AN INVESTIGATION OF SURFACE AND INTERNAL WAVE-INDUCED TURBULENCE IN SHALLOW WATER THERMAL MICROSTRUCTURE James Wesley Powell I DUDLEY KNOX LIBRARY ■AVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA 93940 Monterey, California Isasa Sana ^Smss* Li AN INVESTIGATION OF SURFACE AND INTERNAL WAVE-INDUCED TURBULENCE IN SHALLOW WATER THERMAL MI CROSTRUCTURE by James Wesley Powell Thesis Advisor E. B. Thornton March 1974 U59611 Kfxpfwvzd ^on. pubLLz KclaaAt; dutrUbution uixLlmlttd. An Investigation of Surface and Internal Wave-Induced Turbulence in Shallow Water Thermal Micros tructure by James Wesley Powell Captain, Canadian Armed Forces B.Sc, University of British Columbia, 1964 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL March 1974 <^. t ABSTRACT Measurements of temperature, wave height, and orthogonal water particle velocity were made in May 1973 at the NUC Tower located one mile off Mission Beach, California. A detailed analysis of the tem- perature field was made using digital temperature and isotherm contour plots. Billow turbulence microstructure appeared to be the dominant mechanism with the exception of one run in which evidence of double- diffusion microstructure was found. Spatial correlation lengths, calculated from the plot of the covariances, were of the order of 130 cm and less when the signal was high-pass filtered for waves of 100 seconds and longer. No depth dependence was noticed. Both fre- quency and wavenumber spectra were calculated and a correspondence between the spectra was noted at the frequency and wavenumber of the surface wave-induced particle displacements. The Thornton, Boston, Whittemore model of wave-induced tempera- ture fluctuations was tested and found to model the temperature spec- tra quite well, especially in a narrow band of frequencies associated with surface waves. The turbulent temperature spectrum, calculated as the difference between the actual and wave-induced spectra, had a slope near -5/3 above 0.1 Hz and more negative at lower frequencies. TABLE OF CONTENTS I. INTRODUCTION ----------------- 9 A. HISTORY- ----------------- 9 B. OBJECTIVES ---------------- 13 II. THEORETICAL CONSIDERATIONS ---------- 14 A. MICROSTRUCTURE EFFECTS ---------- 16 i 1. Double-Diffusion and Salt-Fingering- - 17 2. Billow Turbulence- ---------- 19 B. SURFACE WAVE EFFECTS ----------- 22 1. Surface Wave-Induced Temperature Fluctuations ------------- 23 2. Turbulent Temperature Spectrum - - - - 26 C. INTERNAL WAVE EFFECTS- ---------- 27 III. EXPERIMENT AND DATA ANALYSIS --------- 30 A. EXPERIMENT ---------------- 30 B. DATA PREPARATION ------------- 33 C. METHOD OF ANALYSIS ------------ 34 IV. RESULTS- ------------------- 37 A. ANALOG TEMPERATURE RECORDS -------- 37 B. TEMPERATURE FIELDS ------------ 37 C. SPATIAL CORRELATIONS ----------- 48 D. WAVE NUMBER SPECTRA- ----------- 56 E. GRADIENT FIELDS- ------------- 62 F. TEMPERATURE FREQUENCY SPECTRA ANALYSIS - - 69 1. Surface Wave-Induced Temperature Fluctuations ------------- 70 2. Coherence- - - - ----------- 71 3. Phase Difference ----------- 73 4. Turbulent Temperature Fluctuations - - 76 V. CONCLUSIONS- ----------------- 79 LIST OF REFERENCES ----------------- 82 INITIAL DISTRIBUTION LIST- ------------- 84 FORM DD 1473 -------------------- 86 LIST OF TABLES 1. Summary of Experimental Set-up for Minard's May 1973 Experiment -------------- 30 2. Internal Wave - Periodicity ---------- 43 3. Variance and Covariance ------------ 49 4. Correlation Lengths -------------- 53 5. Particle Displacements- ------------ 61 6. Gradient Data ----------------- 62 7. Frequency Spectra Data- ------------ 71 LIST OF FIGURES 1. Schematic of tower and instrument mounting- - - 12 2. Salt-fingering- ---------------- 18 3. High resolution temperature profile of the 310-345 m interval in the Arctic off Ice- Island T-3- ^.----- ------------ 18 4. Kelvin-Helmholtz shear instability- ------ 20 5. Configuration of array for Runs 1,2,3,4 - - - - 31 6. Configuration of array for Run 6-------- 32 7. Analog temperature record, Run 6- ------- 38 8. Analog temperature record, Run 1- ------- 39 9. Digital temperature record, Run 1, Thermistors 1 and 6-------------- 40 10. Isotherm contours, Run 1- ----------- 41 11. Digital temperature record, Run 3, Thermistors 1 and 6-------------- 44 12. Digital temperature record, Run 3, Thermistors 1 and 6-------------- 46 13. Isotherm contours, Run 3------------ 47 14. Non-filtered spatial correlation- vertical array- ---------------- 51 15. Non-filtered spatial correlation- horizontal array- --------------- 52 16. Spatial correlation (vertical array)- ----- 54 17. Spatial correlation (horizontal array)- - - - - 55 18. Vertical array wavenumber spectra ------- 57 19. Horizontal array wavenumber spectra ------ 58 20. Digital temperature record, Run 1, Thermistors 1 and 5-------------- 65 21. Temperature gradient plot, Run 1-------- 66 22. Digital temperature record, Run 7, Thermistors 1 and 5-------------- 67 23. Temperature gradient plot, Run 7-------- 68 24. Actual and calculated wave-induced temperature spectra, Run 6----------- 72 25. Turbulent temperature spectra --------- 77 ACKNOWLEDGEMENTS The author is greatly indebted to both Dr. E. Thorn- ton and Dr. N. Boston for their time and effort in bringing this paper to fruition. The research and experiment was funded under the fol- lowing project: ONR Project No. NR 083-275 I. INTRODUCTION A. HISTORY For the past three years research has been carried out at the United States Naval Postgraduate School (NPS) , Department of Oceanography, to study the shallow water interactions that occur between various oceanographic parameters such as temperature, salinity, surface waves, internal waves, and sound speed. These experiments have been conducted from the Naval Undersea Research and Devel- opment Center's (NUC) Oceanographic Research Tower located approximately one mile off Mission Beach, California. Results of the experiments are presented in NPS