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SOME OBSERVATIONS OF SHORT-TERM HEAT THROUGH THE SURFACE LAYERS OF TIIE TRANSFER OCEAN’ David H. Shonting U.S. Naval Undcrwatcr Ordnance Station, Newport, Rhode Island ABSTRACT Short-term diurnal heating and cooling data from the Tongue of the Ocean, Bahamas, are prescntcd and discussed. The method of acquiring the time series data and the method of presentation arc rcvicwcd. A 7&y diurnal heating study is discussed in relation to the vertical diffusion of heat and its relation to meteorological parameters. Results of a 24-hr diurnal heating cxpcrimcnt arc prescntcd, and thermal eddy diffusion coefficients arc dcrivcd and compared with those obtainccl by other methocls. An approximatc heat budget calculation is performed whcrcby cssentinlly all of the energy input and output arc accounted for, utilizing the diurnal time scrics tcmpcraturc data. INTRODUCTION The data prescntcd here were acquired during an oceanographic sampling program conducted by tho U.S. Naval Underwater Ordnance Station, the University of Miami, and the U.S. Naval Oceanographic Office. The study was in the central part of the Tongue of the Ocean, Bahamas. The purpose of these studies was to dcterminc the time and spatial variability of tempcraturc, salinity, density, and the speed of sound and to relate the observed variability with circulation and meteorological environment. The data present a definitive picture of the thermal response of the surface layer of the subtropical ocean to diurnal solar heating with varying wind conditions. It is the purpose of this study to relate the measurements to a simple theory of diffusion of heat in the surface layers and to consider an energy budget for a specific 24-hr period. In general, the temperature data were obtained under meteorological conditions most favorable for the observation of diurnal heating and for portraying vertical Consequently, the clearly heat diffusion. defined isotherm variations presented arc probably not representative of average spring subtropical diurnal heating conditions, since calm winds, clear skies, and low sea state essential for observing this phenomenon arc not usually present. However, the temperature variations and heat budget probably present a realistic picture of the magnitudes of tho various parameters pertinent to the short-term exchange oE energy at the air-sea interface. OBSERVATIONS The first series of data was collected aboard the USS San Pablo of the U.S. Naval Oceanographic Office while on anchor station in the Tongue of the Ocean (24”40’ N lat, 77’35’ W long) during the period 20-27 May 1962. The data include water temperatures at the surface and at 40 m, measured with protected reversing thermometers during Nanscn casts taken every 3 hr, surface air temperatures obtained with a standard mercurial thermometer, and wind speeds recorded with a Bendix Friez Acrovane mounted on the ship’s bridge. 1 The author wishes to thank the U.S. Naval The second and third series of obscrvaOceanographic Office For the USC of the USS Sun tions included ba thy thermograph records Pablo oceanographic clata taken in May 1962 and the officers and crew of the RV II. C. HazJes for supplemented by air temperatures and their assistance in obtaining the time station clata wind speeds, obtained aboard the RV II. of February and March 1962. The review and C. Hayes under charter by the Naval Uncriticism offered by Dr. E. B. Kraus of the Woods derwater Ordnance Station from Marine IIolc Oceanographic Tnstitution is gratefully apby the This study was supported Acoustical Services of Miami, Florida. prcciatccl. Task Assignment of Naval Weapons Bureau These data were collected during two RU222EOO0/219-l/R004-03-01 to the U.S. Naval periods, 6-8 February 1962, while the H. C. Underwater Ordnance Station for oceanographic H~ZJZS was held by a three-point deep studies. 576 SHORT-TERM HEAT TRANSFER THROUGH ocean mooring system, and 13-14 March 1962, while the vessel was in a two-point bow-stern mooring. During both periods, the sampling was in the central part of the Tongue of the Ocean about 10 km east of Andros Island at about 24’24’ N lat, 77’36’ W long, where the water depth is about 1,500 m. The three-point mooring system held the vessel within a quasi-circular area of about 100 m radius as determined by Dccca “Hi Fix” navigation system. The two-point mooring used in March constrained the H. C. Hayes so that it exhibited an east-west swing of about 1.5km range. The actual speed of the vessel swinging about on these mooring systems was of the order of 1 cm set-I. A discussion of the mooring is given by Shonting, Cook, and Marcy ( 1963). The sampling interval for both series of observations was 15 min. The bathythermographs used had a depth range of 250 m an d were manufactured by Wallace and Tiernan Inc. Each was calibrated bcforc each set of observations at the Woods Hole Oceanographic Institution. Air tcmperatures and wind speed measurements aboard the E1. C. fInyes were made with a standard laboratory thermometer and a Bendix Friez portable anemometer, respectively. During the period 13-14 March, air temperature data were recorded aboard the RV Gercla of the University of Miami, which was moored about 4 km cast of the II. C. Hayes. To supplement the discussion of diurnal heating measurements