature difference of 10°C or of equal temperature and with l%o difference in sahnity, the magnitude of this acceleration is about 1 cm sec~^ or about one-thousandth of the gravitational acceleration. The nature of the equilibrium in a water column is dependent on the oceanographic structure and is shown by the acceleration acting on a small quantum after vertical displacement. The vertical equihbrium conditions that may occur in the ocean and the calculation of the vertical stability that characterize these states have been discussed in detail in Pt. I, particularly in Chap. V, 5. p. 196. It seems sufficient to refer here only to the previous statements. In a system where there are no forces acting other than gravitational acceleration and the internal forces, a dynamic vertical section showing isobars and isosteres allows an immediate estimation of the direction of the water currents produced by the resultant forces due to density differences. Part of such a section is given in Fig. 138; the isobars can be regarded as horizontal and the inclination of the isosteres Water of lower density Woter of greater density /" ..-""^ ^--""K^ ^^-"""^ ---'"^ ^'-P """ 4.--"''' ^ -"''"'' ^\ ^^--'"''' ^-""""' ^.^--^"-^^ I L -^^^ ,^- '"" ' 1-'^ ' . - ' '' .--''"" '^'-"""""'^--^"""""'^-'— "-""""'"'"'' Fig. 138. Dynamic vertical cross-section: p, isobaric; a, isosteric surfaces. Disturbed equilibrium and return to equilibrium state. The Ocean at Rest (Statics of the Ocean) 341 relative to them show that the system is not in static equilibrium (disturbed equih- brium). The water at A is lighter than that at B in the same isobaric level, so that to estabhsh hydrostatic equihbrium the water mass at A must rise and that at B must sink. The forces indicated by the mass distribution (solenoids) show a rotational movement (circulation) which tends to adjust the mass distribution closer to that of static equiHbrium, In the final state the isobars must run parallel to the isosteres; a barotropic mass field is then estabhshed out from a baroclinic one. The direction of the circulation set up is given by the rule that it always proceeds along the shortest path from the mobihty vector B(da8n) to the pressure gradient G(8pldn) (Fig. 138). The strength of the forces and the intensity of the resultant circulation have been dis- cussed in II/5; see Fig. 136c. A more convenient method of characterizing the nature of the equilibrium is by comparison of the piezotropy coefficient of the density yp with the barotropy coefficient Fp (see p. 308). The first determines the behaviour of an individual small element on changes in pressure (depth), while the second characterizes the state of a water mass in vertical direction. If Fp = yp then the mass field is not aff'ected by an interchange of any two small elements. In autobarotropism the equili- brium condition is thus indifferent (neutral), for Fp > yp it will be stable and for Fp < yp it will be unstable. Since in the first case the density diff'erences set up by vertical displacements will tend to return the displaced elements to their initial positions, while in the second, on the other hand, they will tend to displace them further and further from it. Rhythmic (periodic) circulatory movements may be set up in this way, but in the sea, according to their nature, they can hardly persist for very long since the energy of these movements will soon be dissipated by turbulence (Inertia oscillations, see Chap. XIII, 6. Chapter XII The Representation of Oceanic Movements and Kinematics 1. Methods of Observation and Measurement of Oceanographic Currents Two different methods can be used to determine the nature of the currents in the sea. One follows the Lagrange approach and investigates the track which a small element of water follows in time. This gives the trajectory of the water movement from the sequence of points in space through which the water element passes. The other method using an approach closer to that of Euler considers the current from a fixed point, and shows the nature of the current at a fixed point at any particular moment in terms of the current vector, which is variable with time. Graphic representation of the distribution of velocity in space by fines of equal intensity (isotachs, velocity field), or by representing the directional field by means o^ stream lines (see p. 326). The stream lines and the velocity field fix the current field at any particular instant. The trajectories and stream lines must be carefully distinguished; they will coincide only in the case of a steady current and here the stream line will also be the same as the trajectory taken by a small water element. (a) Drift Bottles and Drifting Objects A more or less accurate indication of the direction and velocity of water currents can be obtained by following the drift of objects of all sorts which may temporarily or permanently be floating in the water, whether through change or through having been placed there deliberately by man (Krummel, 1908). It is essential that these drifting bodies should project as little as possible out of the water so as to minimize the important influence of wind and waves on their displacements. The course followed by drifting bodies of this sort, which are subject only to the effect of the currents, gives the trajectories of the water movement. Floating bodies put into the sea especially for this purpose may also be used {drift bottle, bottle post). On account of their cheapness and simple handling drift bottles have been frequently used, and with systematic and methodical work can give useful results. Since the path followed by a drift bottle depends to a considerable extent on chance, unambiguous results are given only by systematic work and by the use and recovery of a large number of such bottles. Large-scale experiments of this type have been made by Prince Albert I of Monaco (1889) in the eastern North Atlantic, by Fulton (1897) in the North Sea and more recently, with particular success, by Carruthers (1954) in the southern part of the North Sea and the English Channel. The ordinary drift bottles usually give only the starting position and the place of 342 The Representation of Oceanic Movements and Kinematics 343 recovery of the bottle ; an approximate mean value for the velocity of the current can be calculated from the path which the bottle is presumed to have taken and the inter- val between the two times. Large errors may occur in both these numerical values. These circumstances have brought the method into disrepute, but as shown by the results of Carruthers and Tait (1930) with the use of care and frequent repetition it may still give a good idea about the system of currents over small areas of the sea. See Thorade (1933fl) for further details. More accurate knowledge of the course of the currents can be obtained by following the course of the drifting body directly by means of continuous triangular measure- ment from three fixed points. Kruger (1911) and Schulz (1925) have used this method for the investigation of the currents in the Jade near Wangeroog and off the Flemish coast and have obtained valuable results. {b) Calculated Displacement The method of determining the course of the currents at the surface of the ocean most used in practice depends on the comparison of an astronomical position with a position given "by dead reckoning". The first gives the true position of the ship found by astronomical observations and the latter gives the position of the ship as calculated from the course steered by the ship and its speed, taking the wind-drift of the vessel into account, and the distance covered according to the log (the position by dead reckoning). Usually this does not coincide with the astronomical position of the ship, since it has been calculated from the apparent speed of the ship in the water. The difference between the two positions is called the ship's displacement and is considered to be due to currents in the time interval between successive positions (usually de- termined at noon). For example, a ship with a noon position 52° 25' N., 42° 16' W. (Fig. 139, point A) has travelled 225 nautical miles in the water in the direction S. 35° W. by the following noon. The triangle AA^C gives the difference in latitude between A and the position by dead reckoning A^, = AC = 184 nautical miles = 184 minutes of latitude. The difference in longitude A^C is 129 nautical miles. Division by the cosine of the mean latitude gives the difference in longitude in arc minutes as 3° 24', while the difference in latitude is 3° 4'. The position by dead reckoning at point Ao is thus: 49°2rN., 45°40'W. Astronomical observation, however, gave 49°44'N., 46°22' W. Thus 35' l2°3gN 2138' N I2»37' 47" 38' 4r3r 47''36 W 47° 35' Fig. 141. Successive positions of the ship, ship's course and circle of yaw at the anchor station 288 of the "Meteor", 27-29 March 1927. ip's positions: 27.iii. 21.45 MGZ 12= 37-8' N. ; 47' 35-2' W. 28.iii. 08.54 MGZ J20 37-4' N. ; 47° (36-7' W.)t 28.iii. 12.00 MGZ 12^ 38-0' N. 47° 35-9' W. 28.iii. 21.47 MGZ 12" 38-0' N. 47° 35-6' W. 29.iii. 08.55 MGZ 12 = 37-5' N. 47° 36-8' W. + The computed longitude of 47° 37-7' W. is very probably an error; 36-7' W. should be the correct value. modilied the method by using a sounding line to obtain a fixed point at the sea bottom, but this can only be used in very shallow waters. According to Witting, the best, fastest and also the most frequently used method is the "'smoothing method''. The current measurements are made from an anchored vessel at the shortest possible intervals and values for a time interval over which the different movements of the ship almost cancel out are combined to give a mean vahie. An interval of about 1 5-30 min seems to be sufficient to eliminate the variations due to the movements of the ship and irregular changes in the current direction and speed. (d) The Scientific Use of Current Measurements The technical refinements of the operative mechanism of the amazingly large number of current recorders used in oceanography need not be discussed here; reference can be made to Thorade (1938^), Sverdrup and co-workers (1946) and particularly to Oceanographic Instrumentation Isaacs and Iselin, 1952). However, the important subject of the scientific use of current measurements will be dealt with here in greater detail. 348 The Representation of Oceanic Movements and Kinematics The individual values obtained from current measurements as discussed above will contain errors due to the simultaneous movement of the vessel, and correction to the true current can only be made if the movement of the ship is known with some accuracy. Since for current measurements in the open ocean only one current meter records on board ship, the correction method of determining the true current cannot usually be used. If the average true current changes only slowly, the smoothing method of ehminating short period movements of the vessel must be appHed. How strongly the observations have to be smoothed has been shown by Thorade (1934) with observations made by the research vessel "Poseidon" in the Kattegat (August, 1931). The Rauschelbach current meter was used here to give continuous records of the current every 10 sec over a long period. Plotting all these current vectors starting from a single zero point of an appropriate co-ordinate system gives a current diagram of the type shown in Fig. 142. The individual current vectors are strongly scattering and Fig. 142. Recordings of the Rauschelbach current meter at the anchor station of the "Poseidon" in the southern Kattegat during I h for each 10 sec. (10 August 1931; 18.30- 19.30 h). The current arrows must be drawn from the point O towards the crosses. The indicated arrow refers to the start of the observations. The dotted line shows the movement of the arrowhead during the following 3 min. The dashed-dotted line indicates the position of the arrowhead after smoothing, the point O shows the mean position during the h h. their end-points form a point cloud covering a relatively large area. It can hardly be assumed by the values given in the diagram that the true current has altered significantly within the half-hour observational time. The dashed line joins the end-points of the vectors for the first three minutes. Even for this short interval the vectors cover almost the entire area of the point cloud. This shows that single current measurements made from an anchored vessel differing widely in the observation time are more or less worth- less. It is rather different, however, if for short observation intervals mean values are taken for more or less long intervals in time. Fig. 143 shows that for the same values as in Fig. 142 the individual means for each minute are rather scattered, but the means for intervals of 5 min, on the other hand, show only small variations during the half The Representation of Oceanic Movements and Kinematics 349 hour. These findings by Thorade indicate that the effect of the movements of the vessel from which the measurements are made and other chance factors can be eUmin- ated by such a smoothing procedure. Instead of using continuous recordings of the current followed by calculation of the mean over a long interval such equipment is used in practice which gives directly mean values for the direction and velocity over R -20 ■30L ,830 1835 1840 1845 1850 |855 19OO IO-2ni-l93l -10 ,E, -20 t- ^"^^"^ + ^' .-"^ 0 — ?~--^— -^ 1830 le 35 16 40 ij j-^s le 50 18 55 19C 0 IO-5fflI-l93l Fig. 143. Upper picture: mean for each minute; lower picture: mean for each 5 min of the north ( \ \ ) and the east component (— O — O— ) of the current measured from the "Poseidon" (see Fig. 142). a longer interval (10 min or more). In deriving the means it should be remembered that they are vectors and in order to reduce them to mean values they must be re- solved into north and east components. The mean obtained in this way is denoted the vectorial mean. Instead of this mean, which is mathematically accurate but in- convenient to calculate, the mean of all the velocities regardless of the direction is often used instead. This is termed a scalar mean of the velocity, and it represents the average velocity of the water displacement. The corresponding simple arithmetic mean of the angle of the flow direction is of no importance especially when the variations in the direction are large. In the characterization of extensive current measurements a further quantity is used to give a numerical value for the variations in direction and speed of the current. The quotient of the vectorial velocity mean and the scalar mean is used for this and is termed the constancy (stability) of the current {Kgl Ned. Med. Inst. De Bilt, 1904, 1908). From the definition of the two kinds of averages it follows that the stabiHty is always a proper fraction. It has the value 1 if the directions of the individual vectors are always the same size, since the vectorial mean is then the same as the scalar. The current constancy is usually expressed as a percentage. 350 The Representation of Oceanic Movements and Kinematics The magnitude of the current constancy is only affected to any large extent by varia- tions in the direction of the flow, variations in the velocity have little influence. Wagner (1932) has found, for example, that if the velocity was assumed to be the same for the individual values and the directions were scattered within an angle of 90°, then the stability was 90-100%, while if the directions were scattered within 180° the current stability was still 60-90% ; individual values with greater velocities could, however, aff'ect these stability values strongly in either direction. A more accurate description of the distribution of a larger number of obser\'ations requires the use of statistical theory (Thorade, 1936). If the measured velocities of the current are m\, w^, Wg, . . ., vv„ for ^-observations and a^, og, a^, . . ., a„ are the corresponding directions (taken clockwise from north from 0° to 360°) then the corresponding ^-components will be m^ = m,\ sin a^ and the A^-components will be Vi = Wi cos ttj, where / = 1, 2, . . ., n. The arithmetic mean of the ^-components will be a, and that of the A^-components v ; then the vectorical velocity is w^^ = u} + y2 and the vectorial mean direction will be u tan a„ = -. V The deviations of the individual values from the vecto^'ial mean are ^i — Ui — u and t^j- = f, — v. Comparison of the frequency distribution with a Gaussian distribution will then allow us to judge whether the deviations are generally random, so that statistical laws are applicable. The mean scatter of the Mj- and ^j-values is then given by the mean error (standard deviation) w,/ = e* and m^ = if. For a case similar to that of Fig. 142 (150 observations over an interval of 10 sec) Thorade found a point distribution given in Fig. 144 for the frequencies of the devia- tions for intervals of 1 cm/sec; the curves show a Gaussian distribution indicating the completely random nature of the deviations, and show that in spite of the small number of observations the deviations approximate very closely to a random distribu- tion. In this way, the direction varies between 270° and 318° and the velocities between 7-4 cm/sec and 21 -8 cm/sec. The vectorial mean gave a current N. 66° W., 14-5 cm/sec, the scalar mean was 14-7 cm/sec, and the current constancy (stability) was therefore 98-6%; m spite of the rather large variations in direction and speed of the current this is a surprisingly high current stability value. The mean scatter gave a considerably better idea of these variations: m„ = ± 2-68, m„ = ±2-64 cm/sec, which indicates that for a random distribution of the deviations about 68% of all the deviations ej ofthe£'-component lie between +2-68 cm/sec and —2-68 cm/sec; analogous conditions apply for the 77^ for the A^-component. According to statistical theory of scattering, the direction and velocity can be charac- terized most accurately by the "mean error ellipse" which must include half of all the individual values. Considerable numerical work is required for calculating this The Representation of Oceanic Movements and Kinematics 351 -8 -6 -4 cm /sec -2 / + 3 + + ~20N^ -15 2 \ \ 4 6 cm/sec 8 ^ East component e- -10 5 + + + + -6 -4 cm/sec 4 6 cm/sec Fig. 144. Frequency distribution of scattering of the north and east component e and -q for the point cloud of the current measurements of Fig. 142 (the full lines indicate the Gaussian frequency distribution). ellipse. In place of it Thorade used the scatter circle the radius of which is given simply by p^ = m^^ + m^. This circle is quite sufficient for the characterization of the scatter of a point cloud of current values. The probabihty that an observation will fall within the scatter circle is with sufficient accuracy about 2/3, that is, about 2/3 of all observed values will fall within the scatter circle. In the case previously mentioned (see Fig. 143). p = 3-76 cm/sec; the actual number falUng within the scatter circle is 103 of the 150 values, which is about 2/3. Elimination of periodic components. The variations in speed and direction of ocean currents often include periodic components superimposed on the mean current {the basic current). The basic current because it is often obtained by elimination of the periodic components is therefore sometimes rather unsuitably called "residual current". The basic current need usually not to be constant either in direction or velocity, but these changes are mostly aperiodic and of long duration and therefore differ consider- ably from the periodic components. The presence of these components is shown par- ticularly well by graphical representation of the individual vectors in a progressive vector diagram. A constant basic current plotted in this way will give a straight line, while a wavy or spiral trajectory indicates the presence of periodic components. Figure 145 shows a case of this type. Generally a water transport occurs directed to- wards west-south-west, but it is not uniform and shows wavy fluctuations to the north and the south (period of these oscillations about 14-15 h). The periodic components can be eliminated by taking a mean over the periods present; the periodic components then cancel out giving the average basic current. Thus, the case of Fig. 145 gives a mean displacement over the entire period of 2-0 nautical miles towards the south and 7-5 nautical miles to the west in 24 h or a basic current of W. 15° S., 16-7 cm/sec. The calculation process for such a separation of observed current values, taken over a long interval into the basic current, and the periodic components can be illustrated 352 The Representation of Oceanic Movements and Kinematics 0 ^ n 4" ■^ . 0" - 2l4 18" •; / 1921 £ 1 2 o - 2053/ > \ ^ . s^ i£^' / '" 10" 14" L 1 1 1 3 1 r:V% y3" S^ ^4^ 'V u 1 Nautical miles 3 1 ' 1 -.1 — 1 1 ,. 1 ' 1 1 Fig. 145. Anchor station of the "Ahair": path of a water particle from 19 June 1938 00.00 to 20 June 1938 14.14 MGZ. (Represented by a successive plotting of the observed current valuesf as a mean between the depth 5 and 15 m). Mean basic current from 19.00 to 20.00 MGZ: north component: —4-5; east component: —161 = W. 15° S., 16-7 cm/sec. t In order to simplify the numbers indicated at each individual point are values rounded off to total hours. Therefore « h is always (« — l)h 48 m, for example, 3 h = 2 h 48 m. by an example. The method given below is mostly used but each case requires indi- vidual treatment. At the "Meteor" anchor station, 16-20 June 1938 (44° 33' N., 35° 58' W., mean depth about 1400 m) the current values were measured for a single interval of 10 min in each hour at eight depths (Defant, 1940Z)). Figure 146 contains unsmoothed values for the N- and jp-components at 5 and 15 m depth for the period from 17 June, 04.00 h to 19 June, 18.00 h (MGZ). This gave a rather irregular, jagged curve, partly because of chance disturbances and partly because of errors in the measurement. Since the tidal currents were ex- pected to be rather strong these were then eliminated by taking continuous means over 24 lunar hours (from one moon culmination to the next). The smoothing showed Tim*- I7 3ZI 4 6 8 10 Fig. 146. Current components at the anchor station of the "Altair" June 1938 (,4> = 44° 33' N., A = 33° 58' W.). — i — i — i — i — , N. and E. components according to the observation; --0 — o--, basic current after elimination of the periodic parts (tidal current and inertia current); 1 1 , basic current + tidal current of the diurnal and semi-diurnal wave + current of the inertia wave (according to the values of the harmonic analysis). The Representation of Oceanic Movements and Kinematics 353 in the present case that the remaining current was indeed very regular but still included a weak periodic disturbance of about 1 7 h. Since it was not improbable that a wave of this type could occur in such current measurements (inertia oscillation) this wave was also eliminated by taking means again over a 17 h period.* Finally, the basic current remains. It has been plotted in Fig. 146 for both components. It changes only slightly with time; the A^-component gradually decreases from 10 to about —4 cm/sec and then remains almost constant, the ^'-component changes from —12 to —17 cm/sec. A more detailed analysis of the periodic components can be made by ordinary harmonic analysis and gives the following equations {t in hours) : 7.TT Iv TV-component: +6-6 cos .- (/ — 17-6 h) + 4-6 cos j^{t — 2-3 h) 277 , + 6-0 cos yj(t - 12-6h). S'-component: +2-7 cos i^{t - 20-4 h) + 3-8 cos -r^ (^ - 5-2h) l-rr + 5-3 cos ynit - 0-Oh). The time ? = 0 corresponds thereby rather accurately to 3 moon hours before the moon passes the meridian at Greenwich (17 June, 1938). The ampUtudes are given in cm/sec. All three waves show almost the same amplitude; the inertia wave also is quite pronounced and, as can be expected from this, can become quite visible in the current. Calculations of the current from both components obtained by the harmonic analysis, and in addition the basic current of the curves presented in Fig. 146, follow the observed values very satisfactorily. However, the differences between the smoothed curves and the observations show that the current measurement is subject to manifold disturbances which are very largely random (or observational errors). From the smoothed mean values for a full period of the single waves current diagrams can be constructed and can be compared with the current ellipses which were calculated from the harmonic values. The left-hand side of Fig. 147 shows this comparison for the semidiurnal tide and on the right-hand side for the 17 h inertia wave. The smoothing of the subsequent values by harmonic analysis is rather obvious * For a curve y formed by the superposition of two harmonic waves of different periods T^ and T2 which has the form y = acos"^ U - ej) + 6 cos -^ (/ - eg) ^1 -'2 if a continuous mean is taken over the period T^ the Tg-wave will disappear completely and there will be left y = ^\ y dt = a —^ sm -;^^ cos ~ (/ - ^i). The amplitude is changed, but not the period and the phase of the T^ wave. If Tj is 17 h and T^ is 24 h then the amplitude of the 17 h wave which was previously a will now be —Q-lla, that is, the Tj-wave is now inverse to the original wave and its amplitude is almost five times less than before. 2A 354 The Representation of Oceanic Movements nad Kinematics in both cases. Reference should be made to Vol. II for further discussion of these current diagrams. It may be mentioned here that the components, phases and ampli- tudes of the inertia waves correspond closely to those given by the theory of oscilla- tions (see Chap. XIII, 6). Since the current diagram deviates only slightly from a circle the fine dashed circle in Fig. 147); for this the amplitudes of both components must be Fig. 147. Current measurements at the anchor station of the "Altair", 16-20 June 1938. Left side: current card of the semi-diurnal tide according to the smoothed values of the individual hours and the current ellipse according to the values of the harmonic analysis. Right side: the same for the 17-hourly inertia wave (the dashed circle indicates the theoretical inertia circle). the same, and furthermore, the phase of the ^-component must lag one-quarter of a period behind that of the iV-component. The observations give an amplitude ratio of M4 and 12-6 h + 4-25 h = 16-85 h as compared with 17-0 h which is a difference of only 0-15 h. These properties of the 17 h wave confirm that it is a pure inertia wave. Decomposition of current data by means of other methods. After the elimination of the periodic components there still remain other more aperiodic effects superimposed on the basic current, which is almost constant in time. These deviations may be due to various causes such as piling up of water at shores (Anstau) or variable wind stress. The wind especially is liable to give rise to drift currents in the surface layers, the direction and strength of which depend on that of the wind and change with it. The observed current in these cases can be looked upon as the resultant of thedrift current and the basic current. If the latter alone is required the two must be separated by a special procedure. Nansen (1902) gave a suitable method for this which was used in the evaluation of the ice-drift observations of the "Fram". If intervals of time, for which the wind resultant is zero, are taken together and the effect of the wind on the water therefore considered very small, then the resultant current for the total interval will be due to the basic current alone. In that way he found by analysis of six rather The Representation of Oceanic Movements and Kinematics 355 typical cases that the permanent current of the deep North Polar Basin flows first at 1-0 cm/sec N. 64° W. and later at 2-1 cm/sec S. 12° W. Brennecke (1921) and SvERDRUP (1928, \93>\b) later used the same method to show from the ice-drift observations of the "Deutschland" and the "Maud" that there is no permanent surface current in either the Weddel Sea or off the North Siberian Shelf. Later, Sverdrup developed another method that makes use of all the available wind and current observations. The vectorial resultant of the current is calculated for wind groups concerning certain directions (for example, four groups with the wind for each quadrant centred on N., E., S. and W.) and divided by the wind strength of each group to give the "relative" resultant current (for 1 m/sec wind). If a pure drift current is present then the resultant of the current vectors of all the wind groups must vanish, since they will be symmetrically grouped around the zero point. If, however, the ob- served current is made up of wind drift + basic current, the resultant of all the groups will not be zero but will represent the basic current. If a coasthne impedes the de- velopment of the wind drift equally in all directions and favours a current parallel to the coast, then the circle connecting the ends of all the current vectors will be replaced by an ellipse (Witting, 1909). Table 114. 'Tram" Expedition: 27 May 1895 - 27 June 1896. Ice drift grouped according to the directions of wind resultants Wind Total drift 1 .^ Wind 1 (without ba drift sic current) quadrant centred at Wind speed V (m/sec) Current Relative intensity current w (cm/sec) wjv Deflection angle ' Relative current Deflection angle N. E. S. W. 3-30 2-86 2-48 2-56 416 j 1-26 9-5° 316 ! 110 53-5° 5-54 2-23 420° 5-92 2-31 200° 1-69 1-65 1-68 1-65 360° 27-5° 24-5° 340° Mean 2-80 _ i _ _ 1-67 1 30-5° Table 114 contains the ice-drift observations of the "Fram" for the period from 27 May, 1895 to 27 June, 1896 (Fig. 148) according to Sverdrup. The diagram on the left of Fig. 148 shows that the end-points of the vectors lie on a circle, but that the centre of the circle is not at the zero point but is displaced in the direction S. 82° W. Vectorial subtraction of the basic current (0-79 cm/sec, bearing 262°) results in the diagram given on the right of Fig. 148. The velocity 0-79 cm/sec refers to a wind speed of 1 m/sec. During the year, however, the mean wind speed was 2-80 m/sec, so that for the period under consideration there was a permanent surface current of 2- cm/sec along a bearing of 262° (direction relative to the 75° E. meridian). Nansen obtained by his method 2-0 cm/sec on a bearing of 256° which is in satisfactory agree- ment. The table shows that the relative wind drift is practically independent on the direction of the wind ; the mean of the four groups shows that a wind with a strength of 1 m/sec gives rise to a surface drift of 1-67 cm/sec deflected 30-5° to the right of 356 The Representation of Oceanic Movements and Kinematics From, May 27 1895- June 26. 1896 Fig. 148. Dependence of the ice drift on the wind direction according to the observations of the "Fram" expedition, 27 May 1895 to 27 June 1896. Left side: the observed total ice drift. Right side: the pure wind drift after subtraction of the effect of the permanent basic current (according to Sverdrup). the wind direction. Also in this case almost identical values were obtained by Nansen's method. Palmen (1930/)) has studied these methods in his work on the currents of the Gulf of Bothnia and the Gulf of Finland more deeply and has used them with success, especially for the observations on wind and currents made at the light ship "Finn- grundet" from 1923 to 1927. 2. The Current Field and its Representation {a) Representation of Mean Current Conditions by Means of Compass Cards To get an idea of the currents in any particular area of the sea the most practical procedure is to tabulate all the available data for the direction and strength of the currents for small areas over which uniform conditions can be expected. These small areas are usually chosen to cover a few degree squares (one, two or more degree squares). The question is thus to count out a large number of observations which can then be presented on a compass card. The prevalence of each direction is then shown by longer or shorter rays from the centre point, and the mean velocity in any direction is shown either by the thickness of this line or by the feathering on these rays. Such a current chart is actually only a graphical tabulation and is very largely free of subjective influences. A personal factor becomes involved only in the interpre- tation of the picture shown by such compass cards. The representation of current conditions by compass cards best satisfies the require- ments of a current chart for navigation, since it gives at a single glance the frequency and strength of currents in each direction and the possibility of representing large variations in the direction and strength of the current. The usefulness of charts con- taining compass cards for scientific investigation of the sea is, however, very limited, because sufliicient observations are available only along shipping routes and there are larger areas of the sea for which cards cannot be constructed due to missing data. The use of compass cards to show average current conditions was previously pre- ferred, and by this a uniform evaluation of the enormous amount of ships reckoning displacements was made. One of the most recent representations using compass cards is that of the Netherlands Atlas for East Asian waters {Kgl. Ned. Met. Inst. The Representation of Oceanic Movements and Kinematics 357 De Bilt, 1935-6). From this atlas the part contained in Fig. 149 was taken; for an explanation of this picture see the legend underneath. A picture of current conditions easier to interpret can be obtained if only a selection of the particularly typical vectors are given as, for example, in the Deutsche Seewarte Atlas containing twelve monthly charts; however, in these the subjective viewpoint of the investigator has a large effect. A different type of representation has been used in the British Admiralty charts. The ship reckoning displacement is not shown by a straight arrow, but by a wave-like arrow with the mean velocity in nautical miles per day indicated by a number underneath. Where there is no displacement the chart is left blank but along the usual shipping routes they accumulate. In practice this method has the advantage that it shows the variations and the uncertainty in the occurrence of the ocean currents and the greater or lesser prevalence of current free regions or only of weak currents. (b) Representation of Average Current Conditions by Means of Stream Lines Instead of giving statistics of individual ship displacements in form of compass cards, these statistics can also be used to give the mean value of the currents in the degree squares. This has been done by the Netherlands Meteorological Institute (1908, 1915, 1919). A vectorial mean for one or two degree squares is taken of ship displacements, and calculations are also made of the scalar means and the stability. The results have been published in tables and charts. This observational material has then formed the basis for a whole series of investigations on ocean currents. Attempts to derive a comprehensive picture of the currents from these mean current vectors are of two types (Schumacher, 1922); one of these represents the current by stream lines broken up into arrows with the feathering or the thickness of the arrows indicating the velocity. The other gives the direction of the current by continuous stream lines and the velocity by isolines (isotachs). To the first group belong the investigations of Michaelis (1923) and Willimzik (1927) on the Indian Ocean, of Meyer (1923) on the Atlantic, a study by Merz (1929) on the Pacific Ocean, and by Willimzik (1929) on the Antarctic surface current and others. The second method was first used in oceanography by Bjerknes and co-workers (1913) for the currents in the Gulf of Mexico. During a renewal of the monthly current charts for the North Atlantic Schumacher (1940) later used another method of representation. The arrows here were drawn to represent not the mean direction and velocity but the most frequent, which is more valuable both for the practical user and in most cases also for scientific purposes. All the available data on observed ship displacement were evaluated on this most frequent value (mode) principle. The quadrant containing the largest number of observations was found for each point; the enormous amount of work required was handled by a punched-card system (Hollerith). The direction separating this quadrant into two halves was then taken as the prevailing direction of the current. The velocity was taken as that usually found in the prevailing direction, that is, the scalar mean of the ship displacements falling within the selected quadrant. Also, the stabiHty was determined as before and was characterized by the probability of a displacement in the selected quadrant, i.e. by the numerical ratio of the number of observations falling within the quadrant to the total number of observations. Four different grades of stability were distinguished. If at least one-third of all observations 358 The Representation of Oceanic Movements and Kinematics The Representation of Oceanic Movements and Kinematics 359 Explanation (to Fig. 149) The current roses are drawn from observations within the areas shown by the pecked Hnes. Arrows indicate direction of current; north arrow current towards N. Velocity of current in nautical miles per day is represented as follows : e-iz '^-^-^ ?5-4b^ 49-72 TSon^ove _ Length of arrows represents frequency, 1 mm 3-7%: j j j i [ . The lower o 50 I °° % figure within the circle gives the total number of observations, the upper figure the per- centage frequency of currents less than 6 miles per day. falls within a quadrant this will be already predominant and its middle line can be regarded as the direction of the prevailing current. If the percentage of the ship dis- placements falling within the quadrant is between 33% and 66% then the prevaiUng current is termed ''variable''. The next grade ''rather steady is reached when at least 33% of all observations fall not only within one quadrant but within one octant. If more than 61% of observations fall within a quadrant and between 33% and 66% within an octant within the quadrant then the prevailing current is denoted "steady \ if both quadrant and octant contain more than 67% of all observations the current is ""'very steady"'. This characterization of stabihty is undoubtedly more illustrative than the ratio of the vectorial and scalar sums of the velocities. An example of this type of representation is given in Fig. 1 50 which shows the chart for August of the surface currents in the North Atlantic as given by Schumacher. The length of the arrows indicating the prevailing direction has no significance here. The velocity is given by feathering or for large values by barbs at the arrow-heads; for the grade of the stability see the explanation on the chart. A similar evaluation of ship displacements has also been given by Schumacher (1943) for the South Atlantic so that modem monthly charts are now available for the whole of the Atlantic Ocean. (c) Current Patterns and their Interpretation Certain definite properties of the current field must be borne in mind in plotting stream lines on the basis of the current vectors. In the j&rst place it should be noted that except at singular points and lines : (1) the individual stream lines are not allowed to intersect; (2) the stream lines are curves that neither start nor finish in the current field ; (3) the stream lines are always continuously curved lines. The stream lines are drawn mostly by vectorial interpolation by the eye. Such a graphical interpolation usually offers little difficulty if the current vectors cover the whole chart uniformly. However, this is usually not the case and the lines must some- times be drawn with a minimum of observational values. For this it is necessary to have some idea of the singularities in the current field (Bjerknes and co-workers 1912, 1913). Because the position of these singularities fixes the general outline of the field and to complete the pattern then offers little difficulty. The simplest singularities and their relationship to the structure of the water masses in the oceans will be described in the following section. Lines of convergence a?id divergence. Figure 1 5 1 shows convergence and divergence from only one and from both sides of the stream lines. In case (a) and {b) there is an infinitely rapid convergence and divergence; cases which are rarely found in this extreme form. An infinite number of stream lines leaves or enters asymptotically 360 The Representation of Oceanic Movements and Kinematics The Representation of Oceanic Movements and Kinematics 361 -Fig. 150. Chart of surface currents for August in the North Atlantic Ocean (according to Schumacher). (Stereographic azimuthal projection accurate at the equator, scale at 0^ N., 30°W. 1: 108.) Velocity 3-0- 8-9 sm/Etm. 90-14-9 15 0-20-9 21 •0-29-9 300-41 -9 420-53-9 540-65-9 66-0-77-9 ( i knots) ( 2 knots) ( f knots) (1 knots) (U knots) (2 knots) (2i knots) (3 knots) — < — <- A- Steadiness Very steady Steady Rather steady Variable rDead reckoning J or taken from 1 other represent- Lations Numerical limits for steadiness: from all ship displacements known in a certain area fall inside the quadrant which is cut by the current direction into two halves (quadrant and octant, respectively) Very steady Steady Rather steady Variable quadrant (%) (up to 45^ to the right and left) More than 67 More than 67 33-66 More than 33 octant (%) (at the most 22^° to the right and left) More than 67 33-66 33-66 Less than 33 Fig. 151. Singularities in the current field: (a) one-sided convergence, ib) one-sided diver- gence, (c) and id) double-sided convergence and divergence ; for explanation see vertical cross-section. 362 The Representation of Oceanic Movements and Kinematics from both sides the hnes of divergence and convergence (case (c) and {d)). Lines of convergence and divergence in most cases represent the boundaries between different water types moving relative to each other. They are generated when heavy water meets lighter water or when lighter water spreads out over heavier water that is sinking. Fig. 151 gives a vertical section showing current conditions on both sides of an inchned gliding surface separating two different water masses. Similar vertical displacements can also be expected for divergence and convergence lines from both sides. In all these cases where there is a velocity component at right angles to the boundary surface the inclined gliding boundary surface cannot be expected to remain stationary. The occurrence of divergence and convergence lines in oceanic current systems is a general phenomenon closely connected with the oceanic circulation. They represent the framework of the circulation and indicate the connecting places between the sur- face currents and the three-dimensional vertical circulation. Some examples will be given later. Rauschelbach ("1931) while making current measurements in the Ost-Friesband Gatje rtielow Emden) took the opportunity to make measurements with a bifilar current meter at a convergence line running through the observation point (an anchored vessel). The convergence line, which was visible as a foam line, ran parallel to a dredging line; it moved the Ems upstream driving with the flood tide, while at the same time it was displaced from the middle of the channel towards the east. It passed the current meter at 1 7 h 3 min 30 sec. Figure 1 52 gives the velocity and direction of the current as measured by the current meter before and after the passage of the convergence line; Fig. 153 shows the distribu- tion of the surface current around it. The course of the boundary surface in the lower layers was not that simple and according to current measurements at a depth of 1-2 m was disturbed by internal waves. 40 20 y I80°0 " 160 140 120 100 (a) ■^ -s/ «o-cr^*^ (b) ^'^S/Su y^^ I \ \ y \ \ fsJ] y rsr 1 l7hQ'" |r 3m ^r Time Fig. 1 52. Evaluation of a convergence line in the Ostfriesland Gatje (downstream of Emden) according to Rauschelbach: (a) current velocities, (b) current directions (clockwise from 0° to 360°). Convergence lines are frequently indicated at the sea siirface by more or less strong agitation of the water and are then recorded in ships' logs as rips. Closer attention has only been paid to them in more recent times. (Romer, 1935, 1936; Schumacher, 1935; Thiel, 1937; Uda, 1936, 1938). It seems to be definitely established that rips in the open ocean are formed at the boundaries between converging and diverging water masses. Sometimes when lighter and heavier water are separated by either a converg- ence or a divergence line, the wind forces the lighter one to move above the heavier, as is often observed. Off the continental shelf and around island platforms there may The Representation of Oceanic Movements and Kinematics 363 also be disturbances of the water movement due to the bottom configuration; the direction of the rips then usually corresponds with the main course of the shelf or of the irregularity in the bottom. In many cases a connection has been shown with the behaviour of the tidal currents in neighbouring oceanic regions. Particularly well known to seamen are the rips in the Straits of Gibraltar and in the Straits of Messina, -1 — \\ \ w \ \\ \ \\\ \ \N "*> "> \\\ \\\ W \ \\\ \ \ \ \\\ \ \\ ^^^ ^^ \ \ \ \ \ \\ \\\ \\\ \ \ \ \N\ \ \ \ \ \ \ V^ \ \ \ \ 1 \ \ \ \ \ \ I \\\\ \ \ I \ \ \ V \ V I \ \\\\ \ I W \\\ \ I \ \ \\\ \ I \\ \\\ \ I w \ \\ \ I \\ \ \\ \\ \\\\\\ I \\\V \ \ I \ \ \ \ \ \ I \ \\ \ \ \ I \ \\ w \ I \ \V \\ \ I \\\\ \ \ I \\\ \ \ \ I \ \ \ \ \ V I \ < \ \ > V 1 \ \\\\ \ I \ \ \ \ \ \ I \\\\\ V I \ \\ \^ V I \ \ \ \\ \ I \ N \ \ \ \ I — I — 1 1 1 1 . ,111 i II I 1 1 1 1 till 1 1 1 1 1 1 1 1 1 1 1 1 ,111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 '''''/l 1 1 1 1 ,111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 till 1 1 1 1 H I I I I mill , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 'I I 1 1 1 1 1 1 1 1,' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 / / 1 1 1 1 1 1 1 1 1 / I I (0) Fig. 153. Current directions and stream lines during the passage through the convergence line in the Ostfriesland Gatje (see Fig. 152). where they are definitely connected with tidal currents carrying different water types and those off the eastern coast of North America in the area of the Gulf Stream, along the west coast of Mexico and in West African waters where they are related to up- welling phenomena. Points of convergence and divergence (Fig. \54a, b). These points represent the inter- section of an infinite number of stream lines. For continuity reasons, movements such as these must always be connected with movements perpendicular to the surface of the sea ; thus a divergence point in a water layer near to the surface indicates up- welling and a convergence point indicates a sinking movement. This need not, however, be the case at greater distances from the surface. The divergence then merely indicates that due to the vertical movement more flows in at one side than leaves from the other; the reverse applies for convergence. The formation of curved stream lines near to the centre point (cyclonic and anti-cyclonic vortices) as shown in Fig. 154 depends largely on the effect of the Earth rotation. If there are different types of water masses in the near vicinity of the vortex they will be drawn into it and combined singularities then occur. Cases of this type are shown in Fig. 154c", d; c represents a cyclonic vortex in the region between a lighter and a heavier water mass. Since the equilibrium state is upset at the boundary between the two water masses the hghter water tends to spread out over the heavier while the heavier sinks underneath the lighter. For such an inward spiraling motion a convergence line forms at the boundary surface ; thereby one part of it will be an up-gliding surface where the hghter water moves over the heavier and the other will be a down-gliding surface where the heavier water sinks underneath the lighter. The lighter water will gradually extend completely over the heavier and will finally give a cyclonic vortex (in the top layer) with a simple convergence point of the form a. 364 The Representation of Oceanic Movements nad Kinematics (Q) / / (b) (d) Fig. 154. Singularities of the current field: (o) and (6) convergence and divergence point; (c) and id) superposition of singularities with convergence and divergence lines ; (c) cyclonic, id) anticyclonic vortex at the boundary between two water masses. The form d represents an anticyclonic vortex in the region between the two water masses. Here the boundary surface sphts up into two divergence Hnes. In this case the anticyclonic vortex causes a concentration of the hghter water in the central part of the vortex. The dynamics of such cyclonic and anticyclonic vortexes will be discussed later (see Chap. XIV, 4). Neutral points (Fig. 155fl, b) occur when currents flowing in opposite direction meet each other and separate again without showing stronger vertical motion. Two asymp- totes to the stream lines then intersect at the neutral point situated in the centre. Singular points of higher order are also possible. The current field is then very com- plicated, see, for instance, Fig. 155a. In place of the second water mass there may be a solid boundary at a coast line where the current divides into two parts. The neutral point then lies on the shore line. In the presence of a wave motion the stream lines take on a special pattern. During the propagation of a wave the individual water elements usually describe elliptical orbital motions in a vertical plane perpendicular to the wave front. The longer axis of the ellipse is horizontal and the smaller is vertical. For such a periodic wave motion it is of course only sensible to plot a stream-hne pattern for a particular phase of the wave motion. Fig. 155c shows that for a propagation of a wave to the right the water masses will converge in front of the wave and will diverge in its rear. At the sea surface this gives rise to a convergence line in front of the wave and a divergence line behind it. Both the singular lines move with the wave at right angles to the wave front. Thus in a wave motion two types of strips occur, so that strips with a movement from left to right alternate with strips moving from right to left. If the wave is propagated to the right the first type of strips will correspond to the wave crests The Representation of Oceanic Movements and Kinematics 365 (c) (d) .^^^^^^^^s:^ (b) (e) 'j (f) Fig. 155. Singularities in the current field: (a) neutral point, ib) one-sided neutral point, (c), {d), (e) and (/) singularities in wave motions : (c) stream lines in a vertical cross-section, (d) stream lines at the surface with a small translation parallel to the wave crests, (e) and (/) the same with a somewhat stronger or a very strong translation oblique to the wave crests (according to V. Bjerknes). and the second to the wave troughs. If in addition to the wave motion there is also a more or less strong translatory motion in the water mass, then the two current fields will be superimposed on each other, and the resulting current field will consist of a system of convergence and divergence lines moving parallel to each other with the wave. Some fields of this type are illustrated in Fig. 155. The singularities are closely connected with the velocity field. Where stream lines intersect the velocity must be zero ; the points of convergence and divergence and the neutral points must therefore be points of zero velocity (places of no motion). The isolines of velocity must be closed around these points. When approaching singular lines there will always be more and more curvature in the lines of equal velocity. This curvature becomes stronger the more the stream lines converge towards the singular line. For weaker convergences this curvature is usually hardly noticeable in the observations. Constructing stream lines usually offers little difficulty, especially if the position of the singularities is fixed first. Usually some of the stream lines running out from the singularities can be drawn in with some certainty and these fix the current field with almost sufficient accuracy. Attention should also be paid, of course, to the velocity field and to relationships with the dynamic phenomena expressed in the distribution of other oceanographic factors (temperature, salinity, etc.). Sandstrom (1909) has given a method for the accurate construction of stream lines. Auxiliary lines termed isogons were drawn in first. An isogon is defined as a line along which the direction of the current is constant, and for each direction there exists only one isogonal curve. If the observed directions are expressed by numbers (usually 16 directions with the numbers 2 to 32) then numbers can be entered on the chart in place of the arrows indicating the direction of the current; the isolines of equal direction are then easily constructed. These are covered with rather short dashes pointing in the direction of 366 The Representation of Oceanic Movements and Kinematics azimuth of each isogen so that the chart is covered complete with short dashes. It is then easy to draw in the curves tangential to these short dashes and these curves are the stream lines. Werenskjold (1922) has pointed out that it is possible to draw in a number of isogons rather quickly by simply using two charts of the eastern and northern components of the current u and v. If a is the azimuth of the current then V tan a = - = k. u Each isogon is fixed by ^ = const. Two isogons can thus be drawn in immediately: for A: = 0 and k = co; they correspond to lines y = 0 and m = 0. Their intersections give the singular points through which all isogons must pass. Since the relation V — ku = 0 is satisfied only at points where m = y = 0 for all values of k. Further isogons are easily found; they can be limited to the eight isogons where ^ = 0, ij, ±1 and ±2; corresponding to these are the azimuths 0°, 26|°, 45°, 63|°, 90° and so on. These usually fix the current field with sufficient accuracy. The stream lines are given by integrations of the differential equation — = A: = ^ dx u (see equation (X. 22) on p. 323). If v and u are given as analytical functions of the co- ordinates X and y, then in many cases an accurate integration of the equation, and therefore also a representation, of the current field is possible. Werenskjold has given a large number of cases of this type and has discussed them in detail. Reference is made to these. Of particular interest are those cases where complex singularities occur; to draw these complicated patterns is usually rather tiresome, but mathematically they are no more difficult than the simple ones. An example will illustrate this. If u and V are given by — u = x^ + (y + ay — r^, V = x^ + (>' — ay + r^, where r^ > a^, then the integration of the diff'erential equation above gives the stream lines represented in Fig. 155a; the isogons u = 0 and v = 0 are circles which are shown by dotted lines in the figure. Their points of intersection give the singular points, one of which is a neutral point and the other is a convergence point ; they are connected by a line of convergence. Such connections of the singularities are relatively frequent in stream-line patterns of ocean currents. (d) Examples of Current Charts Current charts based on these principles have been prepared for many parts of the ocean, usually for mean conditions since there are almost no synoptic data available. They show only surface currents. Various types of presentation have been used. An accurate representation based on strict hydrodynamic principles has been introduced by Bjerknes. Analysis of the current fields and their resolution along the extended lines of convergence and divergence, with more or numerous complex singularities, has shown that the previous conception of large horizontal circulating systems in the The Representation of Oceanic Movements and Kinematics 367 Fig. \55a. Example for a special current field according to Werenskjold (integral curves of dyjdx =■■ v/u = k; circles are curves « = 0 (below) and t; = 0 (above)). Fig. 156. Stream lines south of Africa for May (according to Merz). 368 The Representation of Oceanic Movements and Kinematics currents of the surface layers is untenable, and that the deeper water layers are also involved in the surface current systems. An example of this type of representation is given in Fig. 156 which shows the surface currents to the south of Africa during May according to Merz (1925). A line of convergence runs right across it separating the steady broad west wind drift in the south from the Agulhas Current south of Africa. Charts of this type do not indicate the velocity of the current, its prevalence or the amount of data on which it is based; velocity is mostly indicated by thin dotted lines (nautical miles per day). Because of gaps in the available data current charts such as these, constructed according to strictly hydrodynamic principles, are naturally not certain in all details, but the individual stream lines and singularities support each other by means of their course and position and thus offer the clearest possible picture of the water movement. Another representation of essentially the same type was chosen by Helland- Hansen and Nansen (1909, p. 9) in which the stream lines are represented by a series of short arrows of more or less equal length (Fig. 157). Also here the velocity LT — in ■^k^} J^-x Fig. 157. Mean currents at the sea surface of the European North Sea (according to Helland- Hansen and Nansen). The Representation of Oceanic Movements and Kinematics 369 - ,^ \ I .' E • \ /I v^ I t I t t «..•■••__ • 370 The Representation of Oceanic Movements and Kinematics and the stability are omitted from these charts. This type of representation is usually chosen for current charts which are based less on direct current measurements and more on a qualitative assessment of the horizontal distribution of the temperature, salinity and other factors which the system of currents at the sea surface must reflect. A more comprehensive representation of the currents in an ocean has been used by Meyer (1923) for an evaluation of Dutch observations on the currents in the At- lantic during February. Figure 158 shows a part of this chart. Here also the stream lines were broken up into a series of arrows of equal length but their thickness was used as a measure of the current constancy (stabiHty), the feathering as a measure of the velocity and the amount of data available was indicated inside the breaks in the shaft of the arrow. The singularities in the current field are not shown so clearly by this type of representation and are therefore indicated by special signs, particularly in the case of the more important lines of convergence and divergence. These charts already permit a deeper insight into the nature of the water movements at the sea surface of the ocean under consideration and also allow an estimate of the reliability of the chart at any particular area of the oceanic region. Similar but somewhat modified representations have been chosen by Schott (1926, 1935) and Schumacher (1940, 1943). In assessing the value of a chart and in its use, it is necessary to keep in mind the relatively large uncertainties which still remain attached to them. The number of observations on which the charts are based in the individual degree squares varies considerably and is often so small in some of the squares that chance can be rather important. These difficulties, however, may decrease with time since the number of observations collected by hydrographic institutes increases from year to year and mechanical evaluation of these data by computing machines is much faster than was previously possible. 3. Special Cases of Current Fields Near Land and at the Boundaries of Water Masses (Compensation Currents) The boundaries of the sea, fixed either by coast lines or by the topography of the sea bottom, exert a considerable influence on the pattern of the ocean currents and especially on the form of the current field. For each steady current (potential flow) the water in the immediate vicinity of a solid boundary surface (coast or sea bottom) tends to approach the boundary as closely as possible. The effect of such disturbances is thus shown rather far from the source of disturbance in the current field and also in the distribution of the oceanographic factors (temperature, salinity, etc.). The most simple cases, which occur also in nature again and again can be expressed mathe- matically by the method given on p. 327 ; a few of these can be briefly treated here. (1) Plane flow around a cylindrical obstacle (island) is given by a function F = U{z + {a^lz)). Introducing in r = x + '>' polar co-ordinates, then z — /-(cos (f) -}- i sin ^) = re^'l' and the velocity potential O and the stream function ^ will thus be given by the expressions 0 = t/ jr + -J cos 0 and "F = U Ir - ^-| sin cf,. One stream line is the .v-axis for which sin <^ = 0, another one is the circumference The Representation of Oceanic Movements and Kinematics 371 Fig. 159. Stream lines around a cylindrical obstacle (island). of the obstacle where r — a^jr vanishes. The stream hnes of the potential flow are given in Fig. 159. (2) Choosing F = (fl/2)z2 then 0 = (a/2)(jc2 - j2) and W = axy. The jc-axis and the >'-axis are stream lines (^ = 0) and one obtains in that way the flow towards a straight and vertical coast at which the flow divides into two branches (see Fig. 155 (b)). (3) The function F = Az'^ leads to current cards for bays or around projecting land masses where as a first approximation the boundaries can be taken as straight. Intro- ducing again polar co-ordinates we obtain 0 = ylr" cos ncf) and 'F = Ar"^ sin n^. Parts of the curves V = 0 can be taken as solid boundaries ; this leads to sin «^ = 0 or to the lines ^ = 0 and ^ = Trfn. Putting n = 77/a, then ^ = 0 and — a,2a, . . . can be taken one after the other as the sohd boundary. This gives the irrotational flow (vorticity free) between or off two straight coasts which meet each other at an angle a. Fig. 160 shows some cases which are of interest. The configuration of coast lines and outer boundaries of ocean basins are con- siderably more complex than in the simple cases which are susceptible to mathematical analysis. The simple character of currents that carry water masses from a distance into coastal areas will be disturbed and changed by the coast lines. An important role is ^=180° a--'^5° a--90° ^=270° Fig. 160. Stream lines off a coast as shown on the picture (triangular shape). 372 The Representation of Oceanic Movements and Kinematics played here by the compensation requirement which is a result of the continuity law. Since water is almost completely incompressible it cannot accommodate a widening or contracting of the stream lines by contraction or expansion and movements normal to the flow direction or even counter currents are set up to a much greater extent than in air movements in order to avoid the formation of empty space. The nature of these counter movements can only be fully explored empirically by observations in nature or by special suitable experiments. Experiments of this sort have been made extensively by Krummel (191 1, p. 470) and have been used for a clarification of many phenomena exhibited by the pattern of the ocean currents. The results of that shown in Fig. 161 are particularly instructive. The resemblance of the experimental current system to that in the Central Atlantic can readily be seen ; this system consists of the two wind Fig. 161. Experimentally produced current patterns (simulation of the current system in the central part of the Atlantic Ocean) (according to Krummel). drifts induced at the sea surface by air currents, and the corresponding circulations to the north and the south as well as the (equatorial) counter current between them. At the projecting peak on the left-hand side of the experimental tank representing land (Cape San Roque) the current intensity was surprisingly large (corresponding to the Guayana Current). Standing vortices are formed at coastal bays, in which the flow always shows such a sense of rotation that the current on the seaward side follows the main current while that on the landward side is opposed to it. Hydrodynamically such a vortex can be stationary, but it will always have the same water mass circulating within it and there will be no water transfer from the main current to the vortex. In nature this is usually not the case. Pulsations in the main current will always affect the intensity and the extent of the stationary vortex and will thereby lead to a renewal of the water circulating in it. Such replacement currents in bays and small gulfs are termed "neer currents" and are always present at any reasonably irregular coast consisting of small bays and projecting land. An example is shown in Fig. 162. The compensation requirement need not always to be satisfied by horizontal trans- ports, but vertical movements are also sometimes involved and give rise to very charac- teristic oceanographic phenomena (upwelling). Fig. 162. Sea surface currents in the northern part of Bosporus (according to Merz MoHer) with stationary vortices in individual ba\s. The Representation of Oceanic Movements and Kinematics 373 Fig. 162a. Two streams of water flowing together. Conspicuous phenomena also occur where currents carrying two different water masses flow to- gether and these deserve special attention. If two water masses of different type meet at a sharp land projection or at a motionless water mass there will usually be an appreciable transverse velocity jump at the boundary surface (Fig. 162a). It cannot be expected that separating surfaces of this type will keep for any length of time their simple form, since the state under consideration is highly unstable. Every boundary surface of this sort has a tendency to develop waves and all chance irregularities will thereby grow rapidly and the discontinuity surface will finally dissolve into a number of irregular vortices. These processes are particularly characteristic for the transition from waves to vortices and have been described in detail by Bjerknes (1933) and Prandtl (1942). A boundary surface at which a temporary disturbance of the current field has given rise to a slight bulge is shown in Fig. 1626. This wave-form disturbance will move along the boundary surface with the average of the speeds of the two currents; relative to this wave one of the water masses will move to the right and the other to the left, and with reference to this kind of co-ordinate system the ridges and troughs of the waves will remain in the same place. According to the Bernoulli theory the disturbance in the course of the stream lines will be accompanied by a corresponding transverse pressure disturbance. For a steady state of motion the transverse pressure rise llpidpjds) must be balanced by the centrifugal acceleration c^/r (c denotes the hori- zontal velocity, r the radius of curvature of the stream lines, s the direction of the normal to the stream lines). It becomes obvious that there will be a pressure surplus (+) in the ridges of the waves and a i educed pressure (— ) in the troughs of the wave. This implies that the wave disturbance cannot be stationary but that the water begins to move from the surplus pressure areas to the adjacent areas of reduced pressure; that is, as the wave disturbance becomes stronger it will form current fields similar to those in Fig. 162c, in which the boundary surface will finally be rolled up into vortices, lying one behind the other and all rotating in the same sense. The same phenomenon occurs here in the hori- zontal plane between two water masses with different velocities as in the case of unstable waves at the boundary surface between water masses flowing one above the other (see Vol. n. Chap. XVI, p. 517 Internal waves). Examples of cases such as this are the vortex formations at the boundary between the East Greenland Current and the Atlantic Current in the Irminger Sea, or the vortex-formations at the boundary between the Gulf Stream and the Labrador Current south of the Newfoundland Banks (see p. 471). — >■ ' — >■ — ^ — >■ — >■ ■< — — >J ■* — Fig. 1626. Disturbances in the pressure field due to wave-like deformations of a boundary surface between two currents. 374 The Representation of Oceanic Movements and Kinematics rrrTTTTTrr. Fig. 162c. Formation of eddies behind a sharp edge and their growth. 4. Divergence of the Current Field and the Continuity Equation The current field for a horizontal movement can give information about the place where vertical water movements must occur within the field. Since, on the one hand, in an incompressible medium, divergent and convergent stream lines must be asso- ciated with vertical displacements and on the other hand for parallel stream lines, velocity changes will lead to water accumulations (piling up of water; "Wasser- stauungen") which will also cause vertical movements. Quantitative relationships can be derived from the following considerations. If A A' and BB' in Fig, 163 denote two adjacent stream lines, ds and ds' are elements of these, c and c' are two lines of equal velocity in the current field and 8n as well as 8n' are the parts of these lines between the stream lines, then it is possible to calcu- late the amount of water flowing through the small area ABA'B' — ds 8n in unit time. This outflow per unit area is termed the divergence of the current field and is indicated by div c. It is a measure of the divergence and convergence of the stream lines and also of the velocity. One therefore obtains div c = 1 dsSn [c'B„' - an] = I + I, f = ^ I (^ ««). (XII.I) If the velocity along the stream lines is constant (c — const.) and the small angle be- tween the tangents to the two adjacent stream lines is denoted by 5a then the curve divergence is given by c dSn 8a div c = ^ -^ = c Y-' on OS on (XII.2) The Representation of Oceanic Movements and Kinematics 375 Fig. 163. Divergence of the current field. The divergence is positive if the stream lines move apart and negative if they contract. If the stream lines are parallel {hi = const.) then div c — 8c Ts (XII.3) The divergence here is a consequence of the change in velocity in the direction of the stream lines; a decrease indicates pihng up ("Stauung") and an increase indicates a suction of the water masses. For a given current field the divergence field can be calculated numerically or gra- phically and can be represented on charts; special methods for this have been given by Bjerknes and co-workers (1912, 1913). The general continuity equation (X. 22) can be written in the form dp dt + p div c = 0 for an incompressible water mass this gives div c = 0. (XII.4) (XII.5) If allowances are made for changes in density due to changes in temperature and salinity, then equation (X. 21) applies and for stationary conditions one obtains: dpu 8pv 8pw 8x 8y 8z div c = 0. (XII.6) The total horizontal water transport ("current amount") in a water column from the surface (z = 0) to the bottom of the sea (z = h) is then A/ = pc dz Jo and its components along the x- and j^-axes are given by (XII.7) M, pu dz and My = pv dz (XII.8) 376 The Representation of Oceanic Movements and Kinematics Multiplying equation (XII. 6) by dz and integrating from the surface to the bottom it follows that At the sea bottom w^ equals 0 and further if the vertical elevation of the sea surface above the equilibrium level (positive upwards) is denoted by i then Wq = —{dlidt) and from (XII. 9) follows 8t 1 , -f= divAf (XII.IO) dt po The divergence of the current amount is thus always associated with vertical displace- ments of the sea surface and these can be readily calculated from (XII. 10) if the current amount is known. For a stationary state of the sea surface (C — const.) it follows necessarily divA/ = 0, (XII. 11) that is, at stationary sea surfaces the total current amount must be divergence free. This need not be the case in every layer but in the entire water column an excess in- flow in some of the individual layers must be balanced by a deficit in the other layers, if no effect on the sea-level should appear. Under stationary conditions in the sea there must be in any volume element a con- stant amount of all the dissolved substances in the water besides the constancy in density (see Defant, 1941^/). If the salinity for example is denoted by s and exchange processes are for the moment disregarded, tliis requires ds ds 8s 8s ^s ^ ^ , ,^. ^. = ^ + " ^ + ^ ^ + '^' TT = 0- (XII.12 dt dt 8x 8y 8z Multiplying this equation by p and then adding the continuity equation (X. 31) multiplied by s, it follows that 8 OS 8ups 8vps dw'ps For stationary conditions the first term on the left-hand side is zero and the condition of a constant salinity will be given by the remaining equation integrated over the total volume under consideration. Introducing a space vector S with horizontal components Sx and Sy which is given by the equation •/I S= pscdz (XII. 14) J 0 allows the equation (XII. 13) for stationary conditions to be rewritten in the form div5 = 0 (XII. 14a) S can be termed the salinity amount and the equation states that under stationary con- ditions the vector indicating the amount of salt flow must also be divergence-free. The constancy of the water mass in a given space and the constancy of the characteristic water properties existing under stationary conditions has often been used in the derivation of the current The Representation of Oceanic Movements and Kinematics yjl amount in the considered space. For example, the silicate content is q at three oceanographic stations a, b and c, where the vertical salinity distribution is s. For a prism taken by these stations down to a definite level, there will be current amounts M^, Mj, M3 passing through each side in unit time and a current flow M^ through the bottom surface. If it is then assimied that no water enters or leaves through the upper sea surface (zero precipitation and evaporation) then the constancy of the water volume requires that Ml + M2 + M3 + M„ = 0. If further the corresponding mean amounts of salt and silicate passing through the three surfaces of the prism are indicated by s^, s^, s^ and q^, q sm ^y ; ... ^ \ dp d / 8u antitnptic current :0= ~ — \- — [a ^r p ox cz \ S.v The Euler current will appear for rapid changes in the sea level (storm surges, etc.) ; this is also the relationship on which is based the simple theory of waves, where the water displacements in general have the character of a Euler current. The geostrophic current corresponds to another current constituent of the "elementary" current, namely to the gradient current (deep current), while, during the formation of the wind drift and the bottom current, besides the Coriolis force to a considerable extent fric- tion is also involved. An antitriptic current can be expected in local circulations of small extent, for example, in equalization currents in sea straits where the narrow width prevents an effect of the Coriolis force. 2. Steady Currents in a Homogeneous Sea Without Friction (a) General Equations For a horizontal frictionless water movement, the equations of motion (X.16) for a homogeneous sea (p = const.) (Coriolis parameter/ = 2aj sin ^) will take the form: du ^ \ dp dv \ dp ■ -■• ■ - -7: =>--/; TT. = -/«--/• (XIII.l) dt p dx dt p dy ^ In a homogeneous sea the pressure p at a depth z (counted as positive downwards from the undisturbed sea level r = 0) is given by . p = gp(z + 0, , (XIII.2) 384 General Theory of Ocean Currents in a Homogeneous Sea where C is the elevation of the sea surface above the undisturbed level (counted positive upwards). Equations (XIII. 1 and 2) then give and the condition for non-accelerated (stationary) current is then (XIII.3) (XIII.4) or if the total velocity V = ^y{u^ + y^) and d^fdn is the total pressure gradient {n normal to the lines of equal water level) fdn (XIII.5) For a steady current pressure force and Coriolis force will be in equilibrium. Fig. 165 shows diagrams of the forces acting on such currents for both the Northern and the Southern Hemisphere. The currents follow the lines of equal water level which are at the same time isobars on the level surfaces ("Niveau-Flachen") and it follows the proposition: In the Northern Hemisphere when facing downstream for a steady friction- less water movement the higher water level will lie on the right-hand side of the current direction and the lower water level will be on the left-hand side; the slope of the sea surface is a measure of the current intensity. Such a current is termed a geostrophic current. Lower water level ' o o Gradie ^ Current Higher water level Lower water level Current ^- X Higher water level Fig. 165. Schematic distribution of the forces for a stationary current in a homogeneous ocean without friction (left side: Northern Hemisphere; right side: Southern Hemisphere). Equation (XIII.3) permits integration if the topography of the sea-level is constant in time or unchanged by the current. Multiplying the first equation by u and the second by V, and adding, gives the relation ¥V^= -gdl If a small water particle moves along a level surface from a point where the sea level General Theory of Ocean Currents in a Homogeneous Sea 385 is ^0 above the equilibrium level, to another point where this deviation is ^i, it will acquire a final velocity V^ given by the relation V,^ = 2g(Co - Ci) (XIII.6) if it was at rest at the starting point {Vq = 0). Corresponding values of Fj and ^o — ^i are given in Table 115. Table 115 $Q - ^1 (mm) . . 1 2 5 10 50 100 150 Po ~ Pi (centibars) 001 Vi (cm/sec) ; 14 002 20 005 31 010 44 0-50 98 100 1-50 139 312 If a water element glides downwards without friction along an oblique pressure surface through a short vertical distance, it will immediately acquire a very large velocity. If the water masses were not forced by the Coriolis action to move along the lines of equal water level under stationary conditions, even a very small slope would be able to cause enormously intense ocean currents. Equation (XIII. 5) shows that the forces producing the movement {gradient force) do not, in the stationary case, determine the acceleration of the water movement, but solely, due to the Coriolis force, its velocity. (b) The Effect of Changing Depth and the Spherical Shape of the Earth Equations (XIII.4 and 5) show that the entire water column down to the sea bottom will have the same velocity; it will move hke a solid body with a velocity V in the appropriate direction. This current can only satisfy the continuity equation if the sea bottom is plane. Under stationary conditions {dijdt = 0) according to equation (XII. 16) the continuity equation takes the form dv cu dx 8y = 0. (XIII. 7) It will be satisfied by the values of u and v given by (XIII.4). At constant depth there will thus be no limitation to a geostrophic current. If there are boundaries to the sea in the form of vertical coasts then the boundary condition will require a constant C along them ; the current will then flow only along the coast and there will be no flow perpendicular to the coast. If the ocean depth is variable, conditions will be more comphcated. In Fig. 166 is shown the case where a given uniform slope of the sea surface (Northern Hemisphere) Constant Decreasing Increasing water rtootti water deptti water depth Fig. 166. Deviation of ocean currents for a variable bottom depth. 386 General Theory of Ocean Currents in a Homogeneous Sea from the surface of the figure backwards gives rise to a uniform current from left to right; at first there will be an equilibrium in it between the gradient and Coriolis forces. If the depth of the sea increases in the current direction (bottom slopes downward) then for a constant flow amount, since the current cross-section becomes larger, there must be a decrease in velocity. The equilibrium between the two forces will be disturbed, the lower velocity attained will correspond to a smaller Coriolis force and the current will be deflected contra solem. However, if the depth decreases (i.e. the bottom rises) the velocity must increase; this will give an increase in the Coriolis force and a deflec- tion of the current cum sole. The equihbrium state of equation (XIII.4) will continue for each stream line only when the current follows the depth lines of the bottom. If the depth is variable, (XII. 16) will be replaced by the continuity equation di (dhu 8hv\ Under stationary conditions the equations of motion (XIII.4) will then give the con- dition 8h dC 8h dl This relation states that if the depth varies then steady frictionless currents are only possible if the topography of the sea surface on a relative scale accords with that of the sea bottom. The currents must thus run parallel to the bathymetric curves; the strength of the current is, however, free and depends only on the absolute gradient of the ^-values. If there are coastal limits, the boundary condition requires that the depth should be constant along the outer boundary (the coast). Since the continuity equation for currents in an ocean partly or completely covering the spherical Earth has a diff'erent form (equation (X.27), the conditions for steady currents will also be different. The equations of motion for the meridional and zonal velocity components will now be {R = Earth radius, & = 90° — ^ = zenith distance): g ^l g^l U = — 75-^ 5 ^Y ^"^ ^' = fD aQ- (XIII. 10) fR sm § 8A fR dd For a variable depth // and taking into account that h is always small compared with 7?, the continuity equation will have the form dl 1 Idh sin du dhv\ The condition for a frictionless steady current is then under these conditions 8h 8i 8h 8t 8t The first two terms are identical with the condition for planar co-ordinates (equation XIII.9); they thus include only the efl'ects of variable depth. The third term h tan d{8t,j8X) takes into account the eff'ect of the spherical shape of the Earth; it is largest in the equatorial regions (§ close to 90°) and vanishes at the poles {d = 0°). Some special cases can be selected to illustrate the two efl"ects. (1) If the depth of the sea is constant, the conditional equation is satisfied only if 8l,j8X = 0, i.e., only if zo«a/ currents are possible (along latitude circles). General Theory of Ocean Currents in a Homogeneous Sea 387 (2) The depth shall be a function of the latitude only. Then dhldX = 0 and the topo- graphy of the sea bottom will be symmetrical about the poles. In that case, according to (XIII. 12), there must be either dijdX = 0 or chjcd + /? tangi^ = 0. The first condi- tion leads again to zonal currents ; the second gives on integration h = H cos d where H is the depth of the sea at the poles {d = 0°). In these cases both d^jdd and dl,jcX are free, that is, I, is also free. For a meridional depth distribution of this type (decreasing gradually from a depth H at the poles to a depth zero at the equator) steady currents would be possible in any direction also in an ocean on the spherical Earth; conditions here are then the same as in a sea of constant depth with planar co-ordinates. For this depth distribution both effects balance exactly. It can therefore be deduced that in higher latitudes small changes in depth will be able to compensate the effect of the curvature of the Earth, this effect will therefore be small there. On the other hand, in lower latitudes larger changes in depth will be required to balance this effect and therefore almost only zonal currents will be possible. The critical vertical gradient in meridional direction which will be able to balance the effect of the spherical shape of the Earth is given by {hIR) tan (^. Table 116a gives these critical values for different latitudes and for depths of 3000 and 5000 m. Table 116a Polar distance . Latitude 20° 70° ! 30° 40° 1 50° 60° 50° 40° 60° 30° 70° 20° 80° 10° Critical bottom gradient for h = 3000 m h = 5000 m 1:5810 1 : 3500 : 1:3670 1:2540 1:1780 1:2190 1:1520 1:1070 1:1220 1:735 1:773 1:464 1 13:73 1:224 The discussion of the above equation (Defant, 1929fl, p. 61) leads to an estimate of the two effects on steady currents. Following Ekman (1923), these can be summarized as follows : Up to 3-4° latitude — and when the changes in depth are small, even farther away from the equator — the effect of the bottom relief is rather unimportant for the tendency of the current to flow in zonal direction. Between 10° and 20° of latitude the two effects are equal and in higher latitudes (> 40°) the effect of the bottom topo- graphy gains in importance and the currents tend to follow definitely the isobaths of the sea bottom. The observed fact that in reahty ocean currents do preferably follow a zonal direction in lower latitudes and their direction in higher latitudes is presumably more affected by the bottom topography, appears to be reasonably well explained by the thf oretical results presented above, 3. Eddy Viscosity (Turbulent Friction) in Ocean Currents (a) Mixing Length and Eddy Viscosity (Turbulent Frictional) Coefficient The movement of the water masses in ocean currents is mostly disordered and tur- bulent and part of the strong variations in speed and direction of the flow which are observed in quick-response recordings (see p. 347) can be attributed reasonably to this internal turbulence. More or less large elements of water (water quanta) are continu- ously being carried by these internal turbulent motions into the layers above, below or 388 General Theory of Ocean Currents in a Homogeneous Sea to the side and there is thus an equalization of the momentum (current impulse) in the direction of the strongest velocity gradient. There is also an associated equalization of all the characteristic substances and of the water properties. This equahzation pro- cess has already been discussed in detail in Pt. I, Chapter II (see p. 105). For the property-pair momentum-velocity under conditions of immediate and complete equalization of the flow momentum a general expression for the apparent shearing stress of a turbulent flow has been derived having the form da (XIII. 13) where U is the mean velocity along the x-axis, :: is perpendicular to it, t] is the exchange coefficient for momentum (eddy coefficient or turbulent frictional coefficient). In Chapter II (see p. 329) another expression was derived for the apparent shearing stress occurring in turbulent flow from the analysis of the current variations in it. This was given as T=-pi7^"'. (XIII. 14) The variations in velocity u' and v' are of course connected with the distribution of the mean velocity which varies across the stream lines. To give a practical form to equation (XIII. 14) Prandtl (see especially 1942) introduced the mixing length I, defined as the length which can be regarded as the diameter of the water quanta moving with the turbulent flow or as that distance that such a quantum travels before losing its identity due to mixing with the surroundings. A water element with a mean velocity u(z) at a point z (see Fig. 167) will have a mean velocity u(: + /) = m(z) + l{8uldz) at a distance 777777777777777777777777777777777777777777. Fig. 167. / across the current. If a water element is moved from one layer to another then the magnitude of u is given by u' = u(z + /) - i7(r) = l{du\dz). The variations in velocity v arise from the movements of the water elements entering the place under consideration from different sides, moving one behind the other and approaching or receding from each other with a velocity diff'erence of ll{du\dz) and thus give rise to transverse movements. Thus r' will also have the order of magnitude l(dul8z). Between u' and v' there must, however, be a negative correlation. The water General Theory of Ocean Currents in a Homogeneous Sea 389 elements entering from below will have too small a velocity, those entering from above will have correspondingly too large a velocity as compared with the velocity at the point under consideration ; positive v' will thus occur together with negative u' and vice versa. The product ii'v' is then always negative. The apparent shearing stress is thus always positive and of the order of magnitude p{I(8ilj8zy}. The proportionality factor is here arbitrarily taken as 1 ; this means only a slight change in the meaning of /. To express in this relation that positive cii/cz will accompany a positive shearing stress and negative ciijdz corresponds to a negative shearing stress, the eddy stress must be re-written in the form cii cz = pP cu — . cxin.15) These turbulent shearing stresses change proportional to the square of the velocity and this has been shown experimentally in investigations in hydraulics. The mixing length / is not a constant here, but depends on the conditions in the current and will vary from place to place. At a solid boundary it is zero and increases with distance from the boundary. Comparison of the two equations (Xin.13 and 15) leads to cil = pP dz (XIII. 16) The eddy viscosity coefficient depends not only on the mixing length / but also on the velocity and density and is thus less susceptible to clarity than the concept of mixing length. However, oceanic turbulence problems can only be handled numerically using the quantity t], the eddy viscosity coefficient, especially for a freely developed turbu- lence remote from solid boundaries (coasts and sea bottom). In the layers near the bottom, however, there are considerable advantages in the introduction of the mean mixing length as a characteristic number giving the degree of the turbulence as a function of the distance from the bottom and of its roughness. From relation (XIII. 15) it can be seen that the quantity V P cu l—^ (XIII. 17) has the dimension of a velocity. It is termed the friction velocity (shearing stress velo- city) w,, so that T = puj which as mentioned above gives the flow resistance as a quadratic function of the velocity. The behaviour of a turbulent flow above a rough surface can be judged upon using equation (XIII. 17), making an assumption about the mixing length / (Prandtl, 1942, p. 108). Since /increases with the distance from the underlying surface (z = 0), it can be put equal to kz and if w, is constant, (XIII. 1 7) gives the solution u = u, (-Inr + c). (XIII.18) As has been shown in numerous investigations, the observed profiles are rather well approximated by such logarithmic velocity profiles; for the number k the universal value 0-40 was obtained. If ordinary decadic logarithms are used instead of natural ones, equation (XIII.18) becomes M = 5-75M, log -. (XIII. 19) 390 General Theory of Ocean Currents in a Homogeneous Sea This represents a rather simple connection between the friction velocity and the actual velocity distribution above the bottom. The integration constant Cq can be related to a roughness length or parameter k. It has been found that for small bottom irregularities such as occur on a flat bottom, sand or snow surfaces or surfaces with not too large plants Cq can be given the value Cq = (A:/7'35), where k is the average roughness parameter corresponding to the irregularities. If the bottom irregularities are very large, it is difficult to determine the position of the point where z = 0 for which the mixing length should vanish. It is then best to shift the zero point upwards by a distance Zq and to use z -\- z^m place of r in equation (XIII. 19). This will then mean that in the space within the major irregularities the mean height of which is Zq the turbulent mixing length falls very rapidly to zero. The turbulent eddy viscosity coefficient -q can be obtained from equations (XIII. 16 and 17) rj = phl^ = pU^KZ. (XIII.20) In the lowest bottom layers it will at first increase linearly with distance from the bottom; but above a certain height it is generally assumed to remain a constant. There are very few oceanic observations with which it would be possible to test this logarithmic law for ocean currents above the sea bottom. This would require measurements at close intervals from just above the bottom to a considerable height above it. The measurements made by Merz (Moller, 1928) in the southern entrance to the Dardanelles, which is sufficiently wide for the current to be un- affected by the lateral boundaries, are probably suitable for this. Only the layers just above the bottom need to be considered. Here the rather strongly scattered individual values of the three series of measure- ments gave the following distribution: Height above the bottom (m) . . 