Theses, NPS technical reports and publications. This thesis is based on a study started by Whittemore (1973) and is an extension of the results obtained by Hinard (1973). The Whittemore experiment was conducted at NUC on 8 and 9 June 1972. Measurements were made of salinity, sound velocity, particle velocity, and temperature (at various positions along a horizontal array) for ten pre- selected depths. He concluded that the relationship between temperature fluctuations and surface wave height (for swell type waves) was of the form, T(t) = K(f,z) dT dz n(t) , and that a similar relationship was likely to exist be- tween temperature fluctuations and internal waves. He also found that turbulence degrades coherence between temperature fluctuations in sensors displaced spatially; however, a lack of temperature sensors prevented him from making definitive statements. Reworking the data collected during the June 1972 ex- periment, Thornton (1974) demonstrated a means for sepa- rating turbulent and surface wave-induced vertical veloc- ity spectral components allowing for the quantities to be statistically correlated. The method applied to the wave and velocity measurements allowed the wave energy density spectral components to be converted to velocity spectral components using linear wave theory. The com- puted values compared very well with the measured veloc- ity spectra for the moderate wave conditions encountered and appeared to verify the theoretical spectral transfer function and assumed linear system. Using the same data and techniques, Thornton, Boston, and Whittemore (1974) devised a means for separating tur- bulent and surface wave-induced temperature fluctuations. They assumed that linear wave theory could be used to de- scribe the surface waves; that the temperature isotherms were horizontal; and that the mean temperature gradient was a constant. In this way they developed a stochastic model describing the surface wave-induced temperature fluctuations. This allowed a means for calculating the 10 turbulent temperature spectrum by subtracting out the wave-induced contributions. The measured slope of the turbulent temperature spectra closely approximated the Kolmogorov -5/3 slope over a decade. This supported the result that the turbulent temperature fluctuations are statistically independent of the surface wave-induced velocities . In the present work it is intended to refine the analysis of the data and extend the results obtained by Minard (1973). Minard set out to rectify the lack of sensors noted in Whittemore's experiment and conducted another experiment at NUC on 15 and 16 May 1973. Measure- ments of small scale temperature fluctuations were made in 19 meters of water using a movable array of seven thermistors, with a short (0.015 sec) response time; and accurate to 0.01°C in seawater. A schematic of the tower and instrument mounting is given in Figure 1. In addi- tion to temperature, wave heights and orthogonal water particle velocities were measured simultaneously. Mea- surements were made at various depths with either a ver- tical or horizontal orientation of the temperature array. Minard found that in the presence of internal waves temporal scales were of the order of 4 seconds and spatial scales were of the order of 33 cm, whereas outside the influence of internal waves they were of the order of 10 seconds and 118 cm. It was concluded that in the presence 11 _/^*I BEAM RESISTANCE WIRE WAVE 6UAGE 5?»S (-?■ •'■o'oOooo-'-'oO o o rr^l l;.*-?.^ •■ ^^^^S^^SrO^^OS^QS&^s^^ Figure 1. Schematic of Tower and Instrument Mountinc 12 of internal waves, correlation times were similar for both vertical and horizontal scales and spatial corre- lation lengths were similar for vertical and horizontal scales . B. OBJECTIVES The main objectives of this thesis with relation to the measurements made by Minard in May 1973 are to: 1. Define the spatial and temporal temperature field. . 2. Define the spatial and temporal temperature spectra and determine if an obvious relationship exists between them. 3. Define the temperature gradient field and its variation with time to enable a comparison to be made between the formation of microstructure with and with- out the passage of an internal wave. 4. Test the stochastic model proposed by Thornton, Boston, and Whittemore to calculate the theoretical wave-induced temperature spectrum using these data. 13 II. THEORETICAL CONSIDERATIONS The creation of small scale temperature fluctuations in shallow water is a complex process occurring at all depths and scales, depending on the operative driving function prevailing in time and space. In the near sur- face regime winds over the surface produce tangential surface stresses both from the interfacial stress and from the momentum loss associated with processes such as breaking waves. These result in a downwards momentum transfer with its attendant downward flux of heat and salt from the surface to a depth dependent on the sta- bility of the water column, the gradient of shears pro- duced, and the wavelength of the surface waves. In this way the temperature and salinity in the mixed layer be- come virtually uniform; and, unless the stability or Brunt-Vaisala frequency (defined as N2 = - g 9P p dz ' (1) where g is the acceleration of gravity, p is the density and ■* — is gradient of density) is less than or equal to zero, a thermocline develops due to the increasing con- trast between the properties of the water in and below the mixed layer. If there is substantial surface cooling, there may be a region below the mixed layer in which the 14 stability frequency is less than zero and convective mo- tions can develop extending to considerable depth, parti- cularly in polar waters (Phillips, 1969). The ocean bottom exerts a retarding force on any near bottom water moving over it. The resulting shear zone again causes turbulent mixing. In the absence of dynamic instabilities leading to a complete overturn, the bottom