of 13-14 March 1962, incident and reflected solar radiation data are prescntcd that were obtained aboard the USS San Pablo 10-12 March 1960 while on a three-point mooring in the Tongue of the Ocean at 24’35’ N lat, 77’34’ W long (see Magnitsky and French 1960). The radiation data were recorded with Epplcy pyrheliometcrs mounted on 10-m staffs outbo’ard of the vessel. PROCESSING THE BATHYTIIERMOGRAPH DATA Experience indicates that the bathythermograph is not an accurate thermometer. However, with careful use it does provide OCEAN SURFACE LAYERS 577 a reasonably accurate method of portraying the vertical tempcraturc gradient. During both series of bathythcrmograph and observatioas, that is, in February March 1962, there was a nearly isothermal mixed surface layer about 180 m deep. This depth is normal for the late winter in these waters. This layer provided a constant reference background from which to observe diurnal heating. The serial bathythermograms showed occasional sporadic variations in the temperature profiles, indicating a net translation of the temperaindicated turc axis. This displacement abrupt temperature variations in the isothermal layer of as much as 0.3C. However, simultaneous measurements made with protected thermometers indicated negligible changes (less than 0.05C) in the layer. Rather than discarding these displaced profiles, we applied a smoothing procedure to each series of observations. A depth was chosen where it was assumed the tcmpcrature variation was negligible, and the bathythermograph profiles were adjusted to intersect this point. This smoo,thing remo,ved all abrupt temperature fluctuations, but it did not erase the realistic surface layer temperature variations. The inherent danger in the above procedure is that any horizontal advection of an isothermal water layer with a slightly different tempcraturc would be masked. However, during the periods of measurcment, no surface temperature variation occurrcd that was not attributable to diurnal heating, and since this method of study is approximate, no attempt to differentiate between instrumentation error and advection error was made. In the February bathythcrmograph study, the profile was adjusted to read 24.3C at the 30-m level. The entire trace was essentially isothermal from about 20 to 160 m throughout the sampling period. During the March bathythcrmograph series, simultaneous scmicon tinuous measurcmcnts of temperature were obtained at four depths between 82 and 260 m on a moored taut wire thermistor array positioncd about 1.8 km from the 11. C. Hayes. 578 DAVID TEMPERATURE H. SHONTING t . FIG. 1. Time variation in the temperature file obtained with the bathythermograph. pro- The thermistor at the 82-m depth averaged 24.117C, with a standard deviation of 0.0376C for the entire 60 hr of data points over the 15 hr of bathythermograph observations. Therefore, the reference point of 24.1C at 82 m was used to position the temperature traces for this series. The temperature-depth-time sections were made from the adjusted temperature depth profiles by selecting the 0.2C increments from the individual profiles and plotting them with time. Fig. 1 shows the temperature-depth profiles as they varied at five different times during the February 1962 series. RESULTS The first series of data considered portrays an example of the thermal response of the surface water to the diurnal variation of solar radiation for a period of 7 days. Fig. 2 shows the diurnal march of air temperature ( thin solid line), surface temperature ( heavy solid line), and 40-m temperature ( dashed line) for the period 20-27 May 1962. Above the temperature plot is the wind speed record. The air temperature generally follows the diurnal variation of solar radiation. The sampling interval of 3 hr is so,mewhat longer than desired to observe the rapid rise and fall of the air temperature, but one can estimate the average time of maximum ( 1300 hours ) and minimum ( 0430 hours ) temperatures. The maximum heating for a 24-hr period occurs from 0500 to 1100 and on 21 May had a value of SC. The range of shipboard air temperature for the 7-day neriod is about 5.3C. There is annarent a net rise in the mean axis of the diurnal fluctuations for the 7 days of about 1C. It has been suggested by E. B. Kraus (personal communication) that these air temperature ranges may be biased toward high values, that is, shipboard measurements may give higher air temperatures than occur in the undisturbed air away from the vessel. The time plots of air temperature are to be considered mainly for phase comparison of the time variation of air temperature with respect to water temperature. The surface water temperature (actually measured about 1 m beneath the surface) reflects the diurnal heating with the average time of maximum heating occurring at 1400 hours. The surface water temperature ( and possibly the air temperature) tends to exhibit a double minimum, one at 2300 hours and the second at about 0700 hours. The average diurnal temperature variation for the period is about 0.9C. The apparent net heating of the air over the 7-day interval is reflected in a similar trend in the surface water temperature. The observed increase in surface water temperature for the 7-day period is about 1C. The 40-m temperatures fluctuate about l.lC during the 7-day interval. The