I ^*&.*=-^-'.{ EK5 ■ ■ NOAA Technical Report ERL 384 PMEL 27 A General Model of the Ocean Mixed Layer Using a Two-Component Turbulent Kinetic Energy Budget with Mean Turbulent Field Closure Roland W. Garwood, Jr. Pacific Marine Environmental Laboratory Seattle, Washington September 1976 U.S. DEPARTMENT OF COMMERCE National Oceanic and Atmospheric Administration Environmental Research Laboratories NOAA Technical Report ERL 384-PMEL 27 apWMOSWu.. ^£NT Of A General Model of the Ocean Mixed Layer Using a Two-Component Turbulent Kinetic Energy Budget with Mean Turbulent Field Closure Roland W. Garwood, Jr. Pacific Marine Environmental Laboratory Seattle, Washington September 1976 o U a o U.S. DEPARTMENT OF COMMERCE Elliot Richardson. Secretary National Oceanic and Atmospheric Administration Robert M. White, Administrator Environmental Research Laboratories Wilmot Hess, Director ^Z°\ ^6-19l fo Boulder. Colorado Adapted from a dissertation submitted to the University of Washington in partial fulfillment of the requirements for the degree of Doctor of Philosophy n CONTENTS PAGE NOTATION v ABSTRACT 1 1. INTRODUCTION 1 1.1. Purpose of the Study 1 1.2. Characteristics of the Ocean Mixed Layer 5 1.3. Fundamental Principles and Equations 7 1.4. Course of Action in Attacking the Problem 15 2. REVIEW OF THE LITERATURE 16 2.1. Ekman Depth of Frictional Resistance 16 2.2. Rossby Number 17 2.3. Eddy Transfer Coefficients in a Steady-State Problem 18 2.4. Obukhov Length Scale 18 2.5. Compensation Depth for Shortwave Radiation 21 2.6. Prototype Turbulent-Energy-Budget Model: Kraus and Turner 23 2.7. Adding Dissipation 26 2.8. Role of Mean Kinetic Energy 29 3. CLOSING THE PROBLEM 33 3.1. Net Viscous Dissipation in the Mixed Layer 34 3.2. Net Effect of Redistribution of Turbulent Energy 34 3.3. Shear Production 35 3.4. Need for an Entrainment Equation 36 4. ENTRAINMENT HYPOTHESIS 36 4.1. Entrainment in Earlier Mixed Layer Models 36 4.2. Suggested Turbulent Mechanism 38 4.3. Relevant Parameters and a Dimensional Analysis 41 4.4. Turbulent Kinetic Energy Budget at the Density Interface and the Development of a Theoretical Equation for Turbu- lent Entrainment in the Presence of Mean Shear 43 4.5. Comparison with Moore and Long (1971) Experiment 46 4.6. Completed Model for Shallow Mixed Layers 48 5. BEHAVIOR OF THE EQUATIONS 49 5.1. Nondimensional Form of the Turbulent Energy and Entrainment Equations 49 5.2. Determination of the Constants 50 5.3. Comparison with Earlier Models 52 in 6. DEEP MIXED LAYERS: LIMIT TO MAXIMUM DEPTH 54 6.1. Limiting Dissipation Time Scale 54 6.2. Nondimensional Solution to the Entrainment Function 56 6.3. Simple Hypothetical Cases Demonstrating the Behavior of the Solution 60 6.3.1. Shallow or f - (Ro » 1) 60 6.3.2. Convective planetary boundary layer with order one Rossby number 65 6.4. Filtering Effect of the Storage of Turbulent Kinetic Energy 68 6.5. Interaction Between Forcing Time Scales: Modulation of the Longer-Period Trend by the Diurnal-Period Heating/Cooling Cycle 70 6.6. Simulation of a Real Case 71 6.7. Evaluation of Model Output 74 7. CONCLUSIONS 75 8. SUMMARY OF MODELED EQUATIONS 77 9. ACKNOWLEDGMENTS 78 10. REFERENCES 79 IV ■ a. NOTATION r~\ The horizontal mean (often just capitalized: g = T, etc.) < > The vertical mean over the mixed layer b = B + b Buoyancy B* Parameter for relative importance of H* to Ro -1 for cyclical steady state 5^" Turbulent buoyancy flux ^7 Complex turbulent momentum flux r Specific heat at constant pressure C P C Drag coefficient in quadratic wind D stress law C = U + iV Mean horizontal velocity in complex form c = u + iv Turbulent horizontal velocity in complex form d Ekman depth of frictional resistance d 1 Surface production zone (Niiler); also used as a depth proportional to Ekman depth (d) D Total rate of viscous dissipation of turbulent kinetic energy in the mixed layer D Molecular diffusion coefficient s 1 F = L, m = u2 + v2 + w2 Turbulent kinetic energy (per unit mass) 2 2ri i 2 E * /(hAB) * * * •11 + E 22 + E 33 eTi + Eoo + Eqo Turbulent kinetic energy nondimension- alized with the friction velocity squared, u 2 . f = 21^1 sin* Coriolis parameter F Damping force for inertial motions g Apparent gravitational acceleration G Total rate of mechanical production of turbulent kinetic energy in the mixed layer h Depth of mixed layer h Mixed layer depth if the vertical r extent of the region is stationary or retreating H Mixed layer depth (h) non-dimension- al i zed with the Obukhov length h Compensation depth for radiative heating J Mechanical equivalent of heat, 4.186 x 10 7 9 m cm2 cal sec 2 K Eddy mixing coefficient for neutral conditions K = K Eddy viscosity K Eddy conductivity L Obukhov length scale Integral turbulence length scale o Convective length scale i Rotational length scale m. , n. Model constants i ' K i n Value for integrated flux Richardson number N Brunt-Vaisala frequency p Fluctuating pressure component gp /ax Geostrophic pressure gradient in the g a ocean VI PE + KE * P Q Q' Qo Total mechanical energy per unit mass in the mixed layer Nondimensional entrainment flux Radiation absorption Solar heating function Solar radiation through interface (incident minus reflected) 1 n2 u 2 + v 2 2 q = "~ 2 Ro Ri RF Horizontal component of turbulent kinetic energy Fraction of net production going to potential energy by means of entrainment Rossby number Gradient Richardson number, 9B/8Z |3C/3Z| 2 Flux Richardson number, - bw/ (^ fy + vw |^) Ri Ri Ri. s = S + s t Gradient Richardson number in the entrainment zone Overall or bulk Richardson number, hAB/ | AC | 2 Total Gradient Richardson number 9b/9z (au/8z) 2 Salinity Time VI i (u.) = (U+u, V+v, W+w) U 9' V 9 u*b* W (x..) + (x,y,z) Total instantaneous velocity, the sum of the mean and turbulent components (neglecting any geostrophic com- ponent) Geostrophic components of velocity dh/dt, entrainment velocity Friction velocity, v|cw(o)| Surface buoyancy flux, -bw(o) Wind velocity; sometimes used as mean vertical water velocity Nondimensional entrainment velocity, u /V e Rectangular space coordinates with x. = z aligned vertically upward, parallel to the local apparent direction of gravity -Pq-^p/sS Coefficient of thermal expansion, P n _1 3p/3T (z) 'T 6' i 5C /.B Extinction coefficient for net solar radiation 3B/3Z for Z < (-h-6) = N 2 9T/3Z for Z < (-h-6) Fraction of turbulent energy dissipated Thickness of entrainment zone Excess surface velocity, C(z=o) - Change in mean buoyancy, across the entrainment zone, B(-h) - B(-h-6) vi n AC! Mean velocity drop across the density interface Rate of viscous dissipation of tur- bulent kinetic energy, - _ Q ik(x - c n t) * - V l 9U.8U. Internal wave amplitude on interface = T + Temperature K A M v P p =p ( vv Temperature flux scale, b /eg Thermal conductivity Heaviside unit step function; also used as dissipation length scale Molecular viscosity Kinematic viscosity The instantaneous density of the sea water The density for representative values of salinity, s , and temperature, e Representative density of air p«C„ew o p Turbulent heat flux 1*1 r e Surface stress, p„ u? Dissipation time scale Dimensionless time scale, t/x T I 4> Time constant for damping by F Latitude IX * c*/d-* |aC| 2 E /Ri = ' L— (hAB) 2 * H' Nondimensionalized turbulent kinetic 3 energy, h / (u* t ) Earth's rotation vector a) Angular frequency A GENERAL MODEL OF THE OCEAN MIXED LAYER USING A TWO-COMPONENT TURBULENT KINETIC ENERGY BUDGET WITH MEAN TURBULENT FIELD CLOSURE Roland W. Garwood, Jr. A non-stationary, one-dimensional bulk model of a mixed layer bounded by a free surface above and a stable nonturbulent region below is derived. The vertical and horizontal components of turbu- lent kinetic energy are determined implicitly, along with layer depth, mean momentum, and mean buoyancy. Both layer growth by en- trapment and layer retreat in the event of a collapse of the ver- tical motions due to buoyant damping and dissipation are predicted. Specific features of the turbulent energy budget include mean tur- bulent field modeling of the dissipation term, the energy redistri- bution terms, and the term for the convergence of buoyancy flux at the stable interface (making possible entrainment) . An entrainment hypothesis dependent upon the relative distribution of turbulent energy between horizontal and vertical components permits a more general application of the model and presents a plausible mechanism for layer retreat with increasing stability. A limiting dissipation time scale in conjunction with this entrainment equation results in a realistic cyclical steady-state for annual evolution of the upper- ocean density field. Several hypothetical examples are solved, and a real case is approximated to demonstrate this response. Of par- ticular significance is the modulation of longer-period trends by the diurnal-period heating/cooling cycle. 1. INTRODUCTION 1.1 Purpose of the Study The objective of this study is the formulation of a unified mathematical model of the one-dimensional, nonstationary oceanic turbulent boundary layer. In particular, this model should help explain and predict the development in time of the seasonal pycnocline. Interest in the ocean mixed layer stems from both theoretical and prac- tical considerations. Thermal energy and mechanical energy received from the atmosphere not only control the local dynamics, but the layer itself modu- lates the flux of this energy to the deeper water masses. Conversely, flux of heat back to the atmospheric boundary layer has an important influence upon the climate and its fluctuations. Figure 1 depicts the mechanical energy budget for the ocean mixed layer. U(V) Work by Wind Stress MEAN K.E. ■jJ(\J z +V z )dz -h-S Shear * Production HORIZONTAL TURB. K.E. 2 + v 2 dz -h /// /// /// bw (0) Surface Buoyancy Flux Mixed Layer /j Entrainment Zone Stable, Negl Vert. Flux Interior Buoy. Flux u O Entrainment * Buoy. Flux Redistribution L C VERTICAL TURB. K.E. -h Dissipation Modeled Processes VIquaq 1. Mo.cha.viicaZ Qno/igy budget fan. the. oddan mixQd layoA. In addition to climatological and ocean circulation studies, other applications of a practical model of the mixed layer include investigations of advection and turbulent dilution of dissolved or suspended concentrations of matter such as pollutants and nutrients. Prediction of entrainment of the deeper, nutrient-rich water into the mixed layer has particular pertinence to the estimation of primary productivity. The need to further develop and attempt to improve the one-dimensional mixed-layer model is evident. The failure of earlier models to consistently explain the annual cycle of thermocline development is, for the most part, not attributable to the assumption of one-dimensionality. A refined treatment of the often neglected terms of the turbulent kin- etic energy budget promises to improve the physical understanding of the turbulent ocean boundary layer and make possible the creation of a better- performing model . A detailed review of the literature associated with mixed-layer modeling is undertaken in Chapter II, but Table 1 summarizes the major historical contributions to the understanding of the physics of the mixed layer. The works of primary concern here are those dealing explicitly with equations for the production, alteration, and destruction of turbulent kin- etic energy within the mixed layer. Kraus and Turner (1967) were the first to heed the turbulent kinetic energy budget in a one-dimensional mixed layer model, utilizing the approximately decoupled state of the equations for the thermal and mechanical energies. By neglect of the frictional generation of heat, the vertically integrated heat equation becomes a relationship for the conservation of potential energy. However, viscous dissipation cannot be neglected in the turbulent energy budget, and Geisler and Kraus (1969) as well as Miropol'skiy (1970) and Denman (1973) added it to the basic model. Niiler (1975) showed that in addition to the equations for thermal (poten- tial) and turbulent (kinetic) energy, an equation for the mean kinetic energy should properly be incorporated because entrainment converts some of the mean flow energy into turbulent energy, over and above the parameterized wind- stress production. Further questions remain and limit the general applications of these earlier models: a. The viscous dissipation of turbulent energy has been parameterized as a fixed fraction of the wind-stress production and hence is a function of the surface friction velocity u*. Dissipation may be related to the integral velocity scale of the turbulence, but this scale is not always proportional to u*. Surface heat (buoyancy) flux and entrainment fluxes can contribute significantly to the turbulent intensity. b. Entrainment may also be considered a function of the ambient turbu- lence parameters. Instabilities leading to entrainment are probably induced by horizontal turbulent velocities locally at the bottom of the mixed layer, so the entrainment rate doesn't necessarily correspond to an integral con- straint upon the total turbulent energy budget. Tabic 1. Samma/iy ofi ttU>toKA.a.oJL CoYvUvLbixtioYii, Date Author(s) Cont r i but ion 1905 Ekman 1935 Rossby and Montgomery 19^8 Munk and Anderson i960 Ki ta igorod 1961 Kraus and Rooth 1967 1969 970 1973 197^ Kraus and Turner Ge i s ler and Kraus Mi ropol ' skiy ; Denman Pol lard , Rhi nes and Thompson Ni i 1 er Constant eddy viscosity solution to steady-state momentum equation: Ekman spi ra l ; "depth of frictional influence"; suggested h * W//sin tji . Improved current measurements to demonstrate h (Rossby number ~ constant). u,/f Simultaneous solution of steady-state heat and momentum equa- tions using eddy mixing coefficients variable with Richardson number. Using dimensional analysis, suggested that h length scale. L, the Obukhov Penetration of solar radiation to depth makes steady-state possible for heat equation because of surface heat loss by conduction, evaporation, and long wave radiation; unstable density profile above compensation