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1866 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30

Resonant Wind-Driven Mixing in the Ocean Boundary Layer

ERIC D. SKYLLINGSTAD AND W. D. S MYTH College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

G. B. CRAWFORD Department of Oceanography, Humboldt State University, Arcata, California

(Manuscript received 26 April 1999, in ®nal form 23 August 1999)

ABSTRACT The role of resonant wind forcing in the ocean boundary layer was examined using an ocean large-eddy simulation (LES) model. The model simulates turbulent ¯ow in a box, measuring ϳ100±300 m on a side, whose top coincides with the ocean surface. Horizontal boundary conditions are periodic, and time-dependent wind forcing is applied at the surface. Two wind forcing scenarios were studied: one with resonant winds, that is, winds that rotated at exactly the inertial frequency (at 45ЊN), and a second with off-resonance winds from a constant direction. The evolution of momentum and temperature for both cases showed that resonant wind forcing produces much stronger surface currents and vertical mixing in comparison to the off-resonance case. Surface wave effects were also examined and found to be of secondary importance relative to the wind forcing. The main goal was to quantify the main processes via which kinetic energy input by the wind is converted to potential energy in the form of changes in the upper-ocean temperature pro®le. In the resonant case, the initial pathway of wind energy was through the acceleration of an inertially rotating current. About half of the energy input into the inertial current was dissipated as the result of a turbulent energy cascade. Changes in the potential energy of the water column were ϳ7% of the total input wind energy. The off-resonance case developed a much weaker inertial current system, and consequently less mixing because the wind acted to remove energy after ϳ¼ inertial cycle. Local changes in the potential energy were much larger than the integrated values, signifying the vertical redistribution of water heated during the summer season. Visualization of the LES results revealed coherent eddy structures with scales from 30±150 m. The largest- scale eddies dominated the vertical transport of heat and momentum and caused enhanced entrainment at the boundary layer base. Near the surface, the dominant eddies were driven by the Stokes vortex force and had the form of Langmuir cells. Near the base of the mixed layer, turbulent motions were driven primarily by the interaction of the inertial shear with turbulent Reynolds stresses. Bulk Richardson number and eddy diffusivity pro®les from the model were consistent with one-dimensional model output using the K-pro®le parameterization.

1. Introduction by wind and surface wave forcing, with entrainment ¯ux Atmospheric forcing of the upper-ocean boundary at the mixed layer base dominating the OBL heat budget layer (OBL) plays a fundamental role in regulating the and surface heat ¯ux having a secondary role. As a sea surface temperature of the world's oceans. The most result, the OBL behaves more like a strati®ed boundary direct in¯uence of the atmosphere is through surface layer and cannot be easily described through classic ¯uxes of latent and sensible heat. During conditions of boundary layer theory (Mahrt 1999). strong surface heat loss and weak wind forcing, the OBL The largest changes in the OBL momentum budget behaves much like a well-mixed, convective boundary are caused by the passage of strong synoptic scale layer with turbulent ¯uxes that are in agreement with storms. Because of these storms, the direction of the Monin±Obukhov similarity theory near the surface and wind stress can vary rapidly and force currents that have a mixed layer structure that scales with the surface buoy- a strong inertial oscillation. If the surface winds vary ancy ¯ux (Shay and Gregg 1986; Lombardo and Gregg with a similar frequency and direction, even for only a 1989). Often, however, upper ocean mixing is driven few hours, then these currents will amplify through resonance, leading to signi®cant vertical shear in the OBL. A contrasting situation, when winds are off-resonant, can lead to a reduction in OBL currents and a decrease Corresponding author address: Dr. Eric Skyllingstad, College of in vertical mixing relative to wind magnitude. Episodes Oceanic and Atmospheric Sciences, Oregon State University, Cor- vallis, OR 97331. of enhanced vertical mixing in the thermocline have E-mail: [email protected] been linked to increased vertical shear (e.g., Large et

᭧ 2000 American Meteorological Society

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might not be the case, however, if most of the surface wave effects generate small-scale turbulence that is quickly dissipated in the upper OBL. Thus, determining the importance of the various energy pathways shown in Fig. 1 requires a thorough understanding of the turbulent processes produced directly by surface waves and by mean current shear. Ideally, we would like measurements of turbulent ¯uxes in the OBL to help explain the various energy pathways shown in Fig. 1. However, observational technology is only now reaching a point of development where we can directly observe turbulent eddies in the ocean and produce direct estimates of the turbulent ¯uxes (e.g., Gargett and Moum 1995; Moum 1996; D'Asaro and Dairiki 1997). The majority of turbulence observations in the midlatitude ocean are lim- FIG. 1. Schematic showing main pathways for energy transferred to the upper ocean via the surface wind stress. ited to microstructure data collected during weak to moderate storm conditions. These data can provide estimates of the turbulence strength in a given region, but al. 1986) and can have a dominant effect on the ocean they cannot yield speci®c details on how turbulence was SST in the fall season as cold thermocline water is mixed forced or what instability process created the turbulence. with the relatively warm surface waters created by sum- Our approach for studying turbulence in the upper mer heating. Heat ¯ux estimates based on observed ocean is the application of large-eddy simulation (LES) changes in the OBL temperature have shown values as models. LES models have been used extensively in the high as 7000 W mϪ2 during an 8-h period (Krauss 1981), atmosphere to examine the structure and dynamics of with 1000 W mϪ2 heat ¯uxes observed during some the boundary layer with reasonable success (see, e.g., Paci®c storm events (Large and Crawford 1995). Khanna and Brasseur 1998). Recent oceanographic ap- Build up of momentum in the OBL through inertial plications of LES models include studies of Langmuir resonance has been well documented through obser- circulation (Skyllingstad and Denbo 1995; McWilliams vations and one-dimensional modeling studies (Craw- et al. 1997) diurnal mixing in the central tropical Paci®c ford and Large 1996, hereafter CL; Large and Crawford (Wang et al. 1998), and turbulence in the western Paci®c 1995). What has not been thoroughly explained is how warm pool (Skyllingstad et al. 1999). Here, we apply upper ocean currents create and interact with turbulence an ocean LES model to the problem of turbulence gen- and the strati®ed pycnocline at the boundary layer base. eration by wind-forced inertial currents. Our goal is to A schematic showing the pathways that wind energy quantify each of the energy pathways shown in Fig. 1 follows in driving inertial currents and turbulence is and to link these pathways to speci®c turbulent pro- shown in Fig. 1. In the OBL, energy provided by the cesses through ¯ow visualization and eddy statistics. wind is partitioned between the mean current and tur- Results from LES provide a detailed four-dimensional bulence generated by shear production and wave±cur- dataset that can be used to reveal how mixing processes rent interaction (Stokes production). Some fraction of affect the average structure of the OBL and to determine this energy is removed through turbulent dissipation, ⑀. if existing parameterizations represent mixing processes Another portion goes into vertical mixing of thermocline accurately. One-dimensional models of wind driven water, thereby reducing the OBL temperature and in- mixing (e.g., Large et al. 1994; Mellor and Yamada creasing the potential energy. Additional turbulent and 1982; Price et al. 1986) do not explicitly model turbulent internal wave energy is generated at the mixed layer motions, but rely on empirical relationships between base through the shear of the mean inertial current. En- mixing parameters, such as entrainment rate or eddy ergy from this process is also used to mix thermocline viscosity, and mean ocean properties of buoyancy and water and is dissipated through friction and internal shear. Typically, these mixing models determine shear- wave propagation. Our objectives here are to understand generated mixing rates using some form of the Rich- in detail the physics of these processes and to quantify ardson number, their relative importance in determining the OBL re- N2 sponse to strong wind forcing. Ri ϭ , (1) Accurate prediction of kinetic and potential energy Sh2 changes in the OBL is a critical step in modeling chang- where es in the SST. For example, if surface waves cause in- ␴ץ creased transfer of mean current kinetic energy into tur- g N2 ϭϪ ␪ zץ bulent kinetic energy through Langmuir circulation, ␳ then the sea state may cause more OBL cooling because o of increased entrainment at the mixed layer base. This is the square of the Brunt±VaÈisaÈlaÈ frequency, ␴␪ is the

