DUDLEY KNOX LIBRARY NAVAL P( MONTEREY. C A 93943 NAVAL POSTGRADUATE SCHOOL Monterey, California THESIS THE EFFECT OF INTERIOR MOTION ON SEASONAL THERMOCLINE EVOLUTION by Janice P. Garner December 1983 Thesis Advisor: R. IV. Garwood , Jr. Approved for public release; distribution unlimited T213176 Unclassified SECURITY CLASSIFICATION OF THIS PACE r'#h»n Dele Entered) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER 4. TITLE (end Submit) The Effect of Interior Motion on Seasonal Thermocline Evolution 5. TYPE OF REPORT a PERIOD COVERED Master's Thesis December 1983 6. PERFORMING ORG. REPORT NUMBER 7. AUTHOR^ Janice P. Garner 8. CONTRACT OR GRANT NUMBERf*; » PERFORMING ORGANIZATION NAME AND AOORESS Naval Postgraduate School Monterey, California 93943 10. PROGRAM ELEMENT. PROJECT. TASK AREA a WORK UNIT NUMBERS II. CONTROLLING OFFICE NAME ANO AOORESS Naval Postgraduate School Monterey, California 93943 12. REPORT DATE December 1983 13. NUMBER OF PAGES 85 14. MONITORING AGENCY NAME ft AOORESSf// d///»r«nf from Controlling OKI cm) IS. SECURITY CLASS, (ot this report) Unclassified 15a. DECLASSIFICATION/ DOWNGRADING SCHEDULE 16. DISTRIBUTION STATEMENT (ot tM» Report) Approved for public release, distribution unlimited 17. DISTRIBUTION STATEMENT (ot the ebetrect entered In Block 20, II dllterent /rem Report) IS. SUPPLEMENTARY NOTES It- KEY WOROS (Continue on reveree mldo It neceeeery end Identity by block numbmr) mixed layer modelling, vertical advection, long period interior motion. 20. ABSTRACT (Conllnuo on reveree elde It nocoomory and Identity by block number) The response of the seasonal thermocline formation and mixing to prescribed vertical interior motion is examined. For the annual and shorter period interior motion cases studied, the response was strongest for the annual period. For an oscillatory vertical motion having a 15 m amplitude DD | jam 7) 1473 EDITION OF 1 NOV 68 IS OBSOLETE S/N 0102- LF- 014- 6601 Unclass if ied SECURITY CLASSIFICATION OF THIS PAGE (When Dete Enter: Unclassified SECURITY CLASSIFICATION OF THIS PAGE (Whan Data Entatad) at a 75 m depth, warm sea surface temperature anomalies of up to 2.16 C and cold anomalies of up to 2.04 C are found, depending on the phase difference between the interior motion and the annual heat cycle. The net phase-averaged effect on the mixed layer for annual period interior motion is a reduction in vertical mixing. Higher frequency motion produces a net enhancement of the mixing. S N 0102- LF- 014- 6601 Unclassified SECURITY CLASSIFICATION OF THIS » kG€(Whan Data Entarad) Approved for public release;- distribution unlimited The Effect of Interior Motion on Seasonal Thermocline Evolution by Janice P. Garner Lieutenant, United States Navy B. &., Montclair State College, 1972 M.A., Indiana University, 1974 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN METEOROLOGY AND OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL December 1983 DUDLEY KNOX LIB? ' NAV hoOL > 9394 ABSTRACT The response of the seasonal thermociine formation and mixing tc prescribed vertical interior motion is examined. For th<9 annual and shorter period interior motion cases studied, the response was strongest for the annual period. For an oscillatory vertical motion having a 15 m amplitude at a 75 m depth, warm saa surface tamperaturs anomalies of up to 2.16°C and cold anomalies of up to 2. 04 °c are found, depending en the phase difference between the interior motion and the annual heat cycle. The net phase-averaged effect on the mixed layer for annual period interior motion is a reduction in vertical mixing. Higher frequency motion produces a net enhancement of the mixing. TABLE OF CONTESTS I. INTRODUCTION 11 II. MODEL AND EQUATIONS 15 A. THE MODEL 15 1. Seme General Considerations 15 2. Equation of State 15 9. THE HEAT EQUATION 16 1. Initial Form 16 2. Scale Analysis and Simplification, of the Heat Equation 20 C. MIXED LAYER DEPTH 21 D. INTERIOR VERTICAL MOTION 23 E. POTENTIAL ENERGY AND HEAT 2 7 III. EXPERIMENTS AND RESULTS 32 A. SUMMARY OF PERTINENT EQUATIONS 32 3. RESULTS WITH NO INTERIOR MOTION 34 C. ONE-YEAR PERIOD INTERIOR MOTION 36 1. Effect on Mixed Layer Daoth 37 2. Effect on Mixed Layar Tamperatura .... 39 3. Effect on Heat Contant 41 4. Effect on Potential Energy 44 5. Summary of Response to Ona-Year Period Interior Motion 45 D. EFFECT OF SHORTER PERIOD INTERIOR MOTION ... 47 E. ASYMMETRY IN ENHANCED AND REDUCED MIXING ... 48 IV. CONCLUSIONS AND RECOMMENDATIONS 52 A. CONCLUSIONS 52 B. RECOMMENDATIONS 52 APPENDIX A: FIGURES 55 BIBLIOGRAPHY 82 INITIAL DISTRIBUTION LIST 83 LIST OF FIGURES 2.1 Turbulent Heat Flax Profile 18 2.2 Initial Temperature Profile 19 3.1 Net Surface Heat Flux 33 3.2 Heat Content Example 1 42 3.3 Heat Content Example 2 43 3.4 Two-Layer Ocean Example 50 A. 1 Temperature, No Interior Motion 55 A. 2 Temperature, One-Year Period, 90° Phase .... 56 A. 3 Temperature, One-Year Period, 270° Phase .... 56 A. 4 Mixed Layer Depth, Nc Interior Motion 57 A. 5 Mixed Layer Temperature, No Interior Motion . . 57 A. 6 Heat, No Interior Motion 58 A. 7 Potential Energy, No Interior Motion 58 A. 8 One-Year Period Interior Motion 59 A. 9 h (meters) , One-Year Period Interior Motion . . 60 A. 10 Ah (meters), One- Year Period Interior Mc-ion 60 A. 1 1 TML (C) , One-Year Period Interior Motion .... 61 A. 12 ^TM (C) , One-Year Period Interior Motion ... 61 A. 13 fMU (C) , One-Year Period, Second Year 62 A. 14 ATMU (C) , One-Year Period, Second Year .... 62 A. 15 H (100 *C m) , One-Year Period Interior Motion . . 63 A. 16 AH (100 °C m) , One-Year Period Interior Motion 6 3 A. 17 p (10* °C m2), One- Year Period Interior Motion 64 A. 18 AP (10* °C Di2), One-Year Period Interior Motion 6 4 A. 19 1/2-Year Period Interior Motion 65 A. 20 h (meters) , 1/2-Year Period Interior Motion . . 66 A. 21 Ah (meters), 1/2- Year Period Interior Motion 66 A. 22 fML (C) , 1/2-Year Period Interior Motion .... 67 A. 23 ATML (C) , 1/2-Year Period Interior Motion ... 67 A. 24 TMl_ (C) , 1/2-Year Period, Second Year 68 A. 25 ATML (C) , 1/2-Year Period, Second Year .... 68 A. 26 H (130 °C m) , 1/2-Year Period Interior Motion . . 69 A. 27 AH (100°C m) , 1/2-Year Period Interior Motion 6 9 A. 28 P (10* °C mz), 1/2- Year Period Interior Motion 7 0 A. 29 AP (10* °C m2), 1/2-Year Period Interior Motion 7 0 A. 30 1/3-Year Period Interior Motion 71 A. 31 h (meters) , 1/3-Year Period Interior Motion . . 72 A. 32 Ah (meters), 1/3- Year Period Interior Motion 72 A. 33 fML (C) , 1/3-Year Period Interior Motion .... 73 A. 34 ^TMU (C) , 1/3-Year Period Interior Motion ... 73 A. 35 H (130 °C m) , 1/3-Year Period Interior Motion . . 74 A. 36 AH (100°C m) , 1/3-Year Period Interior Motion 74 A. 37 p (10* °C m2), 1/3- Year Period Interior Motion 75 A. 38 AP (10*°C m2), 1/3-Year Period Interior Motion 75 A. 39 1/4-Year Period Interior Motion 76 A. 40 h (meters) , 1/4-Year Period Interior Motion . . 77 A. 41 Ah (meters), 1/4- Year Period Interior Motion 77 A. 42 fML(C), 1/4-Year Period Interior Motion .... 78 A. 43 ATKlL (C) , 1/4-Year Pariod Interior Motion ... 78 A. 44 H (100 °C m) , 1/4-Year Period Interior Motion . . 79 A. 45 AH (100 C m), 1/4-Year Period Interior Motion 79 A. 46 F (10* °C m*), 1/4- Year Period Interior Motion 80 A. 47 AP (10*°C m2) , 1/4-Year Period Interior Motion 8 0 A. 48 Peak Enhanced and Reduced Cooling Values, Year 1 81 A. 49 Peak Enhanced and Reduced Cooling Values, Year 2 81 ACKNOWLEDGMENT I wish to thar.k Professors R. W. Garwood and A.J. Willmctt fcr their valuable help and encouragement daring the undertaking of this thesis. 