depth (h c ) is source of turbulent kinetic energy produced by convection. Included mechanical stirring (parameterized in terms of wind stress) as well as convective production as important source of turbulence for mixing: turbulent kinetic energy equation and heat equation form two-equation model in two unknowns--T, h; non-steady-state solutions and "retreating" (h) possible; viscous dissipation neglected. First "slab" model in which momentum equation is solved to- gether with turbulent energy equation and heat equation; layer assumed homogeneous in T and U, V and hence moves as a slab; assumed buoyancy flux is fixed portion of mechanical produc- tion—essentially same as Kraus and Turner; applied model to atmosphere with subsidence. Assumed dissipation a fixed fraction of mechanical production; remaining turbulent kinetic energy goes to buoyancy flux down- wards including entrainment; essentially same as Kraus and Turner. Slab model applied to ocean mixed layers; used total mechan- ical energy equation (rather than turbulent equation) plus momentum and heat equations; h, T, U, V--all functions of time. Ignored effects of turbulent energy budget altogether. Re-emphasized the need to use the turbulent kinetic energy budget apart from the total mechanical energy budget; divided region into three sub-regions with no mechanical production in most of the mixed layer. Included turbulent kinetic energy produced by entrainment of zero-velocity water into moving slab, but surface mechanical production minus dissi- pation still parameterized in terms of u.,.. Dissipation is not affected by the additional entrainment production, pos- sibly causing a too-large entrainment rate. c. The use of the total turbulent energy equation and consequently the neglect of energy redistribution among components also results in a somewhat inconsistent method of predicting layer "retreat." The consideration of separate budgets for the horizontal and vertical turbulent energy components will permit a more consistent interpretation of both entraining and retreat- ing mixed layers. In this paper, ad hoc mean turbulent field modeling of the terms in the turbulent kinetic energy equations permits the inclusion of these often salient effects in a generalized one-dimensional model of the ocean mixed layer. 2. CHARACTERISTICS OF THE OCEAN MIXED LAYER The ocean mixed layer is defined as that fully-turbulent region of the upper ocean bounded on top by the sea-air interface. The wind and intermit- tent upward buoyancy flux attributable to surface cooling are the primary sources of mechanical energy for the mixing. The most distinctive feature of this layer and what really defines its extent is its relatively high intensity of continuous, three-dimensional turbulent motion. Vertical turbulent fluxes within the mixed layer can be so much greater than vertical fluxes in the underlying stable water column that the dynamics of the layer are essentially decoupled from the underlying region. (Of course the dynamics of the underlying water masses are probably \/ery dependent upon the mixed layer.) Typically, an actively entraining mixed layer is bounded on the bottom by a sharp density discontinuity separating the layer from a stable, essen- tially nonturbulent thermocline. Minimal stress at the bottom together with high turbulence intensity results in an approximate vertical uniformity in mean velocity and density. This ostensible homogeneity is the root of the term "slab," often used to describe the layer. On the other hand, only small gradients in these mean variables give rise to large turbulent fluxes. There- fore, even the slight non-homogeneity is highly important in the physics of the region and should not be neglected at the outset. The nearly zero-flux state of the underlying thermocline causes the bottom boundary condition of the mixed layer to act almost as a slip condi- tion on the mean velocity. This in turn creates a trap for inertia! motion. Deepening of the mixed layer is accomplished by entrainment of the more- dense underlying water into the turbulent region above. This process entails a potential energy increase and cannot take place without an energy source-- the turbulent kinetic energy of the mixed layer above. A simplified picture of the region is shown in Figure 2. There is an appealing practical aspect to the judicious use of the assumption of vertical homogeneity. This assumption permits the use of the vertically integrated U(V) FlguAe 2. lb [h Idealized model, faon. ocean mixed layen. [ ). Mixed layen. depth. (6) l& the thlc\inebi> oi the Interface on. entnauinment zone. momentum and heat (buoyancy) equations, thus avoiding the turbulent flux (of momentum and buoyancy) closure problem altogether. The depth of the oceanic wind-mixed surface layer is typically on the order of ten to a hundred meters. The horizontal scale size is that of the radius of the circle of inertia—seldom larger than a few kilometers in tem- perate latitudes. These two dominant scale sizes are usually significantly smaller than the horizontal scale sizes of the driving meteorological distur- bances, water mass features, and distance to lateral boundaries. Therefore, the approximation of local horizontal homogeneity for all mean variables is usually accurate and is a basic assumption of this work. The local conse- quence of some lateral inhomogeneity can be parameterized without qualita- tively undermining such a one-dimensional model. For example, a divergence of the horizontal current field results in a nonzero vertical mean velocity which in itself can be assumed locally uniform in the horizontal. A minor rate of loss (compared with the surface flux) of mean momentum by lateral and/or downward radiation of inertial motion may be parameterized also with- out compromising the dominant processes. Substantial barotropic and baroclinic features in the mean fields can be linearly superimposed. The mean fields of concern are therefore the hori- zontally homogeneous components of the total fields. In particular, the momentum equation has the geostrophic component subtracted out, eliminating any lateral pressure gradient. 1.3 Fundamental Principles and Equations The underlying principles employed in studying the mixed layer are the combined conservations of mass, momentum, thermal energy, and mechanical energy. The conservation of momentum and the condition of incompressibility are reflected by the Navier-Stokes equations of motion, invoking the Boussinesq approximation: 2 9u i au i 3 ~ 3 u i p 3t- + p o "j 377 + p o e ijk fi /k + 377" (p o - p)9 6 i3 = ^377377 • (K1) J JO 3U, Because frictional generation of heat is negligible compared with typi- cal magnitudes for the divergence of heat flux, the conservation of thermal energy is decoupled from the mechanical energy budget, and the first law of thermodynamics for an incompressible fluid gives the heat equation. ii + ~ 1L_ = JL- 3 9 . _Q_ (i 7\ »t U j 3x d -p C p 3x j3Xj p C p • U " J ' The conservation of salt mass is of the same form, but it lacks a term analogous to the radiation absorption term, Q. 3S , ~ 3S _ n 3 s /-, „\ 3t + u j 3xT s D S33T3T7- (1 ' 4) A simplified but sufficiently accurate equation of state for local and reasonably short-term application in the mixed layer is given by p = P n [l - eCe - 6 ) + a(s - s )j (1.5) where p = o'(e ,s ) is a representative but arbitrary density at the time and location of consideration. The coefficients (b) and (a) are assumed to remain constant. Taking the scalar product of (uf-f ) with the respective terms of equation (1.1), the mechanical energy equation is formed, U • U • \ „ /U.U.\ U • „~ 1 1 J - _3_ [ 1 1 1.1 9£ 2 / u j 3X- \ 2 / + p Q 3X.J" r\j r>^j r\j buc, = v 3X.3X. «J J (1.6) where the buoyancy (ft) is given by Bf = p - p (1.7) Assuming horizontal homogeneity of mean variables, where the horizontal mean is defined by X X^ 2 r 2 (?) (z,t) = lim— / / f (x,y,z,t) dxdy x ~° x2 U h 2 2 and separating all variables into mean and fluctuating components gives W - U g + u(z,t) + u(x,y,z,t) V g + V(z,t) + v(x,y,z,t) + w(x,y,z,t) T(z,t) + e(x,y,z,t) • Pg + p(x,y,z,t) B(z,t) + b(x,y,z,t) Strictly speaking, the total fluctuating part of each variable includes a component directly attributable to surface and internal wave motion. This 8 component has been removed from the fields depicted above. The virtually irrotational wave velocities are assumed noninteractive except as an external source or sink of turbulent energy due to the net contributions of breaking surface waves and radiating internal waves. In practice, a time mean is employed in data analysis rather than even the horizontal mean. Ideally, the averaging time should be short compared with the time necessary for significant changes in the mean fields but long compared with the integral time scale of the turbulence. At + / ./At At 2 B(z,t) = (t>) ~ 75T I b dt, for example. f At 2 In making the Boussinesq approximation, the hydrostatic component of pressure has already been eliminated. The remaining mean pressure is assumed to be geostrophic: a = o n fV n . (1.8a) ax -o "g 9P g _ = - P n f U . (1.8b) dy ^ g Subtracting the geostrophic equations from the total momentum equation and dropping the negligible viscous terms, the equations for the mean momen- tum become in complex notation (for the sake of brevity) |c. i1fC .|a, (i.9) where C = U + i V and c = u + i v. Using equations (1.3), (1.4), (1.5), and (1.7) and again neglecting molecular fluxes, a buoyancy equation is formed: t^t If + "i iir - .rH • (1 - 10) Equation (1.11) is the mean buoyancy equation. 3B 3t 3bw 3Z p o C p (1.11) The use of a buoyancy equation reflecting the combined conservations of thermal energy and dissolved material is not only more general than a heat equation alone, but it makes more obvious the coupling with the mechanical energy budget. Using the decomposition into mean and fluctuating parts and taking the mean of equation (1.6) yields the mean equation for the total mechanical energy. Where E = u-u. = u 2 + v 2 + w 2 , / U 2 + V 2 . E_\ , d_ 3t I 2 2I3Z f^) 3Z U uw + V vw bw 3U 9u i 9X. 3X (1.12) The viscous diffusion and viscous dissipation of the mean kinetic energy are negligible and have been dropped in equation (1.12). The turbulent kinetic energy equation is formed by subtracting the scalar produce of (U-j) and equation (1.9) from equation (1.12). 1_ 3£ + 3_ 2 at 3z wT^+l — 3U L — 3V uw — + vw — 3Z 3Z = bw - e (1.13) The budgets for the individual components of turbulent energy are formed in the same manner from equation (1.1) by setting i = 1,2,3 without summation: 1 3u : 2 3t - uw 3U 3Z 1 3V : 2 at 3Z 3U_ , „ — 3X + ^ 3 uv - n 2 uw " 3" p 3y a < UV - (1.14a) (1.14b) 10 1 1^.^.8. [wl+.ffil + £-|5+o 2 ui7.^. (1.14c) 2 ?T " uw " 97 \ T T £7 / T p7 37 T " 2 uvv " 3 on Notice that the orientation of the horizontal axes is (x) positive to the east a nd (y) positive to the north^ The rate of viscous dissipation e = v 3ui9ui/3xj9Xj behaves like (E/t e ) and the small size of the dissipati, time constant (x e ^ h//T) compared with the time scale of the meteorological forcing causes the intensity of the turbulence to track along in a quasi- steady state with a continually changing net rate of production. Hence the time rate of change of turbulent kinetic energy is usually much smaller than the other terms of (1.13) and thus may be neglected. Because viscous dissi- pation of energy occurs primarily in the range of wave numbers that exhibit local isotropy (the equilibrium range), dissipation (e) is divided equally among the component budgets (1.14 a-c). The second term in equation (1.13) is the divergence of the turbulent flux of kinetic energy. Over the whole mixed layer, it probably accounts for a net gain of energy. Wind-wave interactions at the surface result in some net downward flux, primarily from breaking surface waves. If the Brunt- Vaisala frequency (N) of the adjacent underlying stable water column is suf- ficiently large so as to be comparable with the frequency of the integral scale of the turbulence, turbulent energy may be lost to the generation of internal waves. One of the most significant aspects of this term is that locally, at the bottom of the layer during occasions of entrainment, a net convergence of flux of energy is necessary to maintain the downward buoyancy flux for a deepening mixed layer. The third term, the rate of mechanical production, is perhaps the dominant source of turbulent kinetic energy. It is the rate of conversion of mean to turbulent kinetic energy by the turbulent flux of momentum down- gradient. The last term on the left, the buoyancy flux, locally within the mixed layer can be either a sink or a source. Usually, however, the mixed region is slightly stable overall, and this term represents the rate of increase of potential energy by fluxing buoyancy downward. During instances of large buoyancy flux up across the surface, this term can become an important source as in the case of strong convective cooling in the autumn. The summation of separate component equations yields (1.13), but one term that is wery important in mixed layer dynamics sums to zero and there- fore appears only in the component budgets (1.14 a-c). This term is the correlation between pressure and rate of strain, p/p 9u a /9x a . Since it sums to zero by^ continuity, it causes only a redistribution of energy among if 2 ", v 7 , and w The individual turbulent energy budgets also have redistribution terms due to rotation of the Earth, but these shall be neglected because of the usually short integral time scale in comparison with one day. Perhaps this 11 effect does become significant in some of the very deep convective mixed layers that are not limited in growth by a permanent pycnocline. Application of the vertical homogeneity assumption to the momentum and buoyancy equations (1.9) and (1.11) gives relationships for the turbulent fluxes in terms of the boundary conditions (specified externally) alone: 57 (z) = cw (0) (!