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potential density, g is gravity, ␳O is a constant reference consistent with the assumption that the spectrum of ki- z) 2, u and ␷ are the netic energy follows a Ϫ5/3 power law for scales belowץ/␷ץ) z)2 ϩץ/uץ) density, Sh2 ϭ horizontal currents in the x and y directions, respec- the model resolution (Ducros et al. 1996). In areas of tively, and z is the vertical coordinate. The condition Ri the ¯ow where the resolved spectra does not follow a Ͻ 0.25 somewhere in the ¯ow is necessary for linear, Ϫ5/3 power law, the method may introduce errors by normal mode instability (Miles 1961; Howard 1961). either removing too much energy or by not removing However, boundary layer entrainment usually occurs at enough. In practice we ®nd that the scheme yields an a somewhat higher measured average Ri, between 0.3 accurate solution for most of the turbulent boundary and 0.5, because the background shear is continuously layer, but tends to underpredict turbulence kinetic en- forced. ergy in regions of strong strati®cation. Another aspect of the OBL that has not been ad- Eddy viscosity in the FSF approach is determined by dressed in mixed layer parameterizations concerns the ®rst ®ltering the velocity ®elds using a three-pass La- role of surface wave effects and Langmuir circulation. placian ®lter and then computing the velocity structure The effects of Langmuir circulation are thought to dom- function from the ®ltered ®elds (see Ducros et al. 1996 inate upper ocean turbulence in conditions of strong for details). In the present application, we applied the winds. However, only minimal observations exist that four-neighbor formulation that uses only the horizontal directly connect wind and waves with mixed layer differences in the horizontal velocity components to growth (e.g., Smith 1998). Our simulations provide a compute the structure function. This formulation is con- means of separating the individual effects of shear driv- sidered the preferred method by Ducros et al. (1996). en OBL growth from entrainment caused by Langmuir We found that the FSF scheme avoided many of the cells. We ®nd that for typical boundary layer depths problems encountered with methods that use the entire (ϳ30 to 40 m) the Langmuir circulation has a relatively velocity ®eld to infer eddy viscosity. For example, the small effect on average boundary layer growth rate, but original structure function model of Metais and Lesieur has a signi®cant in¯uence on when turbulence begins (1992) performed well in the mixed layer but produced to erode the thermocline. As a result, boundary layer excessive mixing in the upper thermocline because of growth in cases with weak waves tends to lag behind increased diffusion associated with large-scale internal the strong wave cases. waves. Similar behavior was noted when using the Dear- The paper is organized with the main goal of under- dorff (1980) subgrid-scale parameterization. The FSF standing the pathway wind energy takes in causing method, however, resulted in lower eddy viscosity val- changes in the average OBL temperature structure. Fol- ues in the upper thermocline because the method is in- lowing a brief discussion of the model and initial con- sensitive to the effects of eddy motions much larger ditions in section 2, we begin our analysis by examining than 4⌬x (i.e., the internal wave velocity ®elds). In ad- the evolution of horizontally averaged OBL properties in dition, we found that the FSF method develops both section 3. We relate changes in the average properties to shear and turbulent kinetic energy pro®les that are in speci®c terms in the turbulent and mean kinetic energy good agreement with Monin±Obukhov similarity theory budgets, connecting the average OBL structure with tur- near the surface. This was an unexpected result that will bulent processes. Section 4 presents a combination of be investigated in a separate paper on applying the FSF ¯ow visualization and statistics to examine how these scheme in geophysical ¯ow models. organized turbulent circulations lead to mixed layer en- The bulk of the experiments are performed on a 160 trainment and strong vertical heat ¯ux. Section 5 dis- ϫ 160 horizontal mesh with 64 vertical grid points and cusses the usefulness of mixing parameters derived from uniform spacing of 2.0 m. We also present results from the LES results, such as gradient and bulk Richardson a single experiment to examine small-scale circulations numbers, and how they might relate to similar indicators that uses a 256 ϫ 256 horizontal mesh with 128 vertical in a typical one-dimensional parameterization [speci®- grid points and 0.75-m resolution. The domain sizes cally, the K-pro®le parameterization (KPP) of Large et considered with the LES model are by necessity much al. (1994)]. Conclusions are presented in section 6. smaller than the Rossby radius of deformation. There- fore, mesoscale adjustments in the ¯ow are not possible and are assumed to be less important than the locally 2. The numerical model accelerated ¯ow. This assumption could introduce errors We apply the LES model described in Skyllingstad that affect the character of turbulence; for example, et al. (1999) and Denbo and Skyllingstad (1996). This large-scale pressure gradients could cause variations in model is based on the time dependent Navier±Stokes the vertical shear of the mean current leading to Kelvin± equations with subgrid turbulence closure provided by Helmholtz instability. the ®ltered structure function (FSF) approach of Ducros Boundary conditions are periodic in the horizontal et al. (1996). Surface wave interaction is modeled using direction with a rigid lid at the model top and an open the Stokes vortex forcing term from Craik and Leibov- boundary condition based on Klemp and Durran (1983) ich (1976). Energy is removed from the resolved tur- at the model bottom. Time step length for the integra- bulent ¯ow ®eld via the eddy viscosity in a manner tions ranged from 0.5 to 2 seconds depending on the

Unauthenticated | Downloaded 10/07/21 06:01 AM UTC 1AUGUST 2000 SKYLLINGSTAD ET AL. 1869 maximum mean current velocity and the Courant±Fried- TABLE 1. Experimental parameters governing the idealized wind richs±Lewy limit imposed by the third-order Adams± and wave forcing. Bashforth method (see Durran 1991). Simulations typ- Case ␻* ϭ ␾ Ϫ f Waves ically required 2 hours of wall clock time per simulated Resonant, 0 ␭ ϭ 30 m, hs ϭ 2m hour on a Hewlett Packard Exemplar computer running weak waves in a 32-node subcomplex. Resonant, 0 ␭ ϭ 130 m, hs ϭ 6m Experiments were designed following the idealized strong waves Off-resonance, Ϫ1.0285 ϫ 10Ϫ4 sϪ1 ␭ ϭ 30 m, h ϭ 2m conditions given by CL that represent typical fall con- s weak waves ditions in the North Paci®c with an initial mixed layer Ϫ4 Ϫ1 Off-resonance, Ϫ1.0285 ϫ 10 s ␭ ϭ 130 m, hs ϭ 6m depth of hm and a seasonal thermocline depth of ht. strong waves Following CL temperature pro®les were initialized as T , Ϫz Յ h mmCoriolis terms are essentially set to zero (except for the ⌬T T ϩ t (h ϩ z), h ϽϪz Յ h Stokes term, see the appendix). As pointed out by CL,  mm m t T(z) ϭ ⌬ht perfectly resonant winds have an effect similar to winds  ЊC with constant direction at the equator. We next per- TmtϪ⌬T ϩ 0.02 (h tϩ z), Ϫz Ͼ h t formed a transformation by subtracting out the average  m surface value of the currents, thereby moving the do- (2) main along with the surface ¯ow ®eld as was done in Skyllingstad et al. (1999). The slight inaccuracy intro- using parameters representing the Ocean Storms dataset duced by these coordinate transforms is justi®ed by the (Tm ϭ 12ЊC, hm ϭ 35 m, ht ϭ 50 m, ⌬Tt ϭ 4ЊC, ⌬ht reduction in advective smoothing and other numerical ϭ 15 m). In all simulations, salinity was held constant artifacts (see appendix). For the off-resonance cases, at 35.2 psu. Wind forcing for the experiments was pre- changing the grid reference does not reduce the effects scribed using of rotation because the ¯ow has a different mean velocity direction at every depth. Therefore, off-resonance t ␶(t) ϭ eAϪi(␻*ϩf )t sin ␲ ,0Ͻ t Ͻ t , (3) cases were only transformed by following the average t s surface current. []s Two surface wave scenarios were applied in the sim- where ␻* ϭ ␻ Ϫ f represents constant angular rotation ulations using a vertical Stokes drift pro®le for a mono- rate relative to an inertially rotating reference frame, ␻ chromatic wave ®eld, is the angular rotation rate relative to the earth's surface, A is the maximum wind stress (set to 1.4 N mϪ2), and 2 khs t ϭ 24 h is the storm duration. We consider two speci®c Us ϭ ͙g/k exp(Ϫ2kz), (4) s 2 cases of wind rotation rates: ␻* ϭ 0sϪ1 [i.e., ␻ ϭ f, ΂΃ or inertially rotating (resonant) winds], and ␻* ϭϪ1.01 where k ϭ 2␲/␭ and hs is the wave height or twice the ϫ 10Ϫ4 sϪ1 (i.e., a case of off-resonant winds; for a wave amplitude. Conditions for the two scenarios, des- latitude of 45ЊN, this scenario corresponds to ␻ ϭ 0, ignated ``strong'' and ``weak,'' were de®ned with hs ϭ or a locally steady wind direction). The inertially res- 6, ␭ ϭ 130 and hs ϭ 2, ␭ ϭ 30, respectively. Strong onant case is similar to the passage of an occluding waves in our experiments denote a deeper pro®le of midlatitude cyclone. Initially the winds are weak north- Stokes drift velocity in comparison with the weak wave easterly. As the storm approaches they increase and veer scenario. We chose these two simple scenarios to make to the east, southeast, and south. The storm passes caus- an initial assessment of surface wave forcing relative to ing the winds to shift to the west and weaken. inertially resonant currents, even though in actual ocean Cases with the Ocean Storms parameters were chosen storms it is very unlikely that these conditions would because they represent the basic case presented in CL. exist for as long as the simulations (ϳ24 hours). For However, because of the strong currents and shear gen- example, in the rotating, resonant wind case, the motion erated in the resonant storm forcing case, numerical of the storm, waves and wind direction would have to effects have a large impact in the simulation at the storm combine such that the wind and wave motion is always peak. In particular, rapid advection over the grid resulted in the same direction as the surface water velocity. In in considerable smoothing and reduction of small-scale future work, the idealized results presented here will be turbulent energy. In addition, rotation of the ¯ow ®eld extended and validated using more realistic surface forc- with periodic boundaries caused disruption of large- ing ®elds. scale coherent motions so that turbulence was concen- Experiments were performed as summarized in Table trated in a middle range of scales. To overcome these 1. Latitude for all experiments was set to 45ЊN, repre- problems in the resonant cases, we used two coordinate senting midlatitude conditions. The resonant wind cases transformations of the model grid. First, we transformed correspond to winds that rotate in an inertial sense rel- the LES domain to a rotating reference frame so that ative to a point located at 45ЊN, whereas the off-reso-