10 I. INTRODUCTION Interior vertical motion can play a significant role in the mechanical energy budget of the ocean boundary layer. Linden (1975) carried out experiments to observe mixed layer deepening in a stratified fluid. Ha found that internal waves are important in the energetics of the mixed layer and that the mixing process may provide a significant source of energy for the generation of internal waves. In this thesis the interior motion is prescribed ana to- affected by the mixed layer processes. However, the modelling of the inter- action of free internal waves with mixed layer processes is an important area for future study. Linden points out that he did not account for waves generated elsewhere which propagate into the region of considsration . Also, the internal waves considered by Linden ware of relatively high frequency. De Szoeke (1960) found wind stress curl- driven vertical advecticn to be as important as the entrainment rate in mixed layer dynamics. Cus hman-Roisin (1981) likewise found wind-driven advecticn to be as important as wind-driven mixing in mixed layer dynamics and temperature front formation. Much work has been done to study the ocean response to storms. Price et al. (1978) , in their work en the mixed layer response to storms, found it necessary to include vertical advection. Price (1981) found that upweiling will significantly enhance the sea-surface temperature response to a slowly moving hurricane. Adamsc at al. (198 1), using a two-dimensional model to study the oceanic response to a hurricane, found a strong interdependence between mixing and advection. Sreatbatch (1983) also found advecticn to be 1 1 important in increasing maximum cooling due to the passage cf a storm. Only recently has much been done to include long period interior motion in mixed layer models. Stevenson (1980,1981) examined the response of the mixed layer when perturbed by linear Bossby waves. He found no mixed layer response during the first half of the heating season while the mixed layer is shallowing and a maximum sea surface temperature response at the end of the heating season and beginning of the cooling season. Stevenson found that the maximum response of mixed layer d=pth to the wave occurred at or near the end of the cooling season. Also, he found that the lowest frequency waves produced the greatest mixed layer depth and temperature responses. This study differs from that of Stevenson in that a finite amplitude prescribed interior mo + ion is used rather than a linear Rossby wave. This produces a phase dependent asymmstry in the mixed layer response to the interior motion. Burger (1982) . successfully introduced tidal period vertical advection into the Garwood (1976,1977) one- dimensional mixed layer model. He found that the interior motion had a significant effect on the near-surface tempera- ture field and that the inclusion of the interior motion car- improve the model for single station forecasting. In this study, the Garwood (1976,1977) one-dimensional ocean planetary boundary layer model is used. Vertical advection was added to the model. The annual heat cycle is a sine function and wind stress is held constant at one dyne per square centimeter. By keeping the surface boundary conditions simple, the effect cf the interior motion can be more readily observed. The interior vertical motion is a prescribed, forced interior wave which is linear with depth and sinusoids! in time : 12 U,t) = _±_ w ,lH Mtrc , . v <1'1> [arte , \ where D is a reference depth, 200 maters, and w0 is the amplitude cf the prescribed vertical motion at depth, D. The periods, T, studied are one yaar, one-half year, one- third year and one-quarter year. The phase difference, , between the interior motion and the annual heating cycle is varied from 0° to 360° at increments of 22.5°. This interior motion is general and may ba representative of a number cf possible long period phenomena, for example, internal Ecssby waves or perhaps Ekman pumping on seasonal scales. The oceanic variables of interest are mixed layer depth, h, sea surface temperature which equals mixed layer tempera- ture, fMl_, the heat content, H, and the potential energy, P. These variables are contoured as functions of time and phase for each cf the perieds studied. Also, contours are made of the differences of these variables relative to the case of no interior motion. "Thesa contours are used to observe the effect of the interior motion on the mixed layer processes. Furthermore, the phase,

, will indicate vertically averaged, mixed layer mean. 2. Equation of State In the Garwood model, "he equation of state has the form : 15 - ft [ » - <*(T-T0) 4- (8(5- 50)] a (2.1) The thermal expansion coefficient, <*■ , is a function of T and is re -computed daily. The constants oe, TQ and Se are typical density, temperature and salinity values, respec- tively. In this study, salinity is fixed (precipitation and evaporation are disallowed) and sat equal to the value of Sp . Equation (2.1) then becomes B. THE HEAT EQUATION 1 • Initial Form The time rate cf change of temperature is given by the one-dimensional heat equation: o)t ^ V / ^VfXV ^ ^ The first term on the right is vertical turbulent heat flux, the second is solar insolation, the third is vertical temperature advection and the last, diffusion. All of the solar insolation will be assumed to be absorbed at the surface {in the first few centimeters) of the ocean. Hence, the solar insolation term will be dropped in favor of a surface boundary condition on the turbulent heat flux. Equation (2.3) then becomes «)t a / ;\ - It . ^ll . (2. ») = - — w' T* ~ w - — + K * z 16 In this model , mixed layer temperature is assumed to be vertically homogeneous. Equation (2.4) then reduces to a T,v,l --(—■) (2.5) for mixed layer temperature, TML. Obviously, fMi_ is equal to sea surface temperature because of the vertical homoge- neity assumption in the mixed layer. The surface boundary condition on heat flux is given by: w frO ~- Q f°c (2.6) where the zero subscript indicates the value of w' T' at z=0. The net surface heat flux, Q/^Oj,, is prescribed to be harmonic in time : _Q_ f'£P 0« PoCi 3. IT X c&<" (2.7) The amplitude of this sine function is set equal to a representative mid-latitude value of 4'10"3"C cm/sec. In this study, heat flux is positive downward. Time t=0 is the start of the "heating season", when Q first becomes positive . The boundary condition on heat flux at the base of the mixed layer is given by the entrainment heat flux. 17 (w'T1 ), = - A (2.8) W e where the subscript, h, indicates the value of w'T' at z=-h, the base of the mixed layer. The variable, We is the entrainment rate and A T is the temperature jump across the entrainment zone at the base of th? mixed layer. The turbu- lent heat flux is linear with depth in the mixed layer (due to the assumption of a well-mixed layer), mcnotonically decreases with depth in the entrainment zone and is zero below. The profile of turbulent heat flux is shewn in Figure 2.1. The net surface heat flux varies from positive Figure 2.1 Turbulent Heat Flux Profile. to negative values according to equation (2.7). The entrainment heat flux term, -Z^T rfa, is always negative or zero. This is because the entrainment rate is positive when the mixed layer is deepening but "turns off" and is zero for non-entraining shallowing cases. The thickness of the 18 entrainmer.t zone, 6 , is assumed zo be much less than ~h depnh of the mixed layer, h. Equation (2.5) now becomes dr. ni_ Q 1* AT We K (2.9) Nets that the advecticn term is absent. Thus, the interior motion does not direc-ly affect tha mixed layer temperature. Instead, the interior motion affects the mixed layer tempera-ure indirectly through the variables h, at and We. £(rvO 6.03 -75" ■- -lAM t approach zero. The resulting equation for retreat depth is 21 a fZS */l (2.11) (Garwood, 1977). In this equation, m, and m i are dimension- lass model constants, is the mixed layer mean turbulent kinetic energy. Also, u^, the friction velocity, is the square root of the surface wind stress divided by density. Throughout this study, it is set equal to one centimeter per second. Since mixed layer mean turbulent kinetic snergy depends on surface boundary conditions, the retreat depth is independent of the prescribed interior motion. In the case of active entrain ment , the mixed layer deep- ening rate in the model depends on the entrainment rate and W\ , the mean vertical velocity at the bottom of the mixed layer due to the prescribed interior motion: — = we - W^ , (2.12) The entrainment rate. We, depends on the entrainment zone temperature jump, AT, and the entrainment heat flux (see equation (2.8)). The entrainment heat flux is modelled by _ .