turbulence will produce a mixed layer similar to that formed in the surface layer and consequently intensify the the thermocline from the bottom. If a thermocline is formed, it will inhibit the verti- cal extent of mixing from either above or below depending on the depth of the thermocline and the intensity of the gradient within it. Turbulence produces changes which tend towards three-dimensional isotropy, and as such will be seriously inhibited by a constraint in any direction. In this case, the vertical motion is inhibited and re- sults in a flow of energy out of the other two components of velocity in an effort to boost the vertical energy. Hence, an increase in stability as defined by an increased degree of stratification or density gradient may well be associated with the degradation of turbulence within the water column, and in particular, that associated with internal waves. Paradoxically, the thermocline, which tends to inhibit turbulent mixing from the vertical boundaries, may itself 15 be a source of increased turbulence. Internal waves propagate along the thermocline and can lead to increased turbulence through shears developed between water masses either above or below the thermocline. In addition, under certain conditions the different rates of diffusion of heat and salt across the thermocline lead to a small scale perturbation in the temperature field. For the purposes of this thesis, the scales to which turbulent temperature fluctuations may be broken down fall into three convenient, though not necessarily inde- pendent, classifications. These are: micros tructure effects, surface wave effects, and internal wave effects. Each of these will be discussed separately. A. MICROSTRUCTURE EFFECTS Much evidence is now available which shows that ver- tical mixing in the interior of stable fluids occurs, on occasion, at scales much smaller than the vertical extent of the water column (Woods and Wiley, 1972) . Although turbulence can enter at any boundary of a fluid, as long as vorticity is generated, two processes have been proposed as capable of producing intense vertical micros tructure activity. The first arises from the differ- ing rates of diffusion of heat and salt (double-diffusion or salt-fingering). The second process is turbulent mix- ing driven either by surface winds or by shear instabili- ties (billow turbulence). It occurs at the vertical 16 boundaries or on internal waves at depth. 1 . Double- Diffusion and Salt-Fingering This process is a local one operating where weak mixing has already increased the vertical gradients of temperature and salinity, and whose vertical scale seldom exceeds a few meters (Gregg, 1973). Considering only a two layer system involving both heat and salt gradients, three distinct situations can arise. The first is an inherently stable system where the upper layer is both warmer and less saline than the lower layer. It is of less interest because both compo- nents act together to produce a stable density step across the interface. The second situation is an inherently unstable system where the upper layer is both warmer and more saline than the lower layer. In a very few minutes there will be a rapid loss of heat in the upper layer making it more dense than the lower layer, which is simultaneously gaining heat and consequently becoming less dense. The result is alternate fingers of 'heavier' water sinking and 'lighter' water rising, hence its name - Salt-Finger- ing (Figure 2) . The pattern of convection will persist only to a limited depth where the descending more saline water undergoes weak overturning to form a well mixed layer of temperature and salinity intermediate to the parent waters. In such a system a series of steps will form beneath the interface with steps of the order of 17 • • • • • • ♦ ♦ • * • « • Hoi ♦ • • Sally • • • » \» 1 i 1 Vj^~^ f-Tv^ • i % * « • « « Cold Less Sally « - 4 « t * • • 1 • • « JL. • 1 * ** V • • / \ • * / • • • iT t . . • • . » • - • • • • • 5^« ./a . •• .*A. / \ . • / \* 7 * \ • ■/ \ / \. • * ' * \ /. A . I . . •/./'■• S* "* -J • J- "" "* •*. % . i • * 1 * \.V-| i*j • \ * / » * * \ / \ • / V * / i • . ^ • . • Figure 2. SALT-FINGERING- The solid lines show the ini- tial and subsequent density profiles when the more rapid diffusion of heat has produced a potentially unstable situation at the inter- face. (From Gregg, 1973) +j CM a Note: The probe is being raised in the lower part of the record illustrat- ing the disturbance caused by the wire and the probe moving through the water ahead of the sensor. 340m 340m 330m Figure 3. High resolution temperature profile of the 310-345m interval in the Arctic off Ice-Island T-3 (From Denner, 1971) . 18 20-30 centimeters and interfaces as small as two centi- meters . The third situation is one in which the layers remain separate but where convective mixing occurs within each. It is characterized by a cooler, less saline upper layer and a warmer, more saline lower layer. Again there is a rapid diffusion of heat across the interface but now it acts to increase the stability between the two layers with intense convective mixing within each of the layers. In a manner similar to that for salt- finger ing the convec- tion is self -limiting and extends into a series of steps above the original interface. Neshyba, Neal and Denner (1971) found as many as 40 steps above a layer of warm saline Atlantic water that had entered the Arctic. Each is about 3-10 meters per step and with interfaces of the order of centimeters (Figure 3) . 