oscillations themselves exhibit larger amplitude during the first 4 days than for the last 3 days. FIG. 2. Seven-clay records of wind speed, air temperature, surface temperature, 40-m temperature, and M2 tidal component sinusoidal curve ( lower curve). SHORT-TERM 6- HEAT TRANSFER THROUGH OCEAN LOCAL TIME -HOURS 12 14 16 18 20 22 00 02 04 06 08 ’ I 1 ( I , I , 1, I ( I , I, I, I,, SURFACE IO 12 14 16 18 20 22 , 1 , I, , , , , , 3 24-,,,,,,,,,,,,,,,,,,,,,,,,,,, FIG. 3. Time variations I of wind speed, air temperature, The frequency of the 40-m temperature oscillations appears to exhibit a semidiurnal component as opposed to the diurnal temperature variation at the surface. To portray this semidiurnal effect graphically, a sinusoidal wave of arbitrary amplitude having a period equal to iU2, the principal lunar tidal component having a period of about 12.4 hr, is plotted under the 40-m curve for comparison. The cloud cover for the entire daylight periods (with the exception of one observation 50 cloud cover at 0500, 23 May) was less than %o and averaged less than ?/lo. The wind speed (Fig. 2) never exceeded 7 m secl and averaged about 3.5 m set-l over the 165hr period. The sea state records for this period never exceeded Beaufort scale of 2 (that is, waves were less than 1 m in height). Fig. 3 includes results of the February bathythermograph study, the measured 579 LAYERS and isotherm I I I I depths for 6-B Felxuary I , , 00 3 I_ 1962. wind speed, the air temperature, and the time-depth-temperature section drawn from the consecutive bathythermograph profiles as shown in Fig. 1. The air temperature fluctuated between 27 and 28C during the period from 1100 to 1700, then dropped sharply at about 1730 hours. The lower temperature between 23 and 24.7C prevailed until the beginning of diurnal heating at 0800 the following day. The peak occurred at about 1100, attaining 31.Z. A trailing-off effect was noted from 1300 to midnight. The cooling effect reflected in the air temperature drop at 1700 was observed within the surface layer from the surface at 1700 hours to about 10 m by 1800 hours. This cooling continued through 1900, with the water surface temperature attaining a low of 24.OC. Between 1930 and 2030 hours, the surface showed indication of heating that attained a maximum at 24.2C. Meanwhile, the 24.2C isotherm reached to about 14 m, and then 580 DAVID IO T iii r” o 0 04 1 [ 06 I [, 08 ,, IO ,,,,(I H. SIIONTINC LOCALTIME- HOURS 12 14 16 18 1 , I 20 , I 22 , I 00 , , 02 , 8- 6r ;I0 27 g 26 - 6 - I2 E 14 w 0 16 I8 2ot1”““““““““““1i FIG. 4. Time variations of wind speed, air temperature, ascended, reaching the surface just after midnight. The diurnal heating commenced at about 1100 with a drop in the 24.4C isotherm, reaching down to 19 m by 1800 hours. The surface water was heated above 24.6C from about 1400 to 1800 hours. By 2300 hours, the diurnal disturbance vanished, and the surface layer was left isothermal at about 24.3C. The winds during this perio,d were moderately strong bcforc the drop in air temperature at 1700 hours. With the cooling of the air, the winds decreased to about 2-3 m see-1 and remained light until the period of diurnal heating the following day. The results of the March bathythcrmograph study are summarized in Fig. 4. The air temperatures were recorded only at 3-hr intervals and thus only approximately incli- and isotherm depths for 13-14 March 1962. cate the heating of the air. The air tempera ture, however, increased markedly from 24.OC at 0800 to 27.OC by 1100 hours, dipped to 25.7C at 1400, rose to 27.3C at 1700 hours, and finally fell to 24.4C by 2000 hours. The first heating of the water appeared at 1000 hours, with the 24.3C isotherms descending and reaching a maximum depth of about 12 m by 1500 hours. This time correponds to the time of maximum heating at the surface in which the temperature exceeded 25.OC. The path of the downward moving isotherms is somewhat steeper than that of the upward trending isotherms From 1700 through to midnight. At midnight, a new heating occurred during the 24 hr, as indicated by the 24.2C isotherm at 6 m. The wind, as indicated in the upper graph SHORT-TERM HEAT TRANSFER THROUGH OCEAN SURFACE 0800 0700 0900 ,581 Jr -a-w--L 00 0600 LAYERS 1000 II00 ---.A 1200 1300 Iwoo 1500 -- ,f 1600 1! If300 1700 LOCAL TME - HOURS I _1 -lo FIG. 5. Two-day average of incident radiation measured in the Tongue of the Ocean, lo-11 Xlarch 1960. in Fig. 4, was relatively strong before 0300; in fact, the winds were recorded at 10-11 m set-l at midnight before the bathythermograph sampling that began at 0300 hours. During the sampling from 0300 to 0700 hours, the winds were moderate, that is, about 9 m set-l at the beginning of the sampling and decreasing to about 6 m set-l by 0700 hours. At the inception of the heating, there was an isothermal layer of 24.1c. From 0700 to 2400 hours, the winds were light, not exceeding 6 m set-l and averaging about 4 m set-l. Fig. 5 ( stepped curve) shows the Z-day average of the incident radiation measured during 10-11 March 1960. During this period, clear skies prevailed. The vertical bars represent the one-half-hr integrals of incident flux in langlies (cal cm-2). The maximum incident radiation occurs about between 1230 and 1330 and is about 1.3 ly min-l. The integral of the incident energy during