2 7 12 17 22 27 II (cm/sec) 0-3 2-8 4-6 5-5 6-5 7-2 These values follow a logarithmic law rather well and lead to the equation It z -= 5-75 log j--;z. u^ 1-32 The representation of the observations by this equation is entirely satisfactory. It is of interest that in spite of the certainly rather pronounced unevenness of the bottom (hence a large value for Cq) the quantity Zq introduced above is apparently zero. This may be because the heights z above the bot- tom are already heights above a "mean" sea bottom and in actual fact already represent z -1- Zq. This dependence of velocity on height appears to apply only up to 25 m above the bottom. As shown by observation the behaviour of u is then higher up completely different. Current measurements near the sea bottom have been made by Mosby (1947) in order to study tur- bulence and friction in the bottom layers. Using a special apparatus he has measured the direction and intensity of the current in the Avaerstrommen (near Bergen, Norway) up to 2 m from the bottom over a period of 3^ h; this gave the following mean vertical distribution of the horizontal velocity: z (cm above the bottom) . . 25 50 75 100 125 150 200 M(cmsec-i) 16 23 27 29 31 31-7 32-5 These values can be represented rather well by the equation ^^ = 5-75 log ^^^. It does not seem to be necessary to consider Zq in the formula. Later measurements (1949) did not show such simple conditions; in the bottom layer (just above the sea bed) the velocity fell off very rapidly to small values. The changes in the «-values with time at different heights above the bottom show clearly the turbulence of the current; it appears to decrease only very slowly towards the bottom. General Theory of Ocean Currents in a Homogeneous Sea 391 {b) Dissipation of Energy by Turbulence The turbulent process mixes neighbouring water quanta; part of the energy is deviated from the direction of the mean basic current, the water masses are flattened out by vortices into thin layers and part of the energy is used up in this, which would otherwise remain in the basic current. The magnitude of the energy dissipation by turbulence can be calculated from the size of the shearing stress (XIII. 13). This shear- ing force acts horizontally ; the relative movement of two water sheets one above the other is dujdz. From this the work done by the turbulence (energy consumption by the apparent friction "Scheinreibung") will be i? = rj(8uldzy. This is that work which must be done in unit volume and unit time to maintain the turbulence against the velocity gradient. (Schmidt, 1919). In the example described above, in the Dardanelles, the velocity decreased from 27 m down to 2 m above the bottom by 6-9 cm/sec. The mean velocity gradient was thus {dujdz) = (1/362). The dissipation of energy per day amounted to 0-6677 ergs per cm^. This appears rather small but over a longer period has an appreciable effect. If t^ = 100 cm~^ g sec~^ then the kinetic energy of a current of 20 cm/sec will be 200 erg/cm^ and this would be entirely absorbed by the turbulence in about 3 days if not continu- ously renewed by other forces. (c) Turbulence and Stratification That the turbulence is dependent on the stratification in the medium is apparent from the following considerations (Ekman, 1906; Schmidt, 1917; Pettersson, 1930, 1935). In the presence of stable stratification the mixing process is affected by the double work required to lift the lower heavier water masses against gravity and to lower the upper lighter ones against buoyancy forces. This hinders mixing and if the density differences become large enough the stability of the water stratification reaches so high a value that turbulence cannot act against it and may cease entirely. In subtropical oceanic regions cases occur in the tropospheric deeper currents in which a thin layer of highly saline water embedded between two layers of low-saline water can spread over thousands of miles without being absorbed in the layers above and below by mixing. The strong stabihty of the vertical stratification of the water masses completely prevents mixing. An example of this behaviour of the subtropical intrusions of highly saline water has been given in Pt. I, p. 169, Fig. 73 and the reader is referred to the discussion at that place. The conditions under which the work expended in the vertical displacement of water elements by turbulence becomes so large that the turbulence is completely suppressed can be found by comparison of the energy dissipation by turbulence and the lifting work done against gravity by mixing. The buoyancy force per unit time and unit mass for a density gradient dpjdz is given by g{/l p)(8pl8z).* The vertical disturbance velocity u'' according to the previous discussion can also be put propor- tional to I{dujdz). From (XIII. 16) and taking into account that for an equilization of the density differences (temperature and salinity), iq must be replaced by the exchange coefficients for the material properties of the water ^4^ (pt. I, p. 103), it follows that the work done against gravity in unit volume and unit time is g(AJ p)(Spjdz). The work * The symbol 8 should indicate the necessary consideration of the changes in density due to adia- batic temperature changes. 392 General Theory of Ocean Currents in a Homogeneous Sea done by the turbulent motion in unit volume is, however, rj(8uldzy. The condition for the decrease of the turbulence in the disordered flov^ and its transformation into an ordered flow is thus that the dimensionless stratification quantity (glp)(Spl8z) (duldzf >l (XIII.21) In earlier investigations it has mostly been assumed that rj and A are numerically equal, i.e. that the mechanism of mixing of a material property is identical with that of the impulse or momentum transport. Then 17 would be equal to A, and since the stabiUty of the stratification would be given by (l/p)(8p/ez) = E (pt. I, p. 196), the condition for the suppression of the turbulence would be gE ;^> 1. (XIII.22) (duldz)' The expression on the left-hand side has been denoted the Richardson number Ri. The upper limit at which all turbulent motion is extinguished is thus given by Ri = 1 ; how- ever, in reality smaller values are sufficient. Referring to the latter statement, theoretical and experimental investigations of Taylor (1931) and Goldstein on small oscillations in a stratified flow with a linear decrease in velocity have shown that the limit can be expected at Ri = 0-25 or |. In oceanography it has usually been found (see pt. I, p. 104) that the ratio rj-.A is of the order of 5 to 20. In the equatorial regions of the Atlantic Ocean in the density transition layer (thermocline) dpjdz is of the order of 3 to 9 X 10-* for a 20 m height interval. The decrease in velocity du/dz should be between 5 and 10 cm/sec for every 20 m, so that Ri must be between 6 and 69 (Defant, 1936c, p. 296 and 363). It is clear that these figures are sufficiently high to prevent the occurrence of turbulence in the tropospheric deeper currents, as has been found by observation. Observations at two stations in the Baltic for which there was almost no turbulence to be observed in the transition layer gave according to Gustafson and Kullenberg (1936) Ri-numbers of 0-59 and 0-95 which are in accord with the hmiting values given by Taylor. Detailed measurements have been made by Jacobsen (1913, 1918) at Schultz's Grund (Kattegat) and in the Randersfjord, which are very suitable for answering the question under consideration. Table 1 17 give as summary ofall the values derived from these measurements. Table 117. Turbulence and Ri-numbers at Schultz's Grund {according to Jacobsen) Depth (m) duldz (cm sec"^ cm~") Salinity gradient per cm 1 dp P dz (gcm- ^ sec"^) Ri A V A 2-5 10 X 10-3 10 X 10-« 7-5 X 10-^ 31 0-3 7-1 111 50 17 15 11-2 3-1 0-4 3-8 7-7 7-5 22 38 28-5 2-7 018 5-9 14-9 100 24 80 600 2-2 005 10-2 43-5 12-5 19 140 1050 1-9 004 28-6 47-6 150 8 111 82-5 3-8 0-2 1250 200 General Theory of Ocean Currents in a Homogeneous Sea 393 At all depths -q has about 10 times the magnitude of the exchange coefficient A determined from salinity measurements made at the same time. The quotient -qjA is almost always larger than the Ri-number and therefore according to the above con- dition is not compatible with turbulence. The Ri-numbers, which vary between 2-6 and 125, are so high that also according to this criterion a turbulent flow can hardly be present. However, the measurements indicated still a small, though very weak, turbulence with a frictional coefficient between 1-9 and 3-8 g cm~^ sec~^. According to these investigations, other factors seem also to be involved in the appearance and maintenance of turbulence (close distance to a solid boundary or the presence of an intermediate layer between the otherwise almost homogeneous water masses above and below). {d) Turbulence and Mixing in the Sea; Statistical Theory of Turbulence The modern hydrodynamic approach to ocean currents has led increasingly to the view that the turbulence of the ocean currents, which finds its visible expression in the oceanic mixing processes, is the basic cause of a number of oceanic phenomena. Oceanography has mostly been concerned solely with the effects of turbulence and mixing on oceanic phenomena; only recently has interest been directed also towards the nature of oceanic turbulence and one has asked the important question : of what kind is this nature ? In laminar flow the velocity can be represented by a simple function of position and time. In turbulent flow the mean velocity, which again can be repre- sented by a simple function of this sort, is superimposed on an additional, irregularly varying turbulent velocity component that changes with both time and space. The sharp distinction between the two types of flow is shown by experimental investigations which indicate that a discontinuous transition from laminar to turbulent flow occurs when a dimensionless quantity, the Reynolds number, exceeds a critical value, the magnitude of which is about 1000. The form of the Reynolds number indicates the cause of this basically different behaviour of the two types of flow. The Reynolds number is given by R = plJL\r], where p is the density, U and L are values for the velocity and the hnear dimension which are characteristic for the structure of the particular current under consideration; r] is the eddy viscosity coefficient (frictional coefficient). It is clear that the current will be turbulent when the momentum (impulse) of the flow pU or the distance L passed through are large; it will be laminar if the viscosity is large. The viscosity is a force carrying neighbouring elements of the medium along the same path. Therefore, it is obvious that large viscosities will have a tendency to smooth the course of the flow. The empirical fact that the current tends to change to turbulent flow even with very small disturbances — i.e. that the laminar flow is unstable — shows that the turbulent flow has in a certain sense to be regarded as the natural form of motion of media with low viscosity. The Helmholtz vortex-laws of classical hydrodynamics show that a vorticity-free current cannot develop vortices spontaneously. Thus no turbulence can occur in it by itself. It can only be produced inside the fluid by friction at solid surfaces, or by similar processes through the forma- tion of vortices at the boundary of the liquid. Once formed it will spread out in the fluid. This is, however, not the case which we meet in the open sea remote from the sea bottom and from the coasts. The ocean currents here usually have a considerable vortex- intensity from the beginning, i.e. from their formation; it is their further distribution 394 General Theory of Ocean Currents in a Homogeneous Sea on vortices of smaller dimensions that has to be regarded as the turbulence of the current. The origin of the oceanic turbulence must thus be traced back to the con- ditions of formation of the ocean current, and this can definitely be considered to have been done, since the conditions which prevail initially during the formation of the current are certainly scarcely of the type that could be described by simple functions of the velocity distribution. On the contrary, everything indicates that during the forma- tion of a current due to the complicated distribution of the shearing stresses of the winds, the ocean current looks right from the beginning rather confused in vertical and horizontal direction, so that a priori there is a very large probability that in the future the resulting current will attain a form which will fall within the general concept of turbulence. Turbulence is not a form of motion that can maintain itself indefinitely. The kinetic energy of the current is continuously converted by the molecular viscosity into heat. If the current is not continuously supplied with fresh energy, it must in time die away. In the ocean, the currents are continually supplied with energy by the tangential shearing forces of the winds so that here steady turbulent currents are possible. This is of particular importance to the nature of ocean currents which are recognized as essentially quasi-stationary phenomena by observations. Turbulence and mixing in vertical direction and also lateral turbulence of the ocean currents were already discussed in § III dand e of Pt. I of this volume. Lateral mixing is on a much larger scale than the vertical ; the turbulence elements are of considerably larger dimension, so that the eddy viscosity and eddy diffusion coefficients are very large. The ratio of vertical to lateral mixing coefficients is of the order of 10^ to 10'. It can be shown both experimentally and by observation that there is a "continuous spectrum" of mixing and turbulence coefficients extending from the molecular vis- cosity coefficients to values for the eddy conductivity of 10^^ (one billion) or more (Richardson, 1926). In a turbulent current where u is the velocity at a certain point and varies with time, the basic velocity is defined as (time interval 7") : 1 r U=^\ u(t)dt and further the supplementary turbulent velocity as u' = u(t) — U, whereby 1 r - u'(t) dt = 0, the intensity of the turbulence is given by 7 = 1/\/{(m')-} and its kinetic energy by E = ip(M')^.* These quantities characterizing the turbulence of the flow depend of course on the length of the time-interval T, and in fact a sufficiently large value for T has to be selected or these quantities lose their meaning altogether. In laboratory experiments in wind tunnels this requirement can always be closely approached, but whether this is also the case for oceanic water masses is difficult to judge. If T is less than a few hours then the »'(0-values will include terms for the small-scale turbulence such as local mixing, while the basic velocity U will include the long-periodic variations The bar above a quantity indicates its mean value taken over the time-interval T. General Theory of Ocean Currents in a Homogeneous Sea 395 in the velocity such as the tidal currents and the annual changes in u' . If T is selected with a value of about a month the tidal currents will also be included in the value of u'{t). If ris chosen for 10 years or more, the seasonal changes will also be included in 11 and only the secular changes will remain in U. From this it can be understood that, in nature, motions in water masses as they appear in the ocean will be much more complicated than, for example, in an experimentally controlled wind tunnel or a water channel. Every size and all different velocities of the turbulent vortices can be expected to occur in oceanic turbulence, and it is not easy to distinguish between the basic velocity and the additional turbulent velocity. These difficulties occurring with turbulent phenomena of the ocean and atmosphere seem to be fundamentally connected with the nature of turbulence. In dealing with mixing processes in the ocean, the simple relationship ds d"s Jt ^ ^8z2 has usually been used, where S{z, t) is the concentration of the diffusing substance and K denotes the mixing coefficient (eddy diffusivity, eddy conductivity), [cm^ sec~^]. This is termed the "Fickian diffusion equation" (see Pt. I, pp. 95 and 104). It is derived by analogy with molecular processes for the larger-scale processes in turbulent currents using simplifying assumptions on the internal nature of turbulence; it does not accord fully with more recent data, and especially not with the fact that the larger the mixing coefficient becomes, the larger the scale of the phenomena under consideration, i.e. with the existence of a continuous spectrum of the diffusion coefficient. With molecular diffusion, as described by the Fickian equation, the movement of each molecule is independent of that of a neighbouring one. In contrast to this, how- ever, in a turbulent current, adjacent elements have increasingly similar turbulent velocities, and in fact the more there are the smaller the distance from each other. The reason for this is easily understood when the behaviour and the effect of the turbulent vortices of all sizes are studied altogether in detail. The distance between two initially adjacent elements is altered only by the smallest vortices ; the effects of the larger vortices cause no significant change in distance, since they give rise only to a simple transport of these elements. If, however, the distance between two elements becomes larger, the effect of the larger vortices is added to that of the smaller ones so that as the distance between them increases the diffusion effect due to the larger-size vortices becomes more and more involved. The most important independent variable cannot be, as in molecular diffusion processes, the position of an element, but the distance from its neighbouring element. This requires that the concentration of a diffusing substance is only a function of the mutual separation of the particles inside this substance and not a function of the posi- tion only. Richardson first showed this difference as compared with molecular diffusion and further investigations have then been carried out to account for this circumstance (Witting, 1933; Sverdrup, 1946; Proudman, 1948). The theory that the concentra- tion of a diffusing substance is not a function of the position of the element which it occupies, but rather of its distance / from the adjacent element leads to the conclusion 396 General Theory of Ocean Currents in a Homogeneous Sea that the diffusion coefficient i^ is a function of the neighbour-distance / and is given by the equation F{1)=^ (/i - kf 2t ' (XIII.24) where /q is the distance between the elements which are at the same distance in the turbulent current at time / = 0, while / is the distance at time /. F can be determined from experimental series-measurements from the values for / and this allows a de- cision as to whether the Fick or the Richardson concept of the internal nature of the turbulence fits the observed data; since according to the Fickian theory F must be independent on / (see also, Ichve, 1950). All the observations made (Richardson, 1926; Witting, 1933; Stommel, 1949; Hanzawa, 1953; Inoue, 1952) show that F is in fact strongly dependent on / and that there exists a definite relationship between them of the special form F(l) = f/4/3. (XIII.25) Figure 167a shows a summary of observed data and it is easily seen that the assump tion of a 4/3 power seems to be fully justified. 10" / o A / 10'° / / / ^ 10^ ' A/ / V 10^ /•//I/ / ■/ V / 10^ / / 7 / / ^ 4 106 1 '' / /> ^ / 1 / / / / 10^ / /, / / l/ / ' / 10^ 105 / / ^ / / V + V / ^} / / y / / / A V > / 102 _/^ / . A / 1 / A/ / 10 0 / / ^ / V / / / / ! ' 1 10"' • / / A / \ 10"' 0 10 10^ 10^ lO" 10^ 10^ 10^ 10^ Fig. 167a. The relation /^(/) = e/*'^ according to observations (logarithmic scale): points, values of Richardson from the atmosphere; crosses, values of Stommel (Blaimore, Bermuda and Woods Hole); triangles, values of Hanzawa. Equation (XIII.25) which has been found inductively has been given a sound theor- etical basis by closer study of the rate of the energy decrease due to turbulent mixing of the large-scale motion. This method of investigation was first introduced by Kol- MOGOROFF (1941) and after some intermediate work Weiszacker (1948) and Heisen- BERG (1948) have brought this statistical theory of turbulence to a certain degree of General Theory of Ocean Currents in a Homogeneous Sea 397 completion. This theory leads to the same 4/3-power law for the turbulent exchange coefficient which was previously derived from observations. With some modifications this theory can be applied to large-scale processes occurring with oceanic currents, and offers the possibility of obtaining a picture of the spectral distribution of energy in oceanic turbulence. It is thus of a considerable interest for oceanography. The semi-permanent wind systems such as the trade winds, the prevaihng westerlies of temperate latitudes, and furthermore, the aperiodic air currents of the extra tropical pressure disturbances, give rise to large-scale movements in the surface layers of the ocean due to the shearing stresses acting on the sea surface. Thereby, these shearing stresses tend to increase the kinetic energy of the currents produced. However the mean kinetic energy of the ocean currents remains largely constant (quasi- stationary conditions) so that finally as much energy is dissipated in heat as is gained by the work done by the shearing stress of the wind. Ocean currents which initially show large-scale turbulence tend to break up into vortices which subsequently degenerate into smaller and smallest vortices. This proceeds until finally the smallest vortices are formed, which are so small that their energy is converted in irreversible processes by molecular viscosity into heat energy. An exact dynamic explanation of the reasons why the large ocean currents break up into turbulent currents, with more or less large vortices of widely varying size, has not yet been given. However, the em- pirical facts of their existence have been shown by synoptic surveys, for instance, in the more recent Gulf Stream investigations. A complete spectrum of vortex sizes certainly exists. This spectrum is necessary for the dispersion of the kinetic energy of the ocean currents continuously supplied by the shearing forces of the wind. In practical oceanography it has long been recognized that the concept of the mean velocity of the oceanic currents is rather dependent on the length of the time interval over which its value was determined. The same applies for space-means of the current intensity. This leads to the expectation that the mag- nitude of the turbulent coefficients also depends fully on what kind of evaluation of the mean has been used. The concept of a turbulence coefficient is absolutely meaningless if the way in which the mean was found is not specified. This can be seen already from the greater magnitude of the turbulence coefficients the greater the dimensions of the movements under consideration; a fact which could not be explained in earher work. The Weiszacker-Heisenberg statistical theory provides information on the fre- quency distribution of the energy in different size-intervals of turbulent vortices, on the way in which the mean velocity depends on the type of mean taken, and lastly on the dependence of the turbulence coefficients on the type of mean taken. If Ln is the side of a square over which the nih. mean is taken then according to Weiszacker the spectral law is, for the turbulent velocity distribution : z7„ proportional to L)^^, for the turbulence coefficient: 7j„ proportional to L,^^ and for the turbulent energy distribution: En proportional to L^J^. 398 General Theory of Ocean Currents in a Homogeneous Sea Weiszacker took a discrete velocity spectrum as the basis of his theory, Heisenberg chose a continuous velocity distribution and provided an elegant mathematical proof (in this connection see also Ichve, 1951). The principal result of the theory, as far as it concerns the exchange coefficients of turbulent motion, is in complete agreement with the 4/3 power law derived from observed data. The more recent statistical theory of turbulence can give a better description of actual conditions in nature than the classical Fickian theory. In par- ticular, the theory gives an explanation for the large differences in size between the turbulence coefficients for small- and large-scale motion, for which there was no ex- planation in earlier time. For small-scale oceanic phenomena the values found for the diiTusion coefficient t] are on the average about 50-100 cm^ sec~^. For large- scale ocean currents, on the other hand, the values were between 10^ and 10^ cm^ sec"^. The ratio between these is about 5 X 10^ to lO**. For small-scale processes L can be taken as about 50 m and for large-scale currents as about 1000 km. The ratio of the L-values is 2 x 10* and for the T^-values should be according to the theory about 5-4 X 10^. The agreement with the values derived from observations is rather good. The question could also be raised, how far the assumptions made by the theory are justified in oceanic conditions. Stommel (1949) has closely examined this question. Not all the sources for turbulence in the ocean are due to air currents, a part is cer- tainly due to the thermo-haline structure of the ocean currents the dependence of which, of course, on solar radiation and evaporation is known. The assumption of a continuous series of vortex sizes with horizontal isotropy can hardly be valid for the large oceanic vortices ; it can be postulated as a first approximation only when they are of smaller dimensions, i.e. for the genuine turbulent vortices of oceanic currents. The changes which should be introduced for oceanic conditions involve the dividing of the vortex sizes into two parts : an anisotropic one, including all the kinematically dissimi- lar, large-scale horizontal movements, and an isotropic part, including all the kine- matically, similar-to-each-other, turbulent vortices. The latter part only appears after a certain nth averaging process. The first part is thus essentially concerned with the advection of different water types. The exchange is only involved in the second, and the statistical theory of turbulence should be fully applicable here. However, in spite of these changes in many of the assumptions the basic idea of the theory remains and offers a solid basis for the study of dynamic conditions of the ocean currents. 4. Steady Currents in a Homogeneous Ocean under the Action of External Forces (a) Introduction The first ideas about the effect of friction on the movement of water masses were based on the assumption that it arose from the roughness of the bottom surface (gliding friction). The frictional force was thus given, as already shown on p. 317, by R = -KpV. GuLDBERG and MoHN (1876) using this principle for atmospheric flow presented a diagram of the forces necessary for a steady motion. It can also be applied to water movements in shallow ocean currents for which the frictional effects of the bottom act throughout the entire water column. In that case the resultant of Coriolis force and frictional force must balance the gradient force. The direction of the current is General Theory of Ocean Currents in a Homogeneous Sea 399 now no longer parallel to the isobars but is deflected at an angle proportional to k. On the right-hand side of the equations of motion (XIII. 1) the components for the frictional force —ku and —kv have to be added. Multiplying the first equation by u and the second by v and adding, gives 1 dW 1 dp 2 dt- p dt For the movement of a water element along an isobar (dpldt = 0) this equation gives K= VQe-'/z)^ (XIII.25) Ittt] From this the two velocity components of the drift current are then obtained M = Fo e--"D cos 1^45° - ^-) and v = V^e--''^ sin (45° - ^-) (XIII.26) D = TT with " \/(2)Dpoj sm(f> \J \pco sm At the sea surface the water in a pure drift current moves with a velocity V^ in a direction 45° cum sole from the wind direction. At increasing depth the angle of de- flection increases while at the same time the velocity of the current rapidly decreases. At a depth D the deflection will amount to a full 180° and the velocity will have fallen General Theory of Ocean Currents in a Homogeneous Sea 401 to e~'" = 1/23 of the surface value. This velocity is already so small that by com- parison with the surface value it can usually be neglected. The depth D can therefore be taken as a measure of the depth of penetration into the sea of a v/ind-generated ocean current on the rotating Earth. It can in general also be taken as a measure of how far downwards the effect of a steadily flowing horizontal layer penetrates into the adjacent water masses. It was termed by Ekman the ''frictional depth'"; for drift currents the additional word "upper" is used in order to indicate that here solely conditions in the top-layer of the ocean are dealt with. According to equation (XIII.26) D can also be taken as a measure of the internal turbulent friction. It should be noted that the shearing stress T is not involved in the equation relating D and rj ; this could be interpreted to mean that the vertical thickness of the drift current should be independent of the wind intensity producing it and maintaining it against friction. This apparent contradiction is clarified by consider- ing that the frictional coefficient increases with increasing wind strength as does also the frictional depth D. Figure 168, according to Ekman, shows the vertical structure of a pure drift current; the arrows projecting from the central column which are also shown in a projection Fig. 168. Vertical structure of a pure drift current (according to Ekman). 2D 402 General Theory of Ocean Currents in a Homogeneous Sea 0 s °' I 0-2 S 0-3 § 04 o i 0-5 ° 06 I 07 E 08 £ 09 .^ 10 y II f. 1-2 -01 0 0-1 0-2 0-3 04 Q5 0€ 07 08 0 9 10 Velocity relative to the surfoce Fig. 169. Vertical current distribution in a pure drift current: (a) in the direction of the surface current; (b) normal to the direction of the surface current. I -4 Fig. 170. Vertical structure in drift currents for an ocean depth J nearly equal or smaller than the upper frictional depth Z) (10 small circles indicate on each curve the end-points of the velocity vectors for the depth 0-0, 01, 0-2 ^ and so on until 0-9 d. The dashed curve at 1-25 D refers to d = 2-5 D, the remaining part coincides with the curve for 1 -25 D). M = \ (m + iv) dz = J 00 General Theory of Ocean Currents in a Homogeneous Sea 403 on a horizontal plane, give a representation of the direction and strength of the current at the surface and at equidistant levels O-ID, 0-2i), etc. The arrovi^ at the peak of the vertical Hne represents the direction of the wind. The arrow-heads he on a doubly curved spiral and the end-points of the vectors on the horizontal plane lie on a logar- ithmic spiral (Ekman spiral). Referring the current components to the direction of the current at the surface and at right angles to it the diagram pictured in Fig. 169 is obtained, which allows one immediately to judge whether the observed vertical dis- tribution of the current carries the character of a drift current. Equation (XIII.26) shows further that the sea surface velocity increases in propor- tion to the shearing stress T but in inverse proportion to the frictional depth D. This is reasonable since, for equal Tthe more water that is set in motion, the smaller must the velocity of the drift current be, i.e. the greater the depth D. The total drift current transport per unit area of the sea surface is given by T 7 that is M^ = (Tjf) and My = 0. The total water transport due to a drift current occurs perpendicular cum sole to the direction of the shearing stress of the wind producing it and since rj is not involved it is independent of the assumption concerning the effects of eddy viscosity. For an arbitrarily chosen co-ordinate system with shearing stresses T^ and Ty in the x- and >Mlirections, the water transports in these directions will be M^ = ^ and My= - j. (XIII.27) Finite water depth. When the depth of the water is about of the same order as D it has a noticeable effect on the drift current. For a depth d the e-functions in the solu- tion will be replaced by hyperbolic functions. At the sea bottom (z = d) u = 0 and V — 0 are assumed as boundary conditions indicating "adhering" ("Haften") of the water on the underlaying surface. It is apparent from this solution and follows also from Fig. 1 68 that as long as the depth of water is greater than the frictional depth D the vertical distribution of the drift current will be unaffected, since the water layers below the frictional depth have an insignificant share in the drift current. When, how- ever, the water depth d becomes smaller than D, the effect of the bottom will be of more influence the shallower the sea. Figure 170 shows the vertical current structure for depths d = 1-25D, 0-50D, 0-25 D and 0-lD. The thin dotted curve near the origin of the co-ordinate system for the curve ^ = 1-25 D shows the deviation towards the curve for an infinitely large depth; thus in practice there is no significant difference between them. The angle of deflection decreases rapidly with the depth of the water and at very small depths, approximately from about d <0-\D, the movement shows almost no effect of the Earth rotation. Other frictional assumptions. In addition, Ekman has given a solution for the case where the frictional coefficient is proportional, not to the difference in velocity be- tween two adjacent layers, but rather to its square. This gives essentially the same results as for a constant -q ; the angle of deflection of the sea surface current is now 404 General Theory of Ocean Currents in a Homogeneous Sea 49-1° and the current dies away at the finite depth of \-25D. It should be pointed out that the relationship between T, D and Vq are somewhat different. The total transport for the quadratic frictional law is, however, also given by (XIII.27) and is thus inde- pendent of the frictional assumption. This can also be shown by strict mathematical treatment. For a variable -q the expression d\u, v) in equation (XIII.23) is replaced by d I d(u, v) 8z [ ^ -d^- see p. 319. Integrating this equation from z = 0 to z = oo or respectively down to a depth at which the drift current can no longer be detected, and considering that the shearing stress is present only at the sea surface, then with the help of equation (XIII. 13) relationships are obtained which are identical with (XIII.27). These, however, were derived for a constant rj. It could possibly be expected that during the transfer of the turbulent wind momentum to the water masses at the sea surface the two horizontal components of the shearing stress (in the direction of the wind and at right angles to it) would be governed by different turbulent coefficients. An extension of the Ekman theory along such lines has been given by Ertel (1937). It leads to deflection angles different from 45° while the vertical current structure becomes a deformed spiral. Another principle applicable both to the wind stress at the sea surface and to the friction at the bottom has been developed by Jeffreys (1923), In conformity with turbulence theory he assumed that at both the sea surface and at the bottom, "gliding" of the water masses occurs in which the friction is assumed proportional to the square of the velocity differences. The boundary condition at the sea bottom is taken as and at the sea surface as - 7] —^ = Kp{u\ v^) ^("'^) V '2 '2^ where p' is the density of the air and u' and v' are the velocity components of the wind relative to the water movement at the sea surface (see p. 317, equation (X.9).) The more recent results of research in turbulence also show that in the vicinity of boundary surfaces the assumption of a constant frictional coefficient leads to current distributions which do not accord with the observed facts. This makes it necessary to introduce turbulent coefficients, wliich vary with the distance from the solid boundary. That such a method leads to results satisfactorily explaining the observed features has been shown by an investigation of Fjelstad (1929) using observations made by Sver- drup on a drift current over the North Siberian Shelf, where there was a strong increase of the frictional coefficient from the bottom to the surface. He succeeded in deriving a functional relationship for these coefficients of the form fZ+ €\8/* 1 =^ Vo ' General Theory of Ocean Currents in a Homogeneous Sea 405 and was then able to obtain a solution for the corresponding equations of motion Fig. 171 presents the vertical distribution of the frictional coefficient as well as of the theoretical current structure, both for a constant frictional coefficient and for a coeffi- cient varying with depth, according to a summary made by Thorade (1931). The observed current values are indicated by crosses. There remains no doubt that agree- ment with the observed data is obtainable only by using coefficients variable with depth. 20 (o) (b) 10 - 'f 5 - / / / / 1 / / / +Onn 0 " / y/- y ilOm /l2m -*r?Oni ^ ■^I5m •0 100 200 300 400 Bottom Fig. 171. (a) Vertical distribution of the turbulent coefficient at a station of the North Siberian shelf, (b) Current diagrams: --0--0--, theoretical distribution for a constant frictional coefficient ; — o — o — , theoretical distribution for a frictional coefficient as in (a); + + + + + + +, the observed values according to Sverdrup. The application of the modern theory for a turbulent flow to drift currents will be discussed later together with its application to gradient currents (see p. 311). Effect of stratification. Assuming a horizontal and stratified sea with a normal density increase with depth, then only minor deviations occur as compared with the case for a homogeneous sea (Defant, 1927). However, essentially different conditions appear for sudden vertical density changes (boundary surfaces between different water masses). Here the stratification affects especially the frictional coefficient, which inside the flow of each more or less homogeneous water mass may remain approximately constant and relatively large but may fall almost to zero inside the density transition layer (thermocline). The effect of the wind is thus confined essentially to the top layer and the drift current in this is transmitted only very slowly to the lower water mass across the transition layer. As a boundary condition at the side beneath the top layer it must be assumed, since the water here meets almost no resistance, that there is perfect "gliding" and the drift current in the top layer will thus be different from that over a solid surface. If ^is the thickness of the top layer (z=d) this boundary condition is given by 7 - = 0 for (z^d). cz ^ ^ Solutions of this sort have been discussed in greater detail by Nomitsu (1933). The shallower the layer of water in motion the stronger is the current produced by the wind and the larger the angle of deflection; a result which is exactly opposite to that 406 General Theory of Ocean Currents in a Homogeneous Sea of the previous case of "adhering" ("Haften") at the sea bottom. For a small thickness an almost geostrophic current is obtained. As the thickness of the layer increases, the structure of the current will of course approach that of the Ekman spiral. (c) Pure Gradient Currents Drift currents in normal form are seldom found to occur in the sea, since the water transport connected with such currents will give rise to piling up of water at coast lines ("Anstau") leading to inclination of the sea surface. In a homogeneous sea the pressure differences produced in this way would extend their influence down to the sea bottom; if there were no frictional effects a geostrophic current would be generated from the sea surface down to the sea bottom. However, friction at the bottom gives rise to disturbances which are of considerable importance for oceanic currents. The equation of motion (X.16) for a steady current will be of the form \ dp 7] 8^u ^ , ^ \ dp 7] 8^v fv- - -/ + i — = 0 and - fu - - -^ + - 5-0 = 0. (XIII.28) p ox p oz^ p oy p cz^ Replacing the pressure gradient by the slope ^ of the sea surface (equation (XIII.2), p. 383) and assuming that there is a pressure gradient only along the j^-axis (dpidx) = 0, then, according to (XIII.5), the geostrophic current will flow in the direction of the positive x-axis and its velocity will be g S^ Considering this in the equations (XIII.28) they can be compressed in the same way as for a drift current into - gZ-2 («+'■") - '/(" + 'i') + //t/ = 0. (XIII.30) To this equation add the following boundary conditions: (1) no wind at the sea surface, that is Su cv ^ forz = 0: =-=0 cz oz and (2) at the sea bottom "adhering" occurs ("Haften") for z = d: u — v = 0, The solution given by Ekman for (XIII.30) is cosh(l - i){7rlD)z u -\- iv = U ^ cosh{\ +i){7T I D)z ^U(l-'h i>P), whereby cosh (7rlD)(d + z) cos (-^iDXd + z) + cosh (nlD)id — z) cos (nlOXd — z), = cosh 27T(dlD) + cos 27r(^/Z)) (XIII.31) General Theory of Ocean Currents in a Homogeneous Sea 407 for ^ the functions cosh and cos are replaced m the numerator by the complementary functions sinh and sin. Thus u = {\ -4>)U and y = 0C/, (XIII.32) D denotes again the frictional depth to which now, since it refers to the sea bottom, is supplemented the additional word "lower". The functions (/> and 1-5 D the structure of the pure gradient current has the form shown in Fig. 173; this is drawn in the same way as Fig. 168. At distances from the bottom greater than D there is a practically uniform velocity at right angles cum sole to the pressure gradient. This is the uniform deep current; it corresponds to the frictionless geostrophic current. The bottom layer is governed by the bottom current, the velocity of which decreases according to a logarithmic spiral down to the sea bottom. For greater depths of the sea the only change in this structure is in the vertical thickness of the deep current ; the bottom current always corresponds to the frictional depth D. Since the deep current runs parallel to the topographic lines ("Niveaulinien") of 408 General Theory of Ocean Currents in a Homogeneous Sea the sea surface it cannot contribute to the equalization of the sea surface slope. This can only be accomplished by the bottom current which always has a component in the direction of the pressure gradient, i.e. a transport of water from a higher to a lower level. This component does the work required to overcome bottom friction. Fig. 173. Vertical structure in a pure gradient current (according to Ekman). The transports (current amounts) M^ and My of a gradient current can be calculated by integration between 0 and d of equation (XIII. 32), after its multiplication by p. In case of no bottom current the current component M^ would be Upd and it becomes smaller due to the velocity decrease at the bottom. One obtains M^ = Upd - U Dp M. Dp Itt For depths less than D the effect of bottom friction is noticeable throughout the entire water layer, and the more so the smaller the ratio djD. The curves in Fig. 174 illustrate the gradient current at depths 1-25 D, 0-5 D and 0-25 D. The angle of deflection d=i-zsD Fig. 174. Vertical structure of gradient currents for ocean depths d nearly equal or smaller than the lower frictional depth D (for more detail see Fig. 172). General Theory of Ocean Currents in a Homogeneous Sea 409 between the current and the gradient direction becomes smaller and smaller as the sea becomes shallower; the effect of the Earth's rotation then becomes less important than that of friction. Other assumptions about friction. The Ekman theory assumes a constant frictional coefficient. It has been used in this form in meteorology and provides an unobjec- tionable explanation of the deflection of the wind direction to the right with increasing height. However, it was found that the lowermost layers of the wind structure follow different laws. These deviations can be attributed mainly to the assumption of a con- stant frictional coefficient in the bottom layers being no longer valid. This fact Ekman (1928) has taken into account by assuming in agreement with the observations a current structure made up of a straight section OA, at A changing into a logarithmic spiral over AB (Fig. 175). Thereby OB is thus the geostrophic wind in higher altitude. The same conditions as for the surface wind must also apply to the oceanic bottom Fig. 175. Vertical structure in a bottom current with a boundary layer above the bottom (according to Ekman). current, and it is already known from current measurements in moving waters and from laboratory experiments that the vertical structure in these, apart from the devia- tion due to the Coriolis force, is somewhat different from that of the Ekman spiral. The velocity curve of Fig. 175 can therefore only be given a physical m.eaning by assuming the presence of a boundary layer just above the bottom in which the velocity changes approximately linearly, and without change in direction from zero at the bottom to the value OA = Vg at its upper limit. The water mass present above this lower boundary layer flows as though gliding over the bottom ; it is retarded only by the slowly moving boundary layer. Ekman assumed a constant frictional coefficient in each of the two layers and investigated the thickness of the boundary layer, the decrease in velocity in it and the angle of deflection which would be able to prove the validity of such a concept. This concept can more or less accommodate the fact that the lowermost layer just above the bottom has a special status, and that in practice the assumption of a constant turbulent coefficient in the water masses above is quite justified. Modern hydrodynamic fluid research approaches the whole problem from the point of view that the variation of the frictional coefficient with distance from the solid underlaying surface changes with its roughness, whereby the entire current structure takes on a different form. The Prandtl theory (see especially 1942, p. 318) starts with the components of the shearing stress T^ and Ty at the bottom. Taking r-positive upwards, furthermore a variable 17, a pressure gradient in the direction of the positive j-axis and taking into account equation (XIII. 13), then equations (XIII. 28) can be transformed into 410 General Theory of Ocean Currents in a Homogeneous Sea T^=f\ pvdz and Ty=f\ p(U - u) dz. (XIII.33) 0 0 Here h denotes the lower frictional depth at which the deviations U — u and v from the geostrophic current vanish. It can further be assumed that T at the bottom has the same direction as the velocity at the bottom, so that = tan a, (XIII.34) Z= 0 where a is the angle between the direction of the resulting T and that of the uniform deep current. These relationships form the basis of the vertical current structure of the bottom current, but further extension of the calculation fails due to the still imperfect knowledge of the laws of turbulent flow. However, by use of the above presented basics for turbulent friction a rather good estimate of the vertical velocity profiles to be expected can be obtained. As a first approximation it can be assumed that in the vicinity of the bottom u varies with the «th root of z u = U (a' Further, near the bottom v=u tan a ; in order that v vanishes at a height z=/7i one has to assume y = M 1 1 — r I tan a. Ill is smaller than h and must be chosen so that the current structure near the bottom is in accordance with that shown by turbulence research (equation XIII. 19), The equation (XIII.33) then gives ,,„ = ^^^' and r. = („ ^ i);p„ ^ 1) PfhU. (Xin.35) For an indifferent mass structure the equation (XIII. 19) gives for the velocity distribution above a rough surface. In Co=(^/7-35) the quantity kisa measure of the roughness height of the bottom. Since for z=hi, u must be equal to U the ratio TJp can be expressed in terms of U • ^^ = {5-75 log (/;i/co)F (XIII.37) and from (XIII.36) 10g(z/Co) /YTTT'58^ " = ^ 1 7rT~\ • (XIII.38) log (hjco) This gives a second equation for u and both must give a curve of the same shape. The most suitable assumption is that both give the same values for the transport General Theory of Ocean Currents in a Homogeneous Sea 411 (current amount) which will lead to the same value for (U — u) dz. From this a relationship between h^ and n is obtained having the form h log — = («+l)loge. (XIII.39) Putting the expressions for T^ equal in (XIII.35 and 37) gives a further relationship between h^ and U A, = 0-160 4?^'^. (XIII.40) n{n + 1) / This relation shows that h-^ is directly proportional to U as was to be expected. With this all the unknowns are determined. Numerical values can be obtained in the following way : for a given value of n, which according to equation (XIII.35) fixes the angle a, and for given latitudes 4> and velocities U, the equation (XIII.40) allows to compute the related h^ and (XIII.39) gives the value for Cq. From Cq the roughness height k can be found quite simply and finally (XIII. 37) gives the value of T^. This then fixes the current structure completely. Table 118 presents corresponding values for different roughness values of the sea bottom as they could be expected to occur in reahty. These values are valid for ^=50° Table 118. Basic values for the structure of the bottom current {according to the Prandtl theory); 0=50°, C/=100 cm/sec n 5 7 9 a 33i= 290 26° Angle of deflection 56i° 61° 64° Frictional depth h ""j Cq >in metres Roughness height k J 174 0-43 3-1 94 0031 0-23 76 00034 0125 Friction velocity u* (era/sec) 6-7 5 0 40 (/= 1-016 X 10~^ sec~^) and for [/= 100 cm/sec. The three roughness values correspond to average conditions. The frictional depths are obtained in a row as 174, 94 and 76 m which are plausible values. The vertical velocity distribution of u and v is shown in Fig. 176 for the first case (« = 5, /?=174m) together with a vectorial representation (uv); for comparison with the values given by the Ekman theory the corresponding curves are shown by the dotted lines. The greatest differences, as would be expected, appear in the immediate vicinity of the bottom; up to about 5 m from the bottom the velocity increases linearly with distance from the bottom as was assumed by Ekman to be the case inside his boundary layer. Similar considerations also apply for the drift current caused by the wind. Here U must be zero in equation (XIII. 37) and in addition the boundary condition at the surface (~=h) must be 412 General Theory of Ocean Currents in a Homogeneous Sea -* « • -t ai = /^i8. Thereby jS was presumed to be the direction of the wind stress. At the same time it should be noted that under the influence of the turbulence of the wind the frictional coefficient is largest at the sea surface and decreases with depth. In order to satisfy 280 - 240 - 200 ' 1 ~ ^ 160 - / t \ 120 - / ' \\ ^ 80 - / / ^ '~'-\ \ - u '<< / 1 1 1 1 1 V ■X 40 - .--- .-'- ,-'■ ^ / / J 02 0-4 0-6 0-8 0-2 0-4 04 S^ ^ 5m ^ ^^2*^ <.°B n 0-2 ■'^ ^ 100' i'^ w '1 w\ \ ^ ;° l6oV rJ- 02 04 Q6 08 Fig. 176. Turbulent bottom current according to Prandtl (full lines: n = 5, /?i = 174 m; dotted lines: bottom current according to Ekman (see Fig. 172). these conditions the vertical current structure in the drift current will diff'er from that in the bottom current where the frictional coefficient converges to zero at the bottom ; it will have a similar form as compared with that shown in Fig. 172. A theory of drift and gradient currents based on similar principles was put forward by RossBY (1932) and later extended by Rossby and Montgomery (1935). This was based on the principles of the newer turbulent flow theories and introduces in place of the earlier used frictional coefficient the Prandtl mixing length. In drift currents this is largest in the surface layers where the intensity of movement is greatest and decreases with depth to vanish at the frictional depth. The theoretical treatment of this assump- tion is very complicated and the results can only be shown by means of tables. Also here the deflection angle of the wind drift comes out to be dependent on both the wind speed and latitude, while according to the Ekman theory it should have a constant value of 45°. The ratio of the velocity of the surface current to the wind speed (wind factor, p. 418) results as equally dependent in a rather complicated way on the same General Theory of Ocean Currents in a Homogeneous Sea 413 quantities. Table 119 gives some values for these relationships. A comparison of these results with those of the Ekman theory and with observational data is given later on (see p. 418). The introduction of a mixing length decreasing with depth and vanishing at the frictional depth should give a correct representation of actual conditions only if the turbulence arises solely from the wind drift and not from other currents which may be present (for instance, tidal currents, gradient currents). If such influences exist, it is necessary to introduce in the theory of wind-drift currents the vertical distribution of the turbulent coefficients which corresponds to the total current. This, however, modifies in turn the results. At present the Ekman theory appears to be a perfectly satisfactory approximation to actual conditions, as long as our knowledge about the vertical distribution of turbulence is not increased. Table 119. Deflection angle and wind factor as a function of latitude and wind speed according to the theory of Rossby and Montgomery (1935) Defl ection an gle in deg rees Wind fac tor VqIw Wind speed w m/sec. 5 10 15 20 5 10 15 20 : 15° 30° 45° 60° 350 38-6 40-6 42-0 38-7 42-8 45-4 46-8 41-1 45-7 48-4 50-2 43 0 480 50-9 52-7 00317 00292 00280 00273 00291 0-0268 00256 0-0249 00276 00254 0-0243 0-0237 00266 00245 00234 00228 (d) The '' Element ar'^ Current In a homogeneous ocean no currents are possible other than drift and gradient currents; at every point the steady current is made up of a pure wind drift and a pure gradient current. These can be superimposed without mutual interference since each component is entirely independent of the other. If the depth of the sea d is larger than the upper and lower frictional depths D' and D", the resulting current system can be separated into three current layers (see Fig. 177, left-hand side). (1) The bottom current from the sea bottom to a height D" (lower frictional depth). (2) The deep current from the level D' (the upper frictional depth) to the level D" (the lower frictional depth). (3) The surface current which is the resultant of the uniform deep current and the pure drift current generated by the wind. 'I Su rface current Deep current D"{~ B( TTTTTTTTTZ Bottom current Wind Fig. 177. Vertical structure of the "elementar" current (according to Ekman). 414 General Theory of Ocean Currents in a Homogeneous Sea This vertical current stratification was termed by Ekman the ''elementar" current. In limited seas the condition of continuity must also be satisfied. For stationary conditions where everything remains invariable with time the inflow and outflow must balance for a given oceanic space. The drift current is determined by the wind, thus the slope of the sea surface and hence the gradient current must be such as to maintain the constancy of the current system in time. The continuity equation and the boundary conditions in this way determine the structure of the "elementar" current. A simple case can be taken to illustrate these conditions (Fig. 177, right-hand side). A wind parallel to a long straight coast will produce a drift current through which a total water transport away from the coast down to the upper frictional depth is initiated. This causes the surface of the sea to lower along the entire coast and will thus produce a gradient current. The uniform deep current extending downwards from the surface to the lower frictional depth D" will run parallel to the coast and thus cannot com- pensate the removal of water away from the coast accomplished by the wind current. This compensation must be provided for by the bottom current which carries water towards the coast in the direction of the pressure gradient. The slope of the sea surface will thus increase continuously, until the removal of water from the coast, due to the drift current, is exactly balanced by the bottom current. The current in the top layer will then be a vector composition of drift and deep current. The angle of deflection at the surface will thus decrease from 45° to 18°. The current vectors are shown in Fig. 177 for depth intervals of 0-2Z), with the same for the bottom current (at D' = D"). The uniform deep current occupying the deepest water layer between surface current and bottom current is shown by the thick arrow; it is non-divergent and because of its thickness is the decisive current component for the water transport in the oceans. Further interesting cases of "elementar" currents in oceanic regions of special shapes will be discussed in the following section. It is of some interest to deal in some detail with the diagrams of forces for the three layers of "elementar" currents. Since the vectors of Coriolis force and gradient force are fixed by the current vector at the point under consideration, and by the sea surface slope the primary task is to fix the frictional vector. This can be done in the following way. If the current vector is denoted by t) (components u and v), the vector of the deep current by 33 {U,V) and the difference vector by lu (h'^., n'j,)=(tu— 3.^). (m— U, V— V), then the equations of motion will have the form ~f^= - Sq^-^ J^x and fu=-g~-i-Ry, whereby -i^(/?a;, Ry) is the frictional vector. However, for the uniform deep current -fV=-g~ and fU=-g^. ■' dx dy Subtraction gives — fWy = R^ and fw^^Ry so that w'x Rx + »*'i/ Ry = 0. This, however, is the necessary condition for the vector of the frictional force ^{Rx, Ry) to be at right angles to the direction of the difference vector tu. Thus the direction of the vectors of all three forces involved are known and therefore a diagram General Theory of Ocean Currents in a Homogeneous Sea 415 offerees for each layer of the "elementar" current can be constructed. Figure 178 shows these force diagrams for always one level of the three current layers. In the surface current the frictional vector is directed to the side of the gradient vector pointing in the direction of the water movement, and rotates in a clockwise direction with decreasing intensity when going downwards and vanishes at the frictional depth. In the deep B. (b) iC (c) Fig. 178. Schematic diagram offerees for three levels of the "elementar" current (Northern Hemisphere): {a) surface current, (Jb) deep current, (c) bottom current. OG, OC and OF vectors of pressure gradient, of Coriolis force and of frictional force; v = velocity vector in the level under consideration; V = velocity vector of the deep current; w = vector of the velocity difference: v — V. current, gradient and Coriolis force balance each other without any frictional effect. In the bottom current the frictional vector is directed to the side of the Coriolis force pointing more or less in the opposite direction to that of the velocity and rotates anticlockwise while approaching the bottom. From this distribution it can be realized that in the surface current the frictional vector corresponds to a driving shear stress which takes its strength at the sea surface from the energy of the wind, while in the bottom current it indicates the retarding effect of the underlaying bottom topography (break on the motion). (e) Drift and Gradient Currents according to Observations; Piling up of Water by Wind {''Windstau'') The two parts of the "elementar" current are never developed in the ocean in pure form and it is to be expected that pure drift currents in the ocean will always be some- what masked by the effects of superimposed gradient currents. It will therefore not be 416 General Theory of Ocean Currents in a Homogeneous Sea easy to test the properties required by the theory. Three consequences of the theory are possibly most suitable for such a test: (1) the deflection of about 45° cum sole from the direction of the wind which is almost independent of latitude (except near to the equator) ; (2) the restriction of penetration of the drift current by the frictional depth D; (3) the dependence of the sea surface velocity of the drift current on the shearing stress of the wind. Angle of deflection. By special selection of oceanic areas, where it would be expected that the wind alone would be decisive in determining the currents, Galle (1910) showed that the deflection required by theory was actually present. For this he used the large amount of data available for the Indian Ocean for all November months from 1858 to 1904 between 20° N. and 50° S. and 10° E. and 130° E. Taking together two degree zones in each ten-degree field, the theoretical deflection to the right was obtained in 77% of all cases in the Northern Hemisphere and in 69% in the Southern Hemi- sphere. Three areas were examined with particular care: the sea between Socotra and the Maldives, the South Equatorial Current and the west wind drift of higher southern latitudes. Table 120 shows average values for larger areas. The mean of all values is about 46° and in fact there seems to be no dependence on the latitude; both these circumstances are in accordance with the theory for a constant frictional viscosity coefficient. Forch (1909) used the survey on wind and current conditions in the Eastern Mediterranean published by the "Deutsche Seewarte" to obtain an estimate of the Table 120. Mean angle of deflection in the Indian Ocean (cum sole) in all cases 5°-20° N. 50-60° E. 60^-70° E. 62° 44° 40°-50°S. - 10°-20° E. 20°-30° E. 30°^0° E. 70°-80° E. 80°-90° E. 55° 41° 42° 41° 43° 10°-20°S. 70°-80°E. 80°-90° E. 47° 51° Table 121. Mean angle of deflection in the Eastern Mediterranean (cum sole) in all cases Area 36=-38°N. 15°-20°E. 34°-36° N. 15°-20°E. 34°-36° N. 20°-25° E. 32°-34° N. 25°-30°E. Annual mean 38-2° ■ 33-1° 52-4° 430° Mean for the four fields Jan. /Feb. Mar./Ap AAV 45° r. May June/July Aug./Sept. Oct. /Nov. 86° 47° 23° 23° Dec. 45° Mean 411° deflection of the current from the wind direction. The differences between wind and current azimuth for the four larger areas are given in Table 121 as annual average values derived from the monthly means. The mean of these rather scattered values is around 42° cum sole. In the annual variation the angle is nearly 45° from December to April, reaches a very high value in May and then during the warmer part of the year from August to November is about 20°. It is possible that the strong surface General Theory of Ocean Currents in a Homogeneous Sea 417 density gradient during the summer gives rise to a strong differentiation in the magni- tude of the frictional coefficients in a vertical direction v^hereby the angle of deflection is reduced. Even more penetrating investigations have been made of the deflection angle in shallow seas (lightship observations). These values have, however, mostly been made in coastal areas or over large banks where disturbances can be expected but these can be eliminated by special grouping of the data. According to the Ekman theory there will be no strong deep currents in any largely enclosed sea (see p. 428). A com- parison between theory and observation can then be made in such a case. For a shallow sea (depth d) the theory requires the deflection to be smaller the smaller the ratio d.D. On the other hand, the thickness D of the drift current will increase with increasing wind strength. It can thus be expected that in a shallow e?iclosed sea, the angle of deflec- tion will become smaller as the wind increases. From data on currents recorded by Finnish light-ships, Witting (1909) found that the angle of deflection was always cum sole and that it could be expressed by the relation a = 34° - 7-5 Vw, where u' is the strength of the wind in m/sec. The strong ellipticity of the current ellipses at the different lightships indicates a preferred current direction caused along the longer axis of the sea which certainly affects the results. Qualitatively, however, it corresponds fully to the requirements of the theory. Also Dinklage (1888) obtained similar results from observations made at the Adlergrund light-ship (Baltic). The question of testing the Ekman theory has been discussed in detail by Palmen (1930 b, 1931) in connection with an evaluation of the currents in the northern part of the Baltic. This was based principally on observations made at the rather openly situated Swedish lightship "Finngrundet" (60-0° N. 18-5° E. at the southern end of the Gulf of Bothnia) for the period 1923-27. Tables 122 and 123 show clearly the relation- ship between wind and current on the one hand for different wind strengths and on the other hand for different wind directions. These correspond rather well to the requirements of the theory. Especially the confirmation of the turn of the current direction with increasing depth deserves our attention because only few observations of that kind are available. After elimination of non-significant disturbances the following corrected values are obtained for wind strengths of 4-5 Beaufort: Vo = 9-2 cm/sec, ao = 35°, KgoiKo = 0-76; K,o = 7-0 cm/sec, ajo = 54°, Aa = 19°. Table 122. Currents at different wind strengths at the lightship ''Finngrundel'' (Gulf of Bothnia, 1923-27) (according to Palmen) Wind strength (Beaufort) 10 20 2-9 3-9 4.9 5-9 6-8 7-8 9 0 9.9 Vq (cm sec) 20 31 5-8 8-4 11-3 12-3 14-7 19-2 22-9 27-3 F^o (cm sec) 1-6 2-2 4-5 6-2 9-6 10-2 130 18-3 19-7 24-1 V,o:Vo 0-87 0-71 0-78 0-74 0-85 0-83 0-88 0-95 0-86 0-88 Deflection a^ 26° 41° 38° 33° 34° 35° 32° 25° 36° 8° O-20 32° 50° 48° 42° 41° 45° 52° 38° 40° 11° "20 - ao ■ 6° 90 10° 90 7° 10° 20° 13° 4° 3° 418 General Theory of Ocean Currents in a Homogeneous Sea Table 123. Currents for different wind directions at the lightship ''Finngrundet" (mean value at 2-7 Beaufort) Wind direction N. N.E. E. S.E. S. S.W. W. N.W. Mean Vq (cm /sec) .... 8-6 11-2 121 8-7 7-5 9-2 7-4 8-2 9-2 K20 (cm /sec) .... 6-8 9-6 11-5 6-8 5-5 7-9 5-7 70 7-6 K20 -yo- 0-79 0-86 0-95 0-70 0-73 0-86 0-77 0-85 0-81 «o .... 30° 35° 41° 41° 40° 38° 22° 34° 35° 020 ... . 39° 46° 47° 46° 55° 50° 41° 47^ 46° 020 — Oo 90 11° 6° 5° 15° 12° 19° 13° 11° The directional turn between 0 and 20 m depth is 19° cum sole and at the same time the velocity falls by about a quarter of the surface value. This turn of the current is in good agreement with the theory; the decrease in velocity is, however, much too small to be explained by a constant frictlonal coefficient ; for a water depth of 23 m and for a 77 about 200-300, it must be about 0-12 instead of 0-81. Only an assumption of a variable r] with depth approximately in the sense of the discussion given on p. 405 could explain such a small decrease. The relationship of wind strength to current strength. According to the theory the surface velocity Vq is given by the relation Ko = ..^ ^ . ,. . (XIII.4]) From this it follows that for constant 77 and p the surface velocity Vq is proportional to the wind velocity w and is inversely proportional to the square root of sin ^: Vq = —-^ w (XIII.42) V(sm (p) A is a universal constant. The quantity VqJw is denoted as the "wind factor". Numerous investigations have been made of this relationship (see especially Thorade, 1914); the following values have been found for A, when Vq and h' are expressed in cm/sec: Mohn 00103 Dinklage C-0127 Witting 00100 Thorade 00126 Pal men 00114 Nansen 00190 Sverdrup Brennecke 00177 00269 The first of these values are in good agreement. For the ice drift, on the other hand, considerably higher values were obtained (Nansen, 1902; Sverdrup, 1928; Brennecke, 1921). See p. 437 concerning these. Usually an almost linear relationship has been found between the wind velocity and the velocity of the surface current. Witting and Thorade, however, arrived at a different result : for a wind force of up to 3 Beaufort a better fit to the observations was obtained by a quadratic relation. Palmen believed, however, that this was due to the uncertainty of the conversion of wind strength from the Beaufort scale into m/sec. For the magnitude of 77 it seems to be also of importance, on what height the wind measurements are based; a better agreement could probably be obtained if also this was taken into account (Exner, 1912; Durst, 1924). General Theory of Ocean Currents in a Homogeneous Sea 419 The shearing stress of the wind and piling up of water caused by the wind. There are two ways in which the wind stress can be determined. The first is afforded by equation (XIII.41). This requires a knowledge of the frictional coefficient iq, but its dependence on the wind strength is not well-enough known. Ekman has indicated a second possi- bility using the piling up of the water ("Wasserstau") by the wind and using the current produced by the wind over a confined sea. If the effect of the Earth's rotation is dis- regarded (/= 0), and if dpjdx is replaced by the slope / of the sea surface, then the first of the equations (XIII.28) for a variable -q gives the equation d I cti] This can be integrated considering the boundary conditions = -T and (m),=) for wind speeds less than Beaufort 3 (about 6 m/sec). All these formulae are of course only approximations, since at the present time systematic current measurements from which accurate values could be derived are not available. Observations on the thickness of drift currents are usually in general agreement concerning magnitude with the values given by formula (XIII. 50). The oceanic struc- ture in the region of the North and South Equatorial Currents in the Atlantic Ocean indicates that the wind current here has a depth of about 150 to, at the most, 200 m and and thus that the frictional depth in these latitudes only barely reaches these values. Towards higher latitudes it decreases. Brennecke (1921) found a frictional depth of about 50 m during the ice drift of the "Deutschland" in the Weddell Sea and Sverdrup (1928) has shown from Brcnnecke's values that there is an increase with increasing wind speed as is shown by the following values : Drift velocity (cm/sec) : 5.52 9.81 14.85 24.60 Frictional depth £> (m) : 45.6 56.2 (39.1) 69.1. General Theory of Ocean Currents in a Homogeneous Sea 423 Using the equations previously derived to calculate 77 gives 77 = 1 -03 vv^ for IV < 6 m/sec, and 77 = 4-3 u'^ for vv > 6 m/sec. The values calculated from these formulae are also to be regarded as only approximate average values; the few directly determined values are widely scattering and indicate a large dependence on the vertical stratification of the water masses, Schmidt (1917) has presented some values : Wind speed (m sec) 1 3 5 7 10 20 ■q (cni-^^ g sec"^) (1) 28 110 220 430 1720. The high values for strong winds apply of course only for the especially intense tur- bulence produced by the wind in the uppermost water layer; below this layer the co- efficient decreases rapidly with depth. An average value for the top layer of the ocean will be between 50 and 100. Its magnitude in the deep layers will be about 1-10, Diagrams of forces for a wind-driven, stratified ocean. With a complete knowledge of the total current and pressure structure of the ocean diagrams of forces for any layer can be derived in the following way (Defant, 1941 b). Denoting the sea surface slopes (of the isobaric surfaces in the deeper layers) in the positive .v-direction (towards east) with i^ and in the j-direction (towards north) with iy, then the equations of motion for a variable 17 are of the form 8 / 8u\ 8 / 8v\ fpv + gpi. + ^, [1 -^.j = 0; -fpu + gpiy+ ^ [r^ j^j = 0. (XIII.52) Integrating these equations from the surface to the depth D with the assumption that the current falls to zero at a depth d and taking furthermore into account that for z == 0 the components of the wind stress are given by cu 8v and vanish when z = d, the following equations are obtained : f7v + g'pi'x+T, = 0 and -f^u -{- gJTy + Ty = 0, (XIII.53) where the integrals (sums) down to the depth d are indicated by a bar. This states merely that for a steady current the Coriolis force must be in equilibrium with the sum of the total pressure force and the total wind stress exerted on the entire layer. The equations (XIII.53) can be evaluated numerically from the absolute topography of the pressure surfaces and of the physical sea-level, as well as from the rather reliable vertical current distribution as measured at two anchor stations in the region of the South Equatorial Current in the Atlantic. Table 1 25 contains all the necessary numerical values and Fig. 179 shows the vertical changes in current- and pressure-gradient quantities for calculation of the integrals. It can be seen that the £'-component of the velocity decreases regularly with depth, while the A^-component changes already in the uppermost layers from small positive values to negative values and then falls back to zero at 100 m. This distribution leads to a turn of the current vector cum sole which must be the case in drift currents. Below this there is only a gradient current 424 General Theory of Ocean Currents in a Homogeneous Sea 0 100 200 300 400 500 /? X dyn cm 0-2 0-4 06 08 10 12 y X ^, "~~^ / ^pA ig N \ \ \ y Ipv ^ / ^^ J V /- -^ pA y \ y -30 -24 -18 -12 -6 p, cm/sec +6 Fig. 179. Vertical changes in the pressure gradients and of the velocity components in the central part of the South Equatorial Current in the Atlantic Ocean. which, however, also disappears at 500 m depth since there the isobaric surfaces become almost horizontal. Table 126 gives integral values for the equations (X1II.53) and the corresponding resultant values of the wind stress; Fig. 180 presents the diagram offerees for this cen- tral part of the South Equatorial Current. The average direction of the south-east trade wind during the observational period was S. 40° to 45° E. and the mean wind force Fig. 180. Schematic diagram of the forces in the South Equatorial Current in the South Atlantic Ocean. about 12 m/sec. This wind direction is in excellent agreement with the direction of the wind stress. The wind stress can be calculated from the meteorological data using equation (XIII.48) or from the oceanographic data using equation (XIII. 37). In the first case wind stress and wind speed lead to a constant value for A' of 2-5 x 10~^ which is General Theory of Ocean Currents in a Homogeneous Sea 425 Table 125. South Equatorial Current in the Atlantic Ocean (approx. 14° S., 20° W. to 8° S., 15° W.) Pressure gradient dyn cm, 100 km Vertical current distribution p (dbars) depth (m) .d/7 100 km U V pu pv in situ Direction (dyn. cm) P'x P'v cm/sec 0 N. 60° E. 0-97 0-86 0-49 -32 + 7 -32-9 +7-2 24-3 50 — . — — -14 -8 -14-3 -8-2 24-7 100 N. 30°E. 100 0-51 0-89 + 9 + 1 -9-2 + 10 25-9 200 N. 20'E. 1-20 0-42 116 -4 -1 -41 -10 27-7 500 0 000 000 000 0 (+1) 00 + 1-0 29 4 Table 126. Diagram of forces in the South Equatorial Current of the Atlantic Ocean (Forces in dyn/cm^) Coriolis force Pressure force Wind stress -/or = +1-77 +fpli= -6-73 SI5°E 6-95 gPl^= +1-49 gpt\ = +3-28 N24°E 3-51 r„ = -3-26 Ty = +3-45 N 43° W 4-74 in good agreement with the known values. Alternatively, taking h-^ (the frictional depth of the drift current) as about 200 m, the roughness parameter Cq as 0-3 and the surface velocity U as 35 cm/sec, equation (XIII. 37) gives exactly the required value of 4-74. These calculations show in any case that the oceanic current conditions are in good agreement with hydrodynamic concepts about the driving forces. The dissipation of the current energy in the ocean. It is probably of some interest to calculate the amounts of energy dissipated in a drift current due to the apparent friction. The energy consumption is of course largest in the uppermost layer and decreases rapidly with increasing depth. If only the /o/a/ energy consumption is required this can be calculated rapidly in the following way. The total work done in the interior of the water must be supplied from the wind at the sea surface. This is, however, given by force x distance. The force is the wind-stress component in the direction of the surface current; the component at right angles does not enter into the calculation. This component is Tcos 45 "" and the distance travelled in unit time is Vq. The energy consumption per second in a vertical water column of 1 cm^ cross-section can then be obtained using equation (XIII.26) (Schmidt, 1919) and is given by W = Vq \/{t]pw sin (/«). The values of tj given by Thorade give the energy values shown in Table 127. In a vertical water column the total work expended should lie between 2 and 40 erg/sec. There is a considerable increase in these amounts with increasing wind speed and the latitude also has an appreciable effect. 426 General Theory of Ocean Currents in a Homogeneous Sea Table 127. Energy Dissipation in Ocean Currents (according to Schmidt) (Values in erg cm"- sec~^) Wind speed w (cm/sec) 4 6 8 10 15 20 10° 4-5 15 35 69 230 550 Latitude 40° 2-3 1-1 18 36 120 290 70° 1-9 6-3 15 29 100 240 (/) The Effects of Coasts on the ''Elementar''' Current The vertical structure of the "elementar" current depends essentially on the direction of the wind relative to the general outline of the coast, since this has a large effect on the equation expressing the condition that for stationary conditions the transport component at right angles to the coast must be zero. Ekman (1923) has presented a solution in two simple and very instructive cases. The first case assumes an extended oceanic region off a long straight coast over which blows a wind of constant force and direction. The water depth d is assumed to be constant and greater than 2D. The sea-level will fall uniformly from the coast towards the open sea and the pressure gradient produced by the piling up of water by the wind ("Windstau") will be at right angles to the coast. With an arbitrary orientation of the co-ordinate system the trans- port components M'^ and Afy will be given by equation (XIII.27). The transport components of the gradient current are given by M'^ = bU^ - BUy and W; = BU^ + bUy, (XIII.54) whereby U^ and Uy are the components of the uniform deep current and (- - S) 5 = V- and b = \ pd If the X-axis is oriented along the coast, then f/,, = 0 and from continuity equation M'y X M"y = 0 is obtained T Bf pcoD sin 0 r.. For a given T and a given angle between wind and coast the drift current and the gradient current is fully determined. Ekman has given a simple graphical method for the construction of the total current structure in this case. Figure 1 8 1 shows this current structure in some special cases. The current arrows have to be visualized as drawn from the point o to the points on the curve and the small points refer to heights of 0-1, 0-2 D etc., above the sea bottom and to depths of 0, 0-1 D, 0-2 D etc., below the sea surface. The wind direction is indicated by the arrow. The cases correspond to angles of /S = 0, +45° and -45°.* There is a considerable difference between conditions when the water flow is un- hindered in all directions or when it is adhered due to any kind of influence. In the * ^ = 0 indicates a wind direction parallel to the coast ; the increase in j3 is positive to the right and negative to the left. General Theory of Ocean Currents in a Homogeneous Sea All Fig. 181. Vertical structure of the "elementar" current for different orientations of the coast relative to the wind (according to Ekman) (the arrow indicates the wind direction). first case only a pure drift current is formed and the effect of the wind is restricted to a relatively thin top layer. At coasts, however, the effect of the current-producing wind extends almost down to the sea bottom due to the generation of deep currents. Their velocity is not insignificant and may be as much as half of that of the surface current. The second case is that of a sea enclosed by land, with a wind of constant direction and constant speed blowing over its entire surface. Here the continuity condition requires that the transport in all directions should be zero, that is, that the total gradient current transport must be the same as that of the drift current and directed oppositely. The boundary condition equations are now ^/■x + ^x = 0 and My + My = 0. Taking the positive j'-axis along the direction of the wind stress, then Ta- = 0 and Ty = T. This gives Tlf-i-bU^-BUy^O and BU^ + bUy = 0 from which it follows that bT . __ BT U.= - and Uy = f(b^ + B') ^' f{b^ + B^) If the angle {cum sole) between the gradient current transport and the pressure gradient is denoted by fi and if Uy — 0, then My^-B "°^ ^^tan-^. This angle is almost 90°, if the depth of the sea is not too small (for djD = 1, 2, 10, ^ is approx. 79°, 85° and 89°, respectively). However ^ = -^=tana, where a is the angle between the direction of the deep current and that of the wind, or a — |7T is the angle between the directions of pressure gradient and wind. Since 428 General Theory of Ocean Currents in a Homegeneous Sea a = TT — ^, this angle will be ^tt — /3 {cum sole). The velocity of the deep current is then U T . ^ 27tT sin p ^ -7^ cos p. bf pfD The gradient current now extends almost throughout the entire water mass, so that even a low velocity of this current is sufficient to compensate the drift current trans- port. The greater the depth of the water, therefore, the lower will be the velocity of the gradient current, and the less will be the effect of the coasts on the surface current given by the resultant drift and gradient current. As shown by the above equation, containing cos ^ and the frictional depth D in the denominator, the deep current V is very weak. Ekman has calculated numerically three special cases {d — 0-5 D, d = \-25 and 2-5 D). Figure 182 shows the vertical current structure in the usual way d='-axis along its crest), and assuming a uniform current U in front of the ridge extending throughout the total water mass (depth of water H) and flowing towards the crest, equation (XIII.29) gives: ?^ dC -^n =f^ and ^ =0; V^O. dy 8x Over this bottom ridge under stationary conditions (duldt = dvjdt = 0) the equations of motion will be ''fx = -^dy-^''=-^^^-''^- If the origin of the co-ordinate system is placed at O vertically underneath the highest point of the ridge, the half-width of which {OA = ^45) is /, and height of which at O is h, then the depth of water will be d=cl,^{hll)x, where the upper sign applies for the forefront side and the lower sign for the rear of the bottom ridge. The equation of continuity requires the same transport through every cross-section, that is UH = u{d^{hll)x]. General Theory of Ocean Currents in a Homogeneous Sea 433 20O"''''^30O Fig. 185. Topographic influence of a submarine bottom ridge on a current flowing normal to the longer axis of the obstacle. Lower picture: vertical profile through the bottom ridge (width, 400 km; height, 200 m; water depth, 4 km; p = 30° N.). Upper picture: stream lines of the main current (U = 50 cm/sec). This gives Over the rise the flow thus is subjected to an acceleration acting along the longer axis of the ridge with a maximum value of —/A/// above its highest point. This acceleration gives rise to a curvature of the stream lines cum sole. To the velocity u is added a transverse velocity v which at a point x = ^ — / (^ is the distance of the point under consideration from point A) is given by IHl H F, whereby /"denotes the cross-section of the bottom surface for the distance from A to ^. The deflection of the current from the initial x-direction will be vju, and for a small bottom slope is given with sufficient accuracy by vjU. The deflection on passing over a bottom ridge is the larger, the smoother the sea, the higher the ridge and the smaller the velocity U. Since in the ocean U is relatively small, it can be expected that the bottom topography will have a stronger eff'ect on the currents. Fig. 185 presents a numerical evaluation of a single case: width of bottom ridge 400 km, its height 200 m, ocean depth 4000 m and 0 = 30° N. while U is taken as 50 cm/sec (somewhat high because of the absence of friction in the current). At the crest of the ridge the deflection will be —37° and in the rear of the rise at its end —55°. The deflection is of course associated with a corresponding change in the sea-level; to the normal slope directed along the crest is now added a slope directed normal to the ridge crest and a corresponding lowering of the sea-level along the .\--direction. If instead of a single ridge the bottom has a series of ridges and troughs 434 General Theory of Ocean Currents in a Homogeneous Sea the vortex formation is repeated periodically corresponding to these bottom waves. Figure 186a shows this case for the Northern Hemisphere ; there is a current curvature cum sole above the ridges and contra solem above the troughs. If the sea surface has an overall slope so that already at a larger distance from the ridge a current at right angles to the ridge is produced then a current field will be formed similar to that shown in (a) (b) Fig. 186. Stream line pattern: (a) for currents crossing a wave-form bottom configuration; (b) for the crossing of a single bottom ridge (Northern Hemisphere, according to V. Bjerknes and co-workers). Fig. 1 86^. The stream lines approach the ridge directly at right angles and pass over it bending cum sole on the forefront side and contra solem in its rear and then finally return to their original direction. This latter curvature in the rear can, however, only occur if there is a convergence on the lee side which is stronger than the divergence on the forefront side. Recently, Gortler (1941) has gone into this problem more carefully taking into consideration the frictional effects also. The mathematical formulation is different as compared with the previous one and shows an improvement in so far as it leads to simpler basic equations which are more likely to be solved quantitatively. The results otherwise agree with those obtained previously. Gortler dealt mainly with a case similar to that above. The bottom ridge was assumed to have a vertical profile ^ = Po{l + cos (2ttII)x} with|jc| < y and h = 0 outside this region. A horizontal projection of the stream lines of the main current is shown in Fig. 187 in the same way as in Fig. 185, but here friction has been considered. For an insight into the frictional effect the dimensionless quantity hrlH is decisive where hr depends on the frictional depth and H is the depth of the sea. This quantity usually appears in the expression G = (Rll)l(hrlH), where R = [///gives the radius of inertia associated with the current velocity U (equation XIII.26), with which the flow approaches the obstacle. The different curves in Fig. 1 87 show for a fixed value of Rjl the effect on the course of the stream lines of the disturbance in the equilibrium between gradient and Coriolis force above the ridge due to the generation of a "secondary" current. When C is 3 General Theory of Ocean Currents in a Homegeneous Sea 435 or greater there is no essential difference as compared with the frictionless case (hr = 0, G = oo). For reasonable values of H and / Gortler estimated the magnitude of G as between 3 and 80, depending on the intensity of U, the latitude and the rough- ness of the bottom. This shows that for actual conditions in nature everything is the same as in the case of no frictional influence. This is important for the practical use of the above results. The effect of the topography of the sea bottom on the course of the ocean currents has been clearly demonstrated for many oceanic regions. Ekman -10 Fig. 187. Upper picture: stream line pattern for a crossing of a bottom ridge depending on friction. Lower picture: vertical cross-section through the bottom ridge. by using these principles was the first to offer an explanation for the striking bending of the current trajectories, of the dynamic isobaths south of the Newfound- land Banks (Helland-Hansen, 1912) which was not understood by simple reasoning. The course of the stream lines is in good qualitative agreement with that given by theory for the changes in depth actually present even if a closer qualitative examina- tion of the phenomenon was not possible. The dynamic evaluation of the observational data made by the "Meteor" expedition in the South Atlantic has afforded a good example of these effects of the bottom topo- graphy (Defant, 1941 b). This example makes it very probable that the large irregulari- ties in the east-west course of the dynamic isobaths that were found in the western part of the convergence zone between about 25° and 50° S. have a fixed position and can be attributed primarily to the morphology of the sea bottom. If the lines of con- vergence and divergence for this disturbance are traced on transparent paper and laid over a depth chart the relationship between the two phenomena shows unmistakably. These conditions are illustrated by a diagram in Fig. 188. The lower part of the figure shows two depth profiles at 30° and at 35° S. extending from the South American continent to 0° W. ; they indicate the course of the bottom irregularities running in a meridional direction as far as the mid-Atlantic Ridge in this part of the South Atlantic. In the upper part are shown the stream lines plotted according to the dynamic isobaths over the area from 30° to 45° S. Every "wave trough" in the bottom corresponds to a bend contra soletn in the stream lines (here the reverse of the conditions as shown in 436 General Theory of Ocean Currents in a Homogeneous Sea Fig. 186, since this is in the Southern Hemisphere). The extremes do not always coin- cide in position but particularly in the eastern part are in excellent agreement. Schumacher (1940, 1943) has indicated further examples. Over the mid- Atlantic Ridge especially, there is often a corresponding bending of the current to observe. The large stationary cum sole vortex off the eastern side of the Azores plateau must also Fig. 188. Upper picture: bottom topography and stream lines for the gradient current in the disturbance region of the subtropical convergence zone in the South Atlantic Ocean (30°-45° S., 50-0 W.). Lower picture: vertical bottom profiles at 30° and 35° S. according to the depth chart of the Atlantic Ocean. be favoured by the bottom topography. In the Equatorial Counter Current the presence of the Atlantic Ridge shows this very typical effect. If the water masses are stratified, there will be corresponding displacements in the isosteres inside the region of influence of the bottom irregularity (see p. 558). If an isolated submarine ridge lies in the path of a current a cyclonic vortex will be formed above it. An example of a vortex of this type is given in the description of oceanic conditions around the "Altair" submarine volcano in the North Atlantic (Neumann, 1940) (see also, Schott, 1939). In discussing the effect of the bottom topography on ocean currents it has always been assumed that the current is more or less uniform from the sea surface down to the sea bottom. In almost all cases, however, the velocity of the current falls off rapidly with depth and in addition there are changes in the direction of the current. In these circumstances it is not so easy to accept a direct effect of the bottom topography on the current in the upper layers of the sea, since these are often separated from the bottom currents by very thick motionless water layers or layers with quite a different type of current. Attention should be drawn to these considerations in any discussion of the effect of the bottom topography on the currents. 