2 . Billow Turbulence Billow turbulence is described as free shear tur- bulence modified by a density gradient and initiated by Kelvin-Helmholtz instability (Woods and Wiley, 1972) which is illustrated in Figure 4. This micros true ture is very complex in form as opposed to the rather simple, symmetrical configuration of the density steps in double- di f f us ion . It appears that a great majority of the flow with- in the thermocline is laminar and devoid of turbulent 19 Figure 4. KELVIN-HELMHOLTZ SHEAR INSTABILITY The upper layer has a higher velocity and a lower density than the lower layer. The instability, generated in a two-layer laboratory tank by rapidly raising and then lowering one end of the tank, caused the two layers to soon lose their coherence and break up into turbulent patches. The diagrams are based on a study by S. A. Thorpe of the National Institute of Oceanography in England. (From Gregg, 1973) 20 motions on any scale (Woods and Wiley, 1972). Occasion- ally, however, shear instability is triggered within small regions by boundary produced shears or internal gravity waves. In this latter case, as one of the wave packets travels along a sheet, shear instability is pro- duced at the crests of sufficient height to cause the local Richardson's number of the billow to fall below a critical value of 0.25 (Woods and Wiley, 1972). The Richardson number for the sheet is given by J = ghAp pAV2 where "g" is gravitational acceleration, "h" is the thick- ness of the shear zone, "Ap/p" is the fractional density difference, and "AV is the velocity difference across the shear. For a given value of J < 0.25 the layer will be unstable to a specific range of wavelengths, of which L = 7.5 x h, will be the fastest growing disturbance (Woods , 1968a) . The condition for termination of turbulent motions is that the Richardson number again increases above the critical value of 0.25 to about unity. When this is reached, the billow no longer extracts sufficient kinetic energy from the shear to supply the billow turbulence and the turbulent motions decay. Woods and Wiley (1972) found that in the sea the approximate scales were 75 cm for the billows, 500 cm for the patch of turbulence, and 300 sec 21 for the lifetime of the turbulence. B. SURFACE WAVE EFFECTS During the past few years considerable effort has been put into understanding the effects of wave motion on the thermocline but relatively little has been done to understand the internal temperature effects such waves produce. Clearly, temperature fluctuations in the near surface region can be caused by a number of factors, in- cluding surface waves and their associated turbulence, internal waves, and advection. Surface wave effects on temperature fluctuations are usually considered only to depths of less than one-half of one wavelength due to the exponential decay of particle motion with depth. How- ever, the energy densities of surface wave-induced fluc- tuations are in the same frequency band and of the same order of magnitude when they are present simultaneously and hence are very difficult to separate. Therefore the spectral separation of the turbulent and wave-induced temperature fluctuations will be considered in greater detail later in this thesis. The temperature field can be considered as a turbulent field having an "energy" containing region, an "inertial" subrange, and an "energy" dissipation region analogous to that hypothesized for velocity fields. In addition, since the rates of thermal and viscous dissipation arc different in the ocean, the smaller scale temperature field will tend to persist longer than the velocity field. 22 1 . Surface Wave-induced Temperature Fluctuations Earlier, mention was made of a paper (Thornton et al, 1974) setting out a relationship between wave-induced temperature fluctuations and the waves. The major portion of what follows is taken directly from that paper and is included here, in a somewhat abbreviated form, for com- pleteness . Thornton (1974) had previously shown that linear wave theory estimated the wave-induced water particle motion very well under small amplitude wave conditions. Since it worked for particle motion, linear wave theory was assumed to estimate the wave-induced temperature fluc- tuations. This implies that the flow is irrotational . It was further assumed that the temperature fluctuations are small, allowing density to be considered constant over short distances. Hence, temperature is considered to be a passive scalar with buoyancy effects neglected. Assum- ing temperature is a conservative property, at least in the body of the flow, the conservation of temperature flux is given by 36 36 36 -r— + U . -5— + V -it— 3 1 1 dx . 3z 1 (2) where 6(x.,z,t) is the measured temperature , u . (x . , z , t ) are the horizontal velocities, v(x. ,z,t) is the vertical velocity, x. are the horizontal cartesian coordinates, z is the vertical cartesian coordinate. The equation 23 considers the individual time change, advection, and dif- fusion of the passive scalar temperature. In terms of their means, turbulent fluctuations and wave-induced fluctuations the quantities are defined as e = 6 + 6 +8* w u.= u.+ (u ). + u . ' 11 W 1 1 V = V + V ' w where the overbar indicates a time-averaged mean, the sub- script w indicates contributions from surface waves, and the prime indicates turbulent motions. Assuming that (1) turbulent and wave-induced motions are sta- tistically independent, (2) mean temperature isotherms are horizontal, (3) mean velocities are zero, and (4) the mean vertical temperature gradient is constant , allows separation of the turbulent motions and Equation (2) reduces to 80 36 -5-^ + (u ). 3-^ + v -_ (0 + 0 ) = 0 dt widx. w3z w (3) where the velocities and the assumption of irrotational flow are as specified in linear wave theory. Temperature fluctuations caused by wave-induced velocities are a priori assumed equal to zero and 24 therefore dQ 89 (u ) . -5—^ + V 3-5L = 0 . w 1 dx . w 3z 1 This leaves a first order, constant coefficient, homogeneous differential equation, if the mean tempera- ture gradient is constant, and therefore d9 r, 5 88 V . (4) dt 9z w This is easily integrated to give q / 4-\ r 38 smh k(d + z), ., „ .. ,_. 6 (x.,z,t) = [- -5 : — - ] a cos (k.x.