+£) + AC (|) jj£ (1.15) bw (z) = bw (0) (1 +|) +| dt Po C p f Qdz 'o w p /. Qdz (1.16) The integral of (1.16) over the mixed layer gives the net buoyant damping for the whole layer. ■/ h-6 bw dz = Jj- AB jj£- bw" (0) p C p •'-h-6 ^z QdA dz . (1.17) Integrating the turbulent kinetic energy equations from z = (-h-6) to z = gives l^(h) = •J-h-6 (-uw 3JU_ az — 8V . t— vw — + bw :) dZ - w(| +B-) 1 Po Po -h-6 (1.18) ^d_ T ^ w = IJ- — dU p d uw + £_ 9Z p 3 n dz - WU' u dz (1.19a) 12 i^(h) = <" VW IF + ^V )dz wv ■if. -h-6 dz (1.19b) ±^(h) = / ^ + ^lf) dz - -h-6 2 p o p o -h-6 / e dz ^h-6 (1.19c) The surface boundary conditions are prescribed functions of time - uw (0) = — (1.20a) _ - v (t) = vw (0) = -a (1.20b) - bw (0) = g [a sw (0,t) - 6 ew (0,t)] (1.21) Also to be prescribed in deriving the system is the radiation absorption function, Q (z,t). The boundary conditions at the bottom of the mixed layer, z = -h, will, on the other hand, conform to the developing situation. To derive these conditions, the equations (1.9) and (.1.11) are integrated over the entrap- ment zone from z = (-h-6) to z = (-h): 6/h U m -h -h-6 3U , 3t dZ [U (-h) - U (-h-6)] ^= AU ^ 13 -h -h-5 1 im 5/h -0 / f V dz = 8UW , i u v t-— dz = - uw (-h) d Z Therefore notation, Also, JL - uw (-h) = AU -nr , and similarly JL vw (-h) = AV -tz , or in complex cw (-h) = AC ||~ . (1.22) bw (-h) = AB ^ (1.23) where AC = All + iAV and AB are the respective jumps in the values of the mean variables across the density interface separating the mixed layer from the nonturbulent region below. The discontinuity need not be a perfect one (o = 0) for the boundary conditions (1.22) and (1.23) to be valid. A suffi- cient condition is for the fluxes of momentum and buoyancy from the mixed layer into the interface zone, resulting in a lowering of the vertical pos- ition of the zone, to be much larger than that portion of the fluxes con- tributing to changes in the momentum and buoyancy profiles of the moving interface zone itself. Integrating equations (1.9) and (1.11) from (z = -h - 6) to (z = 0) provides a form of the equations that includes the effects of the entrainment stress and entrainment buoyancy flux of (1.22) and (1.23). h ^ + AC ^ A = - ifh - c^ (0) (1.24) 14 and h d . AD dh . ir + AB at A = _§£ p o C p J -h-6 Q dz - bw (0) (1.25) where the Heaviside unit step function, A, is dependent upon (dh/dt) r dh^ |dt; lfor&>0 for |<0 (1.26) and < > denotes a vertical mean for the mixed layer: .+6 J = y^ / C dz > etc * -h-6 1.4 Course of Action in Attacking the Problem To lay a foundation and present a perspective of the problem at hand, the literature treating models of the surface mixed layer is reviewed rela- tive to the basic principles and general equations laid down in the previous sections. This approach organizes the historical work in terms of the funda- mentals, and it provides the stepping stones for the development of this research. The turbulent kinetic energy budget is examined closely. The role of the previously neglected redistribution terms is assessed. All of the terms are modeled by use of mean-turbulent-field techniques, permitting the even- tual implicit solution for the turbulent energy content of the mixed layer. The final preparatory work needed to complete the model is treated in a chapter on entrainment. This includes the derivation of an equation relating the rate of entrainment (dh/dt) to the other variables. The final numerical method of solution of the nonstationary, non-linear set of equations permits the solution of hypothetical cases as well as the simulation of field observations. The numerical model requires as input the initial conditions of density and current and the surface boundary flux of buoyancy (heat and/or salinity) and surface wind stress as functions of time. Model outputs include the mean density profile, the turbulent kinetic energy, and the mixed-layer depth, all as functions of time. 15 2. REVIEW OF THE LITERATURE 2.1 Ekman Depth of Frictional Resistance V. Walfrid Ekman (1905) originated the concept of a "depth of frictional resistance" for the upper section of a wind-stressed ocean. This depth (d) comes from the mathematical solution to the steady state horizontal momentum equation, (1.9), in which the Reynolds stress is related to the mean shear by a constant eddy viscosity (K). 3C at ^ = - ifC - 8CW 3Z (1.9) where 4= at u and — v 3C - cw = K - . Then = - ifC + K dfC dz 2 (2.1) If the boundary conditions are - cw (0) = u| + i and - cw (-») = , the solution is C(z W ttZ tt sin ( d~ 4 } / ttZ , ir \ 1 cos ( d~ 4 } TTZ d (2.2) where / 2K (2.3) At the depth (z = - d) the direction of the flow is opposite to the surface current and the magnitude has been reduced to (e -7T ) times the surface magni- tude. Classical thought has suggested that any surface layer mixed by the action of the wind should have a depth that is of the same order as (d). 16 Using a quadratic law relating windspeed (W) to surface stress P uI = C D P a W 2 (2.4) and an empirical relation from observations, | C (o)|-SJfl2LW f (25) /sin (j> Ekman derived a formula for (d) as a function of wind speed and latitude () d = 7.5 sec" 1 W . (2.6) /sin 2.2. Rossby Number Rossby and Montgomery (1935) pointed out that the depth (h) of a surface drift current layer and Ekman' s depth of frictional resistance (d) are not necessarily comparable: the depth (h) has a definite physical meaning, but (d) designates only the theoretical rate of exponential decay for a system obeying (2.1), Rossby and Montgomery derived the formula (h « W/sin ) or, equivalently h - u jf (2.7) where the constant of proportionality is the Rossby number, R = u*/hf. They then presented measurements demonstrating the greater validity of (2.7) in comparison with (2.6.) It should be recognized, however, that Ekman' s result differs from that of Rossby and Montgomery only because of the use of the relation (2.5.) If instead of applying this empirical constraint, the eddy viscosity (K) is modeled in terms of likely turbulent length and velocity scales (K *v* u*h) and is assumed to be constant with depth, then (d = 2ir 2 u*/f) and the quadratic stress law gives 17 So, in spite of suggesting a distinction between the mixed layer depth and the depth of fractional resistance, Rossby and Montgomery obtained a result perhaps more comparable with that of Ekman than with the real situ- ation. This was the case because both derivations considered only the momen- tum budget, neglecting the effects of buoyancy and mechanical energy upon the vertical fluxes of buoyancy and momentum within this turbulent oceanic boundary layer. 2.3 Eddy Transfer Coefficients in a Steady-State Problem Munk and Anderson (1948) first combined the two problems of density structure and current structure into a unified theory on a steady-state thermocline. Like Ekman, they proposed an eddy viscosity (K m ) plus an eddy conductivity (Kh), but these parameters were made variable with the local gradient Richardson number (Ri). Ri = MGSL . (2.9) (8U/8Z) 2 This model therefore included some of the effects of the turbulent energy budget. Vh - (i + c° „ Ri) Vh • (2 - 10) ' v m,H This function for the eddy viscosity and eddy conductivity was chosen because of its asymptotic behavior for small and large values of (Ri): 1 im K u = KO, coefficient for no density gradient Ri -* ™' H 1 im K n = 0, for extreme stability. Ri -v °° ' Because Munk and Anderson assumed steady state and did not recognize the presence of a sharp interface marking a boundary between the fully-turbulent mixed layer and the essentially quiescent stable region below, their results still resembled Ekman' s original solution more than they do the physical real ity. 2.4 Obukhov Length Scale The more recent efforts in modeling the oceanic mixed layer started with a one-dimensional steady-state study by Kitaigorodsky (1960). Assuming that the ocean surface mixed layer was analogous to the constant-flux atmospheric 18 surface layer (as in Businger et al., 1971), Kitaigorodsky concluded by dimensional analysis that the mixed layer depth must be proportional to the Obukhov length, L. For Kitaigorodsky' s assumptions, the momentum and buoyancy equations, (1.9) and (1.11), reduce to 8Z and 8bw ' = (2.11) = . (2.12) 3Z The radiation absorption, Q, was assumed to be confined to the immediate surface layers. Taking the x-direction to be in the direction of the wind, the solutions to equations (2.11) and (2.12) are - cw = constant = u| and - bw = constant = u*b* where (uf) and (u*b*) are the downward surface fluxes of momentum and buoy- ancy. If the depth of the surface mixed layer (h) is dependent only upon the two parameters (u*) and (b*), then u, 2 ( hsg|ew(0)|' (h b ' h = H* L (2.13) where L = uj/b* (2.14) and H* is a constant of proportionality. If the coriolis force is a significant component of the mean momentum budget, then (2.11) is replaced by 9cw 8Z = - ifC . (2.15) 19 Adding the coriolis parameter (f makes H* variable. = 2Q sin ) to the dimensional analysis j* = — (2.16) with a second dimensionless product (b*/u*f). Using data from the NORPAC expedition, Kitaigorodsky found that equation (2.13) with a constant H* was insufficient for cases varying over more than twenty degrees of latitude (see Fig. 3). Using Kitaigorodsky 1 s data, it should be noticed that the Rossby number (Ro) based upon the layer depth and (u*), D fh (2.17) is less variant than H* = h/L for the same data. This can be seen in Figures 3 and 4. \ \ Dota \ Ro dependence: h a W/f \ h a L, Obukhov length: haW 3 /(/3gQ) 23 28 33 38 43 4! Latitude (Degrees North) F-Lgu/ie. 3. kUx&d lay&i dky {I960, 23 28 33 38 43 Latitude (Degrees North) VajQUJih 4. Ulxnd Layoji dzpcndnncz upon both Ro~ l and H* {data {^ftom ICLtaigosi- odUky, 1960). 20 There is a basic flaw in this model arising from the assumption that the ocean mixed layer is analogous to the "constant-flux" atmospheric surface layer. The ocean, in all probability, does have a surface layer over which fluxes are approximately constant and a quasi-steady state does exist. How- ever, the depth of such a layer would be limited to at most the upper ten percent of the vertical extent of the entire mixed layer. With a constant heat flux at the surface, the mixed layer temperature and depth cannot both remain unchanged. 2.5 Compensation Depth for Shortwave Radiation Kraus and Rooth (1961) also conceived a steady-state model based pri- marily upon the buoyancy equation. In their model however, steady state was made feasible by balancing the short wave radiation input Q(z) with a net surface heat loss, p Cp9w(0)>0, by means of evaporation, conduction, and infrared radiation. A compensation depth (h c ) is the depth at which a bal- ance is struck between the surface heat loss, p C p ew(0), and the total radi- ation absorbed in the layer above, Q d z h f, c r If Q = y Qo e YZ and if buoyancy is a function of temperature alone, then (1.11) becomes in steady state 8 6W 8 2 p o c p + -L . (2.19) The depth (z = - h c ) is that level at which the turbulent flux ew goes to zero, with a stable temperature profile below and an unstable one above. Therefore in the region (0 > z > - h c ) turbulent kinetic energy can be con- vectively produced since here bw = pg ew > . 21 Integrating (2.19) from (z = - h c ) to (z = 0) and using (2.18) gives h c = y" 1 In 1 1 - P C p ew(0)/Q (2.20) Figure 5 is a schematic portrayal of the Kraus and Rooth concept. If ew = Kn — , K H ^ constant in z , (2.21) then (h c ) is also the depth of maximum temperature. Using the eddy conductivity closure posed by (2.21), Kraus and Rooth examined the structure of the temperature field from the surface down through the mixed layer and across the interface, and its variation with changes in (Q) and the boundary conditions. Their solutions are only qualitatively useful because not only is (K^,) unknown, but also it is assumed to be con- stant—even across the density interface. -hc c = Y 'An ( h, = Q o^o c p ew(0) ) h > h. / / / / / / / / / z / / / solar radiation through surface > Po c.J^{0) conduction, evap- oration and infra- red radiation ewio) > o 9w 6w(z) = 0w(O)- ° (l-e yZ ) ro g p Vi.QuJm 5. E{)^cX of, the domp2.n6cutlon depth, h c , cutum-ing &tmdy Atate. and Q = YQ e YZ . 22 With regard to the depth of the mixed layer (h), all they could really say was it must reach some depth greater than (h c ). "Dependent upon the intensity of the convective and wind drive turbulence, this convective regime may penetrate more or less deeply beyond the level (h c )." They visualized a steady-state layer depth as being possible only by requiring upwelling (W>o) of sufficient magnitude to maintain the vertical position of the entraining interface. 2.6 Prototype Turbulent-Energy-Budget Model: Kraus and Turner Recognizing the limitations in application of the Kraus and Rooth model--no provision for a possible downward surface heat flux, no account of mechanical production of turbulent kinetic energy, and the steady-state constraint--Kraus and Turner (1967) further improved and generalized this kind of one-dimensional model. Their model was the first instance in which it was recognized that the budgets for thermal and mechanical energies could be considered separately. This is valid because the dissipative rate of heating (p /J»e) is several orders of magnitude smaller than either Q(z) or | p Cp 3ew/3z|. Therefore, their model consisted of two separate equations-- the neat equation and a mechanical energy equation in which the net effect of the work of the wind on the sea surface and the viscous dissipation within the mixed layer are parameterized. This use of a mechanical energy equation together with the buoyancy (heat) equation and the boundary condition (1.23), - bw~ (-h) = AB ^ , (1.23) gave for the first time a closed set of equations whose solution provided h(t). If buoyancy is a function of temperature alone and constant in the mixed layer, equation (1.25) is applicable. Using the radiation absorption func- tion (2.18) it becomes h 4# + AT £r A = -^- (1 - e" Yh ) - ew (0) . (2.22) dt dt p C The turbulent kinetic energy equation, integrated from the top of the en- trapment zone (z = - h) to the surface is f° — G - D = -3g / ew dz (2.23) 23 where r°- w 3U. dz is the total rate of mechanical production and J-h D = / £ dz is the total rate of dissipation within the mixed layer. Neither of these parameters in the Kraus-Turner model is an implicit variable, and each must be specified externally. The heat equation that leads to (2.22) is also used to eliminate ew from (2.23) by integrating between z and 0: m 96W 3Z* P C p ) dz' = or ew (z) = p -f (1 - e YZ ) + ew (o) -z^f (2.24) Integrating ew (z) as prescribed by (2.24) from (z = - h) to (z = 0) gives /.' 