the 24-hr period (dotted curve) attains a value 560 ly. This corresponds closely to the value of the daily insolation for 24.5” N latitude during March indicated by Haurwitz ( 1941). The time variation of the incident radiation exhibited a clear sinusoidal variation with the limits of detectable flux beginning at 0600 and vanishing by 1830. During the period of incident radiation measurement, simultaneous measurements FIG. 6. Albedo of the sea surface calculated from incident and reflected radiation data, lo-11 March 1960. were made of reflected radiation at three positions outboard of the vessel. A plot of the albedo values computed from the records of the incident and reflected radiation is presented in Fig. 6. DISCUSSION It is of interest to note the apparent net heating of the surface water for the 7-day period in Fig. 2, which indicates a temperature increase of approximately 0.1X per day. According to Smith ( 1940)) the average seasonal range of temperature is about 8C in the region of the Bahama Banks. Let us assume that the seasonal temperature T( t ) can be described by a sinusoidal relation of the type T(t) = G sin go, where Ts = the surface temperature range ( 8C ) , n/360 = the ciruclar frequency ( rad day-l), and t = time in days. Let us further assume that the 8C temperature variation attains a minimum in February and maximum in August. Thus, the period of maximum heating would occur in May, the period of the 7-day measurements, or when t equals about 90 days. To evaluate the slope in May, we take the first time derivative of equation ( 1) and upon solving, T = O.lC/day. Since this value is comparable to the observed slope, it suggests that we are observing actual seasonal heating. It is also evident that this heating effect is not readily detectable at 40-m depth. Pronounced diurnal heating and cooling are graphically shown by the plots of air and surface water temperature in Fig. 2. The 582 DAVID H. SHONTINC difference between the thermal and turbulent mixing properties of air and water is shown by the superposition of the air and surface water temperatures, The air cxhibits three to four times the diurnal temperature amplitude of the surface water. It is interesting to note, however, that temperature differences between the air and water are not controlled mainly by conduction This is exemplified by the fact that, in general, the water temperature just after the noonday heating begins to fall sharply, while the temperature of the air ranges Erom about 1.8 to 2.OC above the water temperature. The water temperature starts to fall sharply, catching up to and crossing the descending air temperature at about 1900 hours. It is clear that during the period when Tnila> Twatcr, there can be no net heat transfer from the water to the air by conduction. Other mechanisms must cool the surface water. Wind effects are not to bc neglected, and there is some indication that a diurnal afternoon thermally driven wind or Andros Island sea breeze is generated from the hours of 1400 to late evening. This is apparent in Fig. 2 for the afternoons of 21, 23, 24, and 25 May. This afternoon breeze would greatly contribute to the cooling oE the thin heated surface layer by promoting turbulence that mixes the newly formed thin layer of warm water downward into the essentially infinite source of cooler water below, and by promoting surface evaporation. The surface water does, however, seem to exert a damping effect upon the cooling rate of the air during the evening. Thus, the rate of air temperature decrease rapidly attcnuatcs as the water temperature is passed. The average range between the minimum air and surface water temperatures is about O-SC, while the maxima show an average difference of about 1.8C. This suggests a definite damping and an apparent increase in convective or conducting heat transfer during the nocturnal hours. It is suggested that the optimum conditions for observing diurnal heat effects in the surface layers occur when warm, sunny conditions are supplemented by calm sea conditions (see Stommel and Woodcock 1951). In view of this, it is noted in Fig, 2 that the sharpest rise of sea temperature occurred between 0600 and 1200 hours on 21 May just after a period of calm sea and beEore winds. Unfortunately, the sampling interval of 3 hr is too long to enable one to follow the heating with precision. The temperature curves for the heating an d cooling of the air and surface water are dissimilar (Fig. 2). The complex problem of the thermodynamics of the heat transfer at the air-sea interface is not within the scope of this paper, but it is worthwhile to mention some features of the data. The tempcraturc of the air immcdiatcly above the water is markedly different from that of the sea surface itself. For example, consider the diurnal heating and cooling in these two media. Most of the heating in both the air and water is caused by the This absorption of the solar radiation. means that at the time of maximum heating the energy input to the water and air far exceeds the energy output by back radiation, turbulent mixing, and other less important mechanisms. The air tends to absorb the incoming radiation rather uniformly within its volume, but in the surface water it is ab,sorbcd exponentially with depth. The intensity of this absorI?ion depends on the transparency and