DUDLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOl MONTPREY CA 93943-5101 LIBRARY U.S. NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA THE EFFECT OF WIND UPON THE MIXED-LAYER. DEPTH * * Vlr * * Jack E. Geary -'- -.'< >V ■>'■: V>- Jack E. Geary 1961 AVAL POSTGRADUATE SCHOOL Degree: Master of Science in Meteoi Classification: Thesis: UNCLASSIFIED Abstract: Unclassified Title of Thesis: Unclassified Contains no proprietary information. . FECT OF WIND UPON TH] . •.: D-LAYER DEPTH by Jack E, Geary Lieutenant, United States Navy ►mitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN I CEOROLOGY United States Naval Postgraduate School . California 1961 Library U. S. Naval Postgraduate School Monterey, ( lalifornia EFFECT OF WIND UPON ri\lE MIXED -LAYER DEFT J Jack E. Geary This work is accepted as fulfilling § the thesis requirements for the degree of MASTER OF SCIENCE IN METEOROLOGY from the United States Naval Postgraduate School ABSTRACT The general problem of the mixed-layer depth ot the ocean's surface layers is discussed. Wind mixing and its contribution to mixed- layer depth at ocean station Papa during the year 1958 is evaluated and discussed. A model of mixing by wind-generated wave motion is developed and is tested along with two other models; and the results are discussed. The author wishes to express his appreciation for the advice and assistance given by Profe sot C H. Jung of the U. S. Naval Post- graduate School in this investigation, and also his thanks to the Pacific Oceanic Group for their kindness in making available the necessary oceanographic data. 11 TABLE OF CONTENTS Section Title Page 1. Introduction 1 2. The mechanism of wind induced mixing 3 3. The relationship of wind speed to mixed-layer depth 4 4. Time lag of the mixed layer produced by a given wind 7 5. Random fluctuations of MLD 8 6. Theoretical models of wind mixing 9 7. Fit of mixing models to observed data 11 3. Variation of observed data from computed values 1 9. Conclusions and recommendations for future research 14 10. Bibliography 2 Appendix I. Geary's Model 23 11. Laevastu's Model 26 III. Neumann1 s Model 28 in LIST OF ILLUSTRATIONS Figure Pcige 1. Time Series of Wind and MLD, Ocean Station Papa, June, 1958 16 2. Wind and MLD, Ocean Station Papa, June, through September, 1958 17 3. The Effects of Convection on MLD 18 4. Lag of MLD from Wind 19 5. Mean Hourly Fluctuation of MLD 20 6. Comparison of the Mixing Equations 21 Table 1. Geary's MLD Values 25 2. Lacvastu's MLD Values 27 3. Modified Laevastu MLD Values 27 4. Neumann's Depth of Negligible Uave Motion 28 IV TABLE OF SYMBOLS AND ABBREVIATIONS A wave amplitude D depth to which mixing takes place ni E total energy per unit area of wave surface F fetch length acceleration of gravity .iu significant wave height k wave number MLD mixed-layer depth Ta air temperature T s«_a surface temperature t, duration time of the wind h UQ potent Lai energy per unit area of wave surfaces J_ same as U_ , measured at depth z V wind speed z depth below sea surface P density of sea water CT~ angular wave frequency 1. introduction The [Jiieuometi- mi nag of the surface wat< s of much of the world's ocean area has been studied for many years. Increasing naval operational interest in the deptii of the mixed layer now demands that the groundwork established by many oceanographic investigators be built upon, and that intensive effort be directed toward the goal - a usable forecast of mixed-layer depth over broad expanses of the seas. Mixed-layer depth is the depth below the water surface to which ..lir.itig has established an essentially isothermal temperature distri- bution. The lower boundary of the mixed layer is the thermocline, normally a thin layer or interface of large, negative vertical temper- ature gradient. Large variations of mixed-layer depth occur in space and time. Observation indicates that a full spectrum of time variation ranging fr< m an annual cycle to short term fluctuations of a few minutes duration is to be expected at a fixed geographic location. The ann c^cle is at present the most regular fluctuation of the mixed-layer depth to be identified;, and is in general the largest in magnitude. The annual trend of the mixed layer is a sinusoidal wave which appears to be most closely related to the heat