„■,(„ ) N&/ (2_13) (Garwood, 1977), where mM is a model constant. The entrainment rate can then be expressed as Wj - - - • (2.14) « a K A I 22 Therefore, equations (2.12) and (2.14) give a solution for dh/ and are known. For moderately strong wind stress, mixed layer near. turbulent kinetic energy, , depends primarily on surface wind stress, which is held constant here. The vertical component of turbulent kinetic energy, (wT~2)1/2, depends primarily en surface heat flux and surface wind stress. When the surface heat flux becomes large enough, this term becomes zero, stopping the entrainment process and permit- ting shallowing. These two terms are unaffected by The interior motion. The entrainment rate is affected by the mean vertical motion indirectly as a result of changes in mixed layer depth, h, and entrainment zone temperature jump, AT. D. INTERIOR VERTICAL MOTION Consider for the moment a two layer, deep ocean with the upper layer about 200 m deep ani the lower layer about 5000 m deep. Assuming the fluid is incompressible, the continuity equation is differentiated with respect to z: rr-Z w = " « " - v <2- 15> where 0, V and W are the mean velocity components in either layer. Furthermore, assume that horizontal momentum is vertically well mixed in the upper layer. Thus, — * O ; H. = O (2-16) ^ ' jl 23 Than W must ba linear with depth in the upper layer. With a rigid-lid approximation, I = 0 at the surface. Our interest is confined to the upper 200 m and a linear vertical profile for W is a good approximation for long period interior motion. The functional dependence of W on z is W (i) - - -^r- * » (2.17) where D = 200 m is the reference depth and wD is the interior vertical motion at 200 m given by a sinusiodal function of time, namely + ) . (2.18) The amplitude, W0 , is defined by «"p 1 Z rrs For the one-year period interior motion, W0 = 8 '10-* cm/sec = 69 cm/day. Tha value of W0 will be 2, 3 and 4 times this for the one-half, one-third and one- guarter year periods, respectively. These values of W0 are selected so -hat the displacement amplitude at 75 m is approximately 15 ra. The actual values of the displacement amplitude depend on phase and will be discussed below. Combining equations (2.17) and (2.18) yields the equation fcr the prescribed interior motion, 24 The formula for the particle position due to the interio; motion can be obtained by noting that W = dz/dt, and it given by * = *0 **? -T Wo arr D co-i in t T + ] ~ Cod $ (2.21) For any fixed phase the maximum and minimum values of z are given by m,.^* -A = 20 ^*P -T We - Coi, p (2. 22) and = a „ a « p - T „ <2TT D + t - Ct-3

.« (2. 24) The maximum peak-to-peak displacement amplitude will occur when cos 4> =+1 and the minimum whsn cos 0 =-1. For the vertical mction amplitude given by equation (2.19), a particle with an initial position at z=-75 m will experience a displacement amplitude (half of the peak-to-peak value) between 12.4 and 18.4 m. The model time step is one hour. This is much smaller than the cne-year or even one-quarter year periods consid- ered. Hence, since WB does not vary significantly over a one hour time step, the integral of (2.17) can be approxi- mated by 25 24 s 2, **f (- hii At ) , (2.25) where .at«tt-t, is the time increment (one hour) and W^ is obtained from equation (2.18). In the model, vertical advection is handled by compressing or expanding the vertical grid according to equation (2.25). No significant difference in the results was found between model runs using equation (2.25) and equation (2.21). The vertical increment is initially one meter, but this varies as a result of the compression or expansion of the vertical grid. This variation in the vertical increment is never more than about 25^. A consequence of this scheme is that the lowest grid point is not always at a depth of 199 m as in the initial profile, Figure 2.2. Care must be taken, therefore, when performing vertical integration from the surface to the reference depth, D = 200 m. The integrals that will be calculated in the next section involve the temperature difference, f -T0 . Problems with varying the grid size can easily be avoided by setting the reference temperature, T0 , equal to fj>, the initial temperature at 200 m. This way, T-TD will always be zero at the base of the vertical grid. The grid is initially made deep enough so that the seasonal thermocline never reaches the bottom of the grid; hence, TD always equals its initial value, 6.08°C. Subtracting the reference temperature, fD , from tempera- ture, T, in the integrals in the next section amounts to subtracting the terms: -I -^ .U = - i,D (2.26) "-7> and 26 -!> T,V- <2-27> r 2 J a — — _ Since TD always remains cons-can-, both expressions, (2.26) and (2.27), will always remain constant. Subtracting T5 from f will amount to subtracting constants from the integrals if the next sec-ion. E. POTENTIAL ENERGY AND HEAT The potential energy per unit area for a column of water from the surface to depth, D, is given by p£ p . (2.28) Employing equation (2.2) with T-, as the reference temperature, allows aquation (2.28) to be written as ~7~ ~~l (~ ~ ^\) * ^t ♦" tohsU»t . (2.29) J ( _D The constant in equation (2.29) is t^f 3f»J' .--*-• (2.30, -J> Px, ■z a The integral in equation (2.29) will be used for the compar- ison of the potential energy with and without the interior motion. It will be represented by the variable P with units of °C m2 where P = - f ( T - TD) 2 cU - (2-31) 27 Similarly, heat content will be indicated- by the variable H with units CC m' where M - 1° /- - x (2*32) -D The analysis of the physical effects of the interior motion can be facilitated by developing the expressions for the time rate of change of H and of P. First, differentiating equation (2.32) with raspect to time gives i±x »_ I (t-T,)j,. <2-33' Equation (2.33) can be written as 6\\ (° *T J- (2.34) ** J-i> ^r Substituting equation (2.10) for th? time rate of change of temperature into equation (2.34) yields i±L - _ r°^_/~T=T7\ ._ (°r-,TA2:, (2.35) — ~ - ~ \ — ( w ' T ' ) *U -\ W — - The first term on the right, when integrated, is just the net surface heat flux. Substituting equation (2.17) into the second term on the right of (2.35| and integrating by parts gives 23 -r*n**-¥(-»r>+Lr^)- -x> '-& (2. 36) Using equation (2.26) this can be rewritten as W cU-- IT IT-T^J* = - =? H D (2.37) Equation (2.35) can new be writ- en as g _ v^p H (2. 38) The formula ti on for the time rat 9 of change of the potential energy is similar and is given by & P --I 2. clt (2.39) -i> Substituting equation (2.10) into (2.39) yields P - D (2. 40) The firs- term in equation (2.40) parts: can be integrated by (2.41) -fc -T> The first term on the right of aquation (2.4 1) vanishes because w'T' = 0 at z = -D. Since w'T' is zero below the 29 entrainment zone, equation (2.41) reduces to ^ (2.42) * -— (w' T'\ el* - - j " (*j> T') At . -k This integral may be broken up ovar the mixed layer and the entrainment zone, producing ,o -* (2. 43) (w'T']eli= -k is the vertically avaraged mixed layer mean of the turbulent heat flux. Since wT' is linear with respect to z in the mixed layer, this mixad layer mean can be written -<^TT> = -*-/ -9L. + ATWeV <2-uu> ^ V /°o c P Consider the second term on the right side of equation (2.40). Substituting equation (2.17) for W and integrating by parts yields .wil.u = !*iUi>*T, .zfr^uN (2'a5) using equation (2.27) this can be rewritten as On combining aquations (2.40), (2.43), (2.44) and (2.46), the time rate of change of the potential energy variable now 30 tecom€S t 1 [fi cr (2.U7) T> -k-j Because the turbulent heat flux is mor.otonic across the entrainmsnt zone, as illustrated in Figure 2. 1: w T' ^ ^ \-6 < J A I \>\)e < < Ml W e . (2. 