scattering properties of the water. The cooling processes within the air and water are even more dissimilar than the heating. The major cooling is by back radiation at infrared frequencies by the air an d water, both as approximate blackbody radiators. The air can reradiate its cncrgy with complex interactions but nearly uniformly within its volume, whereas the water reradiates energy at infrared wavclengths solely at its immediate surface. Thus, in considering the different geomctries of the two media as absorbers and as radiators and their different heat cnpacities, it is clear that their heating characteristics arc different. Concerning the tcmperaturc variations in Fig. 3, the 40-m temperature oscillates in SHORT-TERM HEAT TRANSFER THROUGH such a way that it is difficult to envisage a diurnal heating effect at this depth. There is, howcvcr, a semidiurnal oscillation that closely follows tho M2 principal lunar tidal component. May is a poor time of the year at which to observe the vertical extent of diurnal solar heating, because the surface layer tends to bc less isothermal than at other times of the year. This condition is charactcrized by an erratic near-surface tcmperature profile that can mask small tcmpcrature variations caused by surface heating. It is in May that the spring heating disrupts the thick isothermal layer, extending to 150 m, formed during early winter and predominant in February and March. The spring heating brings about a modcrate temperature gradient within the original mixed layer. This is caused by intensified surface heating such that wind mixing cannot make the surface layer isothermal below 20-30 m. During this spring shoaling of the seasonal thcrmoclinc, the layer cxtcnding to 200 m generally becomes disturbed with small temperature anomalies that can be transient in time and that are observed at the 40-m depth, as in Fig. 2. Defant (1961) d iscusses diurnal tcmpcrature variations observed on the Meteor. The 50-m temperature variations detected wcrc less than 0.05C. This stems unlikely inasmuch as this is probably well within the noise level even in an isothermal mixed layer. Considering the gross heat penetrations shown in Figs. 3 and 4, it is unlikely that a diurnal temperature variation is dctcctablc dcepcr than 25-30 m, even with very scnsitive tcmperaturc probes. The degree of heating and cooling in the vertical are to a great extent determined by the wind mixing at the surface and the duration and intensity of the incident radiation rclativc to the amount of blackbody radiation cmittcd from the water surface. It is instructive to examine closely the patterns of the isotherms as a function of depth and time. In Fig. 3, the spacing of the samples prcscnts a jagged pattern both during cooling, bctwccn 1500 and 2400 hours, and during heating, between 1100 OCEAN SURFACE LAYERS 583 and 1200 hours. The vertical extent of the rcsponsc is quite similar in both cases. In both heating and cooling, the downward propagation rate of the isotherms is of the order of 5-10 m hr-r. It is difficult to arrive at definite conclusions from these sections bccausc of the oversimplification resulting from the smoothing and contouring. However, the results indicate the extent of the typical diurnal heat wave that may occur under tropical and subtropical conditions. The thermal response appears to be much faster for cooling than for heating, as shown by comparing the time lag of water temperature with respect to air during the diurnal heating. The heating pattern in Fig. 4 is particularly well dcfincd and shows the time-variable geometry of the isotherms. It appears that the early morning winds ( 0300-0600 hours ) bring about enough vertical mixing to remove any thermal stratification caused by the diurnal heating of the previous day. This, coupled with ideal heating conditions, allows clear observation of internal heating effects. If the daily variation of incident radiation (Fig. 5) is compared with the isotherm pattern, the heating lag is about 3 hr from the time of the maximum incident heat flux to the maximum isotherm penctration. The asymmetry of the isotherm pattern at about 1600 hours is as striking as the symmetry displayed by the variation of incident heat flux. The asymmetry is probably due to the variation in stability of the surface layer as it is heated and cooled through the 24-hr period. Since the thermal stratification is a time-variable quantity, it follows that the turbulent structure of this layer will change. Let us consider criteria of the magnitude of the variation of stability as evaluated by the Richardson number R, given by (2) whcrc g = accclcration of gravity (cm set-2), p = dcnsit y ( g cm-3), and ti = mean current speed at depth x ( cm set-l ) , When surface waters arc heated from 584 DAVID II. any initial isothermal condition, the instability or vertical turbulence is reduced greatly. For example, referring to’ Fig. 4, the variation in the Richardson number at the 6-m level is estimated to increase by a factor of lo” between 0900 and 1400 hours, This was calculated on the basis of the time variation of the temperature gradient, assuming the mean vertical velocity sheer is held constant, This rapid change in vertical stability would account fo,r the lag in the sudden drop in the isotherms between 1000 and 1200 hours, Starting at 1300 