balance of the surface water layers, i.e. mixed-layer depth is inversely proportional to the sta- bility of tne surface water layers. This is not to say that other periodic fluctuations of mixed-layer depth such as lunar -tidal or diurnal heating and cooling are not present, but paucity of data and the magnitude of apparently random fluctuations makes their identifica- tion difficult. -.istantaneous values of mixed-layer depra may be looked upon as the result of various perturbations superimposed upon the mean annual The most significant perturbations appear to be the integral i of meteorological factors operating to alter the sua! Llity of the. surface • it r layers. Winds, evapor; cooling, albedo, moisture coutent of the air, and xnsolatj_on arc a few of these iiieteoro logical tors. Added to the effects of the meterological perturbations arc the a] pan cit L; 'andom motions of internal waves along tuo. thermocline. Geographical variations of mixed-Layer depth are associated with the bathymetry of the ocean basins, the advective and mixing effects of the permanent currents and ocean tides, and the climatological regimes. The enormous problems of data collection and observational density militate against anything but qualitative estimates of the effects of space variation upon mixed-layer depth at the present time. i e purpose of this paper is to evaluate the meteorological factor of i Ln its contribution to mixing of the surface waters at ocean station Papa (50W - 145W) for the year 1958. 2.1 C air-sea Later face by means of wind-driven lotion a cu; ClZJ • It is postulated that . >j tress ice motion o£ the surface layers of watei , giving rise to :s untiL stability tl ther- ms particle notion to such an bat . q i ll.br ium between tl stabilizing forces i3 reached, and ■ . . • proceed no further. I te amount of st zs m to depen a the i i of Lnd, the stability of the air, and the nature of the air-s interface [_2_i • The stability of the I :] (the Limit of . depends up I ■ f '.ress C^Lj » auc; tne beat balance in the mi: layer . Convective mixing is the process by which evaporation and loss of n.^at to tne atmosphere causes the surface water layers to become unstable, resulting in overturning and mixing, l.ind st-'. .a iactor in the flux of water vapor and sensible heat across the a^r-sea inter- face. Convective mixing is, in general, most efficient when h_at losses from the surface mixed layer are greatest. In order to isolate, insofar as possible, the effect of wind- driven mechanical ., Lng for tne purpose of study Ln this section, date from the months of maximum sea-sui .- _ . ... ti records of the pacific Oceanic Group for ocean st Papa, L958, ac s of this process. 3. Ehe rel sh Lp o S] . . '• La; pth .. di ta wei organized : ito time series rai ' nd \ L .<.' spt i, an to scatter 'ams oj the sai variables. S motion depends on duration ol wind rather than instantaneous values, the mean of the eight available daily wind observations was taken as the representative wind for each day. The selection of a representative mixed-layer depth tor a day was a greater problem. In general, only two bathythermograph observations, one at 0200 and the Other at 17uu Greenwich time, were available. In addition, as will be discussed later, considerable fluctuation of MLD about the daily mean value is observed, and there is no certainty that the mean of the 0200 and 1700 observations would be truly representative of the mean for the aay. Therefore, the maximum observed MLD for the day was used. MLD was taken to be the depth below the sea surface at which the tempers ture trace of the wat^r ceased to be isothermal; in general this is the top of the thermocline. This definition permits a zero value for MLD. figure 1 is a time series of mean daily wind speed and maximum observed daily MLD at ocean station Papa for the month of June, 1958. The two curves are very similar, increasing and decreasing together. Close inspection reveals that the MLD curve lags behind the wind speed curve by about 24 hours, and that most fluctuations of wind speed are followed the next day by a similar fluctuation o£ MLD. Another interest- in; characteristic of the two curves is that not only does MLD increase with increasing wind speed, it also tends toward zero as wind speed decreases. Figure 