Unauthenticated | Downloaded 10/07/21 06:01 AM UTC 1870 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 nance cases represent a ®xed wind direction. Simula- temperature and momentum, and in the timing of ther- tions were performed for 24 hours, which encompassed mocline changes. In both resonant and off-resonance the main peak of storm winds as de®ned by (3) and the scenarios, wind forcing creates near-uniform tempera- major thermocline mixing event. ture and momentum ®elds down to ϳ15 m with weak waves and ϳ25 m for strong waves. In the off-resonance strong wave case, surface waves tend to reduce the peak 3. Evolution of the mean pro®les velocities; however the depth of signi®cant mixing ap- Observations of strong mixing events in the open pears greater as shown by the reduced sea surface tem- ocean often consist of time series of temperature as a perature. In general, stronger waves cause relatively function of depth at a ®xed point, for example, as an- greater mixing near the surface, leading to decreased alyzed by Large and Crawford (1995). The importance vertical gradients in momentum and temperature. The of resonant wind interaction is inferred from these data evolution of the average surface temperature for each by examining the end effects of turbulent mixing on the of the cases (Fig. 3e) further demonstrates the relatively mean temperature and current structure of the upper minor role that wave forcing has in controlling the ther- ocean. Our analysis will focus ®rst on this aspect of mal structure, especially when compared with changes inertial resonance by examining the evolution of the forced by resonance. Overall, the behavior of the OBL horizontally averaged velocity and temperature for each as shown in Figs. 2, 3 is consistent with observations of the experiments. We also examine the dependence of of wind forced mixing and the one-dimensional model mean ¯ow evolution on the strength of the surface wave results presented in CL. ®eld. b. Momentum, heat, and energy budgets a. Horizontally averaged ®elds In this section, we analyze the horizontally averaged Time-depth sections of the horizontally averaged ve- budget equations for energy and momentum to dem- locity components, u and ␷, for each of the four forcing onstrate how increased momentum produced by wind scenarios demonstrate how resonance affects the upper- resonance acts to change the OBL temperature structure. ocean momentum structure (Fig. 2). In both resonant Because wave effects were found to have a minimal cases (Figs. 2a,c,e,g), the u and ␷ increase dramatically effect in comparison to resonant wind forcing, the re- in response to the surface wind stress. The faster moving mainder of the analysis will concentrate on results from average currents produced by resonant forcing create the weak wave scenario. Horizontally averaged mo- strong current shear that coincides with the mixed layer mentum is governed by deepening. In contrast, the off-resonance velocity ®elds wץ uץץ Uץ ,Figs. 2b,d,f,h) form a weak, rotating current system) ϭϪ uЈwЈϪK ϩϩ(V ϩ ␷ ) f xץ zץz[]ms΂΃ץ tץ with momentum being removed by the wind stress every half inertial cycle as the currents oppose the surface wץ ␷ץץ Vץ wind. Current shear at the mixed layer base in these cases is greatly reduced in comparison to the resonant ϭϪ ␷ЈwЈϪKmsϩϪ(U ϩ u ) f, (5) yץ zץ zץ tץ cases. []΂΃ Horizontally averaged temperature structure for each where U and V are the horizontally averaged velocity of the four forcing scenarios are presented in Fig. 3. components in the zonal direction, x, and meridional Inertially resonant wind forcing causes markedly stron- direction, y, respectively; u and ␷ are the corresponding ger mixing in comparison to the constant wind forcing total velocity components; subscript s represents the in the off-resonance cases. For all cases, the vertical Stokes drift velocity components; primes denote the de- temperature structure changes only slightly before about viation from the horizontal average; f is the Coriolis hour 5 as the wind stress increases. After this time, the parameter; and Km is the momentum subgrid scale dif- resonant cases begin showing stronger vertical mixing fusion. In the analysis of (5), we focus on two dominant of the thermocline that accelerates rapidly at about hour terms on the right-hand side of each equation. The ®rst 8 with the thermocline expanding in vertical extent from is the stress divergence, which is the sum of the resolved 15mtoϳ50 m by the end of the simulation. In contrast, and subgrid-scale (SGS) terms in square brackets. The the off-resonance cases show only limited changes in second is Coriolis force acting on the mean current, that the thermocline vertical extent, expanding from 15 m is ( fV, Ϫ fU). These are shown as functions of depth to ϳ25 m. We note that the depth of the density mixed and time in Fig. 4. layer depth does not change signi®cantly in the simu- As shown by Figs. 2 and 4, ¯ux divergence in the lations (using ⌬␴t ϭ 0.01 as a measure), in agreement resonant case is closely aligned with the mean momen- with CL and Large and Crawford (1995). tum direction throughout the simulation time period. Surface wave effects are of secondary importance The Coriolis term in the resonant case (Figs. 4b,d) caus- compared with wind resonance in these cases. The effect es the mean currents to rotate at the same rate as the of wave forcing is most noticeable in the near-surface input wind stress and vertical momentum ¯ux. In the

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FIG. 2. Horizontally averaged zonal velocity for weak waves with (a) resonant and (b) off-resonance winds. Horizontally averaged meridional velocity for weak waves with (c) resonant and (d) off-resonance winds. Horizontally averaged zonal velocity for strong waves with (e) resonant and (f) off-resonance winds. Horizontally averaged meridional velocity for strong waves with (g) resonant and (h) off-resonance winds. off-resonance case, zonal ¯ux divergence does not mixed layer current system to oppose the wind stress, change direction, but increases and decreases in re- reducing the transfer of momentum from the wind into sponse to the surface wind forcing (Fig. 4e). As a con- the ocean. The evolution of the meridional accelerations sequence, rotation through the Coriolis term causes the for the off-resonance case are consistent with this pic-

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FIG. 3. Horizontally averaged temperature for (a) resonant weak waves, (b) resonant strong waves, (c) off-resonance weak waves, and (d) off-resonance strong waves. Mixed layer depth as de®ned in the text is plotted as a dark solid line. Also shown is the sea surface temperature for all four cases (e).

ץ ץץ -ture, showing the decrease and then increase in the me ridional ¯ow as the currents rotate inertially (Figs. 2d,h). T ϭϪ wЈTЈϩKTh , (6) zץz΂΃ץ tץ Turbulent transport of meridional momentum tends to redistribute the momentum vertically, removing nega- where w is the vertical velocity, T is the temperature, tive momentum from the near surface layer and depos- Kh is the subgrid diffusion of heat, overbars denote iting it near the OBL base (Fig. 4g). horizontal averages, and primes denote perturbations The horizontally averaged heat budget is described from the horizontal average. Plots of the resolved ver- through a turbulent transport equation, tical heat ¯ux, ␳Cp wЈTЈ, shown in Fig. 5, demonstrate

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FIG. 4. Horizontally averaged (a) zonal stress divergence, (b) zonal Coriolis acceleration, (c) meridional stress divergence, and (d) meridional Coriolis acceleration for the resonant case, and (e) zonal stress divergence, (f) zonal Coriolis acceleration, (g) meridional stress divergence, and meridional Coriolis acceleration for the off-resonance case. Fields are smoothed with a cutoff period of ½ h before plotting. the impact resonant wind forcing can have on the OBL is understandable given that perfect resonance, as as- heat budget. Peak values are about Ϫ7000 W mϪ2 in sumed in the idealized simulations, represents an upper the resonant case and Ϫ2000 W mϪ2 during off reso- bound on the transport of momentum from the wind nance and represent the downward transport of heat to the ocean, and therefore the maximum case for shear stored during the summer in the OBL. These values production of turbulence. are similar to estimates reported by Large and Craw- Analysis of the momentum and heat budgets shows ford (1995), Large et al. (1986), and Krauss (1981) for that vertical turbulent transport is the dominant mech- storm forced vertical heat ¯uxes, although the resonant anism for changing the average properties of the OBL. case is at the upper end of the observed estimates. This To further understand how the average momentum ®elds

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FIG. 5. Horizontally averaged resolved heat ¯ux from the (a) resonant and (b) off-resonance cases. affect turbulent production, we next examine the tur- has a much different behavior with TKE increasing bulent kinetic energy de®ned as steadily over time and deepening with the growing mixed layer. TKE increases until about hour 13 (after 1 3 the maximum winds) and then gradually falls off TKE ϭ uЈ2, uЈϭu Ϫ u , 2 ͸ iiii throughout the rest of the simulation. In the resonant iϭ1 case, TKE continues to increase after the peak wind where the index i denotes the x, y, and z directions, because the inertial currents are still gaining strength respectively. Turbulent kinetic energy is strongly af- from the wind stress. Eventually, the production of TKE fected by wind resonance as shown by plotting TKE as decreases when the momentum input from the wind a function of time (Fig. 6). With off-resonance winds, cannot maintain strong shear and corresponding shear TKE follows the wind stress forcing, which increases production of turbulence, as is shown below in the TKE until about hour 12 and then declines through the re- budget analysis. mainder of the simulation. In contrast, the resonant case The budget of TKE is de®ned using

ץ ץuЈץu ␳Јץץ ii Ä s TKE ϭϪuЈi uЈϪ33uЈg ϩ͗uijЉuЉ͘ Ϫ (uЈ3E ϩ uЈ͗iiuЉu33Љ͘) Ϫ (uЈP) ϩ uЈii,j,kjk(␧ u ␨ )Ј, (7) xץx33ץ xץx3 ␳ojץ tץ I II III IV V

where duction term is the subgrid dissipation term (III), which accounts for the loss of kinetic energy via friction. The uj 2ץuPץ i Ä 2 ͗uijЉuЉ͘ϭϪK m ϩ , P ϭϩq , turbulent and subgrid transport terms (IV) redistribute -x ␳ o3 turbulent energy vertically, but do not change the inץxjiץ΂΃ tegrated energy for a closed system. The budget is ujץ ␨kkijϭ␧ , closed with the Stokes production and pressure transport -xi terms (V). Individual terms in (7) are evaluated by mulץ΂΃ ␳ is the density, g is gravity, q is the subgrid scale tiplying the momentum budget terms by the appropriate turbulent kinetic energy, and P is the pressure (see Skyl- perturbation velocity ®elds before time differencing. lingstad et al. 1999). The ®rst term (I) is the shear pro- This procedure introduces small estimation errors be- duction, which accounts for the generation of turbulence cause the third order Adams±Bashforth time differenc- by instabilities in the mean shear ¯ow. The next term ing scheme smooths the physical mode of the equations