5. Ice Drift The wind drift of the ice in the polar regions (see pt. I, Chap. VIII, p. 243), like the ordinary wind-driven ocean currents, is dependent on three forces: wind stress, in- ternal turbulent friction and Coriolis force; in addition to these it is also affected by General Theory of Ocean Currents in a Homogeneous Sea 437 a resisting force arising from the random movement of the ice which is proportional to the drift velocity and acts in the opposite direction. This ice resistance is the reason why the Ekman theory for the ice drift is inadequate. Nansen had already shown in 1902 from the "Fram" data that the ice resistance cannot be neglected and indicated that one of its effects must be the small deflection angle observed for the ice drift. Brennecke (1921) and Sverdrup (1928) have made important contributions to the clarification of the interrelated forces acting and that of Sverdrup can be regarded as a complete theory of the ice drift (see also, Rossby and Montgomery, 1935). However, the observations of the "Fram" are not suitable for testing this theory, since the ice drift here includes a component due to the permanent surface current (see p. 358), but over the North Siberian Shelf ("Maud" observations) and in the Weddell Sea ("Deutsch- land" observations) the ice drift is free from a basic current and is suitable for this pur- pose. There is, however, one fundamental difference between these two drifts, due to the very different hydrographic conditions under which these drifts occur, and this has a considerable effect on the nature of the pure drift current (without ice). Over the Siberian Continental Shelf the oceanic structure consists of essentially two layers: a top layer of lighter water and a heavier bottom layer separated by a sharp density transition layer (thermocline). In the surface layer the vertical equili- brium state is indifferent (neutral) throughout almost all the year and the turbulence in it is intense. In the discontinuity layer it falls nearly to zero and this therefore has the character of a gliding layer. The entire water mass of the top layer is thus drawn along with the surface current and this, together with the ice masses floating in it, behaves like an elastic sheet. The resistance against the movement thus arises from the effect of varying winds driving this sheet together. In the deep Weddell Sea the oceanographic conditions are different; here there exists no transition layer near to the sea surface and the density increases continuously with depth. A drift current thus develops in the normal way, and also the expected decrease in the velocity of the current and its turn in direction could be observed. In the Weddell Sea it appears necessary to take into account the effect of turbulent friction besides that of the ice resistance. These circumstances require to deal with each of the cases separately. A shallow sea with a density transition layer (thermocline). The wind stress is taken as proportional to the wind velocity u' and thus as equal to cw (c is termed the wind effect); the resisting force (ice resistance) as proportional to the velocity of the ice drift and in opposite direction of it is denoted by —ku (with components — A:m^ and —kUy along the co-ordinate axes). Then as shown by Sverdrup for the case of the North Siberian Shelf, for non-accelerated motions (wind along the positive j-axis) This gives where kUx -/«. = o and kUy + /Wj; = CW n — cfw and ckw 11 "x - " k^+p' Ux f „— J u c . (XIII.60) tan a = — , „,,v. , — — y.-^ Uy k vv / 438 General Theory of Ocean Currents in a Homogeneous Sea Here a is the angle of deflection of the ice drift from the wind direction and r is the wind factor (relative drift velocity, p. 418). Both the angle of deflection and the wind factor increase with decreasing ice resistance if the wind effect is constant. It can easily be shown that the end-points of the vectors of the wind factors must lie on a circle with its centre on a straight line at right angles cum sole to the wind direc- tion. Its radius is /? = c/2/. In Fig. 189 the vectors shown represent the drifts for values k ^ Sf, 3/ and/. Fig. 189. Relation between wind and ice drift for stationary wind conditions and for diflFerent ice resistance (according to Sverdrup) A deep sea with a continuous vertical density increase. Here the equations of motion are the same as for a pure drift current (XIII.23). The boundary conditions are, how- ever, the following (wind along the positive >'-axis) : f(u)u^ and ~ -^ = — F(w)w +f{u)Uy forz = 0: dz P S^ P for z — co: Ux "= Uy = 0. The functions /(m) and F(h') are for the moment unknown. F(vv)vv is equal to the wind stress T. With these boundary conditions a solution for the equations is thus Doj sin , u ^r^^ and r = — w tan a = F(vv)sina. (XIII.61) Doj sin 4> + 71'/(m) vv Doi sin ^ Also in this case the wind factor decreases with increasing ice resistance for otherwise equal conditions, since the angle of deflection a decreases with increasing resistance. As in the previous case, the end-points of the relative drift vectors drawn from the starting point of the wind vector lie on a similar circle as before. The radius is, how- ever, R = {ttF (w)]l{2Dco sin ^). The functions introduced here are not identical with the coefficients k and c used in the previous case, but are in a way similar to them. The function/(M) depends on the state of the ice while F (vv) is related to the turbulence state of the wind blowing over the ice. The observations made during the ice drift allow the determination of both a and r in both cases, and from these the coefficients k and c in the first case and the functions /(«) and F{w) in the second can be determined. For a test of the relations only those periods can be used, of course, in which a quasi-stationary state prevails. These factors are grouped according to increasing wind factor and increasing deflection angle and presented in Table 1 28 ; Fig, 1 90 shows these mean values in a graphical presentation General Theory of Ocean Currents in a Homogeneous Sea 439 for a comparison with those required by theory (see Fig. 189). The theoretical relation is satisfied reasonably well, indeed, but the individual values are strongly scattered — which in view of the possible sources of error is not surprising. With the wind direction almost constant the coefficient of the ice resistance k computed from the "Maud" values decreases from 5-75 to 1-21. In the "Deutschland" values the resistance function Fig. 190. Observed relation between wind and ice drift for a constant wind influence, but for an increasing ice resistance. f{u) increases with increasing wind and drift velocities and in fact so, that a linear function is obtained for/(w). For the ice resistance this gives f{u)u = au^. It is thus approximately proportional to the square of the drift velocity. For the ice drift over the North Siberian Shelf Sverdrup found that the ice resis- tance was directly proportional to the drift velocity. This difference can be explained by the different nature of the ice cover in the two cases. Over the Siberian Shelf the sea is covered throughout the year by a solid connected ice layer, about 3 m thick (Pt, I, p. 273). In the Weddell Sea, on the other hand, the ice cover forms only through- out the winter and also then is not nearly as thick as the Arctic drift ice. Furthermore, in the Weddell Sea even in the winter there are frequent long open spaces in the ice cover ("Wacken") so that even at low wind speeds the ice has a much greater freedom for movement. Table 130. Relationship between wind and ice drift under quasi-stationary conditions {mean values) "Maud" "Deutschland" Group 10^ X r < 1-50 1 •51-200 > 200 102 X r < 2-8 >2-8 a < 30" 3I°-40° > A0° a < 29° > 29° 102 X r . 0-77 1-75 2-07 102 X r 2-32 3-39 a ... 13-8° 36-5° 49-3° a 21-8° 42-8° 10* X yt 5-75 1-90 1-21 lO^a 150 0-7 10« X c 4-56 4-15 3-86 Wb 3-4 31 440 General Theory of Ocean Currents in a Homegeneous Sea According to the observational data the wind function F (w) can be approximately given the form F{w) = b^/w, so that the wind stress T = bw^'"^. By this the results of Palmen are brought in mind because they are in a way similar. The coefficients a and b thus like k and c characterize the strength of the ice resistance and the effect of the wind. The seasonal changes in the relationship between wind and ice drift fit in well with the above considerations. Table 129 shows these changes, together with the calculated variations in the resistance coefficient and in the wind effect. Over the North Siberian Shelf both the relative drift velocity and the angle of deflection show a pronounced minimum in spring and a maximum in summer. This is partly due to the change in the resistance coefficient k and partly due to the wind-effect c. The value of k increases gradually from a summer minimum until the first half of the winter and then rises rapidly to a maximum at the end of the winter in order to fall off again just as rapidly to the summer minimum. These variations can very well be explained by the state of the ice cover during the year. In summer the ice resistance is small due to the numerous open spots ("Wacken") and consequently greater free- dom of movements for the ice. In autumn and at the beginning of winter these open Table 129. Seasonal changes in wind factor, angle of deflection, resistance coefficient and the wind-effect on the ice drift Jan.-Feb. Mar.-Apr. May- June July-Aug. Sept.-Oct. Nov. -Dec. "Maud" 102 X k 1-67 29-4 1-43* 17-9* 1-67 23 0 2-20 40-8t 2-30t 39-4 1-79 30-8 10* X A: 10«c 2-51 4-82 4-66t 6-97t 3-46 612 1-63* 4-76* 1-72 512 2-37 500 "Deutschland" 102 .^ ^ a° — 3-21 418 2-23 300 2-90 3-33 2-85 2-48 <3 00) (2-71) 103 X a 10* X b — 2-6 3-2 7-8 2-7 6-9 3-2 9-5 3-9 (11.2) (40) * Minimum; f Maximum. stretches are covered with fresh ice, and the ice pressure increases the resistance until a maximum resistance is reached at the end of the winter when the ice-cover is strongest and most solid. The annual variation of the wind effect c is more complex. Sverdrup was, however, able to show that it was in full agreement with the turbulent state of the air movement over the ice. In the Weddell Sea also the seasonal changes in a and b are completely analogous. The ice resistance shows, in general, an increase during winter and spring, but the changes from month to month are more pronounced and irregular because of the stronger changes in ice conditions of this broken cover. The coefficient of the wind effect h follows a regular course with the lowest values around the middle of winter and with an almost steady increase towards the end of winter. General Theory of Ocean Currents in a Homogeneous Sea 44 1 This also was shown as at least partly dependent on the turbulent state of the air above the ice. The ice drift thus to a large extent follows regular laws; it is dependent on three forces : the effect of the wind on the ice, the frictional resistance between different ice masses and the dei!ecting force of the Earth rotation. The much greater wind factor over the Weddell Sea than over the open ocean (see p. 449) is due to the fact that the ex- posed surface of the ice is more favourable to the action of the wind than that of the freely moving open sea. Over the Siberian Shelf, on the other hand, the wind factor ob- served was smaller than over the Weddell Sea; this may be due to the thickness and compactness of the Arctic ice cover which must offer a much greater resistance to movement than the ice of the Weddell Sea. 6. Inertia Currents In the preceding sections ocean currents in a homogeneous sea have everywhere been considered as stationary phenomena. Observations show that in most cases this assumption corresponds more or less closely with actual conditions. However, it can hardly be assumed that the forces involved will always be in equilibrium. Any disturbance of the equilibrium must, however, alter the state of motion of the water masses and in this the inertia of the water will play a major role. It is only in more recent times that one has started to draw attention to such phenomena, {a) Inertia Currents as Disturbances of a Steady Current A water mass moving frictionless in a horizontal direction under the action of a gradient force will, speaking completely in general, be subject to the equations of motion (X.16). If the .v-axis is taken in the direction of the pressure gradient (dpjdy = 0), and this pressure gradient corresponds to a steady current (geostrophic current), then 1 cp Fo = ^ V- and U^ = 0 fp ex and one obtains (disregarding frictional forces) 'i/=^^^-^»^ '"^ it = -^"' A periodic solution for an observer moving with current is u = t'o sin// -f Uo cos ft, V = Vq cos ft -f "o sin/r + Fq, or u = Co sm {ft -f ip), V = Co cos (ft + 0) + Vo, where ^0 = Vi^l + '") ar''-component precedes that in the .v-component by one-quarter of a period. These are the characteristic features of a pure inertia movement. It is superimposed on the uni- form gradient current and thus gives an oscillating flow, the period of which depends on the Coriolis force. This period is identical with the period of one revolution around the circle of inertia; numerical values for it are given in Table 1 12a (see p. 316) for differ- ent latitudes. Inertia oscillations are not associated with any large transverse displace- ments of the water masses, since the disturbance velocity c = V — Vq usually remains small. The magnitude of these can be taken from Table 2 for different latitudes and velocities. In the open ocean these transverse displacements are usually of little importance but they are still characteristic phenomena which are quite noticeable in current measurements. If pressure forces are present in a homogeneous sea due to a slope in the sea surface {dijdx = 4; dijdy = iy) the equation of motion (XIII.3) will apply. A steady motion (geostrophic current) is associated with a corresponding slope of the sea surface given by /j. and iy so that - / - / ^x=- V and 'V = - ^ ^• If, further at the time / = 0, there are current components Mq and i\ and slopes ij, 0 and iy o which do not correspond to the condition for a steady state, then the above equations have the following general solution (Fr. Defant, 1940): w = t/ + ("o - U) cos ft + [vo - (glf)ix,o] sin//, V = V-\-{vo- V) cos ft - [uq - (g//)/x_o] sin//, ix = ix + ['x,o — 'x] cos ft — [iyo — iy] sin ft, iy = h + [iy 0 — Iy] COS ft + [/^ „ " I x] siu//. (XIII.63) This set of equations shows that for a completely free initial state, both the current field and the sea surface will perform inertia oscillations around their equilibrium position which, however, will not correspond in all points to the conditions for pure inertia waves. In the current field the amplitudes of the corresponding velocity components will be equal only when the sea surface slope corresponds initially to the steady state. But according to the second pair of equations the sea surface does not General Theory of Ocean Currents in a Homogeneous Sea 443 perform any inertia oscillations at all but will rather remain from the beginning in the stationary position. Thus in the general case the amplitudes of corresponding terms are no longer equal and the motion is then elliptical instead of circular. However, the amplitudes of corresponding terms in the sea surface oscillations are always equal and these are therefore always pure inertia movements. It has been found that currents flowing into a wide area uninfluenced by coastal eff"ects usually follow a wave-form course rather than a straight course. A current with an oscillatory streamline seems to be a more stable type of motion than one with a linear course. Once a bulge is formed in any direction, the centrifugal force draws the water further and further out and the bulge produced by such disturbances will grow steadily; consequently, progressive waves and vortices will be formed in which the current will oscillate about a mean direction. In dealing with problems concerning these oscillating currents it is of course necessary to take the Coriolis force into account (Exner, 1919). It surpasses the scope of this section to penetrate more deeply into the dynamics of progressive waves of this type in an infinitely extended medium ; it rather belongs to and will also be discussed when dealing with the theory of progressive tidal waves (Vol. II) ; for an account of progressive waves with inertia period see Fr. Defant (1940) and Ekman (1941). {b) Inertia Movements Associated with Drift and Gradient Currents In the formation of steady drift and gradient currents the state of motion changes from the first motionless initial equilibrium state into a second state in which there is an equilibrium between all the forces acting. It can be expected that this transfer will give rise to inertia oscillations which will gradually be damped by friction until the new stationary equilibrium state is reached. Ekman (1905) examined in some detail the case of a suddenly starting wind over a deep, extended ocean. A comprehensive treatment of all questions arising has been given by Fjeldstad (1930). The equations of motion (X.16) which stand in question can be combined introducing u -f- iv = w (/ = \/—\)'m order to obtain a single equation dw T) 8^w J, + '>• = I J?- (^"'-^^^ The boundary conditions to be satisfied are for t =Q: vv = 0 and / , dw ^ ^ for all t: If the wind arises suddenly at a time / = 0 with a tangential pressure Tin the direction of the positive j-axis, then the velocity components u and v are given by IttT f^ sin Itt^ and V = pDf Jo V^ "^ \ ^D^ IttT f"^ cos 2tt^ pDf . exp -T7^,U^ (XIII.65) Vi ^^Pi-4^^i^^ 444 General Theory of Ocean Currents in a Homogeneous Sea D is the frictional depth and r = {ftjl-n) is the time expressed in units of 12 pendulum hours. The gradual formation of the drift current can be illustrated by plotting the time-variable velocity vector at different depths in the form of the hodograph curves given by Ekman (Fig. 191). For a suddenly starting wind this curve at the surface will have the form of a damped circular oscillation with a period of 12 pendulum hours which is superimposed on the final stationary state of motion. In the deeper layers the oscillation will at first grow somewhat and then regularly decrease again. The velocity ^=0 Z = 05D z^D Z-- 2D 1^ > Fig. 191. Hodograph curves for different depths of the pure drift current which develops due to a wind beginning suddenly (ocean depth unlimited). components can also be plotted separately along the time axis thereby obtaining the curves shown in Fig. 192. Each component shows a damped oscillation around a stationary final state and both together show very clearly the characteristics of inertia oscillations. At different depths the oscillations have exactly the same phase but with a decreasing amplitude. If the water depth is less than the frictional depth the close distance from the bottom becomes apparent in the curves, but the oscillation is strongly damped only in the immediate vicinity of the bottom; for hjD = 0-25 the current approaches almost aperiodic the steady state. Solutions can also be found for the case in which the effect of the wind is not applied suddenly but only gradually, and also for the case where the wind maintaining a wind drift current either suddenly or gradually ceases. For more detail see Hidaka (1933), Nomitsu (1933), Fr. Defant (1940). The sudden formation of a sea surface slope in a similar way as in the case of a drift current must also give rise for a gradient current to inertia oscillations. Ekman has given the theoretical basis also for this case and has pointed out that for an ocean of greater depth, due to the small frictional effect in the geostrophic current, these inertia oscillations will die away very slowly so that a longer duration of these must General Theory of Ocean Currents in a Homegeneous Sea 445 be assumed. If the pressure gradient, due to a suddenly imposed sea surface slope, acts along the positive j-axis there will be an extra term +/(/ (to be added on the right- hand side) in the equation of motion (XIII. 64), where U is the velocity of the steady gradient current (geostrophic current) corresponding to the sea surface slope. This equation must be solved assuming the boundary conditions that for z = 0 : dwjdz = 0 and for z = h\w = 0 and for r = 0 : u- = 0 and for r = oo : iv = U (stationary state) ; 0-75 0-50 0-25 0 00 0-75 050 0-25 000 025 ooo -0-25 0-25 000 -025 Surface N and E components inunits TDlWu 1 1 1 • i \ ^ ' / *■ ,^ /'"/ /' } i 1 N^ n'\ .>H y ^'^ ^<-^/ -■^ -X ^ -<:x 1/ 1 /' '^r\ ' ' 1 / l\ \ ''V^\ 1 ^~~- ' / \ \ / / "^ V !'- y f*^ ^ ~^\. h/n =li 1 /E n\i Vi / " ->C ^ "- Z--0 ; / -' T \ \ 1 ! ^ '^ s / f ^ ^ — ^ I 1 \ /. -^ N^\ \^__ rJ -'-'-- k.i--l-— ^-- E. _^ ^ 1 : ' 1 1 — 1 — 1 ..1 1 -^ 1 ; ' \ ■ ' 0 4 8 0 4 Pendulum hours Fig. 192. Upper pair of curves: drift current of an ocean of infinite depth for r = 0 (surface). Lower pair of curves: drift current for hID = 1| and in fact for z = 0 (surface), z//z = 0-3 and zjh = 0-6 (north and east components always in units TD/nfj. according to Fr. Defant). the velocity components of the steady gradient current are denoted by Ust and Vst- Introducing again u + iv = w, then the equations of motion reduce to the single relation For stationary conditions (Bwldt = 0, equation XIII. 30) the solution is given by equation (XIII. 31). Under non-stationary conditions a solution is obtained most easily by assuming »*• = n',< - H's^_. "^ fl -.^ '\ • -'-'' ^ , Z//>=09 ;>^- ^^ "^^"^•9^ h-f- i~~ T'" '■ 1 ~- --K" .__,- T I — ' 8 0 4? Pendulum, hr 8 0 Fig. 193. Gradient current in the ocean for an ocean depth hlD = -j, and in fact for z = 0 (surface), z/A = 0-5 and 0-9. (Values for the north and east components for «/C/ and vllJ, according to Fr. Defant). h\D = f . It can be seen that due to small damping the amplitude of the inertia oscillation is still quite large in the mid-depth. Calculations can also be made for the case of a gradually developing sea surface slope; the amplitude of the inertia oscilla- tion produced depends on the rate at which this slope develops but the character of the oscillation is still kept. (c) Inertia Currents in Ocean Currents The preceding discussion leads to the expectation that inertia currents will be of frequent occurrence in ocean currents, but a considerable time passed before their existence was actually proved. This was due to the circumstance that in order to prove the presence of cum sole, turning current variations of this type corresponding to their period, current measurements from an anchored ship over an interval of several days were needed. Measurements of this type are only seldom made and are associated with considerable difficulties which have been overcome only in recent times. The first current measurements in which the presence of inertia currents was suspected, was the long series of measurements made by Helland-Hansen and Ekman (1931) in the trade wind region of the Eastern North Atlantic. At the anchor station with the longest observational period (141 h, 30-2° N., 14-0° W.) there were, besides oscillations with tidal periods and also others with inertial oscillation, periods with 23-844 mean General Theory of Ocean Currents in a Homogeneous Sea 447 hours. This period is only 13 min shorter than the diurnal moon period K^ of 23-94 h which presumably was also present. This difference is, however, sufficiently large to decide out of six wave trains which of the waves is present. Figure 194 shows the diurnal regular oscillation after elimination of the semi-diurnal tide. While during the first three days there was a regular damping of the waves, at the end of the series there 0 12 0 12 0 12 0 12 0 12 0 12 9-2tt 1930 torn II-2IL l2-5nL 135m: wm Fig. 194. North Atlantic Ocean: anchor station of the "Armauer Hansen" 30° 13' N., 13° 57' W. Current measurements in 5 m depth after elimination of the semi-diurnal tide. Full line, north component; dashed line, east component; velocity scale in mm/sec. At the upper rim moon hours. The distance between two vertical lines is very nearly 6 pendulum hours (according to Helland-Hansen and Ekman). appeared to be a phase shift in the meridional component due to a new disturbance ; the oscillations then lose rapidly in regularity. Harmonic analysis for the first three and then for the following three days gave (cm/sec, / in pendulum hours) : cos (27r/12. t - 112^) , N = 1-58 cos (27r/12. t - 102") 115^) ^"^E N = l E = 1-51 sin (27r/12. t 1-29 sin (277/12. / - 97°) These oscillations are pictured by the full and dotted sine curves in Fig. 194. The good agreement led Helland-Hansen and Ekman to interpret these waves as inertia movements. The phase difference between the two components was 12 min more for the first days and for the second three days 20 min less than the theoretical required value of 6 h. The average ratio of the amplitudes was 1-23 as compared with a theoretical value of 1 . The oscillations were thus of the elliptic type with a ratio of 5 : 4. Considering that besides the inertia oscillations presumably the diurnal tide was also present, the results obtained are very satisfactory. An unambiguous proof of the occurrence of inertia oscillations was provided by the current measurements organized by H. Pettersson in the Baltic. As an adjacent sea with- out any significant tides this is particularly suitable for such an investigation. Gustaf- SON and Otterstedt (1932) and Gustafson and Kullenberg (1933, 1936) have made a detailed analysis of the suitable current measurements in the Baltic; in many 448 General Theory of Ocean Currents in a Homogeneous Sea cases there was no doubt of the presence of pure inertia movements. The best example is that contained in the measurements of 17-24 August. The recordings were made between Gotland and the mainland (57-8° N. 17-8° E., depth 100 m) at a depth of 14 m over a period of 162 h. The structure of the sea showed a well-developed density transition layer (thermocline) at 25 m and an almost homogeneous top layer. The variations in direction and strength of the current can be given in the form of a pro- gressive vector diagram which shows the track of a single small water element. On the current directed towards the NNW there is superimposed an oscillation rotating to the right, at first increasing and then decreasing (Fig. 195). The changes Fig. 195. Inertia oscillations in the Baltic in hodograph representation (according to Gustafson and Kullenberg). with time in the two velocity components are shown in Fig. 196. This diagram is particularly reminiscent of the theoretically derived oscillation due to a suddenly starting wind or to a suddenly imposed pressure gradient (Figs. 192, 193). If the first waves of the excitation period are omitted the period of the oscillations is 14-0 h as compared with 14-14 h for the inertia oscillation. The phase difference is almost exactly a quarter of a period and the amplitudes are very nearly equal. The meteorological observations taken at the same time do not permit any deduc- tions about the origin of this inertia wave. The question concerning the horizontal General Theory of Ocean Currents in a Homogeneous Sea 449 extent of this type of inertia oscillation was examined in later current measurements in the Baltic. Recordings from four anchored oceanographic research vessels between the Latvian coast and Oland (along the 56° 20' parallel) in the Baltic all, except for the vessel next to the Latvian coast, showed regular inertia oscillations at 15 m depth (density transition layer) with amplitudes of up to 20 cm/sec. The inertia oscillations (period 14-42 h = i pendulum day) had almost the same phase value but decreased very rapidly towards the coast. The water masses along the parallel investigated thus took part as a whole in the inertia oscillations (Kullenberg and Hela, 1942). Fig. 196. Velocity components of the currents pictured in Fig. 195 (according to Ekman). Specific inertia oscillations were found at the "Altair" anchor station in the area of the Gulf Stream north of the Azores (44-6° N. 34-0° W., 16-20 June 1938). Analysis of current measurements made at this station down to great depths (Defant, 1940 b) showed that besides the semi-diurnal tide there was also a 17 h oscillation; this had a large amplitude at all depths but the phases changed with depth. These phase changes which are related to the oceanographic structure at this station indicate that these inertia oscillations were coupled with internal waves which are the expression of a whole system of inertia oscillations of the surrounding water masses (see Vol. II). The com- bination of the 1 7 h wave with the semi-diurnal tide gives rise to beat phenomena with a period of 14-3 h and a beat interval of 1-86 days. This shows in a typical way the current values at all depths so that there can be no doubt that besides the tides, inertia oscillations were present here. Harmonic analysis gave besides the tidal wave also the value for the 17 h wave presented in Table 130. Division into different layers follows from the similarity in the phase which between 15 and 30 m and between 500 and 800 m shows an abrupt change of about half a period. The 1 7 h wave shows Table 130. Inertia oscillations at the ''Altair' station (16-20 Jime 1938; 44-6= N., 34-0° W.). Period 17 h Depth of layer (m) Current 1 Ratio of ampl. N.:E. Phase N. + 4-25 /; A^-component E-com ponent Difference A-mpl. Phase 1 Ampl. (cm, sec) (h) (cm sec) Phase (h) 5-10 30-100 300-500 800 8-75 14-45 100 7-33 50 9-0 80 20 8-25 1 0-0 50 80 1-85 12-27 14-85 80 106 1-00 1-00 100 1-70 11-58 13-25 6-25 +015 +0-69 + 1-60 + 1-75 2G 450 General Theory of Ocean Currents in a Homegeneous Sea all the characteristics of inertia oscillations. At = 44° 33' the theoretical period is 17-1 h. The amplitudes of the components are almost identical and also the require- ment that the ^-component should follow the TV-component by a quarter of a period (4-24 h) is fully justified. The deviations in the deeper layers must be a consequence of internal waves. Figure 197 shows the course of the N- and ^-components for the layer ,N E *zo -no *io 0 0-10 -10 -20 -20-30 17.2. M.G. Z. 6 12 laa. ia 12 18 io. 0 6 2asi 18 0 6 12 Fig. 197. North and east component of the current at the "Altair" anchor station according to the values of the harmonic analysis for the depth interval 5-15 m (basic current + 17 h period + 12-3 h period). 18 1 1 ■1 1 1 1 I 1 I 1 ■ -T 1 I i 1 1 1 1 1 1 1 1 1 1 I 1 II 1 ; '"\ /"■ -^ E , 'V \ / \ / \ / 1 1 / ^. f --^ .-vC r ^ / \ / / \ \ / / / — ^.,^ rt-~^ / '\ \ 1 1 ^x/^^- y sV/ / ■^ V^ ' ,/ 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 ,1 1 I 1 1 II 1 i 1 1 between 5 and 15 m as given by the values obtained by harmonic analysis. The beats stand out clearly, as does the retardation of the £"-component behind the TV-component characteristic of inertia oscillations. The inertia oscillations at the "Altair" anchor station are of considerable interest in so far as they show that the entire current system, together with the oceanic structure of the surrounding waters, seems to take part in these oscillations following the rhythm of the inertia period. These oscillations which may be initiated by any external disturbance impulses stand out particularly well in stratified waters, since coupled with these oscillations of the flow are corresponding oscillations of the density transi- tion layer and of the system of isosteres which are thus reflected in all layers (see Vol II). Chapter XIV Water Bodies and Stationary Current Conditions at Boundary Surfaces 1. Water Bodies and the Boundary Surface Between Them The theory of ocean currents in a homogeneous sea gives results which allow in many cases its application to actual conditions, although the sea itself is far from being homo- geneous. In changing from a homogeneous to a stratified ocean it is necessary to consider two homogeneous water masses (water bodies) situated side by side and separated by a discontinuity surface (boundary surface). On passing through this, changes in physical and chemical properties occur and also in the state of motion of water masses. This is, of course, also only a schematic model, since in reality the indi- vidual water bodies are not quite homogeneous and the transition from one to the other is seldom abrupt. Usually in Nature there is a rapid "transition layer" between the more or less homogeneous water bodies inside which a steady, rapid change of the properties occurs, while passing through it. The genesis of boundary surfaces of this type is due to the circumstance that in certain oceanic regions specific water types are continuously formed and carried away by the ocean currents together with their characteristic properties. In this way two different water bodies are brought into close contact at singularities in the current field and a boundary surface between them is formed at convergence lines. The prin- cipal changes in the horizontal distribution of a property (such as temperature or salinity and others) occur always in connection with so-called deformation fields of the motion (Bjerknes and co-workers, 1933). The most simple case of a horizontal deformation field is the current field at a neutral point (see p. 365, Fig. 155 a) with hyperbolic stream lines in each of the sectors formed by intersection of the two stream lines in the neutral point (Fig. 198). These straight lines are the principal axes of defor- mation of the field; one of them is an axis of dilatation and the other at right angles to it is an axis of contraction. This deformation field when superimposed on the field of one of the water properties will have a marked effect on the latter. The two full lines in Fig. 198 represent two isolines of a property, such as for example, the temperature. The current field will produce displacements in the position of these lines: all isolines which initially are parallel to the axis of contraction will move away from it and isolines parallel to the dilatation axis will move towards it. It can also be shown that two isolines through the current will always tend towards a direction parallel to the dilatation axis, so that they will first move away from each other to a maximum distance, and then after reaching a certain angle to the dilatation axis will move to- wards it again. In the case of a temperature distribution the effect of the deformation 451 452 Water Bodies and Stationary Current Conditions at Boundary Surfaces field is thus a concentration of the isotherms parallel to the dilatation axis and the horizontal temperature gradient will increase very strongly (according to theory indefinitely). This, therefore, leads to the formation of discontinuity surfaces the inter- sections of which with the sea surface show as discontinuity lines or fronts. Fig. 198. Deformation field and the change of a field of a characteristic water property. a — a, shrinking axis; b — b, axis of dilatation; 1 — 2, isolines of the property in the begin- ning; r — 2', isolines of the property at the end of deformation. The formation of strong horizontal gradients in the boundary regions of water bodies actually occurs most often in association with stationary oceanic deformation fields. However, other circumstances are involved in their maintenance. These are coupled with the effect of the deformation field and may lead to stationary fronts which are particularly characteristic for the horizontal distribution of the oceanographic factors. For an initially meridional temperature gradient and a steady meridional ocean current v the conditions will develop along the following lines (see Pt. I, p. Ill): the temperature &■ at a fixed point will change according to the relation (positive >-axis directed polewards) d^ 1 d^ dt dv If V is directed towards the pole, the temperature at a fixed point will increase since dd'jdy is negative (temperature increase by advection), that is, the isotherms will be displaced towards the pole provided that Q is small. However, due finally to the increase of temperature the first term on the right-hand side will also be increased and as a result all the factors aff'ecting the temperature will maintain an equilibrium state. Although the ocean current is directed towards the pole the temperature distribution will remain stationary. Similar reasoning will also apply to the hori- zontal distribution of other oceanographic factors. Besides stationary fronts of this type there are also frontal formations due to aperiodic occurring processes. However, due to the lack of synoptic observations the course of these usually can- not be traced. An interesting case has been given in Pt. I, p. 182 in the discussion Water Bodies and Stationary Current Conditions at Boundary Surfaces 453 on mixing processes in the transitional area between the North Sea and the BaUic. These are real, progressing hydrographic fronts that are associated with the inter- change of water between the two seas. The parts of the ocean where more or less stationary fronts are found are actually closely connected with the occurrence of quasi-stationary deformation fields with an axis of dilatation in the current system of the oceans deviating as little as possible from the east-west direction. The position of these can be found directly from a chart of ocean currents. In the Northern Hemisphere the most important are : (1) The North At/antic Polar Front which is present with its main part to the south of Newfoundland and there forms the boundary between the Gulf Stream water and the Arctic water of the Labrador Current; its continuation separates the cold low- saline water of the East — and in part also of the West — Greenland Current from the Atlantic water masses. Other parts lie south of Spitzbergen and in the Barents Sea. (2) The North Pacific Polar Front with its main part between the Kuroshio and the Oyashio which can be traced to about the middle of the ocean. These fronts are a consequence of quasi-stationary deformation fields in the current system in this part of the ocean. (3) This is also the case, though less clearly, in the Southern Hemisphere Polar Front which runs right around the Earth. It lies between the West Wind Drift and the Antarctic Current. In the parts where it is particularly well developed (for instance, south of South America and between the Falkland Islands and South Georgia) the connection with the local deformation field is clearly shown. 2. Stable Discontinuity Surfaces If two motionless water bodies are present together in the ocean for a stable equili- brium, the heavier water type must lie underneath of the lighter and the discontinuity surface between them must coincide with a level surface. Two water bodies at rest, situated side by side, will never be in equilibrium, even if each water body by itself has a stable vertical stratification. Since, due to their different densities, the pressure in each water mass will increase with depth at diff"erent rates, pressure differences are created; the resultant water movements will overturn the water bodies and they will only cease when the water bodies are again situated one above the other, separated by a horizontal boundary surface. However, two water bodies side by side can be in stable equilibrium // they are in motion. The form and position of the resulting dis- continuity surface was first given by Margules (1906) following up an investigation by Helmholtz (1888); a more general representation was given later by J. Bjerknes (1921) and later an application to the analogous conditions ia oceanic water bodies has been given by Defant (1929 b). A stationary state of the boundary surface is possible only for a certain definite state of motion in the two water bodies; thereby the boundary surfaces will lie at an angle to the level surfaces, so that the denser water always spreads out in a wedge- form underneath the lighter water. It will be a discontinuity surface for density (temperature, salinity or both) but not for pressure, since otherwise movements would immediately start directed towards the boundary surface. This, however, would interfere with the condition of stationary state. On the contrary, the boundary surface will be a discontinuity surface for the pressure gradient. According to the 454 Water Bodies and Stationary Current Conditions at Boundary Surfaces Hadamard classification (1903)t it is thus a discontinuity surface of zero order for the density and of the first order for the pressure. The horizontal movements in each water body must thus be parallel to the boundary surface since otherwise the surface could not remain at rest. There are kinematic and dynamic boundary conditions that must be satisfied at the discontinuity surface (see p. 324). The kinematic condition (equation X.29) re- quires that (ill — iio) cos (nx) + (vi — V2) cos (ny) + (m'i — u'a) cos (nz) = 0, (XIV. 1) where /; is the direction of the normal to the boundary surface ; u^, v^, w^ are the velocity components of the lighter and U2, v^, Wo are those for the heavier water type. The dynamic condition (equation X.29) requires that the pressure should be the same on both sides of the boundary surface (pressure equal counter pressure) P^-P2== 0. (XIV.2) If w, V, vv are the total acceleration components and X, Y, Z are the components of the forces, the equation of motion for the lighter water body 1 can then be written in the form: dPi = Pi [(A^i - it,) ^x + ( n - i\) dy + (Zi - vi-i)] dz. (XIV.3) An analogous equation will apply for the heavier water body 2. The equations dpi = 0 and dp2 — 0 will then give the equations for the isobaric surfaces according to the motion in each water body while the dynamic condition (XIV.2) will give the equation of the boundary surface [(Pi ^1 - P2 ^2) — (Pi wi — p2 W2)] dx + [(pi Ti — P2 Y^ — (pi Vi - P2V2)] dy + [(Pi Zi - P2 Z2) - (pi vvi - P2 vva)] dz = 0. (XIV.5) In the most general form these are the equations for the slope of the isobaric surfaces in each of the water bodies and for the inclination of the boundary surface. If the water bodies, each in itself, are both homogeneous (pi and pn = const.), the motion is non-accelerated {ii — v — w = Q) and is directed straight along the y-axis (ui = 112 — 0 and \\\ = H'2 = 0), then there will be a static equilibrium in each water body and ^i=/''i, -^1 = ^ and X2=fv2, Zg = g. Further, if the slope of the isobaric surfaces in the (.vz)-plane is denoted by dz\dx == tan ^ and that of the boundary surface by dz\dy = tan y, then the above equations will give f f tan ^^= --^vy; tan /Sg = - - V2, (XIV. 6) t According to the classification of such surfaces introduced by Hadamard (1903), a discontinuity surface at which the velocity and the density (temperature and salinity) change abruptly by a finite amount from one to the other side, is termed a discontinuity surface of zero order. It is defined to be of Ihe first order when the characteristic properties of the water bodies at the surface change continuously but their derivatives normal to the surface are subject to abrupt changes. Water Bodies and Stationary Current Conditions at Boundary Surfaces 455 and tan y f P2V2 — PiVi (X1V.7) g P2— Pi In each water body there will be a gradient current (geostrophic current). The angles /Si and i3o will determine the slope of the planar isobaric surfaces and that of the physical sea level. The slope of the boundary surface is of quite a different order of magnitude. Taking 0. The boundary surface rises towards the centre, in fact more rapidly near the vortex axis and less further out; tan ^, on the other hand, is negative in both layers, that is, the pressure surfaces and the physical sea level rise outwards, more so in the upper than in the lower layer. This is the case of a cyclonic vortex with the upper layer rotating more rapidly. Due to the rotational effect the heavier water accumulates around the axis of rotation while the lighter top layer is forced to the outside. In the central area there is a depression in the physical sea level and the isobaric surfaces. Case b: Ac > 0, for a cyclonic rotation Cg > Ci: tan y < 0. tan /S is negative in both layers and the boundary surface, the pressure surfaces and the physical sea level rise towards the outside; cyclonic vortex with the lower layer rotating more rapidly. The lighter water masses accumulate around the vortex axis and there, as in the previous case, the physical sea level and the pressure surfaces show a depession. In these cyclonic cases the sum of Coriolis force and the centrifugal force act towards the outside and a larger gradient force is required to balance this combined action. The boundary surface slope must therefore be greater than for water bodies arranged in strips. Water Bodies and Stationary Current Conditions at Boundary Surfaces 467 Case c: Ac > 0, for anticyclonic rotation |ci| > [ca]. As long as the term in brackets in (XIV. 10) remains positive, which is always true except in extreme cases, then tan y < 0 and the boundary surface rises towards the outside. Tan ^ is positive in both layers and the slope of the pressure surfaces is less in the heavier water body than in the lighter: anticyclonic vortex with the top layer rotating more rapidly and a central dome-like uplift of the pressure surfaces and of the physical sea level. The rotation gives rise to an accumulation of the lighter water masses around the rotational axis. Case d: Finally, it is possible in an anticyclonic rotation to have /Ic < 0 and then kal > kil- The slope of the boundary surface rises towards the centre since tany is positive (with the same restriction as in case c). The pressure surfaces also rise towards the centre but in this case more strongly in the heavier than in the lighter water layer : anticyclonic vortex with the lower layer rotating more rapidly and a central dome- like uplift of the sea level and the isobaric surfaces. Here the lower heavier water accumulates around the vortex axis. Since in the sea the current velocity almost always decreases with depth, cases a and c will predominate. In a cyclonic vortex the deep water is hfted close to the surface and if the vertical velocity gradient is sufficiently large the boundary layer may reach the surface. Then the vortex centre will be filled with deep water. In an anticyclonic vortex, on the other hand, there is an accumulation of the hghter upper water around the vortex axis that may extend downwards to con- siderable depth. The actual stratification in the sea seldom consists of only two layers; the same laws apply, however, also to a continuously stratified ocean (see Chap. XV). The boundary surface slope is then replaced by the slope of the isosteres and in place of sharp kinks there appears a steady curvature in the isobars. Also here, due to the low velocities and the large radia of curvature of the current trajectories, the centrifugal force is of little importance compared with the Coriolis force for an estimate of the mass field adjustment. Figure 21 1 shows dynamic sections through such cyclonic and anticyclonic Fig. 211. Mass and pressure distribution in rotationally symmetric layered vortices with a decreasing rotational velocity with depth, {a) Cyclonic; {b) anticyclonic rotation. circular vortices in a stratified ocean; in both cases it is assumed that the velocity of the current decreases with depth; for a two-layered ocean they correspond to the cases a and c of Fig. 210. Charts of ocean currents often show more or less extensive vortices in the top layers. They are found mostly in those areas where the wind field also indicates 468 Water Bodies and Stationary Current Conditions at Boundary Surfaces rotational (cyclonic or anticyclonic) motion. The anticyclonic winds around the sub- tropical high-pressure centres thus give rise in both hemispheres to anticyclonic large- scale vortices between the oceanic West Wind Drift and the Equatorial Currents. These are elongated corresponding to the shape of the high-pressure cells and take the form of a broad convergence zone. In the central parts of these anticyclonic vortices there is always a mass distribution corresponding to that in Fig. 211 b; that is, with an accumulation of lighter water in the central part of the convergence area. Condi- tions of this type are particularly well developed in the North Atlantic, where there is an accumulation of warmer water with a corresponding depression of the isosteres to 600-800 m ; the isobaric surfaces and the physical sea level show a corresponding uphft. Large-scale vortices with cyclonic sense of rotation are found in the intermediate region between the oceanic West Wind Drifts and the Polar Currents; that in the North Atlantic between the Polar and the Atlantic Current. Here the actual oceanic structure will be very nearly that pictured in Fig. 2\l a, which shows that the isosteres arch upwards. Such cases will be referred to again when discussing the current con- ditions in particular oceanic regions, A very typical case of a smaller-size cyclonic vortex was observed in the Gulf Stream just north of the Azores above the "Altair" cone during the International Gulf Stream Survey, 1938 (Defant, 1940 b). The centre of the vortex was found in upper layers a little south of the greatest submarine elevation; in deeper layers it appeared directly above the cone. All the vertical oceanographic sections show this vortical disturbance and its vertical structure. Figure 212 presents a somewhat smoothed 100 200 300 E 400 500 600 700 800 900 Fig. 212. Meridional density section through the cyclonic vortex above the "Altair' submarine volcano in the Atlantic Ocean (somewhat smoothed). Water Bodies and Stationary Current Conditions at Boundary Surfaces 469 density section. Although the axis of the vortex is somewhat inclined towards south, current measurements and the mass distribution suggest a subdivision of the total vortex into two parts or systems. (1) The upper system down to 100-150 m depth includes a discontinuity layer at about 25 m. The velocity of the basic current in the top layer is about 15 cm/sec and in the denser lower layer, however, 20 cm/sec. This is thus a strongly stratified cyclonic vortex with a speed of rotation increasing with depth. Under steady conditions the isosteres must therefore dip downwards in the lighter water masses which are con- centrated around the vortex axis. This is shown very clearly by the section given in Fig. 212. (2) The lower system extends through the layers below 150 m, where there is a normal increase in density with depth and a steady decrease in the velocity from about 20 cm/sec at 200 m to about 6 cm/sec at 800 m. This is therefore a weakly stratified cyclonic vortex with decreasing rotational velocity with depth. The required uplift of the isosteres (accumulation of lower denser water around the vortex axis) is again obvious from Fig. 212. Also quantitatively the observed slopes are in a good agreement with that required by theory (equation XIV. 10). Since <^ = 44° 33' N.andtherefore/= 1-023 x lO"* sec-^ equation (XIV. 10) gives for the upper system: a^ = 26-30, q = 15 cm/sec, ag = 26-65, Cg = 25 cm/sec ; the isosteres slope downwards towards the centre by 92 m in 60 km ; observed 70-90 m. For the lower system: ct^ = 26-8, Ci = 20 cm/sec, cto = 27.5, Cg = 6 cm/sec; the isosteres slope upwards towards the centre by 214 m in 100 km; observed 230-290 m. The cyclonic vortex performed pulsations, as was indicated by the observations made at the anchor stations. The period of these pulsations corresponded to the period of inertia oscillations (see p. 472). Sandstrom (1914, 1918), has carried out laboratory experiments to test the effects of cyclonic and anticyclonic air currents on stratified water masses underneath. Reference is made to these in this connection. 4. Up- and Down-gliding Surfaces: Pulsations of Stationary Vortices In systems of moving water bodies for a stationary position of the boundary surfaces there will be no vertical motions according to equation (XIV. 7). If the equili- brium conditions are not satisfied, accelerations will occur and as a consequence vertical motions are generated which will lead to changes in the position of the dis- continuity surfaces. If the slope angle of the boundary surface is denoted by e and differs from that for its stationary equilibrium y, then e will tend towards its equili- brium slope y. If the boundary surface is steeper inclined than in the equilibrium state (e > y), in order to reduce e the upper lighter water must spread out over the lower heavier water and the lower one will intrude underneath the lighter. Above the boundary surface there will be an up-gliding and below it a down-gliding (up-gliding surface). If, on the other hand, for e < y the reverse will apply. In the lighter water type there will be down-gliding and in the heavier up-gliding (down-gliding surface). The processes occurring at the boundary surface can be decisively influenced by the initiated vertical motions. Exner (1924) and J. Bjerknes (1924) have investigated the processes that may 470 Water Bodies and Stationary Current Conditions at Boundary Surfaces occur at arbitrarily inclined discontinuity surfaces. Taking horizontal accelerations into account but neglecting the very small vertical accelerations (w^ = W2 = 0) and if the boundary surface is parallel to the >'-axis having an inclination tan y then equation (XIV.5) gives the relations (^-positive upwards) : (Pi^i — P2W2) =f{pii\ — P2V2) — Sipi — P2) tan € and Pii\ — p.iV2 =f(piUi — p^Uo) (XIV. 11) Near the boundary surface the velocity in each of the water bodies will be tangential to it : Wi = Ml tan e and vt-g = Mo tan e, so that PiH'i — P2H'2 = (pith — P2W2) tan e. (XIV. 12) These equations form the basis of the dynamics of up- and down-ghding surfaces. If in the first of these equations e = y (stationary boundary surface condition), then P2W2 — P2W2 = 0 and from (XIV. 12) it follows that p^Wi = P2^2- On the other hand, according to the second part of the equation (XIV. 1 1) PlVl — P2«2 ^0. This implies that : tip- and down-gliding can also occur at stationary boundary surfaces if the currents are accelerated also in the direction parallel to the gliding plane. If the mutual adjustment between current velocities and stable position of the boundary layer gets disturbed by changes in the velocities, then up- and down-gliding motions must occur along the boundary surface in order to preserve a stationary state of its inclination. Thus when (0 Pi'"i — />2i'2 < 0: piMi — P2W2 > 0 and p^w\ — P2**'2 > 0 and when (2) pjt'i — /Da^a < 0: piu^ — p^Uo < 0 and p^w^ — p<^<2, < 0. In the first case where there is a stronger acceleration in the lower water mass along the positive j'-axis than in the upper, an up-gliding surface is to be expected. In the second case, however, where there is a stronger relative acceleration along the positive j-axis in the upper water mass, there will be a down-gliding surface. These two cases are illustrated in Fig. 213; they apply for the Northern Hemisphere. In the Southern Fig. 213. Stationary up-gliding (to the left) and down-gliding surfaces (to the right) (Northern Hemisphere). Water Bodies and Stationary Current Conditions at Boundary Surfaces 471 Hemisphere the arrow-directions indicating the velocities and the accelerations parallel to the boundary surface have to be reversed. So far the discussion applies only for infinitely extended boundary surfaces. If they intersect the sea surface (fronts) or the sea bottom the up- and down-gliding motions will give rise to horizontal water currents in its vicinity and consequently to changes in the position of the boundary surface. Cases of this type can be found at the oceanic polar fronts. Figure 2 14 shows the polar front between the East Greenland Current and the Atlantic water to the south of the Denmark Strait. The mass distribution requires larger velocities in the polar current towards the south and smaller ones in the Atlantic water as is found by observation. Polar front (a) s. Fig. 214. Oceanic vertical stratification and currents at the East Greenland oceanic polar front. Picture to the left: up-gliding of the polar water and down-gliding of the Atlantic water for an accelerated East Greenland Current : boundary surface progresses towards east. Picture to the right: down-gliding of the polar water and up-gliding of the Atlantic water for an accelerated Atlantic current: boundary surface progresses towards west. In general, there exists a stable equilibrium in the current system between the mass structure and the currents with a stable boundary surface position. If, however, an easterly wind piles up polar water ("Anstau") along the east coast of Greenland, or if other conditions in the North Polar Sea cause an increase in the strength of the East Greenland Current, then the water masses of the current will be accelerated towards the south and the boundary surface will become an up-gliding surface (Fig. 214 a). This up-gliding along the boundary surface in the lighter polar water mass must come to an end at the sea surface ; here it gives rise to a reduction in the inclination of the boundary surface, that is, the extent of the East Greenland Current at the surface will increase and will force the Atlantic water masses seaward. In the opposite case (Fig. 214 6) if the Atlantic water is accelerated towards the north, the boundary surface becomes a down-gliding surface. It thus becomes steeper and the extension of Atlantic water is increased. Pulsations in the basic currents will be associated with variations in the mass distribution. The large-scale aperiodic atmospheric disturbances of these regions must be accompanied by corresponding large changes in the oceanic structure and the ideas outlined above are of major importance in the coupling of these two phenomena. Similar conditions must apply for the much longer polar front in the Southern Hemisphere. Here the temperature is the decisive factor for the mass structure and the boundary surface between the West Wind Drift and the South Polar Current slopes downward towards the north (towards the equator). In order to secure stationary 472 Water Bodies and Stationary Current Conditions at Boundary Surfaces conditions, the West Wind Drift must have a greater velocity towards the east than the South Polar Current to the south of it, which is also directed east. Since here also disturbed meteorological conditions are frequent in this region, the varying influence of the action of the atmospheric flow will sometimes accelerate the oceanic West Wind Drift and sometimes the South Polar Current, and therefore the polar boundary surface will change from an up-gliding to a down-gliding surface and back again and there will be corresponding displacements of the polar front in meridional direction. These processes seem to continue nearly all the time and may be associated with the observed sinking process of large water quanta of sub-Antarctic waters. This process is most probably of a pulsatory character and is definitely the source of the sub- Antarctic intermediate water penetrating far to the north. Variations of the boundary surface can also arise in circular vortices if there are changes in the vertical current structure. If (see in Fig. 215) for example, the boundary Fig. 215. Pulsations of a circular vortex in cyclonic rotation. surface and the physical sea level lie in the position 1-1 under average conditions, then, if the velocity between the upper and lower water bodies increases, there will be greater accumulation of the lower water type around the axis of the vortex and the inclination of the boundary surface will increase (position 2-2). If, on the other hand, this diff"er- ence becomes less, then the accumulation of lower water will be dispersed and the inclination will decrease. Periodic variations in the mass structure will thus occur in the vortex; the boundary surface and the physical sea level will oscillate ^round a mean position and these oscillations will have the character of standing waves (see Vol. II). In the cyclonic vortex over the "Altair" submarine cone in the Gulf Stream north of the Azores (see p. 454) periodic variations of this type were present both in the oceanic structure and in the vertical current distribution. They were very well developed in the upper system and of a period corresponding to the inertia period (17 n). Since the periodic variations in the current amounted to as much as half of the velocity of the Water Bodies and Stationary Current Conditions at Boundary Surfaces 473 basic current, the changes in time of the distribution of the isosteres must have been quite considerable. Figure 216 shows these changes in the vertical current structure in the two layers of the upper system: 5-15 m and 30-100 m. In the lower part of the vortex the velocity is greatest between 2 and 3 h and at the same time least in the upper part. During this time-interval there is thus an increase with depth of the velocity of rotation. In the interval between 9 and 16 h conditions are reversed; at 10 h the 8 16 24 32 cm/sec Fig. 216. Changes in the vertical structure of the current of the upper system in the cyclonic vortex above the "Altair" submarine volcano. •« — , current in the layer between 5 and 15 m depth; <=, current in the layer between 30 and 300 m depth. top layer has the greatest velocity and there is thus at this time a decrease in the rotational velocity with depth. This feed-back of these oscillations of the current field on the mass distribution in the vortex must be extremely strong to give a complete reversal of the current structure. At 10 h there must be an increase of the up-lift of the isosteres and at 2 h an increased depression. These oscillations of the isosteric surface about nodal lines at a certain distance from the vortex centre have been demonstrated by observations of the anchor station. The isotherms and isohalines oscillate around a mean position with an inertia period of 17 h, so that the anchor station must be somewhat displaced towards the outer edge of the vortex, because the isosteres are always lowered at 10-5 h and always lifted at 2 h. The oscillations in a circular two-layered vortex can be accounted for theoretically (Defant, 1940 b) and an estimate can be made of the period of the free oscillations of such a system. If the effect of centrifugal force is neglected (it is always small) then the mean position of the boundary surface in such a vortex will correspond to the follow- ing relation (z positive upwards; centre of the vortex at .v = 0; horizontal extent of the vortex = 21): 474 Water Bodies and Stationary Current Conditions at Boundary Surfaces z = /?2 + S cos (77// )x, (XIV. 1 3) 5 f (p^ih— Pi«i) where o — g Pi — Pi (Margules boundary surface slope). If small periodic variations (disturbance values) are imposed on this equilibrium system in the currents Ui and u.^, then the boundary surface will oscillate about its steady-state position. As a consequence in the most simple case these oscillations will give rise to upward and downward movements in the central part of the vortex with a phase exactly opposite to that of the outer vortex portions. To the equation (XIV. 13) will thus be added an additional periodic term of the form Z — A COS -J- COS a J, (XIV.14) whereby C7„ is the frequency of the free ("Eigen") oscillation (period T = Irrjan', n = 1, 2, 3, ... , gives the number of node-points in the oscillating system). When corresponding boundary conditions are taken into account the equations of motion give an equation for the determination of the frequency a„ of the "Eigen" oscillations of the oscillating vortex as a function of the dimensions of the system. The following equation is obtained where //^ and Ju are the thicknesses of the two layers and 2/ is the total horizontal extent of the vortex. These "Eigen" frequencies depend in a characteristic way on the angular velocity of the Earth. If the Earth were not rotating (/= 0) then the period of the free oscillation would be given by 277 _ 2/ /// cTr ~ n \]\ In 11 llpilh + Pilh g(p2 - Pi) (XIV. 16) This is a period for an internal standing wave in a two-layered water mass of an extent / (see Vol. II). If for large dimensions of the oscillating system the period Tr for a non-rotating Earth is large, then the second term in the equation(XIV.l 5) will be so small as compared with/2 tjj^t jt can be neglected and the longest "Eigen" period of the system will be equal to the inertia period. Ti = ha pendulum day = — ^ . (XIV. 17) If the second expression accompanying/^ in the equation (XIV. 1 5) cannot be neglected, when (r^ > Ti), when (Tr < T,). then it is obtained with sufficient accuracy T= Ti [' ^ ' (SI however, T=Tr [' - m Water Bodies and Stationary Current Conditions at Boundary Surfaces 475 In most cases in the ocean T^ <^ Ti, so that the ''Eigen" period of such an oscillating system will always be close to the inertia period. In the vortex over the "Altair" cone 2/ = 120 km: the mean densities of the upper and lower layer pi and po are 1-0263 and 1-0283 and/- 1-023 x 10"* sec-\ then for hi = 30m and h^ = 1000 m it is found that T, = 17-1 h and the "Eigen" period of the system according to equation (XIV. 1 5) is r = 1 6-76 h. Thus the "Eigen" period of the vortex over the "Altair" cone approaches closely the period of an inertia oscillation, as was found by observation; inertia oscillations are merely the free oscillations of an enclosed sea the equihbrium state of which has been disturbed. They are probably set up by external causes especially by meteorological conditions (hke storms and similar phenomena). In this particular case a storm occurring just before the anchoring of the "Altair" seems to be the cause for the pulsation of the otherwise stationary vortex above the "Altair" submarine volcano. Chapter XV Ocean Currents in a Non-homogeneous Ocean 1. Introduction If all the external forces that may act on the sea are excluded, ocean currents can still be produced by internal forces. Differences in the mass structure will represent an internal system of forces that will act until the resultant mass displacements lead to the establishment of a mass distribution corresponding to that of a static equilibrium. It is customary to denote ocean currents generated by such internal forces as "con- vection currents" although they have nothing to do with oceanic convection pheno- mena. In order to avoid this unsuitable notation it seems to be advisable to call them "density currents", since they depend solely on the three-dimensional difference in the density field. Treatment of these density currents involves greater difficulties than that of drift and gradient currents, in particular, since the external forces (wind and atmospheric pressure) can be regarded as independent from the currents themselves, while the density currents and the density differences producing them influence each other. Furthermore, the density anomalies, being internal forces, are distributed three- dimensionally in space, while wind and atmospheric pressure at the sea surface act only in two dimensions. The beginnings of a theory of density currents goes back to Mohn (1885, 1887) whose work can without doubt be described as "the beginning of a new era in physical oceanography" (Helland-Hansen and Nansen, 1909, Vol. II. 2, p. 390). However, this theory, the aim of which was rather wide-spanned, was incapable of influencing the further development of theoretical oceanography, since it was running far ahead of the development of oceanography, which at that time made its progress mainly along geographical lines and because the defects in it were difficult to eliminate. It was soon forgotten (Thorade, 1925). The foundation for a firmly founded theory of density currents was provided by the application of the Bjerknes theorems of vortex formation and circulation acceleration to oceanographic problems. Thereby it was necessary to leave aside classical hydrodynamics, dealing only with homogeneous media, and to make use of physical hydrodynamics where the media had a full physical reality. Some of the results were later derived directly from the hydrodynamic equations of motion. These derivations are, in part, clearer and more comprehensible, and it therefore seems advisable to discuss the simpler problems first. 2. Relationships Between Current and Density Fields in a Horizontal plane. The law of Parallel Fields A general relationship between density and current fields can be derived quite simply (Defant, 1931). In general, the vertical component of the velocity, that is, 476 Ocean Currents in a Non-homogeneous Ocean All the vertical slope of the stream lines is so small that the current field can be regarded as horizontal. Under stationary conditions the stream lines follow the stream function i/rCxj') = Ci; the horizontal density distribution shall be given by p{x,}') = c^. The angle between the two sets of curves may be y. If the stream lines are at an angle a to the positive .Y-axis and correspondingly the isopycnals at an angle /3, then difj Idip Sp /dp tan a = — K-l^^ and tan j8 = — -- / c.v/ dy dxj dy From this it follows that dip dp dip dp ^ oj^^Z_^y^^ (xv.i) dijj dp dill dp dx dx dy dy If the stream hnes are parallel to the density lines (y = 0), then consequently dj^d_P_djPd_p^^ (XV 2) dx dy dy dx Disregarding for the moment the effects of friction (turbulence), and if there are no physical changes in the water masses due to external circumstances then, for stationary conditions dujdt = dv/dt = 0, the equations of motion (XIII. 1) will also apply for a non-homogeneous sea. Eliminating the pressure p and taking into account the con- tinuity equation and introducing a stream function (equation X.35), equation (XV.2) is obtained. In a non-homogeneous sea stationary conditions require that the stream lines and the isopycnals (isosteres) are parallel. This result is self-evident since otherwise these surfaces would be displaced and this would contradict the condition of a stationary state. The same also applies to isothermal and isohaline surfaces. On the other hand, the following equation can be derived from the equation of motion and the hydrostatic equation (Ertel, 1933) / dpu dpv\ d^p dp d^p dp ^' Y' -dl ~ P""^) = ~ W^ dx-^ ^^z dy- By means of the hydrostatic equation dp -dz=^^P this equation can also be written in the form ■' P dz \vj ^ \dx dy dy dx] If the total velocity V is at an angle x to the ^--axis so that u = V sin x and v = V cos x then If the isobars and isopycnals are parallel in a horizontal plane, then the expression in brackets, D, is zero. The mass field is therefore barotropic and dxjdz = 0, that is, the 478 Ocean Currents in a Non-homogeneous Ocean current does not turn with depth, or the current directions at all depths will lie in one and the same vertical plane. Since for frictionless motion the current follows the isobars and these coincide with the stream lines, D will be identical with equation pCV.2). Except at special disturbance locations (discontinuity surfaces, discontinuity layers and fronts) the stream Hnes therefore will also coincide with the isolines at all depths. If turbulent friction should also be taken into account, it is necessary to go back to the general equations of motion and elimination of p leads to the equation P^e.^a^s,^,^ (XV.4) dx cy By ex j oz^ For a simple potential flowzJ 0 = 0 and the condition of parallehsm of stream lines and density lines still applies. If, however, a vortical motion has to be dealt with, this parallelism will be lost. The angle at which they intersect will depend on the turbulence and on the water depth. It can be shown that now tan y = -^r— , where I, = dvjdx — duldy denotes the vertical vorticity component. If the co-ordinate system is placed in the direction of the average current, then f = 0. At the sea surface assuming a linear pressure gradient (Ap = 0) and a decrease of velocity with depth u = \a z^ (sea bottom z = 0) as well as a depth of water h, is obtained tan y = j— . fpir For fflp = 200 cnr/sec and/= 10-^ sec-i (at about 45° N.) tany=(^) if the depth of water H is measured in metres. For a large water depth y will be almost zero; if the water is shallow (shelf seas) it may reach values of 10-20°. Summarizing, it may be stated that for steady frictionless currents in a non-homogen- eous sea the isolines of the different oceanographic factors and the stream lines must coincide, but in the presence of strong turbulence especially in shallow seas this parallelism is lost. Attempts have very often been made in oceanography to deduce the current field from the distribution of the temperature and the salinity and other factors. In general, such deductions are permissible and the method gives results corresponding reasonably with reality, but deductions from isoline charts should not be taken as more than indications of the rough course of the currents. However, exactly at the point where the current field is of particular interest (near discontinuity surfaces and fronts) the method fails completely (Castens, 1931). These arguments are connected with the "law of parallel fields" (Helland-H.\nsen and Ekman, Ekman, 1923). Comparison of the distribution of the oceanographic factors at different depths shows the striking phenomenon that the isolines at any particular depth are parallel to each other, and moreover that they are parallel also Ocean Currents in a Non-homogeneous Ocean 479 with those in deeper layers. This agreement in the course of these lines also extends to the dynamic isobaths at any depth. It must therefore be concluded that the current vectors are also tangential to all these sets of curves and that there is complete equahty between all these hnes. This law allows deduction according to the Ekman theory of the direction of the deep current outside the upper and lower frictional depth which represents the layers in which the drift current and the bottom current are found. All modem cartographic representations of the horizontal distribution of these factors at different depths confirm the general validity of this law (see, for example, the ''Meteor''' Reports, Vol. VI, Atlas). The basic prerequisites for the vahdity of this law are the same as in the rules derived above for the relationships between the oceanographic factors and the current field in any horizontal plane. These are satisfied for the deep currents except in those areas where they are disturbed by discontinuity layers, or where due to mixing processes there caimot be any stationary spatial density distribution. 3. Horizontal Steady Currents in a Stratified Ocean The dependence of the vertical velocity distribution in a current on the stratification of the water masses in the pressure field is already shown by the behaviour of two adjacent water bodies. In steady state continuous changes in density require also a definite mutual adjustment between the mass and pressure field. If the flow is directed along the positive >'-axis, then for a steady frictionless motion Inserting the hydrostatic equation g = a(8pldz) (z counted positive downwards), elimination of p leads to the relation 8v 8 log a ? 2 log a p- = ^ — p^- - 7- ^^ • (XV.5) 8z 8z f 8x This states that for a given vertical and horizontal mass distribution there will always be a vertical velocity distribution given by (XV.5). Introducing the slope of the isobaric surfaces tan ^ = — {flg)v and that of the isosteric surfaces tan y = — {8pl8x)l(8pl8z) the equation takes the form dv 2 8 log a ^ = -^(tan y - tan j8) -^ . (XV.6) 8z J cz Since 8 log aj8z is always negative, the expression in parenthesis decides about increase or decrease in the velocity with depth. In other words, this increase or decrease in velocity depends on the difference in the slope of the two intersecting sets of surfaces or lines in a dynamic section. Figure 136c (page 331) shows the two possible cases (r is always positive); in that shown on the left-hand side the expression in brackets is always positive, and therefore 8vl8z < 0, or there will be a decrease in velocity with depth. In the case on the right-hand side 8vj8z > 0, and there will be an increase in velocity with depth. When y = ^ then 8vjcz = 0 which is the barotropic case with a constant velocity at all depths. These results can be expressed by the following rule : 480 Ocean Currents in a Non-homogeneous Ocean If the isosteres slope downwards {upwards) from left to right when facing downstream, then a steady current will show a decrease {increase) in velocity with depth {Northern Hemisphere). It can be seen that equation (XV.6) allows a determination only of the vertical velocity differences and it does not give the velocity itself and thus affords only relative velocity difference distributions in vertical direction. This state of affairs recurs in all similar cases and is a consequence of the indeterminate nature of the problem. Equation (XV.6) has been derived from the equations of motion alone; to determine the entire state of motion completely requires the continuity equation. Only then are the conditions uniquely defined. Equation (XV.5) can be written also in another form: dv da oz cz gda fdx This can be used for a step-wise calculation of the vertical velocity distribution from layer to layer (Defant, 1929 b). If at two stations separated by a distance L at a depth r = 0 the specific volumes are tto and a'o and at a depth z = h a-^ and a'^, the following formula can be used for a numerical determination of the velocity difference ^o ~ ^i gh (ai + a'i)ro — (tto + a'o)fi = j^ {a^ a'o + «i — a'l)- (XV.7) Table 133 contains the specific volumes at six depths down to 750 m for the stations 205 and 206 on the section through the Gulf Stream and the Labrador Current south of the Newfoundland Banks (Fig. 202). For 0 = 40° 10' and L = 59 km the equation (XV.7) gives the vertical velocity on the assumption of no motion at a depth of 750 m. Table 135. Calculation of the vertical velocity in the Gulf Stream south of the Newfoundland Banks Depth St. 205 St. 206 L= 59kni h (ag — a'o) + (aj — a'l) V (m) a a' a — a (m) (cm'sec) 0 0-97393 0-97449 56(xlO-^) 50 107(>;10-«) 64-7 50 363 414 51 75 102 59-7 125 312 363 51 125 112 52-5 250 217 278 61 200 101 39-3 450 119 159 40 300 72 20-3 750 0-96973 005 32 00 Werenskjold (1935, 1937) has developed a simple and practical method for the same objective. Neglecting in equation (XV.6) tan ^ in comparison with tan y = /, Ocean Currents in a Non-homogeneous Ocean 481 which is always permissible and integrating it between level ro(po.io) ^iid the level ^liPx.v^ gives i\ = g ./ idp. (XV.8) whereby p,„ is a mean density for the layer r^ — Zq. Denoting the tangents of the slope angles of the isopycnals or isosteres drawn in a dynamic section with intervals A p and Aa, respectively, by /, then equation (XV.8) can be transformed into the simple relation S A p _ _ s Aa t\ = nf p, nf a, ZJ. (XV.9) The summation has to be taken over all the isopycnals or isosteres which cut a given vertical line between levels Tq and z^ and n is the vertical exaggeration of the section. Values of 7 can be read directly from the section using a transparent scale (Fig. 217). If the isopycnals in a vertical section are plotted at intervals of 10~* and the isosteres at intervals of 5 X 10""^ and if the vertical scale of the section is 1 :2500 and the horizontal ;5 \AV ^ ;♦ v\v VA\ 73 \\v \\W n \\\ \V^ 11 \\\ \\W 10 \\ VA\ 9 X\ \\V 8 V \\ . \\\ ^\. \ ^\ N. \. \ 7 $^ 0\\ \^^ \ >. \ N. ^v 6 ^^\^s ^\\ ^\^^\ v^N^N 5 ^^ ^^\ ^ ^$^ 4 ^-^^ :£S^. 3 ^-^^ ^.^£S< ^^■^^---^.^"* -^ ^^/^^^^-^ ■^ ^ ' -^ ■"^ ^*^ 2 --^Zl^ ^^^T:^ — — __ """—*— ^^...^^^^ ■■ ; ____ ~~ — - _^ __ Fig. 217. Tangent scale for the determination of the inclination (according to Werenskjold). 21 482 Ocean Currents in a Non-homogeneous Ocean scale 1 : 500,000, then the vertical exaggeration n is 200 and one obtains for isopycnals and for isosteres 1-885 ^ , , , v^-vi= ^^ 2:y (cm/sec). 4. Ekman's Theory of Density Currents Including Friction Consideration of frictional effects in a stratified ocean is more difficult than in a homogeneous sea for two reasons. First, the mathematical difficulties increase considerably, and secondly, the depen- dence of the frictional coefficients on the stratification is very incompletely known. In a stratified ocean friction should be less than in a homogeneous sea and the intro- duction of a constant frictional coefficient, which must be made, does not fit so well under these conditions as in the case of homogeneous water. Nevertheless, the results obtained on this basis afford some insight into the effect of friction on the formation of density currents. Ekman (1905, 1906) has also dealt with this in his theory of ocean currents and has made important contributions to clarify this problem. A general solution, however, cannot be given. By means of some typical cases only can conclusions be reached, from which the effects of friction can be deduced by comparison with the frictionless cases. A simple case is that where the specific volume decreases uniformly with depth and the isobaric surfaces are thus inclined planes. If, as a consequence of this assumption, there is no pressure gradient at a particular depth d (horizontal isobaric surface), then taking - ~ / = -fV and - - / = + fV p dx ■' p dy ■' (U, V are the components of the geostrophic current) the equations of motion (XII 1.28) give Z)2 d^u Z)2 8^v o^ ^1 + ^ = ^ ^^^ o^ ITS + y=F and ^z tt^ - « = - ^. (XV.IO) Therein D is the frictional depth (equation XIII.26). For a co-ordinate system with the X-axis parallel to the isobaric surfaces (F = 0) and taking as before U = b (d — z) a. solution can be given for (XV.IO). The velocity profile can be calculated for different values oi djD (Fig. 218) from the very complicated equation obtained. The velocity is given in the diagram in units of f//5; they can also be considered as given in cm/sec if the total layer from the sea surface down to the layer of no motion d, of the dynamic section oriented in the direction of the gradient, contains in each 1 km layer a total of 10^cusin(/> solenoids (for 45° there are 51-6 solenoids). The difference from the velocity profiles presented in Figs. 173 and 174 for a homogeneous mass structure is considerable. Wherever the depth of no motion d may be, the motion there occurs nearly in a plane. The friction affects principally the direction of this plane. Table 1 36 gives the largest (amax) and the smallest (auxm) angle of deflection from the gradient Ocean Currents in a Non-homogeneous Ocean 483 X, cm/sec Fig. 218. Velocity profiles in density currents for shallow ocean depths (according to Ekman). The unit of the velocity scale is U:5. Table 134. Frictional influence on density currents in different depth of the ocean dD 0-25 0-50 1-25 2-50 X "max 26" 26° 67" 62" 93" 82^ 91" 86" 90" 90" "surface ^^ 37 74 86 94 100 u direction and in addition the velocity of the surface current as a percentage of the geostrophic current U. The vertical velocity decrease is at first very slow and then becomes almost linear. By this it is shown that the law of parallel fields also applies to a close approximation when frictional effects are present. Simple mass distributions such as these rarely occur in nature. In addition Ekman has also investigated cases in which the eflFect of a homogeneous solenoid field is superim- posed on a gradient current. A lighter stratified top layer spreads out over a homogeneous deep water. The lighter water body may, for instance, be coastal water lying in a wedge- form off a long coast and can be regarded as a mixed layer of fresh water from the land and of deep water. External forces are not taken into account ; at the boundary surface between the top and the deep layer the water movement of the upper density current exerts a shearing force on the deep water which gives rise to an "internal drift current". A closer examination of the case of a boundary layer at a depth d, parallel to a straight coast between a homogeneous upper and lower layer, gives the velocity profiles for different values of dID presented in Fig. 219. The points on each curve refer again to the depths 00, 0-1 D, 0-2 D . . . , below the sea surface. The part of the curve re- ferring to the top layer is shown by a thick line ; the points on the thin part of the curve (deep water) have been omitted for clarity. The unit of velocity is the same as in Fig. 218. If the depth of the top layer is small as compared with D, there will be a strong deflec- tion of the upper current away from the coast. The effect of the deep water lying just underneath the top layer varies according to variations in the depth of the top layer. U d < hD the deep water will in part be dragged out to sea by the water of the top layer so that underneath this there will be a current directed away from the coast and 484 Ocean Currents in a Non-homogeneous Ocean Fig. 219. Vertical structure in a convection current off a long straight shore (x-direction) for a homogeneous top layer of the vertical extent d and homogeneous deep water (D, frictional depth; unit of the velocity as in Fig. 218, according to Ekman). only below this, the current is directed towards the coast. If, on the other hand, d> D, then there will be a normal gradient spiral in the top layer and a corresponding inverse one in the deep water. If the water of the top layer is stratified, the general current structure will be significantly changed (Fig. 220). Now the deep water will be carried along, only to a lesser extent. The deeper the surface layer, the closer will the flow parallel the coast and the lesser will be the eff'ect on the layer beneath. As in the case of Fig. 218 the current is limited to the stratified top layer and its intensity falls near the boundary layer almost to zero. 2 ^=0-25£?. d--0-5L 1 d=0\D / y a -AZ'bD (^ ^^ ^ Fig. 223. The same as in Fig. 219 for a stratified top layer (according to Ekman). Ekman (1928 6) summarized these results and arranged them in a clear manner in Fig. 221. Three alternative assumptions have been made on the thickness (in metres) of the top layer d^\ (1) the top layer is divided into two homogeneous halves with a discontinuity surface in the middle ( — x — x — ^); (2) the top layer is stratified so that in it a density current is generated with a velocity distribution following a cosine-function ( — • — • — ) ; (3) in the top layer the velocity decreases linearly with depth and there is a dis- continuity layer ( — o — ^o — ^). Velocity profiles for the currents produced are shown on the right-hand side of Fig. 221 ; in the upper picture for a top layer the thickness of which is assumed equal Ocean Currents in a Non-homogeneous Ocean 485 to the frictional depth and in the lower layer is assumed as equal to double the fric- tional depth. The thin hnes refer to the lower layer and the thick lines to the top layer. The two arrow-heads at the right-hand edge connected with the + sign represent the vector of the surface current in the case of frictionless motion. For sharper discon- tinuity surfaces and a greater thickness of the top layer the velocity profile, as before, is made up of two Ekman spirals. If the top layer is stratified there is in both cases a Fig. 221. Density currents in a top layer considering friction and for motionless deep water (according to Ekman). current of almost uniform direction and the current velocity will decrease almost linearly with depth. Due to the stratification of the current, intensity in the lower layer (internal drift current) will be strongly reduced, and for a deeper top layer this current will disappear almost entirely. The transport in the deep current will then be insignificant. This leads to the important conclusion that: the sea surface under the influence of external disturbances will adjust itself in such a way that the pressure gradient arising from density dijferences in the top layer has a inaximum value at the sea surface, decreases with depth and will largely or entirely vanish at the lower boundary of the top layer; the deep water will remain practically motionless. The "elementar" current in a vertically comphcated stratified ocean consisting of a stratified top layer and an almost homogeneous deep water will thus, according to Ekman, have the following three current constituents. (1) The physical sea level and the isobaric surfaces of the top layer will be turned in such a way that the pressure gradient has the same direction everywhere and will be proportional at every level to the density; in the homogeneous deep water, however, this pressure gradient will remain constant. The current produced by this mass structure will be a simple gradient current. (2) If the physical sea level and the isosteric surfaces are brought back to the initial position, then an additional current resulting from this mass displacement adds to the gradient current described above. This is called the density current. (3) In addition, the effect of the wind on the sea surface generates a pure drift current. This current will differ only slightly from that in a homogeneous sea if the top layer is sufficiently thick. However, the density current will not be confined to the top layer alone, but when this is reasonably thick, the influence on the homogeneous deep water from above remains small. 