2 is a scatter diagram obtained by plotting maximum daily MLD against the previous day's mean wind speed for the months of June, July, August, and September, 195b. Correlation coefficients for the four months were found to be 0.94, 0.76, 0.41, and 0.68, respectively. Disregarding for the moment the August points, it is seen that the remaining points fall into two groups; those of June and July; and those of September. The August points can be divided into two groups also; those associated with the June and July group, and those assoc- iated with the September group. The separation of the points into two groups is in accord with the two different thermal regimes which are represented. The June and July points represent a period where the water is being actively heated, while the September points represent a period of beginning heat loss by the water; the August points are transitional between the two periods. The basis for this division is that the observed sea surface temperature reaches its maximum for the year in late July and begins its annual cooling trend in mid-August. This difference in regimes may explain in part why the correlation coefficients for June and July are so much larger than the August and September coefficients. In all four of the months increasing winds drive the mixed layer deeper, but only under conditions where excess heat is being supplied is there a tendency for the mixed layer to approach zero under decreasing wind conditions. Linear regression equations were calculated for these scatter diagrams and found to be: MLD = 1.14 V (June) (MLD in meters, V in knots) MLD = 0.94 V : 1.2 (July) MLD = 1.07 V + 26.1, (Sept) respectively. The small difference between the June and July slopes is attributed to the sligl.Lly more stable condition of the July water when heating was at a maximum. The fact that mixing of the surface water can depend not only upon wind speed, but upon convection as well, is illustrated in figure 3. This figure is a scatter diagram of the same variables shown in figure 2, but for the month of November, 1958. Here a correlation coefficient of -0.58 is obtained. In November there is rapid cooling of the surface water, the temperature decreasing some four and one- half degrees fahrenheit in two weeks. In this autumn month, clearly, convection has completely overridden the effects of wind mixing alone, and is independent of wind speed. 4. Time lag of the mixed layer produced by a given wind As pointed out in section 3, changes of MLD lag changes of wind speed. Figure 4 is a time series of wind and MLD observations taken at ocean station Papa on 16 to 19 June, 1958. MLD observations were taken hourly, while wind observations were taken at three-hour intervals. Both curves were smoothed by three-hour overlapping sums, and the MLD curve is plotted with an 18-hour lag from the wind speed observations to obtain good agreement of the fluctuations on both curves. Actually the best agreement would have been obtained by a 12-hour lag fou the shallow MLD (smaller wind speed) with a larger lag for the deeper MLD values. A lag between observed wind speed and the resulting mixing of the surface water is in accord with the theory of wave generation, which requires a minimum duration time for the wind to have blown in order to produce waves of a certain size. In general, the required duration time of the wind increases with increasing size of wave produced [_4_/ > DY analogy, the lag of MLD from wind should increase with increasing wind and mixed-layer depth. 5. random fluctuations of MLD The depth of the mixed layer was observed to vary several meters in a random fashion between hourly measurements. The magnitude of the fluctuations could not be made to correlate with any observed meteoro- logical parameter, but the fluctuations did appear to increase in size with depth. Figure 5 shows the mean hourly fluctuation of ten series of observations plotted as a function of the mean mixed-layer depth of ^.ach scries. The individual series contained from 24 to 48 bathythermo- graph observations, and were spaced at approximately monthly intervals over the year. Schule |_5J and others have observed similar short-term fluctuations of I1LD at other localities and conclude that they are caused by internal waves along the thermocline. The