(II) represents the transfer of energy between TKE and as ui ϩ Ui approaches ⌬x/⌬t. However, this error is potential energy through the buoyant production term. minor; the budget residual is typically much smaller This term represents energy lost or gained by turbulence than the signi®cant terms. in doing work against gravity, lifting cold water or forc- Factors governing the evolution of TKE are revealed ing warm water downward. Following the buoyant pro- by plotting signi®cant terms from (7) as functions of

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FIG. 6. Time±depth section of the speci®c turbulent kinetic energy for (a) resonant and (b) off-resonance cases. time and depth as shown in Fig. 7. Each case displays approximately 0.5 (Smyth and Moum 2000a). In the the same basic pattern for the budget terms, with the present case, however, shear is continually reinforced Stokes drift and dissipation rate having maximum mag- by the inertial response to the surface forcing. Shear nitude near the surface, and the buoyancy and shear and strati®cation are thereby maintained in a quasi-equi- terms peaking near the bottom of the OBL along with librium state characterized by Ri close to the critical a secondary peak in the dissipation rate. The over- value 0.25 (Miles 1961). Stability analyses of the mean whelming difference between the resonant and off-res- currents reveal weakly unstable modes with wave- onance cases is the strength of the shear production term lengths of a few hundred meters (methodological details and corresponding dissipation rate, which are both about are given in Sun et al. 1998). This con®rms that our four times larger in the resonant case. The stronger tur- horizontal domain size is suf®cient to accommodate the bulence produced in the resonant case directly affects largest dynamically important scales of motion. the depth of the boundary layer by forcing entrainment The ®nal component in the energy budget of the OBL at the boundary layer base as indicated by the buoyancy is the conversion of TKE into potential energy, PE, production term. This causes the maximum of shear which is governed by the horizontally averaged PE bud- production to descend as a function of time. get, The shear production term represents the exchange ץץ 1 ץ .of energy between the mean current and the turbulence PE ϭϪ u PE Ϫ K PE ϩ K ␳g zץx 3 hhץ t ␳ץ As such, it is an important link between the average o [ 3 ΂΃ momentum produced by the wind, and vertical mixing ␳ץ forced by turbulent ¯uxes. The dominance of the shear production term, particularly in the resonant case, shows ϩ g␳Јu3 Ϫ gKh , (8) zץ that wind energy entering the system through the mean ] current is converted to turbulence, which leads to bound- where PE ϭ g␳z/␳ 0, ␳ is the density, Km and Kh are the ary layer growth and OBL cooling through the vertical subgrid-scale eddy viscosity and scalar diffusion coef- transport of relatively cold thermocline water. A key ®cients. The PE budget begins with a group of three component of both the shear production term and the transport terms, which redistribute potential energy ver- average momentum budget is the resolved eddy mo- tically but do not change the integrated energy for a mentum ¯ux shown in Fig. 8. Although the momentum closed system. The next term represents the transfer of ¯ux magnitude is comparable for both resonant and off- energy between TKE and PE through the buoyant pro- resonance cases, in the resonant case, which has greater duction term and is of opposite sign from the buoyancy mean shear (see Fig. 2), momentum transport leads to production term in the TKE budget. The ®nal term in a stronger growth in turbulence through shear produc- the PE budget accounts for changes in PE caused by tion. This physical regime is characteristic of entrain- subgrid scale mixing. Estimation of PE is performed by ment via shear instability (e.g., Sun et al. 1998; Smyth examining the change in PE after the advection step and and Moum 2000a,b; Werne and Fritts 1999; Cortesi et then again after the subgrid diffusion. Thus, changes in al. 1999). Shear and strati®cation are continually re- PE resulting from numerical smoothing during advec- duced by turbulent mixing, so the gradient Richardson tion are implicitly included in the advective transport number (1) tends to increase. In unforced ¯ows, insta- term. The buoyant production term contribution is cal- bility eventually ceases and turbulence decays. In this culated explicitly. decaying state, Ri approaches an asymptotic value of Inspection of the vertically averaged PE budget terms

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FIG. 7. Time±depth sections of the largest terms in the resolved turbulent kinetic energy budget for the resonant and off-resonance cases. Units are watts per kilogram.

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FIG. 8. Time±depth cross section of average turbulent momentum ¯uxes: (a) uЈwЈ for the resonant case, (b) uЈwЈ for the off-resonant case, (c) ␷ЈwЈ for the resonant case, and (d) ␷ЈwЈ for the off-resonant case. for the resonant case shows that buoyant production is ancy terms must equal zero at the surface. In simple responsible for most of the change in PE (Fig. 9). The terms, when cold water is forced upward, it must be off-resonance case shows a similar behavior, but with replaced by some portion of the warmer water that is a much weaker amplitude. In general, changes in the being displaced. For the northern Paci®c scenario con- averaged PE mirror the strength of the TKE. Smaller sidered here, the relatively large local changes in PE contributions to the PE budget come from the subgrid represent the exchange of water warmed during the sum- mixing terms and the transport term. In a perfect nu- mer with cool thermocline water that is maintained by merical model, the transport term would be nearly equal horizontal transport and cooling during the previous to zero because of the periodic model domain and rel- winter. Thus, wind forced mixing acts as a catalyst in atively small ¯uxes at the model bottom. However, large moving heat stored near the surface to deeper water. velocity components produced in shear-driven turbu- The distinction between the changes in the integrated lence cause increased smoothing by the scalar advection PE and the local PE are important when considering the scheme, which produces an appreciable change in the response of bulk mixed layer models (e.g., Kraus and integrated PE. This term is considerably smaller than Turner 1967; Davis et al. 1981). Bulk layer models typ- the buoyancy production and may be thought of as part ically predict the integrated energy budget so that chang- of the subgrid-scale mixing since it tends to reduce the es in the potential energy can be related to surface input amplitude of small-scale ¯ow features. of wind energy and heat ¯ux. Bulk layer models assume Vertical pro®les of the PE budget terms during the that entrainment adds a portion of thermocline water to peak of the wind stress (Fig. 10) provide a different the mixed layer without changing the interior thermo- view of how PE is redistributed by turbulence in the cline structure. Local changes in the PE below the mixed OBL. In particular, the advective transport term in the layer, as shown in Fig. 10, cannot occur because of this budget shows a much higher amplitude with increased assumption. Additional mixing through a Richardson PE in the upper portion of the OBL that is almost exactly number criterion, for example, as performed in the Price balanced by decreased PE in the lower OBL and ther- et al. (1986) model, can account for the local PE changes mocline. This pro®le represents the turbulent exchange by modeling the wind-generated current shear and as- of relatively warm surface water with cold thermocline sociated mixing beneath the mixed layer base. water, and is ultimately a measure of the change in sea Our discussion of the energy budget is concluded by surface temperature, since both the subgrid and buoy- examining the primary processes in the energy pathway

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FIG. 9. Vertically averaged terms from the PE budget equation (9) as a function of time for the (a) resonant and (b) off-resonance cases. from surface wind to changes in the PE (Fig. 11). Figure directly cancelled by dissipation and buoyant production 11 shows the cumulative effect of each of the main so that TKE (not shown) is much smaller than the energy sources and sinks of energy including the wind stress budget terms shown in Fig. 11. At the end of the storm term de®ned as Usfc␶/␳, where Usfc is the surface current. in the resonant cases, Fig. 11 shows that ϳ7% of the Focusing on the resonant case ®rst, wind and wave en- input wind energy goes into changing PE. In the off- ergy ramps up following the prescribed wind stress, with resonance case, the rotation of the currents causes the a maximum increase at about hour 14 because of the wind stress term to become an energy sink as the surface stronger average surface currents at this time. About currents rotate to a direction opposing the wind stress. 40% of this input energy is lost via turbulence dissi- This greatly limits the mean current velocity and cor- pation as momentum is transferred from the near surface responding shear production of turbulence. Entrainment throughout the OBL. Most of the shear production is buoyancy ¯ux at the bottom of the boundary layer (re-

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FIG. 10. Time average vertical pro®les of the PE budget terms taken from hours 11±12 from the (a) resonant and (b) off- resonance cases.

sponsible for PE changes) is also reduced because of the overall weaker turbulence. As in the resonant case, dissipation in the off-resonance case dominates the energy budget, mostly by balancing the near-surface Stokes wave forcing and shear production. In both the resonant and off-resonance cases, the near balance between shear production and dissipation in the energy budget suggests that much of the turbulent energy is concentrated in small-scale ¯ow features that are directly affected by the subgrid dissipation. This result may have implications for high-order turbulence closure methods, such as the Mellor and Yamada (1982) parameterization, which attempt to solve a subgrid-scale turbulence energy budget similar to (7). Successful turbulence parameterizations of this type need an accurate estimation of the balance between shear production, dissipation, and buoyant production since the strength of parameterized mixing is typically linked with the pre- dicted turbulence kinetic energy. As Fig. 11 shows, the relatively small difference between shear production and turbulence dissipation explains most of the buoyant production, or change in the PE. The PE gain may therefore be highly sensitive to inaccuracies in the shear production and dissipation terms.