486 Ocean Currents in a Non-homogeneous Ocean Laboratory experiments with stratified water have been made by Sandstrom (1908, 1918) in order to demonstrate experimentally the effect of stratification on wind-generated currents. In the experiment, an air flow over the surface of a multiple- stratified water mass in a narrow rectangular basin immediately produces a current in the direction of the wind. The piling up of water at the windward end of the basin gives rise to a counter current in the lower part of the uppermost layer ; there is a closed circulation in this layer. Friction then produces a somewhat weaker circulation with an opposite sense of rotation in the layer immediately beneath the uppermost one. Further circulations are formed in successive layers beneath this, each with the opposite (direct or indirect) rotational sense to that above it. Sandstrom's experimental results for a narrow basin cannot be applied directly to actual conditions in the ocean. In the laboratory experiment, in the first place, boundary conditions at the outer rim of the narrow basin will play a decisive role, and secondly, the deflecting force of earth rotation will have no effect and thus it is precisely that factor which most decisively influences ocean currents in nature that is left out of consideration. The laboratory experiment is thus apphcable in nature only to narrow confined sea basins and to lakes. 5. Oceanographic Applications of Bjerknes's Circulation Theorem The theory of ocean currents in a non-homogeneous sea received a very strong stimulus from the circulation theorem of Bjerknes, since it opened the road for studying in a quantitative way and for the first time the effects of baroclinic mass fields. There are manifold possibilities to apply this theorem in oceanography some of which will be discussed here in more detail. {a) The Steady State of Motion The most important use of the equation (X.54) is for the steady state in which the circulation accelerations vanish. In this case N=f dfn dt (XV. 11) (here again A'^ = number of solenoids, / = Coriolis Parameter, F^ = area of the projection of curves on the sea surface). The curve s is now made up of the two station verticals AC and BD and of two isobars AB and CD (Fig. 222). The water Ocean Currents in a Non-homogeneous Ocean 487 masses at the upper level move with an average velocity ^o and those at the lower level with an average velocity Dj at right angles to the section. After unit time the water elements, initially at AB, will lie at the line A'B' and those from the isobaric interval CD at CD'. The total surface ABCD transforms into A'B' CD'. The change of the projection of the surface ABCD on the sea surface thus becomes A'B'C'D", so that ciF^ldt = {vq — v^L, where L is the distance between the two stations A and B. Equation (XV. 11) combined with (X.45) gives {Vo = t'l) = Da- Di fL (XV. 12) This equation, which was first derived by Helland-Hansen (1905), forms the fundamental equation of dynamic oceanography. From the difference in dynamic depth of the isobaric surfaces Da — DbS. simple calculation gives the increase in velocity from one surface to the next. Analogous treatment to that on p. 466, however, affords only velocity differences and only the component at right angles to the selected section is obtained. Equation (XV. 12) contains fundamentally the same as equation (XV.7) derived directly from the equations of motion. In the practical appUcation of (XV. 12) it should be noted that /)„ — Di, has to be expressed in units of the potential, that is, in dynamic decimetres when the metre is taken as the length unit. The difference in dynamic depth anomaly, e^ — €{,, can, of course, be used instead of the difference Da - D,. The section to the south of the Newfoundland Banks between stations 205 and 206 can be used again as an example (see Fig. 202). Table 135 contains the dynamic depths, their anomalies and values of €„ — ^6 for selected pressure surfaces down to 750 decibars. In equation (XV.12) <^ = 41° 10' N.;/= 9-60 x 10-^; L = 59 km and the denominator is 5-664. The anomaly differences are multiplied by 10 in order to obtain dynamic dm ; this gives then v in m/sec. The last column gives velocities on the assump- tion that there is no motion at 750 m (see Table 133). If calculations of this type are available for a sufficient number of station pairs it is possible to obtain a complete velocity field at right angles to the cross-section. A comparison of the velocities cal- culated in this way from the mass field with the observed velocities was first given by WiJST (1924) for a cross-section through the Gulf Stream in the Florida Strait. The Table 135. Computation of the velocity profile south of the Great Banks of Newfoundland. St. 205 i St. 206 Pressure (dbar) D Du Depth Anomaly 1 Depth Anomaly t^a ^b (cm/sec^) (cm'sec) (dyn. m) e (dyn. m) e 0 0 — 0— 0— 0— 0— 00 64 50 48-68875 006225 48-7175o 009 lOo 002875 5-1 59 125 121-69188 0-14676 121-75713 0-2120i 0-06525 11-5 53 250 243-2725o 0-2525i 243-40776 0-3877^ 0-13526 23-9 40 450 437-5985o 0-3650i 437-84276 0-6092, 0-24426 431 21 750 728-7215o 0-5020i 72908476 0-86527 0-36324 1 64-1 1 0 488 Ocean Currents in a Non-homogeneous Ocean agreement was very satisfactory; later this kind of comparison has often been repeated confirming the results. If, instead of as in Fig. 222, the vertical section is placed in the direction of the relative velocity Vq — V^, then there will be no component at right angles to the surface, that is, in (XV. 12) ^o — i^i = 0 as well as £)« — D^ = 0 and the dyn. depths in the cross- section must be the same at C and D. If one of these verticals is kept fixed, then the other will move away at the relative velocity Fq — V^ and for every point along its track always applies Da — Dt, = 0. This implies that: curves of equal dyn. depth, which then give the dyn. topography of an isobaric surface relative to another, represent at the same time stream lines of the relative velocity {velocity of one surface relative to that of the other). This theorem is of great importance in the discussion and interpretation of the relative topographies of individual pressure surfaces in the ocean. An example is presented in Fig. 223 which shows the relative topography of the isobaric surface at 750 decibars for the same area containing the section shown in Fig. 202. The indication arrows show the direction and the intensity (nautical miles per hour) of the (relative) velocity of the layer at 750 m depth relative to that of the surface. If the water in this depth is motionless, then they represent the sea surface current. The dyn. isobaths are stream lines for the whole system. .57°W 56 'W 56" Fig. 223. Dynamic topography of the 750-decibar surface south of the Great Banks of Newfoundland according to the observations from 5 to 7 May 1922 (according to Smith). The arrows indicate the computed relative current in nautical miles per hour. Both applications of the circulation theorem have made use of curves in vertical planes, which contain a large number of solenoids. The theorem may also be applied to horizontal curves, which include little or no solenoids. For curves of this type the first term on the right-hand side of equation (X.54) vanishes and there remains only the term expressing the effect of the Coriolis force. On integration it gives -Co=-/(F, Fm 2). (XV. 13) Ocean Currents in a Non-homogeneous Ocean 489 A horizontal circulation free-curve (Cq = 0) will acquire by contraction a cyclonic circulation and by expansion an anticyclonic circulation.* If curves extending as parallel circles all around the Earth and containing an ocean covering the entire Earth are carried towards the equator by the general oceanic circulation, then their projection on the equatorial plane will expand and they will thus acquire a zonal anticyclonic circulation, that is, from east to west. On the other hand, if they are displaced towards the poles there will be a shrinking of the areas enclosed within the parallels and thus there will be a zonal cyclonic movement from west to east. Considerable changes can also occur in the area enclosed by horizontal curves flowing over a submarine ridge thereby causing the formation of cyclonic or anticyclonic circulations. These will be superimposed on the basic current and will give rise to a wave-form character of the current structure (see p. 431). (b) The Sandstrom Theorem In the ocean there exist closed circulations of greater or smaller extent, which are maintained by the continuous supply of heat at certain fixed places and the continuous withdrawal of heat at others. These sources of heat and cold maintain the differences in specific volume. Thus circulation velocity in a frictionless medium will continuously increase, since the circulation acceleration in equation (X.44) has a positive value. In reahty, however, all circulations are affected by frictional forces. Another term R must therefore be added to equation (X.44) containing all the frictional effects. There will be a steady state only when - I adp-\- R^Q (XV.14) that is, in a steady state (disregarding the rotation of the Earth) the work done by the pressure forces (i.e. — /« « dp) is used exclusively in overcoming the frictional forces. This can only be the case when adp<0. (XV. 15) From this controlling equation it is easy to draw conclusions as to how the sources of heat and cold should be located in space inside the circulation in order to allow for a stationary state. The concept of sources of heat and cold must be given in the ocean a * It follows from equation (X.54) dCjdt = fcIF^dt that for an increase in the area dFJdt > 0 there will be an anticyclonic deflection and correspondingly for a decrease a cyclonic deflection. This can, of course, also be derived directly from the equations of motion. For a geostrophic friction- less current these are \ cp \ cp — /f = ^— and + /m = 7— . p (}x p cy By cross-wise differentiation and rearrangement df /div vh + Pv = 0, whereby j3 = -r . If the current is divergent (div r^ < 0) y must be positive; this indicates that the deflection will be anticyclonic or to the right in the Northern Hemisphere ; for a convergence (div vh > 0) u is negative with a corresponding cyclonic turn. 490 Ocean Currents in a Non-homogeneous Ocean somewhat wider sense. In the real ocean differences in specific volume are produced not only by h,eat gain or heat loss, that means thermally, but also by changes in salinity. Evaporation will increase salinity and precipitation, ice melting and the inflow of fresh water (run-off) will reduce it. An increase in salinity has the same effect as a cold source and a decrease in salinity will be equivalent to a heat source. In the following the sources of heat and cold will be taken as including always the combined effects of both factors. In a Camot cycle one single and complete revolution shall now be considered on an [a,/j] -diagram (Fig. 224) consisting of two isobars {dp = 0) and of two adiabatic curves along which there is no addition or removal of heat and changes will occur only due to expansion or contraction. There are two possible cases: Fig. 224. Camot's cycle. Case o: heat source at lower pressure (small ocean depth) than cold source. Case b: heat source at higher pressure (great ocean depth) than cold source. A stationary circulation is only possible in case b, not in case a. (a) Clockwise cyclic process. From 1 to 2 at a constant, but lower pressure {p^ < p^, in the upper part of the sea) there will be a heat input (heat source), from 2 to 3 there will be an adiabatic compression followed from 3 to 4 by a heat removal (cold source) at higher pressure (in the lower part of the sea). Finally, an adiabatic expansion occurs from 4 to 1. Evaluation of the integral (XV. 15) gives, since the isobaric sections of the cycle make no contribution a dp = \ (a a4.i) (ip > 0» f, since both (02,3 — 04,^) as well as dp are greater than zero. The pressure forces are incapable to do work. Any existing circulation will in time be destroyed by frictional effects. (b) Counter-clockwise cyclic process. The heat source works at high pressure I < P2, in the lower part of the sea). In this case a dp (04,1 — aa.a) dp < 0. The pressure forces are capable to do work. If this is so large as to overcome all the frictional forces there will be a steady circulation. If there were no friction, this would be a reversible process and the degree of efficiency of this thermodynamic machine would be given by W = {Q^ — Q-^iQi, where Q^ is the amount of heat absorbed by the medium from its surroundings at the Ocean Currents in a Non-homogeneous Ocean 491 heat source, and on the other hand Q^ is that lost to the surroundings at the cold source. If frictional effects are present, then the process will be irreversible. The machine will give off a quantity of heat Qo, during the course of this process which is greater than in the reversible case {Q'c, > Q2). The degree of efficiency of such a circulation is less than Wand is given by (Qi — Q'^IQ^. In a circulation for which the work done by the pressure forces is exactly sufficient to balance the loss of energy by friction the degree of thermodynamic efficiency will be exactly zero. The Sandstrom theorem thus states: a closed steady circulation can only be maintained in the ocean if the heat source is situated at a lower level than the cold source. Sandstrom (1908) in order to elucidate the content of his theorem has performed a number of very instruc- tive laboratory experiments. Later on Bjerknes (1936) has presented a detailed analysis of all the questions raised when dealing with thermodynamic machines of this type. The two most important of the Sandstrom experiments are: {a) Heat source at a higher level than cold source. Here a single water type is con- tained in a narrow basin but there are two sources (Fig. 225, upper picture). The heat source ("warm") lies at a higher level than the cold source ("cold"). At the beginning n Worm Cold n 1 ^ . ► ^ - — >- — ^ Cold ^ '^ ^^^^z^ Fig. 225. Upper picture: heat source situated above cold source : no circulation and vertically stable stratified water layers. Lower picture: heat source situated below cold source; genera- tion of a stationary circulation in the layer between the levels of the heat and cold source. of the experiment motions will be set up because the heated water will rise in the layers above the level of the heat source and cooled water will sink in the water layers below the cold source. However, when the upper water reaches the temperature of the heat source and the lower water that of the cold source, these water movements will 492 Ocean Currents in a Non-homogeneous Ocean cease and there will be a stable stratification with the temperature decreasing with depth. A state of no motion is created since the circulations previously present will be halted rapidly by friction. (b) Heat source at a lower level than cold source. This is the same experiment as in (a) except that the position of the two sources is inverted. Convectional currents will be set up in this case also, but soon there will form a steady circulation confined to the layers between the levels of the two sources (Fig. 225, lower picture). Above there will be a water movement from warm to cold and below from cold to warm; the most heated water will be above the level of the heat source and the coldest below the cold source. But these layers will not take part in the circulation which is solely confined to the intermediate layers. Later on, Sandstrom modified the experiment in several ways, especially to show more clearly its application to oceanographic conditions; basically, however, these do not give any new results. Jeffreys (1925) has questioned the general validity of Sandstrom's conclusions but Sandstrom's deductions from the circulation principle are undoubtedly correct. The circulations produced by thermo-haline differences are the more intense the greater the vertical distance between the level of the warm and that of the cold source. However, conditions existing in nature in the ocean are not parti- cularly favourable to the formation of any more intense circulations of this type, since the principal heat supply in the ocean is primarily due to the combination of solar radiation and back-radiation from the atmosphere and the loss of heat primarily due to outgoing radiation. These processes operate to a very large extent at the boundary between the ocean and the atmosphere (almost horizontal sea level and evaporation and precipitation also act here. The vertical distance between the location of the heat and cold sources is thus very small. Probably the heat source in equatorial areas lies somewhat deeper than in higher latitudes, but nevertheless the thermo- haline circulation must be limited to a very shallow top layer. Observations provide complete confirmation of the consequences deduced from the circulation principle (see p. 576). 6. The "Reference -level" for the Conversion of the Relative Topography of the Press- ure Surfaces into the Absolute One The relative topography of the isobaric surfaces (relative to the sea level) assumed as plane) can be determined by the methods described on p. 309 and the following pages. Using equation (XV. 12) this also gives the relative velocity differences from layer to layer. In order to obtain a complete quantitative knowledge of the water move- ments it is necessary to convert these relative topographies into absolute topographies. This can be done if the relative topography can be referred to a known topography of any isobaric surface. This determination of the absolute topography would be easy if it were possible to determine from current measurements such a depth level at which the velocity of the current is zero, since at this "depth of no motion" the isobaric surface must coincide with a level surface ("Niveauflache"). In this way, for example, Wiist used the current measurements made by Pillsbury in the Floriaa Strait in oraer to determine the current profile of the Gulf Stream from the mass field. The number of current measurements available for the open ocean is, however, insufficient to fix with some accuracy the position of such a "zero level" Ocean Currents in a Non-homogeneous Ocean 493 ("Nullflache"), quite apart from the fact that short series of current measurements are almost always strongly disturbed by the tides. Thus the essential data needed to decide about the position of the "zero level" is largely lacking. The effort to utiUze the observations as fully as possible and to determine the pressure differences as good as possible, at least in the upper layers, has led to place the zero surface as deep as possible. This choice was also suggested by the generally rapid decrease in the velocity of the currents with depth. Over the entire area under consideration most investigators have thus usually placed the zero level at a constant dynamic depth and as deep as possible (as far as the water depth and the observations available allowed), and from this have derived the absolute topography of the pressure surfaces and that of the physical sea level from the relative topographies. Table 136 presents a summary of all the depths selected for the zero level by different investigators. The differences of more than 1000 m indicate that these are pure assumptions for which there is no firm basis. However, all investigators have been aware of the inadequacy of this procedure and have regarded the selection made purely as a make-shift. The assumption of a zero level at a constant large depth will, of course, conceal all currents in the layers just above and below this depth, and these Table 136. Depth of the ''zero level" {Nullflache'') in the Atlantic Ocean according to the assumption of difl'erent investigators depth depth Investigator Year (m) Investigator Year (m) Bouquet de la Grye . 1882 4000 Helland-Hansen and Nansen 1926 2000 Mohn 1885 550 Jacobsen .... 1929 1000 Zoppritz 1887 2000 Iselin .... 1930 1200 Wegemann 1899 1000 Helland-Hansen 1930 1000 Schott .... 1903 500 Iselin .... 1936 1800 Castens .... 1905 650 are thus falsified if by chance the zero level selected does not correspond with the actual position of such a level. On the other hand, the deeper the zero level is placed, the less will it disturb the pressure conditions at the sea surface. To obtain a correct idea of the deep current, it is not sufficient to assume a constant depth for the zero level. Such an assumption, moreover, does not correspond to the dynamics of the ocean currents in nature and, as has been stressed by Ekman (1939) takes no account of the topography of the sea bottom. These problems of dynamic oceanography have been dealt with by Dietrich (1937 a, c), who has thrown light on a number of aspects of them. The zero level, more suitably could be called "reference- surface" and has to be placed at such a depth where the velocity component at right angles to the dynamic section under consideration is zero. It must, of course, closely adapt to the mass structure of the entire oceanic area, since this is in fact a conse- quence of the currents and is closely connected with them. In these circumstances it is to be expected, especially when larger areas of the sea are taken into consideration, that the reference-level for the reduction of relative into absolute topography must be 494 Ocean Currents in a Non-homogeneous Ocean a surface of locally varying depth. The determination of its form and the different factors that must be considered for fixing its position in oceanic space is not an easy task. It should be stressed that the choice of such a surface is always more or less subjective, and such an assumption can only be made plausible by giving proper weight to all the different view points which are in question. {a) Determination of the Topography of the Reference-Level A first attempt was made by Dietrich in an investigation of the dynamics of the Gulf Stream to introduce a reference-level of variable depth by investigating characteristic features in the distribution of oxygen in order to fix the reference-level. He thus accepted the widely held view that the layers showing the intermediate oxygen minima (see Pt. I, p. 66 and following pages) are at the same time also layers of very weak motion or of no motion at all, and could thus be regarded as motionless boundary layers between individual components of the deep-sea circulation. However, Rossby (1936 a), ISELiN (1936) and especially Wattenberg (1938) and Sverdrup (1938 M have questioned this assumption and have expressed doubts about the suitability of these oxygen minima as reference-levels. In the upper layers of the ocean the oxygen distribution can be regarded, on the one hand, as a consequence of thermal and bio- chemical oxygen consumption, and on the other hand, of the renewal of the water masses by horizontal advection. The intermediate minima are thus regions of parti- cularly strong oxygen consumption and can hardly be regarded as completely motion- less layers. The results obtained by Dietrich for the currents in the Gulf Stream on the basis of this assumption are not such as to give confidence in reference-levels derived from the oxygen minimum. Even the customary division of the water masses of an ocean, pictured by major longitudinal and transverse section> and allowing for the characterization of the different water bodies, is scarcely suitable for the determination of the topography of the reference-level. Even though they may be practical and useful in giving a general qualitative picture of the meridional and zonal velocity components of the ocean currents. Defant (1941 b) has gone a quite different way in order to determine the dynamic reference-level in the Atlantic, which avoids the use of any particular boundary layer between the individual water types and makes use only of dynamic evaluations of observational data, which must be closely connected with the structure of the water masses of the particular area. The differences in dynamic depth of the pressure values between two neighbouring stations give, by means of equation (XV. 12), a relative measure of the velocity difference perpendicular to the cross-section between the sea surface and the corresponding depth. When these differences are plotted in an appro- priate co-ordinate system (ordinate :pressures; abscissa :difference in dynamic depth) they give a relative vertical velocity profile at right angles to the section between the two stations (Fig. 226). This profile cannot be converted to an absolute velocity profile without knowing the zero point on the abscissa. By comparison of a large number of difference-curves for neighbouring pairs of stations it shows in most cases that in each profile there is a layer of considerable vertical thickness in which the differences in dynamic depth are constant or almost constant. If the zero point of the abscissa scale is placed outside of this layer then the entire layer must have a constant velocity. Ocean Currents in a Non-homogeneous Ocean 495 0 400 800 1200 1600 2000 dyn.cm -12 -8 -4 0 +4 +8 -4 0 +4 +8 dyn.cm + 12 Fig. 226. Schematic example for fixing the reference-level by means of the vertical distribu- tion of the dynamic depth of the standard pressures of two neighbouring stations. (The lower "displaced" scale of the abscissa only has its correct position, if the reference-level is assumed in the layer denoted by the vertical arrow; a position of the reference-level at the dashed arrow, for example, would be quite improbable.) while the dynamic structure of the other layers will be divided up in a rather unintelli- gible way. It is more plausible to suppose that this more prominent layer should be motionless, or almost motionless, so that the reference-level should lie within it. Such a layer with obviously low velocities is apparently characteristic not only for the pair of stations under consideration, but is to some extent depending on the pressure field of the entire oceanic region under consideration. The reliability of this method is increased if the individual reference depths, determined from a large number of station pairs, can be combined to give a closed system representing a definite topography of the reference-level. To illustrate the method the difference-curves for the dynamic depths are shown in Fig. 227 for a meridionally distributed set of stations in the Atlantic ; for each curve the vertical extent for which a layer of no motion or only weak motion is most probable, dyn cm 0 1 B 12 16 20 dyn cm Fig. 227. Fixing of the dynamic reference-level for a series of meridionally distributed stations in the Atlantic Ocean. 496 Ocean Currents in a Non-homogeneous Ocean 110° W 100° 90' 100° 90° 80° 70° 60° 50°40°30°20° 10° 0° 10° 20° 30° 40° 50° 60° E Fig. 228. Position of the reference-level for transforming relative topographies into absolute depth (numbers in 100 m units). Ocean Currents in a Non-homogeneous Ocean 497 is marked with a vertical double arrow. The reference-level for conversion of relative into absolute topography must lie within this layer. Already these station pairs show roughly the meridional distribution of the depth of the reference-level in the Atlantic : lower depth in high latitudes (approx. 1500 m or deeper), rising up to 500 m at the equator. The topography of the reference-level can thus be derived for the whole of the Atlantic from a large number of such diagrams. Figure 228 presents the topography determined by this method. The lines are drawn at 100 m intervals (or decibars); for a reduction of the relative into absolute pressure values it is sufficient to know the position of the reference-level to the nearest 50 decibars. It is clearly shown that the assumption of a reference-level of constant depth can never do justice to the dynamic structure of the water masses of the Atlantic Ocean; even over smaller oceanic areas there are appreciable variations in its position. Along each meridian the depth of the reference-level is least near the equator (up to 400 m), in the Southern Hemisphere it sinks uniformly to great depths in high latitudes. But in the Northern Hemisphere conditions are more complex. From the equator it sinks at first to a secondary mini- mum between 5° and 10° N. (about 900 m), then rises again to another maximum between 10-20° N. and from there begins the lowering towards the north-west to greater depths. The irregularity in the northern subtropics has the same form as the asymmetry in the position of the subtropical and tropical thermocline (see Pt. I, p. 120). There is undoubtedly a causal coimection between the two phenomena. In the Gulf Stream region there are considerable deviations from normal. Near to the current core (intense flow) the reference level rises steeply upwards to a depth of 1000 m or less. This phenomenon, which belongs to the characteristic features of this area, must be connected causatively with the inclination of the isosteres in a stratified ocean with intense motion (see p. 331). From the chart shown in Fig. 228, Neumann (1954, 1955) has computed zonal averages of the depth of no meridional motion D (zero level) for the North Atlantic and has plotted them against the latitude (Fig. 228 a). Individual values along the 20°W-meridian were used for the South Atlantic, since the variation in D in the east- west direction is small as compared with the variation of D in a meridional direction. In Fig. 228 a the values of D are marked by circles and the full drawn curves represent the function D= - K?,mcl>^- Kcos &. (XV.16) The constant ^ is different in the Northern and Southern Hemisphere but the increase of D with latitude follows this function closely except in the equatorial regions, where apparently another physical law applies (see Pt. I, p. 120). Excluding the equatorial regions, the relative variation of D with latitude is given by 15 'I = - '^"*- "^^•'" Then, it follows from the Coriolis parameter, f=2w cos d that 1 df -f-^^-- tan^. (XV. 18) 2K 498 N 60° 0 D (mJ! K>00 2000 3000t: Ocean Currents in a Non-homogeneous Ocean 50° 40° 30° 20° 10° 0° 10° 20° 30° 40° 50° 1000 2000 3000 60° S 0 D (m) 1000 -2000 -3000 Fig. 228a. Average depth of the reference-level (layer of no motion) in the Atlantic Ocean (according to Neumann)). Thus for the large scale major oceanic circulations the fundamental relation 8^ D 8& (XV. 19) is obtained. An investigation of the reference level (layer of no motion) similar to that made by Defant was also carried out by Neumann (1942, 1943) in an evaluation of observational data for the Black Sea. For the strong vertical stratification of this adjacent sea the topography of the reference level is more closely connected with the position of the boundary layers characterizing this vertical structure. Figure 229 shows the position of the different boundary surfaces in a longitudinal section near 43° N, It is almost the same everywhere: the lower plankton limit, the maximum density 40° E 41= Fig. 229. Depth of different characteristic boundary layers in a longitudinal cross-section through the Black Sea in 43° N. (according to Neumann). Ocean Currents in a Non-homogeneous Ocean 499 gradient, the upper limit of the H^S-\a.yQV and the reference level are all more or less coincident (except near the coastal areas in the eastern part). All these surfaces join here, forming a single closed system, an almost motionless boundary layer. If in an adjacent sea a density discontinuity layer is found everywhere, the deter- mination of the position of a dynamic reference level is considerably simplified, since the lower limit of the top layer is then usually also the lower limit of the upper flow and the discontinuity layer coincides with a layer of no motion. These methods have already been used by Witting (1918) in his investigations on the continental rise around the Baltic. This simple method can, of course, only be used when the thickness of the top layer is not too great ; it is also possible to apply this method with success to shelf areas, having a sharp subdivision in the vertical into two layers. A new method for the determination of the depth of no meridional motion has been presented by Stommel (1956). It is of interest in so far as it permits a determination of this depth directly from the observed vertical distribution of the oceanographic factors, and because it also shows that there is in actual fact no depth of no motion in the ocean but rather the depth of no meridional motion always coincides with the layer of maximum vertical velocity. From the general equations for a wind driven motion and the continuity equation cross differentiation leads to the following three relations : (XV.20) 8y The quantity pv in the third equation can be ehminated by means of the first equation, giving 8\p^^^ ^g 8p 8^ 8 g dp 1 fcx f ^ 1 X 8z^ ' 1 (^") = g op 1 8^T^ f cy f 8z^ ' 8 - (ph) = 0'^ ^ 1 8 -r PV — -> ^^ / f 0^ 8x where 8z^ f(=) /2 8x 8z^ PCV.21) 8 8x (7-) - If) This function F(z) is more or less indeterminate, but accord'ng to Ekman differs from zero only in a thin upper layer extending from r = 0 to the depth of frictional influence. F(0) is known in terms of the distribution of the wind stress on the sea surface. If the sea bottom is at —d, then the first integral of equation (XV.2] ) can now be obtained : Sz^P^'^^-f ^(-) + C 8F 8z (XV.22) whereby (f'(r) is defined 500 Ocean Currents in a Non-homogeneous Ocean and where C is an integration constant. The meaning of the function 0(z) is easily understood, since for a purely geostrophic flow (from the first equation in XV.20 it) follows pv = 0(z) + C. The constant C is the indeterminate reference velocity and the determination of C can be readily seen to be equivalent to the determination of the depth of no meridional motion, that is, the depth at which pv vanishes. Since for deeper layers F = 0, it follows from (XV.22) 8 ^ (PH') = 0. By this it is shown that the level of no meridional motion coincides with the level of maximum vertical motion. Since the bottom currents are rather weak, the hypothesis dF IFz dF Tz allows the integration of equation (XV.22) between z and —d. Taking F{ — d) = 0 and p\v{ — d) ^ 0, the following expression for pw is obtained pw = J 0(z) dz + C. (- + d) F(z). (XV.23) At the surface, r = 0, pw vanishes; the quantity F(0), according to (XIII. 27) is the net convergence of the wind-driven layer and (XV.23) gives 1 ■^ F(0) - [" jHz) dz The depth at which ) instead of the ^-lines, are based on incorrect reasoning because lines obtained in this way are then no longer stream lines ; they will be intersected by the flow and thus lose their meaning. It is therefore better to retain the g-lines (Thorade, 1937 b). Volume transport charts over more extended oceanic areas have not yet been prepared, although the complete dynamical evaluation of the observational data for such an undertaking would be available. {b) Water Transport in Coastal Currents Werenskiold (1935, 1937) has presented a very convenient method for the calcula- tion of the volume transport in coastal currents, for which, in a cross-section at right angles to the coast, a lighter water is spreading out in a wedge-form on top of a heavier slowly moving and almost homogeneous water. Figure 235 shows a vertical section across a current between two stations A and B. The .v-axis is placed in the sea surface in the Fig. 235. To the computation of the water transport in a coastal current (according to Werenskjold). direction A-^ B and the water depth is denoted by z. In the section there are drawn two isopycnals p and p -[- ^p and two plumb-lines x and x -f ^.v. The boundary surface of the wedge-shaped top layer forms the isopycnal pi reaching the surface at C. The top-layer has a depth z^ at point x, however, the depth Z^ at the station A. At an arbitrary point M on one of the plumb-lines (density p) the component of the velocity of the density current at right angles to the section will given by the equation (VII.8): Vi = fp. i dp- Thereby j is the slope of the isopycnal which is dependent on .y and z. Denoting Sl fPm by b, then one obtains from the relation above Tp -bj dz dx' (XV.25) where the derivative dzjdx has to be taken along an isopycnal, that is, for a constant p. The volume transport at a plumb-line can then be obtained by integration from 0 to r^. By partial integration one obtains S - Vi dz Vi- Zd zdVi. 512 Ocean Currents in a Non-homogeneous Ocean The first expression on the right-hand side is zero, since fj = 0 for z^ and thus using (XV.25) one obtains Pi dz ^ b [Pi dz^ ^ Po (l-"^ 2 \o.dx Pq is the density at the sea surface. The total volume transport through the entire top layer from C to station A is thus finally obtained by integration from x,. to xa St XA h Sdx = ^ Pi dz^ , ~r-dp. The integral of (dz'^jdx) dx is equal to Z^ where Z is the depth of the isopycnal at the station A. Finally, on repeating the partial integration, since Z is zero at the sea surface, we have St-2f '-^^ Pi — P dZ\ (XV.26) If the transport between two arbitrary verticals A and B is required, then the expres- sions (XV.26) are evaluated at both places and the difference is taken. The water transports obtained in this way are subject to the same limitations for the quantity^. It is noticeable that a knowledge of the mass structure at the two stations is sufficient for the determination of the transport through the vertical section between them, without having a knowledge of the distance between the two stations. Werenskiold offered an explanation for this fact by pointing out that the flux in horizontal direction through the section is unaffected by stretching or shrinking of part of this section, because the pressure gradient and therefore also the current intensity are changed inversely proportional to the current width, and the distance between the two stations is eliminated. It seems, therefore, that only the mass distribution of a single station is required in order to calculate the transport through a vertical section by means of equations (XV.26). However, this is not true at all since a knowledge of the stratifi- cation at two stations C and A is required and, furthermore, the water at C is homo- geneous and has the same density as the deep water at A. Since the integration of equation (XV.26) is performed using ordinary metres, the correction required previously for Q is not needed here. Chapter XVI Currents in a Strait 1. Water Stratification and Water Movements in Sea Straits Sea straits connect the open ocean with mediterranean and adjacent seas. By means of the water flux through the connecting straits directed towards the open ocean a medi- terranean sea can often produce considerable effects on the oceanographic conditions in the open ocean. This influence is sometimes so powerful as to involve entire parts of an ocean, changing drastically the oceanic conditions in these parts. Present knowledge of oceanographic conditions in sea straits is only partly satisfactory. The main outlines and the typical features are known but much remains to be explained especially in the details, that will require systematically arranged observations and measurements. The continuous interchange of water between mediterranean and adjacent seas which are completely surrounded by land and the open ocean is controlled very largely by two factors : (1) by the proportion between fresh- water inflow (precipitation and run off (river water and other water)) and evaporation in the mediterranean sea, and (2) by the depth and width of the passage to the open ocean, that is, the morphology of the sea strait. The currents in a sea strait are a consequence of the difference in vertical thermo- haline stratification between the water masses in the adjacent sea and that of the open ocean off the entrance to the strait. Sea straits can be divided on the basis of the currents flowing in them into two groups : (1) Those in which the adjacent sea is surrounded by arid land masses. Here evaporation exceeds precipitation (E — P) > 0. The loss of water due to this excess must be replaced from the open ocean through the strait. (2) If the entire oceanic area lies in a humid climate (E — P < 0), then the excess of precipitation over run-off will flow out into the ocean through the connecting strait. To the first group belong — inside the area of the Eastern Hemisphere rich in evaporation and with little precipitation — the Strait of Gibraltar, connecting the Atlantic with the high-salinity European Mediterranean; the Strait of Bab el Mandeb, connecting the Indian Ocean (Gulf of Aden) with the highly saline Red Sea and the Strait of Hormuz between the Arabian Sea (Gulf of Oman) and the Persian Gulf, To the second group belong — in the northern humid region — ^the weakly saline Baltic Sea which is connected by way o^tiarrow belts and Sounds through the Kattegat and the shelf-like North Sea with the open ocean; the predominantly humid Black Sea connected with the arid Mediterranean through the Bosporus and the Dardanelles; 513 2L 514 Currents in a Strait the White Sea and the Barents Sea with the so-called Gorlo and the Gulf of St Lawrence connected with the Atlantic by the Cabot Strait and others. The interchange currents in all these sea straits occur characteristically on two different levels; there are always two currents in the strait, one above the other. The upper layer always flows toward the sea having greater density, the lower layer in the opposite direction, and between them there is usually a well-developed discontinuity layer in the density field (see Pt. I, pp. 133 and 182-184 (Figs. 56, 83-85) on the general distribution of temperature and salinity in sea straits). Thus in straits of moderate width there are always two water bodies one above the other with a boundary layer between them sloping down from the sea with the greater density towards that with the lesser. The wedge-form of these superimposed water layers along the strait is a characteristic feature of the structure of the water masses in a sea strait. Table 1 38 gives a summary of mean density in the upper and lower water layer and of the slope of the boundary layer for some sea straits in different climatic regions (Vercelli, 1929; MoLLER, 1931). The greater the slope the smaller the density difference, i.e., the slower the interchange movements. In addition to this effect of the density differences other circumstances also control the slope of the boundary layer, particularly the bottom topography of the sea strait, because it affects the continuity requirement of a complete balance between the mass transport in the upper and lower current under stationary conditions. For example, in the Bosphorus, the slope of the boundary layer is strongly dependent on the bottom inclination and because of this the wedge-form character of the lower water is lost there. Table 138. Mean slopes of the boundary layer and mean densities of the upper and lower water in several sea straits Sea strait Mean width (km) Mean length (km) Minimum depth (m) Boundary layer slope (m/km) Mean density Difference Upper water Lower water Danish sounds (Belts) Dardanelles . Bosphorus Gibraltar Bab el Mandeb ca.lQ 4-5 0-7 20 ca. 100 60 30 60 134 6-9 57 37 333 185 012 0-20 M3 4-2 3 0 13-5 180 13-5 26-8 259 23-5 28-8 27-5 28-8 27-4 100 10-8 140 20 1-5 Besides this longitudinal slope there should also be a transverse slope of the boundary layer due to the effect of the Coriolis force. The faster the currents and the wider the strait the greater will this slope be. If the upper homogeneous water mass in the strait has a velocity u^ and the lower one a velocity lu, and if the transverse inclination of the sea surface is given by 8l,^jdy and that of the boundary surface between the upper and the lower current by ^i^i^y^ then, under stationary conditions the equations g-^=-fih and will be valid where (/= 2aj sin Pi- This relation thus fixes the direction of the current in the strait and also give the dependence of the current velocity on the density distribution in the water masses. This can be used to find an approximate value for the current velocity maintained by the thermodynamic forces acting inside the system. According to the circulation theorem, when a = Xjp - i adp = i Rids XVI.6) J abed J abed and since Z), p a dp 0 gives the dynamic depth of the pressure surface p in the water column /, we obtain from (XVI.6) D,- D, = (/?! + i?3)/ + (^2 + ^4)/^. (XVI.7) The integral — j adp is the work done by the pressure forces in the system ; if it is positive, this work can thus be balanced by the work required by the friction. The relation (XVI.6) states that, in the thermodynamic machine the expansion takes place at a higher pressure than the contraction. Since an expansion is associated with an input of heat and a contraction is associated with a heat loss, the heat gain must therefore occur at a higher pressure than the heat loss. Actually, in the model of the sea strait in point there will be a higher pressure and a higher temperature, the latter due to a greater heat gain. Such a sea strait system is thus a true thermodynamic machine in action. The current intensities in a strait can be calculated approximately by means of the above equation (XVI.5). For a channel of length /, if friction is neglected in the vertical part of the circulation, the equation will take the form 2pR/ = g{p, - p,)h. (XVI.8) Currents in a Strait 519 In addition, it is necessary to make an assumption about the dependence of the friction on the current velocity. For a shallow current it is possitble to put R equal to Kpu^ dyn/cm^ (see equation X.9). However, for each horizontal branch and the friction per unit mass of this branch is The total friction is therefore given by /c(2m)2 and the equation (XVI. 8) thus gives an equation for the determination of the mean velocity in one water body 6 Kl p If the dimensions of the strait are known, w can be calculated. Only the value of the Taylor frictional constant requires a little further comment. For a smooth channel K has been found experimentally to be 0-0025. It cannot be expected that the value of K will be as small as this because of the irregular configuration of the sea bottom and sides of an actual sea strait. In rivers, for example, k may be as much as 10 times this value or about 0-03. Considerably higher values of the boundary friction are there- fore to be expected due to the roughness of the bottom in a somewhat wider strait. A proof of this is the frequently observed sharp decrease in velocity in the layer next to the bottom. Choosing mean values for the dimension of a sea strait, for example, / = 50 km, h = 100 m and the difference in density/!/) = 10 x 10"^, according to Table 140, then putting K = 0-03 the equation gives w = 28 cm/sec which accords with the average velocities found by observation. In the Danish sounds (Belts) the velocity of the current is about 10 cm/sec, in the Dardanelles about 25 cm/sec, in the Bosphorus 30 cm/sec, in the Strait of Gibraltar 30-35 cm/sec and in the Strait of Bab el Mandeb about 40 cm/sec. The calculated value fits thus very well in this series of observed values. For a detailed theory of currents in sea straits it is necessary in the treatment of the stationary state to return to the antitriptic equations of motion in which the gradient force and all the frictional forces are always in equihbrium (Defant, 1930). A suitable model is a rectangular channel, depth h^ and length L, connecting two seas with differ- ent thermo-haline structures. Both water types are homogeneous (upper water density Pi, thickness in the middle of the channel h^ ; lower water density pa^ thickness in the middle of the channel h^ — fh over a plane bottom). The co-ordinate origin is placed in the middle of the channel at sea level with the positive r-axis directed upwards. The upper current flows in the direction of the negative .v-axis (see Fig. 239) and the physical sea level must therefore also slope downwards in this direction (pure slope current). The static pressure in the upper layer [z from l,^ to — {h^ — Q] will be 520 Currents in a Strait Fig. 239. To the theory of ocean currents in sea straits. p^= p^-\- gpi(^i — z), however, in the lower layer [z from — {h-^ — i^to — //g] will be P2= Po-\- S(p2 — Pi)(^2 — fh) + ^Pi^i — gp2=- Po is the atmospheric pressure at the sea surface. Putting /Jq = — gPiC then ^ is the displacement of the sea surface produced by an atmospheric pressure p^. The equations of motion in the stationary state, disregarding the Coriolis force and friction on the sides of the channel are then -^8-x^^^ 0 + 7] 8^Ui 0, (XVI.ll) Pi ^ /y y\ P2 Pi ^^2 8^Uo + - — - = 0 P2 Sx p dz^ If Ci and ^2 are small compared with the depth of the strait then, for a linear slope of the physical sea level, u^ and Wg will be independent of x and the continuity equation will take the simple form - /,, - hi i^dz^O. (XVI. 12) Ml dz + U2 Jo J -hi The volume transport of the upper current must be equal to that of the lower current. The boundary conditions are as follows : ( 1 ) If there is no wind, dujdz = 0 when z = 0. The effect of a wind along the channel can be taken into account by the assumption V 8ui az ^1 Pa W'-, where Pa is the density of the air, k^ is the Taylor constant (equation X.9) and n- is the wind velocity relative to that of the water. Taking diijdz = M for z = 0 allows the effect of the wind to be taken into account. (2) At the boundary surface there will be a reversal of the current direction, that is, when z = — hi, then Ui = U2 = 0 (no horizontal motion). (3) At the bottom (z = — h^ three different cases of boundary friction can be Currents in a Strait 521 considered : adhesion to the bottom u^ = 0, ghding du^jQz = 0 and average frictional influence r](8u2ldz) = Kp^ ul. If the roughness of the sea bottom is shght the factor k is of the same magnitude as k^ ; for a rough bottom it has been found in hydrauhcs to be about 10 times greater. Solutions of equation (XVI. 11) can be given for all three cases. For the extreme cases of adhering (haften) and gliding (gleiten) and with uniform atmospheric pressure (I = 0) one obtains Slope of the physical sea level : 2^ 2v Slope of the boundary layer: /g = 7- Velocity of the upper layer : Velocity of the lower layer : aiz^ - hi) + M{z + /?i). .(XVI. 13) adhering : gliding where u., = A(z + h^(z + fh) with A = m (i-) U2 = A [(z2 - hi) + Ih^iz + h,)] with A = 4[\ l-^A Pi . and m = 4a 3M Because A is always negative, the slope of the internal boundary surface will always be opposite to that of the sea surface ; however, because of the density difference (pa — Pi) in the denominator it is always considerably larger. The slope of the boundary surface found by observation is a function of the water interchange between the two seas. The currents in the two water bodies always flow in opposite directions. The current profile in both water bodies is of a parabolic form. In the upper current the maximum occurs at the sea surface; if the wind is in the direction of the upper current it will decrease rapidly with depth, but if the wind is against the upper current the decrease will be small. The upper water in this case will be piled up against the current. If there is a very strong wind at the surface against the upper current, the current maximum may be somewhat below the sea surface. All these theoretical conclusions are in complete agreement with observation. In the lower current the velocity maximum will adjust in variable depth below the boundary surface according to the variable friction at the sea bottom. If there is adhesion it will appear in the middle part of the lower layer, if there is gliding at the bottom it will occur at the bottom itself and for moderate friction it will be situated between the discontinuity surface and sea bottom. Numerical values corresponding roughly to those for the Bosphorus may be taken as an example: length of the strait = 30 km, depth = 70 m; upper layer p^ = 1-013 down to 40 m; lower layer p^ = 1-027, p2 — pi= 14 x 10"^; slope of the physical sea level 6 cm in 30 km, -qj p = 250 cm^/sec, which is about the same as the frictional coefficients for tidal currents; wind = 5 m/sec along the strait. For the slope of the boundary surface (metres in 30 km) the equation gives the values contained in Table 142. 522 Currents in a Strait Table 140. Slope of the boundary layer {given in m/30 km) For adhering For gliding Moderate friction Wind with the upper current Wind against the upper current 44 53 14 17 33 37 The magnitude of these values is similar to those actually found in the Bosphorus which average about 34 m. In the case of a south-west wind the slope is steeper than for a north-east wind, which agrees with the theoretical result. Figure 240 shows the 20 V, cm/sec 40 60 80 100 20 E £40 Q. a> O 60 ■ I — ' 1 1 / ■■ 1 ^.i-:; ^ ^ **»gj _ ^^^ ^ ~'-^=-'=r^ _^^ mmmm ,.-' \ V, cm/sec 0 20 40 60 80 100 20 E i 40 60 I -i — ) ■ 1 — 1 — ^ ^ '^■Mfa-: "^^^ "^^ ?^- ^ y \ ^^^ Fig. 240. Vertical current distribution in the upper and lower current for a certain wind direction at the sea surface of the sea strait (in the lower current: , in the case of clinging to (Haften); , in the case of gliding (Gleiten); , for a medium friction of the water at the sea bottom). current profile in the upper and lower current (omitting signs). These values are also in agreement in all cases with those observed in the Bosphorus and the wind effect was also of a similar kind. The theory is based on two water bodies that are homogeneous over a cross- section at right angles to the strait. In nature they will be stratified and the cross-sectional area can vary considerably along the length of the strait. Furthermore, mixing at the internal boundary surface will tend to spread the discontinuity surface into a density transition layer. The current boundary surface will then no longer coincide with the lower limit of the upper water since there is no longer any sharp boundary. Then conditions become so complex that they can no longer be handled mathematically. Currents in a Strait 523 However, the stratification does not appear to be of decisive importance to the prin- cipal phenomena of the water interchange and therefore the simple case of two homogeneous water types gives the essential outlines. 3. Ocean Currents in Individual Sea Straits {d) Bosphorus and Dardanelles Due to the investigations of Merz and Moller (1921, 1938, with Atlas) these are the straits in which conditions are best known. Systematic surveys along cross-sections and longitudinal sections have given a good understanding of the three-dimensional thermo-haline structure of the water masses and the corresponding currents in both straits and some insight into the detailed mechanism of the processes involved. Over the whole area of water interchange between the Aegean and the Black Sea there is a characteristic stratification with a sharp density transition layer. From a depth of 200-1 50 m in the Black Sea it rises at the entrance into the Bosphorus to less than 1 50 m and in the narrow part it rises rapidly to 20-15 m at Istanbul. It remains at this depth throughout the Marmara Sea until it rises again in the Dardanelles, at first very slowly, then more rapidly in the straits between Nagara and Tschanak to 10 m. At the southern entrance to the Dardanelles it reaches almost to the surface. Figure 241 presents the density distribution in two longitudinal sections along both straits. The wedge-form of the upper water shows clearly in both straits ; in the lower water it is present only in the Dardanelles, since the sea bed in the Bosphorus slopes down- wards towards north as much as the internal boundary surface. At the entrance to the Bosphorus the salinity of the upper water is 16-18%o and at the outlet from the Dar- danelles into the Aegean it is 26-28%o. Of this increase 2%o occurs in the Bosphorus, 5%o in the Sea of Marmara and 3%o in the Dardanelles. Mixing in the straits thus can- not be very effective ; this is also shown by the maintenance of the temperature in- version which is still partly present in the Dardanelles (see Fig. 237). The upper current runs through the channels as a narrow band within limits set by the projections of the coast. In several coastal bays on both sides of the straits numerous standing vortices occur. The current profile shows that the velocity is greatest at the sea surface and decreases rapidly with depth. Due to the wedge-form of the current it increases from north to south; under average conditions it is 40-50 cm/sec at the entrance to the straits and 1 50 cm/sec or more at the other end. The lower current follows the windings of the channel more closely than the upper current and the stream lines of the two currents are therefore not always super- imposed. The lower current is strongest in the central parts of the lower water (in the Bosphorus about 16 m and in the Dardanelles about 45 m above the bottom). The velocity is 100-150 cm/sec in the Bosphorus and decreases from 25 to 10 cm/sec in the Dardanelles. In the straits the boundary surfaces between different currents and between different water types do not coincide ; the first rises from north to south more slowly than the thermo-haline transition layer and they intersect at the narrowest part of the straits. Thus in the northern part of both straits upper water flows with the lower current and in the southern parts lower water returns with the upper current. The changes in the currents due to variations in wind and atmospheric pressure are pronounced. During strong north-east wind the surface current is accelerated, the current core thereby 524 Currents in a Strait Fig. 241. Longitudinal section of the density at. Upper picture: through the Bosporus in Sept/Oct. 1917. Lower picture: through the Dardanelles June/July 1918 (according to Moller). narrows and the standing vortices increase in extent. During south-west winds the surface current becomes weaker and broader and the lower current is accelerated. Figure 242 gives a longitudinal section through both straits showing the currents during a period with stronger north-east winds with a large pressure gradient towards the south-west. This wind influence produces a strong asymmetry in the current structure. For a period with a south-west wind the current conditions are aflTected in the opposite way. These flow conditions, however, no longer represent a stationary state. Currents in a Strait 525 A comparison with the theory presented above can only be achieved by means of current profiles in which the varying effects of changes in pressure and wind are eliminated. A computation of average profiles out of three typical ones for each strait allows a qualitative comparison. Figure 243 shows that excellent agreement can be obtained by suitable choice of the frictional coefficients. A numerical evaluation of equations (XVI. 13) is given in Table 142. The average decline of the physical sea level along the Bosphorus is about 6 cm in 30 km and is greater at the northern end, less at the southern end. Along the 65 km Dardanelles it is only 7 cm; the value of 12 cm in 170 167 161 153 148 Sto).82 .77 69 68 63 61 57 51 180 48 45 38 29 23 17 3 14 35 100 94 89 83 Stat. 75 63 59 30 i5o56 50 39 35 23/27 19 10 Fig. 242. Longitudinal section of the current velocities (cm/sec). Upper picture: Bosporus for N.E. 5 and ^p = 4-5 mm. Lower picture: Dardanelles for N.E. to E. 3^ and /!/> = 3 mm. 526 Currents in a Strait V, cm /sec 20 40 60 80 V, cm/sec 0 20 40 . 40 Fig. 243. Vertical current distribution in the northern part of Bosporus (to the left) and of the Dardanelles (to the right); H \ 1 , according to the observations; . — . — • — , according to the theory. the middle of the strait must be due to piling-up of water in the narrowest part of the strait. Table 141. Sea surface and slope of the internal boundary surface, as well as frictional coejficients in the Bosphorus and the Dardanelles, calculated from current profiles Wind conditions {Sea surface boundary surface . Turbulent coeflRcient (cm^/'sec) Bottom friction k . Bosphorus Northern part NE-SW 101 36 298 0017 Middle part NE.j 5-6 36 371 0-155 Southern part NE and SSW 2-4 cm/30 km 36 m/30 km 485 0 015 Dardanelles Northern part NE.3. 7-6 10 82 0109 Middle part SW 12-2 12 28 0038 Southern part NE.3_, 7-2 cm/65 km 19 m/65 km 420 0-388 The turbulent coefficient is of the same order of magnitude as in tidal currents. The coefficient of bottom friction has a mean value of 0-12 which is very large. The individual values are strongly scattered but are around 50 times larger than the values found for natural channels and about 5 times larger than those found for rivers. The rolls with horizontal axis produced by the very irregular bottom and which cannot be observed by means of current measurements may simulate a bottom friction larger than actually present. (h) Water Interchange Between North Sea and Baltic This takes place in the area between the Kattegat in the north and the Darsser and Drogden ridges in the south. These give access to the Baltic at depths of 18 and 7 m, respectively. The annual inflow of fresh water into the Baltic averages about 500 km^ of which 467 km^ is the inflow from rivers and 206 — 1 82 = 24 km^ is the excess of Currents in a Strait 527 precipitation over evaporation (Witting, 1918). This inflow of fresh water disturbs the equilibrium between the North Sea and the Baltic and gives rise to a water inter- change with an upper current flowing towards the North Sea and a lower current flowing into the Baltic. Knudsen's relations (see Chap. XII. 5) aff'ord an estimate of the water interchange balance. It appears from this that the inflow due to the lower current over the rise on the west side of the Arkona basin is equal to the inflow of fresh water into the Baltic and that the outflow in the upper current is twice as great. Detailed data indicate that the decrease in the water amount being carried by the lower current between the Skagerrak and the Baltic is opposed by a corresponding increase in water amount carried by the upper current. Therefore important mixing processes must always act within the sea straits. Calculation of the proportion of water with a salinity of 33''/oo '" ^^^ lower current (see Table above) shows that until the Fomas section, not less than 67% of the water entering the Kattegat has mixed the water of the upper current and that almost ?0% of the remainder mixes with the upper water before reaching the Arkona basin. Thus only 1° „ of the water of 33°/oo salinity entering the Kattegat in the lower current finally enters the Baltic. The remaining 93 % mix with the upper water and return to the Skagerrak. In the same way a large part of the upper water mixes with the lower current and is carried again towards the Baltic. About a third of the water leaving the Baltic in the upper current w est of the Arkona basin returns to the Baltic and not less than two-thirds of the water in the under current flowing into the Baltic over the rises has come from the Baltic itself, and only one-third is the water witha salinity of 33700 that flows into the Kattegat in the lower current (Schulz, 1930). This applies only for the annual means. For the investigation of the water interchange in individual months the assumption of a constant water amount in the Baltic is no longer valid, since the water level shows an annual variation and other shorter oscillations. In some months the outflow from the Baltic is stronger and in others less. Investigations by Witting for the period 1 898 to 1912 indicate that there are pronounced maxima in fresh-water outflow from February to June, as well as in September. A detailed treatment of the data on currents in the Oresund and the Belts recorded between 1910 and 1916 by Danish and Swedish light-ships has been made by JACOBSEN(1925),who foundforthe period in question good agreement with the annual variation in water outflow from the Baltic found by Witting. In water interchange processes two phenomena must be distinguished. The first is the orderly steady water interchange that takes place in a strait connecting two seas of different thermo-haline structure. This interchange is associated with the two currents which are essentially antitryptic flowing along an inclined boundary surface. In addition to this continuous steady water interchange there is a second phenomenon, the total displacement in both directions of the entire water mass of the strait by the wind or due to differences in atmospheric pressure. In the Bosphorus and the Dardanelles these meteorological influences are of minor importance in comparison with the regular thermo-haline water equalization, but in the connecting straits between the Baltic and the North Sea conditions are reversed. Here the piling up of water by the wind (" Windstau ") and by atmospheric pressure differences is so strong that the regular steady interchange currents are almost completely masked. The main phenom^enon is thus an irregularly occurring, occasional transport of the whole water mass in its total vertical extent more or less in the same direction in spite of its pronounced vertical stratification. The regular steady interchange can only be obtained by elimina- tion of these irregular movements which can be achieved by taking mean values over long periods. Strong tidal effects are also present and must be eliminated by a harmonic analysis. Mean values have been calculated by Jacobsen (1909, 1912, 1913), and the mean structure at four different stations is shown in Table 144. 528 Currents in a Strait Table 142. Mean currents in the Oresund and in the Great Belt (cm/sec) ( + , directed towards the North Sea; — , directed towards the Baltic Sea) Lightship Depth (m) Lappegrunden Drodgen Schultz's Grund Southern Great Belt 0 + 37 2-5 +47-1 + 12-6 +2-4 — 5 + 31-4 + 110 +0-4 + 30 10 -7-5 + 8-8 (7 m) -9-4 + 12 15 -11-4 — -18-2 -2 20 -7-6 — -190 -15 25 -40 (23 m) — -150 -13 30 — — — -13 35 — — — -9 The lightship "Lappegrunden" hes in the most northern part of the sound, the lightship "Drogden" lies in the central part and the Schultz Grund lightship lies in the southern part of the Kattegat at the entrance to the Great Belt. For larger depth the mean upper current is directed towards the North Sea and the lower current towards the Baltic. The current profile corresponds rather well to that deduced theoretically. In the shallow waters of the Oresund ("Drogden") the entire current from the surface down to the bottom is directed towards the North Sea. As shown by Jacobsen, the internal field of force is very weak here and the great width of the channel permits cross-circulations to play an important part. This steady water interchange produced by the internal field offeree is [superimposed on the strong currents] almost always present in this area; which are produced by differences in level between the Kattegat and the southern part of the Baltic due to the piling up of water by the wind and due to differences in atmospheric pressure. These are also antitriptic slope currents and give rise to displacements of the internal front between the water bodies which here are situated side by side (Skagerrak and Baltic water) (see Pt. I, p. 182, Fig. 85). Wattenberg (1941) has made a detailed investigation dynamics of the displacement of these fronts and of the duration of the movements, and has given a basis for the estimation of mixing in the Belts and of the flow of North Sea water into the Baltic. There is a close correlation between the flow through the Great Belt (computed by means of lightship current measurements) and the changes in salinity. A rather close connection exists also with the meridional pressure gradient. The duration of inflow and outflow periods changes, of course, according to the varia- bility in the all-over weather situation over wide limits; the long-period variations are well shown by cyclic variations in the salinity, since the inertia of the water masses weakens or even completely suppresses the smaller shorter-period disturbances. Extreme positions of the internal fronts are due to prolonged inflow and outflow periods (Fig. 244). In the north the front may reach out from the Belt Sea as far as into the middle of the Kattegat. Towards the south under reversed conditions the front may in extreme cases reach the Darsser and the Drogden rises separating the Baltic from the Danish sounds. This difference in behaviour to the north and to the Currents in a Strait 529 south is due to the following facts. During outflow the upper water is not subject to any resistance and may therefore spread out arbitrarily at the surface, while during inflow the more saline water advances towards lighter water in front of it, and in this case the bottom topography exerts great influence. On passing the rises in the south the denser water sinks down to the bottom and the position of the front at the surface remains fixed near the rise. In this way large amounts of highly saline water flow into the basin of the Baltic thus renewing the stagnating deep and bottom water. Such processes are necessarily connected with long periods of weather favourable for inflow, which cause the front to remain in extreme southern position. More recently, Knudsen (Jacobsen, 1936) has organized detailed hydrographic investigations in the area to the south of Denmark. This work has been devoted mainly to the collection of accurate records for the sections between Gedser and Dars and across the Fehmarn Belt, thus providing continuous surveillance of the water inter- change between the Baltic and the Kattegat. (c) The Straits of Gibraltar and Bab el Mandeb In the strait of Gibraltar, instead of a single bottom rise there are three, all west of Cape Tarifa. The first one extends in an arc from the Cabezos reef to Punta al Boassa (maximum depth 320 m), the second one runs from Cape Trafalgar over "The Ridge" (in places only 55 m deep) to Cape Spartel (maximum depth 366 m) and the third one lies about 10-20 km west of the second with a maximum depth of over 300 m. The water interchange between the Atlantic and the Mediterranean takes place in the two channels, one to the north and one to the south of "The Ridge" and follows exactly the same principles as those outlined above. Complete scientific use has been made of the available observational data by Schott (1928 b). Longitudinal tempera- ture and salinity sections are presented in Pt. I, p. 182, Fig. 83 for the transitional period between spring and summer during which more or less mean current conditions prevail. Seasonal variations in the velocity and extent of the upper current towards the east and in the lower current towards the west are quite large. In the winter months (including April) the thickness of the upper current is small, while that of the lower current is rather large. During the summer months (to the end of October) the thick- ness of the upper current increases by 80-100 m and that of the lower current is de- creased correspondingly. During this part of the year the upper current must make up the evaporation deficit in the Mediterranean. From the limiting position of the boundary layer between the two water types it can be concluded that its annual varia- tion west of the rise is of the order of 70-80 m, while east of the rise correspondingly 100 m or little more. The current boundaries also vary by similar amounts. The water boundaries and the boundary between the currents do not coincide, but mixed Mediterranean water is carried back with the upper current over almost the whole of the area. Figure 245 gives a schematic representation of this. According to de Buen (1926), Mediterranean water does not pass westward over the Gibraltar rise in the deep layers, but is piled up on the eastern side and is carried backwards into the Medi- terranean by the upper Atlantic current with an upward motion. Analysis of the ocean- ographic series observations leaves no doubt of the incorrectness of this view of de Buen. 2M 530 Currents in a Strait Currents in a Strait 531 0 100 E 200 . 300 1400 500 Mediterranean seo 0 100 200 E ^- 300 Q. Q 400 500 7 , . • Boundary between different ,^ — water masses -^''"^_ r r /^ .■■' ^-^ . ~ Limit between currents \ of different (jirection Summer Fig. 245. Schematic representation of the water type and of the current Hmit in the inner region of the Strait of Gibraltar (according to Schott). The few sporadic current measurements that have been made in the Strait of Gib- rahar are in good agreement with the currents deduced from the longitudinal sections. The upper current towards the east is particularly strong in the middle of the strait and on its southern side. In the bays on both sides of the strait there are large vortical movements ("neer" currents). The main current is considerably affected by wind and tides and persistent easterly winds may even stop at time the inflow into the Medi- terranean. The mean velocity of the upper current core according to Nares (1872) is 34 cm/sec ; ebb and flood superimpose the mean velocity and this results in a velocity of +57 cm/sec giving an eastward flood current of 91 cm/sec and a westward ebb current of 23 cm/sec. The currents are strongest along the southern edge of the deeper southern channel and may reach as much as 210 cm/sec. The preference for the south- em side, due to the Earth's rotation, can also be seen by means of the thermo-haline cross-sections which show the Atlantic water of low salinity deflected to the right along the African side. Measurements made by the "Michael Sars" expedition (Murray and Hjort, 1912, p. 290) give the vertical current profile shown in Fig. 246. The current boundary lies at a depth of 142 m. Equation (XVI. 13) allows a computation of the average down slope of the physical sea level from the Altantic Ocean to the Mediterranean and one obtains as an average value 0-6 cm in 100 km, while the current boundary surface rises by about 15 m in 100 km. In spite of the simplifying assumptions in the theory there is satisfactory agreement between observations and theory. Direct current 532 Currents in a Strait measurements have been made by Idrac (1928) on the vessel "Pourquoi-pas" in March 1927 to the south of Tarifa. These gave the following values (depth 600 m). Depth (m) 0 100 200 300 400 500 Current towards NE 1/4 E NE1/4E W 1/4 N W1/4N W1/4N W1/4N Velocity (cm sec-^) 72 41 56 62 47 25 The current boundary surface very probably lies at 1 50 m v/hich is about the same depth as that found above. This can be compared with more recent investigations by Menendez (1956). 1/ cm/^ec 1/, cm/sec O 20 , 40 60 80 100 0 20^ 40 60 1 1 y > / ^ ?^ -•^, ^ V 200 250 300 350 Fig. 246. Vertical current distribution in the Straits of Gibraltar and Bab el Mandeb; 1 1 1 , according to the observations; • — • , according to the theory. Conditions in the Strait of Bab el Mandeb are essentially similar. The temperature and salinity distribution are given in Pt. I, p. 182, Fig. 84. Rather early current measurements have been made by Gedge (1898) in the Perim Strait at the surface and at 192 m, and they indicate a strong inflow with a velocity of at least 2-2-75 nautical miles per hour at the surface. The current intensity decreased rapidly with depth and showed a reversal in direction at 130-140 m; the speed of the lower current was variable between 1 and 3 nautical miles per hour. The research vessel "Arimondi" in 1924 made a 15-day measurement at almost the same place and a harmonic analysis of this data was made by Vercelli (1925). The results for the basic current are given in Table 143. Table 143. Velocities of the basic current in the Strait of Bab el Mandeb {March 1924) sea bottom at 175 m; depth of boundary surface at 100 m (+, inflow from the Gulf of Aden; — , outflow from the Red Sea) Depth (m) 5 20 50 100 130 150 Velocity (cm /sec) + 66 + 59 +40 + 1 -30 -68 Currents in a Strait 533 This also gives a good fit between observed values and the theoretical current profile (Fig. 246); the low value at 130 m depth is apparently due to the uncertain elimination of the tides. From the Gulf of Aden to the Red Sea the sea level falls M cm in 100 km and the internal current boundary surface rises about 40 m. Since the sea bottom from the sill out into the Gulf of Aden falls almost steadily from 150 to 0 20 40 N \ - V E 60 80 100 \, "a. - N \ Q - \ 'WM//;;, 1 1 1 I-. 1 , , cm/sec -4 To the South 4 8 12 To the North Fig. 247. Vertical stratification of the basic current in the Strait of Messina (according to the observations of the anchor station of the R.N. "Marsigli", 16-30 August 1922). about 350 m, the internal boundary surface will also decline in the same direction, so that the value given above is merely the deviation from the bottom slope. Because of changes in the direction of the wind from the winter monsoon (east to south-east) to the summer monsoon (north-west to west) the currents in the Strait of Bab el Mandeb are subjected to oscillations with a semi-annual period. In the winter the inflow into the Red Sea is a wind-drift current of strong permanence in speed and direction but larger variations are to be expected during the summer monsoon. {d) Strait of Messina The smallest cross-section in this strait between the Ionian and the Tyrrhenian Sea is at the northern end of the strait. Here it has a cross-sectional area of only \ km'* with a mean depth of about 80 m and a maximum one of about 120 m. From this sill the sea bottom slopes downward, uniformly and rather rapidly in valley form on either side. At the northern outlet the mean depth is 140 m and towards the south it is already about 900 m at about 30 km south of the sill. Since the water of the Ionian Sea is heavier, the current flows from the Tyrrhenian Sea into the Ionian Sea in the upper layer and in the opposite direction in the lower layer. Current measurements over a 15-day interval by the research vessel "Marsigli", at different depths down to 90 m at a section in the narrowest part of the strait, have been analysed harmonically by Vercelli (1926). Figure 247 shows the vertial current profile. Down to a depth of 30 m the current flows to the south, below this, down to the sea bottom, to the north. The velocities are small, in accordance with the low density differences, with on the average about 4-3 in the upper current and about 9-3 cm/sec in the lower current. Strong tidal currents are superimposed on the basic current and there are also strong 534 Currents in a Strait disturbances due to atmospheric pressure and wind variations (velocities of up to 50 cm/sec). The current transport in the strait can be estimated from the Knudsen relations (see Chap. XII. 5). For cross-sections at the narrowest point (Punta Pezzo- Ganzirri) and at the rise of Punta Pellaro to the south of this, the mean salinities are : s = 37-9, z = 38-4 and s' = 38-5, z' = 38-75%o. The equations (XIII. 19) then give as an approximation / = u and /' = m' = 2/. Under stationary conditions the transports will be the same in upper and lower currents, but the transport through the southern cross-section is twice as large as that over the rise. Thus only about half of the water of the lower current entering the southern part of the channel flows over the sill to the north, the other half is carried back in the upper current mixed with Tyrrhenian water. A corresponding calculation for a cross-section to the north of the sill shows that part of the Tyrrhenian water entering the strait from the north mixes with the lower current and is carried back into the Tyrrhenian Sea. There must therefore be large turbulent mixing processes within the strait (see Vol. II). 4. External Influences (Bottom Topography, Tides) on the Oceanographic Conditions in Sea Straits The normal steady current conditions in sea straits may be modified by external circumstances. It has already been mentioned that the atmospheric pressure and winds have considerable influence. Some idea about these influences can be obtained by simple numerical calculations. Besides these there are also other effects, especially that of the bottom topography of the strait and also those of tides, which penetrate from both sides from the open ocean into the sea strait and give rise to special current phenomena there. These latter phenomena will be discussed later in Vol. II, but it seems to be of advantage to mention these processes in connection with the funda- mental phenomenon of water interchange between two seas already here. (a) Disturbances Due to the Sea Bottom Configuration The influence of a wave-form bottom topography on a horizontally flowing current can be understood quite easily by means of theoretical computation, provided the bottom relief can be expressed in the simple form >'=-/? + y cos KX, (XVI. 14) where k = 2irjL is determined by the known wavelength of the bottom waves. The current with a velocity c over such a bottom will also take a wave-form, i.e. all layers from the bottom to the surface will follow the bottom topography but with an ampli- • tude decreasing with distance from the bottom. The sea surface itself will be a stream line and its profile is determined from ^ ^ cosh Kh{\ - (glKC^) tanh k/j) ' (XVI. 15) The denominator will be positive or negative according to whether c ^ {(glK) tanh KhY'K This expression is, however, the velocity of propagation of a wave in motion/ess water of constant depth h. If the dimensions of the bottom waves are large compared with Currents in a Strait 535 the depth, which is usually the case, then this critical velocity of propagation reduces to the value Vgh. The stationary current waves in moving water will have exactly the same form throughout the entire water layer as the bottom wave if c > ■\/sf^'-> if, however, c < \/gh then above a certain height it will be inverted, that is, above a rise in the bottom there will be a depression of the water level and above a depression in the bottom there will be a lift of the water level. If the velocity c is exactly the velocity of free waves then resonance will occur and in this case the frictional forces will be decisive. In all cases occurring in nature \/gh is always several times larger than c and the stream lines show the wave-form of the bottom with decreasing amplitude and only up to a certain height ; above this level of no horizontal motion the wave is inverted, but the amphtude is so small that these waves will scarcely be noticeable. It cannot be excluded that many of the vertical displacements in isotherms and iso- halines, which are always found at the same place, may be due to effects of this type produced by bottom disturbances. In stratified water conditions are different, especially when there are well- developed transition layers. Under certain conditions the disturbance by the bottom relief may be shown in amplified form at a boundary layer; it may even be larger than the disturbance causing it, while the surface of the water remains almost entirely unaffected. Theoretical treatment is also possible in this case (Defant, 1923). If the thickness of the upper layer is hi and that of the lower layer /zg, resonance (enlarged amplitude of the stream-line waves) will occur at two values of the current velocity. If the total depth of water h^ + Ag is small as compared with the wavelength of the bottom disturbance these values are given by the equations c,-Vk(h + h.)} and ., = y[(l-^j)j^J. (XVI.I6) The first value Cj already for small depth is many times larger than any values found in nature. Cg is the velocity of propagation of internal waves at the internal boundary surface (see Vol. II) and may be so small that it can be quite close to the observed current velocities. At these values the boundary surface will show the greatest varia- tions while the sea surface remains almost undisturbed. For example, choosing pg — Pi = 10~^, P2 = 1-028 then for larger h^ and hi = 50 m Cg will be 0-7 m/sec. Values of this order are frequently found in sea straits and it can be expected that at corresponding current velocities there will be large stationary vertical displacements in the density transition layer. The currents in the two water masses in sea straits usually have different velocity values and are of opposite directions. This case can also be treated theoretically. If the thickness of the upper and lower layer is small compared with the wavelength of the bottom wave and their velocities are c„ and Ci, then the conditions for large stationary boundary waves is given with sufficient accuracy by c] hi -{-clhl=(^l- ^^ g hi h,. (XVL17) A good example of this case is shown in the longitudinal density section through the Bosphorus in Fig. 241. The isopycnals clearly follow the outline of the bottom. 536 Currents in a Strait The disturbances are obviously due to this since the equation (XVI. 17) is approxi- mately satisfied. Putting p^ =l-028, Pa — Pi as approximately 15 x 10"^ h-^ = 25 m and h^ = 45 m, and since by observation c„:C}. = 2 then equation (XVI. 17) gives the critical velocity of the upper current as c„ = 1-77 m/sec, while the observed values lie between 1 and 2 m/sec. The upward bulging of the boundary layer in the Strait of Gibraltar and the Strait of Bab el Mandeb is undoubtedly due to the passage of the current over the rise in the middle of the strait. Bulges such as these do not occur in a plane channel. {b) Tidal Effects Since tidal currents entering a sea strait affect the whole water mass from the sea surface down to the bottom, the ebb and flood currents will be superimposed on both, upper and lower currents, either reducing or accentuating them. Since these currents flow in opposite directions the current profile will show rapid changes over a tidal period. An example can be taken of a strait 300 m deep with current reversal at 200 m in which the upper current flows east and the lower current flows west; the upper current is assumed with a surface velocity of 100 cm/sec decreas- ing parabolically with depth, while the lower current is supposed to increase below the boundary surface. The amplitude of the tidal current may be 86 cm/sec and the phase 3 moon hours (ebb towards the east at 3 h and flood towards the west at 9 h). The current structure over a total tidal period is then shown schematically in Fig. 248. At 3 h there is a current directed to the east through the entire water mass with a maximum at the surface; 6 h later conditions are almost reversed and the current is directed towards west with a maximum at the bottom. In addition to this direct influence there is also a second one affecting the boundary surface. This will perform periodic internal vertical displacements initiated by the tidal Fig. 248. Isopleths of the current velocity (cm/sec) in a water column during a total moon period with a superposition of the basic and tidal current. (Type of currents in the Gibraltar Strait.) Currents in a Strait 537 rhythm which will also give rise to variations in the oceanographic factors. It can be shown that the small periodic variations in the slope of the sea surface, produced by the passage of the tidal wave, will be accompanied by waves at the internal boundary layer of corresponding form, but of increased amphtude which will affect the normal water interchange between the two seas. A disturbance of the internal boundary surface in a sea strait due to a periodic displacement of the sea surface (tide) can be treated theoretically in a simple way. The equations of motion for both layers can be obtained from equation (XVI. 11), taking the local accelerations du^'dt and du^ldt, respectively, into account. A periodic displacement of the sea surface can be given the form Ci = ^acosA^exp(/par), (XV1.18) where the variation in the surface gradient has a wavelength A, a period a and amplitude a. These periodic vertical displacements of the sea surface give rise to corresponding variations in the upper and lower currents of the form Ml = v{z)a sin Ajc exp {/ {■r}ihy)at) and u^, = ^{z)y sin Xx exp {/ (r)lh^)at)} (XVI. 19) and these will be associated with a period vertical displacement of the boundary surface $2 = 1^7 cos Aa- exp ( / ^ at) (XVI.20) v_ hlg v{z) and ! / \ 0 y / \ /^ \ / y / \\^ n V^ \ \ , K / 1 \V ^ \ / r-. ^ /] [ ./ ^ / 37 t^ 7 ^/ N \ / \ 376 377 37 8 / / / \N ^y f / J \ / J / \'\ i \ / ■^ITi ' ■ ^, V' 18 20 22 0 2 8X1921 9X 8 10 12 14 16 18 20 22 0 2 lOX Time, hr 152 15-0 MB 146 144 142 o 140 138 136 134 132 13 0 128 o 4 6 Fig. 249. Strait of Gibraltar: periodic oscillations in the mean salinity and mean tempera- ture of the layer 100-200 m according to the observations of the "Dana" St. 1138 (5° 30' W.) (according to Schott). Table 144. Tidal current and periodic variations in temperature and salinity in the Strait of Bab el Mandeb aMOO m depth (Five semi-diurnal moon peroids) Moon hours 0 1 2 3 4 5 6 7 8 9 10 11 10^ nautical miles per hour + 89 +87 +67 -7 -61 -112 -102 -71 -36 + 36 +100 +105 Flood current Ebb current Flood current Salinity 36 "/oo 0-84 0-91 0-96 0-991 o-99t 0 89 0-78 0-69 0-59 0-52* 0-60 0-66 Temperature 25 C 0-28 0-21* 0-24 0-25 0-23 0-25 0-51 0-69t 0 69t 0-55 046 0-39 * minimum; t maximum. oscillation in the sea surface slope as 10 cm in a model of the Strait of Gibraltar between Tarifa and Gibraltar, gave an internal boundary oscillation with tidal period and amplitude of 110 m which is in agreement with the order of magnitude of the observed values. Strong well-developed internal tide waves were also found at the 15-day anchor station in the Strait of Messina. This case is of particular interest because the wave here reaches the limits of stability characteristic for such waves and at times even exceeds it (see Vol. II). 