7— j 1 u? d ew dz = 75- h z ■■ c dt OP - ew(0) h + Yn C rp o p 1 - e - Y h (2.25) Neglecting e" Yh , equations (2.22), (2.23) and (2.25) can be used to give another equation (2.26), which together with j_2.22) constitutes a closed system in h(t) and (t) where G, D, Q and ew(0) are prescribed functions of time at most. h 2 d yT . dh . G - D T -3T + ATh dt A = -W + PoV (2.26) An important contribution by Kraus and Turner was the conceptualization of a model for which a stationary or even retreating mixed layer depth is possible. In such a case where 24 #*»■ A ^ = equations (2.22) and (2.26) are still applicable because of the presence of the Heaviside unit step function. Setting (h = h r ) in the case of a retreat- ing or steady mixed layer depth, these equations reduce to (2.22a) and (2.26a). h 2 - n r d G - D + — r - . (2.26a) dt eg p c pY Neglecting short-wave radiation that escapes the mixed region and eliminating (d/dt) between (2.22a) and (2.26a) gives h = 2(G "_^ . (2.27) eg (Q - ew(0)) Whenever the surface boundary conditions and/or solar radiation adjust to make (h r < h) the mixed layer will "retreat." Of course, the region does not unmix, in accordance with the second law of thermodynamics. The net rate of production of turbulent kinetic energy, G-D, is insufficient to balance the rate of increase of potential energy, r° - I - eg ew dz, J -h required to mix the region all the way to the_ density interface. Conse- quently, as the region warms (notice that d/dt can be only positive at this time), a new density interface is established at z = - h r . Kraus and Turner model G in terms of the friction velocity: G = u* 3 . (2.28) Not knowing the importance of the viscous rate of dissipation, 25 D = / £ dz, J -h they simply neglect it. They find, however, that their model predicts a too- large value for (h), and that the dissipation should possibly be included. Turner (1969) felt that there was a problem with (2.28) as well. From examination of observations of sudden wind speed increase and ensuing mixed layer deepening, he deduced that "a substantial fraction of the part of the work done by the wind which goes into the drift current is eventually used to deepen the surface layer." This statement reflects the need for a more comprehensive model, particularly for unsteady situations. Such a model should reflect the input of energy into the mean velocity profile and the time delay needed to shift some of this energy to turbulence. Again with regard to the Kraus and Turner model, the setting of (D = 0) so that -h u -eg f ew dz = G (2.29) places an unrealistic constraint upon the buoyancy term: it becomes depen- dent only upon the mechanical production. The error in this is most obvious when there is strong surface cooling and (- / ew dz) J -h is less than zero. Kraus and Turner also neglect the effect of entrainment in their turbu- lent kinetic energy budget, equation (2.23). In spite of these deficiencies, this model was a big step in the right direction in its consideration of the turbulent energy budget in recognition of the energy source for mixing and entraining. 2.7 Adding Dissipation MiropoTskiy (1970) and later Denman (1973a) assume that dissipation is a fixed fraction of the shear production, equation (2.30). D = 6' G (2.30) 26 where 6' is an empirical parameter to be determined from observations gives, instead of (2.29), the equation (2.31): This - eg / ew dz = (1 - 6') G (2.31) This does not solve the dissipation problem because 6" cannot have a constant value. In essence, dissipation must be allowed to adjust to the total situ- ation as it evolves. Miropol'skiy also assumed that G°=u|, but a variation of an exercise he uses to deduce this demonstrates perhaps the major source of error in a model like (2.28). G = - L ( uwfl + vw — dz If 3uw/3z = - fV and 3vw/9z = fU, then In general , however. - uw V - uw V -h 3uw _ ... all , fV - w , and 9VW .c,, 3 V JT = fU " at ' giving ^ U: G = uw U - vw V -h (2.32) and thus indicating the importance of the mean kinetic energy and its distri- bution within the mixed layer. As will be shown, even if the wind is steady for long periods of time, the mean kinetic energy can change markedly on a time scale corresponding to the inertia! period. Most recently, the trend in the literature has been to model the mixed layer as a vertically homogeneous moving slab with density and velocity discontinuities at the entraining interface. Geisler and Kraus (1969) were the first to use the slab approach in their model of the atmospheric boundary 27 layer. This problem is almost completely analogous to the oceanic mixed layer problem except that the atmospheric boundary layer is driven by a horizontal pressure gradient rather than by a surface stress. A rigid sur- face boundary rather than a free surface also results in a subtle but impor- tant difference. Nevertheless, the basic setups of the two problems are the same. Simultaneous calculation of the mean velocity together with (h) and (T) permits the implicit calculation of the mechanical production of turbulent energy, an improvement upon the previous (u|) method. The equations (2.33) and (2.34), reflecting the conservations of mean momentum and mean buoyancy are essentially the same as (1.24) and (1.25). The only real difference is that Geisler and Kraus assume a prescribed mean subsidence (analogous to ocean upwelling) in their atmospheric boundary layer. This non-zero vertical velocity (W) can result in a stationary mixed layer depth even when entrainment is occurring. This then is a different mechanism than that developed by Kraus and Rooth (1961) for obtaining a constant (h). h ^^ + AC (~ - W) = - ifh ( - Cg) - cw (0) . (2.33) h ^T + AT ( dT - W) = " ™ (0) • (2 ' 34) In (2.33), (- ifCg) is the kinematic geostroplrk pressure gradient, the source of momentum. The kinematic surface stress, cw (0) is a momentum sink in this case. Geisler and Kraus seem to avoid the problem of dealing with the viscous dissipation of turbulent kinetic energy by prescribing a fixed value for the integrated flux Richardson number, Rfj. .h _ bw dz /„ Rf T = — s ~i —x— ■= n, constant . (2.35) f h / UW 3U " UW 9V\ J Q \ 3Z ZZ) dz This, however, is equivalent to Miropol 'skiy's method. From (2.31) and (2.35), n = 1 - 5' . (2.36) Since (n) is always a fixed positive number, and with no radiation absorption in the model, the buoyancy flux (heat flux) can only be downward. Therefore, this model like that of Ki taigorodsky is restricted in application to only those cases where the mixed layer is stable throughout. 28 2.8 Role of Mean Kinetic Energy Pollard, Rhines and Thompson (1973) apply the slab approach to the oceanic mixed layer, but they complete the entrainment problem with a differ- ent mechanical energy requirement. The time rate of change of the total mean mechanical energy, potential plus kinetic, of the slab is set equal to the rate of work by the wind on the mean flow: 3PE + SKE at at - U uw - V uw z=0 (2.37) This is the same as (1.12) integrated from (z = - h - 6) to the surface if viscous dissipation is neglected and (U) and (V) are constant in (z) within the mixed layer. The turbulent buoyancy flux (bw) gives the potential energy change using equation (1.11): 3PE 3t f rrr _,_ 1 bw dz = h 2 — + 2 n at h AB ah at Neglecting the time rate of change of the turbulent kinetic energy, aKE at 2 at (U 2 + V 2 )h If radiative heating is ignored, (2.37) reduces to * Ri = hAB = 1 U 2 + V 2 Pollard, Rhines and Thompson assume that (2.37) applies as long as (2.37a) S- • (z = 0) Pn - U uw - V uw J is positive. As soon as this rate of work by the wind becomes negative, "energy flow to increase (h) ceases and since the water cannot unmix, (h) must be constant...". Therefore, mixed layer deepening would occur only up until one-half of an inertial period following the_ onset of a steady wind stress. This result is demonstrated by setting cw (0) = AC = in equation (1.24), giving ul and *hf>hi + if ( h ) = U: 29 The particular solution is + i = (iu^/fh)(e -1 ' t -1). Hence x-fr (z = 0) goes to zero when vanishes at time t = -n/f. In this model, energy for entrainment is derived directly from the mean flow. A separate budget for turbulent energy is not considered. That is to say, the intensity of the turbulence is not considered to have an active role in the mechanism of entrainment in this model. In a three-layer model of the ocean mixed layer, Niiler (1974) combined elements of Pollard et al . and Kraus and Turner (1967) in that both turbulent kinetic energy and the mean kinetic energy are considered to be important in the mechanism that determines rate of entrainment. Figure 6 is a diagram of the vertical temperature and velocity structures in this model. The turbu- lently active region was divided into three subregions: (i) a constant-flux surface layer, (ii) the major part of the whole region, and (iii) the en- trainment zone, lying just above a "quiescent abyss." The momentum and buoyancy equations used by Niiler are virtually the same as the "slab" equations (1.24) and (1.25) if B = B(T) only. 1 -d' -h -h-8 T \ Q "Perturbation Energy Production Zone" u T Mixed Layer ^^^^ ^^^^^^ "Perturbation Energy ^ ^^-^^~^ Production Zone" h- "Quiescent Abyss dJ dz " : 0w = uw = vw - w( -^- + -5- ) - "0 c F-cgate 6. IdmLiznd picAmn.2. o{> ocoxm mixed layeA [HUXqa, 1974) 30 h d , . . r dh . £U r — / n \ F —jT— + A A C -rr = lfh - CW (0) . at at p (2.38) h d _ . . T dh _ — / n x ir _AAT dt 9w(0) (2.39) Radiation absorption was assumed to occur entirely in the uppermost layer and was therefore included in ew (0). The additional term (F/p ) is a "damping" force for inertial motions — = |c| C h o (2.40) within the mixed layer and presumably is related to the "'dissipation' of mean motions as well as the radiation flux of momentum from the bottom of the mixed layer." Pollard and Millard (1970) considered such a term as well, but one that was linear and thus resulting in an exponential damping. The time constant (tj) ranged from four to twenty-five inertial periods, depending upon the size of the inertial circle relative to the horizontal scale of the forcing wind system. Chp, T I = If the mean velocity below the density interface is zero and the mean temperature below the interface is given by T(z < - h) = r T z, then the fluxes at (z = - h) reduce to - cw (-h) = . r dh r dh AC dt = dt (2.41) and e^ (-h) = a T ^ = ( + r T h) {£ . (2.42) Since T = and C = for the bulk of the mixed layer, the turbulent fluxes are linear functions in (z): - cw (z) = - cw (0) - I - ew (z) = - ew (0) - Tr cw (0) + dh dt ew (0) + ( (2.43) + r T h) ^J. (2.44) The relative importance of the terms of the turbulent kinetic energy budget (1.13), was hypothesized to vary with subregion. 31 ^_E at 3 ft p 2 p — au L — av UW 97 + VW 3? + bw - E (1.13) The rate of mechanical production, -u n -w aU-j/az, was assumed to be non-zero only in the surface and entrainment subregions. In these two regions of small vertical extent, the buoyant damping was considered by Niiler to be insiqnificant. Thus the balance is between mechanical production, turbulent and pressure diffusion, and dissipation, equation (2.45a): 3_ 3Z w /£_ + £> r aU , — aV n , t— + vw — + e = for a Z dZ > z > - d' -h > z > - h (2.45a) In this model aU/az = aV/az = within the central part of the mixed region, and hence mechanical production is necessarily assumed to be zero. There- fore, here the buoyant damping and viscous dissipation are balanced by diffusion from the two adjacent production layers, giving equation (2.45b). a_ az 2 p o ' bw + e = for {- d'>z> - h} (2.45b) Integratin g equations (2 .45a) and (2.45b) vertically and combining them to eliminate w(E/2 + p/pq) a t (z = - d') and (z = - h) gives ^1- / d °(-f + -§) J aV , vw-\dz • dz + 1I 2 dh = dt /' ^-h bw dz . (2.46) In the manner of Kraus and Turner, Niiler parameterized the sum of the first three terms of (2.46) in terms of the surface stress. - w ■0 p o 2 u.w i dz 32 Integrating (2.42) to give /; bw dz = - 3g /; ew dz = 3g [- 9W i 0) h ♦ \ ( ♦ r T h) $] and using (2.47), equation (2.46) becomes m u| + [ 2 dh 2 dt " Ba - 9W (Q> h + \ ( + r h) — [ l n) dt (2.48) The system of equations (2.38), (2.39), and (2.48) is a closed set of equa- tions in the three unknowns: - h, and . This model more resembles that of Kraus and Turner than it does that of Pollard et al because of the utilization of a parameterized turbulent kinetic energy equation rather than a total mechanical energy equation. The primary difference is the presence of the entrainment production term, || 2 /2«dh/dt, in (2.48) which necessitates the additional equation, the integrated momentum equation (2.38). One aspect of important consequence in Niiler's model manifests the need for an even more comprehensive term-by-term modeling of the turbulent kinetic energy equation. This is the fact that the turbulent kinetic energy produced by entrainment, | 2 /2«dh/dt, must go entirely toward increasing the poten- tial energy. Because of the parameterization, (2.47), dissipation is not permitted to adjust to include either the direct effect of this particular source, or the less obvious effects related to the entrainment or lack of it. The only solution to this predicament would seem to be to model dissi- pation separately from any of the source terms, allowing it to adjust to total turbulent intensity. 