48) This indicates -hat the integral term in equation (2.47) should generally be smaller in magnitude than the first term en the right side, unless of course, the surface and entrainment heat fluxes nearly cancel. Actual values of the terms of squation (2.47) were calculated for one-year period interior motion at two different phases and the second term on the right of the equation was smaller than the first, usually by one ord=r of magnitude. Furthermore, the third term on the right was generally an order of magnitude larger than either of the other two. Equation (2.47) will thus be approximated by 0± Q -2- (°* cf »- — u A T W<2. " -^ Wd 2- -J> P (2.49) 31 III. EXPERIMENTS AND RESULTS A. SUMMARY OF PERTINENT EQUATIONS The equations for time rate of change of mixed layer depth, mixed layer (and sea surface) temperature , heat and potential energy are ^U - w ■ ZT (3-1) The retreat depth is given by hr = " **3 U** *■ **\ (3.5) * 3 G J. ^ cr where (»'!') has bees replaced by aquation (2.6) . The entrain a en t rate is given by 32 AT vv'^ = N(w'M7l<£> '^ Cy K The prescribed net surface heat flux is given by (3.6) a, p? <-? (°nC, a n- t (3.7) and is plotted as a function of time in Figure 3. 1 li/(|»sCf) t°<~ LrH/SeO 4* iO' i— »► t ( a , Figure 3.1 Net Surface Heat Flux. The prescribed vertical motion at the reference depth, D=2 0 0 m, is Wj - W - i'H (^♦0 . (3.8) where W0 is defined by equation (2. 19) . 3y substituting z=-h into equation (2.17), it can be seen that WK varies directly as h and WB . Some additional notation will be useful for the discus- sion of the results. To observe the effect of the interior 33 motion, the values of h, fML# H and ? are compared to those without interior action. Contours are plotted of the difference of mixed layer depth with and without the interior motion, likewise for mixed layer temperature, hsat and potential energy. Therefore, the following new variables are defined: 4k u - (3.9) ■aJ = 0 ATML - H(_ W ^ ^ (3. 10) AH = H " W (3. 11) ap - p - p _ - r> (3. 12) B. RESULTS WITH HO INTERIOR MOTION The mixed layer depth has bean plotted in Figure A. 4. Note that the vertical axis scale is inverted for clarity. The year was started at the beginning of the heating season and the mixed layer retreats for the first quarter year. Then the mixed layer deepens slowly for the r.ext half year and mora rapidly in the last quarter. In tha last few days the mixed layer shallows to the t=0, 1 y ear , . . . ,n-years cyclical steady state value of about 75 m. 34 The mixed layer temperature, r^L, is plotted in Figure A. 5. As expected, the mixed layer temperature increases during the heating season and decreases during the coding season, for the most part. However, note that the peak mixed layer temperature occurs around day 170 and fH1_ starts to decrease ten days before tha end of the heating season. This is tecause the entrain ment heat flux term has begun to dominate equation (3.2). Some entrainment had already started fifty days earlier. By day 170, towards the end of the heating season, AT has reaohed its largest value, 7.8 JC. Cn day 170 the cooling of the mixed layer due to entrainment has reached 0. 021 "C per day while the surface heating has dropped to 0.030 °C per day. The heat content, plotted in Figure A. 6, depends only on surface heat flux for the case of no interior motion (see equation (3.3)). Any heat lost from the mixed layer due to entrainment will be gained by the region below. Since heat is calculated from the surface to 200 m, entrainment will have no effect on its value. Potential energy per unit area is plotted in Figure A. 7. This, too, generally increases with heating and decreases with cooling. However, note that the peak value occurs at about day 305, 25 days into the cooling season. This is because potential energy is also increased by mixing as indicated by equation (3.4) . The entrainment term dominates equation (3 . '4) from day 170 until day 305, when the net surface heat flux is near zero. A close inspection of Figure A. 7 reveals that the potential energy at the end of the year is greater than the initial value. Specifically, the initial value of F is 19608 ;C m2 ; the value at the end of the first model year is 20204 °C m2 and at the end of the second year is 20579 'C m2. This increase in the potential energy is due to the adjustment cf the temperature profile as the ocean approaches cyclical steady state. 35 Figure A.1 is a plot of temperature versus depth ana time. The top of the graph where the temperature contours are vertical is -he nixed layer. The mixed layer structure is quite evident after the first quarter year due to the strong temperature gradient at the base of the mixed layer. Note that the temperature profile at any depth below the mixed layer is left unchanged until mixing reaches that depth. C. ONE-YEAR PERIOD INTERIOR MOTION The zero contours of the prescribed interior motion have teen drawn in Figure A. 8 with upwelling and downwelling indicated. The dotted contours indicate when the magnitude of the interior motion drops to half of the maximum value. During -he first quarter year the mixed layer is shal- lowing. Since the retreat depth depends primarily or. surface boundary conditions, mixed layer depth will be unaf- fected by interior motion at this time. This is borne out by the contours of Ah (defined in equation (3.9)) , Figure A. 10. The first quarter year shows no difference in mixed layer depth. With little or no effect on mixed layer depth during the first quarter year, there will be little or no difference in the value of the first term of equation (3.2) with and without the interior motion. The second term of equation (3.2) will be zero at this time since entrainment stops during shallowing. Hence, mixed layer temperature will also remain unaffected by the interior motion during the first quarter year. This can be seen in Figure A. 12. Heat and potential energy, on the other hand, are influenced by the advection of temperature in the thermocline as indi- cated by the presence of the last :en in equation (3.3) and equation (3.4). Thus, the interior motion will influence heat content and potential energy even during the first quarter year (Figures A. 16 and A. 13). 36 1 • Effect on M i xed Lay_er De gth Equation (3.1) indicates -hat downwelling will enhance deepening, and upwelling will reduce deepening. In Figure A. 9, it can be seen -hat deepening still dees occur after the first quarter y»ar as it did without any interior motion. However, the rate of deepening depends on phase. At the 90° phase, there is upwelling during the middle half of the mcdel year (see Figure A. 2) reducing the rate of deepening. At day 274 downwelling begins and the mixed layer begins to deepen more rapidly. The very rapid deepening in the last quarter of the year for this phase results frcra both entrainment and downwelling working together. Calculated values of the terms of equation (3.1) indicate that entrainment is the dominant term most of the time during deepening. In this example the entrainment rate has been enhanced by conditions set-up by the interior motion. Equation (3.6) indicates that entrainment rate. We, is inversely proportional to mixed layer depth, h, and entrainment zone temperature jump, aT. Compare Figure A. 2 to Figure A.1 (the case with no interior motion) and A. 3 (the 270° phase case) . can be seen that at the start of the last quarter of the year the 90° phase case has a much lower mixed layer temperature than the other two cases (the reason for this will be discussed in the next section) . This lower mixed layer temperature means that the entrain- ment zone temperature jump will be smaller, allowing for a larger entrainment rate. At the start of the las- quarter of the year, the differences in h are not as significant as those in AT. For example, the values of AT, h, We and fc\ have teen calculated for these three cases for day 280. The mixed layer depth, h, is about 50 m for the case without interior motion, 65 m for the 90 c phase and 68 m for the 37 270° phase. Thase values dc not greatly differ. However, the temperature jump for the 90 c phase case is 1.3°C which is less xhan half the 2.9"C value for the case with no interior morion which in turn is less than the u.lcc value for the 270c case. The entrainment rate for the 90J phase case is 90 cm/day compared to 43 cm/day for the no interior motion case and 22 cm/day for the 270 c case. For both phases considered, the interior motion is at least an order of magnitude smaller than the entrainment rate. Nevertheless, at day 280, the deepening in the 90^ phase case is helped a little by downweliing and in the 270° case, hindered a little by upweiling. In the last few days of the year, the