5. Processes in Estuaries (River Mouths) River water flowing into the sea gives rise to compensation currents along the river bed, which show similarities to current processes in sea straits. Ekman (1876) in an Currents in a Strait 539 investigation of Swedish rivers found that the outflow of river water in the estuary was accompanied by an inflow of sea water in the lower layers. Thus, at the mouth of the Gotaelf into the Elfsborgsfjord, there was a strong compensation requirement for the outflowing surface water which could not be satisfied by inflow from the sides. It therefore gave rise to upweUing motions from below. The consequent reverse deep current was clearly shown by the salinity distribution at different depths and could also be shown experimentally by drift buoys. The rising water was both more saline and more transparent than the sewage-laden river water. Figure 250 shows the salinity distri- bution along a longitudinal section; the upstream directed lower current is demon- strated clearly by the 20%o isohahne. Fig. 250. Vertical distribution of salinity in the river mouth of the Gotaelf. (I) 5 August 1875; (II) 19 February 1890. A theoretical investigation of the occurrence of lower currents of this type in river mouths (estuaries) was made by Ekman (1899) using principles similar to those used in the theory of currents in sea straits. He found that under normal conditions there were no currents carrying sea water upstream, but that such a current was formed immediately if there was a tangential force acting on the sea surface. The shallower the water, the greater must be the tangential pressure in comparison with the surface (river) velocity in order to allow for the generation of a compensation current in the deep water. River water entering an estuary flows on top of the sea water partly because of its inertial momentum and partly because of its lower density. It thus exerts the tangential pressure on the lower layer which favours the compensation current. The momentum and the density are apparently, however, of less importance than the density difference between the upper and lower layers and turbulent mixing of the two water types. 540 Currents in a Strait This compensation-current phenomenon probably occurs at the mouths of most rivers, especially those carrying large quantities of water but no accurate systematic investigation has been made of these processes. The situation is different for processes in the sea remote from the mouth of a river. These are easily handled theoretically (Takano, 1954, 1955) and the stratification in the sea, the vertical and lateral mixing and the turbulence of the current can be taken into account. Taking a vertical coast as the j'-axis and at this coast a river mouth where — /<>' y> I: M^ = 0, 541 (XVI.27) where Mq is the volume transport of the river flow at the mouth (which is assumed to be uniform), the solution of (XVI.26) will be given by M„ 0 = i^»<|(^ + /)tan-i- Equation (XVI.24) thus gives + / V - / (v - /) tan-1 X X (XV.28) /^-^!-/|(>- + /)tan-4-'-(.-/)tan-^-^' + 2An y + i y-l -v' - Cv + 0^ '^'' + (y - 0' (XVI.29) H 1 1 1 1 1 1 1 1 h FiG. 251. Spreading of light river water off the mouth in the ocean for different values of the horizontal exchange, (a) R = 1/500; (b) R = 2/500; (c) R = 4/500; (d) R = 8/500; (e) R = 16/500; (/) R = 32/500. Dashed curves: /= 0 (zero Coriolis parameter, non- rotating system) (according to Takano, 1955). 542 Currents in a Strait The vertical density distribution is assumed to correspond to that of the Reid model (1948) P = Po; - ^ ^ z ^ h; p= pa-Ap e^-'^^ (h ^ z ^ d) (XVI.30) where Ap= pd- Po and p = pd {d S z). This corresponds to a homogeneous top layer of thickness h with a lower layer in which the density increases to p^. Then as a first approximation 81: 2Ap 8h dP 5gAp 8h^ — ■ /->-' — and — '-^ ■ — 8x Pq dx dx 2 dx Analogous equations will apply for y and furthermore (XVI.31) /l2=- 5gAp P. (XIV.32) The integrated pressure P can be taken to represent the thickness of the upper homo- genous layer. Putting /= 0 in equation (XVI.29), that is, neglecting the CorioHs force gives Fig. 25la. Schematic representation of the spreading of river water in the ocean off the river mouth. Currents in a Strait 543 ^/ = o = lAnM^ f y + l y-l ^ \x'' + (j + 0' x^ + iy- 0' AAj^MJ i X v2 _ j2 _|_ /2 [.^2 + 0 + /)2] [.x2 + (j - O^]/- (XVI.33) If >'2 — x^ = /2 then h vanishes, that is, the lighter river water fills only the volume between the hyperbolic branches y^ — x^ = P and jc = 0. The river water flows as an upper layer over the lower layer, spreading out laterally between these hyperbolic branches. The first term in (XVI.29) modifies this simple symmetrical spreading of the river water on top of the lower water. This is purely an effect of the lateral and vertical mixing process ; it causes the homogeneous layer to be deeper on the right-hand side and shallower on the left-hand side. The inflow is thus directed to the right in the Northern Hemisphere. Figure 25 1 shows the limits of the river water for the different cases A^ 500 2 500 4 500 500 16 500 and 32 500 where the dashed curve is for / = 0 (non-rotating system). Table 145 2/ in m : a b c d e / 200 5.0 X 106 2.0 X 106 1.25 X 106 6.2 X 105 3.1 X 105 1.6 X 105 600 4.5 X 10^ 2.2 X 107 1.1 X 107 5.7 X 106 2.8 X 106 1.4 X 106 1000 1.26 X 108 6.2 X 107 3.1 X 107 1.6 X 107 7.8 X 106 3.9 X 106 2000 5.0 X 108 2.0 X 108 1.25 X 108 6.2 X 107 3.1 X 107 5.6 X 10' Exchange coefficients for the cases shown in Fig. 257 are contained in the following Table 145 for a corresponding river mouth width 2/ and for/= 10~^ sec~^. The Coriolis force deflects the seaward flow towards the right and gives rise at the mouth of a river in the Northern Hemisphere to a water level sloping from the right bank down to the left bank. For the lateral exchange coeflicients found in practice, 10^ to 10^, and for a river mouth width between about 300 m and 1 km there will be quite a sharp deflec- tion to the right (approximately as in curves d to/). The flow of river water into the sea at the mouth of a river is shown schematically in Fig. 251a and conditions actually found in nature will probably correspond reasonably well to this. Chapter XVII Effect of Wind on the Mass Field and on the Density Current Under stationary conditions all the forces acting must be in equilibrium and the mass distribution must be adapted to this equilibrium if it is to be maintained. In this case it is not possible to distinguish between cause and effect; there is usually a mutual adjustment between the internal field of force and the current present. If there is a change in the field of force then there must also be a subsequent change in the current; conversely if there is a change in the current there must be a rearrangement of the field of force until equilibrium is again restored. These circumstances should be kept in mind for an understanding of the way in which wind influences density currents. 1. A Limited and Stratified sea Conditions in a limited trough-like sea shall be considered first. Work in this direction has been done by Palmen (1926, 1930 a, b and with Laurila as co-worker, 1938) for the Gulf of Finland and the Gulf of Bothnia, principally in particular cases which are only able to give some insight into the mechanism of the processes which occur. The influence on the water stratification occurs as follows : We assume at first no wind at all over a barotropic sea ; the isosteric surfaces and especially the transition layer between the top layer and the deep water will then follow level surfaces (Niveauflachen). If a wind starts, the surface waters are forced to move first in the direction of the wind, but the Coriolis force will soon produce a deflection to the right (Northern Hemisphere) and a piling-up of the water along the sea coasts. In an elongated ocean bay the final result will be a current predominantly occurring along its longer axis. In addition to the wind-generated current in the top layer a gradient (Stau) current is then added in the deeper layers due to the piling up of water which will flow approximately in the opposite direction. Thus a vertical circula- tion in a longitudinal direction is set up and an equilibrium state is present in which the transport due to the surface current is exactly balanced by that of the deep current. This quasi-stationary state of the current is fixed at each level by an equilibrium between the gradient force, the Coriolis force and the frictional force. Since a stronger current is only possible along the longitudinal axis of the bay it follows that the direc- tion of the gradient force usually does not coincide with the direction of the current itself but the deviation will not be great. In addition to the principal gradient in a longitudinal direction in the layers above and below the level of current reversal (layer of no motion) there will also occur smaller components of the pressure force acting at right angles to the direction of the current. These will be largest at the surface and 544 Effect of Wind on the Mass Field and on the Density Current 545 will decrease with depth-changing sign at the layer of no motion. This will modify the mass field which then can no longer remain barotropic. The isosteric surfaces must slope transversally ; the mass field becomes baroclinic. The structure of the associated density current can be computed by means of ordinary methods from this mass field. The primary factor will now no longer be the water stratification but rather the current, while the water stratification can be regarded as a consequence of this current. Palmen investigated data for the Gulf of Finland for steady westerly and steady easterly winds and distinguished between a west type and an east type. He deduced mean mass fields over a cross-section for these two cases from the large amount of data available. In the east type the lighter surface water lies in a wedge-form at the Finnish coast with the isosteres sloping downwards from south to north, while in case of the west type conditions are reversed. Figure 252 shows the distribution of density for the two opposite types. The interpretation is simple: the west wind produces a drift current in which the transport is directed towards the Estonian coast where the lighter surface water will pile up. For an east wind the opposite occurs. Palmen has demonstrated the reality of these changes in sea level between the northern and southern sides out of observations of water level in Hango, Reval and Helsinki. For the east Estonio Finland J 100 Fig. 252. Normal density distribution in the cross-section Aransgrund-Kokskar (Fennic Bay, 25"' E.); at, values. , east type; , west type (according to Palmen). 2N 546 Ejfect of Wind on the Mass Field and on the Density Current type the difference in water level was 4-4 cm and for the west type this difference is 3-4 cm. The absolute velocity of the wind-generated surface current will thus be for the east type 6-9 cm/sec towards the west and for the west type 5-3 cm/sec towards the east. Current measurements give 7-5 and 6-0 cm/sec, which is in good agreement. The relative changes in velocity with depth can be calculated by ordinary methods (equation XV.20) from the mass field and can then be converted to absolute velocities using the surface velocities given above. Table 148 containing these values shows clearly the division of the current structure into two layers; at the middle of the Gulf of Finland the current reversal is at a depth of approximately 27 m. It changes in a corresponding way towards the Finnish and Estonian coasts. The calculated values are a little too large, since friction has been neglected, but otherwise are in satisfactory agreement with observed values. In some special cases for a strong wind and steeper inclination of the isosteres in the transverse section, the velocities are much greater (for instance, 7 October 1936; surface velocity 23-5 cm/ sec) and the layer of no motion occurs at greater depth (about 35 m) in full agreement with the observed values. Table 146. Current stratification for different wind directions in the Gulf of Finland (according to Palmen) (positive sign towards west; negative sign towards east) Depth (m) 0 10 20 30 40 50 60 70 Velocity (cm/sec) For east type For west type . + 7-3 -5-3 + 51 -3-7 + 1-8 -11 -0-9 + 1-3 -3-3 + 3-7 -4-3 +4-6 -5-3 + 50 -5-3 + 5-3 When the wind is in a direction other than directly east or west only the eastern or western component will have any effect. The inclination of the isosteres in the trans- verse section will therefore be correspondingly less and the number of solenoids will thus be reduced and must therefore show a dependence on the wind direction. The rearrangement of stratification caused by the wind in an elongated oceanic region will thus proceed in the following way: (1) A steady wind with a component along the longitudinal axis of the sea will originate a vertical circulation; this will be made up of a drift current in the top layer and a corresponding gradient current in the deep water. (2) This current system will produce a vertical transverse circulation which in turn will give rise to an inclination of the density transition layer and of the isosteric surfaces, that is, the longitudinal circulation produced by wind will give rise to a solenoid field at right angles to this circulation. The strength of this field will be a function of the wind influence. When an equilibrium state is reached this cross circulation will vanish. (3) A transverse slope in the physical sea level will develop at the same time and its intensity will also be dependent on the wind. (4) From the solenoid field and the transverse slope of the sea surface the current structure in a transverse section can be calculated. In a steady equilibrium state the slope of the internal boundary surface in a two-layered sea will be greater than that Effect of Wind on the Mass Field and on the Density Current 547 of the physical sea level in the ratio Pi:(p2 — Pi). It is easily shown that this slope is given by •_ ^ g{p2. — Pi)hi where pi and p, are the densities of the top and lower layers, respectively, h^ is the thick- ness of the top layer when the system is at rest and T is the shearing stress of the wind. The deep water is assumed to be motionless. This relationship has the same form as the equation (XIII.45) which gives the piling up of water by the wind (Windstau) in a homogeneous sea except that p is replaced by the density difference (pa — pi). Hellstrom (1941) showed that in a stratified sea with two layers the piling up of water by the wind differs markedly from that in homogeneous water and that the effect of the wind is larger. The wind stress calculated from equation (XIII.45) (p. 419) is much too large, and the less the depth of the discontinuity layer the greater is the error. Palmen's investigations, however, showed that the changes in water level in the Baltic due to the effect of the wind are almost independent of the water stratification. This contradiction was resolved by Palmen (1941) by estimation of the time required to establish an equilibrium state. This time required is very large, of the order of several days, while only a few hours are needed to produce a piling up of the water similar to that for homogeneous water. Usually, the wind direction does not remain invariable for a longer time to allow the slopes of the discontinuity layer and the sea surface to reach a steady state. Initially, the piling up of water by the wind in a stratified sea is approximately the same as in a homogeneous sea. However, the longer the duration of the wind the closer is the approach to the Hellstrom values. The equation (XIII.45) can thus be used in almost all cases for the calculation of the wind pressure, although strictly it is valid only for homogeneous water. Fjelstad (1946) has made a thorough theoretical examination of steady currents in a stratified water contained in a wide channel and has obtained results in complete agreement with the observations. The transverse circulation is usually connected with another important pheno- menon. In a sea of sufficient width a strong wind may produce an inclination of the density transition layer sufficient to bring the deep water to the sea surface. A rapid fall in temperature will then occur and an increase in salinity in a long band along the coast to the left of the current (Northern Hemisphere). The phenomenon of "cold upwelling water" along an extended coastline has previously been regarded largely as a direct result of an offshore wind (land wind) (Sandstrqm, 1922; Krummel, 1911, p. 536 and following), forcing the deep water upwards to the surface at the lee coast while the surface water is forced downwards to deeper layers at the windward coast (luv- coast). Besides this direct effect, the effect of earth rotation in the above senses, seem however, of more importance. In the Gulf of Finland and in the Baltic (Mae, 1928) the upwelling of cold water found during strong persistent longitudinal winds gives support to the importance of the indirect wind effect, 2. General Conditions in the Open Ocean These are essentially the same as in channel-form elongated oceanic regions. The efiFect of the wind is mostly restricted to a more or less broad band of the sea surface, and outside this area the water is either motionless or subject to the effect of a wind 548 Ejfect of Wind on the Mass Field and on the Density Current from another direction. Thus, for example, in a broad band of an oceanic region with vertical increase of density and forming a channel around the earth in the Northern Hemisphere, conditions will be more or less as follows. If there is a persistent wind in the direction of the channel the immediate effect of the drift current (westerly wind) is to transport lighter surface water to the right (south) side of the channel. In the top layers the isosteric surfaces can no longer be horizontal and will adjust with an inclination from north to south in order to corres- pond with the accumulation of lighter water on the right-hand side of the wind. A solenoid field of this type will, however, produce a density current in the direction of the wind in which the velocity will decrease with depth corresponding to a similar decrease in the slope of the isosteres. At the same time, water will be piled up on the right-hand (south) side of the channel and this will give rise to a gradient (Stau) current in the direction of the wind. Its velocity will remain constant down to the lower frictional depth. In this way the stratification will lead to a considerable complication of the conditions and even more so if changes due to other factors (heating, cooling, evaporation and others) must, too, be taken into consideration. It is doubtful whether a gradient (Stau) current will be generated in such a current system. The displacement of the water masses in the top layer, where the solenoids are numerous and which is superimposed on deep water where the solenoids are few, may proceed so that the isobaric surfaces in the deep water remain horizontal (see discussion on p. 483 and following pages). If the effect of the water accumulation (rise in physical sea level) occurring on the right-hand side of the wind direction (Northern Hemisphere) on the pressure field of the deeper water is compensated exactly by the baroclinic mass distribution of the top layer there will be no gradient (Stau) current. In actual practice, the relationship between the topography of the physical sea level and the mass structure of the upper layers is usually satisfied so that any deep reaching slope current is improbable. A complete theoretical treatment of the problem of currents in a baroclinic ocean offers considerable mathematical difficulties, since it must take into account vertical frictional effects, lateral mixing processes and boundary-surface conditions. In con- nection with an investigation on the circulation of the antarctic circumpolar waters, SvERDRUP (1933) has discussed the possibility of formation o{ o. steady drift current in the presence of a baroclinic stratification of the water masses. He showed, in agree- ment with the results of Ekman, that steady vertical circulations can hardly develop in the ocean if only the effect of wind is taken into account. Due to the non-uniformity of the wind field (divergences and convergences), and due to the boundaries between different water bodies and the coasts, vertical circulations will be formed and will produce changes in the mass field. However, since the density distribution in the sea is usually a stationary one and apparently steady circulations still occur, it follows that the effect of the vertical circulations produced by wind must be compensated by other factors which affect the density. This gives emphasis to the great importance of these factors for the development and maintenance of the oceanic circulation. Heating, cooling, evaporation, precipitation and other factors thus take part indirectly in the formation of the oceanic circulation. The convective sinking of cold waters in higher latitudes plays an especially important part for the maintenance of vertical oceanic circulations. Ejfect of Wind on the Mass Field and on the Density Current 549 Ekman (1931) has drawn attention to a special effect of the wind on a given solenoid field. In a top layer (the place where density currents occur) the isosteric surfaces are assumed to rise from south to north (Northern Hemisphere; approximately the conditions found in the Atlantic between 40° to 50° N. and 30° to 40° W.). In the absence of wind there will be a density current directed towards the east. If now a steady persistent wind gives rise to a drift current, thus altering the mass field, then, for a northerly wind the total transport of the drift current will be directed to the west and for a southerly wind to the east. The basic current therefore will be either retarded or accelerated. An east wind blowing against the current will produce a transport of the upper water to the north and will thus tend to even out meridional density differ- ences, and in this way to decrease the velocity of the density current. If the wind blows towards the west (as in the Atlantic over the Gulf Stream), then the upper layer will be driven towards the south and the slope of the isosteric surfaces will increase. As long as only the total system of surfaces without internal change is displaced towards the south the strength of the density current, which is largely fixed by the horizontal distances between the isosteres, will remain unchanged; however, under certain con- ditions changes in inclination of these surfaces will also occur and the density current will increase its strength. This is especially the case when the upper lighter water is displaced by the wind, while the lower one remains unaffected. The wind blowing in the direction of the density current, in addition to the generation of a drift current, also has the effect of localizing the density current and may transform an otherwise broad and slow current into a narrow rapid one, still with the same transport. Ekman saw in this process an explanation for the narrowness to which the Gulf Stream is confined in this part of the Atlantic. This peculiar phenomenon of a "river in the sea" is in any case an argument in favour of such wind effects. Another example of wind effect on the mass field is the boundary surface found throughout the interior of the entire Antarctic Ocean which appears at the sea surface of the ocean as the Antarctic Convergence Line (Southern Hemisphere Polar Front). This boundary surface separates the heavier, colder, Antarctic water to the south from the lighter but more saline water of the oceanic troposphere to the north. The boundary surface has a slope corresponding to the density and current conditions. It behaves like a solid wall (continental slope) and makes an Antarctic vertical circulation possible. Figure 253 (Sverdrup, 1933a) shows a meridional density section at 30° W. derived from the observations of the "Discovery" expedition. The boundary surface meets the sea surface at 50° S. in the Antarctic convergence line. The topography of the physical sea level and the 1000 decibars surface (both relative to the 3000 decibars surface) are shown in the diagram above. These isobaric surfaces slope downwards from north to south corresponding to the current flowing eastward in both water bodies; this current must be stronger on the northern side than in the Antarctic water to the south. The cause of the formation of a discontinuity surface is not immediately apparent, since the current flows exactly towards east in all latitudes and meridional current components are required in order to produce and to maintain it. Two factors favour the occurrence of a northward component in the Antarctic water. (1) The prevailing westerly winds, and 550 Effect of Wind on the Mass Field and on the Density Current Fig. 253. Vertical section of density (a,) in the Atlantic Ocean along 30" W. between 24° and 58° S. Above: topography of the physical sea level and of the 1000-decibar surface (relative to the 3000-decibar surface assumed as plane). A.C., Antarctic convergence (oceanic polar front). (2) the continuous supply of water with low salinity which is produced by melting of the northward drifting pack-ice. This second factor requires the presence of a thermo-haline circulation directed at the surface from an area with high specific volume to another one with a low specific volume. A circulation of this type is certainly present but the wind conditions are probably the main cause (Deacon, 1934; Sverdrup, 1934Z)). In latitudes between 40° to 65° S. the prevailing wind is always westerly and gives rise to a drift current and a consequent surface water transport to the north. According to meteorological obser- vations the strongest surface wind in higher latitudes occurs between 50° and 60° S. The water transport to the north is thus greatest between 60° and 50° S. and north of 50° S. is comparatively smaller. This gives rise to the formation of a convergence line and a discontinuity layer in the mass field. The wind and its differentiation in a meridional direction may also be considered the main reason for the intensification and concentration within a narrow strip of the density current which would otherwise spread out over a wider area. 3. General Relationships Between Wind and Currents The investigation of steady currents produced by wind in a baroclinic top layer is easily handled, since the deep water can be regarded as essentially motionless and the wind field as quasi-permanent showing no changes with time or position. This allows the eff'ects of both the vertical and horizontal eddy viscosities to be taken into account. The equations of motion (XIII. 52) must then include terms for the horizontal eddy viscosity, denoted briefly by h^ and h^. Integration of these equations over the entire depth d and introduction of //. j: /^. dz, H, = h„ dz and P pdz (XVII.2) Ejfect of Wind on the Mass Field and on the Density Current 551 gives ^+/M, + r, + //, = 0, ~^-fM, + Ty + Hy = o. (XVII.3) Therein M is the vector of the mass transport (equation XII. 8, p. 376). To these must be added the continuity equation for an incompressible fluid. For a given value of T and ignoring the effects of the horizontal components of the eddy viscosity the three equations (XVII.3 and 4) can be regarded as equations with three unknowns P, M^ and My. Thus, in such a baroclinic current the total pressure P and the mass transport M can be represented as functions of the wind stress. Ehmination of P by cross-differentiation, taking into account equation (XVII.4) and putting ^ = df jdy gives (f-i)+^-.+rf-t)-- According to this vorticity equation the wind-stress vorticity must be balanced at every locality by the vorticity of lateral mixing and by the term /SMy, which is the effect of the change of the Coriolis parameter with latitude. This equation is reminiscent of the equation (XIII. 59a) derived by Ekman who designated the term ^My the planetary vorticity. SvERDRUP (1947) and Reid (1948) have applied this equation to the equatorial currents of the eastern Pacific Ocean which correspond closely to the above conditions. The X-axis is taken pointing eastward and the >'-axis pointing northward. For the trade wind belt it is possible to put dTy/8x = 0 so that neglecting lateral mixing, (XVII(.5) gives ^My == - ^' (XVII.6) and with (XVII.4) and M. = .-^(?^' tan 0 + i? ^^) (XVIL7) 2ajcos0\ej ^ 8y^ / cP — dT^ ox dy and dP ^ 8^T^ ^=-^^^^^^^'^ + ^- Thus for X == 0, (at the north-south vertical boundary), M^ = 0 (integration limits 0 to Ax). The bars denote average values of the stress derivatives. The mass transports Mg and My can be found directly from (XVII.3) if dP/dx and 8PJdy are known. 552 Effect of Wind on the Mass Field and on the Density Current These equations have been tested by the "Carnegie" and "Bushnell" observations of corresponding areas (approximately between 160° to 80° W. and 10° S. to 20° N.) and showed good agreement with the values derived from the observations. The theoretical values were calculated from the distribution of wind stress obtained from the wind field given in oceanic climatological charts; thereby use has been made of formulae (XIII.48 and 49). Figure 254 shows the excellent agreement between the ob- served and theoretical meridional distributions oi APjAx and M^. It should be kept in 25M 25 V^' fy\ i ^'^ -^._^^ 20 - ^ ^\ ■ /;(r1=-ff ton ^ jp- + r. ^ \ 4^0ctNov grodients \ JXCarnege stations ) ^ J/'t>Jov-Mar Qfodienls 10 - / jr Carnegie a SusHnell '^ y stations ° 0 / /^5 - ° ( 0 1 1 1 I 0 ||'a/f(r)(dyn.cm-^ 1 1 -20 -1-5 • -1-0 0 / ° 05 10 0 y -5 - ^ -10 - Fig. 254. Picture to the left: theoretical and observed values APjAx in two sections of "Carnegie" and "Bushell" stations. Picture to the right: Latitude dependence of the longi- tudinal mass transport computed by two independent methods. (M^ = eastward mass transport in tons per sec through a column of 1000 m depth and 1 m width). mind that the theoretical values are derived from mean wind conditions while the ob- served values are based on some oceanographic stations made at different times of the year. From these results it can be concluded that mass structure and mass transport of the currents in the eastern equatorial areas of the Pacific can be regarded as a con sequence of the average shearing stress of the air currents on the surface of the sea. This conclusion should also be valid for the equatorial currents in other oceans. 4. Velocity Computations of Oceanic Surface Currents in the Equatorial Regions from Wind Data The currents in the equatorial regions can, as a first approximation, also be regarded as the result of a drift current and a gradient current of the type described by Ekman. However, at the equator itself the two components are indeterminate and the geo- strophic approximation gives infinitely large values. In dynamic calculation these areas must therefore be excluded. The question of how to calculate the currents in the immediate vicinity of the equator from oceanographic data has been dealt with by Weenink and Groen (1952), which gave an exact solution to the problem and by Effect of Wind on the Mass Field and on the Density Current 553 TsucHiVA (1955fl, b) who made a second approximation to the geostrophic current equation for/ = 0. For the surface velocity of a drift current and a frictionless gradient current the equations (XIII.26 and 31) give Tcos(ifj — 77/4) - - ("A rsin(iA-7T/4) V(fpoV) ~^ fp< r (XVII.6) where 0 is the angle between the wind stress and the direction of the ^-current; the subscript zero refers to the sea surface. Indeterminate solutions are obtained from (XVII. 6) for the equator. If an exact solution is required the eddy viscosity cannot be taken as insignificant by comparison with the pressure gradient and the Coriolis force. Only in this way there is an equilibrium between the wind stress, the pressure force and the vertical friction in the equatorial belt. The simple equation of motion (corresponding to (XIII. 23fl) and (XIII. 30) is now where V = Vjc ^ iVy and p ~ Po- The boundary conditions are L^] =.-T=-iT, + iTy) and y(z = O)) = 0 (XVII.8) (XVII.7) is identical with ry9 " av^b, (XVII.9) cz where a = — and d = [^ + ' ^ 7] 7] \dx cy If b{z) is known from observations then, taking equation (XVII.8) into account and since a is independent of z this can be solved. To determine b{z) Weenink and Groen used the Reid model (1948) which gives a good approximation for the equatorial regions. This postulates a homogeneous layer of thickness h below which the density of the water increases with depth according to an exponential function (see XVI. 30). For this model (as in XVI. 31) one obtains ldp\ Ap8h /8p\ Apdh and the solution of (XVII.9) at the surface (z = 0) will be „„ = Jl _ *« (, - 1+^%-H A (XVII.IO, 7]^/a a \ 1 + h\/a J When the value of h\/a or of /is large the expression in brackets will equal 1 and (XVII.IO) will be nearly equal to (XVII.6). It is thus apparent that at a latitude of 2° to 554 Ejfect of Wind on the Mass Field and on the Density Current 3° the value of h\/a is already large enough to allow equation (XVII.6) to be used instead of (XVII. 10). For a sufficiently narrow belt on both sides of the equator expansion into a power series with respect to h\/a gives, neglecting higher order terms t^o = :^ (^ - lb A + Ab,h^ + . . . (XVII. 1 1) If lateral mixing is neglected (// = 0) the equations (XVII. 3) become T = AP-^ifM (XVII. 12) and (XVII. 1 1) with (XVI.31) becomes Vo - 4boh^ + — ^ + . . . (XVII. 13) Po Since M remains finite at the equator this gives finally by means of (XVI.31) and (XVII. 12) %Th Vo=--F~ • (XVII. 14) The behaviour of v^ can be illustrated in the following way. If the first term on the right-hand side of (XVII. 10) is the drift current and the remainder of ^o is taken as the slope current, then both components tend to infinity on approaching the equator, but due to the coupling between these two components they behave in such a way that their sum remains finite and approaches the vector (XVII. 14) as a limit of zero latitude. The surface current, the wind stress and the surface pressure gradient all have the same direction at the equator. Figure 255 illustrates their behaviour near the equator. wind Fig. 255. The two components (vwind and Wgrad) of the current velocity (ftot) somewhere near the equator. Exactly at the equator the vectors of the current velocity, the pressure gradient Ap and the wind stress T fall all in the same direction. More recently Yoshida (1955) has shown that the model used by Weenink and Groen apparently leads to a solution involving a discontinuity in the vicinity of the equator. This singularity originates in the assumptions of the model. A modification of the model which seems more realistic in the light of recent observations appears to give a reasonable solution. Ejfect of Wind on the Mass Field and on the Density Current 555 The method of Tsuchiva is simpler. The equations of motion of the geostrophic current are where D is the geodynamic depth. All the quantities in these equations can now be expanded into the Taylor series with respect to y and equation of terms of the same power of V gives, putting /S = df jdy, (-)^^0;,.„^-(P)^.„. (1)=0; .^0. (XVn..) The distribution of D is easily found from oceanographic data. The east-west com- ponent ?/o of the current velocity at the equator can therefore be obtained from the second equation (XVII. 16) and the north-south component Iq is zero. At the same time {8D/cx)o and (cD/8}^o must be zero. The oceanographic data show that these conditions are fairly well satisfied in most cases. Values ot u and v near the equator can be obtained by substitution of higher-order derivatives of u and v into expansions of these quantities. In a later paper Tsuchiva has also dealt with the effects of the inertia and frictional terms but these do not seem to alter the previous results. In the immediate vicinity of the equator the east-west velocity component of the current is determined by the curvature of the isobaric surface in the meridional vertical section and not by the slope. The geostrophic approximation for the ocean currents can be used much closer to the equator than has so far been done. The method used by Tsuchiva is purely mathematical and not founded on any physical basis. Cliapter XVIII Basic Principles of the General Oceanic Circulation 1. Introduction The ultimate cause of all movements in the sea is the supply of energy by solar radiation. The meridional variations in the energy supplied lead to regional differences in the structure of the oceans. The oceanic circulation modifies, however, the distri- bution of temperature and salinity, which are basically determined by the climate, and also affects the distribution of dissolved gases in the sea; it therefore has an indirect influence on the distribution and accumulation of marine life. The general oceanic circulation is therefore the fundamental problem of oceanography. The transformation of solar radiation into heat in atmosphere and sea takes place mainly in the layers close to the interface between air and land, between air and water, respectively. Other important influences from the hydrosphere on the atmosphere and the reverse are also localized at the sea surface and in this way the sea surface becomes one of the most important interfaces of the earth; it is the starting point of both the atmospheric and the oceanic circulation. The principal factors involved in these, such as the solar and sky radiation, outgoing radiation, evaporation, precipitation, melting of ice and the wind stress on the water exert their major effects here. In com- paring the atmospheric and oceanic circulation the special circumstance should be kept in mind that the interface (sea surface) which is decisive for the initiation of vertical motions is situated below the atmosphere but above the sea. Therefore, in order to start a vertical circulation in the atmosphere air must be lighter than the surrounding air masses (rising motion), while in the ocean water as compared with the surrounding waters must be denser (sinking motion). The variable position of this interface, from which the vertical circulations originate, causes corresponding differences of the circulation system (Defant, 1929). According to the general causes, mentioned above, of steady water movements in the sea, two fundamental factors stand in question: (1) the internal field of force of the mass structure, and (2) the external field of force due to the winds. Other less important external forces such as the supply of water by precipita- tion or its removal by evaporation are less effective than the wind forces (see p. 572). These two basic factors act quite differently on the water movements and an under- standing of the general circulation can only be based on the resultant of the two effects. Most investigations have been limited to the components of motion of the 556 Basic Principles of the General Oceanic Circulation 557 circulation in a meridional plane with only supplementary extensions to three- dimensional space. This has no doubt been unavoidable in the past due to the lack of sufficient observations, but a complete understanding of the oceanic circulation can be obtained only in terms of spatial phenomena. The magnitude and the complexity of the problems makes it understandable that a solution in full detail has not yet been obtained and probably will not in the near future, but the accumulation of further data and the advance of theoretical knowledge will lead closer to a comprehensive elucidation of the mechanism of the general oceanic circulation which is the aim of oceanography. The permanent oceanic currents can be divided into three groups according to their genetic origin: (1) currents produced by thermo-haline convection, mainly due to cooling of surface water in higher latitudes; (2) currents produced and maintained by the transfer of wind energy to the sea surface ; (3) currents maintained by the excess of precipitation over evaporation, or vice versa occurring in special oceanic regions. Each of these types of flow shows a different physical behaviour and acquires on the rotating earth an individual form, which is also strongly influenced by continental slopes acting as barriers for the oceanic movements. 2. Oceanic Sea Surface Currents (a) Charts of Sea Surface Currents It has taken quite a long time until data on sea surface currents were that numerous as to allow a reliable representation of the currents over the entire ocean surface. Charts of currents presented in ordinary atlases are seldomly based on critically tested observations and are often constructed making hypothetical assumptions. As amount and density of the observational material (current measurements) increased, charts of current conditions over smaller oceanic areas could gradually be extended until finally world maps of ocean currents could be constructed. At the suggestion of Neumayers (1898), Schott prepared a world chart of ocean currents. A new edition of this was published in 1942 incorporating in an excellent manner the oceanographic progress of the last 40 years. This chart (Schott, 1942), Deutsche Admiralitatskarte no. 1947, 2 sheets, 1942) shows the total earth for the Northern Hemisphere winter and an inset map for 30° N. to 20° S. shows seasonal variations for the tropics during the Northern Hemisphere summer. North of 50° N. the chart represents more summer conditions for which the data are more numerous. This current chart is reproduced in Plate 8 on an equal area projection. The use of current arrows has been simplified in places: velocities are indicated at \ knot intervals with a lower limit of 12 nautical miles in 24 h and an upper limit of 36 nautical miles in 24 h. Differences in velocity are indicated by the thickness of the arrows and the constancy of the current by the length ; the last factor was expressed in four degrees : variable, fairly steady, steady and very steady corresponding roughly to 25, 25-50, 50-75 and 75% flow displacement in the direction of the arrow. Naturally in such large-scale charts only a somewhat general representation of the currents can be given and some subjective interpretation is always possible. Details in the infrequently navigated parts of the ocean are, of course, 558 Basic Principles of the General Oceanic Circulation highly deficient and must be supported by theoretical deductions. For details in parti- cular areas of the ocean, reference must be made to special charts; the literature sources will be indicated below. As is apparent from the current charts in Plate 8, the more schematic distribution of oceanic currents known from earlier work is really present to a large extent in all oceans. Northern and southern equatorial currents characterize everywhere the tropical surface circulation and are usually separated by an equatorial counter current flowing in the opposite direction, while the surface circulation of higher latitudes is composed principally by the West Wind Drift and the Polar Current. Separation of these current regions gives convergence and divergence lines which are specially indicated in the current chart. They are rarely clear-cut lines; instead they are usually rather wide areas intruding between individual currents. It is often difficult to deter- mine their position accurately since they move backward and forward periodically in time. The connection of this surface current system with the currents of the deeper layers lies in these singularity areas, and they are thus of great importance. In the following sections a brief description will be given of the surface-current conditions in the individual oceans and of their seasonal variations. The dynamics of single currents will be dealt with later. {b) The Surface Currents of the Atlantic Ocean The backbone of the system of currents present in the Atlantic is formed by the two equatorial currents; that in the Southern Hemisphere is the stronger one and is more constant and of greater extent. During the whole of the year this current crosses the equator from west of the island of St Thome until the South American coast. The meridional distribution of the current intensity shows a double current core for nearly all months; one of the two just north of the equator at about 1° to 2° N. and the other one at about 4° to 5° S. (especially between 20° to 30° W.). Between them along the equator is the equatorial region of divergence which belongs to the tropospheric deep sea circulation (p. 595). This divergence coincides with the tongues or island of cooler water that are shown in temperature charts, particularly in the period from June to August and indicate the upwelling of deep water accompanying the diver- gence. In the central part (8° to 40° S.) the South Equatorial Current is most intense from June to July and hardly drops below 20 nautical miles in 24 h. The southern current core divides into two parts at Cape San Roque — one turning south and be- coming the Brazil Current, and the other joining the northern current core in the latitude of the Amazon estuary to form the strong Guiana Current flowing along the South American coast. The Northern Equatorial Current is less constant in extent and strength. Its northern boundaries fluctuate, but from about 20° N. its itensity decreases and it passes into an extensive region of weak and variable currents with frequent motionless areas. South.of 20° N. its average intensity is about 15-17 nautical miles in 24 h. Schumacher's monthly charts (1940) which give greater detail show the eff"ect of the bottom topo- graphy on the current system where it passes over the mid-Atlantic Ridge (see p. 435). During the winter months when the equatorial counter current is very weak the North and South Equatorial Currents flow together along a convergence line from about 20° W., 4° N. to approximately 50° W., 11° N. but during the summer months Basic Principles of the General Oceanic Circulation 559 when the counter current is more strongly developed this only occurs between 50° W., 10° N. and 60° W., 14° N. From here a combined current runs in a westerly direction towards the West Indies throughout the whole year; this is the source for the surface currents in the West Indies and therefore also for the Gulf Stream (Dietrich, 1937 b; 1939), which is in agreement with the results of Brooks (1930, see also, Shaw and Hepwort, 1910) showing that the fluctuations in the south-east trade winds are more closely connected with water and air temperatures in Western Europe than are those of the north-east trade winds. The Equatorial Counter Current lies between the two equatorial currents. Table 147 presents its position in different seasons. During almost the whole of the year it is divided into two parts; the "western" counter current weak and not very broad, found particularly during the first winter months and the "eastern" counter current which is present all the year round. Only in the summer months do they join, thereby forming a mighty counter current. The origin of this lies west of 50° W., near the American coast, its width covers the area between 10° and 3° N. showing considerable speed and constancy. During the period of its greatest extent the central area of the current is characterized by a convergence region towards which water flows from both sides. An attempt has been made by Schumacher (1940) to show a connection between the temporary interruptions in the counter current above the mid-Atlantic Ridge and the topography of the rise. Table 147. Extent of the Equatorial Counter Current in the Atlantic Ocean (according to Schumacher) Region with Western Counter Current Eastern Counter Current no currents (deg. lat.) January 53' W. 10° N. until 37° W. , 6°N. 26°W., 7°N. ^ 19° 5° 11 February 49° 90 until 41° 6° 22 March 53° 10° until 47° 7° 20° 4° 27 April 52° 90 until 37° 0° 24° 4° 13 May 47° 6° until 33° QO** 28° 5° 5 June 51° 9° until 38° 3°** 36° 5° until the 2 July 51° August 56° 90 10°* — ^African coast 0 0 September 52° 10°* — 0 October 53° 10°* — 0 November 54° 10°* until 32° 8° 31° 7° 1 December 51° 90 until 30° 6° 29° 6° J 1 * Starts presumably farther north-west,** with interruptions. Northern Hemisphere. The combined equatorial currents enter the Caribbean Sea between the Antilles and spread over almost its entire width as the Caribbean Current; this flows almost due west with its greatest velocities in the southern part. In some months large vortices are formed off" the coast of Costa Rica, Panama and Colombia. 560 Basic Principles of the General Oceanic Circulation 100° 30 20 - 100 30 20 - !00 Fig. 256. Schematic picture of the sea surface currents in the Gulf of Mexico (according to Schumacher). The current then enters the Gulf of Mexico through the Yucatan Channel with veloci- ties of up to 3-7 knots at the current core. The currents of this mediterranean sea are shown in Fig. 256 (Schumacher, 1940). The major part of the stream lines leaving the Yucatan Strait tend to circle or cross the Gulf clockwise following the shelf line. The branch that flows directly to the Florida Straits is stronger and is steady only during the winter months. The eastern branch of the Yucatan Current forms the Florida Current the water transport of which is the main source of the Gulf Stream. No other ocean current has been so intensively investigated as this. An enormous amount of literature has been accumulated on the subject that is impossible to cite here in detail. The water piled up in the Gulf of Mexico flows out through the Florida Straits towards the north as a gradient current (Florida Current) against the prevailing winds. This current becomes stronger where the channel narrows off Bimini and may have a velocity of over 60 Basic Principles of the General Oceanic Circulation 561 nautical miles in 24 h with up to 80-100 nautical miles in the current core. These values correspond to about 1 •5-2-5 m/sec which is hardly reached even in the down- stream parts of big rivers. According to Krummel (1911, p. 576), the axis of the stream under steady conditions is: 35 nautical miles in the Yucatan Channel (east of Contoy Island), 25 nautical miles north of Havana (85° W.), 11 nautical miles east of Fowey Rocks (Florida 25-7° N.), 19 nautical miles east of the Jupiter light tower (Florida 27° N.), 38 nautical miles south-east of Cape Hatteras. At the edges, particularly on the western side, the current shows often variations in direction and strength. Not infrequently there is a counter current flowing in a south- westerly or westerly direction along the Florida Keys into the Gulf of Mexico and is well separated from the basic Gulf Stream. It is connected with the counter current always found further north off the east coast of America. In the most narrow parts of the channel the current has a width of about 30 nautical miles, off Cape Canaveral (28-5° N.) about 60 and off Charleston a width of as much as 120 to 150 nautical miles. In general, the western border of the blue coloured warm water of the current follows the continental slope. To the west of it on the shelf the cold green water of the "cold wall" is usually travelling slowly to the south; (see Pt. I, p. 144, Fig. 60). The Florida Current is joined here by the important Antilles Current flowing north-west to the north of the Bahamas. Before the junction (27° N.) it is narrowed in the con- vergence region of the Sargasso Sea, whereby it becomes of some importance (see Nielsen, 1925; Wiisx, 924). North of Cape Hatteras the Gulf Stream turns farther and farther away from the continental slope, possibly due to offshore winds, Coriolis influence and the increasingly strong cold coastal current of low salinity. This is the beginning of the second part of the Gulf Stream. Its left-hand boundary remains sharply separated from the coastal waters but the right-hand edge is extremely blurred. Here, due to the deflection of the stream lines a counter current is formed which, although narrow, weak and variable is a characteristic phenomenon of the eastern flank of the main current, but because of its narrowness it can rarely be detected by means of ship displacements ; however, the farther to the north-east the stronger and more frequent this current appears. Only mean positions of the current can be deduced by evaluation of the average physical conditions at the sea surface. Better results can be obtained by systematic recordings of the sea-surface temperature at short time intervals ; these then give a more accurate indication of the mean position of the warm Gulf Stream core and also of its northern and southern limit (see Pt. I, p. 144, also FuGLiSTER, 1947). Determinations of the Gulf Stream position obtained by different methods can be combined to give an average picture (Neumann and Schumacher, 1944) but it should always be borne in mind that the boundaries of the warm- water belt cannot necessarily be regarded as identical with the boundaries of the current. From about 55° W. the left side of the Gulf Stream is flanked by the cold and weakly saline water of the Labrador Current. At this polar front the cold water masses sink below those of the Gulf Stream and thereby numerous vortices are formed. To the south of the Newfoundland Banks the Gulf Stream turns sharply towards the south (p. 421) and again back towards north and from here gradually widens and splits into 20 562 Basic Principles of the General Oceanic Circulation current branches of varying strength and of varying temperature. From Cape Hatteras to the Irish coast its direction remains mainly eastwards or north-eastwards ; the average velocity falls from 15 to 5 nautical miles in 24 h and its constancy from 70 to 30%. The almost synoptic surveys of the International Gulf Stream Expedition of 1938 showed that the Gulf Stream to the north of the Azores is no longer a single current, but is broken up into several branches flowing to the north-east as warm and highly saline intrusions between cold, weakly saline water masses moving slowly in the opposite direction. Neumann (1940) has shown that this finger-like ineraction of 38° W 36° 32° 26" 48° 48» 46° / •7 .44' fe f.m ]■ / <^ 42° ^^ Z ^ Jl. / t^=^ S^::^ 40" ^'^ :^ ^ i:^ y 42° f -^ rr\rM-P.S 0 „ . ^-^ ^^ F\^ > r 40° olOO J Azores o^ ^ 38° 36< 38° yy 36° 28° Fig. 257. Most probable course of the Gulf Stream north of the Azores in June 1938. (The open arrows indicate the assumed position of the cores of individual branches of Gulf Stream.) Basic Principles of the General Oceanic Circulation 563 different water types was no chance phenomenon present in June 1938 but is a per- manent feature of the current in these regions (see Fig. 257). In the eastern half of the ocean the Atlantic Current divides into two main branches at about 20° W. ; one of these flows north-east past Ireland and with a reduced strength and moderate Constance through the Faeroes — Shetland Channel into the Norwegian Sea and along the Norwegian coast. It is still noticeable in the Arctic Ocean. The weak and variable second branch turns east-south-east towards the French and Spanish coasts (the Portugal Current). The stronger and also more steady Canaries current in the south-eastern North Atlantic cannot be regarded as a continuation of the Gulf Stream (Thorade, 1928). It seems to be advisable to refer to the whole current from the Florida Straits to the Norwegian coast as the Gulf Stream System but to distin- guish six separate parts of this system (Iselin, 1938); the most important are: (1) the Gulf Stream close to the coast or the Florida Current (from the Gulf of Mexico to Cape Hatteras) ; (2) the Gulf Stream in the open ocean (from Cape Hatteras until north of the Azores) ; (3) the Irish Current (from the splitting point until the Faeroes — Shetland sill); (4) the Atlantic (or Norwegian) Current (along the Norwegian coast). A side branch of the Irish Current flowing from the south of Iceland to its conver- gence with the East Greenland Current is called the Irminger Current. Helland- Hansen and Nansen (1909) deduced the sea surface currents of the Norwegian Sea from an analysis of temperature and salinity in charts and vertical sections (Fig. 157, p. 368). North of the Lofoten the Atlantic current divides into a branch flowing towards north and north-west (towards Spitzbergen) and another one flowing north- east into the Barents Sea (Schulz, 1929). Towards Greenland the East Greenland Current is still wide and strong north of the Denmark Strait. In the central part of the Norwegian Sea there is an extensive area of extended vortices apparently connected with the topography of the sea bottom. Southern Hemisphere. The Brazil Current is a continuation of the South Equatorial Current from Cape San Roque southward. Between 15° S. and 20° S. it is still inside the region of the south trade winds. Off" Cape Sao Thome and Cape Frio the main current flowing south-westwards shows a contraction from its eastern (left) side during most months ; from here it follows the continental shelf line fairly close, probably due to the influence of the Coriolis force. Over the shelf a counter current exists which can be regarded as a branch of the current along the Patagonian shelf (Falkland Current). Off the La Plata estuary the eastern part of the Brazil current turns south-eastwards working into each other in a finger-like fashion with the Falkland Current flowing from the south-west. Near the coast the Falkland Current intrudes to the north and north-east as far as 35° S., deflecting the Brazil Current to the east. Between the two opposing currents there is thus a sharp convergence line formed which is clearly shown by the distribution of the oceanographic factors. This gives rise to vortices found in this part of the ocean. The interaction between Falkland and Brazil Current form a southern hemisphere counterpart to the Labrador Gulf Stream system in the Northern Hemisphere, but the first ones are less well developed and of less intensity. The area of the West Wind Drift includes the whole of the southern part of the South Atlantic Ocean between about 35° and 63° S. It belongs to the large circumpolar 564 Basic Principles of the General Oceanic Circulation i;0° 100° 90° 80° 70° 60° 50' 50° 40° 30° 20° 10° 0° 10° 20° 30* 40° 3U- \W 120° Kk)° 90° 80° 70° 60 50° 40° 30° 20° 10° 0° 10° 20° 30° 40^ 60° E Fig 258 Singular lines in the current field of the sea surface in the Atlantic Ocean. (A) 'in the system of the tropospheric circulation: (1) the divergence region m the area of the Cap Verde Islands (7° to 15^ N.); (2) the equatorial divergence region; (3) the con- vergence region in the Equatorial Counter Current. In the region of the tropica thermoclme these singular lines correspond to inverse ones. (B) the divergence region of the Benguela Current (C) , subtropical convergence; , polar and equatorial limits of the subtropical convergence regions. (D) , the oceanic polar front (Arctic and Ant- arctic convergence). Basic Principles of the General Oceanic Circulation 565 current which keeps the water masses constantly in motion around the earth from west to east. It is of much greater strength and constancy than the corresponding West Wind Drift in the North Atlantic. South of 35° S. and east of 20° W. it flows mainly in a north-easterly direction. There are widely differing opinions about the position of its northern boundary in the area of the subtropical convergence; the southern boundary is found at about 63° S. but is not sharply defined either. At the core of the West Wind Drift lies the boundary between two quite different water types, the subantarctic water of middle latitudes and the Antarctic polar water. In the Atlantic this latter water type has its origin almost entirely in the Weddell Sea. A small part only comes from the Pacific through the Drake passage. The boundary between the two water bodies is denoted the South Polar Front {Antarctic Convergence) on both sides of which the currents flow between east and east-north-east but the velocity is greater on the northern side. For the dynamics of this front see p. 549. The Polar Current in the Southern Hemisphere flows in the coastal regions of the Antarctic carrying cold polar water westward until the Weddell Sea where it turns in a great arc around a central almost motionless region and flows towards north or north- east to become the southern part of the West Wind Drift. East of 10° W. the course of this Antarctic polar current coincides almost entirely with the mean pack-ice limit of the southern summer. The framework of the circulation system of the sea surface formed by singular lines and regions inside the current field is shown in Fig. 258. In the tropical and subtropical circulation the divergence lines stand out clearly in the eastern parts of the North and South Equatorial Currents. In almost all months there is a narrow area of divergence off the West African coast in particular between the Canaries and the Cape Verde Islands that extends towards the south-west beyond 35° W. as a two-sided divergence line and forms the southern boundary of the North Equatorial Current. This is connected with the upwelling of cold water off the West African coast. Its counterpart in the Southern Hemisphere is the extended divergence line in the area of the Benquela Current off the coast of South West Africa; the upwelling of cold water also occurs here (Defant, 1936a). Reference has already been made to the divergence line along the equator between the northern and southern branches of the Equatorial Current (p. 559) and also to the convergence line in the Equatorial Counter Current. The Cape Verde divergence line, the equatorial divergence line and the con- vergence line that lies between them are all part of the tropospheric circulation system and are associated with contrary singularities in the lower layers of the troposphere (p. 595). The oceanic regions between the Equatorial Currents and the West Wind Drifts in both hemispheres contain weak and variable currents. Stream lines deflected to the right from the Atlantic Current and from the North Equatorial Current together form the region of subtropic convergence. This extends across the Atlantic from 75° to 20° W. but is not a continuous uniform convergence line. Vortex formations are the charac- teristic type of motion with the existing slight density differences. In these vortices warm water sinks to become part of the warm-water mass of the troposphere in this region. This convergence is always indistinct and shows everywhere large seasonal variations (Felber, 1934) and is therefore more appropriately called a subtropical convergence region than a convergence line. In this convergence region the interaction 566 Basic Principles of the General Oceanic Circulation between highly saline and warm water from lower latitudes with weakly saline and colder weater from higher latitudes lead to vortical movements of large extent. Similar conditions are found in the subtropical convergence region of the South Atlantic. There are rather different opinions about the question how far the West Wind Drift reaches equatoward depending on whether the subtropical convergence is fixed according to ship displacements or if it is derived by means of the distribution of oceanographic factors. The position given by Deacon (1937), deduced mainly from the temperature distribution, is always about 6° to 10° further south than that obtained from current measurements. According to Bohnecke (1938, p. 201) the "subtropical convergence" (of the currents) should be carefully distinguished from the "subtropical boundary" (deduced from temperature and salinity). The former in a rather charac- teristic way coincides with the tropic boundary and the latter with the polar boundary of that large disturbance region which extends between the southern limit of the Equatorial Current and the West Wind Drift (p. 564) as is found during the dynamic preparation of serial observations. Also here it seems more appropriate to speak of a convergence "region" between the two bordering water types being the place for subtropical vortex formations. The Southern Hemisphere Polar Front (Antarctic convergence line) has been dis- cussed on p. 549. The Northern Hemisphere Polar Front is sharply developed between the Labrador Current and the Gulf Stream near the Newfoundland Banks but gradually fades towards the north-east, reappearing again as a frontal zone between the East Greenland Current and the Irminger Current. Larger and smaller vortex formations with corresponding vertical movements are also found along this con- vergence line. (c) Sea Surface Currents in the Indian Ocean Ships displacements available for other oceans are much less numerous than in most parts of the Atlantic and current charts are therefore correspondingly more uncertain. Reference to analogous conditions as in the Atlantic will usually permit briefer description here, but the Indian Ocean has a single particular peculiarity in its northern part where the wind system changes character completely every six months, correspondingly causing similar changes of the ocean currents. This is the best possible proof that the winds are decisive for the generation and maintenance of ocean currents. A full cartographic description of the currents here requires monthly charts (British Admiralty 1895; Deutsche Seewarte 1908; Dallas and Walker, 1908; MoLLER, 1929) but charts for the summer monsoon and for winter are usually con- sidered sufficient. The currents during the time of the north-east monsoon (north-east trades) corres- pond best to the general system of ocean currents. They resemble those of the Atlantic and the Pacific except that the Equatorial Counter Current lies between about T S. and 8° S., that the Northern Equatorial Current moves partially into the Southern Hemisphere; during this part of the year the thermal equator is always south of the equator. In the north the North Equatorial Current (monsoon drift) runs almost due west. It is strongest to the south and south-west of Ceylon where the cross- section through the current is narrow. In the Bay of Bengal there is an anticyclonic vortex. The strong north-west to north-east winds over the Arabian Sea produce a Basic Principles of the General Oceanic Circulation 567 drift current towards west-south-west or west. Thereby a current boundary is formed beginning north-west of Cape Comorin and can be followed along about 10° N. westwards until 60° E. It carries the character of a convergence line between water from the Arabian Sea and water masses of the main current flowing from the east. ScHOTT (1928fl) has mentioned the great contracts in surface salinity here. Part of this water transport into this region enters as a very strong current into the Gulf of Aden and continues through the Strait of Bab el Mandeb into the Red Sea. The other part forms a strong south-west current flowing along the Somali coast to about 7° S., where the Equatorial Counter Current starts rather abruptly having a direction towards east. South of the counter current flows the broad South Equatorial Current and shows large seasonal variations in velocity and constancy caused by the annual variation of the south-east trade winds. The current core lies near the northern boundary of the current at about 10° S. to 15° S. in both summer and winter (Michaelis, 1923). The irregularities in the South Equatorial Current due to Madagascar have been investi- gated by Paech (1926). In the Southern Hemisphere summer a "Stau" current flows as a southward current along the African coast starting at 10° S., the Mozambique Current, with a tributary current from the east coast of Madagascar. Both form the source for the Agulhas Current at about 30° S., which continues closely to the conti- nental shelf until it swings out from the shelf around the Agulhas Bank at the southern tip of Africa. The northern part of the core, however, still keeps to a very large extent over the contmental shelf. From the southern end of the Agulhas Bank part of the current then flows north-west as the Benguela Current and part turns back into the Indian Ocean forming a series of large vortices. The complicated nature of the currents in this part of the convergence zone between the Agulhas Current and the west wind drift is clearly shown in an analysis of the current field which has been prepared by Merz (1925). The atmospheric pressure and wind distribution over the Indian Ocean north of the equator changes drastically during April. Almost immediately the sea surface currents react to this change in the wind direction and at the same time there is a redistribution of the water piled up at the coasts. The South Equatorial Current still remains in the Southern Hemisphere (south of 5° S.) but is considerably intensified. The counter current disappears and over the entire northern part of the ocean except the coastal zones a fairly constant eastward current appears, the South-west Monsoon Current. The convergence line between the South Equatorial Current and this monsoon current is well developed along the total width of the ocean and broken only in the extreme west where a strong branch turns northwards from the South Equatorial Current between 5° S. and 0° and flows along the coast into the Arabian Gulf as the Somali Current. It follows closely the steep pressure gradient off" the coast between the region of piled up water ("Anstau"-Gebiet) between 5° and 10° S. and the area from which water has been removed by the monsoon current between 5° N. and 10° N. This is accompanied by upwelling just off" the African and Arabian coasts (Puff, 1890). The Somali Current possesses mostly an extreme intensity, so that speeds here are greater than in the Florida Current (often more than 100 nautical miles in 24 h) (Fig. 259). The formation of anticyclonic vortices to the south-east of Ras Hafun and the marked concentration of the current core into a narrow coastal belt is characteristic and accords with the increase of the Coriohs force towards north. 568 Basic Principles of the General Oceanic Circulation 55° yM 36 ^i^ h^^J C^^^<^,^^^J^U.^^I^ '^42 ,3p>t. \ 22-^ "-^v/ ^ O 3Jr- ,-i~ \CP lol -"^ \ -.._--V^ 7 18 . „, ^ 6?^ Fig. 259. Current displacements in the Somali Current at the time of south-west monsoon. The southern boundary between the current branches of the South Equatorial Current and the West Wind Drift is again a long convergence line at about 40° S. For its position see Willimzik (1929) and the alternative interpretation by Schott (1925, p. 163). South of the convergence region and especially in higher latitudes the West Wind Drift has a very low constancy corresponding to the variable winds of this region. The non-uniform character in the current is already shown by the rapid decrease in constancy as the number of observations increases. The Antarctic Comergence runs right across this broad current gradually receding from 48° S. in the west to about 54° S. In this area the West Wind Drift flowing east-south-east meets the cold coastal Antarctic water flowing west-north-west and north-west (Willimzik, 1927). (d) Sea Surface Currents in the Pacific Ocean The principal currents of the Pacific are again the North and South Equatorial Current. Because of the great width of the Pacific they are almost purely east-west Basic Principles of the General Oceanic Circulation 569 currents. Since the thermal equator remains in the Northern Hemisphere throughout the whole year these currents are not symmetrical about the geographical equator. The southern boundary of the North Equatorial Current lies between 6° N. and 7° N. in winter and between about 9° N and 11 ° N. in summer. It is much stronger in winter. At its southern boundary the current at each location has a purely zonal direction and constant speed, while its velocity increases steadily towards the west. Off the Philli- pines (north of Mindanao) the strong current divides : one branch flowing northward to become the Kuroshio and the other turning sharply southward into the Equatorial Counter Current. Off the east coast of Mindanao it flows southwards with a 100% constancy (Schott, 1939, see also Puls, 1895). The South Equatorial Current covers the wide south-east trade wind belt between about 5° N. and 40° S. The greatest velocities and constancy again lie along the northern border between 5° N. and 5° S. and, as in the Atlantic, a double current core is occasionally present. By this a long and narrow tongue of extremely low tempera- ture is caused in the thermal field in the eastern part of the Pacific west of the Gala- pagos Islands. These areas of cold water are associated with the occurrence of eastward ship's displacements within the South Equatorial Current. Similar ship's displacements are occasionally observed in the Atlantic. West of New Guinea and the Solomons the South Equatorial Current during the northern summer is a torrent current extending almost as far as Halmahera ; it supplies the main water mass of the counter current. Off the east coast of Australia the South Equatorial Current bends and is called from thereon the East Australian Current which corresponds to the Agulhas Current in the Indian Ocean. All the year long a well-developed counter current is inserted between the two Equatorial Currents. During the northern winter it is weak and narrow, except in its starting area in the west, but during the northern summer especially during August and September it flows with great Constance from Mindanao-Palau-Halmahera to Panama (almost 8000 nautical miles) with a width of about 300 miles between 5° N. and 10° N. It is separated from the Equatorial Currents by well-defined bound- aries especially on the northern side. The Kuroshio is a continuation of the North Equatorial Current and in many respects an important phenomenon for Eastern Asia. A review of what is known of this current and a comparison with the Gulf Stream system with numerous references has been given by WiJST (1936^, see also, Uda and Okamoto 1930, 1931 ; Uda, 1933). In summer it starts flowing northward east of Formosa with a velocity of 24-36 nautical miles in 24 h and a width of about 300 nautical miles. Then it runs west of the Ryukyu Islands between the Ryukyu Ridge and the East China shelf with decreas- ing width and correspondingly increasing speed (36-48 nautical miles in 24 h) until it branches south of Japan ; one branch, the Tsusima current enters the Sea of Japan and flows north-north-west, the other, the proper Kuroshio, flows with a reduced width along the south-eastern coast of Japan. Between 31 ° and 35° N. it is only about 150 km wide but its velocity rises to 48-56 nautical miles per day. Its left-hand boundary is sharply defined but the right-hand one (oceanic side) is blurred. Here, like the Gulf Stream, it has a weak counter current. It turns abruptly eastwards towards the open ocean at 36° N. off the Boso Peninsula with an almost invariable width but with gradually decreasing velocity (48-24 nautical miles per day). This deflection of the 570 Basic Principles of the General Oceanic Circulation current has been regarded by Hidaka (1927-28) from experimental evidence as due to the change in direction of the north-east coast of Japan, but Wiist believed that topographical factors south of the Boso Peninsula were responsible. The Kuroshio extends out into the open ocean as a relatively strong current along 34-36° N. as far as 175° E. a distance of about 1,600 miles. Only for a short distance along the coast the current keeps the north-east direction. Figure 260 shows a schematic representation of the main current cores of the Kuroshio system during the summer as given by Wiist. Table 148 gives a comparison with the Gulf Stream system. Table 148. Comparison between Kuroshio and Gulf Stream (Mean values for summer, according to Wiist) Current section Width (km) Direction Speed (cm/sec) Nautical miles/24 h Temp. (°C) Salinity /oo Kuroshio 23°-24°N. 27°-28° N. 31°-33°N. About 36° N.t . 300 230 150* 150 N. to E. N.E. N.E. E. 51-77 77-103 100-120 51-100 24-36 36-48 48-56 24-48 5^22-28 34-8-34-9 Gulf Stream 23°-24°N. 27°-28°N. 31°-33°N. About 36° N. . 110 140 180: 180 E. N. N.E.toN. N.E. 100-120 140-160 About 120 About 100 48-66 66-75 About 56 About 48 I 22-28 360-36-4 * After separation of the Tsusima Current. t 400 km east of the Japanese Coast. 1 After confluence with the Antilles Current. The Oyashio flows south-west to south-south-west in the dead angle between the north-west coast of Japan and the north-western branch of the Kuroshio as far as 37° N. It is a relatively cold current with a very low salinity (33-5"/oo). It does not reach as far south in summer as in winter. According to Uda the boundary between the Kuroshio and the Oyashio as a convergence region consists of numerous vortices similar as at the boundary between Labrador Current and Gulf Stream. Differences in temperature and salinity across this convergence line in winter will be at least as large as those off the Newfoundland Banks. Driven by the strong northerly and northwesterly winds the Oyashio takes its cold water supply from the Sea of Okhotsk near the Kuriles and in part also from the Bering Sea. The water is mostly in slow motion between the cold boundary which runs east- wards a little north of 40° N. and gradually fades away and the subtropical conver- gence which begins in the west at 20° N. and turns northward, at first only slowly, to reach 35° N., remaining in this latitude until about 138° W. This continuation of Kuroshio is termed the North Pacific Current. It main part turns southward between 150° and 135° W., part joining the California Current and part mixing with the water from the North Equatorial Current along the subtropical convergence. 120 1%0° 20'u — -- 200m„ — — ,IOOOm_-._~..2000m Mansyu stations May- June I925-? Monsyu stations Jan. Feb 1927 lao* '20° 130° ^HQ' E Fig. 260. Main current branches of the Kuroshio system (according to WUst). ( 1 ) Kuroshio (main current); (2) Tsusima Current; (3) Korean side-branch; (4) northern branch of Kuroshio; (5) Oyashio; (6) Liman Current; (7) Counter currents of the Kuroshio. At R position of the Riu-Kiu section, at S position of the Shiono-Misaki section. Basic Principles of the General Oceanic Circulation 571 The northern part of the North Pacific Current turns northward and flows in an anticlockwise direction around the Gulf of Alaska; it is a well-developed current and is fairly constant, particularly near to the coast. This Alaska Current flows along the Aleutians and extends into the southern Bering Sea through all the passages between the islands. In the eastern part of the Pacific the southward movement off the Cali- fornia coast is denoted the Californian Current (Thorade, 1909; Warmer, 1926). It replaces the water which is carried westward by the north-east trade winds. The north-east to south-west direction of the current indicates the presence of an ofi"- shore movement, giving rise to the upwelling of cold water along the greater part of the Californian coast. This upwelling occurs mainly during the warm part of the year. The northward to north-westward movement of water along the entire western coast of South America is called the Humboldt Current after its early investigator. Where it runs close to the Chilean and Peruvian coasts it is called the Peru Current and this current and its variations have been described in a detailed monograph by ScHOTT (1931). A later evaluation of the available data has been given by Gunther (1936, 1936a). Figure 261 shows the probable field of motion according to Schott for the two seasonal extremes. During the period of intensified trade winds in the Southern Hemisphere winter (Chart a, Aug.-Sept.) the Humboldt drift current and its con- tinuation, the South Equatorial Current, intensify considerably. The strength of the current rises from 0-5 to 0-7 knots along the coast of northern Chile and Peru and increases to 1 and occasionally 2 knots where it flows north-westwards in a wide region around the Galapagos Islands. Further out to sea it turns westwards. The W-Lq W-Lg Fig. 261. Most probable current pattern in the region of the Humboldt Current and north of it (according to Schott): {a) for the Southern Hemisphere winter (August/September); (6) for the Southern Hemisphere summer (February/March). 572 Basic Principles of the General Oceanic Circulation coast as far as 5° S. is thus a one-sided convergence line and as a consequence up- welling occurs along its entire length. The other extreme of seasonal variation is at the end of the Southern Hemisphere summer (Chart b; Feb.-Mar.). Conditions in the equatorial region at this period are very complex and unstable and are subject to the influence the more or less pronounced development of the Equatorial Counter Current and the North Equatorial Current. The Humboldt Current is now weaker and about 4° C warmer at the coast. The unstable character of the current is due to simultaneous instability in meteorological conditions in the entire area between the Cocos Islands, the Galapagos and the coast of Ecuador and Peru. In many years the thermal equator and the associated zone of minimum atmospheric pressure are dis- placed into the Southern Hemisphere, so that the south-eastern trades along the Peruvian coast are then disturbed and rainy north and north-west winds occur in northern Peru. These disturbances of atmospheric and oceanic conditions are, how- ever, usually not too powerful, but in general conditions are so unstable in northern Peru that abnormal developments frequently occur. The warm weakly saline water of the Equatorial County Current can then easily advance into the area of the Humboldt Current. This warm water is then carried southward by the northern and north- western winds (most often at Christmas time). This current in contrast to the Peru Current is regarded as a "counter current"; it is called "El Nino". Normally the changes are not very great but occasionally when the disturbances are particularly well developed there may be torrential rains followed by flood catastrophes in coastal areas of northern Peru which are adapted to a dry climate. The simultaneous change in the character of the water masses off" the coast in addition has disastrous conse- quences for the guano birds which are suddenly deprived of food. Detailed descriptions have been given for years when these disturbances have been particularly well marked, for 1925 by Zorell (1928); Murphy (1926) and Schott and for 1891 by Schott (1931). The wide area of the Pacific covered by the essentially eastwards flowing West Wind Drift extends south of the subtropical convergence which is more a "con- vergence region" than a line. The available data on this current, especially in the thirties and forties, is rather uncertain. Near 40° S., off" the South American coast there exists a zone of remarkably low salinity (34%o) apparently originating from western Patagonia (Schott, 1934). Corresponding to this distribution the West Wind Drift must swing sharply north to north-westward, that is, to the left. The Antarctic Con- vergence runs through the West Wind Drift at about 55° S. It was encountered in every profile recorded by the "Discovery" Expedition and is the only convergence line circling the entire earth in the Antarctic region. 3. Currents Caused by Excess of Precipitation and Run-off Over Evaporation The possibility of the direct formation of ocean currents due to the flow of excess water from the precipitation areas and those with run-off" from rivers into evaporation regions, was first investigated in detail by Ekman (1926) using his classical theory of deep and bottom currents. For a circular oceanic region he obtained after considerable simplifications a final equation of the form 277 curl K -^{P- E), (XVIII.l) Basic Principles of the General Oceanic Circulation 573 where V is the velocity of the deep current produced, p is the average density of the bottom current layer D and (P — E) is the difference between precipitation and evaporation in the area under consideration. Estimation of the velocity in some actual oceanic regions gave maximum velocities of the "evaporation currents" of not more than 1-2 cm sec"^, but probably only fractions of this value are reached. This is valid for open sea surfaces. For partly enclosed basins the quantity (P — E) may be of exceeding consequence ; the current processes occurring with water inter- change in sea straits have been already discussed before (Chap. XVI, p. 513). Besides the water transport through the sea straits also the salt transport stand in question. If the inward water transport is A/,, the outward water transport Mq and the correspond- ing salt transports are Si and Sq, then under stationary conditions the two equations MiSi = MoSo and M, = Mo-(P ~ E) (XVIII.2) are valid, and thus Mi =(P-E) ^^\ . (XVIII.3) This formula is identical with the simple Knudsen relations (p. 379). For example, when the inflow through the Straits of Gibraltar is about 1-75 x 10^ tons sec-\ the average salinity of the inflowing water about 36-25%o and of the outflowing water 37-75%o, then for the Mediterranean Sea according to the formula (XVIII.3) the quantity E — P results to 0-07 x 10^ tons sec-\ which is in good agreement with other estimates. More recently, Goldsbrough (1933) has dealt with ocean currents produced by the given distribution of precipitation and evaporation. Already before that Hough (1897) in his famous theoretical study of tides on a rotating globe has dealt with this problem of currents produced by a zonal distribution of precipitation and evaporation. Since he ignored frictional effects, he found a uniformly accelerated system of purely east-west geostrophic currents as a consequence of these distributions. From the impossibility of finding a steady state solution he concluded that precipitation and evaporation cannot be a significant cause of ocean currents. Hough did not accept any meridional boundaries in the ocean. Goldbrough took instead a model with precipi- tation predominating in one hemisphere, evaporation in the other and assumed meridional boundaries in the ocean. This model gave a steady current field, provided that the integral of the precipitation-evaporation function taken along each parallel of latitude between the two boundaries, vanishes. This is a very severe restriction which no natural distribution of precipitation-evaporation necessarily fulfils. Figure 262 shows the current system produced in this case for one hemisphere; the other hemi- sphere will be the mirror image of this. The field of pressure, the elevation of the free surface and the flow will be steady. The horizontal velocity components will thus be entirely geostrophic, and the current will flow along the isobars. The vertical component will be zero at the bottom and will increase linearly from the bottom up to the sea surface where it will equal the precipitation-evaporation rate. At the eastern edge of the precipitation hemisphere there will be two low-pressure cells, and at the western edge two high-pressure cells. At the poles the flow is directed from the region of evaporation into the region of precipitation; however, in the opposite direction in 574 Basic Principles of the General Oceanic Circulation EVAPORATION PRECIPITATION Fig. 262. The steady circulation of Goldsbrough type driven by precipitation over one-half of a hemisphere and evaporation over the other half. Only one hemisphere has been pictured, for the other hemisphere applies the reflected image. The curved lines with attached arrows are isobars. The centres of high- and low-pressure cells are to the right resp. To the left of the middle line. subtropical and tropical regions. The geostrophic current will everywhere be directed towards the equator in the precipitation hemisphere and towards the poles in the evaporation hemisphere. The current towards the equator will require a horizontal divergence, that towards the poles will require horizontal convergence. This diver- gence (or convergence) distribution must be suflficient everywhere to absorb (or supply) the water locally precipitated (or evaporated). The solution in Fig. 262 is valid for an entire hemisphere but it is evident that a coastal barrier could be placed along any complete isobar without affecting the solu- tion. Thus, meridional barriers can be placed tlirough the centres of the precipitation and evaporation hemispheres, and also the equator itself can be selected as such a barrier. This schematic representation of Goldsbrough's results has been discussed here in some detail, since Stommel has used it as a basis for a discussion of the fundamental principles of ocean circulation (see Chap. XXI). 