-a.t)(5) w 1 dz smh kd 111 where V (x. ,z,t) is the wave-induced vertical water-par- w 1 ticle velocity specified by linear wave theory, 'a' is the surface wave amplitude, k is the wave number, 0 is the radian frequency, d is the depth, and z is the nega- tive downwards displacement from the mean water surface. In terms of the spectral transfer function, Hfl (0) (the square bracket in Equation (5)), and the sinusoidal surface displacements, r)(x.,t), Equation (5) reduces to ) (x.,z,t) = Hfl (a) nU. it) (6) W 1 U X Equation (6) satisfies Equation (3) exactly, which means that moderate swell type waves do little or no mixing, but only serve to pump the thermocline up and down. 25 2 . Turbulent Temperature Spectrum In a manner similar to that discussed in Thornton, (1974) the paper by Thornton, Boston, and Whittemore (1974) discusses the separation of the turbulent temperature spectrum from the wave-induced temperature spectrum. For completeness the arguments are repeated here. The wave-induced temperature spectrum, SQ (0) , is w calculated from the wave spectrum, S (a) , using the com- plex transfer function, Hfl(a) (given in Equation (6)), in the form sfl (a) = |h„ (a) | " s la) . (7) w In terms of the total temperature spectrum the turbulent temperature spectrum is given by sQl(a) = sQ(a)+se (a)-2Re{HQ (a) «sQ (a)} . (8) w n If the cross-spectrum is expressed in terms of its co- and quadrature spectra S0 (a) = CQ (o) + iQQ (a) n n n then Equation (8) becomes s01(o) = sQ(a) + se (o) - 2Hg(o) Cg (a) (9) w which gives the turbulent temperature spectrum in terms of measurable quantities. 26 Assuming statistical independence between turbu- lent and wave-induced temperature leads to further sim- plification, since SQ (a) = SQ (a) n wti and HA(a)sfl (a) = s (a) = | h„ (a) | s (a) '9 l"'"6 wn w n This allows the turbulent temperature spectrum to be given simply as the actual temperature spectrum minus the wave-induced temperature spectrum, or, s01 (a) = sQ (a) - |hq (a) | s (a) (10) Further the coherence between temperature fluctua- tions and the waves is then given by 2 se-(a) -i ^e (a) " [1 + irHzY] n sQ (a) w sQ (a) * (11) w C. INTERNAL WAVE EFFECTS Internal waves are subsurface waves existing between layers of different density or within layers where a ver- tical density gradient exists. They occur often due to a- variety of causes such as flow over an irregular bottom, atmosphere disturbances, tidal forces, and shear flow. 27 The theory of the existence of internal waves implies perfect coherence of plane waves over space; whereas in practice naturally occurring internal waves are not so coherent. In addition, the ocean contains a number of generating sources which may reinforce or cancel each other (LaFond, 1966) . In shallow water, at the same site as this experiment, LaFond (1966) found internal waves of amplitudes ranging from 2 to 23 feet with mean amplitude of 5.6 feet. In general, the magnitude of the shorter period waves was found to be inversely proportional to the gradients in which they were found. Theoretically, internal waves have periods ranging between the lower inertial and higher Vaisala or stability frequencies. LaFond found periods ranging from 4 to 10 minutes superimposed on longer diurnal cycles and four and one-half day cycles. He also found that internal waves moving towards the shore had an average speed of 0.31 knots and ranged from 0.11 to 0.60 knots. Physical intuition would lead us to suspect that tem- perature fluctuations associated with internal waves should exhibit a linear relationship between the temperature gradient and the wave amplitude. Therefore a reasonable expression for the temperature fluctuations associated with internal waves would be 6. (t) 1W ii a z Vt: 28 where the overbar indicates the time-averaged mean and A (t) is the instantaneous amplitude of the internal wave. o The temperature fluctuations due to internal waves are usually easily identified because of the narrow range of frequencies where significant energy is present, and the presence of a valley, in the spectral energy density, separating the surface and internal wave regions. Further, energy contained in the turbulent temperature fluctuations is small relative to the energy of the internal waves, and is, therefore, only a minor feature at these low frequen- cies. :>•■ Ill . EXPERIMENT AND DATA ANALYSIS A. EXPERIMENT Measurements were made at various depths with either a vertical or horizontal orientation of the temperature array as summarized in Table I. TABLE I. Summary of Experimental Set-up for Minard's May 1973 Experiment Run Number Date Time Depth to top of frame (m) Array disposition Internal waves Digi- tized Tide 1 15 May 1300 2.7 vertical no yes ebb 2 15 May 1437 5.2 vertical yes(l) yes slack 3 15 May 1607 11.3 vertical yes (2) yes flood 4 15 May 1722 2.7 vertical no yes flood 5 16 May 1017 2.7 horizontal no no ebb 6 16 May 1050 2.7 horizontal yes(l) yes ebb 7 16 May 1244 8.5 horizontal yes (1) yes ebb 8a 16 May 1410 13.1 horizontal no yes slack Figure 5 illustrates the array configuration for the first four runs, conducted on 15 May, where the placement of the thermistors was vertical. Figure 6 illustrates the configuration of the array on the second day, 16 May, when the thermistors were in a horizontal line. Run 6 was conducted in this configuration but in all subsequent runs the array was rearranged such that Thermistor 1 was removed and placed 24 inches (60.8 cm) below Thermistor 5. 30 Baylor Wave Gauge 21 cm 71 T 84 cm I I I ± - 0 cm - 7.6cm -15.2 cm -30.4 cm 5 8.9 cm Flov/Mefer 60.8cm -96.5 cm ^ < Q s -162.5 cm Figure 5. Configuration of Array for Runs 1, 2, 3, 4 31 r-« CM CO ** =*: 1 =fe o «o CM ■~- ■-•■■— ^ il.rv-7; :;•-£ , I.I- Thermistor 1" * ■ ■ 1 1 ■ ■ « ■ r — t 9 t » ■ v s * ■ ■ f v 1 1 r- i t . . ■ ■ -. I . -> i If >" tit- t-hrri Thermistor "2~" ii^"^^^^"n< f* r~r "b Wrj 'T ' m&vtl^^SB _^J^*^SS^-r^^■^^-,•y^v^ -^^xv-awv^r Thermistor 3. H ! 1 h jj J.^tg;£pc:-]'K ±-4;2^S - .Thermistor 4. 64 sec Thermistor 5 P"" Thermistor 6 ■ rlii* i-ii i i~i j i . "rr',!l Thermistor 7 » f'^Aw^/'-J^^Vv., $0.35°C / ! ■^^^ LUjuv^^T W i : • ... ..,•...