4. Spatial structure of the turbulence Understanding boundary layer dynamics involves more than quantifying the average budgets of heat and FIG. 11. Cumulative vertically integrated input wind energy, total kinetic energy (KE), PE, and turbulence dissipation rate for (a) res- kinetic energy. For example, it is important to know onant and (b) off-resonance cases. whether turbulent eddies span the entire mixed layer

Unauthenticated | Downloaded 10/07/21 06:01 AM UTC 1880 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 depth or are small-scale features that only cause localized mixing. In this section, we present visualizations of the turbulent ¯ow ®elds from the resonant case, focusing on how increasing mean current shear produces strong boundary layer entrainment through injection of horizontal momentum into the upper thermocline. The boundary layer evolves into a three layer system with large-scale coherent motions near the surface, small- scale, shear-driven eddies at the boundary layer base, and nonturbulent internal waves in the thermocline. Close examination of the results from the coarse-resolution simulations described in sections 3 and 4 showed that the largest circulations were ϳ100 to 150 m in scale and extended through the central section of the OBL. Near the OBL base, however, turbulence scales were found to contract so that most of the entrainment mixing was performed by eddies ϳ5 to 10 m in scale, indicating that our resolution was too coarse for detailed analysis of the turbulence structure. In this section, we use results from a simulation using a smaller horizontal domain size, 192 m in the horizontal and 96 m in the vertical, but with higher resolution (0.75 m). Because of the higher resolution and larger number of grid points, this simulation required about eight times more computation FIG. 12. Horizontally averaged root mean square downwind u, cross wind ␷, and vertical w velocities at hour 11. than the budget simulations. Therefore, we initialized this case with horizontally averaged pro®les of u, ␷, and T from hour 8 of the budget calculation and limited the eddy ®eld via convergence of the stress component cor- duration of the experiment to 3 hours. The results de- responding to vertical shear of the downwind current. scribed in this section therefore correspond to a ``snapshot'' of the ¯ow ®eld taken at hour 11 from the budget experiments. Only the resonant, weak wave scenario a. Flow visualizations was considered. Results are presented in an inertially Cross-sections of the vertical velocity ®elds, hori- rotated coordinate system where the x axis is aligned zontal perturbation velocity vectors, and horizontal per- with the surface wind stress direction, so that u and ␷ turbation temperature provide a closer view of the wind- represent the instantaneous downwind and cross-wind driven OBL turbulent structure (Fig. 13). At this time, velocity components, respectively. the depth of active mixing extends to ϳ60 m, so the We begin the discussion with a look at the vertical cross sections shown in Fig. 13 represent the main areas pro®les of horizontal, root-mean-square averages of the of turbulent production described in the TKE budget velocity components (Fig. 12). To facilitate interpreta- discussion. At 10-m depth, Langmuir circulation is in- tion, we conceptually divide the water column into three dicated by the organized, linear structures in the ¯ow layers, with boundaries at depths 25 m and 60 m. Below ®eld. The largest eddies behave much as in previous 60 m, the ¯ow consists of weak gravity waves. These simulations of Langmuir circulation (Skyllingstad and motions play a minor role in the processes of interest, Denbo 1995; McWilliams et al. 1997), with scales of and will not be discussed in detail. Between 60 m and ϳ30 to 50 m, or roughly equal to the mixed layer depth. 25 m, the ¯ow is dominated by downwind motions. This Regions of strong upwelling tend to correspond with form of anisotropy in the energy-containing scales is upwind velocity perturbations (e.g., x ϭ 110 m, y ϭ 50 typical of shear-driven turbulence (e.g., Smyth and m), whereas regions of strong downwelling tend to cor- Moum 2000a,b). In this layer, mixing of near-surface respond with downwind velocity perturbations (e.g., x water with thermocline water is driven by the shear at ϭ 125 m, y ϭ 150 m). These circulations have a rel- the base of the wind-driven surface current. In the upper atively long downwind coherent structure, although they 25 m, the structure is more complex. Between 10 m and are not oriented as in previous studies that show a con- 25 m, we see anisotropy suggestive of Langmuir cells; sistent orientation to the right of the wind (e.g., Smith that is, downwind motion is weak compared with cross- 1998). Here, the orientation of the circulations appears wind and vertical motion. As the surface is approached, to be affected by a larger-scale circulation (roughly do- these motions become primarily crosswind. We also ob- main scale) that is noticeable in the near-surface per- serve an increase in downwind velocity near the surface, turbation temperature ®eld (Fig. 13c). Where the large indicating a coupling between the wind and the ocean eddies force positive (negative) ␷, the small scale vor-

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FIG. 13. Horizontal cross sections of vertical velocity at (a) 10 m and (b) 40 m, along with horizontal perturbation velocity vectors and perturbation temperature at (c) 10 m and (d) 40 m taken from hour 11. Velocity vectors are plotted every sixth grid point. All perturbation quantities are departures from the horizontal average. tices are aligned to the right (left) of the large eddies. entation in these simulations probably differs from past We also noticed this behavior in the simulation used in studies because of the intensity of the wind forcing and the budget calculations and in an additional coarse res- upper ocean currents, although the overall scale is sim- olution experiment with horizontal domain size of 720 ilar to observations of 2±200 m Langmuir cells reported m (not shown), indicating that the small domain size by Smith (1998) and Plueddemann et al. (1996). used in this case was not a determining factor in the Throughout most of the OBL, large-scale currents Langmuir cell behavior. The large-eddy structure ori- re¯ect the form of the Langmuir cells. However the

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FIG. 14. Vertical cross section along x ϭ 120 m showing perturbation potential temperature and perturbation velocity vectors from every fourth grid point as a function of y and z at hour 11. coherence of these eddies deteriorates approaching the tween x ϭ 150 and 192 m and the ejection between x bottom of the boundary layer (Figs. 13b,d). Between 20 ϭ 50 and 75 m. Coincident with the injections and and 40 m, mean current shear starts to control turbulent ejections are regions of warmer and colder water, re- processes, causing strong horizontal u perturbations at spectively, as indicated by the temperature perturbation 40 m (cf. Fig. 12). These horizontal velocity pertur- ®eld. Strong negative temperature perturbations in ejec- bations are associated with large excursions in the tem- tions extend upward from the thermocline to ϳ35 m perature ®eld and may de®ne the exchange of water depth, while corresponding positive perturbations are between the OBL and the thermocline. The role of the stronger below ϳ45 m. large eddies in transporting downwind momentum is Similar weaker horizontal current perturbations at shown clearly by the w component and horizontal vector ϳ30 to 40 m depth appear to be more linked with eddies plots; areas of downwelling have downwind velocities driven from the surface. This region has a number of that are ϳ0.4 m sϪ1 larger than the surrounding water small scale (ϳ10 to 25 m) overturns that are actively (e.g., between y ϭ 100 and 150 m). These eddies are mixing thermocline water into the mixed layer. The also the primary mechanism for moving cold water from Langmuir region is characterized by strong downwelling the bottom of the OBL upward to the surface as shown pulses linked to the surface forcing, for example, at x by the perturbation temperature. Because of the mo- ϭϳ25 m and ϳ110 m. These vertical jets extend down- mentum transport, shear is enhanced beneath the downward from about5mto25mandoccasionally combine welling circulations, causing a slight increase in tur- with circulations in the entrainment zone, indicating a bulence intensity. Both the w and TЈ ®elds show this possible enhancement of entrainment when small-scale effect at 40 m with regions of slightly enhanced small- Langmuir cells are particularly strong. Vectors in the scale structure centered beneath the downwelling center Langmuir cells show a tendency to point more toward at y ϭϳ150 m. positive x, indicating the vertical transport of positive A cross section at x ϭ 120 m (Fig. 14) illustrates the u momentum. complex form of the large-scale circulations. Dominant In comparison to the downwelling cross section, the features include a cool upwelling zone near y ϭ 60 m, ¯ow ®eld in the upwelling branch of the large eddy and a more diffuse, warm downwelling region at y ϭ circulation has a more subdued turbulent structure. Eddy 120 m. Turbulent processes associated with these down- velocities in the entrainment zone are also fairly weak welling and upwelling zones are visible in cross sections in comparison to the downwelling case, with less ver- at y ϭ 120 m and y ϭ 60 m, respectively (Fig. 15). We tical excursions in the perturbation temperature ®eld. focus ®rst on the downwelling region. In the entrainment Near the surface in the Langmuir zone, vertical veloc- zone at z ϭϳ45 m, u velocity shear has a strong in- ities are dominated by upwelling, with vertical transport ¯uence on the horizontal eddy circulation, for example, of negative u momentum indicated by the backward as shown by the injection of positive u momentum be- pointing vectors.

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FIG. 15. Vertical cross sections along (a) y ϭ 120 m and (b) y ϭ 60 m showing perturbation potential temperature and perturbation velocity vectors from every fourth grid point as a function of x and z at hour 11.

The ¯ow visualization presented in Figs. 13±15 pro- identifying the importance of speci®c ¯ow features, vides a snapshot of the processes responsible for trans- such as downward directed injections, is the application porting horizontal momentum and entraining thermo- of quadrant analysis (Lin et al. 1996; Sullivan et al. cline water into the OBL. Large-scale eddies, forced 1998). Quadrant analysis divides the heat ¯ux using near the surface by Stokes drift interaction with the conditional averages, separating the heat ¯ux into four mean current and shear production, produce regions of possible groups, wϩT ϩ, wϪT ϩ, wϩT Ϫ, wϪT Ϫ, where T active downward transport of zonal momentum. Mo- is the perturbation from the horizontally averaged tem- mentum ¯ux convergence near the base of the mixed perature and the superscript ϩ and Ϫ refer to positive layer leads to intermittent injections and ejections of and negative perturbations, respectively. This method momentum that extend into the upper thermocline, en- provides a way to discern systematic patterns in the hancing the local exchange of thermocline water with entrainment ¯ux, for example more cold water moving the colder OBL. Because of the coherent large-eddy upward on average. turbulence, areas of strong momentum and heat ¯ux are Application of quadrant analysis to hour 11 (Fig. 16) localized beneath the downwelling portion of the eddies. shows that heat ¯ux in the shear-driven OBL has a complex structure for the negative ¯uxes, while positive b. Quadrant analyses of vertical ¯uxes ¯uxes are nearly equal. Because the ¯ow is stably strat- Flow visualizations suggest that entrainment is pro- i®ed, the negative heat ¯ux dominates, giving a total duced by a wide range of ¯ow scales. One method for ¯ux with a maximum at ϳ40 to 50 m depth. The peak