4. The Thermo-haline Circulation The general atmospheric circulation is produced solely by heat differences in meridional direction, ultimately caused by the sun radiation. By analogy to the Basic Principles of the General Oceanic Circulation 575 atmospheric conditions it was assumed at an early date that there would be a simple water circulation in a vertical plane between the equatorial zone and the polar oceans. This opinion was first expressed by Humboldt (1814, 1845, p. 322) who also offered a more detailed reason for it. He pointed out that the very low temperatures in the deeper water layers at low latitudes could only be regarded as a consequence for the cold water transport in the deeper layers from the poles towards the equator, which would also imply a surface water transport towards the poles. The entire mass of the oceans between the equator and the poles including the water at very great depths would thus be in constant motion, Humboldt considered the differences in density between equatorial and polar water masses as the cause for this closed circulation system. Since the circulation is in accordance with the given temperature distribution, he concluded that the distribution of salinity was not such as to disturb the thermally produced circulation. Humboldt's ideas were adopted by many investigators and for three-quarters of a century formed the basis of a generally accepted view on ocean circulation. Lenz (1847) found that already for small depths, temperatures in the equatorial regions are much lower than in the subtropics, and he concluded that the almost horizontally flowing deep current coming from higher latitudes must assume an upward directed component near the equator. He deduced from this that, sym- metric to the equator, there must therefore be two major vortices in a vertical plane, one on either side of the equator with the cold deep currents rising and merging in the equatorial region; cold deep water would thus be found nearer the sea surface here than further north or south. He found support for his conclusion in the salinity minimum of the equatorial zone. Ferrel (1856) took the Coriolis force into account and proposed a modified form of Lenz's vortices limited not in the polar regions, but only in middle latitudes, but followed by another vortex in the polar regions of each hemisphere with a rising movement near the poles. The analogy between the atmospheric and the oceanic circulation is particularly evident in Ferrel's model; he completely ignores the differ- ence due to heating of the ocean from above, and of the atmosphere from below, and also disregarded the effects of the salinity distribution and winds. The wide adoption of the thermal circulation theory is due to the circumstance that it has been included in an important oceanographic work of that period by Maury, The Physical Geography of the Sea (1st edition, New York, 1856). Croll (1870-71, 1875) refused it, but took another extreme viewpoint, since he regarded each vertical circulation as produced by the wind. Also Carpenter (1870-77) tried to conclude from the "Challenger" observations that a thermal circulation was present. Both agree on the existence of a major vertical circulation and differ only on its cause. Detailed analysis by Buchanan (1885) and Buchan (1895) of data from the "Challenger" expedition showed that the actual spatial distribution of temperature and salinity is incompatible with a vertical circulation of the type suggested by Lenz. In all oceans there are alternating layers of different temperature and salinity under- neath a relatively shallow top layer. This excludes the possibility of a single closed circulation system with two vortices syrmnetrically placed on either side of the equator. According to Sandstrom's proposition (p. 491) a thermo-haline circulation is substantially promoted and intensified if the heat source is at a lower level than the cold source, particularly when the effects of heat conductivity and turbulence are of 576 Basic Principles of the General Oceanic Circulation minor importance as is the case in the ocean. It was mentioned that in the ocean these heat and cold sources are at approximately the same level and that therefore conditions are not favourable for the development of powerful circulation systems. In any case they can be only of small vertical extent and they will be entirely incapable of filling the whole of the oceanic space from the poles to the equator. Conditions along a meridian will be more or less the following: Latitude 60" 50° 40° 30° 20° 10° 0° Predominance of heat loss due to out- going radiation Heat gain due to incoming radiation Predominance of salinity decrease (P—E > 0, melting of ice) Salinity increase (P — E < 0, Salinity decrease through predominance of evapora- precipitation and run-off tion) Since a salinity increase is equivalent to a heat loss and a salinity decrease to a heat gain, the thermal and haline circulation will act in the same direction in the region between the equator and in the Ross latitudes (0° until 30° N. and 30° S.). North and south of the subtropical regions, however, they will counteract each other. A powerful thermo-haline circulation can thus be expected only in the tropics and subtropics. The water transport occurs towards the poles in the uppermost layer and toward the equator underneath with an upward motion in the equatorial regions and a descending one in the subtropics. This circulation can, however, develop only in a thin top layer and the Lenz schematic circulation is restricted to this kind of shallow circulatory water movement. The circulation of this tropical and subtropical top layer is dealt with in Chapter XIX. 5. Wind Effects and the Current System in a Hydographic Circular Vortex That the wind system of the atmosphere is also involved in the development of the ocean circulation was not excluded by many investigators, but no agreement was reached about the importance of its effects as long as the properties of wind drifts were still unknown. The significance of atmospheric currents as a cause of the ocean circulation was considerably clarified by Ekman's investigations. Probably the most important result was to show that the wind affects directly only a top layer of not more than 100-150 m thickness. The piling up of water at a coast by the wind will, however, give rise to a slope in the physical sea level and to gradient currents reaching down- wards to greater depths. In stratified water, mass compensation between upper and lower levels (pp. 485 and 548) seems to prevent the development of deep-reaching gradient currents. This remarkable compensation principle is readily illustrated by a two-layered oceanic model. If in such a water mass (upper layer: pi, hi; lower layer: p., and /72 — hi; Fig. 263) a current V is generated along AB in the upper layer, then the physical sea level along AB will adjust itself to give a state of equilibrium between the gradient and the Coriolis force. The deviation of the physical sea level from a level surface ("Geoid") is denoted by Ci. Displacements of mass in the upper layer will also disturb the equilibrium in the lower layer with a resultant mass transport in the direction from D towards C, the internal boundary surface will decline {CD'), but Basic Principles of the General Oceanic Circulation 577 in a direction exactly opposite to that of the sea level. This displacement of the internal boundary surface will automatically reduce the pressure gradient imposed on the lower layer from above. In the final equilibrium state of the lower layer there will be no pressure gradient and therefore no motion. If io is the deviation of the internal boundary surface from a level surface, the condition for this new state of equilibrium is given by ^' Ci. (XVIII.4) P2 P\ This simple relationship will always be present if sufficient time is available. Con- ditions at the outer boundaries of the current aX AC and BD will be considered later (p. 622 et seq.). Fig. 263. Position of the physical sea surface and of the internal boundary surface of a two-layered ocean for a forced movement of the upper layer in the interval AC-BD. The total effect of air currents on the ocean surface can be suitably illustrated by the simple case of an ocean uniformly covering the entire earth (no continents). This ocean can be assumed to have two layers, an upper troposphere and a lower stratosphere, separated by a clearly defined density transition layer. To correspond to actual conditions in the tropics and subtropics it can be assumed further on that the troposphere in these regions is subdivided by a transition layer at about 100 m depth separating the top layer from the subtropospheric water masses beneath. Only zonal (east-west) currents will be present in this hydrosphere covering the total earth and it can be regarded as a circular vortex as described by Bjerknes (1921), centred around the axis of the earth. The movement of the water masses in this vortex will be east- west, and the adjacent stream lines will not influence each others. The hydrosphere will be affected only by the atmospheric currents at the sea surface, that is, by the trade winds between the equator and the Ross latitudes (30° N. and S.), by the west winds in middle latitudes between 30° and 60° N. and S., and by polar east winds polarwards 60° N. and S. The oceanic movements in the individual zones of the circular vortex and the position of the boundary surface will then be a consequence of these effects. Since conditions are symmetrical around the rotational axis it is only required to consider a meridional section through such a wind-generated circulation. Fig. 264 578 Basic Principles of the General Oceanic Circulation gives a schematic representation of the water movements expected according to these theoretical considerations. Between 30° N. and 30° S. the north-east and south-east trade winds give rise to the broad North Equatorial and South Equatorial Current of the Northern and Southern Hemisphere, respectively. The maximum intensity is reached in the regions where the Polar current Polar, front West winddrift Norttiern subtropicol convergence North equatorial current Soutti equatorial current Souttiern subtropicol convergence West winddrift Polor front Polar current Fig. 264. Schematic representation of the hydrosphere as a circular vortex. Current zones and position of the main boundary surface and of the isobaric surfaces (with a strong exaggeration of the vertical scale). {W, current towards west; E, current towards east). trade winds are most strongly developed ; their intensity decreases toward the regions of high atmospheric pressure in the subtropics and also towards the equator. They are deflected 45° cum sole from the wind direction and must be associated with a water transport towards the poles. Water will therefore be piled up at their polar boun- daries (in about 30° latitude) and therefore a pressure gradient will be generated in the troposphere towards the equator. Sea level and the isobaric surfaces will be de- pressed at the equator and will rise from here towards the poles. If there is no motion Basic Principles of the General Oceanic Circulation 579 in the water masses of the stratosphere the boundary separating it from the tropo- sphere will slope in the opposite direction in accordance with the compensation principle mentioned above. At this internal surface there is a stratospheric ridge at the equator and a trough in Ross latitudes. Thus in the Atlantic the boundary is at 300 m depth at the equator and at 700 m depth in Ross latitudes: ^2 = 400 m at 30° latitude. With the observed values pj = 1-0260, pa = 1-0275, equation (XVIII. 1) gives the rise in physical sea level from the equator to 30° latitude as approximately 58 cm; an order of magnitude which agrees with the dynamic computations of the absolute topo- graphy of isobaric surfaces. At 20° latitude where the physical sea level has a rise of 35 cm and p^— pi = 25 x 10~^, the decline of the tropospheric transition layer is 140 m, also in good agreement with observation. In this circular vortex there is no circumstance which would give rise to an equatorial counter current. Winds in the atmospheric West Wind Drift are of rather variable character; but only in the general average westerly winds predominate. In the top layer they produce an oceanic West Wind Drift and a consequent piling up of water cum sole towards the subtropics, which counteracts the accumulation of water associated with the equa- torial currents. There is thus an accumulation of water from both sides in a belt around the earth. On the equatorial side of this belt water flows westward, on the polar side eastward. This is the subtropical convergence region, one of the most important bound- ary lines of the oceanic circulation. Corresponding to the downward slope of the physical sea level towards the poles there is an upward slope in the internal boundary surface between the troposphere and the stratosphere from its deepest position in the subtropics to the surface of the ocean at the polar front {polar convergence). This is the 60° N. and S. it must rise 700 m over 30 degrees of latitude. When pi = 1.0265 and P2 = 1.0275 the physical sea level will have a slope of 68 cm according to equation (XVIII. 1). If the physical sea level at the equator is taken as zero, it will have an eleva- tion of 58 cm in the subtropics and a depression of 10 cm at the polar front. The prevailing easterly winds around the polar caps produce a westward drift current (polar currents) and there is a corresponding rise in the sea level from its lowest position at the polar front. Although the circulation system shown in Fig. 262 is only schematic, it shows the main features of the surface circulation system clearly, particulary as in the Pacific and in the circumpolar Antarctic waters where it is not strongly disturbed by the presence of continents. With a circular vortex of this type under stationary conditions no vertical movements are to be expected. A deep-sea circulation will therefore not develop and the three horizontal current zones (the Equatorial Currents, the West Wind Drifts and the Polar Currents) can be explained as solely caused by winds. The topography of the physical sea level, of the internal boundary surface between the troposphere and the stratosphere and of the tropospheric transition layer of the tropics and subtropics are coupled with these zones. 6. The Influence of Meridionally Oriented Coasts on the Oceanic Circulation The oceans are bounded everywhere on their western and eastern sides by conti- nents which act as meridional barriers to the oceanic circulation and prevent the formation of a simple circular vortex around the earth. At the meridional barriers the equation of continuity must be satisfied, and in order to allow the conservation of 580 Basic Principles of the General Oceanic Circulation mass, meridional currents must develop that will determine the nature of the circula- tion. It appears that these boundary conditions are more easily fulfilled for a sea with a meridionally oriented eastern coast than for one with a meridionally oriented western coast. {a) Conditions West of a Meridionally Oriented Coast SvERDRUP (1947) has shown that a steady state solution can be found for a density- layered ocean by starting at a meridional boundary and working westwards even when frictional effects are neglected. In the vorticity equation (XVII. 5) the wind stress vort- icity must be balanced by the planetary vorticity alone and, as shown already in XVII.3 second of the major boundaries of the oceanic circulation. To reach the surface at the boundary conditions and the equation ofcontinuity(XVII.4) determine the currents westward from the meridional boundary (east coast). For a purely zonal wind {Ty = 0), the mass transports (omitting the first term of (XV1I.7) ; lower latitudes) will be given by My = --^^' and M^^j-^. (XVIII.5) Assuming in a schematic way according to actual conditions in the ocean (equator to 30°: easterly winds; 30° to 60°: westerly winds) T = a sm -r-y. (XVIII.6) where / is the distance from the equator until 60°, then -^.^-jT^^njy. From this it is easy to derive the following table of signs of the different quantities for an eastern or western meridional barrier. Barrier to the east Barrier to the west y 0-1/ il-il 1/ - 1/ 0-il y - ii ^/-f/ T, _ _ + — + + Ax — — - c » + + + + d^TJdv^ + + — — 1 + + — — M. . • ! ~ — + + c 1 ► + + ~ " Possible case Impossible case West of the barrier, T^ and M^ are, according to (XVIII.5), both positive or both nega- tive. However, east of the barrier they are of opposite signs, which is impossible. The equations (XVIII.5 and 6) give a steady state solution only for a sea area to the west of the boundary. The foUov/ing example can be taken as an illustration of such a solution. Selecting T^ = — 0-4 sin 6(f> dyn cm"', gives M. 2-4 2 cos 6(f) ; Mx 14-4 Zljc 2Rw cos sin 6(f) and tfj = — 2-4 Ax 2co COS(f> cos 6(f). Basic Principles of the Geiieral Oceanic Circulation 581 Fig. 265. Stream lines of the flow representing the field of mass transport; differences of the values of the stream function between two stream lines represent the net mass transport in 10® metric tons per second flowing between these stream lines (from the surface down to a depth of no motion). Figure 265 shows stream lines of flow representing the field of mass transport. The principal troughs and ridges are accounted for by the wind stress function. Off the coast in the east the currents are weak and the meridional component is directed south- wards in middle latitudes. The integrated equations give no information on the distribution of vertical motions in the deep oceanic layers. A better comprehension of these currents can be gained by accurate calculations for the very simple model of Sverdrup. Stommel (1957) has recently given a very instructive description of such a case, in which zonal wind stress was assumed to act on a homogeneous ocean surface with an eastern coast line. Figure 266 shows the solution. At the surface there is a zonal wind stress with a similar distribution as that shown in Fig. 265. The stream lines will therefore also be similar to those in the diagram. The transport in the thin Ekman layer, indicated by the upper arrows, will produce a vertical downward velocity in the central part of the diagram. Outside the zonal belt of westerly winds the vertical velocity will be directed upwards. These vertical components from the bottom of the Ekman layer to the bottom of the ocean decrease linearly to zero. The divergence and convergence system of the meri- dional components of geostrophic velocity are coupled with this vertical velocity field. At the latitude of maximum westerly wind, where there is no impressed vertical velocity, the geostrophic flow will be entirely zonal and will decrease linearly towards the eastern coast. The topography of the physical sea surface, which determines the pressure field associated with the geostrophic flow, is also shown in Fig. 265. 582 Basic Principles of the General Oceanic Circulation If in addition bottom friction of the type described by Ekman is taken into account, the current field will be slightly altered; now the bottom current must aiso contribute in order to satisfy the convergences and divergences appearing in the current field of the Ekman top layer. Fig. 266. Sverdrup-type solution in a homogeneous ocean of uniform depth, bounded by a meridional coastal wall on its eastern side. The wind system with sinusodial pattern is indicated by shaded arrows hovering above the surface. The curved lines with arrows are isobars and give the direction of the geostrophic horizontal flow (independent of the depth). At a number of subsurface depths the velocity components are shown by solid arrows (according to Stommel 1957). Considerably more complicated models of this type can, of course, be developed, but they will all show that the boundary conditions at any coast to the west cannot be satisfied except by taking into account processes involving the dissipation of energy. (b) Conditions East of a Meridionally Oriented Coast In the western part of the oceans, and particularly along the western boundary, the vorticity related to lateral friction must also be taken into account with an additional term in order to satisfy mass conservation and space continuity conditions in the vorticity equation (XVII. 5). With this equation Stommel (1949, 1951) was the first to give an explanation of the westward intensification of ocean currents. He took the case of a symmetrical anticyclonic wind circulation over a closed rectangular oceanic area in the Northern Hemisphere. The wind stress vorticity is thus negative over the entire ocean. The effect of the wind stress can be expected to cause an anticyclonic circula- tion in the sea. The horizontal eddy viscosity will tend to counteract the effect of wind stress. In the western parts of this ocean the anticyclonic flow will transport water northward, in the eastern parts southward; in equation (XVII. 5) the planetary vorticity effect is therefore negative at the western side of the ocean and positive at the eastern side. This is a consequence of the conservation of angular momentum or, what amounts to the same, of the variation of Coriolis parameter with latitude. Basic Principles of the General Oceanic Circulation 583 If the absolute numerical values of the three vorticity terms in (XVII. 5) are denoted by a, b and c, then for a symmetrical wind system, {a) would be negative and would have the same numerical value in both east and west. For an equal velocity, a sym- metrical oceanic circulation would require an equally great frictional vorticity; (Jb) would thus be positive and have the same numerical value in the east as in the west. The planetary vorticities in the west and in the east would also have the same numerical value but are of opposite signs. Thus Off the western boundary — « + Z7 - c =0 Off the eastern boundary — a + Z) + c =0 These requirements are satisfied only when c = 0, that is, when there is no meridional transport, and are therefore incompatible with the conservation of mass. This is a qualitative explanation (1) of the impossibility of a symmetrical circulation in association with a sym- metrical wind field, (2) of the impossibility, mentioned above, of deriving a suitable circulation off the western coast of an ocean without accounting for frictional influences. As shown by Stommel, an anticyclonic circulation is possible in the case just discussed only when the water transport off the western boundary is substantially intensified and the lateral shearing stresses consequently, of course, increased corres- pondingly. To illustrate this, Stommel gives some arbitrary values for the vorticity terms in an asymmetric circulation. These are shown in the following Table 149. Table 149. Vorticity tendencies in an asymmetric circulation Strong northward Southward flowing flowing currents current over the in the western edge rest of the ocean Wind stress (a) . Frictional (b) Planetary (c) - 10 + 100 - 90 -10 +01 + 0-9 Total 00 00 Among the interesting consequences of this theory are : (1) the fact that although energy is added to the oceans by work done by the wind over the entire surface, it is dissipated primarily in the strong western currents ; (2) that a good representation of the circulation in the zonal currents of westward or eastward direction can be obtained independently of friction from a know- ledge of the wind stress field alone. MuNK (1950) was able to evolve a comprehensive theory of a wind-driven ocean circulation by combining three new concepts : (fl) the introduction of lateral stresses associated with the horizontal exchange in large eddies (Defant, 1926; Rossby, 1936a), (6) the possibility of computing currents in baroclinic oceans from the known wind stresses (Sverdrup, 1947), and 584 Basic Principles of the General Oceanic Circulation (c) the consideration of the variability of Coriolis parameter with latitude (Stommel, 1948) which makes it possible to explain the westward intensification of a wind-generated ocean circulation. This theory accounts for many of the major features and some of the details of the general ocean circulation on the basis of known mean annual winds. Briefly the fundaments of this new theory are: The vorticity equation (XVII. 5) can be put into a practical form by the introduction of expressions for the lateral frictional forces. According to (XL 13 and 14) these frictional forces have the form (d^u dhi\ , IdH cH\ ^- = '^ (a? + 8/) ^°^ "' = ^ (a? + if) ■ (^^"") A is the lateral eddy viscosity pertaining to horizontal shear v*'hich is presumed to be constant and horizontally isotropic, neglecting variations due to differences between zonal and meridional motion of large horizontal vortices on a rotating earth. Intro- duction of these expressions into (XVII. 5) with the stream function according to (XVI. 25), gives the differential equation for mass transport AV^ - iS ^\^ = - curL T, (XVIII.8) where V^ is the biharmonic operator (see XVI.26) and curl, Tis the vertical vorticity component of the wind stress. It can be shown, in accordance with the relationship lateral stress curl + planetary vorticity + western solution + wind stress curl = 0 ^r (XVIII.9) central solution J that in the central and eastern oceanic areas the planetary vorticity and the wind- stress curl have opposite signs, resulting in balance in which the lateral stress plays a negligible part. Along the western boundary the planetary and the wind-stress curl have the same sign, and the lateral-stress curl balances both, planetary vorticity and wind-stress curl. It can be verified that in this region the wind-stress curl is numerically unimportant although it is, of course, the primary cause of the circulation. To equation (XVIII. 7) must be added the boundary conditions ^- = 0; (yj-O, (XVIII.IO) boundary \ / boundary where v is normal to the boundary. The first equation states that the boundary itself is a stream line, the second that no slippage occurs against the boundary. Munk assumed : (1) a rectangular ocean extending from x = 0 to .v = r and from y = —s to y = -\-s. The boundary conditions will then be 0 = dijjjdx = 0 for ;c = 0 and x — r "\ rxVTlT 1 H 0 = dxltjdy — 0 for_y = —s and y = A^s j Basic Principles of the General Oceanic Circulation 585 (2) a zonal wind circulation (T y= 0); for this the stress on the ocean surface in the interval —s < y < -hs can be given as a Fourier series, a general term of which is T^^^ = c + aco^ny + b sin ny with n = j^-; (j =1,2,...) (XVIII.12) The solution of (XVIII. 8) which satisfies the boundary conditions is 0 = — rXfS-'^ curl, T whereby / 1 \ r 2 ikx / 2 -*A^---/V3_ _. ^ J 1 kr kx — e-''(^-^) 1 west ^ , ' j.(XVIII.13) central ^ , ' east Here k is the "Coriolis friction" wave-number which has the vale ^(fijA) and is assumed to be constant. The solution is valid as a first approximation when y = (njky <^ 1 and g-''" < 1. When ^ = 0-016 km-^ and r = 6000 km the value of the stream function ip will be accurate within 10%, if y < 0-25, corresponding to a minimum zonal wavelength, lir/n, of about 1500 km. Since for the mean annual stress distribution the shortest wave length of the important north-south variations, the distance between the northern and southern trade winds amount to 4000 km, the approximation leading up to (XVIII. 13) therefore appears to be valid for a study of the general ocean circulation in relationship to the general atmospheric circulation. A knowledge of the wind distribution over an ocean thus permits a direct quanti- tative calculation of the current field in the ocean. It was calculated by Munk for the North Pacific, first as an approximation for a rectangular ocean, and later for a tri- angular ocean (Munk and Carrier, 1950), which gives a better representation of actual conditions. The solution (XVIII. 13) shows in the first place that the zonal wind system divides the ocean circulation into a number of gyres. The dividing lines between them lie in the latitude of maximum west wind, in the northerly and southerly trade winds and in the doldrums. The latitudinal axis of each gyre may be defined by d^TJdy"^ = 0. The Atlantic Sargasso Sea is associated with the inflection point in the mean wind stress curve between the westerly winds and the north-easterly trades. The inflection points between the doldrums and the northern and southern trades determine the boundary of the equatorial counter current. When Xis computed from (XVIII. 13), it is found that the equations fall naturally into three parts, each of which dominates in a given sector. At the western edge of the ocean x <^ r, and becomes Xwest = \ e-^- cos (^ ^^ - ^) + 1 (XVin.l4) representing slightly "underdamped" oscillations with a wavelength given by 586 Basic Principles of the General Oceanic Circulation A remarkable feature is a counter current east of the main current, with a magnitude of 17% of that of the main one. There can be little doubt that such counter currents exist, although this fact has been obscured in some instances by the smoothing of data. This theoretical result has in fact been shown to be in agreement with observa- tions (see p. 536 et seq.). The total transport of the western current and counter current is found by putting numerical values of A' into X.14) giving "Av M7/-;8-icurl, r. (XVIII. 16) The resulting expressions are independent of A and the transport can be computed with a relatively high degree of accuracy; the uncertainty is of the same order as that in the calculation of wind stress. Table 1 50 gives a comparison between the transport values of some western currents determined from oceanographic observations, and those computed from the zonal wind stress using equation (XVIII. 14). The two sets of values are of the same order of magnitude, but the calculated transport values differ from the observed values by a factor of as much as two ; the discrepancy is not surprising when it is considered that amongst other uncertainties the wind and current data are not for the same year, nor necessarily for the same time of the year. Another source of error may be due to possible underestimation of the wind stresses at low wind speeds. It can be assumed, in accordance with views held at the present time, that the dependence of wind stress on the wind velocity is given by /c = 0-0026 at high wind speeds and K r-^ 0-008 at low speeds, with the discontinuity at Beaufort 4 (see p. 421 and especially MuNK, 1947). This assumption, however, does not appear to be absolutely certain and further investigations are required. Table 150. The mass transport of some western currents determined from the wind stress and from oceanographic observations Current Lat. 1013^ (cm-1 sec~^) (km) 101° {8T,ldy) (g cm-2) 101-^ by wind stress (g sec-i) Ocean, obs. (g sec-i) Gulf Stream . Kuroshio Oyashio C. Brazil C. 35° N. 35° N. 50° N. 20° S. 1-9 1-9 1-5 2-2 6500 10000 5500 5500 70 50 -15 -20 36 39 -6-5 -5-8 74* (55)t 65* -1% -5 to- 10* * Sverdrup et al. (1942), pp. 605, 761. t Adjusted for a supposed southward motion of 19 x 10^^ g of slopewater. + For August (Uda, 1938). Away from both boundaries the stream-line function X reduces to -^central = 1 " which gives the central oceanic drift; this is a broad constant drift that compensates for the swift shallow western currents. Equation (XVIII. 17) also gives (XVIII. 17) (XVIII. 18) which agrees with the relationship derived by Sverdrup (see p. 580, equation XVIII. 5). Basic Principles of the General Oceanic Circulation In the eastern part of the ocean, the eastern solution is valid in the form X J_ r kr Xe 1 \ — g-k(r-x) 587 (XVIII. 19) It represents an exponential slippage zone with a width of approximately rr/k. If A 10' cm^ sec-\ the width will be about 200 km. The complete circulation of an ocean shows pronounced east-west asymmetry. The westward intensification of ocean currents is an effect of the planetary vorticity. The asymmetry may be expressed by either of the ratios : My (west, cur.) —0'55kr or V3 kr (XVin.20) My (cent, cur.) "' x (west. cur. axis) that is, by either the ratio of the maximum western current to the central drift, or by the ratio of the width of the ocean to that of the western current. The asymmetry increases with r, decreases with A and ; for the Atlantic kr ^ 100. Along the western coasts of the continents there are relatively strong seasonal ocean currents (California Current, Benguela Current, Peru Current), which cannot be explained by the simple assumption of zonal winds. To cover these currents which are also essentially dependent on winds, the theory must be expanded by the introduction of corresponding meridional wind stresses. This solution also has been given by Munk together with a general solution in which is introduced a general field of wind stress associated with the large-scale atmospheric circulation. To demonstrate the ability of this theory of the general ocean circulation to express the actual mean current conditions in an ocean, a theoretical solution for the Pacific as an approximation for a triangular ocean is given for comparison with a recent representation of currents based on observations in Figs. 267 and 268 (cf. Munk and Carrier, 1950). It can be clearly seen that all the essential features of the current patterns are covered by the theory. There is no doubt that the Stommel-Munk theory of ocean circulation explains the large-scale geographic picture of the horizontal ocean currents in all oceans as a direct effect of the permanent wind system over these oceans. There is very good qualitative agreement between the water transport computed from wind distribution and that Fig. 267. The computed mass transport in an ocean of triangular form represented by stream lines. Between two neighbouring stream lines 6 million tons of water flow in the direction of the arrows per second. 588 Basic Principles of the General Oceanic Circulation Fig. 268. The oceanic mass transport of the North Pacific Ocean, derived from data available. Between two neighbouring stream lines 6 million tons of water flow per second. (1) Kuroshio; (2) Oyashio; (3) Alaska Current; (4) California Current; (5) Sub-Antarctic Current; (6) North Pacific Current; (7) East Pacific Vortex; (8) North Equatorial Current. deduced from oceanographic observations, and this agreement is confirmed by all investigations that have been carried out along the lines of Munk's computations. HiDAKA (1950, a, b, c, 1951) has dealt in particular with the wind-generated ocean circulation of the Pacific and has obtained an overall climatological oceanic circula- tion, that fits admirably with that deduced from ship's displacements. His mathe- matical treatment of the problem differs from that used by Munk only in taking higher order terms into consideration and in using infinite series for the solution of the differential equation, in some instances with spherical co-ordinates, while Munk and his collaborators have used planar co-ordinates. More recently, Hidaka (1955) has presented a detailed numerical theory of the general circulation of the Pacific which he regards as a purely wind-generated phenomenon. He uses the assumption that the vertical velocity vanishes exactly at all points. Further, he gives the horizontal distribution of the stream lines for different subsurface levels. These circulation patterns are all similar to the sea surface circulation. The only noticeable difference is a general reduction in intensity of the movement with depth. It may be already as little as half the surface intensity in 250 m depth. His numerical results are, however, difficult to interpret on a physical basis, and appear insufficient for an explanation of the vertical mass transports necessary for continuity. Hansen (1951, 1954) treated the circulation problem as a boundary value problem ("Eigen" value problem). His method is equally suitable for finding the volume trans- port and the form of the sea surface in an enclosed part of the ocean from the known wind field. Hansen calculated the volume transport and the sea surface topography for the equatorial part of the Atlantic from the average August wind field, and obtained a satisfactory agreement with results based on observations of ship's displacements and of the density distribution. While for all methods the agreement is very good qualitatively, this is not always so quantitively. Munk, for instance, obtained transport values for the Atlantic and the Pacific which were only half as great as those computed from observational data (36 and 39 x 10^ m^/sec for maximum transport by the Gulf Stream and the Kuroshio, respectively, against observed mean values of 55 to 74 and 65 x 10^ m^/sec, res- pectively). It is not improbable that the discrepancy arises from the fundaments of the theory, possibly from the use of the mean wind stress based on climatological wind Basic Principles of the General Oceanic Circulation 589 charts without taking into account the deviations. It might also be due to the imper- fections in the present knowledge of the relationships between wind velocity and wind stress (see pp. 421 and 586) or due to the use of plane co-ordinates instead of spherical ones for the calculation of conditions on the curved surface of the earth. It is note- worthy that Hidaka has obtained good numerical agreement for transport in the Kuro- shio Current using spherical co-ordinates. The most probable reason however is that the actual dynamics of the strong western boundary currents (such as the Gulf Stream and the Kuroshio) are left essentially unexplained by the Stommel-Munk theory. In order to explain the narrowness of these boundary currents it is necessary to take an eddy viscosity so large that the eddy sizes would be comparable to the width of the cur- rent. This can never be the case. Pressure inertia and the variations of Coriolis para- meter with latitude all seem to play an important part in the dynamics of these boundary currents (see p. 550). It is striking that there is no indication of a "westward intensifi- cation" of ocean currents in the Southern Hemisphere; the Brazil Current and the East Australian Current for instance are not so strongly developed along the east coast of the continents as the Gulf Stream and the Kuroshio. It wouid be expected that if the planetary vorticity were the only cause of the westward intensification in the oceans of the Northern Hemisphere it would show the same effect in the South Atlantic and South Pacific. It appears however that the vertical structure of the ocean also plays a role in the theory since the depth d is correlated with the oceanic structure and the magnitude of d cannot be chosen arbitrarily, d denotes the depth over which an integration has to be performed in order to eliminate the effect of the vertical oceanic stratification and of internal vertical friction. Usually the depth of no motion has been taken as d and only the horizontal velocity of the water movement has been taken in- to consideration; the vertical velocity is presumed to be zero or so small that it can be neglected. This assumption is certainly incorrect and may lead to an entirely false picture of the horizontal circulation. Stommel (1956) has given a detailed discussion showing that the existence of a level of no motion in the ocean where all the three velocity components vanish cannot be substantiated; in fact the maximum vertical velocity occurs at the depth of no meridional \Q\ocity (see p. 499). A paper by Neumann (1955) is of interest here. He has re-examined the theory for a horizontal wind-driven ocean current taking into account the spherical shape of the earth the average vertical density stratification and the variable depth of the lower boundary of the circulation system. The latter assumption is the same as the assumption that the depth d is the depth of the layer of no meridional motion. Integration of the usual equations of motion for the geostrophic wind taken over the depth _ between +^ and —d and with P = p(x,y,z) gives the equations of transport dP 81, cd, ■^ -^ cy ^^^^ cy ^^ ^ dy CP CL dx. (XVIII.21) Introducing ' T+d p(-) dz; P(-d) = gp(i + rf) and P = p dz = 2f (? + df I gP 590 Basic Principles of the General Oceanic Circulation and taking into account that the divergence of the total mass transport is zero and C <^ d, one obtains /dxdp dddp\ Idxdl dddr\ ^ ^ ^ and from the second equation fM, = Igd' ^ + gpd^^ (XVIIL23) Equating M^ in (XVIII.22) and (XVIII.23) gives /^ 1 8d\ 8C/Pd 8d\ I8p nSC\ 8p\ 8d _ \f~ ddy) dx-^ [jl" d^rpdx^ [ddy-^ -p dyjdd " ^- (^^111.24) In the case of a homogeneous ocean (p = const.), equation (XV1II.24) reduces to B 1 8d\8^ 1 8C 8d This equation states that in the case of a constant depth d only zonal steady currents are possible, because the first term will vanish only when 8l,j8x = 0. When the depth J is variable, all current directions are possible, if d satisfies certain conditions according to (XVIII.24). If the depth d is a function only of y (the latitude), then, provided that 8d/8y ^ 0 B 1 8d This equation is identical with (XVI. 19) and states that for stationary currents the decline of the lower depth d of the current system towards the poles must follow a law d — K?.m 4>. In a stratified ocean (p = p{x,y,zy) the interrelationship is more compli- cated. Equation (XVIII.24) shows, however, that for a constant depth t/ of no horizontal motion, there can be no meridional mass transport due to frictionless currents, since when d = const., the equation reduces to (XVIII.27) On substitution in equation (XVIII.23) it is found that M^ = 0. It has been shown above (p. 497) that in the Atlantic Ocean the zonal mean of the depth of no meridional motion follows the above equation. This can be interpreted to mean that the planetary vorticity {^My) is compensated by a corresponding balanced topography of the lower boundary of the current system. This is frequently the case in the South Atlantic, and here the westward intensification, which of course is a consequence of the planetary vorticity effect, is only weakly developed. In criticism of Neumann's arguments, Stommel has questioned the assumption that the depth f/ is a depth of no motion, and has pointed out that on the contrary, the greatest vertical velocities occur at this depth. Neumann's equations can also be derived from the basic assumption that the potential vorticity in the large-scale Basic Principles of the General Oceanic Circulation 591 oceanic circulation is constant (see p. 336); that is, dldt{t, -'rf)/dc, — 0, where ^ is here the relative vorticity. Since generally ^ < /, this equation reduces to dy \ d I 0 for stationary predominantly zonal currents. From this it follows, since C <^ /, that / \8d I 8f p ^~ const, or -^ =--/=-„ (XVIII.28) d ddy fdy f which is equation (XVI1I,26). However, the assumption of constant potential vorticity is valid only for horizontal geostrophic currents, but does not hold when vertical velocity components are also present. Stommel's objection seems then to be justified and Neumann's equations are valid as a first approximation only when the vertical velocities are small compared with the horizontal ones. Chapter XIX The Tropospheric Circulation 1. The Position and Structure of the Oceanic Troposphere The important subdivision of the oceanic space into troposphere and stratosphere is due primarily to the climatic influence of the atmosphere on the water masses of the uppermost ocean layers. More or less constant conditions in weather and radiation at the ocean surface give rise to the development and maintenance of water types of diff'erent character in different climatic zones. Broadly speaking there are two principal water types which are constantly being formed in large quantity and with a rather constant internal structure; they correspond to the two great zones of contrasting climate, the tropical and subtropical regions, and the polar regions. These two water types are: (1) the tropical-subtropical water type which is warm due to the excess of incoming radiation and has a high salinity due to evaporation, and (2) the cold weakly saline water type of the subpolar and polar zones. The former is lighter, the latter heavier, and this difference is the cause of con- tinuous large-scale movements. These movements follows the fundamental principle that each water type tends to flow by the shortest route, by vertical or horizontal dis- placement to the depth in the ocean at which it will be in a stable equilibrium corres- ponding to its density; here it will spread out as a layer. The heavier subpolar water type therefore sinks to greater depths, and spreads more or less horizontally to fill in this way the deep lower layers of all the oceans. The lighter tropical and subtropical waier type, on the other hand, remains in the upper layers of its original zone as the lightest water type. The subdivision in the structure of the oceans is thus a con- sequence of circulation. It is to be expected already from the history of formation of the two main oceanic subspaces, that they will have essentially separate circulations; these will be called tropospheric and stratospheric. This does not imply that there is no connection between the two circulations; on the contrary, at certain places inter- actions occur and the water masses of both type undergo transformation by turbulent mixing and manifold atmospheric influences so that tropospheric water becomes stratospheric and vice versa. The thermo-haline structure of the troposphere has been explained in pt. I, Chapter III, §4, p. Ml et seq. and IV §3, p. 165 et seq. The most important phenomenon is the layer of discontinuity in the vertical distribution of temperature and density which is always sharply defined in the tropics and subtropics and is associated with a charac- teristic salinity distribution. An example is shown in Fig. 70 of pt. I. Beneath the dis- continuity layer which acts as a barrier to upward and downward movement, is the subtroposphere which is occupied by little differentiated and nearly motionless waters. 592 The Tropospheric Circulation 593 It is usually difficult to fix a definite boundary between the troposphere and the stratosphere. In the vertical density profile it appears as a slight intensification of the vertical gradients; but often it is quite indistinct because of the very great distance between observation levels at these depths. It should probably be referred to only as a boundary layer. An approximate boundary can be obtained using the oxygen content as a criterion (Wust, 1936Z?, see pt. I, p. 66 et seq.); it is then defined by the inter- mediate oxygen minima. The method is based on the assumption that these minima indicate layers where the air supply is least, that is, those localities where the renewal of the water masses is particularly slow and where horizontal movement of the water is entirely missing. It has frequently been pointed out (p. 494) that in the uppermost layers the position of the oxygen minima is affected by biological processes. However, oxygen minima can be used at greater depths to specify approximatively the different circulations. In the Atlantic the oxygen minimum extends across the 1 10 degrees of latitude (from 45° S. to 55° N.) between the oceanic polar fronts of both hemispheres; its mean depth along a meridional section is given in Table 151. From the Southern Hemisphere polar front the lower limit of the oceanic tropo- sphere sinks rapidly down to 600 m in the southern convergence region (between 35° and 25° S.), and rises again to about 300 m in the tropics. Just north of the equator, it is at first somewhat irregular and then sinks gradually down to about 950 m in the northern convergence region (30° to 40° N.). Reasonably accurate data are available for the tropospheric circulation which extends throughout the space between the sea surface and the lower boundary of the troposphere. Defant's (1936c) representation of conditions in the Atlantic also includes subsurface data over the whole area. For the other oceans the series observa- tions are sufficient for interpretation only along single meridional or zonal sections. No major differences between the oceans in the principal features of circulation are to be expected. Table 151. Lower limit of the troposphere in the Atlantic Ocean (Determined from the position of the oxygen minimum. Depth in metres.) Section 50° 45° 40° 35° 30° 25° 20° 15° 10° 5° Equa- tor r (1000) 850 830 820 770 550 280* 350 Western section 400 IS 1 _ ■i — 400 500 550 600t 580 450 300 280* r 450 790 IIQ 830 880t 870 680 470 380 i 330* Central section 400 IS — (100) 320 500 600t 580 550 420 300* 400 r (900) (900) (900) (900)t (250) 820 680 (550) 520 400 Eastern section -. 350* IS — 300 470 530t 510 450 380 300* 390 400 N. Northern Hemisphere; S. Southern Hemisphere * Minimum values; f Maximum values. Q 594 The Tropospheric Circulation 2. The Tropospheric Circulation of the Tropical and Subtropical Oceans The tropical and subtropical circulation of the oceanic troposphere is dominated by the enormous water transports of the North and South Equatorial Currents. They determine dynamically the position of the tropical and subtropical discon- tinuity layer. Its depth in the Atlantic between 25° N. and 25° S. is shown in Fig. 269. From a depth of more than 200-300 m in western Ross-latitudes of both hemispheres Fig, 269. Depth (m) of the tropospheric discontinuity (thermocline) in the Atlantic Ocean between 25° N. and 25° S. the discontinuity layer rises towards the southeast to a depth of 40 m in the Northern Hemisphere and towards the north-east to a depth of 20 m in the Southern Hemi- sphere. Between the equator and about 6°-10° N. these rising slopes are separated by an east-west depression extending into the Gulf of Guinea. This striking arrangement of the topography of the discontinuity surface is a direct consequence of the equatorial currents on either side of the equator; because of dynamic reasoning these currents also determine the rise of the discontinuity layer towards the equator. Up to about 6° to 10° N., the depth of the density transition layer is associated with the Equatorial Counter Current and its further extension (the Guinea Current). For a connection between the state of motion of the water masses above and below the discontinuity and the topography of the discontinuity layer see p. 463 et seq. Further information on the conditions of motion in the individual layers of the oceanic troposphere can be gained by investigation of the striking salinity maxima near the discontinuity layer, The Tropospheric Circulation 595 that intervenes between the homo-haline and weakly saline top layer and the deeper lower salinity layers with an equally low salinity. Study of the position of these maxima and their development showed that they intrude under the less saline top layer from the extensive subtropical accumulations of highly saline water to the north and south. These intrusions spread along preferred paths, the location of which throws some light on movements within the middle and lower layers of the troposphere. This spreading and its dynamics have been discussed in pt. I, Chapter IV, p. 166. There Fig. 72 (p. 168) shows that the salinity maxima are present everywhere except in two narrow bands in both hemispheres where the density transition layer comes closest to the sea surface. Evidently, the horizontal extension of the highly saline intermediate layer is cut short in this region, and here the water masses must be deflected upward. The region between the two bands without salinity maxima lies in the Equatorial Counter Current. Here the supply of water that forms the salinity maxima comes from the west, from regions which are not reached by the bands free from the salinity maxima and are fed here from north and south. From these facts it is possible to derive a three-dimensional system of currents in the oceanic troposphere of the tropics and subtropics, that is illustrated schematically by the meridional section in Fig. 270. 20° S 15 20° N 25° 200^ Fig. 270. Schematic representation of the zonal and meridional velocity components of the tropospheric circulation in the Atlantic Ocean (the topography of the thermocline is exaggerated in the vertical scale by about 1 :1 million; that of the physical sea surface even more); W, current towards west; E, current towards east. Where the stream lines are divergent in the top layer they are convergent in the dis- continuity layer; the two bands with a low salinity are thus regions of upwelling water. The zonal components of motion do not appear in the meridional section and it should not be forgotten that these are considerably more important. Compared with these the transverse circulation is rather weak. This transverse circulation is primarily a thermo-haline circulation and is the consequence of the internal forces of the mass distribution (p. 575). It involves only the top layer down to the density transition layer and in the strong zonal motions of the wind-driven equatorial currents it can hardly be detected. It is, however, responsible for the pronounced vertical and horizontal salinity distribution that is characteristic for the uppermost layers of the tropical oceans. The water masses beneath the density transition layer (in the subtroposphere) are very uniform and colourless and the water movements here must therefore be very weak. Since they lie beneath the barrier, they can be only slightly aff'ected by turbulence 596 The Tropospheric Circulation and convection and they have an extremely low concentration of oxygen which is largely due to the almost total stagnation and also due to biological causes. The internal forces, providing the motive force for the entire current system of the tropics and the subtropics, are produced, on the one hand, by the wind system present in these zones and on the other hand, by the internal pressure field set up by the thermo- dynamic conditions. Figure 271 shows the absolute topography of the physical sea level in the Atlantic Ocean pictured by isobaths drawn at intervals of 5 dyn/cm between 35° N. and 35° S. and at intervals of 10 dyn/cm outside this area (Defant, 1941^). The direction of this stationary gradient current, which corresponds to this pressure field is indicated by arrow-heads on the dynamic isobaths. Comparison of this topo- graphy in the tropical and subtropical area with that of the tropospheric density transi- tion layer (Fig. 269) shows that they are almost mirror images; in deeper layers the Fig. 271. Absolute topography of the physical sea surface (dynamic isobaths drawn from 5 to 5 dyn cm, 10 to 10 respectively). The Tropospheric Circulation 597 pressure surfaces are of the same form as the sea surface but the pressure gradient decreases rapidly with depth (Fig. 272). The lower limit of the tropical and subtropical circulation must lie at the 500 decibar surface where the pressure gradient is almost zero; already at 200-300 m depth the velocity of the currents is very slight and the Equatorial Counter current does not reach nearly as deep as this (approx. down to 1 50 m). A comparison of the topography of the physical sea level and the gradient ra° W Fig. 272. Absolute topography of the 100-decibar (upper picture) and 500-decibar surface (lower picture) of the subtropical and tropical region of the Atlantic Ocean (dynamic isobaths are drawn from 2-5 to 2-5 dyn cm). 598 The Tropospheric Circulation currents at the sea surface derived from it (see ''Meteor''' Report VI §2, supplement 22) with current charts derived from observations shows that the trade winds are the main cause of the currents in the uppermost layer of the sea. These give rise to a total water transport at right angles cum sole of the wind direction. In the Northern Hemisphere the water '^flows towards west-north-west and in the Southern Hemisphere towards west-south-west. Along the east coasts of continents and also at the eastern boundary of the strong water displacements, which are directed from north to south along the coast lines, water is accumulated and piled up and thus a pressure gradient is created to the south-east in the Northern Hemisphere and to the north-east in the Southern Hemisphere. This is shown clearly by the topographies of the pressure sur- faces and of the sea surface, respectively. In the trade-vv-ind region the resultant ocean current is then no longer solely due to the effect of the permanent air currents charac- teristic for these latitudes, but is also affected decisively by the mass distributions in the uppermost layers. A diagram of forces for the central part of the South Equatorial Current according to the "Meteor" observations, has already been discussed (Fig. 180, p. 424). It allows an estimate to be made of the effect of the individual forces in the formation of this major current. It is of particular interest that the water masses in the equatorial currents^ow against the slope of the physical sea level and the pressure surfaces, that is to say, uphill. Part of the force transferred to the water by the winds is used in overcoming this gradient, so that the velocities of the water displacement are correspondingly somewhat reduced. The pressure field associated with the Equatorial Counter Current is clearly shown in the topography of the physical sea level (Fig. 271) and in the topography of the isobaric surfaces (Fig. 272). This current is undoubtedly an essential feature necessary for the stability of the tropical current system. Its asymmetry about the equator is a consequence of the displacement of the thermal equator into the Northern Hemi- sphere and of the accompanying asymmetry of the atmospheric circulation (see p.463). The main contributions to the theoretical explanation of the mode of formation of an Equatorial Counter Current have been primarily due to Sverdrup (1932); Defant (1935, 1941); Thorade (1941) and Palmen and Montgomery (1940). For an atmos- pheric circulation assumed symmetrically about the equator, the Equatorial Counter Current can be readily explained as a compensation current produced by the distur- bances of the pressure field by a meridional continent opposing the wind drifts corresponding to the North and South Equatorial Currents. It flows eastwards as a gradient current in the direction of downward sloping sea level and is retarded only by friction at the lower boundary surface and at both sides of the current. Stockman {\9A6a-d) has attempted to consider also the baroclinic mass field, though without taking into account the dependence of the Coriolis parameter on latitude. According to this explanation the accumulation of water carried westwards and piled up by the equatorial currents is the most important factor in the formation of the counter current. The asymmetry of the counter current about the equator would then be due to the asymmetry of the atmospheric circulation. Presumably for the Atlantic this explanation of the counter current can be considered as an adequate one, but for the considerably more extended Pacific it is doubtful whether the effect of the water accumu- lation piled up in the west is sufficient in order to give rise to a counter current as a very narrow band over such a great distance. The Tropospheric Circulation 599 Evidence against this conception of the equatorial counter current as a pure gradient current has been accumulated by Sverdrup (1947) and Reid (1948), who showed that the main features of the baroclinic mass distribution in the tropical and subtropical Eastern Pacific are due entirely to the effects of the mean wind stress distribution in these regions. A method for the determination of the mass field and the mass transport of the currents from the given wind field has already been described on p. 550 and following pages. By means of Fig. 254 it has been demonstrated that the mass structure and the currents of the equatorial region of the Eastern Pacific are only effects of the wind stresses. In these investigations full account was taken of the dependence of the Coriolis parameter on the latitude, but the influence of lateral friction and of thermodynamic effects such as radiation and evaporation and others was neglected. The good agreement between theory and observations is an indication that the latter effects are of secondary importance in the dynamics of the equatorial counter current. Figure 273 presents diagrams of forces for the equatorial currents and Coriolis force Wind stress (b) Wind stress (c) Pressure grodient Windstress Pressure gradient Pressure yadient Equatorial Counter current Coriolis force Coriolis force Fig. 273. Diagrams of forces: {a) for the North Equatorial Current; {b) for the South Equatorial Current; (c) for the Equatorial Counter Current. for the counter current. Basically there is no difference between them; since they are each produced and maintained primarily by the wind in a sea with a baroclinic mass structure. A comprehensive representation of the oceanic structure and circulation in a section along the middle axis of the Atlantic is contained schematically in Fig. 274. It is self- evident that this picture is of a schematic nature only, however, an attempt has been made to include all the characteristic features of the tropospheric oceanic structure as well as the corresponding three-dimensional circulatory movements. This circulation in its zonal extent is largely a consequence of the air currents over the sea surface. The 600 The Tropospheric Circulation South equ current Potar front Convergence *fE--* — pE— E-T^ ,w-«- *- y-*-' w— H-*^r-^ Po'ar front Fig. 274. Schematic picture of structure and circulation in the troposphere of Atlantic Ocean in meridional direction. Limit between the tropo- and stratosphere. Position of maximal density gradients. Tropical-subtropical thermocline. |^:-!v';v>?.-l Layers of extremely low oxygen contents ( < 1-5 cm^/1). Position of tropical-subtropical salinity-maxima. W, E Zonal velocity component (W towards west, E towards east). meridional components of motion, on the other hand, are a consequence of meri- dional variations in radiation and evaporation-precipitation difference and are there- fore only weak. The lower currents stand clearly out in salinity sections of the Pacific and of the Indian Ocean as tongues of high salinity. They originate and spread out again from the subtropical accumulations of highly saline water. A meridional salinity section through the central part of the Pacific Ocean (Pt. I, p. 172, Fig. 76) shows that the intrusion of this water from the South Pacific is the stronger one reaching as far as 12° N. in a depth of 150-250 m. The northern branch, however, is present only between 22° and 25° N. In the east these intrusions seem to be still weaker (see the vertical section in the Eastern Pacific given by Schott, 1935, p. 182); contrary in the west Pacific region they are stronger. The southern undercurrent shows as a spectacular phenomenon (see Fig. 275, Wust, 1929) though again, the northern branch is only weakly devel- oped. 10° N Fig. 275. Longitudinal section of salinity through the subtropical deep current in the West Pacific Ocean (according to Wiist). The Tropospheric Circulation 601 The equatorial currents are particularly well developed in the Pacific. As in the Atlantic the counter current lies in the Northern Hemisphere throughout the whole year and especially far from the equator during the northern summer. The surface velocities reach values of more than 2 knots. The structure of the water masses was first pictured in a "Carnegie" section (at about 140° W.) in October 1929 (Sverdrup et al. 1 942., p. 709). Figure 276 shows the temperature and salinity distributions between Stot 159 LcrtiO"! S 300 Horizontal velocity, cm/sec Fig. 276. Temperature, salinity and computed velocity in a vertical section in the Pacific Ocean between 10° S. and 20° N. (according to the "Carnegie" observations; arrows indicate direction of the north-south flow; E. and W. indicate flow towards east and west respectively) (according to Sverdrup, 1942). the sea surface and 300 m as well as the velocity distribution calculated on the assump- tion of no motion at the 700-decibar surface. The equatorial counter current hes between 5° and 10° N., and in correspondence with the sea surface slope flows downwards in the calm belt between the trade winds. The maximum velocity at the surface is a little over 50 cm sec-^ in good agreement with observed values. The "Carnegie" section gives an eastward transport by the equatorial counter current of approximately 25 million m^ sec-^. The character of the transverse circulation is evident from the distribution of salinity, oxygen, phosphate and also silicate and is quite similar to that shown in Fig. 269 derived from observations in the Atlantic. A detailed theoretical treatment of the circulation in the top layer of the equatorial parts of the oceans has been given by Yoshida, Mao and Hoover (1953). They start out with the steady-state equations involving the Coriolis force, the pressure gradient and horizontal as well as vertical mixing. For the mean wind-stress distribution and 602 The Tropospheric Circulation the mean density distribution they took Reid's model (1948) which is generally applicable in equatorial regions. The wind drift and gradient current were super- imposed correspondingly considering the boundary conditions, and finally the vertical velocity in the upper mixed layer was calculated using the continuity equation. Figure 277 shows the dependence on the latitude of the horizontal velocity components u and V at the sea surface and the horizontal wind stress Tx, acting only in zonal direction. It is evident that a strong equatorial counter current is formed between — •(Equatorial Counter Currency -35 -30 -25 -20 -15 20 25 30 0€ 04 0'2 OO DYNE/Cm2 Fig. 277. Latitude dependence of the horizontal velocity components u and v at the sea surface and the horizontal wind stress T^ acting only in zonal direction on the ocean surface. 2-5° and 1 1° N. in the area of weak westward wind stress between the strong north- east and north-west trade winds. All the velocity components decrease somewhat with depth down to the lowermost boundary of the upper mixed layer, the w-com- ponent of the equatorial counter current decreases least so that almost uniform values are found throughout the entire top layer. The vertical velocity resulting from the continuity equation is shown in Fig. 278. Its distribution is rather noteworthy. It shows : (1) very strong upwelling at or near the equator, this is the equatorial divergence; (2) strong sinking at the southern boundary of the counter current; and (3) fairly strong upwelling at the northern boundary of the counter current. The vertical velocity is of the order of lO-* and 10"^ cm sec"^. Farther to the north the velocities are small and irregularly distributed. Considering that the Reid model is only a crude approximation of true conditions and especially that the wind-stress distribution with zonal components only can hardly correspond to actual conditions, The Tropospheric Circulation 603 sjstauj U! MidaQ Depth in meters 604 The Tropospheric Circulation the similarity with the vertical velocity field shown in Fig. 269 is remarkable. It should be noted that the vertical velocity component does not vanish at the lower boundary of the upper mixed layer. The current does not follow the inclined surface of this boun- dary unless the divergence of mass transport in the upper mixed layer is zero. This does therefore never correspond to the conditions shown in Fig. 269. Also in the Indian Ocean conditions are similar with the same much weaker develop- ment of the phenomenon in its eastern parts (see Pt. I, p. 172, Fig. 75). Since the thermal equator remains here always south of the equator the tropospheric circula- tion is again rather asymmetrical and, as in the Pacific, the southern hemispheric branch is the stronger one. However, while the conditions in this branch are almost unchanged throughout the total year, complications must appear in the Northern Hemisphere due to the seasonal changes in the current system of the sea surface. During the summer months the strong south-west monsoon current extends down to the lower layers of the troposphere and the subtropical undercurrents are suppressed. The available sections do not show the nature of this change. Probably the highly saline water masses of the southern hemispheric lower currents extend into the Nor- thern Hemisphere and partly enter the influence region of the wind drifts of the south- west monsoon. The Equatorial Undercurrent. Cromwell, Montgomery and Stroup (1954) discovered an Equatorial Undercurrent in the Central Pacific in a zone between the equator and latitude 1° N. and at a depth of 50-150 m. It is found as a narrow east- ward current both in the lower part of the top layer at the equator and in the upper part of the thermocline in this zone, where the South Equatorial Current extends into the Northern Hemisphere. Its position in the vertical and horizontal circulation of this area is sketched in Fig. 279. Fofonoff and Montgomery (1955) have shown that the Equatorial Undercurrent agrees with a simple application of the vorticity equation E 0 U ATOR \ \ \ EQUATORIAU CURRENT / 1 T 1 T "seasuRFace ' eOUATOftua. W«0£RCU«W£hT Fig. 279. Meridional section showing idealized currents in the surface layer (about 100 m deep within about 3° of equator, reader looking west). The flux components in the plane of the section are indicated by broken arrows. Zonal components of velocity at the top and the bottom of the layer are indicated by diagonal arrows drawn in perspective. The Tropospheric Circulation 605 (X.68). In a cross-section through the meridional circulation the water flows towards the equator in the part of the top layer beneath the drift current and rises at the equator. The zonal component of the surface current can be taken as uniform and the relative vorticity/o being zero. If a water layer moves without friction from an initial state ^o> /o' /?o to a new state the vorticity equation gives the relationship For water sinking from the surface ^q = 0, and if the thickness is assumed to remain constant during the displacement, and if all the water is assumed to have started from the same initial state, the distribution of the zonal velocity component can be found by integration of f+C=fo- (XIX.2) For a predominantly zonal current dii 1 8ii ^ 8y R defy and for low latitudes the solution can be written in the simpler approximate form u-Uo = Roj( - cf^of- (XIX.3) If, in the South Equatorial Current, the surface water has a velocity of 0-5 knots with no lateral shear and sinks from latitude (f)Q — 2-1° and flows without friction or changes in thickness to the equator, it will reach the equator as the east undercurrent with a velocity of 2 knots. The component of the velocity directed towards the equator in waters moving from latitude 3^ to the equator can also be calculated. The zonal velocity component is given by the equation %=fi-gi.,^ (XIX.4) where /^.^ is the longitudinal slope of the sea surface at latitude (/>. Its existence is made possible by the presence of the continental barriers. Since du dii fv - -jj ^fv -V ^ ={f+ i)v =foV (XIX.4) can be written in the form ^ = 7 ix, ~ ^^—r ^x,4>- (XIX. 5) /o 2cu9o If ^0 = 3°; /o = 7-6 X 10-*' sec-i and i^^ = — 3 x 10-^ (see Montgomery and Palmen (1940) and Jerlov (1953)), the velocity component v towards the equator is —4 cm/sec or 2 nautical miles a day. The Equatorial Undercurrent is consistent with the flow towards the equator in the lower part of the top layer close to the equator, if this flow is approximately friction- less so that absolute cyclonic vorticity is conserved. Continental barriers which permit a longitudinal component of the pressure gradient, are essential for any extensive development of the undercurrent. 606 The Tropospheric Circulation 3. Other Currents of the Oceanic Troposhere (a) The Guiana Current and the Current Conditions of the American Mediterranean The stream lines of the tropospheric undercurrents of the Southern Hemisphere converge from the whole of the South Atlantic towards the area off Cape San Roque on the east coast of South America and the water of the South Equatorial Current flows into the Northern Hemisphere at this point. The subtropical salinity maximum of 36-7%o at about 120 m depth can be followed far to the north (as far as the West Antilles and beyond) in a salinity section following the course of the Guiana Current north-westwards along the South American coast. The character of this water remains almost unchanged from the area of South Equatorial Current in the Southern Hemis- phere to the Antilles. For the most part the current axis remains over the broad shelf off the mouths of the Amazon and the Orinoco. The corresponding pressure gradient could be determined so far only from very few stations. The direction of the pressure gradient in a gradient current must be reversed on passing from the Southern to the Northern Hemisphere. This can be seen in the sea level topography given in Fig. 271. South of the equator the higher pressure occurs at the coast with the lower pressure farther out; north of the equator this is reversed and here the Guiana Current is accom- panied along its right-hand edge by a narrow ridge of high water level with a down slope towards the coast which in accord with the great strength of the current is quite considerable. The Guiana Current, together with the southern part of the North Equatorial Current, flows into the Caribbean through the passages between the Lesser Antilles (sill depth less than 1000 m). The observational data for this sea has been evaluated principally by Parr (1935, 1937«, 1938a); see also Seiwell (1938) and Rakestraw and Smith (1937) on chemical aspects and a review of these conditions by Dietrich (1939). The tropospheric currents between 100 and 200 m are very clearly shown by the salinity maxima of the undercurrents which are a continuation of those of the North and South Equatorial Currents. Figure 280 shows a chart of surface Fig. 280. Distribution of salinity in the core of the subtropical salinity maximum in the American Mediterranean (according to Dietrich). The Tropospheric Circulation 607 currents during the spring. The large salinity diflferences which appear where the under- currents of the Equatorial Current join off the Antilles soon disappear in the eastern Carribean. There is a striking uniformity in the Caribbean and in the Yucatan Channel due to lateral mixing. The weak inflow through the Windward Passage (sill depth about 1600 m) makes little change. The differences in the Gulf of Mexico are larger. The extended areas with vortices in the north-eastern and the western parts of the Gulf which are very pronounced in the surface currents remain outside the circulation of the tropospheric layers. Investigation of [r,5']-curves in the water masses of the South Equatorial Current, the Sargasso Sea and the Yucatan Channel allows to estimate how much of the inflow water through the Antilles takes part in the water passing through the Yucatan Channel. Iselin (1936) found that of a total transport of about 26 million m^/sec through the Yucatan channel approximately 6 million originates in the South Atlantic. For the deeper layers the eff'ects of the inflow through the Wind- ward Passage and the Virgin Passage are of greater importance (see Pt. I, p. 133). The uniformity of the distribution of the oceanographic factors over the area shows the effect of the mixing processes which are stronger here than the pure transport processes. Dynamic evaluation of the data for the latter should give greater informa- tion (Parr 1937Z?). Figure 281 gives the dynamic topography of the physical sea level relative to that of the 1200-decibar surface for the Caribbean and for the Cayman Sea. The mean current core running from the Antilles through the Yucatan Strait to the Florida Strait is clearly marked. The course of the dynamic isobaths shows that the water flows uphill to reach the Yucatan Channel (see also Sverdrup, 1939). Similarly as in both the Equatorial Currents, the water transport here is also largely due to the air currents (prevailing wind to the east-north-east with a mean velocity of 10 m/sec). Thus to a very large extent these currents are also gradient currents in a baroclinic sea though they are subject to significant modification by the wind. {b) The Gulf Stream and its Internal Structure Although the Gulf Stream is the largest and the most important current of the Northern Hemisphere a more dynamic investigation of its course has only recently been started. The first current measurements in it were made by Pillsbury in 1885-9 from the "Blake" which was anchored in very deep water. Further investigations were begun in 1914 by the oceanographic survey vessel "Bache" (Bigelow, 1917, four transverse profiles through the Florida Current and the Antilles Current). More recently a systematic survey has been started by the oceanographic survey vessel "Atlantis" (Woods Hole Oceanographic Institution). The first dynamical evaluation of some transverse profiles in the Florida Current was given by WtJST (1924) using the "Blake" measurements. This and subsequent work have aff'orded a more or less complete description of the vertical structure of this current from the Florida Strait to the Newfoundland Banks. Special mention should be made of the work of Jacobsen (1929) on the Sargasso Sea using "Dana" observa- tions and that of Iselin (1936) giving a detailed review of the comprehensive results collected by "Atlantis". Dietrich {\92>lb, see also WiJST, 1930a) has given a detailed analysis of numerous sections to show the process of formation and the dynamics of the Gulf Stream. The thermo-haline structure of the Gulf Stream is immediately apparent from the set of six success profiles given by WiJST (Figs. 282, 283). Profile I 608 The Tropospheric Circulation 85° VV ^0° 75° 70° 85° W Fig. 281. Dynamic topography of the physical sea surface (relative to that of the 1200 decibar surface) for the Caribian Sea and the Cayman Sea. (Lines of equal dynamic anomaly drawn with an interval of 005 dyn m.) is in the Yucatan Channel, profile II north of Cuba, profile XII at the narrowest part of the Florida Strait, profile IV at the exit from the Florida Strait just before the junction with the Antilles Current, profile V at Cape Hatteras and profile VI from the Newfoundland Banks in southward direction. The temperature profiles show that the Gulf Stream is by no means a deep-reaching current of high temperature. It differs only little in the thermal structure from the neighbouring Sargasso Sea. The steep oblique slope of the isothermals and isohalines is characteristic and the narrower the section the more rises the lower, cold and weakly saline water at the left-hand boundary. This baroclinic mass distribution is connected with the current velocity and direction and is more pronounced the stronger the flow. It is thus more prominent in the narrow sections to the south. Profile V shows the Gulf Stream beyond the junction of the Florida Current and the Antilles Current where it has its greatest vertical thickness, about 1000 m, and has the considerable core width of about 50-70 km. Its left-hand The Tropospheric Circulation 609 New Foundtand Bonk Sargasso Sea Fig. 282. Cross-section of temperature through the Gulf Stream (profiles I, lla and V according to Jacobsen; profile VI according to Helland-Hansen; profile II and IV according to Wiist). 2R 610 The Twpospheric Circulation New Foundlond Bank rrhernportof^x <;, AvVV"^.". Sorgasso Sea 100 200 km 300 Fig. 283. Cross-section of salinity through the Gulf Stream (see remarks below Fig. 282). The Tropospheric Circulation 611 edge is sharply defined and keeps about 100 km off the coast. The right-hand edge is diffuse and differs little from the water farther to the east. Where it swings eastward the current spreads out and loses thermal and haline thickness by mixing with colder surrounding waters. To the south of the Newfoundland Banks it begins to break up into a number of branches ; profile VI shows only the northern branch which borders on the Labrador Current. The further branching of the current in the east has been discussed on p. 562, Fig. 257. The Gulf Stream is only slightly more saline than the Sargasso Sea and in the deep layers there is no difference. The salinity maximum lies in the subtropical undercurrents which enter through the Antilles as part of the North and South Equatorial Currents into the American Mediterranean and from there across the Gulf of Mexico into the Florida Strait. It is thus a long-range effect of the tropospheric circulation of the tropical and subtropical Atlantic. The Antilles Current also shows this highly saline intermediate layer; but here it is in direct connection with the highly saline top layer of the Sargasso Sea. Further along the course of the Gulf Stream this salinity maximum comes at times up to the surface, but in the North Atlantic Current it dips beneath the weakly saline surface layers. It can be traced well into the Norwegian Sea (see pt. I, p. 171, Fig. 74). The salinity profile also shows another long-range effect of the Atlantic circulation: this is the last traces of the weakly saline subantarctic intermediate water which can still be seen at a depth of between 700 and 1000 m {S < 34-9%o) as far north as 25° N. in the Florida Strait; in the Sargasso Sea, however, it reaches only to 10° N. The dynamics and the water transport of the Gulf Stream are derived primarily from velocity profiles. Several such profiles are available at the present time; they are based partly on direct-current measurements and partly on dynamic calculations from the mass field. The cross-section is not everywhere completely occupied by the current; particularly where the current flows out of the Florida Strait into the open ocean. The current flows as a jet through the narrow part of the strait and follows the direction imposed on it for a considerable distance. The velocity distribution in the cross- section is related to the mass field and the agreement between the calculated and observed current profiles is generally good. Beyond the junction of the Florida Current and the Antilles Current the weak counter current between them disappears completely, but the counter current on the right-hand side of the main one is retained. In the cross-section off Chesapeake Bay shown in Fig. 284 it lies just outside the profile. A deeper insight into the dynamics of the current can be obtained from the absolute topography of the isobaric surfaces and of the physical sea level. These are parti- cularly dependent on the choice of the reference-level. In the narrows of the Florida Strait this lies near the bottom where the velocity decreases almost to zero. Further north it lies in the Sargasso Sea at about 1900 m depth (corresponding to Fig. 272) and rises steeply from the right-hand side of the Gulf Stream to 1000 m depth or even less. Over the current core the physical sea level rises steeply from left to right and at Cape Hatteras this rise amounts to about 100 dyn cm. It remains more or less of the same order up to the Newfoundland Banks, but gradually spreads out horizontally so that the actual gradient falls to about a third. The right-hand side of the Gulf Stream is associated with a high-pressure ridge which can be traced from the Bahamas to the 612 The Tropospheric Circulation Coostal stream Gulf stream Fig. 284, Velocity profile (cm/sec) across the Gulf Stream off Chesapeake Bay, 20-22 April 1932. south-west of the Newfoundland Banks. Eastwards from here there is a counter current steadily broadening to the south. The absolute topography of the 500 decibar surface still shows clearly the same pressure gradient as at the sea surface but it is rather weakened. The 800 decibar surface shows a rise across the current of at the most 20 dyn cm; and the pressure gradient has fallen to about a quarter. The 1400 decibar surface is almost plane and the lower limit of the current system must there- fore lie between 1000 and 1200 m. A detailed analysis of the origin and the transformations of the Gulf Stream water as it flows from the Florida Strait to the Newfoundland Banks were investigated by Dietrich (1937) with the aid of distribution of oxygen content in numerous profiles. He was able to show that the water masses of the Florida Strait and of the Antilles Current to the north of the Lesser Antilles were made up partly of tropical South Atlantic water and partly of subtropical water from the western North Atlantic. The Gulf Stream water reaching Cape Hatteras has, however, undergone changes making it almost completely identical in its properties with the water of the western North Atlantic. This transformation was attributed by Dietrich to the transverse circulation and to mixing. From the distribution of the oceanographic factors such a transverse circulation seems not unlikely, but it is not possible to determine it from the pressure field because of the low velocity and probably also because of its variability. The amounts of water and heat carried by the Gulf Stream are enormous. The Florida section shown in Fig. 284 gives a water transport of about 25 million m^sec. It can be assumed that this will also be the transport in the currents through the Carib- bean and the Yucatan Channel, since the precipitation and the inflow of river water (run-off) are small compared with this very large quantity. Some idea of the enormous quantity of water involved is given by the estimate that it is twenty-two times as much as is carried by all the rivers of the earth together. The amount of water carried by the Gulf Stream further north is much larger than this and the transverse profile off Chesapeake Bay gives a transport three times greater (82 million m^sec). It can be assumed as a first approximation that the amount of water carried by the North The Tropospheric Circulation 613 Equatorial Current, together with that carried by the Guiana Current and passing between the Lesser Antilles will be about the same as the total transport of the Gulf Stream through a cross-section off Cape Hatteras. From this it follows that the part of the Gulf Stream that passes through the Florida Strait makes up only about a third of the total transport. According to WiJST the Antilles Current carries 12 million mVsec and the Florida Current about 37 million m^/sec. From here the current enters regions with larger depth and there occurs a rapid increase in the water transport because the current absorbs water masses with a temperature of less than 8 ° C from the lower layers of the south-western Sargasso Sea. Further along to the north and north-east the Gulf Stream is subject to a velocity decrease and an increase in width, but the water transport remains nearly constant. However, it becomes more and more difficult to distinguish its limits from the surrounding sea. Iselin has attempted to divide up the Gulf Stream velocity profile at Chesapeake Bay (Fig. 284) into individual inflow components (Fig. 285). The area A contains water warmer than 20° C and the c 200 400 600 800 1000 1200 1400 1600 1800 ^ 0 Fig. 285. Subdivisions of the velocity profile across the Gulf Stream off Chesapeake Bay, 20-22 April 1932. The figures give the transport (in mill, m^ sec"^) for the different parts of the current (according to Iselin). velocity of this gives a transport of 10-6 million m^sec. The same layer in the Florida current according to the WUst profile corresponds to 13-1 million m^sec and in the Antilles Current to 4 million m^sec. The sum of these two is greater but no more so than could be due to differences in the homogenity of the material. The area B contains only water colder than 8° C, most of which was absorbed by the Gulf Stream in the section with a larger depth. According to the velocity profile tliis area corresponds to 12-7 million m^/sec and only a very small part of it can possibly be assumed to have its origin in the Florida Strait. Water is also drawn into the main current along both edges by friction and mixing. If these areas in the profile are limited by the isoline of 20 cm/sec, these areas C and D will correspond to a transport of 0-7 and 12T million m^sec respectively. These figures indicate that water is drawn into the current on the right-hand side much more strongly than along the more sharply defined left-hand boundary. The remaining area E corresponds to 46-1 million m^/sec. In the Wiist profile for the Florida Current and the Antilles Current this area corresponds to 614 The Tropospheric Circulation about 26-4 million m^sec. The transport in the current core has thus grown to twice its magnitude in a distance of about 600 nautical miles. This very large increase from 26 to 83 million m^sec, where the current passes into a region with larger depth, can be attributed to three principal sources. The smallest of these is due to the Antilles Current which brings the total transport up to 37-1 million m^sec leaving 45 million to be accounted for from the other sources. This is supplied, on the one hand, by water drawn in from the south-western part of the Sargasso Sea and on the other hand, by water fed by the counter current coming from the area of the Newfoundland Banks and mixed with the Gulf Stream by means of numerous vortices. In this way Iselin derived the schematic outline of the main sources and of the course of the Gulf Stream shown in Fig. 286. Each line represents a water transport of about 12 million m^sec. This may seem somewhat schematic, however, it gives an instructive idea about the origin and composition of the water masses transported by the Gulf Stream. 