increase in mixed layer depth begins to take over, particularly for the 90° case. This inhibits further entrainment and the mixed layer then rapidly shallows to the cyclical steady-state value of 75 m (Figure A. 9). In the 270° phase case (Figure A. 3) there is down- weliing during the middle half of the year, which increases the deepening of the mixed layer. By the start of the last quarter of the year, the mixed layer is warmer than in the case with no interior motion or in the 90= phase case. This leads to a larger value of Z\T and a reduced entrainment rate . Comparing Figures A. 8 and A. 9 it can be seen that mid-year upweiling is associated with a reduced rate of deepening and mid-year downweliing is associated with an increased deepening rate. Looking at Figure A. 10, it can be seen that the actual decreased or increased values of mixed layer depth lag the start of upweiling or downweliing (respectively) by 90°. The time rate of change of mixed layer depth tracks along with the interior motion in accordance with equation (3.1). However, actual mixed layer depth is the integral of this and integrals of periodic 38 functions lag them by 90° . Also note -hat the strong enhancement of and strong hindrance to deepening at the end of the year (Figure A. 10) lags the strong decrease and strong increase in mixed layer temperature (Figure A. 12) by about 90c. 2« Effect on Mixed Lay_er Temperature Consider now the mixed layer heat equation (3.2). The net surface heat flux is prescrioed and unaffected by the interior motion. The eatrainmsnt heat flux, given by equation (3.6), is inversely proportional to mixed layer depth. The other terms in equation (3.6) depend primarily on surface boundary conditions. So, the first term in equa- tion (3.2) is inversely proportional to mixed layer depth, h, and the second, inversely proportional to the square of h. This implies that an increase in h will reduce the magnitude of mixed layer temperature change; a decrease in h will increase the change in T|VV<_. This is because, with a deeper mixed layer, a given amount of heat added to or removed from the mixed layer must be spread over a larger mass cf water. This effect is illustrated in the examples shown in Figures A. 2 and A. 3. In Figure A. 2, the mixed layer is shallower than in Figure A. 3 from about day 130 to about day 300. This resulted from the mid-year upwelling for the 905 phase case and the aid-year downweliing for the 270^ case. The shallower mixed layer in Figure A. 2 allows for a much more rapid temperature drop than occurs with the deeper mixed layer of Figure A. 3. This can be seen in the enhanced gradients in Figure A. 2. After day 300, the rapid deepening of the mixed layer for the 90° phase case and slow deepening for the 270° case reverse the situation. To see the effect of the interior motion for all of the phases, the T ^L contours should not be compared directly 39 to the interior motion contours but rather to the contours of the mixed layer depth differenc=, dh (defined by equa- tion (3.9)) . Between the 0' and 13 3° phases, after day 180 there is a region in Figure A. 10 where the mixed layer is shallower with the interior motion than without.' This causes the mixed layer temperature decrease (the second half of the year is the ceding season) to be greater, resulting in the negative values of /_iTML in the last quarter year at these phases (Figure A. 12). Between the 180° and 360r phases the opposite occurs. At the end of the year the mixed layer depth returns to 75 m. Mixed layer temperature does not return to its initial value. The second year rML data is plotted in Figures A. 13 and A. 14. The reduced temperatures for phases 0° to 180" and increased temperatures for phases 180° to 360° are continued into the second year. The mixed layer depth follows the same pattern in the second year as it did in the first. This causes the FKu pattern begun in the first year to be enhanced in the second year. Continued pumping of the ocean by this period interior motion enhances this structure. If permitted, it would eventually cause the upper 200 m to become isothermal (in the region of the 90° phase at the end of the year) , followed by deepening beyond the limits of what the model can handle. It would not be realistic, however, to have this single period interior motion continue for the several years necessary for this to happen. In reality, the interior motion would probably be damped by interactions with the boundary layer mixing processes. There should also be more than one period present. Furthermore, surface heat flux should be influ- enced ty sea surface temperature. These three possibilities open up avenues for further research. 40 3. Effect 22 1L2.&1 Content Equation (3.3) indicates that change in the heat content depends on net surface heat flux and on pumping of the ocean by the interior motion. Tha net surface heat flux is prescribed and is independent of the in-erior motion. Subtracting equation (3.3) for the casa of no interior motion from the same equation with interior motion gives c>AM w (3.13) — - - __£ h , where AH is defined by equation (3.11). Equation (3.13) indicates that upwelling will cause AH to decrease and downwelling will cause AH to increase. This can be observed by comparing Figures A. 8 and A. 16. The input to heat content due to net surface haat flux alone causes H to increase during the first half of the year and decrease during -he second half as shown in Figure A. 6. The combina- tion of surface heating with the affect of the interior motion produces the saddle-shaped curve shewn in Figure A. 15. The affect of different phases of the interior motion can te demonstrated by the following two simple exam- ples (Figures 3.2 and 3.3) . No mixing is permitted and a two-layer ocean is assumed for simplicity. In the first example shown in Figure 3.2 thara is initial downwelling followed by heating, then upwelling followed by cooling and final downwaiiing to tha initial mixed layer dapth. "When downwelling precedes heating, the heat is distributed ever a thiclcar mixed layer. A fixed amount of heat will produce a smaller increase in T than wcuid have resulted without the downwaiiing. If this is followed by upwelling before cooling, the thinner mixed U1 Heat (-C in) ii-> H 1 1 1 1 r ^T-T^LM initial 100 .itr*^ ——I 1— i — > T-~i)LJ':-' dcwnwailing 120 heating 150 upw€lling -zc ac^ 1 ■ ' ■ — -r — i — > i - 1 1) (a<0 100 cooling ■2.0 iL-~\ 3.s i i i i H > I - i0 (.*<-} -io". 70 ?(Vk) 3.3 downwellirig -Zd T--r*C°0 87.5 Figure 3.2 Heat Contsnt Example 1 U2 -15 Heat (°C ra) initial -(ml -i > 1 1 r- + T-TD ("O 100 upwelling •itrvO 1 I 4 -i — t- -i — > T-T^O 80 heading 4i(.r^> -2.C - — i — t — i — i- -* T -ut'c") -v }v»-"c 110 -« — ' — < — i — *-H — * T- > & C'O dowr.welling 165 coding upwelling -30 -> • Hf-i .-^ T-T-jCc.) — ^o JC m UM ■*— • — ' — ' — -+-< — »— * t-t, (*c> 135 1 12.5 Figure 3.3 Heat. Conxant Example 2. U3 layer will reach a lower temperature for a given amount of heat removed. At the end of the complete cycle the mixed layer temperature will be lower. Recall that the heat content is defined by Therefore the lower mixed layer temperature produces a lower heat content. Upweiling preceding the heating season distributes the heat ever a thinner mixed layer. This allows a greater increase in I than would otherwise occur. If this is followed by downwelling preceding the cooling season, a fixed amount of heat removed will produce a smaller drop in T. The net effect will be a higher temperature in the mixed layer and a higher heat content. Because of the asymmetry, as illustrated in these examples, heat is net conserved for a particular phase of the interior motion. This can be seen in Figures A. 15 and A. 16. At some phases the year-end heat content is higher than it was at the beginning of the year, at other phases, lower. However, if the heat content is averaged over all of the phases, the year-end value equals the initial value. The average heat content for sixteen equally spaced phases from 0C to 360° was tabulated. The phase-averaged value of H was 0.07rC m less at the end of the year than it was initially. This small error can be attributed to the