importance of the random f luctuationu is two- fold. A single measurement of MLD will not necessarily be representative of the mean IiLD for any given day. A series of observations should be taken each day when verifying computed values of mixed-layer depth. The oti.er important aspect of internal wave motion along the thermocline is that internal waves may provide one mechanism of vertical mixing. Ball {3ZI describes an experiment in which internal waves were induced along the density discontinuity between two water layers. The waves were observed to become sharp-crested and filaments of the denser water were drawn out from the sharp wave crests and mixed into the u.>per layer. The reverse, mixing downwards, was not observed to occur. 6. Theoretical models of wind mixing At the start of the present investigation a search of the literature produced only one theoretical model to explain the mechanism of \^ind mixing. Hunk and Anderson 0_J developed this model incorporating the concept of vertical Austausch coefficients in the ocean, solving a system of five equations to obtain the relationship of wind speed to the depth of the mixed layer; they obtained results to within an order of magni- tude of observed data, and concluded that convective mixing was at lec si; as important as wind mixing. This is not surprising in view of Laevastu's observation Cl~2 that the vertical Austausch coefficient varies by more than three orders of magnitude with space and time in the oceans and is at present neither measureable nor predictable. The author approached the problem of wind mixing from the stand- point of particle motion produced by wind-driven waves. Essentially, what is required is a way to link wind speed to the motion of water particles in the surface water layers resulting from wind-produced waves. having accomplished this, a limit remains to be set on the effectiveness of the particle motion to produce mixing. The relation- ship of wind speed to the various surface wave parameters (significant wave height, period, and length) was obtained using Neumann's spectrum S~J\ _j t and the subsurface particle motion was in turn calculated from the theory of simple Airy waves. The maximum limit of mixing due to particle motion was assumed to be the depth were the buoyant force of the denser water below the thermocline exactly opposed the dowiward force of the denser particles in circular orbit. An equation (hereafter referred to as Geary's model) of the following form was derived: The depth at which equilibrium will be established depends upon the stability of the thermocline (tiie density difference across the tnermoclinc) and upon the characteristics of the surface wave produci g tiie motion. The necessary wave characteristics can in turn be obtained as a function of wind speed alone, assuming fully-developed seas. During the present investigation the author became aware of the fact that Laevastu j__l_J had approached the problem of wind mixing from a similar viewpoint. The essential difference between the two models is that Laevastu set an arbitrary limit to the effectiveness of subsurface particle motion in accomplishing mixing. .lis limit was the depth where the diameter of the particle orbits was ten centimeters or less. Neumann [_ 7 J , in an extension of his surface wave spectrum to subsurface motion, derived a relationship giving the ratio of potential energy at some depth to that of tiie surface waves, and proposed chat at the depth where the ratio was five percent or less, subsurface' pa.:ticle motion could be considered negligible. Neumann did not apply this concept to the problem of mixed-layer depth, but the level of wave energy present at any depth must have some influence on the work done in mi;.ing at that depth. Gonsecpaently, Neumann's relationship was evaluated along with the models of Laevastu and Geary. 