3. CLOSING THE PROBLEM The equations (1.19a-c), (1.24), and (1.25) do not by themselves consti- tute a closed system of equations for the mixed layer. Vertical integration over the mixed layer simplified the equations but introduced yet another unknown, the mixed layer depth (h). 33 3.1 Net Viscous Dissipation in the Mixed Layer f -h-6 3U. 8U. v ax-9xT dz A dissipation time scale (x e ) is defined by e = /x e . For fully-turbulent geophysical flows having large Reynolds numbers, viscous dissipation of the turbulence occurs primarily in the small eddies that are locally isotropic. As explained by Tennekes and Lumley (1972), an inviscid estimate of dissipation may be made by taking the rate at which large eddies supply energy to small eddies (equal to the rate of dissipation) to be proportional to the reciprocal of the time scale of the large eddies. If the time scale of these large eddies is pr opor tional to the mixed layer depth divided by the rms turbulent velocity / , then an integral model for dissipation in the mixed layer, independent of viscosity and the small scales is •0 F -h-6 d z = mi 3/2 (3.1) wh^ere (m^ is a constant of proportionality. For those situations where = u|, equation (3.1) is the same as that used by Miropol'skiy (1970) and Denman (1973). An important concept in modeling dissipation is that of local isotropy. Turbulent kinetic energy generated at the largest scale (^h) is transferred without much additional production or dissipative loss through the inertial subrange to the larger and larger wave numbers (smaller eddies) by vortex stretching. Dissipation is significant only at the lower end of this iner- tial subrange. Because there is no preservation of the original orientation, the small eddies of the inertial subrange are "locally isotropic." Therefore, dissi- pation draws approximately equally from all three turbulent energy compon- ents. Of course, the existence of an inertial subrange is dependent upon a large Reynolds number, and this is certainly the case for the oceanic mixed layer. 3.2 Net Effect of Redistribution of Turbulent Energy R = / B- r-2- dZ . a / P n 3X 4-6 ° a 34 As previously discussed, Ri + R 2 + R3 = 0, but R a may be an important source of sink term for the individual turbulent kinetic energy budgets. Following the early lead of Rotta (1951), but in agreement with the dominant term of the rational closure technique of Lumley and Khajeh-Nouri (1974), R = -m 2 /e> (3 ) (3.2) In addition to dimensional consistency, the concept leading to (3.2) is that of a "return to isotropy." In other words, the correlation of pressure and turbulent rate of strain tends to redistribute energy equally among the three components. 3.3 Shear Production -h-6 (uw 37 + vw az } dz 6C, AC dh dt (3.3) where |6C | is the "excess" surface mean velocity in the direction of the wind stress. Notice that in this instance the inhomogeneity of the mean velocity field cannot be neglected. 0* ■*•«>] -h-6 d z = nr 3 u* 3 where uj = |cw (0) | . If I A\ dz = m 3 u 3 + ' 2 dt - ^ > This is basically in concurrence with the work of Kraus and Turner, Denman, and Niiler. 35 3.4 Need For an Entrainment Equation Using the suggested relationship (1.24), (1.25), and (3.2), equations (1.14 a-c) become (3.5 a,b). The two horizontal component equations have been ad ded t og ether t o giv e an equation for horizontal turbulent kinetic energy = + , together with the equation for vertical turbulent energy : 1 it (h 7 ^ ) = m3lj3 + lA T i oT A " m ^ { ^ " ^ p7) - 2m 1 F 3/2 - (3.5a) 1 d 2 dt {h7 ^ ] = 2 bw~ (0) - AB & A - -^-Q dt p L + m 2 / ( - 2 ) m i = 3/2 3~ (3.5b) where Q -h-6 z Qdz) dz and : If h (t) is unknown, equations (3.5a,b) together with (1.24) and (1.25) are a n incomplete system of four equations in five unknowns: h, , , . An entrainment hypothesis will provide the fifth equation needed to close the system. 4. ENTRAINMENT HYPOTHESIS 4.1 Entrainment in Earlier Mixed Layer Models In this study the entrainment velocity, u e = dh/dt, will be modeled explicitly in terms of the other free parameters of the system. However, in much of the literature treating models for the ocean mixed layer, (u e ) is a consequence of various assumed constraints placed upon the mechanical energy budget for the layer as a whole. In the first really tenable model of the mixed layer that was capable of simulating a growing mixed-layer, Kraus and Turner (1967) assumed that all of the turbulent kinetic energy produced in the mixed layer goes to increase the potential energy of the system. After taking into account any surface 36 buoyancy flux, the balance of the potential energy change went to entrap- ment — mixing the requisite amount of underlying denser water uniformly throughout the homogeneous mixed region. Their resultant entrainment velocity is 2 (G - D) + wb(0)h + I-H-) 0/h u = ^P/ (4 1) u e hAB { * mi) where (G-D) is the net rate of mechanical production of turbulent kinetic energy minus the rate of viscous dissipation for the whole layer. The surface buoyancy flux wb(0) may either increase or decrease the entrainment velocity, depending upon its sign. Again, not knowing how to deal with the dissipation (D), Kraus and Turner ignored it. They set G = u|, so (4.1) becomes u «£ [4. la) e hAB where the solar heating function (0/) and the surface buoyancy flux have been combined in defining a buoyancy flux scale: §221- wb (0) p n C n b* = -^ (4.2) Geisler and Kraus (1969), Miropol'skiy (1970), Denman (1973) and Niiler (1974) all assumed that a fixed fraction of the mechanical production of the turbulent kinetic energy would be dissipated. Therefore, their resultant entrainment rates are the same as (4.1a) except that some constant smaller than 2.0 would precede u|. Pollard, Rhines and Thompson (1973) did not explicitly consider the turbulent part of the mechanical energy budget, and therefore a conceptually different relationship results from their use of the total mechanical energy equation: (hAB - |aC| 2 ) ^jj- = u*b*h . (4.3) 37 If (b* = 0) this reduces to Ri* = 1, or h> AC AB The (>) sign was added to prevent the mixed layer from retreating when mean kinetic energy is removed in the second half to the inertial cycle. This approach at first seems to give a plausible result based upon the stability of the mean flow. The problem is that the mixed layer is already turbulent, and the system has an excess of turbulent kinetic energy, some of which may be available for mixing at the interface, regardless of the value of the overall Richardson number, Ri*. It is granted that Ri* may be con- strained to having a value greater than some critical value by reason of a mean flow instability, but the fact is that in the laboratory and in geophys- ical cases, measurements indicate that entrainment occurs even though Ri* is much larger than one. The entrainment experiments with mean shear of Kato and Phillips (1969) and Moore and Long (1971) (Figs. 7 and 8) show no direct relationship between Ri* and a maximum layer depth. More importantly, Turner (1968) and others have conducted experiments with growing mixed layers that had only turbulence and no mean shear (Ri* = °°). All of these laboratory results when examined together strongly imply that the bulk Richardson number Ri* is not the parameter most relevant to entrainment rate. Instead, a similar nondimensional number, using the tur- bulent kinetic energy (E) rather than |aC| 2 , is suggested. As shown by Niiler (1974), the mean kinetic energy can influence entrain- ment rate by increasing (E) in a growing mixed layer. This in turn increases the average position of the bottom of an oceanic mixed layer undergoing a sequence of entrainment and retreat due to varying wind stress (uj) and heat- ing/cooling (u*b*) cycles. Figure 9 therefore indicates some statistical de- pendence upon the vertically-averaged Richardson number and hence Ri* as well 4.2 Suggested Turbulent Mechanism Benjamin (1963) shows that three basic types of instabilities are pos- sible for a system where a flexible solid is coupled with a flowing fluid. At the interface between the mixed layer and the denser water beneath, a so- called class "A" instability will arise if where (k) is the wave number of the interfacial disturbance, 10"' - -2 _ \ 1 1 1 1 1 1 1 1 | 1 1 1 oo * .v\ * +« '" ■ "V 5s\ ° - * ^«- V"» - \^ o 1.92 3.84 7.69 i£ X l0 3 cgs \ 0.995 □ ■ O 1.485 0. h 2.120 • + - 2.750 * 1 X - 1 ! 1 1 ! i : i i i hAB Ri* a (E*)~ Vlgaxe. 7 . Rata o£ zntnjovinmcint v6. hAB/ul [Koto and PhMip*, 1969) VlgaAd taken in.om Kantka [1975). Ftgu/ie. S. Rate, o£ y 2AB However, the class "A" instability is dependent upon energy dissipation in the lower fluid, and this is likely to be small compared w ith inferred rates of convergence of energy flux, - 3/3z [w(p/p + E/2)]_ n , at the inter- face. For geophysical flows of this type having large Reynolds and Pe*clet 39 DEPTH flvERACCC RIOARCSCN NLTBER. 15.8-26-2 METERS DEPTH flVERRCED RICHARDSON NUMBER. 26.2-46-2 METERS 111 FlguAz 9. Vzpth-avz/uxgzd gradient RichaAdi>on numbeA, AzAB/|AC| 2 , v£/l6o4 time, at two dupth nxmgz& [buoy mcaAuAments comttuy o& Vavld HatpoAn) . \Jatuzi> gucatcn. than 100 weA& defined ai> equal to TOO. Vaik&d kofhizontat line at 1.0 izptieAzntA the. condition ^on. maAgtnat dynamic i>tabiJUXy. numbers, the class "C" instability is therefore most likely to be the domin- ant mechanism leading to observed rates of entrainment. The specific mechanism that is envisioned in the destabilization of the interface and the resulting entrainment is a "local" Kelvin-Helmholtz (K-H) instability. The shear needed to trigger such an instability is provided by the local turbulent eddies. The mean shear contributes to the instability but cannot in itself generate a critical Richardson number. The mean gradi- ent Richardson number Ri : _ 9B/3Z (3U/3Z) 2 (4.4) would not be likely to achieve a critical value because the total instan- taneous Richardson number, 40 R . = j^z_ (4>5) 1 (3U/3Z) 2 is the relevant parameter. The minimum value of the envelope of Rij(t) at the interface would determine the advent of any appreciable interface insta- bilities. The onset of the K-H instability and its exponential growth rate is pre- dicted by linear two-dimensional wave theory. As individual wave packets achieve a significant amplitude, the nonlinear and three-dimensional effects of the turbulence field prevail by distorting the wave shapes and advecting parts of the exposed cusps of denser water up into the mixed layer. There- fore, only the initial stages of the instability are strictly of the K-H type, where the induced suction at the crests of a perturbation wave on the interface is large enough to overcome the restoring buoyancy force. 4.3 Relevant Parameters and a Dimension Analysis Since the mechanical energy needed to continue this mixing process and thus provide for a significant entrainment velocity must come from the turbulent eddies, the rate of supply of turbulent kinetic energy, - d/dz [w(p/p + E/2)]_., just above the interface should determine the value of (u e ) for a given buoyancy jump (aB) and mean velocity drop | AC | across the interface. As Long (1974) summarizes experimental studies, the nondimensional entrainment velocity W* is found to depend upon the first power of the dimensionless parameter E* based upon the buoyancy jump across the interface, the depth of the homogeneous layer (h), and the intensity of the turbulence . Where Long assumes a uf, and W = — aE . (4.7) The relationship (4.7) together with an integrated turbulent kinetic energy equation to provide (t) plus the mean buoyancy equation could be used to close the problem. The equation (4.7) is appealing because it depends strongly upon that which is accomplishing the erosion of the interfac e, t he turbulent eddies having a length scale (Mi) and velocity scale (^ / ). However, its use based only upon laboratory evidence raises some questions about what is 41 really happening in the turbulent kinetic energy budget and why the mean velocity jump |aC| does not appear to be important at a first glance. Resorting to a dimensional analysis in which the relevant parameters are h, u , AB, |aC|, and gives v (W*, E*, Ri*) = (4.8) where and w* = e E* = hAB Ri* hAB AC Here, the physical significance of (h) is its assumed proportionality to the length scale of the turbulent motion responsible for the interface instabil- ities. Re-writing equation (4.8) so as to solve for (u e ), u -vG&f-te , -^-) . (4.8a) e V hAB |AC| 2 / The experimental relationship (4.7) suggests that Ri* may be of negli- gible importance compared with E*. Other investigators besides Long, doing different experiments have found varying relationships between W* and E* (See Kantha, 1975). The question of why W* would tend to be linearly related to E* in many but not all cases may be answered by closely examining the turbulent kinetic energy budget in the entrainment zone. 42 4.4 Turbulent Kinetic Energy Budget at the Density Interface and the Development of a Theoretical Equation for Turbulent Entrainment in the Presence of Mean Shear At (z = - h), at the top of the entrainment zone and within the fully- turbulent mixing region, the turbulent kinetic energy equation is = - — 3U , — 9V uw — + vw — U 9Z 3Z + bw -h -h dz W(P P.. + v - e -h -h (4.9) At (z = - h), the turbulent fluxes are cw -h A r dh AC dt (4.10) and bw -h = " * B oT (4.11) The convergence of flux of turbulent energy at the interface is re- sponsible for the entrainment buoyancy flux, bw(-h). bw" (-h) - I- 3_ 3Z w(£- + I) Pn 2 -h The problem is to estimate the time scale (r e ) required to transport some of the turbulent energy to the vicinity of the entraining interface. 9Z -h The mixed layer depth (h) or a length scale proportional to (h) is the distance over which turbulent energy must be transported by the vertical component of turbulent velocity (w). Therefore, (x e ) is taken to he pro- portional to (h) divided by the rms vertical velocity scale, / , giving the entrainment hypothesis, equation (4.12). This equation is a refinement o f what Tennekes (1973) has suggested. The difference is the use of / rather than 3 / 2 . 