coarse resolution of averaging only sixteen phases over the entire 360 ' . **• Effect on Potential Energy Time rate of change of potential energy is given by equation (3.4). From this equation it can be seen that surface heating tends to increase potential energy, surface 44 cooling tends to decrease it. The net surface heat flux is independent of the interior motion so the effect on :'i V due to this term depends on Ah. If there is an increase in mixed layer depth during heating, the buoyancy added will be mixed deeper, increasing potential energy. If there is an increase in mixed layer depth during cooling the effect will be opposite. The entrainment heat flux term has a positive sign in equation (3.4). Increased mixing moves buoyant water down and less buoyant water upward, increasing potential energy. Comparison of values of the terms of equation (3.4) indicates that the third term on the right is generally dominant. The exception is when the interior motion goes to zero, twice a year. The first term of equation (3.4) will not differ greatly with and without interior motion unless there is- a large difference in mixed layer depth. Recall that entrainment heat flux is nearly inversely proportional to mixed layer depth (equation (3.6)). Thus, the second term will net vary significantly. So, the potential energy closely follows the interior motion. This can be seen by comparing Figures A. 8 and A. 18. During downwelling, potential energy increases, during upwelling it decreases. The physical reasoning behind the effect of the third term of equation (3.4), the pumping term, is that downwelling pushes buoyant surface water downward, whereas, upwelling is in the same direction as the buoyant forces. 5. Summary of Response to One^Year Period Interior Motion During the first quarter year there is shallowing and the interior motion has little effect on either mixed layer depth or temperature. Mid-year upwelling reduces deepening and mid-year downwelling enhances deepening. The 45 shallower mixed layer resulting from upwelling will ccol faster; the deeper mixed layer resulting from downwelling will cool slower. A cooler mixed layer has a reduced jT allowing for a larger entrainment rate and more rapid deepening. A warmer mixed layer deepens more slowly. Potential energy and heat are both directly affected by the interior metier. Beth are increased by downwelling and decreased by upwelling. Before examining the mixing response to higher frequencies, an observation can be made regarding horizontal structure produced by the interior motion. As noted in chapter one, phase may be related to horizontal displacement by substituting , iff x u into equation (2.20) to obtain Vv * , , / .zrr t , rr * \ (3. 15) The wavelength, L, might be on the order of 300 km, large enough for the one-dimensional assumption to be valid. Figures A. 1 2 and A. 14 show the development of hori- zontal bands of positive and negative mixed layer tempera- ture anomalies- which would result from a one-year period interior wave. Because temperature is assumed to be verti- cally homogeneous in the mixed layer, fHL also represents sea surface temperature. Figures A. 11 and A. 13 show a time sequence of the horizontal sea surface temperature field resulting from the interior motion, the heat cycle and mixing processes. The entire surface warms during the heating season and cools during the cooling season. During U6 the first half of the first year, there is no significant difference in sea surface temperature along the horizontal axis. after the first half year, the region where there is upwelling at the start of the cooling season becomes and remains cooler than the region where -here is downwelling at the start of the cooling season. D. EFFECT OF SHOBTEE PERIOD INTERIOR MOTION Comparison of Figures A. 19 and A. 21 shows that as before downwelling increases deepening and upwelling reduces deep- ening. This can also be observed by comparing Figures A. 30 and A. 32 and by comparing Figures A. 39 and A. 41. The mere rapid variation in mixed layer depth does not yield the strong effect on temperature found for the one-year period interior motion. This can be seen in the weaker gradients of AT",ml °f Figures A. 23, A. 34 and A. 43 compared to that of A. 12. Hence, the higher frequency interior motion does not have as strong an effect on the mixed layer as the lower frequency mcticn does. The second year temperature data was plotted for the half-year period interior motion in Figures A. 24 and A. 25. Notice that the same structure seen in the first year reap- pears in the second, but the cold spots have not grewn significantly colder nor the warm spots warmer as in the case of the one-year period motion. As before, the heat and potential energy differences, /\H and AP, track along with the interior motion as seen in Figures A. 27, A. 29, A. 36, A. 38, A. 45 and A. 47. The resulting contours of H and P appear more complex. This is because the values cf H and P without the interior motion follow a one year cycle (Figures A. 5 and A. 7) and the higher frequency cycles due to the pumping by the interior motion are superimposed on the one year cyole. 47 Fcr the cne-year period interior motion, upwelling lasts long enough to enhance cooling significantly (at phases around 90°) and downwelling lasts long enough to inhibit cooling significantly (at phases around 270°). The shorter periods dc not maintain upwelling and downwelling long enough to produce an effect this strong. E. ASYMMETRY IN ENHANCED AND REDUCED MIXING In Figures A. 48 and A .49 the values of peak enhanced cooling and peak reduced mixed layer cooling are plotted. Both first and second year values are considered (in Figures A. 48 and A . 49 , respectively). Negative values of &Tni_ indicate enhanced coding due to the interior motion. The maximum absolute value of all of the negative AfML data points was determined to indicate the peak enhanced cooling. Positive values of Afjnu indicate reduced cooling. The maximum value of all of the positive aTml data points indi- cates the peak reduced cooling. The positive values dc not generally indicate enhanced heating, because the wave has very little effect on the mixed layer temperature during the heating season. The first obvious observation that can be made from Figures A. 48 and A. 49 is that the largest peak anomalies occur with the Dne-year period interior morion. For the cne-year period interior motion the peak enhanced and peak reduced mixed layer cooling is 2.04 and 2.16°C, respec- tively. For the one-half year period motion "-.hese values are 1.35 and 1.15°C, respectively. Also, for each of the periods studied, the difference between the peak enhanced cooling and the peak reduced ccoiing is greater in the second year than in the first. All of the periods studied except the one-year period have grearer peak enhanced cooling than peak reduced cooling. 48 The phass-averagsd values of the variables .Ah, ATKL, AH, and A? were computed for sixteen equally spaced phases from 0° to 360° . The phase-averaged values at the end of the first and the end of the second model years are listed in Table I. These values indicate the net effect of the interior motion after each year of interacting with the mixing processes. Consistent with the results shown in Figures A. 43 and A. 49, tha results in Table I indicate that the one-year period interior motion has produced a net reduction of mixed layer cooling. The higher frequency interior motion has produced a net enhancement of mixed layer cooling. For all of the periods studied there has been a net reduction of the mixed layer depth. Heat is conserved, hence the phase-averaged value of AH is zero at the end of each year. The effect of the one-year period interior motion was to decrease the potential energy. The higher frequency interior motion produced a net increase in potential energy. A net1 increase (decrease) in potential energy indicates that the interior motion has enhanced (reduced) the overall mixing. The amount of deepening required to produce an equivalent increase in potential energy is instructive. This can easily be calculated for the simplified example of a two-layer ocean, illustrated in Figure 3.4 with initial mixed-layer temperature, T0 and depth, ha . This mixed layer is deepened to h, with new mixed layer temperature, T, . Heat is conserved, so, using the definition for the heat variable, H, given in equation (2.32) : (t, -td )K= (rc - rD Ho <3-16> Using equation (2.31), the change in the potential energy is given by 49 Pi - po = C T. - ~D ) C _ <• T* - td Vh£ . (3.17) X Substituting aquation (3.16) into equation (3.17) yields P. - = ± ( T '») U( W, - k.) (3. 