10 7. Fit of mixing models to observed data The appendices contain evaluairoas ot the three mixing models for is wind speeds. Curves 1, 2, 3, and 4 of figure 6 are plots of tables 1, 2, 3, and 4 respectively, from the appendices. The stippled i :ea represents the field oi scatter- of the observed d ta for the months oi June, July and the first half of August, 1958, some 76 obser- vations in all. Curve 2 is the Lcevastu equation for iiLD using nis equation ML, pp. 7Cfj tor computing signifies at wave height. it can be seen that curve 2 forms an upper limit on tac observed scatter of points. Curve 3 is the Laevastu equation lor MLD, but using {4j for the calcu- lation of significant wave height. it is evident that the method of calculating surface wave characteristics has an important effect upon the values obtained for mixed-layer depth. Curve 1 is Geary's equation, and it is in surprisingly good agreement with curve 3. An explanation for this agreement cannot be suggested at the present time. Curve 4 is the KLD equation based on Neumann's ratio of potential energies, and it can be seen that this curve fits the observed data better than the others. It may well be that a calculation of energy at depth is a better measure of mixing efficiency than are approaches using particle acceleration or the geometry of motion. Of some interest is the fact that all of the curves deviate more from the trend of the observed scatter points at Lovlz. wind speeds, which may indicate deficiencies at low wind speeds in the wave generation equations. Munk and others have proposed a "critical wind speed" (about 13 knots) , ..bove and below which wave generation by wind proceeds differently. 11 0. riation of observed data from computed /alues Factors operating to cause variation of pbserved wind and MLD data from theoretical and empirical relationships are of three kinds. First arc those that have to do with c'.ata collection and processing. Bathythermograph observations at ocean statioi Papa were not actually taken at a fixed location, but over an area of many square miles. The basic grid is ten nautical miles on a side, and observations are reported from several different grid positions in the course of a month. The best accuracy to which the bathythermograph lata could be ..cad is on the order of - one meter. It is entirely possible that the differ- ences in geographical location at which the observations \vorc taken could account tor the scatter of observed data from computed vales. A second factor, concerning only the mi: Lng models, stems from oversimplification oi the vertical mixing process and from imperfections in wind-produced wave theory. The author's assumption (appendix I) of a two-layer system is an oversimplification of actually existing density gradients. That systematic part pf the variation of the mix- ing models from observed data may be ascribed to: 1) wind may not build waves exactly according to the theory used: 2) use of a 24-hour mean wind instead of an integrated wind could produce a systematic error of the tyre observed between the scatter points and curves on figure 4. The third factor is internal wave motion. Random hourly variation of MLD was found to exist (section 5) and was calculated to be an increas- ing function of mixed-layer depth. Mean hourly variation of MLD increased from nearly three meters at shallow depths to about five or s?:: meters at large MLD values. Also, the standard deviation of MLD from the 12 monthly mean uas found to be a function of \.iud speed, depth, and h c balance. It is semi that internal wave motion, in addition to observational factors, could account for all of the random scatter of the plotted points. 9. Conclusions and recommendations lor future research Wind is an important factor in determining the depth to which the surface waters arc mixed. The mechanism of mixing b, driven Wi motion must exist throughout the seasons, but is dominant only when the raixed-layer depth tends toward zero in the absence of mixing, in gen- eral, this coi.dition will occur during the surfer months when heating of the surface water is maximum, and convective mixing is at a minimum. Equations incorporating the subsurface particle motio:. of wind waves cm be called upon to hindcast much of the observed daily and longer -interval changes in mixed-layer depth at ocean station Papa during the summer of 1958. Although turbulent mixing due to the vertical velocity shear of drift currents also must cake place, this effect seems to be masked by wave mixing, since the equations for wave mixing generally give values of MLD greater than those actually observed at wind speeds greater than about 12 knots. It is possible, however, that mixing by drift currents accounts for