43 9_ 3Z w (I- + I) = mi / J-h (4.12) Two possibilities are suggested for how the mean shear at the interface should adjust. The first is according to a gradient-diffusion model having an eddy viscosity scaling with the integral scales of the mixed-layer turbu- lence. The mean shear becomes 3Z - m. cw i-/ (4.13) Then if local dissipation is negligible compared with the flux divergence at the interface, equation (4.9), using (4.10) - (4.13), becomes m 5 l A C| 2 au 2 (— ) ^dt j h / ■ djr 'df m^ y ab m + — u = o (4.14) For those cases where is proportional to , (4.14) can be reduced to simply * p-i * m h * * (W ) 2 - — - (W ) + — E Ri = mg mg (4.15) in non-dimensional form. Solving (4.15) for W* gives * Ri W = — 2m k Iff -m 5 m C * * — E Ri mg or w* = «i * r- 2m i _ /m^* 5 L (4.16) where 4>* - mC ™5 rT* • (4.17) 44 If * < k, the binomial expansion gives /l - 4 cf>* = 1 - 2 4>* - 2 * 2 -4 <}>* 3 - and therefore (4.16) may also be written as W* = mC E* (1 + <(.* + 2 * 2 + . . . .) (4.16a) For small <)>*, (4.16a) rapidly converges and only the first term may be needed. For more general applications, where is not always proportional to , the equivalent solution to (4.14) is dh dt m^ / Rab mi + m 5 |AC| 2 1 + + (hAB) 2 (4.18) A second possible way for the mean shear at the interface to adjust is so as to maintain a gradient Richardson number of critical size, Ri 5 . Then in place of (4.13), the thickness of the interface (6) adjusts so that 3C dz AB 1 -h " TaC Ri (4.19) where Rl - - , 6AB ., (4.20) In this case, equation (4.9) becomes M _L_ \ a r dh + m k l^l - n (1 _ RT~ } AB dt + h^ ° (4.21) 45 or dh dt m^ / FaB (4.21a) If Ri^ is a constant of order one, then (4.20) predicts, independent of the entrainment equation (4.21a), 6 _1 h ~~ Ri (4.22) 4.5 Comparison with Moore and Long (1971) Experiment Figure 10 is a fit equation (4.16) to the results of Moore and Long (1971). The constants (m£) and (1TI5) are determined by the best fit. Notice that in this experiment (E*)" 1 happens to be proportional to Ri*. This is probably due to the experimental method and has no general significance for the ocean mixed layer. Also, W* can be interpreted only as a dimensionless entrainment flux for this experiment because the experiment involves two turbulent mixed layers, each entraining the other equally, resulting in a motionless interface, giving VIquah 10. lk.10n.QXLQ.0Ji o,ntA.ain- mo.nt cujivq. v6 . data ofi Uoon.0. and Long [1971). An -inAtab-LLittj u> pn.QdX.oXQd at E* = 0.036. ThQ btopQ appn.oao.hoj> (-7) qj> . /(39.4E*) 46 w* = -bw(-h) _ -uw(-h ) AB /<£> AU / The lowest order term of (4.18) agrees with the general theory of Kraus and Turner (1967) and the experiment of Kato and Phillips (1969) for those instances where / a, m iu 3 - u+b+h * u *' The higher order terms of (4.18) contribute to the entrainment process by a feedback mechanism that seems to explain the large-scale instability observed by Moore and Long for small Ri*. Equation (4.18) predicts an instability when Ri* < (Ri*) cr where ( Ri *) C r = 4 m ^ 5 HaF ' (4 * 23) For the Moore-Long experiment, (E*/Ri*) cr = 0.105, or < Ri *>cr ■ 9 - 55 m • < 4 - 23a » Although this instability seems to be possible in the laboratory, it is not clear whether it can ever occur in the oceanic mixed layer because the ex- pected turbulent entrainment attributable to the turbulent flux convergence (4.12) alone is sufficient to increase (hAB) at a more rapid rate than Ri* may decrease. The instability observed tends to verify the gradient-diffusion hypo- thesis, equation (4.13), at least for small Ri*. However, the constant -Rig hypothesis, equation (4.19) seems to be a better fit for 9U/3z(-h) over most of the range of E*. Figure 11 is a plot of the observed (6/h) versus the values predicted by equation (4.22). Of course, the side walls in flume experiments of this type prohibit pushing the results too hard, but it is fairly clear that the Moore and Long experiment as well as the others mentioned verify the lowest order term of (4.18a) as well as (4.21a)— no matter how 3C/3z(-h) is modeled. Therefore this shall be taken as the entrainment equation: 47 :? 0.6 :- :: s s s s s ♦ / s s s / s / Figu/in 7 1 . Vn.ddu.ctdd InteA^aae 6lope and Mooie. and Long meat- uAe.me.nt6 vqaaua dJjnem>iontUi> buoyancy ^lux.. 06 0.8 10 1.2 1.4 aBaU xlO- dh ., ^ / (4.24) 4.6 Completed Model for Shallow Mixed Layers Equations (1.24), (1.25), (3.5a,b ) and (4.24) constitute a closed set of five equations in the five mixed-layer variables: h ^- = - AC |jjr A - 1f h - cw(0) . (1.24) h dfB> __ . AB dh A + _M dt dt p„C Qdz - bw(0) . (1.25) o"P -h-6 1 d AC I 2 dh 2mi i dT (h ^ >] - m3u3 + 2~ dT A - m2/ (< ^ «ni 3/2 - 2) j- (3.5a) ( h ) = - ${u*b* + AB ^ A) + m 2 v^ET ( - 2 ) - J- 3/2 • 2 dt dt ,, mi f / dT : hAB (3.5b) (4.24) 48 In addition to Q (z,t), other specified (external) conditions are the surface fluxes u*b* and cw(0). Starting with initial conditions on , , , , and h, the model is solved as an initial-value problem in time. Also to be specified separately are the initial profiles of B and C below the bottom of the mixed layer. These two variables contribute to AC and AB, which are important for a deepening mixed layer: AB = - B (-h -6) ; (4.25) AC = - C (-h -<5) . (4.26) 5. BEHAVIOR OF THE EQUATIONS 5.1 Nondimensional Form of the Turbulent Energy and Entrainment Equations Using the surface flux scales u* and b*, new dimensionless variables are defined: * H = u*b* h 2m 3 u* 3 D* - . hAB dh o 3 dt ' (5.1) (5.2) 2/3 mi ' "r E*.. = (— ) 2£ ; (5.3) mi 2 /3 u* E*33 - (i7 ) ^. (5-4) H* is the ratio of buoyant damping (production) from surface heating (cool- ing) to wind-stress production. P* is the ratio of energy lost by entrain- ment (potential energy increase) to wind-stress production. Invoking the quasi-steady state assumption for the turbulent energy budget, the entrainment and turbulent energy equations become 49 and P * = r- E *ii / ^ ' < 5 - 5 ) = l+^r-p 2 (E*.. -3E 33 ) /E^T-fCE*^.) 3 / 2 , (5.6) = - H * - P* + n 2 (E*.. - 3E* 33 ) /E*77 - I (E^.) 37 * , (5.7) m^ m 2 where Pi = — and p 2 = — . (5. 8), (5. 9) 5.2 Determination of the Constants The ratio m 2 /mi = P2 is equivalent to (18 £j/A) where Ui/A) is the redistribution to dissipation length scale ratio of Mel lor and Herring (1973). From boundary layer data, they suggest — = 0.05 ± 0.01 . A Hence p 2 is of order one and will be taken to be equal to one in this anal- ysis I8l 1 P2 = ~r-^ 1 • (5.10) The ratio m^/mi = Pi may be determined from the asymptotic case of pure convection, H* ■»■ - °° and Ri* = ». The equations yield Pi = (iTf) /, E* 33 /E* 1i t 5 - 11 ) E*. 33 1 ? where -^ — = t (1 + irr~ ) for large Ri*. (5.12) t .. j jp 2 The fraction (r) of turbulent energy converted to potential energy by entrap- ment is found to be 0.036 ± 0.031 by Farmer (1975) from measurements under the ice. Using p 2 = 1 and r = 0.036, equation (5.11) gives (pj = 0.1). 50 P* may now be solved as a function of H* and Ri*. Figure 12 shows P* (H*) for Ri* = « and Ri* = 1. The one remaining constant needed to complete the model is m 3 . This may be determined from the Kato and Phillips results, Figure 7, together with P*(H* = 0) from Figure 12. P *(0) = hAB j}JL (b* = 0) = 0.0227 2m^u dt (5.13) 3 U * hAB jj£ (b* ■ 0) = 2.5 u* 3 . (5.14) This gives m 3 = 55. However, this value seems to be quite high for an oceanic mixed layer. The conceptual model for the mixed layer developed earlier, equation (3.3), implies that m 3 = |5C |/u, F-iguAe 12. Ikod^Z notation ^on. AkaJtlow mixzd Jtcuf2SU>: pot^ntlaZ moAQtj note, ofa incA.e. . A more moderate value of m 3 ^ 10 is expected. In the laboratory experiment, the production was enhanced by the side-wall boundary layers, resulting in a too-large value for m 3 lother method of determining (m 3 ) which is suggested by Fig- than using neutral (H* = 0) stability situations (storm )nly approximately neutral by virtue of a large u* 3 compared There is another ure 12. Rather Situations are OFi i j uppiuAimai-ciji ncunai uy vii luc ui a laiyc u^ L-UMipc with u*b*h ), cases of limiting strong stability where P* = and h = h, could be examined H* = 0.4 = y*^*!i , so max o a 2m 3 u* 3 m 3 = u * b * h . (5.15) 0.8 u 3 The effective buoyancy flux u^b*, given by equation (4.2) should properly be determined by measuring Q(Z), and the mixed layer depth h = h r should be ascertained to be in the "retreat mode": dh/dt 5 0. This method remains to be attempted. 5.3 Comparison With Earlier Models The dotted straight line in Figure 12 is equivalent to the solution for P* (H*) if equation (4.1a) is calibrated to behave properly for both the re- treating (H* = H* max ) and neutral (H* = 0) situations. The greatest differ- ence between the two models then occurs as H* becomes increasingly negative. This inability to calibrate the earlier model to both stable summer and convective fall situations has been demonstrated by Thompson (1974). Nearly equivalent to the Kraus-Turner type model for entrainment, equation (4.1a), is Tennekes's model which suggests in effect that Pi q/o P* = r (E*..) 3/2 (5.16) 52 rather than (5.5). In such a model, the total turbulent kinetic energy equa- tion, the sum of (5.6) and (5.7) is used: - 1 - H* - (1 - j^) P* - (E^-*) 372 • (5.17) The redistribution term does not appear at all in (5.17) but this very term is the source of an ambiguity as H* + H* ma x = 1. As H* gets larger, E*-ji and hence P* go to zero for the set of equations (5.16) and (5.17). If rather than using (5.17), we consider the component equations together with (5.16), no unique solution exists for H* when P* = 0. Considering the budget for the vertical component only reveals that (H* max ) must be less than one and that E*-jj must be greater than zero when P* = 0. This, however, violates (5.16). The alternative equation (5.5) does not have this problem and it suggests a more consistent mechanism for layer retreat. As H* approaches its maximum value, more and more of the vertical com- ponent of turbulence is damped. Notwithstanding the redistribution effect, eventually redistribution cannot keep up with the combined losses due to buoyant damping and viscous dissipation. Virtually all of the energetic large-scale turbulent motion is two-dimensional, and the vertical flux of turbulent energy (-wT) into the interface ceases and therefore entrainment stops. If (b*) is further increased (or u* decreased), (h) must retreat since H* cannot exceed H* max % 0.4. Locally, at the base of the mixed layer, this would correspond to the flux Richardson number (Rf) surpassing a criti- cal magnitude (- 0.2). Summing up, a new mixed-layer bulk model has been proposed that suggests that the energy redistribution between vertical and horizontal components of turbulence must be accounted for in a general model used to treat the entire range of stability—from pure convection to layer retreat. The proportion of vertical turbulent energy to total turbulent kinetic energy, E* 33 /E*ii of Figure 13. varies most significantly in the typical oceanic range of possi- bilities, H* = 0.4 to * - 1. Layer retreat is predicted to occur for a smaller value of stability than is required with the earlier models because viscous dissipation remains important as long as u* > 0. The model, as it stands, is incomplete. The above applies only to the relatively shallow boundary layer associated with summer conditions. An additional feature is required for year-round application of the model because of the importance of the rotational time scale in the physics of deep mixed layers. 53 ^33 Efl Figu/Ld 13. Ratio ofi veAtlcal tuA.bule.nt kinetic enexgy to total tuA.buJLe.nt kinetic ene/igy cu> a function. 6. DEEP MIXED LAYERS: LIMIT TO MAXIMUM DEPTH 6.1 Limiting Dissipation Time Scale The need for a constraint upon the maximum depth of the mixed layer has been demonstrated. Niiler has pointed out that models of the Kraus-Turner type will continue to deepen incessantly during each winter cooling period. This is the unavoidable consequence of requiring some fraction of the net production of turbulent energy to be allotted to increasing the potential energy by means of entrainment. Thompson (1974) also found this to be true. In doing a year-round integration, the model required an unrealistic value for mean annual surface heat flux yielding a net heat storage in order to achieve a realistic cycle. Outside of the tropical latitudes and away from the continental shelves, lateral advection and upwelling are insufficient to balance this heat stor- age. The most noticeable failure of the model is at mid-latitude regions having a permanent pycnocline. The model for dissipation is still e ^ However, the dissipation time scale (t e ) was previously (in equation 3.1) governed by the depth of the mixed layer and the turbulent velocity scale, giving a strictly convective time scale: Ul> / 54 As planetary rotation begins to rotate the mean shear profile, and hence the geometrical aspects of the integral scale, there arises a new length scale, associated with the coriolis time scale (f _1 ). l 2 = / f . It is now increasingly clear from such studies as those by Arya and Wyngaard (1975) and Sundararajan (1975) that the coriolis time scale (f -1 ) plays an important role in the internal structure of the convective planetary boundary layer or mixed layer. The concern here is more with the bulk pro- perties of the region and less with the details within the mixed layer. However, it is suggested that this time scale has an important role in the overall turbulent energy budget above and beyond the turning and decay with depth of the mean velocity in the classic Ekman spiral. In reality the two effects are inseparable because of the link through local shear production. Rather than to simply replace the convective scale (&i) by the rota- tional scale {12)9 the suggestion is to use £ -1 = i^ 1 + i^ , giving equation (6.1), T e »-^ + f. (6.1) Replacing (3.3) in the bulk (vertically integrated) formulation is equation (6.2) i {E* u )* /2 + Ro" 1 (E*.