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water upward near the top of the OBL (thereby increasing the local PE), and warm water downward in the entrainment zone (thereby decreasing the local PE). Comparison of the quadrant analysis results with Sul- livan et al. (1998) shows a signi®cant difference between entrainment produced by shear and that produced by thermal convection. Sullivan et al. identi®ed relatively large differences between each of the condition- ally averaged heat ¯uxes so that the net result was that most of the entrainment ¯ux could be attributed to warm plumes moving upward into the atmospheric inversion layer. Here, we see that entrainment is dominated more by forced convection, that is, warm water being forced downward against buoyancy. In the convective boundary layer studied by Sullivan et al., the main energy for turbulent eddies comes from buoyant plumes that extend through the depth of the mixed layer. For the wind driven OBL, turbulence in the entrainment zone is dominated by a balance of shear production and dissipation, with buoyancy production having a lesser role. Another major difference between the convective boundary layer and the wind driven OBL is the relative strength of the total heat ¯ux and the individual quadrant ¯uxes. Sul- livan et al. found that heat ¯ux quadrants in the convective boundary layer tended to cancel each other so that the net heat ¯ux was smaller than all of the individual quadrant ¯uxes. Here, the largest ¯ux is typically the total ¯ux, indicating that in shear-dominated mixing less water recirculates adiabatically than in convective mixing. FIG. 16. Quadrant analysis from hour 11. Superscript ϩ and Ϫ denote conditional averaging with positive and negative perturba- Quadrant analysis of the zonal momentum ¯ux, uЈwЈ (Fig. 17), yields pro®les consistent with the roughly tions, respectively. Fluxes are multiplied by ␳Cp. linear growth in mean momentum as a function of time shown in Fig. 2. Because the wind stress is positive, values for the negative and positive ¯uxes are not lo- almost all of the vertical momentum transport occurs cated at the same depth; negative ¯ux has a maximum through uϩwϪ and uϪwϩ. In the entrainment zone, uϩwϪ value at ϳ50 m, whereas the positive ¯ux peaks at ϳ55 is slightly larger that uϪwϩ, indicating that downward m. One interpretation of this is that, on average, warm injections of positive momentum have a slightly larger parcels are forced downward against gravity at ϳ50 m role in momentum transport in comparison to ejections (wϪT ϩ), and tend to rebound upward at ϳ55 m (wϩT ϩ). of negative zonal velocity. A somewhat different mo- However, horizontal mixing and diffusion prevent a mentum ¯ux behavior was noted in Lin et al. (1996) completely adiabatic process so that heat contained in for a neutral atmospheric boundary layer. Their average transient eddies is diffused laterally, resulting in a net pro®le was similar to Fig. 17, with a peak negative vertical heat ¯ux. Additional cancellation of the positive momentum ¯ux near the boundary but with reversed and negative heat ¯ux quadrants is produced by internal transport since the atmospheric system is a mirror image gravity waves, as identi®ed above in the lower portion of the OBL. Momentum transport in the ABL was dom- of the OBL. Internal waves act to lower the depth of inated by ejections (uϪwϩ) from the slow moving sur- the peak positive and negative heat ¯uxes relative to the face layer, whereas our results did not show a strong depth of the total heat ¯ux. preference between uϩwϪ and uϪwϩ near the upper Focusing on just the negative ¯ux quadrants, we note boundary. that the larger fraction of the negative ¯ux in the entrainment layer is generated by relatively warm water c. Turbulence spectra being pushed downward. This corresponds to injections of momentum as shown in Fig. 13d. In contrast, above The scales of turbulence produced by inertial current ϳ30 m the negative heat ¯ux is dominated by cold water shear vary signi®cantly as a function of depth in the being forced upward. This separation of the heat ¯ux boundary layer. This variation is quanti®ed by exam- is consistent with the PE pro®les shown in Fig. 10, ining the horizontally averaged velocity and temperature namely, that wind driven turbulence tends to force cold spectra from three model depths representing the Lang-

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solved by the model domain. This is consistent with previous LES results for Langmuir turbulence away from the surface (McWilliams et al. 1997; Skyllingstad and Denbo 1995) and indicates that energy enters the turbulence ®eld at the largest eddy scales, with very little injection at larger wavenumbers. Spectra in the upper entrainment zone at 25 m show a tendency to follow a Ϫ1 power law for u, but not for w and ␷, implying that turbulence at these depths is gaining energy from the mean ¯ow across a range of scales that are oriented with the shear axis (x direction). This behavior is somewhat similar to wind tunnel experiments for a rough-wall boundary layer performed by Antonia and Raupach (1993), who found a Ϫ1 power law for the streamwise velocity, although they also noted a Ϫ1 power law for the ␷ component. Below ϳ40 m, the model u and w spectra do not display a signi®cant range with a Ϫ5/3 dropoff, sug- gesting that the model has insuf®cient resolution in part of the entrainment zone. Following Skyllingstad et al. (1999), we apply two commonly used length 3 1/2 scales, the Ozmidov scale Lo ϭ (␧/N ) and the buoyancy length scale Lb ϭ w/N, to see where buoyant strati®cation is acting to suppress vertical eddy di- mensions (Fig. 19). These two length scales estimate the distance that turbulent motions can travel before being strongly affected by strati®cation. As Fig. 19 shows, strati®cation begins to control vertical eddy scales at depths below ϳ45 m or in the lower half of the entrainment zone. For the coarse resolution budget FIG. 17. Quadrant analysis of the u component of momentum. calculation, the effects of buoyancy become important at an even shallower depth of about 25 m. Lack of a muir zone, middle of the mixed layer, and from the top well-resolved inertial subrange does not necessarily of the entrainment zone (Fig. 18). Also shown are lines mean that the energy containing scales are inaccurate. representing the Ϫ1 and Ϫ5/3 power laws, which can As Skyllingstad et al. (1999) demonstrated, even be thought of as describing the ``production'' and ``in- though LES produced dissipation rates were much ertial'' subranges of turbulence for shear ¯ow (Kader smaller than observed values, the resolved heat ¯ux and Yaglom 1991; Katul and Parlange 1995). The gen- was still quite accurate. This is because dissipation erally accepted theory, based on wall-layer ¯ow, is that occurs at much smaller scales of motion than ¯uxes. energy injected at small wavenumbers through eddy Wang et al. (1998) also found that LES simulations of asymmetries follows a Ϫ1 power law. At low wave- equatorial turbulence gave good results in comparison numbers, the ¯ow is asymmetric and maintains a ``mem- to observations, even though their simulations were ory'' of the process that initially formed the eddy, for also under-resolved in the strati®ed region of the ¯ow. example like the horizontal jet shown in Fig. 13d at x Nonetheless, comparisons of velocity spectra from the ϭ 180 m, y ϭ 125 m that was produced by vertical coarse resolution budget calculation (not shown) and advection of the mean shear. As energy cascades to the high resolution case presented in Fig. 18, indicate small scales, the memory of injection events is lost and that turbulence energy levels in the low resolution sim- the ¯ow becomes isotropic. The transition to isotropy ulations were reduced. The net effect of the stronger is accompanied by the transition from a Ϫ1toϪ5/3 turbulence in the high resolution case is increased tur- power law and the establishment of an inertial subrange bulent mixing, however, the relative importance of the as presented in Kolmogorov's (1941) theory. TKE and PE budget terms was not signi®cantly altered Temperature spectra exhibit a Ϫ5/3 slope over at least by the increased resolution. a decade of wavenumbers. Velocity spectra tend to fol- Poor resolution in the thermocline as indicated by Lo low a Ϫ1toϪ5/3 power law, although it appears that and Lb may also help explain the relatively weak internal the inertial subrange in the entrainment zone is not well wave ®elds shown in our results. The model is unable established (e.g., at 40-m depth). In the Langmuir zone to produce small-scale eddies in the strongly strati®ed at 10 m, spectra for all velocity components follow an regions of the ¯ow. These eddies would likely generate inertial subrange starting from the largest scales re- small-scale internal waves that could transport momen-

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FIG. 18. Horizontal spectra of (a) T, (b) w, (c) u, and (d) ␷ at 10 m, 25 m, and 40 m at hour 11 as a function of

wavenumber, k y ϭ 2␲/␭ where ␭ is the wavelength. Spectra for u, ␷ , and w at 25 and 10 m are offset by 10 and 100, respectively. tum away from the mixed layer. Simulating these waves 5. Mixed layer parameters: Comparison to KPP would require much higher resolution than is currently model feasible. We also note that the simulations do not show signi®cant internal wave growth via Kelvin±Helmholtz Martin (1985), Large and Crawford (1995), and Large instability. Inclusion of a domain-scale pressure gradient and Gent (1999) have shown that simpli®ed one-di- (as would be forced by geostrophic adjustment) could mensional models are capable of reproducing many of cause a more unstable velocity pro®le to evolve. We the average changes that occur in the wind-forced OBL. have not attempted to examine this problem here, but These models typically represent mixing rates through hope to investigate the effects of large-scale pressure some combination of bulk stability parameters and em- variations in future work. pirically derived mixing coef®cients. Here, as an ex-

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(B Ϫ B(z))z Ri ϭ r , (9) b 22 |VrtϪ V(z)| ϩ V (z)

where B(z) ϭ g␳Ј/␳, Br and Vr are the average surface layer buoyancy and velocity, and Vt is a measure of the turbulent velocity scale. The shallowest depth where Rib exceeds a critical Ric is considered the boundary layer depth. Using observations of OBL depth and surface

parameters, Large et al. found that Ric ϭ 0.3 gave the overall best mixed layer growth, although good results

were noted with a range of Ric between 0.25 and 0.5. Plots of Rib estimated from the coarse resolution results (resonant winds, weak waves from section 3), using horizontally averaged buoyancy and velocity ®elds (Fig.