the larger iiLD v ..lues observer at shallower depths, i.e. for wind speeds less than 12 knots- r so. The models of wave mixing should be considered <;s indices of mixing efficiency, not as precise physical descriptions of the vertical mixing process. ..'o one has been able to describe the exact process, but the wave mixing approach appears to be the most promising thus far advanced. It is difficult to say from one test which model of wave mixing as presently written, if any, will prove to be useful in a forecasting scheme. Laevasta's equations give the closest approximation to an upper limit on the observed mixing. Neumann's energy ratio fits the observed distribu- tion better than the other equations. however Geary's model takes into 14 account thermocline stability, a necessary parameter Li othi conditions locations are ever to be investigate The time lag between observed wind and resulting mixing is a provident phenomenon from a forecasting standpoint, and must be considered when verifying calculated mixed-layer depths. dourly and other short-term random fluctuations of MLD about a mean value prevent the forecasting of exact, instantaneous mixed-layer depths, and will have to be considered for forecasting and investigative pui poses . The most important factor requiring investigation is the problem of convective mixing. At present, this process has no model to des- cribe its effects, yet it appears to be the singula! mechanism domina- ting the depth to which mixing takes place during most of the annual cycle. The use of a high-speed computer for a running computation of the heat balance in the surface waters may prove to be of valuable assistance in describing short-term fluctuations of mixed-layer depth. Future research should also be applied to improving and testing models of wind mixing. 15 00 • v.^ P ON gH o ■■cm « •O 0} (3 c CO ? »"3 •c o • •^ 05 > P cc «H Ph o -'- o CO Q T-l vn ^-1 +J ON u a vo r- oj +> •••-I CO C/2 • o ft) A c E «3 0 •h ca h> *«$ — xn 1.6 00 fit s o •p & in x: o % P •"3 (0 P (X, p 'O CO co P co c o u E O P 0) C/2 & & P C (i o « • • a P at P P •"3 0^ 0 o •en to h CVJ 4) VA •*> «J CM 0) 2 o VfN • • • o o & « • C\J" C\iJ £1 S8 (s^ou:>i) WW m • • • • • • • • ►VP, 17 e o 0 o 0 0 0 e o 0 o -■CO . » o u s 0 O a ° 0 • 4> c £ p ° o o «M H ■•NO £-■ o 'XT\ I I I I o o o o \rc cfr r> ^ cm 18 20 knots £2 10 1 8 1 1 1 00 12 17 June 00 12 18 June 00 Figure k Lag of MLD from Wind 19 120 100 80 160 40 20 I 3' I f5 «6 * I •5 '6 Mean Fluctuation Figure 5 (m) Mean Hourly Fluctuation of MLD ■ ua l-Ge g ry 2-Leevestu 3-Modified Laevastu ^-Neumann Observed Values Figure 6 Comparison of Mixing Equations Xl 30 2i u U 0) 2£ tS e c £ 15. 10 1 1 1 10 15 20 Wind (knots) * 3b 21 BIBLIOGRAPHY 1. Laevastu, T., factors Affecting the Temperature of the Sui ace yer of the Sea, Societas Sci« ii. run Fennica, Connnentationes Physico-Mathetnaticae XXVI, IS 60. 2. Fleagle, R. G., Note on the effect of nir-sea temperature cilJfccence on. wave generation, Trans. Amer . Geo;;hys. Union, vol.37, no. 3, pp. 275-277, 1956. 3. Hunk, W. H. and E. R. Anderson, Notes on a theory of the ther- mocline, Journal Marine Research, vol.7, no. 3, pp. 276-295, 1943. 4. i'ierson, U. J., Jr., G. Neumann, and 1C. W. James, Practical Methods for Observing and Forecasting Ocean Waves by Means of Wave Spectrum and Statistics, U. S. uydrographic Office Publication no. 603, 1955. 5. Scnul^:, J. J., Jr., Effects of weather upon the thermal structure of the ocuan, U. S. tlydrographic Office Misc. Publication, 1536J, 1952. 6. Ball, F. K. , Control of inversion height by heating, Quart. J. .;. M t. Sue, vol. 86, no. 370, pp. 483-494, 1960. /. Jeumana, G., On wind generated wave motion at subsurface levels, ans. Amer. Geophys, Union, vol. 36, no. 6, pp. 905-V92, U55. 22 Al ; I GEARY1 S MOD] Mechanical i Df water particles across the t her mo c line must Lted when the vertical forces of particle motion are oppoi i buo .it force of the dens I water below the taurmoclinc. Based upon tills consideration it is possible to establish a simple model for mechanical mixing by wave with the following assumptions: 1) A homogeneous layer of water with density yO (the nixed layer) overlies homogeneous water with a greater density, jO . The difference in density between the two layei ed solely by their temperature difference. 