■ Arrow indicates a temperature increase Figure 7. Analog Temperature Record-Run 6. 38 vf ft "•-•-•■■ -p* "T«t>;^i!Vr*tM--?;r» ^; v-*v*r.T~n ~r"tif.7i~ 7]"'7'"~ -"— t .Thermistor 1 - ' - - - ----- ._4o.T7°c ... .;.— — w— — ir*H /•'■ / — v'1 jm w r ■** .._:*::.■■ 64 sec -: ;■ '-:— r .£";.-.">-::! -•_'- "j - 1 -.L_L4_.;_-L- v-A./^' r* V\w. Th e rrai sior ,._2 X — frv-V1*^ ---I: --t — ♦ — r — * ! ' '-: !" ■ -! -:- I i : ■: I ■■■■■■•-.■■■: i A.^^-v^V^^- T h e r rax s-t'o r - 3 -*- :T" ;.:-41^ FT; IT, :;_Th.ernuLstx).r...4 .l _• ---UV-- /> Thermistor 5 1* ^^^^-^Wlj^ lit' j'VJH!}, Thermistor 6 fa Thcrmis tor 7 . . $0.7°C "xy^-YYi $0.'3 5°C ■Vjr"' \y ■• Arrow indicates a temperature increase Figure 8. Analog Temperature Record-Run 1 39 o VO o r^ T3 C nj 0) o U o 0 vo -p w •H e Q) o o + Eh in „ "u iH CD CO C 3 CD tf e o •H ^ o Eh ■0 ^r -a- o o u CD « cu n +J rtl 0) s (1) Eh (0 +J ■H 01 •H Q 0) ^1 3 C7i •H fa +. (D0) oan^ejDduioj, 40 2-lto O N Q L"> (uio) q^daa i — r~r O <0 CM r^ in O o 10 u-i 01 en o o en (N • CO iH kU C 3 Pi <& U vD 7i in + 0 -P C) a a) 0 tU C) 1 — in (1) g in R M ■sT ■H ll> Eh +j 0 m H o ^r 1 n rH O u xs c rd Q) U c rd •H M (0 > H H H W m Q) O n3 C tt) Ifl >H •H rd — «-l >in ■H o fa H W X CO d) fO OCN a a u i id o x: ■H »-» tn *H ■h cd EC > 0 O 0) U c rd ■H M id T3 > CD ^-s M IT) (1) O +J rH H CD •H U X h C 1 rdCN c •■H U o ^ o 2 rd — > ■ o u en CO • • ro ^f rH o cn CN cn CO ro cn ro LO cn CO o rH CO ro co o CO iH CN o r- • • « • o • • r> o o r> ro "sT ID CM ro LT> ro M1 i-H lO cn cn cn o rH in CO rH i-H H rH ld ^ -^ in <* m r> oo <£> ^o •& o cn n r~- i£> co co m co r-^ cn cn p- co co t-~ cx> oi co co co co r^ in CD u g c ^ Cn rd e r^- rd -P U • rH~ CO —' CN •r) Q ■P CU CD )H Q 0 P 1 U) •rH E r-\ Vh cu c x: p En « O CN \D CN •rj< CO . o rH CN CN CN CN CN rH ■* ■* in in 'j in o> ^ cn t cn cn cn in vD CO VD rH CN o co cn ro <* cn cn m vd oo cn cn o o rH cn i-H i-H rH CN CN CN .rH e CN in .c p Ch| m n id iD -3" ro cn rH H rH ^ cn in ro cn in co o cn m.co cn rH o rH f- 4 9 ^ -H in iD • • -* £> (N o oi ci ro ro in 01 n f l1 01 in r\i n 01 on 01 in H n ro ro ■=* ro i-i iH cn id i£ id n m (^ in in ro o CO ^t1 01 H 0\ M V0 <* CN ro CO O m CO ro rH co in "31 cn .H .h m cn ■=? O oi ro Oi cn ro co T ro cn ro cn cn r-H cn H o r-» cn o in CN ro co ro 01 o m cm id id h n oi m H O lO H H 01 in m in id ro to id co m h o co io o c» i-H rH rH CN rH CN ro r~- O UO cn co o r- rH o o id n h oi id o n in oi ■* in h in ^f rH CN rH ****** c o 03 O p o c e CM p CM I G O -H -P rd .H rd ft co 0) 0) ■M <-{ -H fc< I C o S3 CD M 00 +- UOT^PX3^J°0 POZTTPIUJON 51 0 U c id rH (0 >i < ra u •H -P U 0) > c o ■H -P ffl cH Q) U U O u rd •H 4-> rd & 1 ■P o N •H 5-1 O o •p iH (1) H M O U ■P 0) •H UOT^T?T3JJ03 pDZTTICUIJON vertical array at similar depths. However, this result is not conclusive since horizontal and vertical array runs were made on different days. Further examination of Table III shows that the runs with significant internal wave activity have higher energy as evidenced by the higher numerical values of variance and covariance. This trend is present even when the signal was filtered for waves of 100 sec and greater except that Runs 2 and 3 are not significantly greater in energy than Runs 1, 4, and 8a. This indicates that Runs 6 and 7 contain significantly more energy at all frequencies than Runs 2 and 3. Also the difference in covariances between signals not filtered and filtered show that Runs 1 and 8a contained much less energy at periods greater than 100 sec than any of the other runs. D. WAVENUMBER SPECTRA The normalized wavenumber spectra were calculated by Fourier transforming the spatial correlation functions (Figures 16 and 17) calculated as explained in Section III.B. The spatial correlation was calculated at equal incremenr.s along the length of the array by linearly in- terpolating between the calculated points on the array. Figure 18 is a composite of the normalized wavenumber spectra when the array v/as vertical (15 May) ; and Figure 19 is a composite when the array was horizontal (16 May). 56 o 2 1.0 £ o \ u t, .1 >1 ■p •H 0) c « Run 8a -5/3 -2 -3 -4 I J. -2.30 -1.G0 -0.90 -0.20 LOG FREQUENCE IH2 Figure 25. Turbulent Temperature Spectra 0.5 77 temperature fluctuations and that the natural preference in nature is for a near -5/3 dependence of turbulent spectral energy on frequency in the inertial subrange. Seitz (1971) has suggested that in a stationary, homogeneous, isotropic field of turbulence the energy spectra will have a dependence on a -2 power law at low frequencies and. a -5/3 power law at high frequencies pro- viding convection velocities are moderate. The transi- tion from one power law to the other should be at some intermediate frequency and would be a function of the magnitude of the convection velocity. Following this line of reasoning then it can be seen in Figure 25 that all runs exhibit inertial subrange character in the energy spectra above 0.1 Hz„ Runs 1 and 8a, which had no apparent internal wave activity, exhibit similar slopes of -7/3 below 0.031 Hz with transition between these fre- quencies. However, Runs 6 and 7, which did show signifi- cant internal wave activity, exhibit slopes of -3 and -4.5 respectively below 0.031 Hz. This would tend to reinforce the notion that internal waves contain much more energy at the lower frequencies and are primarily responsible for the time series being non-stationary. 