20), are in general agreement with the range of Ric suggested by Large et al. (1994), although the depth

where Rib ϭ 0.3 appears to yield a boundary layer depth somewhat shallow in comparison to the depth of the peak heat ¯ux. It should be noted, however, that the value used in KPP is tuned to work with a coarse vertical grid and an assumed functional form for eddy diffusivity, which may produce a more rapid boundary layer

growth for a given Ric. A bulk mixing coef®cient estimate can be derived from FIG. 19. Horizontally averaged Ozmidov length scale (Lo) and

buoyancy length scale (Lb) from model hour 11. the LES results by assuming a simple eddy viscosity budget equation for temperature and solving for Kh, ץ K ϭϪwЈTЈ T. (10) zץample we examine the Large et al. (1994) KPP model. h ΋ Eddy diffusivities in KPP are modeled using empirical pro®les derived from atmospheric boundary layer mea- This calculation is consistent as long as the vertical tem- surements. Accurate boundary layer prediction using perature pro®le is not completely mixed. Figure 21 shows these pro®les requires a robust estimate of the boundary Kh from the LES model along with Kh calculated using layer depth. For KPP, boundary layer depth is estimated KPP as presented in CL. Comparison below ϳ15 m through a bulk Richardson number de®ned as shows good correspondence, with KPP values remaining

FIG. 20. Contours of bulk Richardson number values 0.4 and 0.3 overlaid on the resolved turbulent heat ¯ux, ␳CpwЈTЈ, for the resonant case.

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winds cause strong acceleration of the velocity ®eld, and a corresponding increase in shear-produced vertical mixing. In contrast, off-resonance winds develop a much weaker current system and much less vertical mixing. We found that the effects of wave forcing or Lang- muir cells were mostly con®ned to the initial stages of boundary layer growth. With strong wave forcing, vertical momentum transport was more rapid, causing entrainment mixing to start earlier than in the weak wave cases. Overall, the effects of surface wave forcing were minor in comparison to the impact of resonance for the cases considered here. Through our simulations, we were able to quantify the energy pathways shown in the schematic represen- tation in Fig. 1. Wind stress forcing initially generates an upper ocean inertial current that immediately begins to transfer energy into turbulence via the shear production and the Stokes term in the turbulence kinetic energy equation. Most of this energy is dissipated through a balance between the shear production, subgrid dissipation rate, and buoyant production term. This balance prevents the turbulence kinetic energy from growing signi®cantly so that most of the integrated growth in FIG. 21. Horizontally averaged vertical mixing coef®cient derived turbulence energy is through vertical expansion of the from the coarse resolution, resonant weak wave case using the re- OBL. After ϳ6 h, mixing in both the resonant and off- solved heat ¯ux and average temperature gradient between hours 14± 18, along with the equivalent mixing coef®cient from the KPP model. resonance cases reaches the top of the seasonal thermocline and begins entraining cooler thermocline water into the OBL. In the resonant case, the buoyancy ¯ux within ϳ30% of the LES prediction. The shape of the associated with entrainment is signi®cant, amounting to pro®le is somewhat different in the LES results, with about 7% of the input wind energy (this percentage slightly larger values below ϳ60 m depth and smaller should not be confused with previous mixing ef®ciency estimates that considered the wind energy at 10 m; here, Kh in the main entrainment zone between 30 and 50 m. In general, however, the curves both have roughly linear input wind energy is that imparted by the wind stress pro®les in the entrainment zone, and the differences be- and waves). The off-resonance case develops a much tween the two estimates may have more to do with the weaker current system as the wind acts to remove energy timing of mixing than with errors in the pro®le shape. from the ¯ow after ϳ¼ to ½ inertial cycle, which limits We note, however, that qualitative comparison between the OBL deepening to about 50 m. In contrast, turbulent the LES temperature time depth plots shown in Fig. 3 mixing in the resonant case extends to ϳ100 m depth. and KPP results reported in CL show that KPP is less During the initial period of OBL growth, three dis- diffusive and tends to maintain a sharper gradient of tem- tinctive layers can be identi®ed in the simulation as perature in the thermocline than the LES. The differences shown in Fig. 12. In the upper ϳ25 m, turbulence is shown in Fig. 21 are consistent with these qualitative dominated by vertical motions associated with wave in- differences and may indicate that the parameterized K duced Langmuir cells. Langmuir cells with scales up to several hundred meters are produced in this zone, with pro®le or Ric used in determining OBL depth need adjustment for shear-driven entrainment mixing. smaller scale features having structure that mirrors the largest scale circulations (e.g., strong downward jets located in the large-scale downwelling regions as shown 6. Discussion and conclusions in the schematic). The effects of shear in the Langmuir The effects of resonant wind and wave forcing on the layer are minimized in comparison to the surface wave ocean boundary layer were examined using a large-eddy Stokes drift forcing and the vertical advection term. In simulation turbulence model. Two scenarios were ex- the entrainment zone, shear begins to dominate in the amined, one with constant or off-resonance winds and production of turbulence. Here, the eddy ¯ow ®eld is one with inertially rotating or resonant winds. As a sec- punctuated by strong horizontal injections of energy pro- ondary issue, the distinction between weak and strong duced by overturning eddies. The injection of energy surface wave forcing was also investigated. The evo- from the mean ¯ow into turbulence causes a maximum lution of average temperature and momentum pro®les in the turbulent heat ¯ux in this layer, which is responsible was consistent with previous observations and one-di- for cooling the surface waters and warming the upper mensional model predictions showing that resonant thermocline. Below the turbulent boundary layer lies the

Unauthenticated | Downloaded 10/07/21 06:01 AM UTC 1AUGUST 2000 SKYLLINGSTAD ET AL. 1889 internal wave zone. In this region, motions are produced wind forcing with high temporal resolution as originally by propagating internal waves and do not lead to sig- suggested by Large et al. (1986). Wind variations on ni®cant mixing. the scale of hours can make huge differences in the Analysis of the PE budget demonstrates how local strength of upper ocean currents and subsequent mixing changes in the PE are affected by eddy transport, with from shear production of turbulence. Also critical is the most of the local changes in PE attributed to water ex- vertical resolution in one-dimensional representations of change that has no net impact on the integrated PE or OBL mixing. Estimating the shear is crucial for deter- TKE. Locally, however, the change in PE is much greater mining the strength of turbulence production, and is and represents the vertical exchange of water heated during directly related to the amount of vertical smoothing im- the summer season with cold thermocline water estab- plicit in coarse vertical resolution models. lished during the previous winter. Overall, our results are consistent with the one-dimensional modeling experiments Acknowledgments. We are pleased to acknowledge performed by Crawford and Large (1996) using identical the supercomputer time provided by the National Center forcing and initial conditions. We found that bulk Rich- for Atmospheric Research, which is funded by the Na- ardson number criteria were consistent with values used tional Science Foundation. We also thank the two anon- in the KPP one-dimensional model, demonstrating that ymous reviewers for very useful comments and sug- boundary layer depth is well-represented by nonlocal mea- gestions. This research was supported by the National sures such as the bulk Richardson number. The wind- Science Foundation through Grant OCE-9711862. driven OBL is only slightly stable, so it behaves much like a neutrally forced boundary layer, although strati®- APPENDIX cation leads to a somewhat lower mixing coef®cient. As Model Domain Rotation Fig. 21 shows, the effective Kh from the LES results is smaller than the KPP value throughout the depth of the There are two main reasons rotating ¯uids are dif®cult entrainment layer. However, the depth of the K pro®le in to model in Cartesian coordinates. First, rotating cur- KPP is limited by the bulk Richardson number, which rents can cause ¯ow oscillations that result from the de®nes a shallower mixed layer depth in comparison to model resolution effectively changing as the mean cur- the LES results. One possible modi®cation to the KPP rent direction passes from grid parallel to grid diagonal pro®le would be to use the local gradient Ri to adjust the directions. This causes variable numerical smoothing depending on the ¯ow direction. The second effect of value of Kh and increase the critical Rib to allow for a deeper boundary layer. In future research, we plan to in- rotating currents is produced by misalignment of co- vestigate this possibility by performing a series of exper- herent vortices such as large scale convective rolls. Vor- iments with KPP and the LES model so that the relation- tices are forced to have orientations and scales that ®t the periodic domain rather than a natural scaling. ship between Kh and local stability can be better determined (e.g., Wyngaard and Brost 1984). For resonant ¯ows, these problems can be avoided to Without a surface heat ¯ux, the wind driven OBL is some degree by rotating the domain with the mean ¯ow, essentially a strati®ed, entraining boundary layer, which which for perfect resonance is equivalent to setting f ϭ has few counterparts in other geophysical ¯ow problems. 0.0 sϪ1. However, setting f ϭ 0.0 sϪ1 is not an appropriate Probably the closest analogue turbulent ¯ow is encoun- assumption for cases with a surface wave Stokes drift tered in the nocturnal atmospheric boundary layer (ABL) because of the Stokes drift Coriolis term. The solution when surface strati®cation decouples the free atmosphere is to retain the Stokes drift Coriolis term, as is shown by from the surface. Winds increase above the stable bound- a straightforward analysis. We assume a simpli®ed set of ary layer in response to synoptic pressure gradient and equation for a layer averaged velocity ®eld, can eventually generate turbulence that entrains down- ␶x ␶y ward toward the surface (this is also one of the few cases utsϪ f (␷ ϩ ␷ ) ϭ , ␷ tsϩ f (u ϩ u ) ϭ , (A1) where the ABL does not need to be ¯ipped over for ␳␳ comparison to the OBL!). Stable boundary layers in the where u and ␷ are the average horizontal velocities, us atmosphere can be divided into two classes depending and ␷ s are the Stokes drift components, and ␶ x and ␶ y on the intermittancy of turbulence (Mahrt 1999). With are the surface wind stress components. These equations strongly stable boundary layers, turbulence is very in- can be combined using complex notation, U ϭ u ϩ i␷, termittent with long periods of quiescent ¯ow interrupted Us ϭ us ϩ i␷ s, and ␶ ϭ ␶ x ϩ i␶ y, yielding by bursts of turbulence. Weakly stable boundary layers are characterized by almost continuous turbulence that Ut ϩ if(U ϩ Us) ϭ ␶. (A2) acts to decrease the background strati®cation through en- Applying a wind stress with rotation, ␶ ϭ ␶Ä exp(Ϫift), gulfment of cooled surface air. We ®nd that the latter Ä Ä Us ϭ Us(t) exp(Ϫift), and assuming a solution U ϭ U(t) category is most applicable to the storm-driven OBL, exp(Ϫift) gives with strati®cation being maintained by entrainment of d thermocline water rather than surface cooling in the ABL. UÄÄϭϪifU ϩ ␶Ä , (A3) Our results again emphasize the importance of using dt s