18) The term T0 -TD is just the initial entrainment zone tempera- ture jump. The increase in potential energy due to deep- ening with heat conserved is directly proportional to the increase in mixed layer depth for the two-layer example. » T fr I— -9 I after mixing initial profile Figure 3.4 Two-Layer Ocean Example. Consider now the phase-averaged potential energy anoma- lies at the end of the first year, shown in Table I. Suppose the initial mixed layer depth is 75 m and the initial temperature jump is 4 °c. The net reduction of potential energy for the one-year period interior motion would then te equivalent tea reduced mixed layer depth of 50 0.73 m. The net increase of potential energy for the 0:2- half, one-third, and one -quarter year period interior action is equivalent to an increase of mixed layer dspth by 1.6, 1.8 and 1.8 m, respectively. The anount of dcwnweiling and upwelling are the same over the whole year and for all phasas. Yet, the net reduc- tion of mixing and mixed layer cooling does not equal the net enhancement of the same. The net effect of *he one-year period interior motion is to hinder mixing. The net effect of the shorter period interior motion is to enhance mixing. TABLE I Phase-averaged Hean Anomalies Phase-averaged mean of: Day &h A'Th. AH A? Period (m) (CC) (>C m) ("C m ) (years) 0 0. 0. 0. 0. 365 -0.43 ♦ 0.C6 0. - 10 9. 1 730 -0.71 + 0. 22 0. -645. 0 0. 0. 0. 0. 365 -0. 16 -0.05 0. 23 9. 1/2 730 -0. 16 -0. 06 0. 33 1. 0 0. 0. 0. 0. 365 -0.23 -0. C5 0. 26 7. 1/3 730 -0. 23 -0. 07 0. 435. 0 0. 0. 0. 0. 365 -0.31 -0.05 0. 27 5. 1/4 730 -0. 31 -0.07 0. 45 1. 51 IV. CQNCLDSIQNS ANJ? RJCOMHEMDATIOHS A. CONCLUSIONS Seasonal-scale interior motion has a significant effect on boundary layer mixing processes. This effect is most pronounced whan the period of the interior motion is one year. For this pericd, upwelling between the heating and cooling seasons can strongly enhance mixed layer cooling and subsequent deepening. Downwelling can strongly reduce the same. This effect may produce a significant, observable horizontal pattern in sea surface temperature. Although there is negligible effect on sea surface temperature and mixed layer depth during the beginning of -he heating season, the interior motion does affect the potential energy and heat content at this time. Throughout the year the potential energy and heat are directly related to the pumping by the interior motion. Upwelling decreases and downwelling increases both potential energy and heat content. Higher frequency interior motion does not have as strong an effect. The cycling between upwelling and downwelling during the cooling season prevents the strongly enhanced mixed layer cooling from occurring. The effect of the interior motion is asymmetric. Generally, the enhancement of mixing and mixed layer cooling exceeds the reduction of the same. The exception to this occurs for the one-year p=riod interior motion. B. RECOHHENDATIONS One aspect of this research which departs from reality is in prescribing a single wave for the interior motion. This could be changed quite easily by prescribing 52 w 0 - H ^y, -> i v\ I ^rr t I T„ +- *. (4.1) The amplitudes, W^, and periods, T^ , could be based on the first few dominant frequency components observed in actual wave spectra. Another way that reality could be more closely approached is by allowing the net surface heat flux to vary depending en the sea surface temperature. This should dampen "he strongly enhanced or reduced cooling of the mixed layer temperature seen with the one-year period motion. An area open to further research is to use a finite depth ocean and an appropriate vertical wave structure. An appropriate wave structure for planetary waves is given by Willmott and Mysak (1980) . In their work the vertical motion is linear in the mixed layer: c^ W -- C , (4.2) but, below the mixed layer the vertical motion is given by o W o h C, W r 0 (1.3) where N2 is the Brunt-Vaisaia frequency and c^ represents the speed of propagating long waves in the non-rotating cass for mode number n. The solutions of equations (4.2) and (4.3) must be matched at the base of the mixed layer, z=-h, to ensure continuity cf the vertical velocity. The boundary conditions are rigid lid and no normal flow through the ocean bottom, so, w=0 at z=0 and at the ocean bottom. At each time step the mixed layer depth is recalculated and so the vertical wave structure problem would also have to be solved for the new w (z) . The combination of this sort of 53 vertical wave structure with the Garwood model for boundary layer mixing processes could provide an interesting area for future study. In this study the interior motion was prescribed and acted on the mixed layer. The interior motion was not allowed to he changed by the mixed layer processes. If, for example, the wave motion enhances mixing, the increase in potential energy should be balanced by a decrease in wave energy. Although the frequencies considered here are much lower than those in the work by Bell (1978), the processes involved could be sinilar. 54 APPENDIX A FIGURES Solid contours indicate positive values. Dashed contours indicate negative values E- , Q_ a S o 10' oo ^26 o H 10 10 U Uj te. 10 Jo. I 0 30 60 90 120 150 180 210 TIME DAYS 240 270 300 330 360 Figure A. 1 Temperature, No Interior notion. 55 60 90 120 150 180 210 240 270 300 330 350 TIME DRY5 Figure A. 2 Temperature, One-Yaar Period, 90° Phase 30 60 90 120 150 180 210 240 270 300 330 360 TIME DAYS Figure A. 3 Teapsrature, One-Year Period, 270° Phase. 56 1 — ' I I I — ' I ' ' — I — I — I — I — I — p — I 0.0 30.0 60.0 90.0 120.0 150.0 180.0 210.0 240.0 270.0 300.0 330.0 360.0 TINE DAYS Figure A. 4 Mixed Layer Depth, Ho Interior Motion. 0.0 30.0 60.0 90.0 120.0 150.0 180.0 210.0 240.0 270.0 300.0 330.0 360.0 TIME DAYS Figure A. 5 Mixed Layer Temperature, Mo Interior Motion 57 I I — I- 1 — I — I — I — 1 — I I I — I I I — ] — I I I I I I I — I — [■ 0.0 30.0 60.0 90.0 120.0 150.0 180.0 210.0 240.0 270.0 300.0 330.0 360.0 TIME DAYS Figure A. 6 Heat, No Intsrior Hotion. "* * i ■ ■ i ' ■ i — ' — • — i — ' — ' — i — > — i — i — i — i — i — i — > — i — i — i — i — i — i — i — i — i — i — i — i — i 0.0 30.0 60.0 90.0 120.0 150.0 180.0 210.0 240.0 270.0 300.0 330.0 360.0 TIME OflYS Figure A.7 Potential Energy, No Interior Hotion. 58 0 "'••% ^^ "'-._ ~ >W <<,* x X *-*, .. % X vJ \w^ ' * , •v . * '* ■>>- <. '*j - s. - " % v-\ ~ - _ *% o •■v * v % "'• K-. S"" ^S. "*N, ^V o- , . 1 3 c . i ;i5 Figure A- 8 One-Year Period Interior Motion. 59 I I I ' I 1 I — I ' I ' ■ I I 0 30 60 90 120 . 150 180 210 240 270 300 330 360 TIME DRYS Figure A. 9 h (meters), One-Year Period Interior Hotion. CD oj a en Q_ to LO. "i r— — i r 0 30 60 90 120 150 180 210 240 270 300 330 360 TIME DAYS Figure A. 10 Ah (meters), One-Tear Period Interior Motion, 60 30 T 60 T 90 120 150 180 210 TIME DAYS 240 270 300 360 Figure A. 11 T(Vll_ (C) , One-Year Period Interior Hotion. a o en az X in 0- 2 30 Figure A. 12 TIME DAYS ATM (C) , One-Year Period Interior Hotion 61 365 395 425 455 485 515 545 575 605 635 665 695 725 TIME DAYS Figure A. 13 TMt_ (C) , One-Tear Period, Second Year. 365 395 425 455 485 515 545 575 605 635 665 695 725 TIME DAYS Figure A. 14 £*"„,_ (Q, One-Tear Period, second Tear. 62 i 1' i ■ i ■ i 1 — ^— r Q 30 60 90 120 150 180 210 240 270 300 330 360 TIME DRYS Figure A. 15 H (100*Y5 Figure A. 19 1/2-Year Period Interior Hotion. 65 30 T 60 90 120 150 180 210 240 TIME DAYS . 270 300 330 Figure A. 20 h (asters), 1/2-Year Period Interior 360 Motion. — to dj cm a en en O)' in TIME DAYS Figure A. 21 £h (meters), 1/2- Year Period Interior Motion. 