2) Subsurface particle motion is in accordance with simple Airy wav< try for deep water, i.e, the particles move Ln circular or'. Lth Their diameters decreasii onentially with increasing depth. 3) Penetration of the thermocline interface stops when the . tical forces of a particle in motion are exactly balanced buoyant force. The balance of forces in such a system are given by: jo 9 +/>***- e~*'*/>'S (2) Where^OO is the gravity force, yOMO Cr is the rimum force due to the vertical component of acceleration of the particle in orbital motion, and /O Q is the buoyant force . The force balance cf (2) is correct, but it must be understood ■2. -/?2 that the acceleration term sH O" c3 applies to laminar 23 conditions of • rid is used here only as an approximation to the magnitude of ver-ical particle acceleration in what must be actually turbulen- n. The laminar acceleration term is utilized in an attempt to resolve in a general way the difficulties of this simple, mechanistic approach. Lquation I in be evaluated implicitly for depth as a function of wind speed alone ii -tability of the thermocline is assumed constant, i.e., P P = constant, for all depths and the various P surface wave characteristics are calculated according to Neumann's spectrum of wave generation by wind. The stability tersru P ~~P , was evaluated from the data at ocean P station Papa for June and July of 1958, and was found to be approx- imately between the lin )f 0.0001 and 0.0004. Using £j\~^\ to obtain the wave charactf- tics T" , t— , and A/ for fully developed seas, table 1. gives the computed values of MLD for various windspeeds and two stability values- Expressing ( 2 ) as (3) ~y *p /^ a sample calculation follows: for a given windspeed / , Z_,/7are speci fied as above. Assume/0 r = 0.0004, A = ±1 j CT= OH , k= 4^ I solve ( 3 ) for :2 . which represents the MLD. 24 10 12 14 16 1C 20 22 24 26 Table 1 Geary s MLD V.:UiuS Wincispe d (] to eel-Layer Depth (meters) ^p . = 0.0001 ^H^'- *-* 0.0 8.8 6.9 12.8 10.0 17.6 14. 0 23.3 18.4 29.7 23.6 36.3 29.4 45.0 36.0 54.0 43.0 63.5 51.0 73.7 59.4 25 APPENDIX II LAEVASTU'S MODEL LaevasLu £0 a tentative formula for the depth of mixing by waves; unfortunately, he does not give the derivation, but states that he arrives at rhe expression by using, the relations for trochoid waves for computing the velocities of water particles at various wave height and at various depths, the depths given by Neumann Q5 J , where the total wave energy ha? decreased to five percent of its value at the sea surface and assuming that the mixing by waves is negligible at approximately a depth where the diameter of the orbital paths is smaller than 10 centimeters. The formula /) = /Z.5 Nk . •*-VT>l s (4) Laevastu uses the following equation for computing significant wave height: "s = Q.0008 V'ESO+CK-Ta)] < /=- s< j>6 / (5) It is interesting to note that (5) incorporates the sea-air tempera- ture difference parameter, while other formulas for significant wave height do not. Depth of the mixed layer calculated from (4) and (5) is given in table 2, assuming Tw - Ta = 0, F = 100 kilometers, and t - 24 hours. a MLD obtained from formula (4) is greatly influenced by the equa- tion used to calcul.. ant wave height. As an illustration of this point, table 3 give,;. MLD for various wind speeds using formula (4), but with significant wave height computed according to Neumann's equation, 26 Table 2 Lac 3 MLD Values Wind speed Mixed-Layer Depth l5 sec _ i kno t s 3 5.3 3.9 l: 7.3 6.6 5 9.7 9.9 7 . .6 17.6 8 15.5 22.3 9 17.5 26.3 10 19.4 31.3 12 23.3 42.3 Table 3 Modified Laevastu MLD Values Wind s^eed uixcd-Layer Depth s sec-1 knots meters 3 5. 1.4 7.3 2.3 5 9.7 5.0 7 13.6 15.5 16,0 17.5 21.5 10 19.4 17.9 12 23.3 44.1 27 )IX III ! --pectrum of wav< I ration by win L J on for the ratio of average potential .rage potential energy of the sur. s as a functi ■ ;nd speed and depth: a =fj+4ja» +£g] e'4^. I I Uo i- ' - - ~ - (6) e vertical displacement of a particle at id by substituting the following I ^ ~ 9P ' (7) S (3) ann cons the ratio of £ becomes equal to 0.05, e notion i Solving (6) for the condition where u -^- is ble 4 is obtaii Table 4 r Depth of negligible Wave Motion Ln . _^_£ hixed-Layer Depth ■•s sec " cers 2 0.42 5 5.7 10 22.9 15 52 20 92 25 143 23