hours, the strong gradient formed by the close spacing of the 24%24.6C isotherms increases the vertical stability by decreasing the density of the surface layer and tends to dampen vertical mixing. By 1800 hours, there is virtually no direct solar energy being added to the water column. At this time, the isotherm distribution is at the mercy of the ambient turbulent characteristics controlling the vertical mixing of the water. This turbulcncc is a function of the wind-imparted momentum at the surface and the mixing across the relatively warm isotherms into the colder isothermal water at 24.1C. Surface cooling due to evaporation and conduction is also associated with wind speed. The wind speed was quite constant during the period 0930-2400 hours, averaging about 4 m set-I. Assuming that the wind stress on the water varies as the square of the wind speed, this would give a variation of the stress by a factor of about three. It is clear that even with quasi-steady wind conditions, the wind-imparted turbulence may have an appreciable time variation, giving rise to complex interactions with the isotherm distribution. Apart from variable wind effects, the newly formed stability of the layer apparcntly causes a much slower return of the isotherms to the surface. THERMAL EDDY DIFFUSION The plot of the iso therms as a function of time and depth allows one to approximate the diFfusion properties after making certain assumptions concerning the nature SIIONTING of the diffusion mechanism. Let us assume that the process by which heat is transferred vertically downward can bc described by the Fourier heat equation for heat flow in a solid in one climension, that is, --dT =Ka”T (3) at ax2 ’ where T is tcmperaturc, t is time, and K is the thermal eddy diffusivity, which is assumed to be constant with depth x. For the case of diurnal heating, let us assume a surface time-variable boundary condition T (0,-t) = To cos wt, (4) where To is the amplitude of the tcmpcraturc wave at the surface and w is the frequcncy. The sinusoidal time variation has often been used in describing diurnal heating within the atmosphere (XX, for cxample, Lettau 1951 and Haltiner 1958). According to Sutton ( 1953), this form is a good approximation of the daily heating for clear skies. A solution to equation ( 3) agreeing with equation ( 4) is T (x,t) = Toe-+ cos [w&] * (5) The first factor in expression (5) is the amplitude of the temperature wave at a given depth and thus is independent of the time of maxima and minima. Following Dutton and Bryson ( 1962), let us define the function representing the total range of temperature at a depth (that is, twice the amplitude) as T,&(x) = 2Toe-“& Writing P=-, the frequency 211 we obtain w (6) in terms of period, - It is clear that the range of temperature at a given depth increases with the period of heating. The variation of Tlz versus depth as calculated using the data in Fig. 4 is given in Table 1. - SIIORT-TERM HEAT Solving for the eddy diffusivity tion (7), we obtain ‘1ltANSFER TIIROUGII in equa- OCEAN SURFACE calculated TR K ( cm3 xx+) (“c) 1.0 1.0 0.8 0.7 0.6 0.5 0.3 0.1 0.0 0 Upon comparing equation (3) with this more general form, we observe the contribution of the product term of the vertical gradient of K’, and the temperature gradient. It is indicated that in a region in the water column whcrc equation (8) shows a rapidly changing value of K with depth, 585 1. Vales of the thermal eddy diffusivity from equation (8) for various depths TABLE Depth z ( m ) and hence an estimate of the thermal diffusivity at various depths. Table 1 gives the values of K calculated from equation (8) for the case of diurnal heating described in Fig. 4, where P equals 24 hr (8.64 x 10’) see ) . The range of calculated values of K is comparable with values given by Defant ( 1961) for eddy diffusivities derived from various oceanic tcmperaturc measurements made within the upper 300 m. It is to be noted that equation (8) is a statement of the constancy of K if (3) holds true. The variation of K with depth indicates that equation (3) does not hold true. IIowevcr, the extent of the applicability of a type of equation such as (3) (with, albeit, a K varying with depth) is approximately indicated by the values of the eddy thermal diffusivity that are quite represcntativc as a first-order estimate of values obtained by other methods. As a check on how realistic the heat equation is in describing the vertical heat transfer, equation ( 1) was evaluated using finite difference quantities from Fig. 4 (the region from 13X5-1345 hours and 4-12 m depth). The calculated value of K is about 155 cm2 set-l. It is reasonable to assume the rate of vertical diffusion (that is, the intensity of turbulence) should be a function of depth. This is expressed by an equation of the form LAYERS z G 8 9 10 12 1.4 111 102 89 61 25 10 0 the nonlinear term in the right-hand side of equation (9) has become important, owing to a large value of ~3K’,/&z. Equation (9) explains nothing of how or why K’, varies with depth. The difficulty in postulating the exact nature of the diffusivity is that little is known of the parameters controlling diurnal heating in the surface waters of the ocean. The same situation exists in the atmosphere. Haltiner ( 1958) describes solutions to the heat diffusion equation (9) with the diffusivity