66 a to- ri C3 «■ o o cc 0- 2" u">. to a -r- 30 60 120 150 180 210 240 TIME DAYS 270 300 330 360 Figure A. 22 T,^ (C) , 1/2-Tear Period Interior Motion. CD r\] f, 1 CM a O uj 2" en en x in 30 TIME DAYS Figure A. 23 /\TtM_ (C) , 1/2-Iear Period Interior Motion 67 C3 oj o o en - b o a CD b o CO O CO I 'I 1 1 I I ' I ' I 365 395 425 455 485 515 545 575 605 635 665 695 725 TIME DAYS Figure A. 24 T^,_ (C) , 1/2-Year Period, Second Year TIME DAYS Figure A. 25 ATin, (C) , 1/2-Year Period, Second Year. 68 I I I II I I I I ■ I ' ' I ' I ■ 0 30 60 90 120 150 180 210 240 270 300 330 360 TIME DAYS Figure A. 26 H (100 C a) , 1/2-Year Period Interior Motion. 0 30 60 90 120 150 180 210 240 270 300 330 360 TIME DAYS Figure A. 27 &H (100'C a), 1/2-Yaar Period interior Motion, 69 i i i r ' D 30 60 90 120 150 180 210 240 270 300 330 360 TIME DRYS Figure A. 28 P (10* C »2) , 1/2-Year Period Interior Motion, 0 30 60 90 120 150 180 210 240 270 300 330 360 TIME DAYS Figure A. 29 zip (10* °C m*), 1/2-laar Period Interior Motion 70 IS2.0 71 1*6 P*W5 2tn Figure A. 30 1/3-Iear Period Interior Motion. 71 3D 60 -r 90 120 150 180 210 TIME DRYS 240 270 300 330 360 Figure A. 31 h (aeters) , 1/3-Year Period Interior Motion. 8" CS> CM a o cj SB- CD a: a= ^ 30 Figure A. 32 TIME DRYS Ah (meters) , 1/3-Year Pariod Interior Motion, 72 I 'I 1 ' I 1 ' I 0 30 60 90 120 150 180 210 240 270 300 330 360 TIME DAYS Figure A. 33 TMU (C) , 1/3-Year Period Interior action, LJ r\j a o LJ 2 en cc X in Q_ to tf>_ 30 TIME DAYS Figure A. 34 ^TMl_ (C) , 1/3-lear Psriod In-erior Motion 73 TIME DRrS Figure k. 35 B (100 *C a), 1/3-Year Period Interior Motion. ° Q 30 60 90 120 150 ^Q 2lo 240 270 300 330 360 TIME DAYS Figure A. 36 AH (100°C a), 1/3-Year Period Interior Motion 74 TIME DAYS Figure A. 37 P (10*'C m2) , 1/3-Year Period Interior Motion, TIME DAYS Figure A. 38 AP (10* °C a2), 1/3-laar Period Interior Motion 75 365" Figure A. 39 1/U-Year Period Interior Motion 76 150 180 210 TIME DAYS Figure A. 40 h (Meters), 1/4-Tear Period Interior Motion. T 90 120 150 180 210 240 270 300 330 360 TIME DAYS Figure A. 41 Ah (Meters), 1/4-Year Period Interior Motion 77 150 180 210 TIME DRYS 270 300 330 360 Figure A.42 f„t (C) , 1/4-Year Period Interior Hotion. o o ^ TIME DflrS Figure A. 43 ATML (C) , 1/4-Iear Pariod Interior Motion 78 150 180 210 240 TIME DRYS r 270 30G 330 360 Figure A. 44 H (100 "C m) , 1/4-Iear Period Interior Motion. TIME DRYS Figure A. 45 AH (100°C m) , 1/4-laar Period Interior Motion 79 30 60 90 120 150 180 210 240 270 300 330 TIME DAYS 360 Figure A. 46 P (10* C «z) , 1/4- Tear Period Interior Motion. TIME DAYS Figure A. 47 & P (10* "C i2), 1/4-Tsar Period Interior Motion 80 MO L -' I -t ft ^*\ *.£ JL A K^Heti A v ^ -0-"-- P*' Figure A. 48 Peak Enhanced and Reduced Cooling Values, Year 1. to «t \ 3 \\ I e r\ "V <- *<-t <>' Vs. ) \ % r J . ( «n_ ? J. ___£__ - -*. . * ! >\ = I ^ e *.<■ _ Figure A.49 Peak Enhanced and Reduced cooling Values, Year 2. 81 BIBLIOGRAPHY Adamec, D., R.L. Elsberry. R.W. Garwood and R.L. Haney, 1981: An embedded mixed- layer--ooean circulation model. Dynamics of Atmospheres and Oceans, 6, 6 9-96. Bell. T.H., 1978: Radiation damping of inertial oscillations in the upper ocean. Journal of Fluid Mechanics, 88, Burger, R.J., 1982: Oceanic mixed layer response to tidal period internal wave motion. M.S. thssis. Naval Postgraduate School. Cushman-Rcisin , B. f 1981: Effects of horizontal advection on upper ocean mixing: a case of front ogenasis. Journal of P£Ll§ical Ccjano^ra^hj, JO, 134 5-1356. De Szceke, R. A. , 1980: On the effects of horizontal vari- ability of wind stress on the dynamics of the ocean mixed layer. Journal of Physical Oceanography, 9, 1439-14 54. Garwood, R.W. . 1976: A general model of the ocean mixed layer using a two-component turbulent kinetic eneray budaet with mean turbulent field closure. Ph.D. thesis, University of Washington, (NOAA Tech. Rep. ERL 384-PMEL 27). Garwood, R.W., 1977: An oceanic mixed layer model capable of simuli ' 7, 45 simulating cyclic states. Journal of Physical Oceanography, "5-468. ~~ Greatbatch, R.J., 1983: On the response of the ocean to a moving storm: the sea-surface temperature response (unpub- lished manuscript) . Rang, Y.Q. and L. Magaard, 1980: Annual baroclinic Rossby waves in the central north Pacific. Journal of Physical Q£§£££2I!.Eh.I ' 10. > 1159-1167. Linden, P.?., 1975: The deepening of a mixed layer in a stratified fluid. Journal of Fluid Mechanics , 7, 385-405. Price, J.F., C.N.K. Mooers and J.C. Van Leer, 1978: Observation and simulation of storra-ir.duced mixed-layer deepening. Journal of Physical Oceanography, 8, 582-599. Price, J.F., 1981: Upper ocean response to a hurricane. j2°.!11.2§i 2i Physical Oceanography, 11, 15 3-175. Stevenson, J.W. , 1980: Response of the surface mixed layer to guasi-aeostrophic cceanic motions. Ph.D. thesis, Harvard University. Stevenson, J.W., 1981: The seasonal variation of the surface mixed layer response to the vertical motions of linear Rcssby waves (unpublished manuscript) . Willmott, A.J. and L.A. Mysak, 1980: Atmospherically forced eddies m the northeast Pacific. Journal of Phy sica 1 Oceanography, 10, 1769-1791. 82 INITIAL DISTRIBUTION LIST 7. Professor E.L. Elsberry (Code 33Es) Department of Meteorology Naval Pcstaraduate School Monterey, CA 93 943 8. Professor K.L. Davidson (Code 63Ds) Deoartmer.t of Meteorology Naval Postgraduate School Monterey, CA 93943 9. Professor G.L. Geernaert (Code 63) Department of Meteorology Naval Postaraduate School Monterey, CA 93 943 10. IT J. P. Garner Naval Oceancgrachv Command Center, Guam Box 12 FPO San Franciscc, CA 96630 11. Director Naval Oceanography Division Naval Observatory 34th and Massachusetts Avenue NW Washington, DC 20390 No. Copies 1. Defense Technical Information Center 2 Cameron Station Alexandria, VA 22314 2. Litrary, Code 0142 2 Naval Postgraduate School Monterey, CA 93943 3. Professor C.N.K. Mcoers (Code 68Mr) 1 Chairman Deoartmer.t of Oceanography Naval Postgraduate School Monterey, CA 93943 4. Professor R.J. Renard (Code 63Rd) 1 Chairman Department of Meteorology Naval Postgraduate School Monterey, CA 93943 5. Professor R.W. Garwood (Code 583d) 1 Deoartment of Oceanography Naval Postgraduate School Monterey, CA 93943 6. Professor A.J. Willmott 1 Department of Mathematics University of Exeter North Park Road Exeter EX4 4QE Enaland 83 12. Commander Naval Oceanography Command NSTL S -.at ion Bay St. Louis, MS 39522 13. Cera man din g Officer Naval Oceancgraphic Office NST1 Station Bay St. Louis, MS 39522 Ccramandir.g Cfficsr Fleet Numerical Cceanoara Dhy Center Monterey, CA 93940 14. 15. Ccmraandina Officer (Attn: S. Piacsek) Naval Ocean Research and Development Activity NSTL Station Bay St. Louis, MS 39522 16. Co mm an dine Officer Naval Environmental Prediction Research Facility Monterey, CA 9 3 940 17. Chairman, Oceanography Department U. S. Naval Academy Annapolis, MD 2 1402 18. Chief of Naval Research 800 N. Quincy Street Ar ling- on, 7A 2 2217 19. Cffice of Naval Research (Code 420) Naval Ocean Researc1 800 N. Quincy Stree Arlington, VA 2 2217 v-/ i. _ _ v, — vjj. [l c; » a i i^oci»_'^ii y v*. v> u. w -* i- J ) Naval Ocean Research and Development Activity 800 N. Quincy Street 20. Scientific Liaison Office Office of Naval Research Scripps Institute of Oceanography La Jclla, CA 92037 21. Library ScripDS Institute of Oceanography P.O. 3ox 2367 La Jclla, CA 92037 22. Library Department of Oceanography University of Washington Seattle, WA 98105 23. Library CICZSE F. 0. BOX 4803 San Ysidro, CA 92073 24. Library School of Oceanography Oregon State University Corvallis, OR 97331 25. Ccmmander Oceancaraphy Systems Pacific Box 13 90 Pearl Harbor, HI 96860 84 26. Richard J. Greatfcatch GFDI, NCAA P.O. Box 308 Princeton, NJ 0 8 5U2 27. Professor Peter Muller Department of Oceanography Umveristy of Hawaii 1000 Pope Rd. Honolulu, HI 96 822 28. Professor R.L. Haney Department of Oceanography Umveristy of Hawaii 1000 Pope Rd. Honolulu, HI 96822 85 Thesis G19^3 c.l Garner The offect of in- terior motion on sea- sonal thermocline evo- lution. 206375 Thesis G19U3 c.i Garner The offect of in- terior motion on sea- sonal thermocline evo- lution. thesG1943 The effect of interior motion on seasona 3 2768 002 01059 7 DUDLEY KNOX LIBRARY