80° 70 60 50 4 0 20 10° W Fig. 286. Schematic representation of the main sources of the Gulf Stream waters (broken lines) and the pattern of the Gulf Stream system (continuous lines). In the western half of the ocean each stream line represents a water transport of approximately 12 x 10® m^ sec~^ (according to Iselin, 1935). The systematic survey of the Gulf Stream between Montauk Point and the Bermudas carried out by the "Atlantis" from June 1937 (Iselin, 1940) showed that the mass transport of the Gulf Stream varied between 93 and 76 million m^sec. There was a definite annual variation with two maxima in early summer and in winter and two minima in October-November and April-May. The differences in the sea-level across the current are closely related to these variations and can be deduced from them. This annual variation is probably due to variations in the intensity of the atmospheric The Tropospheric Circulation 615 circulation over the southern part of the North Atlantic. In winter the strong anti- cyclonic circulation over the ocean increases the inflow into the Gulf Stream and in the summer there are more frequent southerly winds and a greater part of the water masses of the North Equatorial Current is blown directly into the Gulf Stream without passing through the Carribean and the Florida Strait. Both of these effects intensify the Gulf Stream. It is not improbable that the aperiodic variations from year to year wil provide an extremely good indicator of the variations in the intensity of the atmos- pheric circulation over the Atlantic. The most recent investigations of the Gulf Stream have the main goal to obtain accurate detailed surveys of the current at short successive intervals, that is, to obtain quasi-synoptic surveys of an extended part of the current. Such methods of investiga- tion need in the first place the rapid gain of the structure of the water masses down to great depths, while the survey vessel is under way whereby the position of each station has to be fixed with accuracy. Both of these conditions can be satisfied by the more recent methods used on board of the oceanographic survey vessels. Quasi-synoptic surveys of this type have been made for the Gulf Stream down to 275 m depth between Cape Hatteras and the Newfoundland Banks but at the present time only few of them exist. They give a very clear picture of the complicated course of the current and show particularly the very considerable local variations in form of meandering wave patterns of large amplitude at both sides of the current. Occasionally a water mass in one of the amplified troughs and ridges is cut off from the main current to form finally a large vortex which will be cyclonic on the southern side and anticyclonic on the northern side. These vortices are different from the smaller size eddies in the shearing zones of a turbulent current that also occur in the Gulf Stream (Spilhaus, 1940). Furthermore, the synoptic surveys have shown that the current velocity in the core may be intensified up to about 4-5 knots over a relatively narrow band (about 10-15 miles wide) a little inside the left-hand boundary of the current; in the counter current the velocity reaches 3-4 knots. It is not surprising that the approximate and mean values obtained by the previous methods of investigation gave only low velocities. The first multiple ship survey of the Gulf Stream area between Cape Hatteras and the Newfoundland Banks was made during 6-23 June 1950. Six oceanographic survey vessels took part in this "Operation Cabot" and they obtained an almost synoptic survey of the Gulf Stream down to 275 m which gave a clear picture of the compli- cated nature of the current. Figure 287 presents the course of the current as character- ized by the mean temperature of the upper 200 m layer. According to this survey the Gulf Stream is a remarkably narrow band about 40-60 km wide and sharply separated at the edges from the surrounding water masses. The early view of Franklin of the Gulf Stream structure was confirmed, and certainly in the sector between Cape Hatteras and the Newfoundland Banks the Gulf Stream resembles a "river in the ocean" rather than a broad diffuse ocean current. The current does not, however, follow a straight line, but instead flows in long waves which are usually of small amplitude but take occasionally quite a large amplitude. Successive surveys have shown that these long lateral waves move slowly eastwards with increasing amplitude. Figure 288 shows the position of the Gulf Stream at the beginning (8 June) and the end of the operation 21 and 22 June). The current core therefore tends towards a meandering behaviour of a pronounced character. The amplitude of these meanders may increase so much 616 77?^ Tropospheric Circulation 66° 55° 5,„. ^^^ ^°°M§^^ "s= ^^^ ^glls V J- "'"'^-V '»^ ^ 5^ '■,,-73°' n>> ^: .... \, 72° 74° J '73° MJ J ::;,--S? fi 4 *^'" A M Ships t racks -~-,,,^-- ^ W ULK Current airecTions -♦■ 36" 38» 740 730 72° 71° 70° 69° 68° 67° 66° 65° Fig. 287. Mean temperature (°F) in the upper 200 m layer of the Gulf Stream, 8 June 1950. while moving eastwards that large sections of the current can be cut off. This process results in the ejection of a water mass from the current and the formation of large cyclonic vortices on the southern side of the main current. This cut-off process is similar to processes involved in polar jet phenomena in the upper atmosphere which are of major importance in the dynamics of these air currents. The process can be followed clearly in successive charts from 16 June to 21 June. On 17 June this process reaches its maximum stage (Fig. 289). The cyclonic vortex clearly stands out in the band of temperature concentration and in direct current recordings. It was at first a strong vortex but gradually weakened during the following days and finally vanished. A further characteristic phenomenon is the break-up of the Gulf Stream into several separate branches. Usually there are three, sometimes separated by counter currents. Figure 290 shows the current velocity and temperature distribution usually found at the sea surface. Consideration of these recent results shows that there are three principal questions on the internal dynamics of the Gulf Stream that require an answer. (1) Why is the current asymmetrically developed and why is the current core displaced to the left-hand side (looking downstream) ? ,72° 71° 70° 69° W 68° 67° 66° 65° Fig. 288. Position of the Gulf Stream. Mean temperature (°F) of the upper 200 m layer for 8 June (full lines) and for 21 and 22 June 1950 (dashed lines). The Tropospheric Circulation 617 Fig. 289. Mean temperature (°F) in the upper 200 m layer on 17 June 1950. Current direction from geomagnetic electrokinetograph (GEK) (according to Arx, 1950). (2) Why does the Gulf Stream keep such a concentrated narrow form over a long distance sometimes taking on a meandering character? Why does it break up into several smaller branches separated by motionless bands or weak counter currents? (3) Why is the total energy of the current concentrated in a relatively thin top layer and why does the current not penetrate down to the deeper layers when it flows out over regions with larger depth? Research on these questions is in progress but more fundamental results have been obtained only for some individual questions. It appears that these strong oceanic boundary currents are analogous in many respects to the "jet streams" of the strong westerlies in the upper atmosphere and are especially characteristic for the dynamics of free jets. (c) To the Dynamics of the Gulf Stream RossBY (1936, 1937, 1938) in a series of papers has advanced some new ideas on the theory of ocean currents which are of some interest. These arguments have been applied primarily to the Gulf Stream System between the Florida Strait and the area south of Newfoundland. But their use is not limited to these currents and in many respects they can also be appHed to all boundary currents flowing parallel to a coast 618 The Tropospheric Circulation a u The Tropospheric Circulation 619 (Kuroshio, Peru Current) and others. Rossby's theoretical investigations are put forward mainly along two lines. The first deals besides the vertical also with the lateral frictional effect which is of influence on the horizontal velocity profile in currents. The second deals with currents of constant momentum (impulse) transport and in particular applies the theory of free jets to ocean currents. Since the exchange co- efficients of lateral mixing are of considerably greater magnitude than those for vertical exchange (see Pt. I, p. 103 et seq.) Rossby considered it absolutly necessary to account for frictional forces due to lateral mixing and put strong emphasis on these forces. The usual equilibrium conditions in a geostrophic current for mass elements along a vertical line primarily determine — as always^the vertical velocity distribution. By introduction of the lateral shearing forces this condition will not be changed in any great extent, but the lateral shear imposes a definite transverse velocity profile to which little attention has been paid in the past. A linear current in the positive v-direction with a mean velocity v will be fixed by the geostrophic equilibrium between the pressure gradient —(1/ p)(dp/8x) and the Coriolis term —fv. As a result of the horizontal turbulence, however, the individual mass elements will have a movement at right angles to the mean direction of the current and the equations of motion (XIII. 1), will apply for its horizontal components u and v. If the deviations of w and v from the mean velocities « = 0 and I; are denoted by u' and v' then: dv' ^ , du' 1 dp T,=-^" and ^=A.' w,th -^£-/S = 0. (XIX.6) From (X.39) the lateral shearing stress is T = - pTlT. (XIX.7) Introducing the Prandtl mixing length / of lateral mixing (p. 388) allows (XIX.7) to be rewritten as = p/«' (/+ ^)- (XIX-8) For a uniform horizontal current the lateral shearing stress will not approximate to zero except when Under stationary conditions the lateral mixing imposes a definite horizontal velocity profile, and indeed there must be a velocity decrease towards the right-hand edge of the current (Northern Hemisphere). This is quite large and in middle latitudes (43°) amounts to 1 cm/sec in 100 m).* Since such large transverse variations in velocity are hard to observe it must be presumed that the right-hand edge of the current always tends to accelerate the left- hand side even when the right-hand side has a lower velocity. This effect ceases only when the condition (XIX. 9) is satisfied. * Against this conclusion the objection has been raised by Priebsch (1943), that besides the lateral turbulence across the gradient current, also that in the direction of the current should be taken into account. If this is done, it is found that the effect ot the earth's rotation mentioned above no longer exists. On the average the effects in the two directions balance exactly. 620 The Tropospheric Circulation The second part of Rossby's arguments concerns the problem of a straight accel- erated turbulent current. In such a current the horizontal pressure gradient will not be equal to that corresponding to the meari basic current and not balance completely the Coriolis force in stationary equilibrium. This gradient of the stationary current was termed the Coriolis pressure gradient by Rossby and for this the following relations apply -^^-p/i; and -^^ ^ + pfu. (XIX.IO) A numerical value can always be found for given u and r. The turbulent accelerated motion, however, will be subject to other equations: du dp St„„ and dv ^ ^P , ^'^vx where r^y and Ty^ are the x- and jF-components of the lateral shearing stress. Intro- ducing p = Pc -\- Pr then by means of (XIX.IO) and dv dp,. dxy^ ^dt^~d^'^ '8^' The movements which correspond to these equation occur under influence of "residual pressure gradients" Pr as though the earth was not rotating. The continuity equation du dv ai + a7 = ° *^"''" fixes the current field u, v, while (XIX.IO) gives the Coriolis pressure gradient and the corresponding mass field. Since /?(. is usually considerably greater thanpr it is clear that the general pressure distribution is of secondary importance in considering accelerated currents. The mass field which is determined by the mean steady current field gives no information on the cause of the currents. However, according to Rossby /j^ should, dynamically, be more important than p^. Against these considerations Defant (1937) and Ekman (1939) have raised doubts affecting more particularly the practical usefulness of the above equations. But nothing can be said generally against the main lines of the basic argument if one remains in agreement with actual conditions. For an application of the above equations to Gulf Stream problems Rossby took into account the phenomena that occur when the flow of a medium takes the form of a jet. The theory of free jets (Prandtl, 1926) has been further developed by Tollmien, 1926; FoRTHMAN, 1934; Ruden, 1933. For a steady state (du/dt = 0) in a laterally restricted current the first of the equations (XIX. 12) together with the continuity equa- tion (XIX. 13) gives pu^ dy = constant, (XIX. 14) The Tropospheric Circulation 621 that is, in a current of this type the momentum {impulse) transport through a current cross-section is constant. Neglecting dp^jcx (which is permissible) and introducing the shearing given by '»-'fy according to equation (XII. 15), then for a mixing length / = ex (proportional to the distance travelled) a complete solution can be found that fixes the horizontal current profile in the free jet. The very good agreement between theory and experimental results for the current profile in a free jet, is a consequence of the assumption made for the mixing length which is completely valid only for limited dimensions. Whether it is also applicable for the very large dimensions of ocean currents is questionable. One consequence of the assumption is also that in a free jet with constant momen- tum transport the mass transport increases downstream, and is in fact proportional to the square root of the distance travelled. Due to the incorporation of surrounding water the current cross-section will increase downstream while the mean velocity will decrease. Since the energy remains the same, the mass transport will increase. Condi- tions are somewhat different if the inflow through the initial cross-section does not start from a point source but has a finite width. The velocity profile in Fig. 291 is 15 \-0-\ 0-5 1-0- 1-5- FiG. 291. Velocity distributions in a jet (Freistrahl, according to Ruden). D, nozzle diameter, all lengths are given as multiples of D. based on experimental values for the velocity at different distances from the outlet of a nozzle through which there is a constant inflow. In a free jet there is a core in which the initial velocity and the other properties of the medium remain unchanged for a relatively long distance from the nozzle. The formation of a core region and a surroun- ding one of mixing are characteristic of the phenomena occurring in the ocean under similar conditions. These results apply in the absence of rotation. According to Rossby, the principal effect of earth rotation is the formation of a different mass distribution (according to (XIX. 10)) corresponding to the Coriolis pressure force; the velocity profile, however, will not be disturbed further by it. The stationary properties of the current, that is that the stream lines, isobars and contours of the physical sea level coincide, remain more or less unchanged. The deviations from a geostrophic current occurring in the interior of the free jet that are produced by the shearing stress, will be accentuated by the deviations due to inertia. There will thus be an overall dynamic equilibrium. According 622 The Tropospheric Circulation to this concept it is the residual pressure field even though it is weak that provides the driving forces. This is the basic idea of the Rossby theory. It is undoubtedly attractive but whether it actually corresponds to reality is impossible to say. In any case it deserves considerable attention. The further phenomena that occur when the medium in which the free jet is formed, is stratified, can be fairly readily dealt with. If there are two layers in the medium the velocity of the upper layer will affect the sea surface slope and also the position of the internal boundary surface between the two water masses. The sea surface slope and the internal boundary slope are given by equations (XIV. 6 and 7) (p. 455). If the lower layer is assumed to be motionless then the velocity of the free jet gives the mass distri- bution in a transverse section. This is shown schematically in Fig. 292. In the current the Motionless 1 Jet current 500 E a 1000 Q 1500 Motionless ibb'^>.^ Fig. 292. Cross-section through a jet (Freistrahl) current in a two-layered ocean. boundary surface will slope downwards, in the Northern Hemisphere from left to right, and the thickness of the free jet will therefore vary across the current. The total mass transport through a cross section will be M = lpu{D, + Ci + y dy. (XIX. 15) where Dq is the mean thickness of the top layer and ^^ and i^ are the deviations of the physical sea level and the boundary surface from their positions when the system is at rest. Evaluation of this integral gives the result that the difference. Drigiu — ^left between the two sides of the current must increase downstream as long as the mass transport increases. This has several effects on the course of the current. The inflow of water from the surroundings into the free jet will be asymmetric because of the asymmetry of the system. On the left there will be only a shallow water layer available, but on the right the water can be drawn in from greater depths. Under steady conditions the transverse velocity must therefore be greater on the left-hand side than on the right. The surrounding water masses can be assumed to be stationary, but this state can hardly be expected to persist under the given conditions. At some distance from the current boundary the thickness of the layer D in the motionless water will be somewhat greater than Dieit and Dri^ht at the left- and right-hand edges. On the way from motion- less water towards the boundary and into the interior of the free jet the water columns drawn into the current will undergo deformations, which will be associated with hydrodynamic vortex formation at the current boundary. The theoretical form for a cross-section through a free jet of this type is that shown in Fig. 293. This requires The Tropospheric Circulation 623 a counter current at the left-hand edge of the free jet and a current in the direction of the main current at the right-hand edge. Due to the increased sea level difference between the right- and left-hand sides, the counter current on the left-hand side will increase in strength downstream, while on the contrary the other current will become weaker on the right-hand side until it finally vanishes. The effect of the free jet and the counter current must thus increase steadily downstream and therefore tend towards impossible unstable conditions. The effect of the vortex formation will give rise to a water move- ment through the main body of the current from right to left, and since the left-hand 500 1000 1500 Deep woter (motionless) Fig. 293. Cross-section through a jet (Freistrahl) current in a two-layered ocean with a full development of a counter current and compensation current in the adjacent water masses (according to Rossby). edge and the counter current are shallow there must also be a transverse current in the lower part of the top layer in the opposite direction in order to compensate the upper transport. This gives a cross-circulation as was assumed by Dietrich. In addition to his earlier work, the processes occurring at the edges of a jet-form current penetra- ting a motionless water body have been discussed in two later papers by Rossby (1937, 1938). Thereby, he assumed that the initiation of the current from a state of rest was due to a wind field whose action was restricted to a band-like oceanic region. Particular attention was paid on the one hand to processes at the edges of the current, on the other hand to oscillatory processes which occur while the current tends towards a steady state. In such cases counter currents are formed on both sides of the basic current; in homogeneous water they are broad and slow, but in a two-layered sea narrow and intense. The zones between the basic current and the counter current are dynamically unstable and show a tendency to break up into large horizontal vortices. The depth to which a surface disturbance may penetrate into the lower layer down to the sea bottom, and the time required for the restoration of stationary conditions, are of particular interest and are especially important in dynamic oceanography (see Chap. XXI. 4). Without question the theory has applications to the Gulf Stream between the Florida Strait and the Newfoundland Banks, and several theoretical consequences are un- doubtedly realized in the actual behaviour of the Gulf Stream. The criticism on this theory expressed by Ekman is concerned not so much with the theoretical fundaments, but more with the question of the extent to which the Gulf Stream actually keeps the character of a free jet and contains the energy (momentum of motion) required by the 624 The Tropospheric Circulation theory. By means of approximate calculations he was able to demonstrate that the current leaving the Florida Strait will probably have a kinetic energy, so that already half of this energy would be able to carry the water against frictional resistances of various types exactly as far as a wind of 3 to 4 Beaufort could do blowing from the Florida Strait until Cape Hatteras in the current direction. Over this section of the current the theory should be able to make the most important characteristics of the Gulf Stream understandable. However, for the section of the current from 60° to 20° W conditions appear to be rather different, and in this section the initial velocity of the water seems to be only of minor significance. Ekman therefore came to the conclusion that for most of the ocean currents the theory is of limited usefulness only and can be applied solely to very fast currents (such as the Florida Current and its immediate continuation, see also, Thorade, 1938). The Rossby theory, due to its consequent and careful style, had a very stimulating effect and has lead to a better understanding of a number of phenomena displayed by the Gulf Stream between Cape Hatteras and the Newfoundland Banks. A satisfactory theory of the Gulf Stream must take into account a further important fact that has already been referred to by Dietrich (1937a). Determination of the mean sea level along the North American coast from Florida in the south to Nova Scotia in the north by means of precise trigonometric measurements has shown that the sea level rises along this total route to the north with a mean slope of 13 cm in 1000 km. The strongest slope occurs just north of Cape Hatteras (see Table 152 according to An VERS, 1927 and Rappleye, 1932). The Gulf Stream thus shows an w/7H'(7r(/ motion along this section like the Caribbean Current where according to Parr and Sverdrup (p. 607) there is a slope of about the same magnitude. However, as it was shown by Dietrich, that the Gulf Stream in contrast to the Caribbean Current does not show this slope when the physical sea level Table 152. Average Mean Water Along the North American East Coast. (Zero point relative to Florida-Georgia) Location Mean water Distance along the coast (km) Anvers Rappleye Average per 1000 km St Augustine, Fla. Femandina, Fla. Brunswick, Ga. 1 ] 0 0 0 0 6 Norfolk, Va. 4 7 6 1000 Cape May, N.J. Atlantic City, N.J. Fort Hamilton, N.J. 16 24 20 1400 35 13 Boston, Mass. Portland, Me. } 25 30 28 2000 12 Halifax, Nova Scotia 1 35 35 2600 topography is calculated from the mass distribution along the continental slope. Dietrich took the oxygen minimum layer as reference-level but recalculation for a The Tropospheric Circulation 625 deeper reference level changes the results very little. There is thus a contradiction between the "geodetic" and the "oceanographic" levelling which requires explanation. A plausible explanation was indicated by Sverdrup (and co-workers 1946, p. 578) based on the following assumptions. (1) That the geodetically determined gradient of the sea level is actually present in the coastal waters just off the coast and that corresponding to this there is a coastal current flowing southwards. (2) That in the neighbouring waters the physical sea level slopes down seawards until the left-hand edge of the Gulf Stream which causes a current to flow southward due to the piling up of water. This gradient current would be one part of the large elongated vortex on the left-hand side of the Gulf Stream while the second part flows along the left-hand edge of the main current and in the same direction. (3) Corresponding to this current and the adjoining Gulf Stream, the physical sea level rises steeply seaward from the coast (p. 607). The depression showing the deepest water level thus would follow the continental slope rather closely so that the south- ward flowing branch of the vortex lies over the shelf. The topography in a transverse section across the Gulf Stream thus has some similarity with that shown in Fig. 203 1231 1230 1229 1228 1227 1226 500 1000 1500 2000 „Atlontis" _ April 1932 Fig. 294. Density distribution (at) and position of the lower limit d of the current system in a cross-section through the Gulf Stream. "Atlantis", April 1932 (Chesapeake Bay, Bermuda). (p. 460) south of the Newfoundland Banks where on the other side of the depression in sea level the Labrador Current flows eastward. According to the Rossby theory the elongated vortex between the Gulf Stream and the continental slope is a dynamic necessity. The Gulf Stream now would flow downhill in accordance with its mass structure and the surface slope would be directed southwards only at the coast. This piling up of water over the continental shelf was regarded by Sverdrup as due to the prevailing wind over the North Atlantic. The south-west wind over the northern part of this ocean maintains a high water level along its northern borders and maintains 2S 626 The Tropospheric Circulation in this way a decline of the physical sea level along the eastern and western sides. This would be the geodetically determined rise between Florida and Nova Scotia. All cross-sections through the Gulf Stream show a strong stratification in the upper layers but beneath this where the current is weak it is less pronounced. It is to be expected that there will be a layer of no motion just beneath this layer. Figure 294 given by Neumann (1956) shows the position of the zero level d in an "Atlantic" cross- section through the Gulf Stream. The latter one indicates clearly the form given in Fig. 292 with shallow depth along the left-hand edge of the current, a strong down- ward slope below the maximum transverse density change and uniform larger values at the right-hand edge. Tliis distribution is characteristic of all sections through free jet currents in the ocean. Neumann has also shown that over the whole of the moving layer from the surface down to the depth of no motion d there are only slight changes in the mean density distribution. There exists thus in a first approximation no trans- verse density gradient. This means that the entire current system is an equivalent to that of a two-layered model in which there are two water bodies, one on top of the other with an internal boundary surface between. Thus as a first approximation the Gulf Stream can be regarded as an equivalent-barotropic system in which the boundary layer slopes downward from the left-hand to the right-hand edge. Figure 295 shows the 1231 1230 1229 1228 1227 1226 500 1000 1500 2000 50km Fig. 295. Velocity distribution in the "Atlantis", section Chesapeake Bay-Bermuda, April 1932 {d, lower limit of the current system). velocity distribution calculated from the mass field (Fig. 295) for a cross-section at the lower limit of the current system d. For the vertical shear under equivalent-barotropic conditions as a first approximation one obtains (XIX. 16) dv g dp dz f p 8x' where p is the mean density of the current layer. If as on p. 608 t, denotes the surface The Tropospheric Circulation eii of the physical sea level and —d and p_a are the depth and the corresponding density of the layer of no motion, then from (XIX. 16) when m_• f=fo + Ky-yo) which gives finally after some calculation and which is valid for all values of y and ip. The equation (XIX.23) and (XIX.24) then give the final equation 8i/j f I /dG dh g*h g* \ bijj Its solution, subject to the boundary conditions h = h{y) and ') is (XIX.29) (XIX.30) The velocity v is obtained as a function of ijj and y from the equation (XIX.24) and the values of X corresponding to 0 and y are given by the equation dx = hv (XIX.31) The Tropospheric Circulation 629 which, only requires a numerical quadrature; the boundary condition here is (/» = 0 at ;c = 0. The application of this theory put forward by Chamey starts with the deter- mination of the two constants in equation (XIX. 26). Taking 0 = 0 at the coast, then i/< is the volume transport of the current. The zero point for y is midway between the Florida Strait and Cape Hatteras ( y = y^), that is, 700 km from both sides. The calculated geostrophic transport in the Florida Strait is approximately 30 X 10^ m^ sec~^ and the increase from here to Cape Hatteras is approximately 50 X 10« m^ sec-i. Hence ^o = 80 x 10^ m^ sec~^ and y has the value 2-55 x 10"^ msec~^ Further- more, in (XIX.27), /o = 0-84 x 10"^ sec-^ and /3 = 1-8 X lO-^^ m-^ sec-^. If we postulate that h = 0 when x — 0, y = yo and tp = 0, then substituting these values in equation (XIX. 30) gives ^„ = /^0„y' =820m (XIX.32) which compares well with the observed mean value of 900 m given by Iselin (1936). The results of the integrations are shown in Fig. 296. This gives in perspective the calculated position of the boundary surface h by contours of h (full lines) at 100 m intervals and on this surface the stream lines (broken lines) of the volume transport for each 10 million m^ sec~^. On top are given calculated velocity profiles for several cross-sections through the Gulf Stream. Comparison of the position of the internal boundary surface with the observed mean depth of the 10° C isotherm, which gives approximately the lower limit of the Gulf Stream, shows that they are in excellent agreement. The characteristic way in which the current swings away from the coast in the northern part of the region considered can also be seen. This takes place away from any projection of the coast line and is found both in the Gulf Stream and in the Kuroshio. The current profile shows towards higher latitudes an increasing concentra- tion of the current energy towards the left-hand edge (westward intensification). The velocities along the left-hand edge are probably too high in the north but would be reasonable since boundary friction was neglected. The theory takes a simple form if a quasi-geostrophic approximation is made, that is, when both M and V are assumed to be geostrophic and when h varies linearly with y, then h = ho + H-iy - yo)- With the condition A = //, at x = 0 (at the coast) the solution is h = 7i( >•) -(h- h,)e-x'x. (XIX.33) The width of the current is given approximately by Since at the right-hand edge the lateral velocity at the outer boundary is | i7 1 = (g*lf)H; one obtains A = V(mIP). (XIX.34) It is apparent that the Gulf Stream is a phenomenon that depends essentially on the variation of the Coriolis parameter with latitude. Observed values of u and j3 give a value for A of about 50 km. V decreases laterally to a quarter at a distance of about 70 km which is in accordance with the down- slope to the right shown in Fig. 294. The geostrophic approximation predicts roughly the character of the current but does not predict all the details. 630 The Tropospheric Circulation The Tropospheric Circulation 631 The theory of the Gulf Stream and similar boundary currents requires further development. The double-layered model must be replaced by one with continuously stratified water and the effects of friction in both vertical and horizontal directions must be taken into account. Lateral friction against the coast should give a reduction in the velocity of the current at the left-hand side as is shown by observations. The boundary current theory attributes the ocean boundary currents of the general oceanic circulation, in so far as they have the character of a free jet, to the effects of pressure and inertia and to the variation of the Coriolis parameter with latitude. It has been pointed out above (p. 580) that the Sverdrup solution starting from an eastern continental boundary and working westwards is unable to satisfy the boundary conditions at the west coast of the ocean. Only by including the effects of a strong lateral friction (mixing) Stommel and Munk have been able to satisfy the boundary conditions at a western boundary and to give a general theory of a wind-driven ocean circulation. However, along the eastern side of a continent (western side of oceans) the currents apparently do not correspond to this theory. They are narrower and more intense than would be expected from the general theory. The Charney theory gives the explanation for this and yields in this way a western continuation to the Sverdrup solution, without the addition of strong frictional effects but taking into account the effects of inertial terms and the variation of the Coriolis parameter with latitude. The density stratification of the water and the lateral inflow into a meridionally directed jet current have been found to be of particular importance in the formation of these boundary currents. These provide the connection with the western transport of the zonal wind currents of lower latitudes. {d) Further Aspects of the Dynamics of the Gulf Stream Associated with the questions raised on p. 617 another one stands out concerning the total current energy in a relatively thin top layer. This energy concentration in a narrow current band occurring in the very upper layers persists for more than 2000 km, from Cape Hatteras to the region east of the Newfoundland Banks while beneath this top layer the velocities remain small. This remarkable phenomenon is probably explicable by an association between momentum losses in the lower portion of the current and the upper energy concentration. It should be stressed that the zonal width and the high speed of the upper Gulf Stream layers rather definitely exclude an inter- pretation of the current in this part of the Atlantic as the result of momentum added locally by the prevailing winds. Rossby (1951) has attempted to find out what kind of verticaly velocity profile would be formed in an immiscible stratified current subject to momentum losses through contact with the underlying surface or at lateral boun- daries. It would be of particular value to know the nature of the special velocity profile corresponding to a minimum value of the momentum transfer in unit time across a vertical plane normal to the current axis. It is reasonable to assume that this profile represents a limiting state which would be gradually approached by any stratified current subject to momentum losses but unable to escape to the sides. In a straight aparallel current of this type in which the water is considered to be incompressible and the density varies with depth, the momentum transfer across a vertical strip normal to the current axis is given by 632 The Tropospheric Circulation MT = r (pm2 + p) dz, (XIX.35) where z is counted upward from the bottom and where p is the water hydrostatic pressure. Assuming that the mass transport in every infinitesimal isopycnic layer remains constant during the variation process, then puz da = pUq Zq da = v{a) da, (XIX. 36) where the subscript 0 indicates initial conditions. Here a is a new independent variable which determines the vertical density distribution and i = dzjda. With these quantities (XIX.35) gives MT = f /-^ + pz\ da. (XIX. 37) With the fundamental hydrostatic equation one obtains finally f = - gP^ (XIX.38) aa MT^ - \ i^ +-\ da. (XIX.39) The variation problem is the determination of the particular function p of a which reduces MT to a minimum value for the given distribution of v with a. The variation of p vanishes at the sea surface and it can be assumed that it also vanishes at great depths. Under these circumstances the minimum value of MT is then given by 8{MT) =[ \(^-^-^~-]8p-^ 8p] da = 0. (XIX.40) This is true for arbitrary values of 8p provided the function p satisfies Euler's equation gp da P^ gp. 0 (XIX.41) which on substitution reduces to du^ = p da, (XIX.42) where a is the specific volume. To determine the final velocity distribution from the initial mass transport distri- bution it is necessary to combine (XIX.42) with (XIX. 36) or pu dz = v{a) da. (XIX.43) Rossby has discussed several models with special density distributions according to this principle; only those more or less directly concerned with the Gulf Stream will be considered here. For a uniformly stratified current with speed Uq and depth D^ that is flowing on top of a homogeneous bottom layer of density p^, in which the volume transport is zero and that is allowed to readjust itself to a minimum momentum transfer current The Tropospheric Circulation 633 profile, a determination of the density and velocity distribution in the final state can be made by taking P = p,(l + Iko) and p, = p,{\ + 2k), (XIX.44) where Ps is the surface density and p,, is the deep water density. For a uniform initial stratification (subscript 0) it follows that With the continuity requirement, the basic equation gives as a good approximation Further when o^^j^ — 1 one obtains D I 3«2 \i/3 „ 3 m\ It follows that the current must become shallower and the bottom layer will increase in thickness whenever Wq falls below the critical value, Mo.crit defined by Wo < "o.orit - J^^\ (XIX.48) The end of the adjustment process can be illustrated by means of a numerical example. Initially the upper moving layer extends down to 600 m (Z)o = 600) and Wq = 0-75 m sec"^ In the Gulf Stream region an adequate value of the total range in CT< is 4-5 so that to a close approximation Ik = 4-5. Thus D results to 300 m and for u^ one obtains 2-25 m sec~^ Figure 297 shows a graphical representation of this case. It is clear that the dimensionless quantity F defined by P - IT \i X n (XIX.49) {(pb - Ps)/pb}gDo has the form of a Froude number in which the gravitational acceleration is reduced in proportion to the total percentage density range of the fluid. It can be seen that this new number determines the nature of the baroclinic movements of a current subject to momentum losses due to frictional influence. If the "internal Froude number" is less than a certain critical value (in the above case ^) the current will be concentrated in the lighter top layers. Apparently, oceanic currents usually have subcritical values of F. They then have a tendency to develop a strong shearing motion with increasing velocity and increasing stability near the sea surface and decreasing velocity and stability lower down. In the Straits of Florida and in the Gulf Stream region as far as Cape Hatteras the range in CT( is smaller than it is further downstream and there is no homogeneous deep water to facilitate a separation of the current from the bottom. After the current leaves Cape Hatteras, however, the momentum it gains due to direct action of the wind on the narrow strip exposed at the atmosphere is presumably incapable of balancing the losses 634 The Tropospheric Circulation which result from interaction with the deeper water masses or are due to lateral mixing. The current thus tends to become more and more superficial ; this process maintains the high surface velocities. The cause for the horizontal meander-like oscillations of the narrow current band of the Gulf Stream after leaving the continental shelf is not entirely clear. These meanders occasionally become unstable and then complete cut-off vortices are formed ; Om lOOm 200 m- 300 m 400m 500m 0 U 600m mps 1-0 o- Fig. 297. Transformation of a uniform current with a constant vertical density gradient into a flow characterized by a minimum value of the momentum transfer. The initial uniform velocity distribution is given by the heavy broken line, the final velocity distribution by the heavy full line. The density distributions before and after adjustment are given by lines marked by a (initial) and a (fmai)- Note that the depth of the final current is one half of the initial depth. The total percentage density range has the value 00045. this has been discussed already on p. 616. Recent investigations on the vertical strati- fication in the Gulf Stream (Arx, Bumpus and Richardson, 1954) using stations with little distance from each other have shown that the narrow current band has a filamen- tary structure. It is composed of thin layers of high velocity alternating with layers of lower velocity. This extraordinary stratification is possibly connected with gliding processes imposed by external circumstances on the individual water layers of the Gulf Stream and can be assumed to be a consequence of turbulence processes, which are imposed from outside. The meandering of the narrow current band of the Gulf Stream appears to be a common phenomenon. These meanders show v/avelengths of about 200 km and their speed of propagation is about 1 1 nautical miles a day, which is about a tenth of the speed of the current itself. Stommel (1953) has given a simple meander theory for a The Tropospheric Circulation 635 wide current in a stratified ocean in which he showed that the stability of the waves depends on whether C/2 > g:^D. (XIX. 50) P Here f/is the velocity of the basic current, D is the thickness of the upper moving layer and J p is the density difference between the lower, homogeneous and motionless layer and the homogeneous upper layer. The upper inequality sign results only in stable waves and the lower one only in unstable waves. For W = g(Aplp)D there is a single "just unstable" wave, the wave-number of which is given by k =f/{U\/2). This wave always remains stationary. Choosing a surface layer 200 m thick moving at 200 cm sec^^ and having a density ratio zJp//3 = 2 x 10~^ the wavelength of the "-ust unstable" perturbation is 180 km. All other wavelengths are stable and do not grow. It is remarkable that this wave- length corresponds closely to that observed. Some objections can be raised against the application of the Stommel perturbation theory to the meanders actually observed in the Gulf Stream and it would be desirable to test the Stommel model somewhat more closely and to specialize some of his assumptions. In order to handle the problem of the meandering behaviour of the Gulf Stream in a more comprehensive way, the problem may be looked upon as intimately connected with the way in which the stability of a narrow geostropliic current is changed when this flow is subjected to external perturbations. In a deeply penetrating way the latter question has been dealt with by van Mieghem (1951) for atmospheric currents. He assumed a straight geostrophic flow in hydrodynamic equilibrium in any direction on the rotating earth allowing for horizontal (transversal) and vertical wind shear. On this current he imposed a disturbance acting in lateral (transverse) as well as vertical direction and attempted to find the conditions under which the disturbance decreased in time (stable state) or increased in time (unstable state). In the stable case the chance disturbances vanish with time; in the unstable case they grow into meanders and may even degenerate into independent vortices. If the positive x-axis is chosen in eastward direction, the >'-axis normal to it (to the north) and the r-axis positive towards the zenith and if the geostrophic current flows along the j'-axis (w^ = 0, iiy ^ u(x,z), u^ = 0), then the equilibrium values of the pressure P = P(x,z) and the specific volume a = a(x,z) are only functions of jc and z and the equation of motion as well as the quasihydrostatic equation leads to the Margules equilibrium condition of the geostrophic current : cPca_cPca^^^ (XIX.51) ox cz dz ex where oj^ and a»y are the horizontal and vertical components of earth rotation vector (coj. = ojy = oj cos ) and N is the number of solenoids in the cross- section {x,z) (baroclinicity). For a small fluid particle in the interior of the water mass which is at the co-ordinate origin at time t^ and at that instant is subject to a transverse impulse, its velocity components relative to the earth at the same instant will then be V, = u-\- r„ Vy = Vy, V, = V,. '. (XIX.52) CU CU (^z CZ _u 2<^x ex 636 The Tropospheric Circulation Assuming that the specific volume a^ of the disturbed particles is conserved, then the equations of motion for the displaced particles will take the form : dv where dt + 2a)yV^ = ijjx dv^ dt — 2cOyl\ = ^y 4'x = - - a^^x - a^^z. •A. = - - a,^x - a-^^z. } (XIX.53) (XIX.54) X and z are the displacements of the small particles in the x- and 2-directions and may be positive or negative. The coefficients a^x, ^xz and a^^ are given by \ dxf dx 8x du --/(/♦-^l-^. = +/* / (^•-9 dP8a dx dz dP8a Tzd^ (XIX.55) J with axz= ^zx and/* = 2a; cos ^. It can then be shown that at a point in a geostrophic current at which there acts a transverse disturbance, conditions will be stable, neutral or unstable according to whether the quadratic form (Kleinschmidt) : x2 + 2a, + a. ^0. (XIX. 56) The sign of Q is determined firstly by that of the discriminant a ^ a^, — arr. a 'XX "xz (XIX.57) and secondly by the sign of one of the coefficients of the quadratic terms in Q (for instance a^^). The condition (XIX. 56) thus becomes a-0 or idu _ daldx ^\ , ^>Q \dx daldz' 8z) ■' < ' (XIX.58) The last equation can be re-written with the help of (XIX.55) and by neglecting terms of lower order one obtains 4:^1 /(/+ir"- (Jadz (XIX.59) The expression - ^ p dz is the static stability (z-positive upwards; p. 196) and f{f-\- dujdx) is the expression for the inertial stability. The equation (XIX.59) gives a hydrodynamic measure in as far as the geostrophic equilibrium in the current under consideration is hydrodynamically stable or unstable when subject to external impulses acting normal to the direction of the flow (in transverse or vertical direction). The Tropospheric Circulation 637 The application of these equiHbrium conditions to the Gulf Stream requires an esti- mate of the order of magnitude of the individual terms. These can be obtained approximately from the "Atlantis" sections for concentrated boundaries of the current and one obtains the following values given in the [cm g sec] -system: 10- du ex cu cz 10 da dx ' 8a Fz gda a dz 10- 10-^ to 10-8 10-= /• f (f- a . 10-* . 10-1* . 10-8 Introducing these values in equation (XIX. 59) shows that in the Gulf Stream, in spite of always secured static stability and in spite of the almost always secured inertia instability, hydrodynamic instability may still occur provided the vertical shear in the flow reaches excessive values. This can be illustrated by an example taken from the "Atlantis" section shown in Fig. 294. (Chesapeake Bay-Bermuda, April 1932). Along the left-hand side of the Gulf Stream in the region of largest vertical and horizontal shear (depth 220 m) one obtains du cu — ==0-47 X 10-2sec-i; ^ cz ox 0-33 X lO-^sec-i and cz 0.33 X 10-«. With these values and with/ = 0-85 x 10-* [fj^^ =0•16xl0-l^ while ' CxJ a CZ The current and density stratification is thus, of course, hydrodynamically stable as could be expected since at this part of the Gulf Stream the current shows no tendency to meander. Hydrodynamic instability would only occur if the vertical shear in the flow would reach values four times larger. Further to the north, in the section between Cape Hatteras and the Newfoundland Banks, conditions might be diff'erent and may readily be so that the current system becomes hydrodynamically unstable; these small horizontal wave formations will soon grow into large meanders and finally lead to the formation of vortices. Strong vertical current shear and low static stability are required for this. It can be understood that a strong acceleration of the flow in the top layers of the Gulf Stream caused by the direct action of a strong westerly wind acting on the sea surface will provide the necessary vertical current shear to give rise to hydro- dynamic instability in the current system and to lead to the formation of meanders. Haurwitz and Panofsky (1950) in a study of the stability and meandering be- haviour of the Gulf Stream have attempted to show that especially favourable con- ditions for the development of unstable waves occur when the Gulf Stream is not too 638 The Tropospheric Circulation close to the continental shelf. The tendency towards a formation of meanders appears only after the Gulf Stream leaves the continental shelf, but probably there are other factors that will decide about the development of meandering motion than the distance from the continental shelf. As yet no fully satisfactory explanation has been given for the observed split of the Gulf Stream into a number of branches. Hansen (1952) has demonstrated that under certain conditions a northwards flowing current while turning towards the east can break up into several branches; but his solution is of more formal character and no actual reasons can be offered for this phenomenon. {e) The Kuroshio The three-dimensional structure and the dynamics of this current have been investigated by Uda (1930), Sigematsu (1933) and Kisindo (1934) on the basis of series observations made by the hydrological department of the Japanese Marine and the Imperial Fisheries Experimental Station in Tokyo (since 1925) and also by the oceanographic survey vessel "Mansyu". A number of transverse profiles have been prepared and critically worked with by Wust (1936a) in a comparative study of the Kuroshio and the Gulf Stream and further valuable work has been performed by KOENUMA (1939). Wiist has dealt with a cross-section at right angles to the chain of islands, the Ryu-kyu, from 27° to 29° N., just before the Tsusima current splits into branches and with another cross-section farther north (little to the south of Shiono at Misaki, the south cape of the projecting Kii peninsular at about 30° to 34° N.). See Fig. 261 for the position of these sections. The inclination of the isolines of the oceanogi-aphic factors forced by the water movement appears clearly in all cross-sections through this strong current. A com- parison with conditions in the Gulf Stream shows that there is an almost identical thermal structure but considerable differences occur in the salinity distribution; the Kuroshio has a low salinity 34-32 to 34-98%o and a very weak vertical salinity stratifi- cation, while the Gulf Stream possesses considerably higher salinity (34-97-36-65%o) and a pronounced stratification. The Kuroshio region also shows an intermediate salinity minimum at 500-800 m depth resulting from an intrusion of the weakly saline sub- Arctic intermediate water flowing in from the north (p. 172). Figures 298 and 299 show the temperature and salinity distributions in the Ryu-kyu section (Feb. 1927) and in the Shiono-Misaki section (Jan. 1927). Disregarding the top layers, the sections for the summer months show entirely similar conditions. These sections have also certain similarities with those through the Gulf Stream (see Figs. 282, 283). The Ryu-kyu section corresponds closely to that through the Florida Strait, the Shiono-Misaka section to the Chesapeake Bay transverse section. It is also apparent from these sections that the Kuroshio is throughout the entire vertical extent a weakly saline current as compared with the Florida Current; the highly saline core layer can again be explained as a distant effect of the tropospheric circulation of the subtropics and tropics. The velocity distribution calculated from the mass field of the Ryu-kyu winter section shows maximum intensities of 61 cm/sec below the sea surface at 150 m depth. In summer highest values of about 90 cm/sec occur at the sea surface. The weakening and downward displacement of the current maximum in winter is in The Tropospheric Circulation 639 439 438- 4-_ - ^>22^2j^ f" 11 /\ m — :ir~~" Fig. 298. Cross-sections of temperature through the Kurochio (R, Riu-Kiu section at 28^ to 29° N., "Mansyu" stations; S, Shiono-Misaki section at 34" to 30" N., "Mansyu" stations, January 1927) (according to Wiist). Station KDOO Fig. 299. Cross sections of salinity through the Kuroshio, section S (Shiono-Misaki) (see remarks below the Fig. 298). 640 The Tropospheric Circulation correspondence to the piling-up effect ("Aufstau-Effekt") of the winterly north-west monsoon. These values are in good agreement with direct current measurements at a station in the current core. The total amount of water transported through this section amounts to 21 million m^/sec in winter and about 23 million m^/sec in summer. The Kuroshio and the Florida Current thus carry about the same amount of water. The Shiono-Misaki section has been evaluated both by Wiist and by Koenuma. WUst thereby placed the reference-level at the upper limit of the weakly saline inter- mediate water, at about the depth of the 10° isotherm; Koenuma on the other hand, bases his calculations on velocities of 16 cm/sec of the intermediate water observed in coastal areas moving there to the north-east and for larger distances from the coast he assumed that the intermediate water was transported to the south-west at 5 cm/sec. The two vertical velocity profiles independently found by both methods thus do not agree. The velocity distribution obtained by Koenuma is in good agreement with actual current measurements while the values obtained by WUst are somewhat too low. The Kuroshio here keeps closely to the coast with velocities of 160-180 cm/sec and extends seawards for 140 km. As is true for the Gulf Stream, there is a counter current to observe towards the south-west on the right-hand side with maximum velocities of up to 20 cm/sec. Here also a downstream increase in the water transport can be noticed, but the counter current on its right-hand side with its higher velocities compensates the outflow towards the east to a considerable extent. There is so far no proof whether there are any seasonal changes in the amount of water transported (see also, in this connection the works of Ichiva, 1953-54). The Kuroshio does not show such pronounced characteristic properties as to be termed without more ado as a free jet current in the sense of the Rossby theory. It lacks especially the jet-like outflow from a narrow sea strait; it is formed instead by the gradual deflection of the stream lines out from the North Equatorial Current and only at a later stage forces its way into the relatively narrow channel-like region between the shelf and the submarine ridge of the Ryu-kyu Islands. By the further weakening due to the separation of the Tsusima branch its quasi-jet character is entirely lost. The continuation of the Kuroshio out into the Pacific from about 35° N. onwards (see p. 570), according to vertical sections (Uda, 1935), possesses the character of a relatively narrow current which, however, like the Gulf Stream in the central parts of the Atlantic, has a tendency to break up into single-current branches intermittently separated by vortices and counter currents. The one branch turning north from the Kuroshio meets the cold water masses of the Oyashio, and there in dynamic respect similar conditions occur as are present when the Gulf Stream meets the Labrador Current off the Newfoundland Banks. Table 153 finally presents a survey about mean water, heat and salt transports according to Wiist for the Gulf Stream and the Kuroshio. About 22 times as much water passes through the Kuroshio section and even about 33 times through the Gulf Stream as is carried by the water transports of all the rivers and glaciers on the earth (run-off from the continents on the average about 1-2 million m^sec). Even more spectacular are the enormous amounts of salt carried through these cross- sections, corresponding roughly to loads of 79,000 and 121,000 rail-road goods wagons respectively, each of which takes 10 tons. The question thus arises, why the climatic effect of the Kuroshio on the eastern Pacific and on the neighbouring continent The Tropospheric Circulation 641 Table 153. Mean water, heat and salt transports of the Gulf Stream and of the Kuroshio between 27° N and 37° N. Water amount 10* m^/sec . Heat amount 10^° kg cal/sec Salt amount 10® tons/sec Gulf Stream (Florida and Cheapspeake section) Kuroshio (Ryu Kyu section) Ratio between (Kuroshio : Gulfstream) 1 : 1-46 1 : 1-44 1 : 1-54 is so much weaker than the corresponding effect of the Gulf Stream on the Eastern Atlantic and on Europe, although the heat transport is not appreciably less. This difference must be governed by topographical conditions (Dall, 1881, Koppen, 1911). After leaving the Japanese coast at 35° N. until it diverges northwards and south- wards on the eastern side of the ocean the Kuroshio water travels about 8000 km, while the Gulf Stream water after leaving the American west coast travels only about 5000 km. Beneath the Kuroshio waters there is weakly saline, cold sub-Antarctic water, but beneath the Gulf Stream the water is warmer and more saline and con- tinuously renewed by the outflow of the highly saline European Mediterranean waters (see p. 529, Fig. 245). The Gulf Stream water is thus protected from consider- able heat and salinity losses downwards. The greater efficiency of the Gulf Stream must be attributed to the much longer conservation of its properties over the considerably shorter distance it travels and to the favourable conformation of the European coasts. (/) The Agulhas Current This current is due to the outflow of the water piled up by the South Equatorial Current of the Indian Ocean along the coast of South Africa and Madagascar and as such is a typical gradient current. A detailed dynamic evaluation of the observational data available from the different expeditions has been carried out by Dietrich (1935). For the surface currents see p. 567 ; for the structure and dynamic of it see p. 470, Figs. 205-7. As subtropical and Antarctic water masses are situated side by side the three-dimensional mass distribution is a rather complex one. Everywhere along the African continental slope as far as the latitude of Capetown there is a steep rise of heavier water (cold, but weakly saline) towards the coast. Towards the Agulhas Bank the slope is flattened out and on the shelf itself is occasionally superimposed by lighter water brought in from the south and south-east by the wind. To the south of this heavy water mass there is found a relatively lighter (warmer, but more saline) water mass of subtropical origin in a trough-like fashion bordering on the denser sub- Antarctic water which moves eastwards in the south. Figure 205 shows the distri- bution of the specific volume anomaly in a cross-section oriented from Capetown in south-westerly direction. All cross-sections through the current are of similar nature as this one. The depth of the trough-like confined mass of the lighter water body (corresponding to the schematic picture of Fig. 204) is about 1000 m. Underneath this, weakly saline sub-Antarctic intermediate water spreads out everywhere, in which the salinity minimum weakly follows the trough-form and the rise towards the coast. 642 The Tropospheric Circulation Since the sub-Antarctic water forms an almost zonal boundary to the lighter water mass in the south, the trough of lighter water is narrowed towards west by the African continent, until it finally takes almost a wedge-form at the southern peak of the Agulhas Bank. In the further course this wedge then splits into three separate branches with simultaneously occurring vortex formations; the southernmost of these intrude into the heavier sub-Antarctic water and the northernmost intrude into the sub- tropical water of the South Atlantic. The lighter water thereby decreases considerably in thickness. A dynamic interpretation of the above-mentioned section running south-west of Capetown has been attempted in Fig. 206 ; similar scientific evaluation of the other sections gave results in agreement with this. The nature of the current is shown more clearly by the dynamic topography of the isobaric surfaces. Figure 300 shows the dynamic depth anomaly for the 200 decibar-surface relative to that of the 1000 decibar- surface; the first one can be taken as an approximation to the absolute topography of the 200 decibar-surface. According to this the Agulhas Current at the 200 m depth flows with intense velocities along the continental coast as far as the southern tip of Africa. However, it thereby diminishes rapidly its mass and velocity and finally loses its current character forming three large quasi-stationary vortices, the cores of which are identical with the three branches of lighter water mentioned before. According to Dietrich about three-quarters of the water masses of the Agulhas Current, transported at the southern tip of Africa into the South Atlantic, is drawn into these vortices and after mixing with the current of the higher latitudes returns to the Indian Ocean. Analysis of the pressure distribution in the current interior shows it to be the resul- tant of two components. The first is an effect of the internal pressure determined by the mass distribution, and corresponds to the normal pressure distribution in a system in which a lighter motionless water mass is embedded between two denser moving water bodies. The second component corresponds to a ridge of high pressure occurring in the boundary region between the two currents flowing in opposite direction and is due to the piling up of water. Since the Agulhas Current in the northern part of the current system as well as the broad oceanic West Wind Drift in its south both give a total water transport towards left. In the boundary region between them water accumulates giving rise to the second pressure component. In combination with the first a total pressure distribution is generated which is characteristic for that found in the Agulhas Current. Especially typical is the circumstance that the two adjacent currents of opposite direction face each other with their faster moving parts. The large lateral shearing forces thus formed give rise to large vortical movements (p. 570) in which most of the flow energy is dissipated. Dietrich, 1936 has given a comparative discussion about the structure and move- ment of the Gulf Stream and of the Agulhas Current and reference is made to this investigation here. 4. Upwelling Phenomena A characteristic phenomenon occurring in the narrow oceanic strips off" the western coast of the continents in middle latitudes is the observed cold coastal water, wliich due to its influence on the atmosphere is of considerable climatological importance. Until recently the investigation of these phenomena had to be based on surface data only. The Tropospheric Circulation 643 o t: "3 fc 60 O o. o H 644 The Tropospheric Circulation which was not enough to afford any insight into the inner mechanism of this phenome- non. Some data for the area off Chile and Peru have been obtained by the last "Car- negie" cruise (Sverdrup, 1930) and the "Meteor" expedition during the spring of 1937 made six profiles at right angles to the coast with the objective to study the upwell- ing water phenomenon off the north-west coast of Africa (Defant, 1936a). Detailed systematic investigations of the strong upwelling phenomena off the Californian coast have been made since 1937 by the Scripps Institution of Oceanography (Sverdrup, 1938a, Sverdrup and Fleming, 1941). These cover the development of upwelling phenomena in successive surveys and have provided some understanding of the dynamics of the upwelling process. Some comments might be made here on individual regions with upwelling. A summary for the oceanic regions off south-west Africa has been given by Defant (1936a), see also, Bobzin, 1922). The surface temperature conditions are given in the charts of the "Meteor" Report, vol. v. Atlas. In all months the low temperatures occupy the total width of the shelf (about 100 nautical miles), at the continental slope occurs the rapid rise to the higher temperatures in the west. During every month the temperature anomaly is highest at the coast with maximum values of — 8°C to — 10°C. The area of maximum anomaly moved in a meridional direction during the course of the year: in the summer (January) it occupies its southernmost position and is strongest between Table Bay and Luderitz Bay (32° S to 23° S.); in winter it moves furthest to the north (between the Luderitz Bay and Walvis Bay, 27° to 14° S.). During the entire year the current system of the sea surface shows a particularly characteristic one-sided divergence line which extends along the coast from about 30° to 20° S. or even more. In the south its distance from the coast amounts to about 160 nautical miles; in the north, however, 300 to 360 nautical miles. The region to the east of this divergence line is the region of cold upwelling. Where the unilateral divergence is most strongly developed, also the temperature anomaly is greatest. The anomaly at the coast vanishes north of 20° S., where the divergence with a de- creasing intensity turns westwards and gradually fades away. The uniform rise of the isopycnals from west to east (towards the coast) is a particularly marked feature of the thermo-haline structure of the upwelling region. Off the coast especially in the north there is a well-developed transition layer, and all the isolines immediately beneath this transition layer off the coast show a surprisingly sharp downward deflection to a depth of 350 m. This is only explicable as an effect of piling up of water at the conti- nental slope whereby in the depths lower than 30 or 40 m the water masses are pressed downwards. Similar conditions apply also to regions with cold water upwelling off the north- west coast of Africa. From January to May especially this region can be visualized by a tongue of cold water extending from higher latitudes southwards along the coast. Figure 301 shows this temperature anomaly for April ; it occupies the entire area between the Canaries and Cape Verde in which the anomaly already on the average is increased to almost — 7°C just off the coast and for individual cases reaches values of — 10°C or more (see Schumacher, 1933). Here also a sharp density transition layer can be found extending along the edge of the shelf until just off the coast. Particularly well-developed upwelling phenomena occur in the region off the western coast of North America between about 46° N and 25° N., especially off Cali- fornia with extreme conditions at Cape Mendocina (north of San Francisco). An The Tropospheric Circulation 645 analysis of the thermal conditions in this oceanic region has been carried out by Thorade (1909) and McEwen (1912, 1934). The onset of upwelling phenomena usually occurs in March and reaches its maximum during the summer months (July to August). The culmination coincides with the maximum frequency of the north-west winds. It is absent during the autumn and winter although off-shore south-easterly winds are not 40° 35" 30 25° 20" 15 10" 5° 0° 5° Fig. 301. Mean anomaly of the sea surface temperature off the north-west coast of Africa for April (drawn from means of two degree squares of the Atlantic Ocean). uncommon. The cold upwelHng water off the South American coast has been dealt with by GuNTHER (1936) (see p. 571). The west coast of Austraha is not entirely free of cold coastal water as has been shown by Schott (1933) and rising water sometimes occurs off the north-western coast. Occasional observations of cold upwelling water have also been made along many other coasts, for instance, off the Somali coast during the summer months during off-shore winds and at the southern tip of Ceylon and others. In considering the dynamics of the phenomenon it should particularly be remem- bered that for a current in stratified water the mass field adjusts baroclinic, so that 646 The Tropospheric Circulation under stationary conditions the lower and cold as well as nearly always weakly saline waters are lifted on the right-hand side of the current core in the Northern Hemisphere and on the left-hand side in the Southern Hemisphere. If there is a parallel coast along this special side of the current the water off the coast already for this reason alone will be colder and will have a lower salinity than further out. This state does not represent an upwelling phenomenon, but rather a state of long duration dependent on the nature of the vertical water stratification and on the current strength. Most of the anomalies appearing off the coasts are due to such a simple effect on the mass field produced by the currents. Upwelling of cold deep water occurs only if in a wind-driven current with a flow component parallel to the coast a water transport away from the coast sets in. The continuity condition then requires a rising water movement at the coast. In a first attempt in order to explain this phenomenon Thorade, 1909 used this theory, and later on particular interest has been devoted to the determination of the vertical velocity profiles in the rising water (McEwen, 1912) and to the determination of the depths in which the upwelling phenomenon starts out (Sverdrup, 1930). It was soon found out from the thermo-haline structure in the upwelling region, that these depths could not be large and that due to the inclination of the isothermal layers off the coast an upward water movement of only a few hundred metres would be sufficient to explain the observed sea surface anomaly. The formation of a one-sided divergence line running more or less parallel to the coast is the characteristic feature of the current field. The occurrence of rising movements at divergence lines in the case of non-stationary discontinuity surfaces and vortices is, of course, understood theoretically (p. 469) and water movements of this type are shown definitely by numerous observations of the vertical and horizontal distribution of the oceano- graphic factors (for example, equatorial cold tonges in the Atlantic and Pacific (pp. 558 and 569); boundary regions at the oceanic polar fronts, p. 471). In the upwelling regions off the west coasts of continents all upwelling phenomena are of a similar type as discussed above. From the analysis of the mean oceanic state off the coast of South West Africa Defant (1936^) has derived the schematic diagram shown in Fig. 302 of the structure and the water movements in a cross-section at right angles to the coast. Essentially the cross-sectional movement consists of an elongated vortical motion around a horizontal axis which is superimposed on a much stronger and uniform current parallel to the coast. The water beneath the axis of the transverse vortical motion flows in the lower part of the top layer, in the density transition layer and beneath it towards the coast and gradually rises just off the coast. The upwelling phenomenon is very largely confined to the narrow strip between the divergence line and the coast. It rises up to the sea surface from a depth of only 100-200 m and as a consequence of the current field the temperature distribution, observed in vertical direction remote from the coast, is twisted around and changes its position into a horizontal one; so to say is projected on the horizontal sea surface. A necessary consequence of this circulation is the destruction of the density transi- tion layer in the upwelling region off the coast. This is clearly shown by the "Meteor" cross-section (1937) over the shelf off the north-west African shelf. The gradual break down of the transition layer, which at times is also strongly developed in the area The Tropospheric Circulation 647 Divergence Horizonol temp, distribution, °C ■^15° 14° 13° 12° 11° -3° -4°-5°-6°-7^jemperature anomaly, 0 -100 --200 Q -300 500 400 300 200 Distance from coast in Sm 100 Fig. 302. Schematic cross-section normal to the coast of south-west Africa. Full lines, isopycnals ; arrows, zonal and vertical velocity components (the length of the arrows can be taken approximately as a measure of the speed) ; letters, meridional velocity components and in special ; A^, parallel to the coast towards north ; S, parallel to the coast towards south (the size of the letters can be taken approximately as a measure of the speed) ; wavy lines, axis of the vertical current vortex (vertical exaggeration 1 :2300). nearest the coast, is a consequence of internal tidal waves which gradually become unstable as is definitely shown by the series of observations. This is thus a precondition for the upwelling of deep water (see vol. ii, p. 581). SvERDRUP (1938a) in the evaluation of the almost synoptic surveys made by the Scripps Institution of Oceanography, La Jolla, from March to June 1937 along a transverse section off and at right angles to the Califomian coast from Port San Luis (35-2° N., 120-7° W.) has obtained good insight into the dynamics of the up- welling processes. Figure 303 presents two topographies of the physical sea level as well as the 100 and 200 decibar-surfaces relative to that of the 500 decibar-surface. In the time between the two surveys typical mass displacements have occurred. The changes in the profile occurring down to the 200 decibar-surface can only be interpreted by a water transport away from the coast and by the piling up of the lighter surface water near Sts. 4 and 5. These movements can be looked upon as a consequence of the winds which blow with little variation for long periods, on the average from N. 23° W. at about 6-7 m/sec, almost parallel to the coast. According to the Ekman-theory under these conditions a transport directed away from the coast can be expected. This trans- port can be derived from the change in the course of the density lines between the two surveys. These surface waters are carried outwards and piled up about 100 km off the coast. From the analysis of all the fields Sverdrup has derived the mean current field shown in Fig. 304 during the period between the surveys. The calculated maximum transverse velocity seawards thereby amounts to 1 1 cm/sec, in good agreement with the velocity of the wind drift. Between the coast and the water piled up further out 648 The Tropospheric Circulation ST NO I 040 t-0-35 0 0-90 085 -oeo 0-75 u 060 - 5 100 D-BAR. OVER 500 D-BAR < z 0-55 o 200 D-BAR. OVER 500 D-BAR. DISTANCE FROM COAST IN KM. 50 100 150 200 250 Fig. 303. Topography of the physical sea surface and of the isobaric surfaces (relative to the 500-decibar surface) for the oceanographic surveys. I, 25-26 March 1937, and II, 5-6 May 1937, of the profiles through the Califomian region of upwelling water (according to Sverdrup). STNXii Fig. 304. Computed mean vertical circulation for both profiles I and II in the cross-section through the Califomian region of upwelling water (according to Sverdrup). The direction of the motions is indicated by the thick lines with feathers; the horizontal velocities are given by the thin lines. The region indicated by -f -f -f + + shows a zone with stronger flow parallel to the coast and directed into the picture. The Tropospheric Circulation 649 there is a partly closed circulation down to a depth of 80 m. In the upper half of this circulation the water flows away from the coast, in the lower half towards the coast. Near to the coast the water rises and in the region remote from the coast it sinks along a boundary layer. This outer boundary layer itself moves away from the coast and as a compensation a replacement has to be made from below (from depths of not more than 200 m). In other cases dealt with by Sverdrup conditions are somewhat more complicated but the essential characteristics are retained. In a study of the large amount of observational data, on the Californian upwelling region, collected by the Scripps Institution of Oceanography in La Jolla, Defant (1950, 1951) it has been shown that the piling up and upwelling processes are associated with characteristic displacements of the sea surface and of the internal boundary layer which gradually develop under wind influence and adjust with simultaneously formed and normal to the coast occurring circulations. They finally tend towards a stationary state. These condition can be illustrated by two opposite cases. During the first cruises (28 February to 15 March 1949) it was found that the wind component towards the coast predominated over the entire region with a maximum of 5 m/sec and caused considerable piling up of water along the coast. During the second cruise (27 April to 15 May 1949), in contrast to the first case, the water was driven away from the coast where as a consequence upwelling occurred. Cruise 1 thus is a typical example for a water accumulation along the coast, while cruise 2 is typical for coastal upwelling. Figure 305 shows the dynamic topography of the ocean surface represented by lines of equal positive and negative deviation from the basic distribution produced by the Californian Current flowing south. This basic distribution has been obtained by elimination of the disturbances caused by tide waves and internal waves (Defant, 1950). The two cases show completely opposite trends. First of all it may be noticed that the channels of positive and negative deviation (shown by the contours) are more or less parallel to the coast following the wave-like form of the disturbance, thereby forming a marked regular pattern. In cruise 1 the coastal strip shows a pronounced positive deviation — with maximum values at the coast. Outside this there is a strip of negative deviation, then farther out a strip of postitive deviation, and finally a second negative strip forms the western border of the region. Cruise 2 gave the same pattern with the signs reversed. In cruise 1 there is undoubtedly a piling up of water at the coast ; it was fully developed at the beginning, but during the remainder of the cruise (about two weeks) it could be maintained to this extent only if the tangential wind stress towards the coast exactly balances the pressure gradient of the sloping physical sea surface. The water masses piled up on the continental shelf are drawn from the oceanic strip just off the conti- nental slope; there the sea level consequently lies somewhat deeper (trough-like form). This disturbance then develops wave-like oscillations farther westwards and generates the adjoining disorders. Exactly the same applies to cruise 2 but instead of piling up of water a depression in water level occurs. Consequently, to these primary disturbances the adjacent displacements in the sea level thus take place in the reversed order. The dynamics of the processes of upwelling and removal of water as a surface drift requires that the rise and fall of the physical sea surface should be accompanied by a corresponding fall and rise in the density transition layer. In these processes (close to a 650 The Tropospheric Circulation Fig. 305. Position of the physical sea surface and of the internal thermohaline boundary surface and the corresponding circulation cells of the upper layer during the cruises 1 and 2. In the first case: "Anstau" at the coast (piling up of water); in the second case: upwelling off the coast. The inclinations of both boundary surfaces are strongly exaggerated, that of the physical sea surface by far more than that of the thermocline. Stationary equilibrium) in a sea composed of two layers, the displacement of the physical sea surface is always inverse to that of the internal discontinuity surface. However, the fluctuations of the internal discontinuity surface is many times greater (inversely proportional to the difference in density of the two water masses). Figure 306 shows a schematic cross-section for cruises 1 and 2. The effect of the wind on the sea surface gradually builds up to such a stage where the wind effect is exactly in balance with the developing pressure gradients. While approaching this stage circulations have developed mainly in the mixed layer, and must take the form shown in Fig. 306. On cruise 1 the water accumulation at the coast causes a downward circulation here and a sinking of the density transition layer. Upwelling occurs in the trough forming outside this region of accumulation. In contrast to these conditions, during cruise 2 the water is driven away from the coast, where upweUing thus takes place and the water masses sink down in the accumu- lation region away from the coast. These primary circulations at the coast are followed further out by successive secondary circulations of diminishing intensity. The Tropospheric Circulation 651 652 The Tropospheric Circulation To the Dynamics of UpweUing There are a number of causes for the vertical water movements in the ocean. For continuity reasons these vertical motions are closely connected with the divergence and convergence of the surface waters, and there is no doubt that the upwelling and sinking of oceanic waters is primarily connected with convergence and divergence regions occurring at the sea surface. The cause of these divergences and convergences in most cases lies in the distribution of wind stress exerted by the prevailing wind on the sea surface. A totally satisfying explanation of upwelling at continental coasts has not yet been given, and is probably not possible at all since the total process is composed of a number of substages each of which is always controlled by other factors. Coastal upwelling is confined to a narrow strip close to the coast (less than 100 km) and must therefore be regarded as a boundary phenomenon. It is a known fact that winds blowing at a suitable angle to a coast will carry light surface waters away from it and the water mass transported away must be replaced near the coast by heavier subsurface water. Defant (1952) gave a theoretical explana- tion on the assumption of a sea composed of two layers with different density; previous to this a more general investigation was made by Jeffreys on the effect of a steady wind on the surface of a homogeneous ocean near the coast. The application of a theoretical model as simple as this showed that the stationary wave disturbances at right angles to the coast take their origin from the piling-up region or the upwelling region ("Anstau oder Auftriebsgebiet") near the coast (see Fig. 306) and gave results in good agreement with those obtained by observation. A theory of the upwelling produced by a wind parallel to a coast has been given by HiDAKA (1954) whereby the effect of the earth's rotation and the frictional forces due to both vertical and lateral mixing have been taken into account. He deals only with a case of a steady state. The equations of motion, together with the equation of continuity and the boundary conditions which must be satisfied at the sea surface and along the coast, give a rather complicated solution to the problem. Calculation of the magnitude of the off-shore currents and the upwelling velocity for a numerical example allows the results to be compared with values estimated correctly from obser- vations. Figure 307 gives the solution in the form of stream lines in a vertical plane perpendicular to the coast. Upwelling develops close to the coast and there is no off-shore movement of the water in the upper layers of the sea directly beneath the surface swept by the wind. The upwelling is confined to the strip until 0-5Z)„ from the coast and the sinking process occurs outside the wind zone. If the vertical mixing co- efficient ^4^, is chosen with a value of about 1000 then the vertical Ekman frictional depth Z)^, will be 162 m at 30° N. For a horizontal mixing coefficient A^ = 10^ the horizontal frictional depth will be about 162 km. Estimation gives the width of the coastal upwelling region as ID^ = 339 km. From this the average velocity between the surface and the layer 0-2Z),, can be calculated as 3-35 cm/sec (off-shore the maxi- mum upwelling is 2-7 m/day upward or approximately 80 m/month). Sverdrup (1938) obtained a similar large value for the upwelling velocity off southern California. The depth at which the upwelled water originates is about 200 m which is also in fair agreement with observed values off the southern Californian coast. Hidaka has also investigated the cases arising when the wind is at certain angles to the coast. If the wind is at right angles to the coast, then the induced circulation has a rather complicated The Tropospheric Circulation 653 Fig. 307. Upwelling as induced by a wind parallel to the coast illustrated by the stream lines in the vertical plane perpendicular to the coast. In the numerical example D^= 162 m and D^ = 162 km; the width of the coastal wind belt is about 340 km. Structure with two vortices in the upper layers, one of which is situated close to the coast and the other near the outer boundary of the wind belt. The upwelling due to a longshore wind (Fig. 307) is far more effective in lowering the temperature of the coastal region than that induced by an off-shore wind, since the former one brings a larger amount of colder water to the surface from deeper levels than the latter. This theory put forward by Hidaka deals only with the stationary case; no attention is paid to the water stratification which as shown by observations plays a decisive role for the processes involved before a steady state is reached. The process of upwelling is shown by observations to be variable with time. If the duration of the wind is as short as a few hours, the off-shore component of surface water transport will not be very large since drift currents will not fully develop. If the winds are more or less steady for several hours up to as much as a day, the drift currents may develop but they will not be followed by considerable upwelling because of oscillations of the thermoline. However, the process will be different if the wind continues for several days up to a week. If the wind continues for a longer time- interval than about a week, the surface currents will reach a steady state with an inter- mediate stage for a wind lasting a few days up to a week during which the geostrophic equilibrium is approached. This latter section of the process has been dealt with theoretically by Yoshida (1955) using the conditions in Californian waters as a guide. In his model the .v-axis is directed eastwards, the >'-axis directed northwards and repre- sents the coast line. The r-axis is chosen positive downwards with z = 0 being placed along a mean sea level. The conditions were taken as constant in a north-south 654 The Tropospheric Circulation direction. In addition at this stage only small-scale processes, i.e., processes extending over a period of several days to a week and over a distance of up to 10 km, were considered of interest. The equations of motion are then -fv = - I (XIX.60) dv 8i 8 / 8v\ r„ A is the eddy viscosity, Ty is the northward component of wind stress, T and H is the average thickness of the mixed layer. The corresponding vorticity and divergence equations are dt, fwn curl^ T dt H H (XIX.62) /^ - g, (XIX.63) where Wf, is the vertical velocity at z = /? (depth of the thermoline). The equation of continuity and a condition for the quasi-isostatic adjustment with g* = g(Ap/p) give 1 8p w,^-,^. (XIX.64) The mutual adjustment between the pressure and the current seems to be completed within a period of one to two days, so that the above equation is reasonable for up to about a week after this first stage of adjustment is over. From the equations (xix.62- 64) is obtained where k =fl\^{g*H). The boundary condition along the coast {u — 0) will require ( 8w\ _A:2 with the condition w = 0 when x = -co the solution of (XIX. 65) will be k^ .^j fh Cx ro Ty e^a^-f)^! + Ty e-^(^-f) di + e^^ Ty e^^ d^ 0 J — CO J — CO (XIX.66) It can be shown that __ 1 ej; ^^ ~ y8x for values \kx\ > 1 and along the coastline we have Wo = y [" Ty e^^ dx . (XIX.67) A uniform northerly wind over off-shore water will give rise to a coastal upwelling given by -« = ^, • (XIX.68) The Tropospheric Circulation 655 The upwelling velocity will be proportional to the intensity of the northerly wind but is not directly dependent on the latitude. When g* = g{Apjp) — 2-5, i/ = 40 m = 4 X 10^ cm and Ty,Q = —0-5 then H'a-^o = — 5 X 10"^ cm sec ~^. In five days this upwelling will give an upward displacement of the thermoline of 22 m. This upward movement of the thermocline off the coast will continue until an equilibrium is reached in about a week and according to observations seems then to be maintained for about one or two months. The region where this coastal upwelling occurs is confined almost entirely within a narrow strip close to the coast. With the numerical values introduced above, k will result to '^0-7 X 10~^ cm~^; at a distance of 40 km, w will be reduced to 6% of that at the coast and to only 3% of the coastal H-value at 50 km. The process is practically limited to a distance of 40-50 km from the coast. The effective width of coastal upwelling is given by a characteristic length Yoshida also investigated the changes in surface conditions which were derived from the above model of a transient state of upwelhng. He found that the variations in surface characteristics were largely confined within the narrow coastal regions. The coastal upwelling is associated with considerable changes in surface conditions within the coastal waters of width L, while upwelling or sinking outside this strip will not give rise to such significant changes during a period of only a week or two. In the succeeding stage of the upwelling process, in which now the isostatic adjustment can be con- sidered a complete one, the laterial mixing process in the inshore regions stands out as the most important factor. The dynamic equations are now - A- = - I , (XIX.69) ft^ = ^ + A,-^„ (XIX.70) where A,, is the coefficient of lateral mixing. The upward movement of the thermoline, due to the ascending motions, will produce a sharp horizontal density gradient and when conditions are variable in an oscillatory way, as is usually the case, internal waves will originate and cause intense mixing across the thermocline. The equation for the conservation of mass will now become or, approximately w ^ - An dx" The boundary condition at the coast gives Tq = 0 so that finally ^^ = -g-dx' ^^^^-^^^ 656 The Tropospheric Circulation The equation for w will become the same as in the earlier state and the vertical velocity distribution will therefore remain unchanged throughout the whole period of upwelling process as long as the wind is kept steady. During this period the ascending water movement will be subject to mixing with the surrounding waters and the thermo- line will not be raised to any large extent. From equation (XIX. 71) it follows that at this stage the vorticity in the surface layer will be proportional to the vertical velocity. Upwelling will thus be associated with cyclonic vorticity in contrast to the initial inshore increase in negative vorticity produced by the coastal upwelling. This approach developed by Yoshida undoubtedly appears to give a deeper insight into the dynamics of the upwelling process, but a more specific representation in detail of these processes would be desirable. 