-) (6.2) 2m 3 u* 3 where Ro-* - (^) 1/3 * (6.3) m 3 u* and, as before, E *.. = (—) 2 / 3 11 x m 3 „2 55 6.2 Nondimensional Solution to the Entrainment Function Equations (5.6) and (5.7) will now change to reflect the new term, giv- ing (6.4) and (6.5). = 1 + £f* - P 2 (E* t1 - 3 E* 33 ) /E*77 - | E*.. (/E*77 + Ro" 1 ). (6.4) = _ H * - P* + p 2 (E*.. - 3 E* 33 ) /E*77 - i- E*.. (/DTT + R -i). (6.5) The entrainment equation, (5.5), remains unchanged. Pi P* = -r E *ii / ^ * (5 - 5 > The solution to P is now a function of two variables, and is shown in P* = P* (H*, Ro" 1 ) (Figure 14) Specifically, several aspects concerning this nondimensional solution should be observed: a. The solution for Ro -1 = is identical with the earlier solution (Figure 12). This limit applies to both the case of shallow (h«u*/f) mixed layer and the case of \fery low latitude, <\> « u*/|ft|h. b. For a retreating (P* = 0) mixed layer, h r is now a function of both the Rossby number and the stability H*. Actually, h r is uniquely determined by the product of these two parameters, making it a function of * b B = -2*. fu, A fairly good analytical approximation to this function is the hyperbolic relationship (6.6): 56 Entrapment Buoyancy Flux -bw(-h)h _ p * 3 Layer Stability -bw(Q)h u * (Ro)-'=^ TIqojkl 14. GmvwJL solution to zwtAalwmnvvt and tuAbulnvvt kin&tLc zneAgy . (0.5 + H ) (1.725 + Ro" 1 ) = 1.552 (6.6) which can be put in terms of h r , giving equation (6.7) h = 1-725 d- + 0.5L n r - 2 1 + 2.76 1.725 d' + 5L (6.7) 57 where d ' is proportional to the Ekman depth of frictional resistance and L' is proportional to the Obukhov length scale. The approximate solution for retreat, equation (6.7), has limiting cases of d' ■* « and L' -* « that are in agreement with the earlier works of Kitaigorodsky (1960) and Rossby and Montgomery (1935). L* = h p /H* . (6.8) d* = h r /Ro _1 . (6.9) Physically, the reason for the cessation of entrainment is ultimately the same for the convective boundary layer whether or not there is signifi- cant surface buoyancy flux. Dissipation that is approximately equally divid- ed among the three components of the turbulent kinetic energy (IF + v 2 + w 2 ") causes to be consumed more rapidly than it can be replaced by the pres- sure redistribution or "return to isotropy" effect. When this happens, the vertical flux of all properties, including the turbulence itself, ceases at the depth h = h r . Buoyant damping or production modifies the overall budget, changing the relative value of h r f/u*, but dissipation plays the ultimate role. Buoyant production may be sufficiently strong (H* < -0.5) that dis- s i pa t ion cannot limit maximum mixed layer depth. Then the production of is too great to be overcome by dissipation with increasing depth. c. A cyclical steady state is now possible. The surface buoyancy flux may go through an annual cycle having no net heat storage and (h) will not increase without bounds at the end of the cooling phase. The simplest steady state is the neutral stability point represented by the coordinates (Ro" 1 , H*, P*) = (1.38,0. ,0. ) . This is the situation for a neutral boundary layer, (bw(0) = 0), where h r = 1.38 d'. A simple cycle having a sinusoidal variation in bw(0) and a constant u* is qualitatively depicted by Figure 15. This depicts the locus_of coordin- ates for (Ro -1 , H*) a_s_ the cycle progresses through a heating (bw(0) < 0) and then a cooling (bw(0) > 0) phase. The initial [0] and [4] points coin- cide at the neutral steady-state point. Between [0] and [1] buoyancy flux is directed downwards and is increasing, and the mixed layer retreats to its minimum depth which occurs at [1]. Between [1] and [2] there is active entrainment but the rate of entrainment is slow because of significant buoy- ant damping in the stable mixed layer. This would correspond to the summer warming period, culminating in maximum surface temperatures at about [2]. Cooling begins at [2] and reaches a maximum at [3]. Not until the end of the period does the rate of entrainment become significant because the heating period [0]- [2] causes a potential energy deficit in the system which must be replaced before deep-ending may proceed rapidly. In the time period just 58 FiguAe. 7 5. Cyclical. Atzady AtaXe. In tkz (H*,Ro l ) plant. 59 after [3] erosion of the thermocline is enhanced by both increased system potential energy and by buoyant production which remains strong even though (bw) is decreasing. Of course, the real oceanic system is not in an exactly repeating annual cycle. However, year-to-year variations resulting in long-term changes in heat storage can be predicted accurately only by a model that can simulate the hypothetical cyclical steady-state. Any net annual buoyancy flux across the surface that is balanced by advection is not of concern in studying the model in the one-dimensional form. In any case, a large seasonal variation in bw(0)--and u* 2 --and the resultant cyclical response are dependent upon the proper model response for the more simple situation. 6.3 Simple Hypothetical Cases Demonstrating the Behavior of the Solution 6.3.1 Shallow or f = (Ro » 1). (1) Initial linear stratification (a) bw = 0, u 2 . = constant The result is the same as that predicted by the Kraus-Turner type model as well as by the Kato and Phillips experiment. See Figures 16 and 17. dji u^ dt " hAB giving t 1/3 The reason these models are in agreement is that with b* = 0, u* is the only turbulent velocity scale. <* a u£ . (b) Free convection: bw(0) = constant > 0, u| = 0. This situation is out of range for Figure 13 because H* -* - °° . How- ever, if the model equations are nondimensional ized on the buoyancy flux velocity scale (u^b^h) 1 / 3 , rather than on (u*), the problem may be solved. In this case, h = t*/ 2 , which is also in agreement with the Kraus-Turner type models. GO -4.00 to 2 -8.00 -12.00 -16.00 20.00 19.90 O CP a> Q 19.80 19.70 19.60 ro 'g 0.800 X 0.400 'o h o -0.400 O en -0.800 Q CM I O no iu/ifiacc heat faux and a constant uxind i>txa>i> Lt> imposed at time Z2A0. :: Deg C (c) Cyclical surface buoyancy flux: constant. bw(0) « sin wt, u* = Figures 18 and 19 show the response of the mixed layer to a simple hypothetical diurnal heating and cooling cycle. The magnitude of the heat flux was chosen to be particularly large in order to accentuate the features in a relatively short time span. The initial stratification is the same as for the earlier neutral (bw(0) = 0)run, l°C/20m. The wind stress is also the same, 0.2 dynes. Over the five-day period there is an overall cooling trend because of the entrainment process and lack of net surface heat flux. However, the daily maximum surface temperature is greater than the initial surface tem- perature for all five days. Starting with the fourth day, there is larger retreat corresponding to the decrease in the Obukhov length scale relative to the value of h. In spite of midday retreat, the mixed layer more than recovers the temporarily decreased vertical extent. Without eventual enhancement of dissipation, deepening would continue without bound. A comparison between Figures 18 and 16 demonstrates another important difference, in addition to the presence of a strong diurnal layer temperature change. Although there is no net surface buoyancy flux for either case, the situation having the diurnal cycle has a distinctly slower average rate of deepening. This holds even for the first three days when there is no re- treat. The explanation for this phenomenon lies in the nonlinearity of P* (H*) (Figure 12) . 62 in a> a; O -4.00 -8.00 -12.00 20.80 20.40 Q 20,00 19.60 0.15 (/) CO 0.10 E u 0.05 -r T" -r ~r "r t - - 2 i i i 1 1 ! 20 40 60 80 Hours 100 120 140 FIquac 18. Solution to hit] {on coac with cyclical iunfiacc heat {lux-, ew[0) = 0.015 bin (wt). CompaAc with VIqvjkl 16. (d) General: u* 2 f 0, bw(0) f Since the earlier model projects a linear P*(H*), only two points (at two values of H*) may be made to agree with this model. For example, if summer "retreat" and the neutral (bw(0) = 0) cases are coincident in both models, then the winter deepening rate will be greater for the earlier model. This is the mathematical difference in the basic responses. 63 VlquJin 7 9. TrnpoAcutuAQ. pn.o{ltz {on. cai>tn.cuti.{i- coutlon, with cyctlcai 4uA{ace. hzcut {lux. CompaAe. mJjfi FiguAz 17. Deg C The physical difference may best be seen from the Tennekes (1973) model in which -bw ] -h 3/2 and "r 3/2 a , , <* m u£ - u^.b^h . This model is identical in response to the Kraus-Turner model and is sui tabl e for comparison here because it identifies the turbulent velocity scale /. Then retreat (dh/dt < 0) can occur only when ■> 0! Tennekes did not intend for this formula to be extrapolated to the strongly stable situation. Instead, he suggested it as an interpolation between the neutral and free convection cases—the typical atmospheric boundary layer range. Nevertheless, the above is equivalent to recent oceanic applications of the Kraus-Turner type model. The particular problem with applying this method to the stable, retreating mixed layer is that there must still be significant turbulent energy ( > 0) above z = - h r . Conse- quently, the fraction of turbulent energy going to increase the potential energy cannot be independent of stability, H*. This explains the curvature of P*(H*) and the slower rate of damping for the hypothetical case with an oscillating surface buoyancy flux. 64 (2) Cyclical surface buoyancy flux No 1-D steady state is achievable because for f = 0, h can be limited only by the Obukhov length scale, therefore requiring a net downward buoyancy flux (or heating). Then bw(0) could be balanced only by advection. Gill and Niiler (1973) have shown that advection is insufficient for lati- tudes greater than about 15°. 6.3.2 Convective planetary boundary layer with order one Rossby number (1) Neutral steady state With zero surface heat flux, a steady state is predicted (see Fig. 14 at the limit h -> 1.38 u*/f ) . Starting from an initial linear stratification and h < u*/f, steady state would be approached only after a relatively long time. This is not likely to be achieved in geophysical flows, except perhaps by the atmospheric boundary layer during the polar winter. See Businger and Arya (1974). (2) Cyclical steady state This situation has already been qualitatively described and illustrated by Figure 15, but the relative importance of the rotational and surface buoyancy flux scales needs to be estimated. For the sake of studying the relative response of the mixed layer, all cycles will be assumed to be repetitive (in steady state). A sinusoidal surface buoyancy flux and a constant wind stress are used to drive the model . T = p Q U* 2 . bw(0) = - | bw(0) | sin ait . The layer response is found to be a function of the parameter B*, the ratio of the buoyancy flux scale to the rotational scale. B* = 1^(011 . {6A0) f u* 2 Figure 20 shows the cycle of h(t), nondimensionalized on u*/f as a function of B*. For large B*, which is typical of the annual cycle for temperate oceanic regions, the low minimum mixed layer depth at cot = tt/2 is attributable to the strong influence of the buoyant damping corresponding to a small positive Obukhov length scale. 65 E — flgu/tc 20. Cyclical 6tcady-6tatc mixed- lay qa n.ci>pon6c to 6-inu- i>oidal AuA&acc buoyancy &lux and constant wi,nd 6&L2A6. t=p u* 2 ; HuJ[0)=-\'EuJ[0)\&4.n[tat) . In addition to the change in the range of h with B*, there is an important change in the shape of the function h(t). For the case of weak heating and cooling, the variation in h is almost in phase with bw(0). In the H* - Ro" 1 plane (Fig. 21), the locus of the cycle would be small and elongated. For increasingly larger B*, the heat and therefore buoyancy is stored during the "summer" at an increasingly shallower depth (in comparison with h max ^ u*/f). This creates a substantial potential energy deficit which ViquAQ, 21. Cyclical i>tcady-i>tat, could be neglected in the solutions. However, fluctuations in the surface boundary conditions u* 2 and bw(0) of sufficiently short period require the consideration of these terms. A wery simple model, which may be solved analytically, provides the answers to the question concerning the importance of this unsteadiness and how it might be treated. In the simple model only the two largest terms, shear production and dissipation, are balanced by the time rate of change of the turbulent kinetic energy. # + ^=^(l + asin.t) (6.11) The prescribed shear production is a constant plus a sinusoidal ly-varying component (a < 1). By defining dimensionless time, .* - = t/- (6.12) VIqixaq. 24. SuAfcace. trnpeAatuAn T(2=0,t) faon, the. hypothetical coie Mtth htonmi,'- CojiH A, modeJL without dluAndl cycZe.. Phase of Annual Cycle 68 and turbulent energy, u/* = h (6.13) (6.11) is transformed to (6.14), W 1 3x^ + y* = 1 + a sin 0J*T*, (6.14) where to* = T 10. (6.15) For (a = 0) the steady-state solution is simply v* = 1. For finite (a), the complete solution, neglecting the initial transient, is y* = 1 +i j* 2 + 1 (sin w*t* - a)* cos w*t*) (6.16) There is a phase shift that increases with (u>*), but the important aspect of (6.16) is the relative magnitude of the response associated with the fluctu- ating component. Figure 25 is a plot of this versus to*. For to* < * 0.5 (slow fluctuation period compared with x e ), the situation of quasi-steady state is satisfied. Depending upon the Rossby number and H*, this corresponds to a minimum fluctuation period of several hours up to about a day. FiguAo. 25. ReAponie. o& the. mixzd layeA to fcluctuationA in the. AuAfaace. boundary condition*: lilteAlng d^zct. Summer : shallow, fast -responding mixed layer .