K’, varying periodically with time and exponentially with height. I& solves the equation in a finite difference form, utilizing a high-speed digital computer. This approach should prove fruitful when applied to oceanic diurnal heating data if one for can stipulate a proper form for K’,(x) the ocean. HEAT BUDGET CALCULATION Let us consider a vertical column of water of unit cross section and x m in depth measured from the surface, in which we wish to measure the heat budget for a 24-hr period. To perform this calculation, we must have estimates of the following quantities: the net heat gain or loss within the water column QN, the incident solar radiation upon the sea surface Qr, the reflected radiation Q n, the energy transmitted across the air-sea intcrfacc by conduction and evaporation QzI, and the cncrgy reradiated from the sea surface at infrared wavelengths QItn.Any heat advected in or out of the region by horizontal currents is ncglccted. 586 DAVID The heat budget equation QN = Qr - Qre - QT II. is written - (10) Qnn. In applying this equation to the diurnal heating measurements made 13-14 March 1962 ( Fig. 4)) th e various components of equation (10) wcrc derived as follows: The net heat gained z?y the water, QN-The quantity QN must represent the total energy absorbed in the water column. In the present case, one can estimate the heat intake to a depth limited by the thermal sensitivity of the bathythermograph (that is, to the nearest O.lC). The average depth of the diurnal thermal perturbation is approximately 10 m. For convenience, assume that the quantity of heat intake Q,V is made LIP of two quantities, Qls and Qx, the first of which is the energy calculated within the isotherm layers followed by the bathythermograph. From Fig. 4, it can be seen that this is the amount of energy in the layer bounded by the surface and the 24.lC isotherm. The second quantity is that radiation that was absorbed below the mean depth of the 24.1C isotherm. Its computation will be discussed later. The amount of heat absorbed in the region above the 24.1C isotherm can be calculated by the following relation: Qra = C, z L,( T,-1 + 0.05” - 24.1”), (11) n-1 where C, = specific heat of seawater, 0.93 cal g- 1 ‘C-l for 36sG salinity, L,, = the thickness of the layer bctwcen two adjacent isotherms of cross-sectional arca of 1 cm2, and T = +Ln tPmncwahlrp nF the nth isotherm. SI-IONTING culated as +105 ly day-’ ( the plus sign indicates heat taken into the layer), It is of interest to use equation ( 11) to calculate the heat required to produce the isotherm distribution as it occurred at 1500 hours, the time of maximum depth penctration. The amount of energy ncedecl is about 550 ly. Fig. 5 shows that by 1500 hours only about 480 ly were incident on the sea surface since sunrise. There is thus an indication of the absolute error in the temperature or depth sensors (or both) of the bathythermographs. An error of 1 m in the depth of the isotherms can change the 1500 hours heat content calculation by 85 ly, giving a 15% error. Error in the estimation of thickness of the upper isotherms becomes critical to the total integral of the heat content, because the upper layers contain the most absorbed heat. The incident s&r w&don, Q1-This value is found from Fig. 5 by integrating the hillf-hourly values of incident radiation. Although these data were collected a year before the March 1962 measurements, they were taken at the same latitude and under clear skies similar to those on 13-14 March 1962. The integrated 2-day average is 562 ly day-l. The reflected solar radiation, QIt-This quantity was measured directly, simultancously with the direct solar radiation. As a check on the reflected energy values, a plot oE the calculated diurnal variation of albedo is included in Fig. 6. The time variation of the seawater albcdo with the sun’s altituclc and amount of reflected cncrgy for the whole day agree well with values given by Svcrdrup, Johnson, and Fleming ( 1942). -n ,.. x ?... . . SHORT-TERM HEAT TRANSFER THROUGH OCEAN TABLE 2. 7 SURFACE 587 LAYERS Twenty-four-hour energy butlgct Incident radiation QI = 560 ly = 25 ly Energy reflected QR Energy reradiated + Energy conducted + = 405 ly QT Energy of evaporation 1 Energy absorbed Qrs = 105 ly Energy left over = QI - ()I< - QT - VI.4 = 25 ly FIG. 7. The half-hour integrals (Curve A) and the 24-hr integral ( Curve B ) . of QT + Qm of QT + QM where R = Bowen ratio, A = 1.5 X lo-” cm-Z set”, H = 585 cal g-l, u = wind speed, m set-l, r = relative humidity, and e,,. = saturation vapor pressure at the ambient air temperature. According to relations derived by Brunt ( 1944)) Dorsey ( 1940), and Budyko (1956), the infrared radiation emitted from the sea surface is approximated by Qrrlc = 0.9&TA4 lO-Z&.) ( 0.39 - 5.04 x ( 1 - 0.6n”), ( 13) where (J = Boltzmann’s constant = 8.17 X lo-l1 cal cm-2 min-l , T-4 = absolute temperature, “K, and n = cloud amount in tenths. Combining these equations we have: QT + QItlr = 0.985 TK4( 0.39 O.O5dre,,) ( 1 - 0.6n2) + (1 + R)(l-r)5.26 x lo-“ue,. (14) The value of e,, is determined from TK, and thus the energy transmitted by long wave radiation, conduction, and evaporation is reduced to a function of only surface temperature, TK, the wind speed, u, and the relative humidity, r. Kraus and Root11 ( 1961) computed the value of QT + QRlr against surface temperature with varying equation wind speeds. For evaluating ( 14)) let n = 0, r = 0.8, and R = 0.1, values representative for the period 13 hlarch, 0000-2400 hours. The half-hour in- tegrals of QT + Q RIGare shown in Fig. 7 ( Curve A). The value of the 24-hr integral QTI + Q1<,: (Curve B ), is about 405 ly. In summary, the 24-hr period energy budget is given in Table 2. There is a moderate excess of 25 ly. The value of excess energy may be the result of our inadequacies in estimating the magnitudes of the variables considered in the budget. In particular, the value of the net energy absorbed in the water by 2400 hours (that is, +105 ly ) may be in error by as much as 15-20% owing to error in estimation of depth of the isotherms. On the other hand, the imbalance could be attributed to two other factors, advection of warmer water into the region or the previously mentioned energy Q-r that was transmitted through the surface layers into the region where the heating was insufficient to detect with the bathythermograph. It is apparent from data collected in the general area during this period (see Shonting et al. 1963) that the horizontal temperature gradients were small, that is, less than O.OlC km-l and not monatomically varying in any one direction in the Tongue of the Ocean. However, not enough is known of the current pattern or speed to assess properly the magnitude of the advective effects. To estimate the energy which, according to Beer’s law, would reach a depth below 10 m (the average depth of the 24.1C isotherm), Beer’s law is used in the form Qz = Q,,c (I~, (15) where QZ = energy reaching at depth z (m), Q. = energy penetrating the surface water, and a = extinction coefficient, m ‘. Using a value of N of 0.05 m ‘, which is appropriate for subtropical Sargasso sea- 588 DAVID II. SIIONTING -. 1960. Teplovoi halanz zcmmio poverwater similar to that found in the Tongue blcn osti cjidromcteorologichcskoc izdatel’stvo. of the Ocean, it is found that about 70% Leningrad. 255 p. of the energy should be absorbed within DEFANT, A. 1961. Physical oceanography, v. 1. the first 10 m. On the basis of 100 ly day-l Pcrgamon, New York. 730 p. absorbed above this level, about 45 ly day 1 DORSEY, E. 1940. Propcrtics of ordinary water substance. Am. Chcm. Sot. Monographs. would bc absorbed below this level. By Reinhold, New York. 220 p. varying tither the extinction coefficient or DUTTON, J. A., AND Ii. A. BHYSON. 1962. Heat flux in Lake Mendota. Limnol. Oceanog., 7 : the depth in the Beer’s law calculation, we could easily arrive at nearly the 25 ly day-l HALzIi;z*C: , . 1958. The diurnal temperature cxccss energy found in the previous calcuwave’ wi& a cocfficicnt of diffusivity which lations. This value of the excess energy is varies periodically with time and exponentially with height. J. Meteorol., 15: 317-323. almost negligible in comparison with inHAURWITZ, B. 1941. Dynamic meteorology. Mcstrumental errors and horizontally advected Graw, New York. 36s p. heat . KRAUS, E. B., ANT) C. RoovRi. 1961. Tcmperaturc and stcacly state vertical heat flux in the These budget calculations appear to rccocean surface layers. Tcllus, 13 : 231-238. ommend the use of the temperature-depthLETTEAU, H. 1951. Theory of surface temperatime section as a working tool with which turc heat transfer oscillations near a level to observe mixing and turbulent processes ground surf act. Trans. Am. Geophys. Union, 32: 189-200. in the ocean. The results demonstrate that MAGNITSKY, A. W., AND H. V. FRENCH. 1960. an accurate estimate of the oceanic distriTongue of the Ocean research experiment. bution of solar energy is feasible using Tech. Rept. No. 94. (Unpublished manuscript.) U.S. Navy Hydrog. Office, Washingand meteorological data. oceanographic However, as a method of studying turbuu s to:AT;:* l~~E~;OrRAPEIIC 9 OFFICE. 1962. lent eddy mixing in the ocean, it must be ’ * Oceanogiaphic Station Data, USS San Pablo considered as an empirical approach. To Cruise No. 31947 of May ancl June, 1962. ( Unpublished manuscript. ) 150 p. analyze heat diffusion on the ocean surface D. H., G. S. COOK, AND C. F. MOREY. layers more properly, it is necessary to have SHONTING, 1963. Oceanographic data report, Tongue of knowlcdgc of the dynamic motions ocof the Ocean, NUOS Cruise 3, March 1962. NUOS Consec 349. (Unpublished manucurring within the regime of diffusion. script,) U.S. Naval Underwater Ordnance This aspect has generally been neglected in Station, Newport, Rhode Island. 150 p. thermal studies in the ocean, but it might sMyIITII c . L . 1940. The Great Bahama Bank. lead to a better understanding of oceanic I.’ General hyclrographic and chemical fcaturcs. J. Marinc Rcs., 3: 147-170. diffusion problems. REFEXiENCES 1944. Physical ancl clynamic metclb<UNT, D. orology. Camhridgc, London. 570 p. RUIIYKO, M. I. 1956. The heat balance of the Hydromcteorological Instiearth’s surface. (Translated by Office of tute, Leningrad. Technical Scrvicc, U.S. Department of Commercc. ) STOMMEL, II., ANU A. WOOIXOCK. 1951. Diurnal heating of the surface of the Gulf of Mexico in the spring of 1942. Trans. Am. Geophys. Union, 32: 563-571. McSUTTON, 0. G. 1953. Micrometeorology. Graw, New York. 333 p. SVERDIXJP, H. U., M. W. JOHNSON, AND R. H. PrcnticcFLEMING. 1942. The oceans. Hall, Englewood Cliffs, N.J. 1087 p.