Unauthenticated | Downloaded 10/07/21 06:01 AM UTC 1890 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 showing that the Stokes Coriolis term must be retained torial ocean model using large eddy simulations and observa- when rotating the model domain to accommodate in- tions. J. Phys. Oceanogr., 29, 449±464. , J. C. McWilliams, and P. P. Niiler, 1986: Upper ocean thermal ertial resonance. response to strong autumnal forcing of the northeast Paci®c. J. Phys. Oceanogr., 16, 1524±1550. , , and S. C. Doney, 1994: Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization. REFERENCES Rev. Geophys., 32, 363±403. Lin, C.-L., J. C. McWilliams, C.-H. Moeng, and P. P. Sullivan, 1996: Antonia, R. A., and M. R. Raupach, 1993: Spectral scaling in a high Coherent structures and dynamics in a neutrally strati®ed plan- Reynolds number laboratory boundary layer. Bound.-Layer Me- etary boundary layer ¯ow. Phys. Fluids, 8, 2626±2639. teor., 65, 289±306. Lombardo, C. P., and M. C. Gregg, 1989: Similarity scaling of viscous Cortesi, A. B., B. L. Smith, G. Yadigaroglu, and S. Banerjee, 1999: and thermal dissipation in a convecting surface boundary layer. Numerical investigation of the entrainment and mixing processes J. Geophys. Res., 94, 6273±6284. in neutral and stably-strati®ed mixing layers. Phys. Fluids, 11, Mahrt, L., 1999: Strati®ed atmospheric boundary layers. Bound.-Lay- 162±185. er Meteor., 90, 375±396. Craik, A. D. D., and S. Leibovich, 1976: A rational model for Lang- Martin, P. J., 1985: Simulation of the ocean mixed layer at OWS muir circulations. J. Fluid Mech., 73, 401±426. November and Papa with several models. J. Geophys. Res., 90, Crawford, G. B., and W. G. Large, 1996: A numerical investigation 903±916. of resonant inertial response of the ocean to wind forcing. J. McWilliams, J. C., P. P. Sullivan, and C.-H. Moeng, 1997: Langmuir Phys. Oceanogr., 26, 873±891. turbulence in the ocean. J. Fluid Mech., 334, 1±30. D'Asaro, E. A., and G. T. Dairiki, 1997: Turbulence intensity mea- Mellor, G. L., and T. Yamada, 1982: Development of a turbulence surements in a wind-driven mixed layer. J. Phys. Oceanogr., 27, closure model for geophysical ¯uid problems. Rev. Geophys. 2009±2022. Space Phys., 20, 851±875. Davis, R. E., R. deSzoeke, and P. P. Niiler, 1981: Variability in the MeÂtais, O., and M. Lesieur, 1992: Spectral large-eddy simulations of upper ocean during MILE. Part II: Modeling the mixed layer isotropic and stably-strati®ed turbulence. J. Fluid Mech., 239, response. Deep-Sea Res., 28A, 1453±1475. 157±194. Deardorff, J. W., 1980: Stratocumulus-capped mixed layers derived Miles, J. W., 1961: On the stability of heterogeneous shear ¯ows. J. from a three-dimensional model. Bound.-Layer Meteor., 18, Fluid Mech., 10, 496±508. 495±527. Moum, J. N., 1996: Energy-containing scales of turbulence in the Denbo, D. W., and E. D. Skyllingstad, 1996: An ocean large-eddy ocean thermocline. J. Geophys. Res., 101, 14 095±14 109. simulation model with application to deep convection in the Plueddemann, A. J., J. A. Smith, D. M. Farmer, R. A. Weller, W. R. Greenland Sea. J. Geophys. Res., 101, 1095±1110. Crawford, R. Pinkel, S. Vagle, and A. Gnanadesikan, 1996: Ducros, F., P. Comte, and M. Lesieur, 1996: Large-eddy simulation Structure and variability of Langmuir circulation during the Sur- of transition to turbulence in a boundary layer developing spa- face Waves Processes Program. J. Geophys. Res., 101, 3525± tially over a ¯at plate. J. Fluid Mech., 326, 1±36. 3543. Durran, D. R., 1991: The third-order Adams±Bashforth method: An Price, J. F., R. A. Weller, and R. Pinkel, 1986: Diurnal cycling: Ob- attractive alternative to leapfrog time differencing. Mon. Wea. servations and models of the upper ocean response to diurnal Rev., 119, 702±720. heating, cooling, and wind mixing. J. Geophys. Res., 91, 8411± Gargett, A. E., and J. N. Moum, 1995: Mixing ef®ciencies in turbulent 8427. tidal fronts: Results from direct and indirect measurements of Shay, T. J., and M. C. Gregg, 1986: Convectively driven turbulent density ¯ux. J. Phys. Oceanogr., 25, 2583±2608. mixing in the upper ocean. J. Phys. Oceanogr., 16, 1777±1798. Howard, L. N., 1961: Note on a paper of John W. Miles. J. Fluid Skyllingstad, E. D., and D. W. Denbo, 1995: An ocean large-eddy Mech., 10, 509±512. simulation of Langmuir circulations and convection in the surface mixed layer. J. Geophys. Res., 100, 8501±8522. Kader, G. G., and A. M. Yaglom, 1991: Spectra and correlation func- , W. D. Smyth, J. N. Moum, and H. Wijesekera, 1999: Upper- tions of surface layer atmospheric turbulence in unstable thermal ocean turbulence during a westerly wind burst: A comparison strati®cation. Turbulence and Coherent Structures, O. MeÂtais of large-eddy simulation results and microstructure measure- and M. Lesieur, Eds., Kluwer Academic, 388±412. ments. J. Phys. Oceanogr., 29, 5±28. Katul, G. G., and M. B. Parlange, 1995: The spatial structure of Smith, J. A., 1998: Evolution of Langmuir circulation during a storm. turbulence at production wavenumbers using orthonormal wave- J. Geophys. Res., 103, 12 649±12 668. lets. Bound.-Layer Meteor., 75, 81±108. Smyth, W. D., and J. N. Moum, 2000a: Length scales of turbulence Khanna, S., and J. G. Brasseur, 1998: Three-dimensional buoyancy- in stably strati®ed mixing layers. Phys. Fluids, 12, 1327±1342. and shear-induced local structure of the atmospheric boundary , , 2000b: Anistropy of turbulence in stably strati®ed mixing layer. J. Atmos. Sci., 55, 710±743. layers. Phys. Fluids, 12, 1343±1362. Klemp, J. B., and D. R. Durran, 1983: An upper boundary condition Sullivan, P. P., C.-H. Moeng, B. Stevens, D. H. Lenschow, and S. D. permitting internal gravity wave radiation in numerical meso- Mayor, 1998: Structure of the entrainment zone capping the con- scale models. Mon. Wea. Rev., 111, 430±445. vective atmospheric boundary layer. J. Atmos. Sci., 55, 3042± Kolmogorov, A. N., 1941: The local structure of turbulence in in- 3064. compressible viscous ¯uid for very large Reynolds numbers. Sun, C., W. D. Smyth, and J. N. Moum, 1998: Dynamic instability Dokl. Akad. Nauk SSSR, 30, 301±304. of strati®ed shear ¯ow in the upper equatorial Paci®c. J. Geo- Kraus, E. B., and J. S. Turner, 1981: A one-dimensional model of phys. Res., 103, 10 323±10 337. the seasonal thermocline. II: The general theory and its conse- Wang, D., J. C. McWilliams, and W. G. Large, 1998: Large-eddy quences. Tellus, 19, 98±105. simulation of the diurnal cycle of deep equatorial turbulence. J. Krauss, W., 1981: The erosion of a thermocline. J. Phys. Oceanogr., Phys. Oceanogr., 28, 129±148. 11, 415±433. Werne, J., and D. C. Fritts, 1999: Strati®ed shear turbulence: Evo- Large, W. G., and G. B. Crawford, 1995: Observations and simula- lution and statistics. Geophys. Res. Lett., 26, 439±442. tions of upper ocean response to wind events during the Ocean Wyngaard, J. C., and R. A. Brost, 1984: Top-down and bottom-up Storms Experiment. J. Phys. Oceanogr., 25, 2831±2852. diffusion in the convective boundary layer. J. Atmos. Sci., 41, , and P. R. Gent, 1999: Validation of vertical mixing in an equa- 102±112.

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