__J I I L 1 month 1 day 1 hour 1 min Winter : deep, slow-responding mixed layer ____J I I J 1 month 1 day 1 hour 1 min 69 For faster fluctuation having a period less than about an hour, the phase shift (lag) in the response approaches 90° and, most importantly, the amplitude of the response is negligible. In other words, the high frequen- cies are filtered out. This becomes a possible cause of error in using observed winds to drive a model having an integration time step smaller or comparable with r e . If the quasi -steady state assumption is made to facilitate solution, but the surface boundary conditions are not properly smoothed or filtered, an incor- rect high-frequency response will not only be present but will bias the mean trend. 6.5 Interaction Between Forcing Time Scales: Modulation of the Longer-Period Trend in the Diurnal -Period Heating/Cooling Cycle As already shown in comparing Figure 18 with Figure 16, if H* is per- turbed by a diurnal-period heating cycle, then the mean P* over the cycle will be less than P* of the mean H*. P* (H*) < P* 'WJ (6.17) An atypical situation was used in Figure 18 to demonstrate the effect of (6.17) during a relatively short time period. Over a year, however, the consequences can be significant. This is shown by Figures 26-28— case "B." Case "B" is the same as case "A," Figures 22-24, except for the superposition of a diurnal cycle on top of the annual heating/cooling cycle. Phase of Annual Cycle 5 7 13 15 17 19 21 23 25 Deg C Tiqusie. 26. Mixed layeA depth, hit) ^oh. hypothetical annual cycle., Ccu>£ 8: model with dluAnal heat ^tux cycle, icompa/ie. u)lth &Lg. 22] YIquac 27. Mid&ummeA tempeAatuAe. pn.0' \UUl, cahe. 8: Model. wJbth diu/inal cycle.. [Compasie. with {)lg. 23) , 70 Z3 23 1 1 1 1 1 1 1 1 21 ^ 19 / \ 17 \ o S 1 15 — / \ 13 \ II \ 9 / V 7 c 1 1 1 1 Vigun.2 28. SuA^ace. timpoxatuJid, ecu 2 B: Model, with duxAnal cycZn [com- paAd \KsiXk {jig. 24) . l-rr Phase of Annual Cycle Throughout the year, the principal effect is the decrease in mixed layer depth. The decrease in upper ocean potential energy, higher surface temper- atures, and lower heat storage at depth are directly due to the modulation of the longer-term entrainment rate. The basic shapes of the two solutions to h(t) are similar. Without a noticeable difference in phase, there is the possibility that the effect may be absorbed in the constants of calibration. In other words, a one-day or longer time step might be possible if computation speed were of the essence in a numerical model. 6.6 Simulation of a Real Case An experiment conducted by Hal pern (1974) provides a unique record of 1-D mixed layer evolution in that wind, currents, and thermistor chain meas- urements were made simultaneously at one location (47°N, !28°W) for the period of a month. During this time one "strong" and several lesser storm events occurred. Figures 29-33 show the model results. Aug 4 Sep 8 12 16 20 24 28 I 1 -16 -24 -32 1 1 i — J 1 - - J J ' / f / ', V , - 5 Cent Deg_ TiquAd 29. Sequence. o& t2mp2K.atuJ\.2 pfio^iZoJb pK.2di.2t2d by thu modal. [SuLactetsive. pno^-HeA o^oA by 1 .25 do.gti2.2J> ) . 71 if 15 T~T th r Thermistor Record Model Prediction 5 Aug 10 15 20 25 30 1971 Sep FlguAe. 30. Composition beArtzzn obi>eAve.d and predicted tmpeAj&tuAeA out 9.6-meXoA depth. Here the 31-day records of temperature at 9.6 and 23.4 meters are com- pared with the model prediction. The actual wind speeds are used to compute u* for the model, but surface buoyancy flux is rather crudely assumed to be constant plus a constant-amplitude diurnal component. Nevertheless, the gross features of the record are reproduced: (1) A weak storm on August 11 deepens the layer from 13 to 18 meters. (2) A period of low winds from August 13 to August 19 resulted in a retreating layer and warming at the 9.6-meter depth. During this period the guessed-at surface heat flux gave a too-shallow depth of retreat that can be seen in the August 16 temperature profile. Therefore the predicted temperature at 9.6 meters is too low until the layer deepens suffi- ciently on August 17. 72 10 Thermistor Record Model Prediction 5 Aug 10 15 20 25 30 1971 Sep Vi.Qan.2. 37. Compa/uAon boJM2.2.n obAQAvzd and pfiedicLted tompoJuaAuJiQA cut 23.4-m&£2A duptk. (3) The relatively strong storm of August 20 deepens the layer to over 24 meters and the temperature at 9.6 meters drops considerably. At this time the 23.4-meter thermistor for the first time lies within the mixed layer and registers the higher temperature. (4) Subsequent warming and then cooling after August 30 is due to another storm. This last storm is nearly as strong as that of August 20, but its work is not as obvious because it starts with a relatively deep layer. Figure 32 shows another possible model output--the mixed layer wind- driven current. As shown by Figure 12 5 the mean current at the base of the mixed layer has only a small effect upon the turbulent energy budget (even for Ri* ^ 1). The depth of the mixed layer, on the other hand, directly influences the amplitude of C. Therefore, any surface current prediction requires knowledge of h(t). 73 12 13 14 August 1971 22 F-LguAe 32. A mea&uAe ol ient-ible heat ^Zjulx at the ocean AuA&ace by buoy obi>eAvatto\xb (couAteAy ofi Vavld HalpeAn) . Qu)[0) « waT; W = uoind 6peed at 2 mete/u above 6u/i&ace.; AT = AuAfcaae wateA tempexatuxe mlnm> aix tempexatuJte. 6.7 Evaluation of Model Output This simulation of a real case is not a strict test of the model. The surface buoyancy flux is not known sufficiently well for this run to be used either for calibration purposes or as a test of model behavior. Inasmuch as the model was designed to have general applicability over all seasons (wide range of values for H*), the proper calibration and testing procedure requires an integration over a full year. This remains to be done at a future date under a more operationally-oriented program of research. Figure 32 shows a measure of the sensible heat flux at the surface (by a bulk parameterization) versus time for the first half of the total period of observation. (Air temperature was not available after the storm of 21 August.) Because relative humidity was not available, the relatively larger latent heat flux could not be computed. Because the latent heat flux is very roughly proportional to the sensible heat flux (assuming an approximately constant Bowen ratio), Figure 32 indicates the large magnitude of the net 74 Model Prediction VIquat>iv2,K.vatiom>. -8 -6 -4 ■2 2 Kilometers 6 8 diurnal heat flux compared with the mean daily values. The mean daily heat flux and the diurnal component were assumed to be constant over the entire period. This was obviously not true. Nevertheless does have didacti sponse to realist particular case i ment and retreat tant in determini guess at bw(t) wa the diurnal cycle integration using , this particular run, as illustrated by Figures 29 to 33, c value in demonstrating the character of the model re- ic boundary conditions. The overall conclusion for this s that the wind stress controls the timing of the entrain- events. The degree of surface heating or cooling is impor- ng the extent of the retreat or the deepening. Although the s only marginally satisfactory, an important result is that prescribed gives a much better prediction than does an a constant heat flux alone. 7. CONCLUSIONS This model for the ocean mixed layer consists of a number of hypotheses and assumptions regarding the mechanical energy budget, originally shown in Figure 1. In arriving at these hypotheses, mean turbulent field techniques were used, but this was done without losing the simplicity of the bulk model concept. The particular details of the structure of the mean fields within the boundary layer were not of highest priority as a model output. In any case detailed geophysical data revealing this structure is unavailable to test such a prediction. Of highest importance in this model was the ability to predict the year-round evolution of the ocean mixed layer depth, together with the bulk properties. There are several models, including those by Wyngaard et al. (1974) and Sundararajan (1975), which provide structural predictions quite nicely, but these boundary layer models are not directly 75 concerned with a changing layer height. An entrainment model is necessary to provide boundary conditions at the moving density interface. The atmospheric and oceanic boundary layers present nearly identical problems. However, an important difference arises in the surface heat flux boundary condition. The atmospheric boundary layer is predominantly unstable because of minimal absorption of solar radiation within the layer. On the other hand, most of the solar radiation is absorbed near the ocean surface and hence turbulent heat flux downward in the oceanic boundary layer is about the same as flux upward in the course of an annual cycle. Therefore, the stable situation is of relatively greater importance in the ocean. As shown by Figures 12 and 14, the nonlinearity of P* (H*) is greatest for H* > 0, the stable region. The treatment of the oceanic case must be made with emphasis upon this point since the linear extrapolation of the unstable situation will not suffice. In addition to the nonlinear dependence upon stability, a Rossby number dependence for the entrainment rate has been found in the nondimensional solution, Figure 14. This makes possible a cyclical steady state for the boundary layer without resorting to unrealistic values of upwelling or lateral advection in a long-term integration. In the short term, on the order of days, the upper ocean, even at higher latitudes, exhibits significant three-dimensional baroclinic activity. A case in point is well documented by Gregg (1975) in which only lateral advec- tion can explain the change in heat content after the passage of a storm. In doing the ensemble or horizontal averaging, such phenomena as well as the short-period internal waves are treated as noise in the essentially one- dimensional mixed layer dominated by the strong vertical turbulent fluxes of heat, salt, and momentum. This does not eliminate the practical difficulties of analyzing and simulating actual observations made at a single point, as shown by the case using real winds to drive the model. The importance of shorter-period fluctuations in the surface buoyancy flux in modulating the longer-term response has demonstrated the need to know the typical daily heating/cooling cycle: Solar radiation, evaporation, con- duction, and back-radiation as a function of season and geographical coordin- ates. The detailed features of this diurnal cycle were not of importance in conducting a qualitative evaluation, but accurate quantitative results from an operational model would be impossible without them. For the same reason, the radiation absorption function should be known. This last suggestion may not be possible without the incorporation of a primary productivity model in which case the physics becomes directly dependent upon the biology! Some of the most fruitful applications of this model lie in the link-up with other models. The region below the mixed layer has been over a short time period successfully treated as a zero-flux "quiescent abyss." This was done primarily for simplicity when the point of focus was the overlying turbulent boundary layer. The first natural joining of models is between such a boundary layer model with an ocean general circulation model. But why stop there? An atmospheric general circulation model may be combined with 76 the ocean general circulation model --but only through two boundary layer or mixed layer models. It has been demonstrated here that the sea-surface tem- perature, which is so important for weather prediction, cannot be predicted with accuracy without the proper consideration of the ocean mixed layer. Climate modeling must also pay heed to the oceanic boundary layer. Anomalous events in the evolution of the mixed layer may induce a delayed response by the atmosphere when the mixed layer again deepens sufficiently. The interaction between diurnal and annual time scale events has been particularly emphasized, but other time scales even longer than a year pose some interesting possibilities, particularly for inactive models. 8. SUMMARY OF MODELED EQUATIONS ENTRAPMENT BUOYANCY FLUX 2 m^ ■bw (-h) = £ (1) BUDGET FOR HORIZ. COMPONENTS OF TURB. KINETIC ENERGY Id /. -y—, — ?■ x o h bw (-h) !r> s 2jt (h ) = m 3 u* 3 2 R ]» (2) 2m! -m 2 ( - 3 ) " 2 - - T - (" 2 + fh) BUDGET FOR VERTICAL COMPONENT OF TURB. KINETIC ENERGY ^d_ (h ) = llM^hl + h_M0i (3) + m 2 ( - 3 ) 2 - ^- ( 2 + fh) - Bw (-h) - Bw (0) + -SS- / Q dz (4) dt p o c p J . h 77 h ^^ = cw (-h) - cw (0) - if h (5) 'JUMP CONDITIONS" AT BOTTOM OF MIXED LAYER - CW (-h) = AC jj: - bw (-h) = AB ^ NOTATION L/2 L/2 C = U + i V (") = I™ L. j j ( ) dxdy -L/2 -L/2 E = u 2 + v 2 + w 2 u* 2 = |cw (0)| <( )> = £ f ( ) dz -h B = (p - J) g/p B = P _1 3P/3T hAB Ri* = AC 9. ACKNOWLEDGMENTS This dissertation and the research leading to it were accomplished with the assistance and encouragement of many. Dr. David Halpern, as Chairman of the Supervisory Committee, provided support and advice in a manner that permitted the evolution of this work in a most creative atmosphere. Professors Joost Businger and Frank Badgley, in addition to participat- ing on the Reading Committee, helped lay the foundation of my understanding of turbulent boundary layers. As a visiting professor, Dr. John Wyngaard introduced me to the techniques of mean-turbulent-field modeling. 78 Dr. Eric Kraus, by inviting me to participate in the N.A.T.O. Advanced Study Institute at LJrbino in September 1975, gave me the benefit of his own insight as well as that of other primary workers in the field at a most opportune time in the development of this work. Drs. Pearn Niiler and Raymond Pollard were among this group. Mr. Francis Way must be mentioned because it was he who initially steered my interests toward the ocean. My parents, Roland William and Geraldine Patrick Garwood, have lent their support for the longest time. It has been greatly appreciated. Last, but most of all, thank you to my wife Marilyn. With her support this task was made easier, and the entire period was made to be a most en- joyable one. 10. REFERENCES Arya, S. P. S., and J. C. 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