78 V. CONCLUSIONS The temperature field is extremely complex in the shallow waters near the NUC Tower. There were indica- tions that double-diffusion type microstructure was present as well as the more normal, for this area, billow turbulence microstructure which appeared to be suggested by the temperature stratification for the majority of the runs. Temperature fields with no apparent internal waves may contain turbulent oscillations suggestive of internal waves or their remnants. The basis for this statement is that the oscillations in Run 1 appeared to be periodic of period three minutes which is close to that reported by LaFond (1966) for internal waves. Significant internal waves present on the two days of the experiment were periodic with approximately 20 minute periods. In form they were more like solitary waves rather than sinusoids but this is probably due to the fact that they were shallow water waves at this location. Judging from the temperature profiles the internal waves appeared to have positive displacement from a mean position with little or no negative displacement. The positive displace- ments occupied less than one-half of one period. Internal waves appeared to be the major source of tur- bulence during this experiment under the conditions of 79 light winds and small ampitude waves. This conclusion is based on the higher variances in the runs with internal waves . When internal waves were present there was signifi- cantly more energy at all wavenumbers and not only at low wavenumbers. This statement is predicated on the fact that even after filtering out waves of period 100 seconds and greater the runs with internal waves still had more energy than runs without internal waves. Internal waves may possibly break, thus contributing a large amount of turbulent energy to a localized area and thus assist mixing. With respect to spatial correlation lengths, the length of the array (64 inches = 1.6 meters) was too short to measure the scale length of the internal waves. The high- pass filtered time series had correlation lengths of 130 cm and less. There did not appear to be any depth dependence on the length but there may have been a dependence on whether the array was vertical or horizontal. With respect to the gradient field, it is confirmed that temperature gradients taken over short distances are extremely variable over time and space as expected for a turbulent phenomenon. Wavenumber spectra may be significantly altered at higher wavenumbers depending on the type of interpolation used in determining the spatial correlation function to be Fourier transformed. Further, the vertical and perhaps 00 the horizontal particle displacements at the frequency of maximum coherence in the frequency spectra appear as changes of slope in the wavenumber spectra. The conclusions reached in respect -to the spectral model proposed by Thornton et al (1974) were that: 1. The model is not always realistic in the speci- fication of a constant mean gradient. The gradient required to be used in the transfer function to achieve consistency between the actual temperature spectrum and the wave-induced tempera- ture spectrum was much less than the observed or calculated mean gradients. 2. The wave-induced temperature spectrum of the model showed great similarity to the actual tem- perature spectrum over a narrow band of frequen- cies associated with regions of high coherence between surface waves and temperature spectra. 3. The model describes reasonably well the wave- induced temperature fluctuations in shallow water (less than 18 m) . 4. The model illustrates that moderate waves do little mixing and serve only to pump the thermocline (and temperature field) up and down. 5. The temperature field lags the wave field by ap- proximately 180°; but the lag is modified by as yet undetermined factors and depth. The spectral analysis yielded the above with respect to surface waves and the results of the isotherm plot in Run 3 indicate the same for internal waves. 6. The model demonstrates the high probability of statistical independence between turbulent and wave-induced temperature fluctuations. 7. The turbulent temperature spectra agrees very well with inertial subrange theory in that the slopes of the log/log spectra are very near -5/3. That the slope is nearer to -2 may indicate that the time series was non-stationary. 8. The low frequency end of the turbulent spectra appears to have different slopes depending on whether the individual run had internal waves present . nl LIST OF REFERENCES 1. Black, C. F., The Turbulent Distribution of Tempera- ture in the Ocean, The Bissett-Berman Corporation, Report MJO 1049, p. 1-8, December 1965„ 2. Denner, W. W., "The Layered Microstructure and Acoustic Propagation in the Arctic Ocean." U.S. Navy Journal of Underwater Acoustics, Vol. 21, No. 1, p. 45-51, January 1971. 3. Gregg, M. C, "The Microstructure of the Ocean." Scientific American, Vol. 228 , No. 2, p. 65-77, February 1973. 4. Gregg, M. C, Cox, C. S., and Hacker, P. W., "Vertical Microstructure Measurements in the Central North Pacific." Journal of Physical Oceanography, Vol. .3, p„ 458-469, October 1973. 5. LaFond, E. C, "Internal Waves." The Encyclopedia of Oceanography, Encyclopedia of Earth Sciences Series, Vol. 1, Edited by R. W. Fairbridge, p. 402-408, Reinhold Publishing Corporation, 1966. 6. Minard, J. E., A Study of the Effect of Internal Wave Induced Turbulence on Small Scale Temeprature Structure in Shallow Water, M. S. Thesis, Naval Post- graduate School, September 1973. 7. Neshyba, S., Neal, V. T. and Denner, W. W., "Tempera- ture and Conductivity Measurements under Ice-Island T-3." Journal of Geophysical Research, Vol. 76 , No. 33, p. 8107-8120, November 1971. 8. Phillips, O. M., The Dynamics of the Upper Ocean, Cambridge University Press, 1969. 9. Chesapeake Bay Institute Technical Report 72, Results of a Field Study using the 3-Axis Doppler Shift Current Meter, by R. C. Seitz, Reference 71-6, The Johns Hopkins University, September 1971. ' 10. Thornton, E. B., Separating Turbulent and Wave-Induced Fluctuation: Part I, Water Parcile Velocities, Naval Postgraduate School, 1974. 11. Thornton, E. B., Boston, N.E.J. , and Whittemore, M.A.N. , Separating Turbulent and Wave-Induced Fluctua- tions: Part 2, Temperature, Naval Postgraduate School, 19 7 4. 82 12. Whittemore, M.A.N. , Small Scale Temperature Fluctua- tions near the Sea Surface, M. S. Thesis, Naval Post- graduate School, March 1973. 