5. Processes at the Polar Boundary of the Subtropical Convergence Region The subtropical convergence regions are oceanic areas where the oceanographic factors show large local and time variations (p. 575). They can be interpreted as con- sequences of vortex formations between the two somewhat different types of water on the polar and the equatorial sides of the convergence region. On the one hand, there are intrusions of warm highly saline water from lower towards higher latitudes, and on the other hand, intrustions of cold and weakly saline water occur in the opposite direction. All the isolines of the oceanographic factors and the isolines of the dynamic topography of the pressure surfaces thus show a wave-like structure. Whether all the deviations from a smooth curved pattern are of an aperiodic nature propagated in one direction along the boundary region between the two water types and in time dying out, cannot be decided without a rapid succession of synoptic surveys. Since series-observations, made in the convergence region at quite different times, can all be combined without excluding any large number of individual values into closed comprehensive representations ; it may be safely concluded that the disturbances are often quasi-stationary vortical disturbances whose position and extent are probably determined by external factors. These wave-form disturbances are particularly well developed in the convergence region of the South Atlantic. The topography of the physical sea level between 25° and 50° S. (Fig. 308) shows the irregular wave-like patterns in the course of the dynamic isobaths. This starts suddenly off the broad Patagonian-Argentinian shelf and extends across the total width of the Atlantic to the region south of Africa. According to the topographies of the deeper levels these wave-form disturbances reach down to con- siderable depth but their intensity decreases rapidly with depth. They can hardly be detected in the topography of the 1400-decibar surface. Their greatest intensity is always found in the top layers where they must originate and therefore the reason for their formation must be looked for here. The entire oceanic structure is shifted towards the poles and the equator, respectively, by the interacting intrusions of different water masses in a strip-like manner, and thereby differently stratified oceanic spaces oppose each other side by side that would normally be found arranged in a zonal fashion. Then inside the resultant vortical formations of both water types, heavier water sinks down at the boundary surface extending to more southern latitudes, while the lighter water at the same time is lifted and extends further towards the poles. The sinking process of the heavier waters apparently does not take place everywhere along the The Tropospheric Circulation 657 2U 658 The Tropospheric Circulation extended more or less zonal boundary surface, but rather in form of individual mass intrusions {quantum-like) at different places whereby as a consequence mixing is con- siderably increased. The nature of the processes involved can be illustrated by putting side by side successive stages of the oceanic state in a meridional section (Defant, 1941Z>), and one obtains thereby all the characteristics of the disturbances which occur. The bottom topography in this part of the South Atlantic was earlier assumed (p. 435) to be the cause of the wave-form current pattern appearing in the region of the sub- tropical convergence (Fig. 187). It should be emphasized, on the other hand, however, that the vortical disturbances originate on the shelf of the South American continent between 45° and 35° S. far in the west, and from here extend as a continuous chain of regular vortices throughout the entire area as far as the southern tip of Africa. This source region or birth place, is the region where the denser water of the Falkland Current meets the lighter water of the Brazil Current and where the tendency for a vortex formation is extremely large. Here a strong solenoidal field is continuously regenerated, which can be considered as the necessary condition out of which vortices are formed and the disturbance field then stretches far out into the Atlantic. A probable explanation of these wave-form disturbances can be derived by means of the arguments put forward by Rossby and co-workers (1939) in a discussion of the sinusoidal disturbances in zonal atmospheric air currents. In a wave-like disturbance, which is superimposed on a pressure field that decreases to the south (Fig. 309, P+2 W ^+ P+2 P + l Fig. 309. Wave flow for a uniform towards south decreasing pressure field. Southern Hemisphere) the water transport through the cross-sections A and C where there is an anticyclonic curvature of the isobars will be greater because of the occurring centrifugal force than that through section B where there is a cyclonic curvature. There will therefore be a horizontal divergence and pressure fall between sections B and C and a horizontal convergence and pressure rise between A and B. The wave disturbance will thus move eastwards and since the centrifugal force is larger when the curvature is greater the shorter waves will travel eastwards faster than the longer ones. In addition to this effect, there will be a pure latitude effect which originates from the relation of the geostrophic flow to the pressure gradient. Due to the Coriolis force the mass transport across the section 5 in a lower latitude will be greater than that across sections A and C in higher latitudes. This gives rise to convergence and pressure rise between A and B. This latitude effect which is independent from the wavelength causes a west- ward movement of the wave. Both effects are of the same order of magnitude and it is easily understood that for a particular wavelength the wave disturbance will be The Tropospheric Circulation 659 stationary. The mathematical basis extended by Haurwitz (1940) affords a relation between the wavelength L, the latitudinal extent D of the stationary disturbance and the velocity of the basic current, U, in the form 4772^ 1 +L^ID^' whereby j8 = 8f/R8)/R is the change of the Coriolis parameter / with latitude and R the earth radius. Analysis of wave disturbances in the South Atlantic convergence region gives an average disturbance length at latitude circle 38° S. of 10-0° or 880 km. The latitudinal extent averages 15° or 1650 km. With these values the velocity of the basic current U is obtained as between 26 and 28 cm/sec. This means that the wave disturbance within the zonal basic current (oceanic West Wind Drift) can be stationary only if such a mean velocity towards East is present. Current charts show an average surface velocity of 25-30 cm/sec. It is thus very probable that the stability of the stationary wave system in the convergence region is due to an equilibrium state between the action of the latitudinal dependence of the Coriolis force and the effect of the curvature of the current trajectories on the horizontal mass transport. The strong solenoidal fields at the boundary between the Brazil and the Falkland Currents may be responsible for the formation of the eastward following series of vortical disturbances inside the general oceanic West Wind Drift. If this is so then the topographical effect of the bottom configuration will be only a supplementary effect which may intensify and probably modify these disturbances. Similar phenomena may also develop in the North Atlantic. In the oceanic strip of the North Atlantic Current to the north of the subtropical convergence region there are marked pulsations that also stand out clearly in the charts of the dynamic topo- graphy of the individual isobaric surfaces and in that of the physical sea level. The results of the International Gulf Stream Survey (1938) to the north of the Azores enabled a study to be made of the oscillations in the current system in this particular region. The oceanographic work of the "Armauer Hansen" in 1909, 1925 and 1935-6 in the Norwegian Sea off the coast of Norway (Helland-Hansen, 1934, 1939) showed that vortices with vertical axes probably played an important role in the interior of the Atlantic Current. They are also associated with considerable variations in mass transport. It is rather obvious that such variations at fairly long intervals cause reactions in the oceanic phenomena in the Arctic and take influence on climatic conditions in the Scandinavian countries. At present, however, the investigation of these phenomena is only at the very beginning and systematic and synoptic surveys are required in order to obtain a deeper insight into the mechanisms involved. An unusual theory of the variations of the surface circulation in the North Atlantic, especially of the current branches off the coast of Europe, has been given by Le Danois (1934) in his Atlantic Transgressions. He distinguished between three water types in the Atlantic: the tropical, the polar and the continental. The latter has an extremely variable salinity and remains at shallow depths in a relatively narrow band along the coasts. His "transgressions" are periodic movements of variable amplitude carrying Atlantic water of tropical origin, in temporary intrusions into water masses of polar and especially continental origin. The water of the transgressive masses always has a 660 The Tropospheric Circulation salinity greater than 35%o. From a large number of individual cases Le Danois has attempted to derive definite rules according to which these trangressions move to the north-east. These instrusions of Atlantic water into north-west European waters are discernible only in their effects on the "continental" water masses over the shelf. Here the warm transgressions at the surface over the continental plateau always are preceded by highly saline transgressions in the deeper layers. The transgressions appeared nearly always to follow the course of the valleys of the submarine relief. The direction of spread is mainly to the north-north-east, so that the speed of this spread of the intrusions is the less the more it deviates from this direction. By following these phenomena in the sea off the coast of France for a large number of years Le Danois has found certain periodicities in the occurrence of the transgressions, which superimpose each other in the same manner as waves. However, it appears difficult to follow the Le Danois theory of these transgressions, since he uses several arguments quite contradictory to the established fundamentals of dynamic oceanography (Schubert, 1935). Chapter XX The Stratospheric Circulation 1. Introduction Beyond the polar convergence (oceanic polar front) towards the poles the oceanic stratosphere reaches upward to the sea surface and is here subject to the full influences of the atmosphere (radiation, evaporation, precipitation, freezing processes and others). The water types continually formed by the climatic conditions here are heavier, due to their low temperature and in spite of their low salinity, than the waters of the adjacent convergence regions of the oceanic troposphere. Thus, in relation to these latter water types they tend to sink, intruding beneath the oceanic troposphere, until they reach a depth corresponding to their density. The sinking, strongly favoured by the thermo-haline structure, reaches down to great depths. After sinking, the almost horizontal spread of the water underneath of the troposphere causes a layered leaf-like structure in the oceanic stratosphere. When this structure is sufficiently well developed it is therefore possible to tell from it something about the path of spread of the water masses and gain thereby an insight into the stratospheric circulation. This is the method that has been used up to the present time in the study of the water movements inside the stratosphere. In the absence of sufficient direct current measure- ments, however, the results of such investigations were largely only of a qualitative nature. Preparation of the observational data according to dynamic methods can provide further insight into the nature of the stratospheric oceanic flow, but at the present time only a few investigations of this type have been made. All these methods, of course, give mean conditions only. Over large parts of the ocean, however, especially for the deeper layers the basic prerequisite of stationary movements will be satisfied. But aperiodic disturbances of shorter or longer duration and of greater intensity un- doubtedly occur. By means of the observations available at present, and also due to the manner in which they have been gained, it seems hardly possible to draw any conclusions about the nature of these disturbances. The surface layers of the oceanic stratosphere poleward (the polar fronts) are, of course, subject to wind influence, so that also in the polar and subpolar seas wind- driven ocean currents are generated. The complicated orographic configuration of the continents in the Northern Hemisphere affects the nature of these surface currents and exerts strong influence during their transformation into gradient currents. In this way, piling up (Stau) phenomena play the principal role, and meridionally oriented coasts in higher latitudes form excellent guiding channels for southward outbreaks of the cold polar water masses. The zonal polar circulation obtains in that way meridional components, so that on the eastern sides of polar land mass water flows south, while on the western sides mainly water of subtropical origin flows north. 661 662 The Stratospheric Circulation 2. Polar Currents of the Northern Hemisphere Phenomena similar to those found in the subtropical convergence region can be expected also to occur at the polar convergences. These will be even more intensive there, since a much sharper density difference exists between the adjacent water masses. External factors will, at many places, cause the formation of vortices between the warmer highly saline waters of subtropical origin and the cold weakly saline polar waters. These travel along the boundary zone, continually forming and disappearing and thus giving rise to a continuous mixing of the two water bodies. For these reasons, in the Northern Hemisphere, the left-hand border of the polar currents is not sharply developed and here polar waters and water masses of subtropical origin work into each other. This is shown to be true for all currents, especially for the East Greenland Current along its boundary region against the Irminger Current to the south of Iceland and for the Labrador Current where it encounters the Gulf Stream. Some insight into the processes involved in the vortex formation in the region of interaction between two adjacent water masses, especially as found in this part of the ocean, has been obtained from the almost synoptic surveys made by U.S. Coast Guard vessels (see the bulletins of the U.S. Coast Guard, International Ice Patrol, Washington). The sea around Greenland (Greenland Sea, Labrador Sea, Davis Strait and Baffin Bay) has been well surveyed oceanographically by numerous expeditions, and from the entire data available it is possible to obtain an idea about extent and course of all the currents. This is especially true of the East Greenland Current which can be follov/ed along its entire course from the Denmark Strait to Cape Farewell and from thereon as the West Greenland Current until it finally disappears (see Defant, 1936^ for references). Little information is available on the East Greenland Current from its origin near the Spitzbergen Rise to the Denmark Strait but there are appreciably more data to the south of this strait. All cross-sections show a similar structure. The polar water layer always has a cold core in which the temperature is almost at freezing point. Figures 3 10 and 3 1 1 show two cross-sections through the East Greenland Current in the Denmark Strait and off Cape Farewell, The analysis of 37 sections of this type through the East and West Greenland Currents has enabled the course of the polar water flowing around Greenland to be followed in detail. In Fig. 312 an attempt has been made to show the boundary separating polar water from Atlantic water; in addition, the average minimum temperature in the core layer of the polar water is indicated in this figure which is usually at a depth of 80 m. The minimum temperature in the core layer gradually rises from — 1-7°C in the Denmark Strait to about — 1-0°C at Cape Farewell. Past the southern tip of Greenland, where the current turns sharply around, the core layer rises towards the surface; its temperature increases rapidly and from about 61° N. on is usually no longer negative. The East Greenland Current from the Denmark Strait southwards where the width of it is more than two-thirds of the width of the strait remains entirely over the shelf; where the shelf is broad the current is also wide and where the shelf is narrow (for instance between 62° to 63° N.) its width is very small and does not exceed 25 to 30 nautical miles. The lens of cold water forming the current core at first extends well to the east, but becomes smaller towards south and shrinks from the Denmark Strait to Cape Farewell under the impact of the warm water of the Irminger Current. It is, however, still present and shows that the polar The Stratospheric Circulation 663 Hdll2 Hdlll ^19 Hd63 0o4432 -^ZB Hd30 Hd3l Hd32 Hd33 Hd34Hd35 29 30 31 K 31 32 32 33 34 38 100 200 300 400 500 9519 Hd63 , pa4432 o528 Hd30 Hd3l Hd32 Hd33 Hd34Hd35 I 2 2 T 12 21 I 2 4 10 Fig. 310. Vertical cross-section through the East Greenland Current for the region of the Denmark Strait at about 67" to 65" N.; below, temperature; above, salinity. water is an uncustomary water type in the oceanic space under consideration and is maintained only by continuous renewal. The intrustions of the Atlantic water occurs in the form of vertical vortices which break through the polar front, broaden and deepen and if the inflow weakens soon disappear. (Defant, 1930a; Bohnecke, Hentschel and Wattenberg, 1930-32). From Cape Farewell the current bends northwards still keeping also here over the shelf. At first the cold core layer is still present but its temperature rises rapidly indicating stronger mixing with the Atlantic Water penetrating northwards along the continental slope. From about 64° N. the current weakens more and more and near the Davis Strait (about 66-5° N.) there are only traces of the cold core layer found off the Greenland coast. In this region, in all 664 The Stratospheric Circulation profiles, another core layer at about 80 m depth shows, which must clearly be fed from the north-west by cold polar water that flows in through the Davis Strait with the southward along Baffin Land directed current and finally joins the Labrador Current. As yet no dynamic preparation has been made of all the available data for the East Greenland Current. Topographies of the physical sea level and the isobaric surfaces in this region are also contained in Fig. 271 (see also Fig. 200). The downward slope of the isobaric surfaces from the Greenland coast towards the open sea is quite large and shows clearly the entire system of the East and West Greenland Currents. This current system can no longer be seen in the 800 decibar surface; the stronger current intensity is thus confined to the top layers. The main cause for the development for the East Greenland Current must lie in the wind- and atmospheric-pressure conditions over the North Polar Basin. At all times of the year due to wind and atmospheric pressure the water is driven eastwards and water laden with pack-ice and drift-ice is carried towards the coast of north-east Greenland. Here they find, supported by the wind turn towards south, a guiding channel in the form of the Greenland coast. The pressure due to the piled up water in combination with the action of the deflecting force of the earth's rotation produces a southward gradient current. It could be expected that these cold weakly saline waters on penetration into the warm but highly saline Atlantic water masses would soon be dispersed by mixing. This is not the case and they still show, only slightly weakened, as far as the southern tip of Greenland. They are maintained only by the continuous advection of polar water from the north and by the climatic regime which maintains the inland ice in Greenland. The polar climate generated by the inland ice, together 3 4 5 e ico- 200 300- 400- 500 600 700- 800 200- 300 400- 500 600 700 800 Fig. 311. Vertical cross-section through the East Greenland Current somewhat north of Cape Farvel (about 60' N.); left-hand side, temperature; right-hand side, salinity. The Stratospheric Circulation 665 45° 40' Fig. 312. Spreading of water masses of the East and West Greenland Currents derived from 35 oceanographic cross-sections. , limit between the east and west respectively of Greenland Current and the Atlantic water. , limit of the core layer of the Baffin water — 1-8'C: minimum temperature in the core layer of the polar water. with the continous transport of cold inland air which spreads well out over the sea, produces a belt around Greenland in which the temperature is lowered so much that also there a polar climate prevails. Within this belt the East Greenland Current maintains itself as a polar current as far south as 60° N. An excellent monograph on the water masses of the oceanic region between Green- land and North America with numerous temperature and salinity sections and velocity profiles calculated on the basis of these sections is that by Smith, Soule and Mosby (1937). For a general understanding of conditions here any of the cross-sections can be selected from each current section since the main features are very similar in all of them. Figure 313 shows these conditions in a cross-section through the Davis Strait. The different water types moving through the strait are clearly shown, in particular by the temperature distribution. Water with a temperature of less than — 1 °C keeps well towards the Baffin Land side and forms the core of the Baffin Land Current; its centre is found at about 100 m depth and here as shown by the velocity profile the current direction points towards south. On the western side of the strait there is a warm weakly saline top layer flowing northwards with a small velocity that represents the last branching remnants of the West Greenland Current. There is a core of warm and highly saline water at about 400 m depth; this is Atlantic water that moves northwards within the lower layers of the West Greenland Current along the conti- nental slope. Along the 600 nautical miles that this water travels in about 3 months from Cape Farewell its temperature falls by 4°C and the salinity by 0-50°oo due to mixing. The Labrador Current, after reinforcement by the inflow through Hudson Strait, also keeps close to the continental coast and as in the case of the East Greenland 666 The Stratospheric Circulation _ CM f^ ^3" in 1^ fv. r^ r^ ^- 30-91 30-85 Fig. 313. Cross-section of temperature (above) and of salinity (below) through the Davis Strait ("Godthaab" stations 168-175, 17-19 September 1928). 32-10 32-34 34-42 32-53 34-41 0 40 80 I : 1 : ! miles 0 100 60 6-9 6-7 6-8 7-0 6-3 70 6-4 7-1 -r ■^\ ST^ $ O) " 200 -V. a-y^ 400 _ N \J \ .3 44 3206 32-48 33-02 0 31-95 32-72 32-72 33-61 400 Fig. 314. Above :'cross-section Oi of temperature (to the left) and of salinity (to the right) through the Labrador Current near to the Belle Isle Strait ("General Green" stations 1333- 1341, 7-8 August 1931). Below: Cross-section Pi of temperature (to the left) and salinity (to the right) through the Labrador Current near White Bay ("General Green" stations 1229-1238, 6-7 July 1931). The Stratospheric Circulation 667 Current the shelf forms the main path of the southward-flowing cold, weakly saline water. Figure 314 shows this in a marked way and is characteristic of all the cross- sections from Davis Strait to the Newfoundland banks. A calculated dynamic topography of the sea surface relative to that of the 1500- decibar surface based on a dynamic evaluation of all the observational data (1928-35) is shown in Fig. 315. This gives some idea of the current conditions in the very upper layers, since it should correspond rather well to the absolute topography. The trough- like depression of the water level between Greenland and Labrador stands out parti- cularly well in this figure, with an even narrower continuation reaching southward as far as the southern end of the Newfoundland Banks. The strong concentration of the dynamic isobaths and a high coastal water level off south-west Greenland indicates the West Greenland Current and off the north-east coast of America the Labrador Current, while the strong rise from the southern peak of the Newfoundland Banks towards the north-east is due to the Gulf Stream. According to the topographies of the 600- and 1000-decibar surfaces, the strength of the currents decreases very rapidly with depth. In the region of the Labrador Current there are differences in water level of about 30 dyn. cm at the sea surface while over the same distance the difference in sea level at 600 decibars is only 3 cm and at 1000 decibars is not more than 1 dyn cm. These currents are thus typical density currents and are confined to the top layers. Volume transports calculated from the velocity profiles for several different cross- sections are given in Table 1 54 which also gives a rough budget for the water and heat exchange amounts in the Labrador Sea. The pure gain in water is about 7-5 milhon m^sec while the outflow along the Labrador Coast amounts to about 5-6 million m^sec. Both values refer to a transport down to 1500 m depth. This gives a difference of 1-9 million m^/sec from which the authors assume that it is the water of the West Green- land Current that sinks down to depths below 1 500 m and very probably flows out of the Labrador Sea in the deepest layers. Figures for different seasons and for different years vary considerably; for instance the transport of the Labrador Current was 1-31 milHon m^sec in 1930 and 7-60 in 1933. From this it must thus be concluded that Table 154. Exchange of water and heat in the Labrador Sea {after Smith, Soide and Mosby) Exchange of Water X 10® m^ sec-^ Heat X 10» kg cal Inflow West Greenland Current (average at Cape Farewell) Baffin Land Current ..... Hudson Bay discharge ..... Total . Outflow West Greenland Current to Baffin Bay Labrador Current (average South Wolf Island) Total 50 20 0-5 7-5 10 4-6 5-6 17-5 -1-2 0-5 16-8 0-5 14-6 151 668 The Stratospheric Circulation W 00 ^ 80 70 60 50 40 30 20 10 60 W Fig. 315. Dynamic topography of the physical sea surface in the region of the Davis Strait and the Labrador Sea, relative to that of the 1500-decibar surface (Mean for the period 1928-35, according to Smith, Soule and Mosby). The Stratospheric Circulation 669 the out-flow from the Labrador Sea is subject to large variations dependent on a number of diff"erent phenomena occurring at the sea surface of the polar regions (see also KiiLERiCH, 1939). Table 156 also contains a heat budget for the Labrador Sea. The heat gain amounts to 1-7 X 10^ kg cal/sec. If the mean temperature of the waters which sink below 1500 m is taken as 3-2°C then the heat flux with the outflow mentioned above will be about 6-1 X lO^kgcal. This then gives for the Labrador Sea a heat deficit of 4-4 X 10^ kg cal. It is not improbable that this heat deficit according to its magnitude is totally compensated by the heat absorption of solar radiation in the water during the summer. It can be calculated that of the total radiation from sun and atmosphere about 20 X 10^ kg cal reach the sea surface of the Labrador Sea. Of this then more than 40% (8 X 10^) is lost by reflection; the remaining 12 x 10^ kg cal goes to radiation, evaporation and absorption. Since the radiation is probably not very eff"ective, about two-thirds of this goes to evaporation and one third or about 4 X 10^ kg cal to absorption. This quantity is of the same order of magnitude as the quantity given above, but due to the uncertainty of the calculation this result should only be accepted with reservations. 3. The Processes which occur at the Antarctic Convergence Zone The causes for the formation of an Antarctic convergence within the broad oceanic West Wind Drift of higher latitudes in the Southern Hemisphere were discussed on p. 549. This discontinuity layer in the thermo-haline structure of the upper water masses appears in the pressure field as a discontinuous step in the meridional slope of the isobaric surfaces and the physical sea level (Fig. 253). This can also clearly be seen in representations of the dynamic topography of the isobaric surfaces constructed by Deacon (1937) according to the data obtained by the "Discovery" for the broad ring of water surrounding the Antarctic continent. Figure 316 shows the dynamic topo- graphy of the physical sea level (relative to that of the 3000-decibar level) for this oceanic region. The downward slope of the pressure surfaces towards south at all meri- dians is not uniform and a discontinuity extends all around the earth that makes the meridional gradient much stronger in a belt coinciding with the Antarctic polar front. This frontal zone is also shown to exist in the topographies of the isobaric surfaces for larger depths; but corresponding to the much smaller gradient it is less strongly developed in the deep sea. From the analysis of a series of vertical sections between Antarctica and South America (partly in the Drake Strait) as well as at 30° W. in the South Atlantic between 36° and 50° S. based on the observations of the "Discovery" Sverdrup (1933a) has deduced the vertical circulation in the Antarctic Convergence Zone. Since conditions around the Antarctic continent are very uniform, the results should be typical for the whole of the circumpolar region. The essential details can be seen in the temperature, salinity and oxygen sections at 30° W. shown in Fig. 317. According to all such meri- dional sections and also according to those for the other oceans the water masses of the upper layers south of the Antarctic Convergence sink down along the boundary layer. In the salinity distribution this is clearly shown by a tongue of weakly saline water. At the polar front at first the water sinks immediately down to 400 m and then spreads almost horizontally to the latitude of the subtropical convergence region where 670 The Stratospheric Circulation Fig. 316. Dynamic topography of the physical sea surface relative to that of the 3000- decibar surface (according to Deacon). The figures are anomalies of the dynamic depths referred to a homogeneous ocean of 0°C and a density a^ of 2800. it sinks again rapidly to 800 m or more. The temperature sections show a tongue of relatively warm water beneath the Antarctic water of the uppermost layers that forms an intermediate layer between 400 and 800 m and must be interpreted as a returning current flowing back towards south. Since this warm intermediate layer is found every- where it seems to be a general phenomenon. Above it, in all sections (in summer), a tongue with a lower temperature is found directed northwards at a depth of between 80 and 200 m. This stratification is no effect of a northward water transport but is rather a remainder of the cooling which has been effective during the previous winter (seePt. I, p. 137). The Stratospheric Circulation 671 In the deep layers the salinity distribution indicates a deep flow from north to south between 1800 and 3200 m and, beyond 40° S, gradually rising to 1000 m, while in the far south just off the Antarctic continent the cold Antarctic water sinks to the bottom layers of the ocean. The following sections of this chapter are devoted to these processes. Inside the water masses south of the convergence there is thus found in the upper layers a vertical circulation that extends down to about 1000 m which occurs in an anticlockwise sense when looking towards east. The uppermost layers are carried northwards by the wind, sink down at the Antarctic convergence and form the main constituent of the subantarctic intermediate current. Part of this water mass, however, mixes with deep water and returns southwards in the Antarctic circumpolar ocean as a warmer intermediate current. The top layers of sub-Antarctic water are rich in plant 675 1000 20CHD 3000 36° 38° 40° 42° 44° 46° 48° 50° 52° 54° 56° 675 673 671 668 666 663 661 1000 2000 3000 Fig. 317. Distribution of temperature (upper picture) and of salinity (lower picture) in a vertical section at 30° W. from 34° S. to 58° S. in the South Atlantic Ocean (series measure- ments of the "Discovery 11", end of April 1931, according to Sverdrup). 672 The Stratospheric Circulation and animal organisms. Dead organisms sink downwards and decompose and therefore the water of the returning intermediate current is also rich in phosphate. The lower oxygen content in and just beneath the returning current indicates strong oxidation of organic matter. Since only a part of the water transported in the uppermost layers returns to the south there must be a compensating poleward component in the deep layers in order to replace the cold polar waters sinking down in the very southern latitudes along the Antarctic continental slope. Beneath the vertical circulation of the upper layers there should therefore be a somewhat weaker one which rotates in a clock- wise sense looking east. These vertical circulations are superimposed on a general basic movement towards the east so that the resultant motion occurs in form of elon- gated spirals. In these circulatory motions of the water the water properties are altered in the upper layers by influences from the atmosphere above while in the lower layers changes occur due to mixing. The water of the higher southern latitudes is thus made up partly of water of the returning intermediate current and partly of deep water from lower latitudes. The schematic block diagram presented by Sverdrup that is shown in Fig. 318a shows the meridional components of motion in the Antarctic Circumpolar Ocean. A.C. / Fig. 31 8o. Schematic representation of the meridional circulation inside the Antarctic Circumpolar Current (according to Sverdrup). This concept of the circulation character occurring in these higher latitudes of the Southern Hemisphere differs somewhat from the ideas expressed in elder investigations. Merz and Wust (1928), for instance, interpreted the warm and highly saline water of the intermediate layer of the higher southern latitudes only as the last traces of Atlantic Deep Water reaching the sea surface in this region. According to Clowes (1933), this water should be of Pacific origin and should reach the Atlantic only by way of the zonal circulation. Both suppositions are only partly tenable. Sverdrup attempted to estimate also the magnitude of the meridional velocity components, on the one hand, from the shearing stresses of the wind leading to an estimate of the resultant total water trans- port, and on the other hand, from the steady-state condition in the temperature field of the returning intermediate current. For the upper layers a mean value was about The Stratospheric Circulation 673 2-5 cm/sec. The time required to perform a single complete cycle in the upper vortex with a horizontal axis in the area of Drake Strait thus amounts to at least a year when the above mean velocity value is used. This transverse circulation is, however, undoubt- edly stronger here than elsewhere in the Antarctic Circumpolar Ocean. South of the oceanic West Wind Drift the physical sea level and the isobaric surfaces rise again towards the Antarctic continent. An indication of this rise in the Atlantic Ocean can be seen also in Fig. 316. Near the continent easterly winds prevail, and the currents flow towards west. In this flow along the continent there will thus occur a vertical circulation similar to that appearing in the oceanic West Wind Drift except that it performs a clockwise rotation when looking east. There are indications of such a circulation found in the observations of many Antarctic expeditions. In this connection, Sverdrup also drew attention to the transport of lighter water by the wind towards the Antarctic shelf where it is strongly cooled. The wind thus has a tendency to pile up the lighter surface water against the shelf and produces stronger and stronger solenoidal fields, which are of no consequence, however, since the water simultaneously is cooled there. Both effects thus work in opposite sense and prevent the development of strong solenoid fields and also of stronger currents which would otherwise be formed solely by the action of the wind. 4. Dynamics of the Antarctic Circumpolar Current It is of interest to investigate the extent, in a broad current which encircles the whole earth, to which the wind stresses acting on the sea surface are balanced by frictional stresses against the outer boundaries of the ocean basins. For most oceanic currents the computed transports diff'er as was shown by Munk (1950), by a factor of not more than 2 from the observed transports. Munk and Palmen (1951) have made a similar calculation for the Antarctic Circumpolar Current. They considered the Antarctic Circumpolar Current as an eastward flow on a plane tangential to the earth at the south pole. The flow is induced by the constant eastward winds and depends only on the distance r of this plane from the pole. The balance between the wind stress T and the lateral friction is expressed by the relation where A^ is the lateral kinematic viscosity and M is the eastward mass transport across a normal vertical plane of unit width extending from the sea surface to the sea bottom. For a solution in which M vanishes at the Antarctic continent {r = r^), and at some other latitude (r = r^) the total mass transport of the flow will be 18AV^ '« r, + ro'"/-oy Putting r = 2 dyn cm-^, A^ = 10^ cm^ sec-^ and /"o = 70° S., r^ = 45° S. one obtains M = 5 X 10^*^ g sec"^ whilj the observed transport is at least 1-5 x 10^* g sec"^. This discrepancy is not materially altered on taking spherical co-ordinates or allowing for the variation of the wind with latitude. The transport M is inversely proportional to Ah and only values of 10^°t or more can give an agreemenwith the observed facts. Values of Ah as large as this are. however, improbable. Munk and Palmen attempted to reconcile the two values by taking into account the friction at the bottom especially 2X M - I Mdr= ,^^ I ri^ - r^^ - :^^ In -| . (XIX.2) 674 The Stratospheric Circulation there where the major submarine ridges lie as transverse obstacles in the path of the current. If the wind stress should be completely balanced by the frictional stresses along the sea bottom, then the Antarctic Circumpolar Current must extend deep enough to reach the sea bottom. It is certain from the vertical oceanic stratification in these latitudes that the current reaches down to very large depths; this is clearly indicated by the dynamic topographies of the individual isobaric surfaces. However, the velocities decrease very rapidly with depth and at depths of more than 4000 m the flow intensity of the Antarctic Circumpolar Current is extremely small. Corres- pondingly, the frictional stresses at the sea bottom will also remain rather small. By making the most favourable assumptions Munk and Palmen showed that the retarding pressure of the submarine ridges against the deep current might still be able to balance the wind stress on the surface. HiDAKA and Tsuchiya (1953) have recently taken up the problem again and attempted to find a hydrodynamic solution. From the equations of motion and the continuity equation with the corresponding boundary conditions they derived for planar co-ordinates, a complete solution in the form of infinite series giving the total mass transport, the surface slope and the vertical velocity distribution. Their calcula- tions using some arbitrary numerical values of the lateral and vertical eddy viscosity {Ah and y4„) give the same results as those of Munk and Palmen. For A„ = 2 x 10^ and Ah = 10^" cm-^ g sec"^ they found a total mass transport of 9-3 x 10^^ g sec~\ a surface slope of 3 m per 25° lat. and directions and strength of the currents in good agreement with those observed. But also in this case choosing values of ^4^ less than 10^^ would give impossible conditions. In a more recent treatment of this problem Takano (1955) introduces a special vertical and meridional density distribution corres- ponding approximately to the observed ones. The rather complicated mathematical solution led to the following conclusions: if the Ekman frictional layer is disregarded then the geostrophic approximation can be safely applied for the small velocities near the sea bottom. However, in order to obtain agreement with the observed values of the surface velocity, of the surface slope, of the density diff'erences at the sea surface and of the mass transport, it is necessary to take ^4^ = M x 10^". This is again the same large value that was found to be a necessity in the investigations mentioned before. There must thus be yet another source of energy dissipation in order to have a complete balance in the sense put forward by Munk and Palmen between wind stress and frictional stress. This can probably be obtained by taking into account the effect of the boundary friction, not only at the sea bottom but rather along the extended continental slope of the Antarctic continent which was previously neglected. An essen- tially different explanation of the dynamics of the Antarctic Circumpolar Current has been given recently by Stommel (1957). While Munk and Palmen and all others who have treated the problem regarded the Antarctic Ocean as an example of an ocean without meridional barriers for which a Sverdrup type solution could not be con- structed, Stommel believed that while the circumpolar ocean was indeed a continuous ring of water around the earth, it was so strongly narrowed at Drake's Passage between Grahamland and the southern tip of South America that a pure zonal flow could hardly develop in this section. On this basis the Antarctic Circumpolar Current is amenable to treatment by the Sverdrup theory and is essentially frictionless except The Stratospheric Circulation 675 in a short section after its passage through Drake's Passage. The entire energy dissipa- tion and all the other disturbances occur at this point; in all the other sections of the current course it is a simple frictionless geostrophic current. Stommel developed a simple model (Fig. 318Z7,(af) consisting ofa homogeneous ocean of uniform depth surrounding a schematic Antarctic continent and only at one place (indicated by the heavy radial line) a barrier closes Drake's Passage completely. The zonal wind system assumed is also shown in Fig. 3186,(<^) with trade winds from the equator to 30° S., westerlies from 30° S. to a little over 60° S. and further south a nar- row zone of easterlies. The Ekman drift current transport is northwards in the westerlies and southwards in the easterlies. Therefore a divergence zone exists between about 55° and 50° S. and a convergence zone further north. Since there is a complete barrier it is not difficult to maintain a wind-driven circulation. The meridional components of this circulation are indicated by arrows in Fig. 3186,(fl) and the entire circulation is shown in Fig. 318Z),(^)- At the western coast of the ocean (the eastern side of the barrier) an intense western boundary current will be set up and this simple cir- culation will be characterized by two immense gyres around the earth parallel to the latitude circles. Stommel calculated that the transport in the southern gyre would be somewhat more than 100 x 10^ m^ sec"^, and somewhat less in the northern gyre. In fact, however, the northern gyre is broken up by the African and by the Australian- New Zealand land mass. If now the barrier between South America and the Antarctic is broken in the manner indicated in Fig. 3 1 86, (c) then the transport lines will run through this opening and a circulation to the east will develop at the southern rim of the barrier. The flow through the passage still remains unexplained but without doubt models can be devised in order to describe it. Stommel's explanation of the dynamics of the Circumpolar Current is quite different from the previous explanations. He also attempted to make this explanation more plausible by embedding this current system under consideration into the system of the sub-Antarctic-Antarctic circulation. 5. The Sub-Antarctic Intermediate Current The most important facts concerning the spread of the subpolar Antarctic inter- mediate water as far as they can be deduced from the distribution of the oceano- graphic factors have been described already in Pt. I, p. 173. This water type forms the uppermost part of the oceanic stratosphere. The sinking at the polar convergence is shown by all meridional salinity sections (see Pt. I; Fig. 62 for the Atlantic, p. 147, Fig. 75 for the Indian Ocean and Fig. 76 for the Pacific, p. 172). The fact that this process at the Antarctic convergence (see p. 669) occurs with about the same intensity all round the earth indicates that at all places the sinking and the subsequent spread of this water type are caused by the same factors. In the Northern Hemisphere the morphological configuration of the continents interferes with the formation of an Arctic intermediate current and traces of it are found only along the western side of the Atlantic. The weakly saline intermediate current in the Atlantic is consequently not symmetrical about the equator and we may only speak of a sub-Antarctic intermediate current here. In the Pacific the northern current branch is almost as strong as the southern one and therefore in the region of the thermal equator (6° to 8° N.) very similar water types come in contact with each other. In the Atlantic the Antarctic branch is, however, so strongly 676 The Stratospheric Circulation developed that it extends past the equator and can be traced almost as far as 20° N. It is noteworthy that the thickness of the intermediate water at first is about 1000 m and later on diminishes in wedge-form, and that it is found across the entire width of all cross-sections through the ocean (see Pt. I, Fig. 77 p. 174). A detailed analysis of the sub-Antarctic intermediate current in the Atlantic — which is the only ocean for which this is possible at the present time — using the core layer method and the [r5']-relationship has been given by WiJST (19366). By a deter- mination of the percentage with which the original water type can be found south of the Polar Front at each place in the entire space, and how much of it has been lost due Fig. 3186. {a) The schematic southern ocean. Antarctica is the full black circle. The meri- dional barrier projecting out from Antarctica is represented by the full heavy black line. The schematic wind system (purely zonal) is depicted by the heavy arrows on the lower left. Latitudes of Ekman convergence and sinking at the surface are indicated by 0, latitudes of divergence and upwelling are indicated by ®. The direction of the required meridional geostrophic flow is indicated by thin radial arrows. (Jb) Transport lines of the solution for the model depicted in Fig. 3186,(«) The western boundary currents are to be interpreted schematically. (c) Hypothetical form of the solution, that results from rupturing the American-Antarctic barrier in such a way as to permit water to flow throughout, to obstruct all latitude circles (according to Stommel 1957). 77?^ Stratospheric Circulation 677 90' 80° 70° 60° 50° 20° 10° 0" 10° C0° 3C W 120' 110° iOO° 90° 80° 70° 60° 50° 40° 30° 20° 10° 0° 10° 20° 30° 40° 50° 60° E Fig. 319. Absolute topography of the 800-decibar surface (smoothed representation). (Dynamic isobaths north of the subtropical convergence region drawn from 1 to 1 dyn cm, otherwise from 5 to 5 dyn cm.) 678 The Stratospheric Circulation to mixing, one obtains a rather good insight into the mixing process going on in the total space of spreading (with reference to these conditions see Pt. I, p. 212 and following pages and particularly the Figs. 100-102). In general, the diagrams indicate a uniform spread towards the north taking place over the whole cross-section almost immediately after the sinking at the polar convergence, but further north there is a preference for the western half of the ocean which must be due to the effect of the Coriolis force. Here close to the South American continent the spread possesses current character. The entire width of the layer across the total ocean gets its supply, then from the western side by lateral turbulence and by occasional occurring large intrusions but the salinity distribution shows only the final stage after lateral mixing has been effective and does not give information about the nature and way with which the lateral mixing process operates. Since a current is formed on the western side along the South American continent these processes can be regarded as a case of free turbulence (Defant, 1936c) and the ratio [exchange: velocity] can be determined along the entire spread of this water type. This then gives some idea about the current character of the spread of the Antarctic intermediate water. In order to find the pressure forces that give rise to this water transport it is necessary to determine the dynamic topographies of the isobaric surfaces at these depths. The absolute topography of the 800-decibar surface which corresponds north of 40° S. closest to the core layer of the sub-Antarctic intermediate water is shown in Fig. 319 for the region of 20° N. South of 40° S. the zonal course of the dynamic isobaths indicate the downward extension of the large Antarctic Circum- polar Current flowing eastward; but at this depth the meridional pressure gradient is only half of that observed at the sea surface. Also, the broad high-pressure ridge in the subtropical convergence region is present only with a somewhat diminished intensity and in the convergence regions still vortical disturbances appear extending down to these depths. North of the high pressure ridge the isobaths run also from east-north-east to west- north-west, but beyond 25° W. they turn towards the north and finally run along the South American continent as far as Cape San Roque. The pressure gradient here is thus directed towards the east but this gradient does not extend very far out from the coast; the broad area from about 20° S. to 20° N. as far as the African coast shows almost no gradient. Already downward from 500 m the water movements in this large region must be extremely weak and there is no indication whatsoever of a circulation. The water displacement corresponding to the absolute topography (see Fig. 320) on the northern side of the subtropical disturbance zone in the Southern Hemisphere is directed first to the west-north-west and then to the north-west and finally extends in a narrow band along the South American coast as far as the West Indies and con- tinues into the Gulf Stream region. The velocities everywhere remain small, between 6 and 12 cm/sec in the core layer, falling rapidly to weak intensities towards the eastern edge. An analysis of the salinity distribution in the Intermediate Current gives values for the ratio [exchange :velocity] of 0-8 to 2-3 at the upper and lower edges, respectively. This leads to exchange coefficients of about (5-10 g cm"^sec"i) which is in good agreement with the order of magnitude found by other methods at such depths. The Stratospheric Circulation 679 680 The Stratospheric Circulation 6. The Polar Bottom Current The second water type originating at the sea surface of the Antarctic ocean is the Antarctic Bottom Water. It is formed all along the Antarctic continental shelf and especially in the area of the Weddel Sea which is the place of formation for this coldest and thus heaviest water type; it sinks along the continental slope down to the greatest depths and extends northward following the bottom topography of the ocean as an Antarctic Bottom Current. As it spreads it is subject to continuous mixing with the water masses above. Its spread is hindered by transverse ridges which the current must pass and limits are set to spread by meridionally oriented rises; the deep passages through these zonally and meridionally oriented ridges thus form important guiding channels for the bottom currents. The extension of antarctic bottom water in the individual oceans as deduced from the thermo-haline structure has been described in detail during the discussion of the temperature distribution in the bottom layers of the ocean, so that the reader is only referred to this here (see Pt. I, p. 149). The spread of the bottom water is shown in Plate 4 by lines of equal potential temperature and from these the course of the bottom currents can be readily followed. The generation of bottom water in the Antarctic is so enormous that the same process in the Arctic is by comparison quite insignificant. In the Atlantic one can hardly speak of any proper Arctic bottom current, since the high upward extending ridges between North America, Greenland, Spitzbergen and further to the south between Greenland, Iceland, the Faeroes and Scotland almost completely block the outflow of bottom water from the Arctic Basin. Bottom water with a characteristic potential temperature of between —0-2° and — 1-5°C passes over the above mentioned submarine rises into the open ocean in only very small amounts. Recent investigations of the flow near the bottom across the Iceland-Faeroes Ridge have been made by Dietrich (1956, 1957). All the five cross-sections over these rises have shown that the warm North Atlantic Water and the cold sub-Arctic water are in contact over the ridge forming a narrow frontal zone. The heavy sub-Arctic water lying underneath the lighter north-east Atlantic water always covers a large part of the summit plateau of the Iceland-Faeroes Ridge, and sinks down immediately on its western side because of its higher density keeping thereby close to the slope. In spite of mixing with warmer water of smaller density its density remains still higher than that of the surroundings, and consequently it sinks to form the bottom water in the north-east Atlantic at depths below 3000 m. The velocity of this downward directed bottom current on the western side of this ridge can be determined using a formula given by Defant (1955) and results to about 35 cm sec"^ For a thickness of the sinking water of 50 m and with a total width of the passage of 150 nautical miles the water transport will amount to about 50 x 10^ m^ sec"^ Like a waterfall these waters flow out in individual bursts and may be observed at any time of the year at the Iceland-Faeroes Ridge. Oceanographically they have a greater importance than the sinking movements caused by winter cooling over the shelf of the Bay of Biscay and elsewhere along the continental shelf and slope which can occasionally be observed (see Cooper and Vaux, 1949 and Cooper, 1952). The main mass of North Atlantic Bottom Water thus originates outside the Arctic. WiJST (1943) termed this water type with a potential temperature of between 1° and 2°C as the sub-Arctic bottom water and the current fed by it the ^'sub-Arctic The Stratospheric Circulation 681 bottom current.'''' This sub-arctic bottom water comes mainly from two source regions: (1) from the north-western Labrador Basin where the colder bottom water with a temperature of less than 1-2°C is formed (Wattenberg, 1938; Smith, Soule and MosBY, 1937) and (2) from a region of formation extending all along the 3000 m depth of the south- east Greenland continental slope into the inner angle of a bay; this source was already referred to by Nansen (1912). From these two main centres the sub-Arctic bottom water spreads out towards more southern regions. Influenced by the bottom topography, this spread, however, keeps close to the western side along the foot of the continental shelf off the Labrador coast as far as 50° N. A Labrador submarine rise here prevents its further southward spread. The second centre of formation in the Irminger Sea is obviously less productive; since already in about 55° N. this water type has mixed with warmer waters and has lost its characteristic cold temperature. A small Arctic bottom current also occurs in the Pacific; cold bottom water in moderate amounts penetrates over the boundary rises of the Okhotsk Sea into the open ocean. However, this is likewise only of sub-Arctic origin and its productiveness remains small. The ratio [exchange: velocity] can also be derived from the analysis of meridionally oriented temperature and salinity sections (see Pt. I, p. 153) and stream lines of the water transport can be constructed in order to obtain a representation of the current course in its core (Fig. 321). The stream lines follow closely the bottom topography. Over the crests of the ridges values of the above ratio lie between 2 and 3, in the depres- sion between 5 and 6. For the same values of exchange the current intensity shows a proportion of about 2-5:1. Wattenberg (1935) by keeping track of chemical processes at the sea bottom and in the layers just above it found an exchange of about 4 cm~^ g sec"^. With this value, the velocity of the bottom current on the western side Brasilian Basin 9000 10000 Guyana-Basin 14 000 15000 16000 3000- D 5000 40» S 3 Fig. 321. Stream lines and values of the ratio between exchange and velocity in the core of the Antarctic Bottom Current in the Atlantic Ocean (evaluated from the temperature and salinity distribution in a longitudinal section of the Western Atlantic Trough). 682 The Stratospheric Circulation of the Atlantic should be of the order of 0-5-2 cm/sec. In order to flow through the distance from 50" S. to the equator, Antarctic waters would thus require about 10-30 years and would have lost 40% of its characteristic water properties on reaching the equator. Variations in these properties occurring at a certain moment in the area of formation of the water types could only be noticed in the bottom layers at the equator after appreciably long time and with a considerably diminished intensity. WiJST (1957) has recently made a dynamic investigation of the "Meteor" profiles and has thereby extended the determination of the absolute topography of the physical sea level and the isobaric surfaces made by Defant to the layers between 2500 m and the deep-sea bottom. He based his computations on the topography given by Defant for the dynamic reference surface (p. 496) and continued the calculations from this surface to the sea bottom. These topographies were used to determine the velocity components at right angles to the profiles. Figure 322 shows the resultant chart of the xS-^S i-tW. .♦ v^-y T ^;^m^u.-. hi::- />:V"7' \ *s\ ■'■■ Northward current component ^I^Southword current corr.ponent ■A Axis of Antarctic bottom current — Core of Antorctic bottom water 75° 60° 45° 30° 15° 0° 6° Fig. 322. Current distribution in the Antarctic Bottom Water of the Atlantic Deep Sea (in a depth of more than 3500 m) computed from the mass distribution taking as a basis the reference level of Defant (according to WiJST, 1957). The Stratospheric Circulation 683 bottom currents. The Antarctic bottom current in the Southern Hemisphere shows measurable velocities (>3 cm sec~^) only close to the western side of the West Trough, that is, at the foot of the continental slope and about 1000 m above the level of the proper deep-sea bottom. With few exceptions only very weak velocity components were found in the east. These results derived from dynamical computations agree well with the above described ones. With these new velocity values the water masses of the bottom would need about 5-5 years in order to travel from the southern rim of the Argentine Basin (48° S.) to the northern rim of the Brazilian Basin (5° S.). 7. The Deep Currents in the Middle Part of the Oceanic Stratosphere of Individual Oceans In a fully symmetrical ocean there would be in each hemisphere a subpolar inter- mediate current in the uppermost part of the oceanic stratosphere and a polar bottom current in the lowermost part of it. These water transports directed towards the equator for reasons of a compensation require an additional poleward water trans- port in the middle part of the stratosphere. These compensation movements are called the "deep currents" of the oceans. In this way the scheme of the meridional components of the stratospheric circulation (Fig. 323) thus consists of two closed circulations in each hemisphere; one circulation in the upper part of the oceanic stratosphere containing the intermediate current and the upper half of the deep current and moving in a clockwise sense when looking east in the Northern Hemisphere, and a second circulation in the lower part of the oceanic stratosphere that includes the polar bottom current and the lower half of the deep current and moves in an anticlockwise sense. It should be borne in mind in looking at Fig. 323 that only the meridional flow components of the two circulations are shown which are always weaker than the zonal ones. The rather varying character of the polar components in the actual oceans gives 0 1000 2000 3000 4000 5000 fionn PC E C P ^ V^V^-:- S- -^ -^ - -\C - ^ ^ - ^ -'A^'u i i ' 1-- -J -1 ; -t - 1 ,..,(, 1 ! 1 1 60° 40° E0° 0° 20° 40° 60° Fig. 323. Schematic representation of the meridional components of the oceanic circulation in a symmetrical ocean. -« < , circulation of the troposphere; •^— , subpolar intermediate currents; < , polar bottom currents of the stratosphere; < , mean deep currents of the stratosphere; , limit between oceanic tropo- and stratosphere; P, polar front (polar convergence); C, subtropical convergence; E, equatorial counter current. 684 The Stratospheric Circulation rise, of course, to large differences in the development of the deep currents. The closest approach to the ideal case pictured in Fig. 323 is found in the Pacific. The meridionally oriented sections, although they are based on insufficient data and often do not reach right to the bottom show the approximately symmetrical arrangement of the subpolar intermediate currents about the equator. Warm water sinks in the convergence region of both these currents as it is required in Fig. 323 ; such downwards motions are indeed indicated by a downward bulging of the isothermal layers in the meridional temperature sections. At greater depths the deep layers of the Pacific are almost uniform and there is no special differentiation to indicate any particular motion (Wust, 1929, \9Z0b). This is supported also by the absence of any temperature inversions which are very characteristic of the Atlantic and the Indian Ocean. The marked asymmetry of the polar components in the Atlantic Ocean due to the almost complete absence of the Arctic current branches gives rise to a strong develop- ment of the southward directed North Atlantic Deep Current. This provides the only compensation here for the Antarctic water carried north by the intermediate and bottom current. Disregarding at the moment the water layers from about 1000 to 1500 m between 50° N and 20° N. (particularly on the eastern side) the oceanic spaces underneath are filled with relatively salinity- and oxygen-rich waters. The structure of these waters indicate by their vertical structure a sub-Arctic origin. Its principal characteristic is the oxygen content of the core layer and the distribution of this shows clearly its origin from the area east and south-east of Greenland and from the boundary zone between the East Greenland Current and the Irminger Current south-west of Iceland as well as from regions in the north of the Labrador Sea. These are the same regions that form the source of the sub-Arctic bottom water (p. 680). WiJST termed this sub-Arctic bottom water as the "Lower North Atlantic Deep Waters" as opposed to the "Middle Deep Water" occurring above. In these regions mentioned above the almost homogeneous structure of the sea during autumn and winter allows the surface waters to sink to great depths forming there the source of the more or less horizontal southward water transport between 1 500 and 2500 m depth. In the "Meteor" cruise made in late winter 1935 the kind of conditions were found along a profile south of Greenland which are required to allow the autumn and winter convection to proceed to great depths. The oxygen distribution along this profile (Fig. 324) clearly shov/s this downwards tendency of the surface layers (Wattenberg, 1938). Below 1000 m the source for the middle North Atlantic Deep Water is formed here. When this water moves further to the south the transport obviously keeps closely to the western side of the ocean due to the influence of the Coriolis force, but even after crossing the equator it still prefers the western side and the effect of the Middle Atlantic Ridge is clearly noticeable. The upper layers of this southward water movement show the effect of mixing with Mediterranean water (see later) since the [TS]-relationship for the core layer at 35° N. shows a definite reversal point (see Fig. 325); apart from this the curve as far as 50° S. is almost a straight line and indicates gradual mixing with the water types above and below. Beyond 30° S. these waters enter the deep-reaching circumpolar flow of the very southern latitudes and under the influence of this are deflected to the east. The pressure conditions in the core layer of the North Atlantic Deep Water are best indicated by the topography of the 2000- decibar surface. Figure 326 shows immediately that the main course of the isobaths is in The Stratospheric Circulation 685 Fig. 324. Distribution of oxygen (in percentage) in a section from the southern tip of Greenland to the Great Banks of Newfoundland (according to Wattenberg). Fig. 325. Standard curve of the [r5]-relation in the core-layer of the mean North Atlantic Deep Water. its main features in agreement with the spread of this water type deduced from the thermo-hahne structure. The source of this water transport is in the north-west of the Atlantic and from here it flows southwards principally in three branches (see Fig. 327). The western branch keeps close to the North American continental slope, passes through the North Ameri- can Basin and enters into the Southern Hemisphere to the east of the Antilles. The middle branch follows the eastern slope of the middle Atlantic Ridge as far as 5° N. 686 The Stratospheric Circulation 120 » W 00° 40" E 60° Fig. 326. Absolute topography of the 2000-decibar surface in a somewhat smoothed representation (dynamic isobaths are drawn from 2 to 2 dyn cm). The Stratosphere Circulation 687 ^ 0) ti u O ° o ^ ^ o g . U C <^J — tf J '*- ^ ->■ ?i U -3 •£ o 3 cs t: o ^ ^^ o o •• o 5 o C 3 1 "•"' %w «u o o t*- o OJ '^ R a f^ -5 ^ h •T3 o <^ ., & o c ~ o *. u f^ « >> OJ :3 J= ■B o 1 2 ^ c £2 £ c 60 'So a. «3 o + ^X) I "^ 05 s :^ e (l; T3 P^ g t: p 11 CM c V. ■~ y^T-, J= x: « ■*-• i« 3 w^ o O & r-^ k. J= 3 a T3 Wi 0 u U M b £ J= 688 The Stratospheric Circulation and then breaks up into vortices. The third much weaker branch meanders along the East Atlantic Trough past Madeira to the Canaries and the Cap Verde Islands and a side branch of it seems to enter the Guinea Bight. The course of the first two branches under the influence of the Coriolis force used apparently the bottom morphology as guiding limits for their spread. The westernmost and most important branch keeps also in the Southern Hemisphere at first close to the continental slope until about 25° S. and then bends towards east-south-east and fills from here as a broad water transport the total oceanic space between 25° S. and 40° S. Finally, it passes south of Africa into the Indian Ocean. The velocities in the Northern Hemisphere branches of the current are seldom more than 2 cm/sec. Where the current concentrates along the South American continental slope it reaches about 3-4 cm/ sec until 15° S. and at Cap San Roque it reaches maximum speeds of 8-12 cm/sec before falling off to 0-5-1 -5 cm/sec further south. The spread of middle North Atlantic Deep Water as deduced from the oxygen content of its core layer is shown in the left-hand chart of Fig. 327 ; the arrows in this figure indicate the principal branches of spread determined from the dynamic topo- graphy of the pressure surfaces. The agreement between the results of the two methods is remarkable. Sverdrup (1930) has given a diagram showing the deep currents in the southern part of the South Atlantic based on the "Carnegie" observations that fits well in the topography of the 2000-decibar surface. WiJST (1957) has calculated the corresponding velocities at right angles to the "Meteor profiles" for the current course of the Atlantic Deep Water in the area between 10° N. and 30° S. The distribution of these velocity components is presented in Fig. 328 and shows obviously good agreement with the distribution in the right- hand diagram of Fig. 326. In the core the velocity (reduced to the "true" direction) is now 9-2 cm sec~^ with individual values varying between 2-1 and 17-4cmsec"^. It should especially be noticed that also here the flow is concentrated towards the west just off" the American continent while the eastern parts of the oceans are completely inactive. In the Indian Ocean a deep current stands out between 2000 and 3000 m marked by a highly saline deep layer and a pronounced temperature inversion. Its strong development is due primarily to the large density differences between the equatorial and polar water masses which are continuously renewed by the supply of salt from the Red Sea and the Persian Gulf (Pt. I, pp. 183 and 529). A deeply penetrating detailed analysis of some oceanographic series observations in the Indian and Pacific Oceans has been made by Helge Thomsen (1933, 1935). From the [rS'J-diagrams it appears rather doubtful whether there is actually a deep current in the Indian Ocean between 2000 and 3000 m similar to that in the Atlantic. On the other hand, the Intermediate Current and the Bottom Current are well developed as well as the effects from the Red Sea are easily followed far to the south. 8. A Survey of the Water Transports in the Individual Layers of the Atlantic Ocean The total amounts of the water transport in meridional direction in the South Atlantic total space (between 5°S. and 35° S.) which Wiist has derived from mean velocity values calculated from the individual profiles of the "Meteor" expedition are of great interest. The most important results arc summarized in Table 157. The figures The Stratospheric Circulation 689 Fig. 328. Current distribution in the lower Atlantic Deep Current (3000 m depth) com- puted from the mass distribution taking as a basis the reference level of Defant (according to Wust, 1957). given in this Table show considerable scattering due to random errors and inaccuracies in the basic data. Nevertheless, they give a rather good idea of the budget of the water transports in the South Atlantic space which is valuable in many respects. The final budget of the meridional transports (current amounts) is practically perfect with a discrepancy of only O-I million m^ sec~^ A complete balance between northward and southward transports in each of the two troughs cannot be expected. In the Western Trough the North Atlantic Deep Current with 27-5 million m^' sec^ towards the south is the main circulation component ; this is very largely confined to the narrow strip along the South American coast. The transport in the uppermost part and with the Bottom Current together is only 9-0 million m^ sec^ In the Eastern Trough the transport towards the north in the bottom and deep currents is exceedingly small. 2Y 690 The Stratospheric Circulation There is no current here which can be continuously followed through carrying water in large quantities to the south. Probably only very weak spreading and mixing processes operate here in variable direction. The deep sea circulation of the Western Trough is thus dominant and sets the basic pattern for the whole of the South Atlantic oceanic space. Table 155. Mean values of the meridional water transport in the total space of the South Atlantic Ocean {between 5° S. and 35° S.) given in units 10^ m^ sec ^ Current constituents Water transports throughout entire width of the ocean Through the Western Trough towards 1 Through the Eastern Trough towards Towards the Towards the north south north south north south Sea surface current "1 Deeper currents > Intermediate current J Deep current Bottom current 22-7 3 0 25-6 70 20 27-5 15-8 } 4-8 — 9. The Effects of the Subtropical Adjacent Seas on the Deep Sea Circulation. Analysis of series measurements in mid-latitudes of the eastern North Atlantic led already at an early stage to the recognition of a warm highly saline water type with little oxygen content, the principal characteristics of which point towards the Straits of Gibraltar which can therefore be considered as effects on the waters of the Atlantic, of the water flowing out of the European Mediterranean. The significance of "Mediter- ranean water" in the Atlantic deep-sea circulation was first pointed out by Jacobsen (1929). A detailed investigation and review of the phenomenon was then given by WiJST (1936) in the "Meteor" Report; he termed this water type "upper North Atlantic Deep Water". It is characterized by its high salinity which is in sharp contrast to the Antarctic intermediate water above it. Off Spain the core layer can be found at about 1000-1250 m and lowers down towards the equator reaching a depth of 2000 m between 10° S. and 20° S. From the salinity distribution in the core layer it is immediately obvious (see Fig. 329) that the spread takes its origin from the waters off Spain, and that it obtains its high salinity content of 36-4%o or more by way of the Mediterranean water flowing out through the Straits of Gibraltar in the lower layers (p. 529 et seq., see also, pt. I, p. 182). This water sinks to about 1000 m where it finds a corresponding density and then spreads out in a fan-like fashion under the action of turbulence and Coriolis force. Figure 330 impressively shows the great distances to which still an effect of the Mediterranean Water can be traced. It extends northwards past 50° N. and it reaches particularly pronounced directly across the entire Atlantic as far as the Ameri- can coast. Towards the south the last traces can be followed even to the higher latitudes of the Southern Hemisphere. The percentage of Mediterranean water present at any point can be determined from a standard curve for the [rS'J-relationship in the The Stratospheric Circulation 691 iOO" W 80 Fig 329. Spreading and depth of the upper North Atlantic Deep Water (Mediterranean Water) The thin dashed Unes indicate the depth of the core layer in metres (according to Wiist). 692 The Stratospheric Circulation core layer of this upper North Atlantic Deep Water (Fig. 330). In the western part of the North Atlantic there is still a content of 25-30%, at the equator 20-18% in the South Atlantic the Mediterranean content gradually falls to below 2%. The form of the [r^l-relationship which is nearly a straight line indicates that the changes in the core layer are due essentially to a simple mixing process. The great effect of the water flowing out from the Straits of Gibraltar on the compo- sition of the water masses in the Atlantic is at first sight astonishing. A rough calcula- tion shows, however, that it is of the right order. According to Schott (1939), about 366 Fig. 330. Standard curve of the [TiS] -relationship in the core layer of the upper North Atlantic Deep Water (Mediterranean Water). 52-000 km^ of water a year flows out from the Mediterranean into the Atlantic. For a mean velocity of spread of about 2 cm/sec it would require about 6 years to spread over the area between 45''N to 15° N. During this time the Straits of Gibraltar will supply 312-000 km^ of Mediterranean Water, which, distributed evenly over a layer of 500 m thickness from 45° N to 15° N., would mean a contribution of about 3-4%. The layers inside the Spanish bay will, of course, show a considerably higher percen- tage.* IsELiN (1936) has not quite agreed with the idea of an extension of Mediterranean Water to the higher latitudes of the Southern Hemisphere. On the basis of "Atlantis" observations he investigated the deviations of individual values from the standard value for the whole region using the Helland- Hansen anomaly method (see Pt. I, p. 114). A positive anomaly is present at 1200 m depth only as far as about 20° N. (until the North Equatorial Current), while farther south deficits appear due to the effect of mixing with Antarctic intermediate water. According to Iselin the effect thus extends no further than 20° N. This difference in viewpoint can be explained by differences in the definition of the "Mediterranean Water"; the fact at least remains that traces of Mediterranean Water can be followed far into the South Atlantic. The process of spread of Mediterranean Water through the Straits of Gibraltar and out into the Atlantic is certainly of a twofold nature. During the first part of the outflow and sinking of the heavier Mediten-anean Water, until it reaches the shelf and the continental slope and until it finds the depth of equal density inside the Atlantic, * These percentages refer to the water present between 600 and 700 m depth west of the Straits of Gibraltar which has a temperature of 11-9° C and a salinity of 36-5 %„ and was termed "Mediter- ranean Water" by Wiist. If absolute values are required of the proportion of Mediterranean Water from east of the Straits of Gibraltar then the given values must be reduced by half. The Stratosphere Circulation 693 the Mediterranean waters flow with considerable velocity and due to the influence of the Coriolis force keep especially in the Spanish Bay to the northern side. Finally, they pass around Cape San Vincent while steadily sinking and still keeping close to the Portuguese coast past Cape Finisterre as far as the Bay of Biscay. Observations show that this is the first stage of spreading; and the whole process of spread behaves exactly in the way described on p. 524 et seq. and in Fig. 251a. The second stage of spreading starts from this tongue of Mediterranean Water off the Portuguese coast. Due to the much lower velocities the Coriolis force is no longer effective and the influence of lateral and vertical mixing becomes dominant. The picture presented in Fig. 330 is thus an effect of mixing processes and Defant (1957) has shown that a lateral eddy viscosity coefficient of about 5-5 x 10'' cm^ sec~^ is quite sufficient to explain the lateral spread. A precise account of the whole process, however, requires systematic series observations and current measurements along suitable sections. The Indian Ocean also shows in all meridional salinity sections, starting from the Gulf of Aden in the north-west, an unmistakable effect of the highly saline waters spreading out from the Red Sea through the Gulf of Aden into the Indian Ocean. In this case also the effect of this outflow is of decisive importance for the stratospheric circulation. It is only to be expected that there will be seasonal variations in the extent of the spread of the water from the subtropical adjacent seas, since the outflow in itself is known to be subject to rather strong variations of this period (p. 503). The observa- tional data available at the present time do not allow to show the influence of such seasonal fluctuations in the open ocean. Investigations of the extent of spread of the Mediterranean Water show the great importance of the subtropical adjacent seas for the deep-sea circulation. Due to the high density of the water masses flowing with the deeper currents into the open ocean the layers of the stratosphere will have a sinking tendency and form a source for the onset of large-scale circulations. This source is at least as important as the convection acting from the sea surface downward in polar and subpolar seas. These inflows are also important because of another reason. Mixing of the water masses transported by these currents with tropospheric water masses above causes an interaction between the oceanic troposphere and stratosphere, and direct exchange between the two main layers of the ocean is probably restricted to these places. While the Atlantic and Indian Oceans are affected by subtropical adjacent seas, the outflow from which considerably intensifies the circulation in the uppermost part of the oceanic stratosphere, there are no adjacent seas of this type connected with the Pacific. Consequently, the Pacific lacks the large meridional contrast in salinity of the deeper layers which provides the driving force for a stronger circulation. Chapter XXI The Main Features of the General Oceanic Circulation and Their Physical Exploration 1. The Oceanic Circulation in the Atlantic The results obtained by numerous expeditions in the Atlantic allow a complete and, in itself, closed picture to be built up of the tropospheric and stratospheric oceanic circulations. Knowledge of the circulation systems in the other oceans is not so precise, but the conditions in them should not be so very different as is confirmed clearly by the available observations. An attempt has been made in Fig. 331 to picture the entire circulation system of the Atlantic in a somewhat schematic meridional section in order to summarize its main characteristics. This representation applies mainly to the western side. It can be seen that the main water movements are confined to an extremely thin layer. The circular representation shows especially the enormous horizontal extent of the oceanic troposphere. Its vertical thickness is, however, small so that in spite of the large vertical exaggeration in scale it is difficult to picture the internal circulation properly in the figure. All the main currents and singular points of the current system of the sea surface are indicated at the edge of the figure. It should be remembered that all extensive ocean currents are mainly surface currents and belong essentially to the oceanic troposphere; they extend down to the water masses of the oceanic stratosphere in only a few places and to a limited extent. This is especially so in the tropics and the subtropics. As compared with the large horizontal extent of the oceanic troposphere the source regions for the stratospheric water types appear small, nevertheless they remain the regions of origin for the water movements inside the extended space of the oceanic stratosphere. In these regions also the forces must be contained for a renewal of the stratospheric waters and their movements. The effect of the European Mediterranean which can be regarded as a lateral intrusion from the east appears of no less impor- tance. The small arrows in the diagram indicate the direction of spread of the individual water types; the current-like spread is thereby mostly indicated by full arrows while convectional spread is shown by wavy arrows. The figure shows only the meridional components of the water movement and deals only with mean conditions. The zonal components surpass by far the meridional ones especially in the southern part of the South Atlantic and in middle latitudes in the North Atlantic. The characteristic asymmetry of the Atlantic circulation and the great importance of the Antarctic for the stratification and movement of the water masses throughout the entire Atlantic 694 Main Features of General Oceanic Circulation and their Physical Exploration 695 tZ*^ QOO^O'' >o\ dWe' rqerice Equatorial counter current Equotoriol divergence South equo/o Current i^e/o '^^-e^ Fig. 33 1. Meridional vertical cross-section from pole to pole through the Atlantic Ocean. Schematic representation of the tropospheric and stratospheric oceanic circulation. — ocean bottom, , boundary layer between tropo- and strato- sphere from Northern polar front to Southern polar front. Salinity distribution: Fw>n>1 , >36-0%o, B>i?l , 36-0-34-9?4, ^^^ , 34-9-34-6%„, 1^^^ , <34-6%o, — >-, current-form spreading,-'- ,,', convection-like spreading and convection-like sinking, exaggeration about 1 :400. stand out particularly in this diagram. In the north the effects are more sub-arctic due to the bottom topography, but their influence on the stratospheric water movements is still extremely important. If we ask for the driving forces of the stratospheric oceanic circulation it must be stated that only differences in the thermo-haline structure of the water masses can be the cause for these circulations, and these contrasts can only be maintained by atmos- pheric influences affecting the regions north of the oceanic polar fronts and are so regenerated again and again. Thermodynamic machines of this type can only do work when the compressions of the medium set into motion occur at a lower pressure than the expansions (see p. 489 and following pages). The water in the upper circulation branch is set in motion from a region of smaller to a region of greater density, and in 696 Main Features of General Oceanic Circulation and their Physical Exploration the opposite direction in the lower branch. The meridional density sections show that this condition is satisfied and the dynamic evaluation of the observational data has given proof of the internal forces acting in the pressure field and resulting from the three-dimensional mass structure. In the troposphere the thermo-haline circulation in a meridional direction is less important as compared with the effects of the wind. The air currents therefore set the characteristic pattern for the circulation here and determine its more zonal direction. The western and eastern boundaries set by the continents to the oceans, due to the surface accumulation of water (piling up; Anstau), give rise to gradient currents which besides the wind drift determine the character of the tropospheric oceanic circulation. WUst chose a different type of representation to show the oceanic circulation. The surface currents and the deep-sea circulation of the Atlantic were shown in form of a block-diagram in order to arrive at a three-dimensional representation and to elucidate thereby the internal completeness of the circulations (Fig. 332). This survey of the oceanic circulation teaches that the basic causes of the entire oceanic circulation lie in the atmosphere. They are due partly to the vv/>7^ which transfers energy to the water, and partly due to climatic effects on the water masses, especially in polar and subpolar oceanic regions. These then give rise in the first place to the water movements in the deep layers. 2. Summary of Present Individual Theories and the Prospects of a Comprehensive Theory of the General Circulation Including the Deep Layers The existing theory of the wind-driven circulation in closed oceanic basins has been found applicable to individual parts of the ocean, but a comprehensive theory of the wind-driven circulation covering all oceanic parts is so far still missing. It has already been pointed out (p. 583 et seq.) that the highest advanced theory of Munk and Carrier (1950, led at least qualitatively to very reasonable results. Criticism has been expressed primarily on account of the high value of the coefficient of lateral eddy viscosity required in order to explain the intense currents along western coasts. Morgan (1956) in attempts to overcome this drawback has examined the necessity of the inclusion of the lateral eddy viscosity for balancing the wind torque on the water surface. The ocean can be represented on a different model from those used previously. In this it is divided into a northern and a southern part, and attention is paid only to the southern one which in itself is subdivided into an interior region and a boundary region adjacent to the western shore. Figure 333 shows these three oceanic subdivisions and the boundaries between them. The figure contains a typical stream line of the circulation, most of which or perhaps all of the stream lines can be expected to pass through all three regions. The equations of motion given for spherical co-ordinates are formally integrated over the depth both for a homogeneous ocean and for a two- layered ocean. From these the approximate equations are derived applicable to the interior region /^ of the currents, that is, to a region sufficiently remote from any coast. They show that all terms which are non-linear in the velocity components as well as the terms giving the contributions of the lateral eddy viscosity are negligibly small there. This is the same result as obtained from the Sverdrup solution. Wind and Coriolis forces are the principal forces in this region. For the boundary region /,, Main Features of General Oceanic Circulation and their Physical Exploration 697 3 "a 0) o o • — r- C 60 '-direction. As a consequence, the problem is thus one-dimensional and the equations are considerably simplified. This gives two equations which permit a study of the reponse of the physical sea level and the internal boundary surface to the variable shearing stress of the wind. It is interesting from the mathematical point of view that these equations can be combined to give an equation with a single variable without raising the order. It is of the fourth order and has the form 1 R T' A k Rxxxt 72 ^xttt P^xt "T HA k Rxx 7^ ^tt ^= ~7~ • (XXI. 1) R has a fixed numerical relationship to the displacement of the sea surface and the internal boundary surface and can have two values, 7?^ and i?2- In the same way k has the numerical values ki and k^ corresponding to the values R^ and Ro. Moreover, ^^ (= gDJf^) where Dg is the equilibrium thickness of the lower layer and A is a quantity termed by Rossby the "deformation radius". The solution of the differential equation (XXII. 1) gives the "normal values of motion" (equation of normal modes) and makes it possible to determine all the desired quantities of the model such as the displacement of the boundary surfaces and the velocity in the different layers. This equation can be used to derive the free waves of the system and their depen- dence on the dimensions of the system, when the wind stress is omitted in the equation. A knowledge of the free waves is of considerable value because of its great importance, since in view of resonance phenomena they may have considerable influence on the forced waves which are generated by the action of the wind. Assuming a normal mode of the form Ri = Si sin (Ix + ojit ) (/ = 1 , 2) ; (XXI.2) that is, in form of a wave progressing in the negative x-direction with a frequency coi, then the equation (XXII. 1) transforms into (/)' fl\f with [j-^ +(l+^.)_!+^ = 0 (XXI.3) The solution of this algebraic equation (three positive roots) gives the frequency 8000 Plate 2. Schemaiically simplified world-map of ocean dcpih. (.S