13. Woods, J. D., "Wave-Induced Shear Instability in the Summer Thermocline . " Journal of Fluid Mechanics, Vol. 3_2' Part 4, p. 791-800, 1968. "" 14. Woods, J. D. and Wiley, R. L . , "Billow Turbulence and Ocean Microstruc tur e . " Deep-Sea Research, Vol. 19 , p. 87-121, Pergamon Press, 1972. 83 INITIAL DISTRIBUTION LIST 1. 2. 3. 4. 5. 6. 7. 8. Defense Documentation Center Cameron Station Alexandria, Virginia 22314 Library (Code 0212) Naval Postgraduate School Monterey, California 93940 Department of Oceanography, Code 58 Naval Postgraduate School Monterey, California 93940 Dr. E. B. Thornton, Code 58 Department of Oceanography Naval Postgraduate School Monterey, California 93940 Dr. Noel E. Boston, Code 58 Department of Oceanography Naval Postgraduate School Monterey, California 93940 Oceanographer of the Navy Hoffman II 200 Stovall Street Alexandria, Virginia 22332 Naval Oceanographic Office Library (Code 3330) Washington, D. C. 20373 Staff Officer External Affairs Defence Research Establishment Ottawa Ottawa, Ontario, Canada K1A OK2 No. Copies 2 10. Staff Officer External Affairs Defence Research Establishment Pacific Esquimalt, British Columbia, Canada Staff Officer External Affairs Defence Research Establishment Atlantic Dartmouth, Nova Scotia, Canada 84 11. Commandant Canadian Forces Fleet School C F B Halifax Halifax, Nova Scotia, Canada 12. Captain J„ W. Powell 1072 Indian Village Road Pebble Beach, California 93953 13. Director of Defense Research & Engineering Office of the Secretary of Defense Washington, D. C. 20301 ATTN: Office, Assistant Director (research) 14. Office of Naval Research Arlington, Virginia 22217 ATTN ATTN ATTN ATTN (Code 480) 3 (Code 460) 1 (Code 102-0S) 1 (Code 105) ■ 6 15. Director Naval Research Laboratory Washington, D. C. 20375 ATTN: Library, Code 2620 6 16. Commander Naval Oceanographic Office Washington, D. C. 20390 ATTN: Code 1640 1 ATTN: Code 70 1 17. NODC/NOAA 1 Rockville, Maryland 20882 18. National Defence Headquarters Ottawa, Ontario, Canada K1A 0K2 ATTN: DGITP 1 19. SI0 Library 1 University of California, San Diego P. 0. Box 2367 La Jolla, California 92037 20. Department of Oceanography Library 1 University of Washington Seattle, Washington 98105 21. Department of Oceanography Library 1 Oregon State University Corvalis, Oregon 97331 85 SECURITY CLASSIFICATION OF THIS PAGE (Won Data Entered) REPORT DOCUMENTATION PAGE READ INSTRUCTION'S BEFORE COMPLETING FORM REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER 4. TITLE (and Subtitle) An Investigation of Surface and Internal Wave-Induced Turbulence in Shallow Water Thermal Microstructure 5. TYPE OF REPORT & PERIOD COVERED Master's Thesis March 1974 6. PERFORMING ORG. REPORT NUMBER 7. author^; James Wesley Powell 8. CONTRACT OR GRANT NUMBERfsJ Project Order No.P04-0121 ONR 55859 Research Project RR131-0301 9. PERFORMING ORGANIZATION NAME AND ADDRESS Naval Postgraduate School Monterey, California 93940 10. PROGRAM ELEMENT. PROJECT, TASK AREA 4 WORK UNIT NUMBERS Program Element 61153N Task NR 083-275-2 II. CONTROLLING OFFICE NAME AND ADDRESS Naval Postgraduate School Monterey, California 93940 12. REPORT DATE March 1974 13. NUMBER OF PAGES 87 14. MONITORING AGENCY NAME 4 ADDRESSf/f different from Controlling Office) Office of Naval Research Arlington, Virginia 22217 15. SECURITY CLASS, (ol this report) Unclassified 15a. DECLASSIFICATION/ DOWN GRADING SCHEDULE 16. DISTRIBUTION ST ATEMEN T (ol this Report) Approved for public release, distribution unlimited. 17. DISTRIBUTION STATEMENT (ol the abstract entered In Block 30, II dltlerenl from Report) IB. SUPPLEMENTARY NOTES Master of Science Thesis supervised by Dr. E. Thornton and Dr. N. Boston, Associate Professors, NPS 19. KEY WORDS (Continue on reveroe aide If necefiemry and Identify by block number) Internal Waves Temperature Microstructure Temperature Fluctuations Temperature Gradients Temporal and Spatial Spectral Analysis Turbulent Temperature Spectra Wave-induced Temperature Fluctuations 20. ABSTRACT (Continue on reverae aide If necemaary and IdentMy by btock number) , Measurements of temperature, wave height, and orthogonal water particle velocity were made in May 1973 at the NUC Tower located one mile off Mission Beach, California. A detailed analysis of the tem- perature field was made using digital temperature and isotherm contour plots. Billow turbulence microstructure appeared to be the dominant mechanism with the exception of one run in which evidence of double- diffusion microstructure was found. Spatial correlation lengths;, DD ,FJAT7J 1473 (Page I) EDITION OF 1 NOV 65 IS OBSOLETE S/N 0 102-0 14- 6601 | '.ECUHITY CLASSIFICATION OF THIS PA0« (Whan Dal* Bnlmfd) 86 CliC-UWITY CLASSIFICATION OF THIS PAGEf»7i«n Dntm Entered) calculated from the plot of the covariances, were of the order of 130 cm and less when the signal was high-pass filtered for waves of 100 seconds and longer. No depth dependence was noticed. Both fre- quency and wavenumber spectra were calculated and a correspondence between the spectra was noted at the frequency and wavenumber of the surface wave-induced particle displacements. The Thornton, Boston, Whittemore model of wave-induced tempera- ture fluctuations was tested and found to model the temperature spec- tra quite well, especially in a narrow band of frequencies associated with surface waves. The turbulent temperature spectrum, calculated as the difference between the actual and wave-induced spectra, had a slope near -5/3 above 0.1 Hz and more negative at lower frequencies. DID Form L473 (BACK) , , 1 Jan 73 . . S/N 0 1 02-0 1 1 -660 1 SECURITY CLASSIFICATION Ot THIS PAGL'»7,«'i O.I. f„ffd) 87 74 2 3 5 0 6 150551 Thesis P7635 Powell m cl An investigation of surface and internal wave-induced turbulence in shallow water ther- mal microstructure. 2 35 06 74 "1 Thesis - 'u P7635 Powel 1 c.l An investigation of surface and internal wave-induced turbulence in shallow water ther- mal microstructure. of solace and intema1 P* IIP