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 flow in a box, measuring ∼100–300 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 45°N), 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 profile. 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 ∼7% 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 ∼¼ 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 profiles from the model were consistent with one-dimensional model output using the K-profile parameterization.

1. Introduction

Atmospheric forcing of the upper-ocean boundary layer (OBL) plays a fundamental role in regulating the sea surface temperature of the world’s oceans. The most direct influence of the atmosphere is through surface fluxes of latent and sensible heat. During conditions of strong surface heat loss and weak wind forcing, the OBL behaves much like a well-mixed, convective boundary layer with turbulent fluxes that are in agreement with Monin–Obukhov similarity theory near the surface and a mixed layer structure that scales with the surface buoyancy flux (Shay and Gregg 1986; Lombardo and Gregg 1989). Often, however, upper ocean mixing is driven by wind and surface wave forcing, with entrainment flux at the mixed layer base dominating the OBL heat budget and surface heat flux having a secondary role. As a result, the OBL behaves more like a stratified boundary layer and cannot be easily described through classic boundary layer theory (Mahrt 1999).

The largest changes in the OBL momentum budget are caused by the passage of strong synoptic scale storms. Because of these storms, the direction of the wind stress can vary rapidly and force currents that have a strong inertial oscillation. If the surface winds vary with a similar frequency and direction, even for only a few hours, then these currents will amplify through resonance, leading to significant vertical shear in the OBL. A contrasting situation, when winds are off-resonant, can lead to a reduction in OBL currents and a decrease in vertical mixing relative to wind magnitude. Episodes of enhanced vertical mixing in the thermocline have been linked to increased vertical shear (e.g., Large et al. 1986) and can have a dominant effect on the ocean SST in the fall season as cold thermocline water is mixed with the relatively warm surface waters created by summer heating. Heat flux estimates based on observed changes in the OBL temperature have shown values as high as 7000 W m−2 during an 8-h period (Krauss 1981), with 1000 W m−2 heat fluxes observed during some Pacific storm events (Large and Crawford 1995).

Build up of momentum in the OBL through inertial resonance has been well documented through observations and one-dimensional modeling studies (Crawford and Large 1996, hereafter CL; Large and Crawford 1995). What has not been thoroughly explained is how upper ocean currents create and interact with turbulence and the stratified pycnocline at the boundary layer base. A schematic showing the pathways that wind energy follows in driving inertial currents and turbulence is shown in Fig. 1. In the OBL, energy provided by the wind is partitioned between the mean current and turbulence generated by shear production and wave–current interaction (Stokes production). Some fraction of this energy is removed through turbulent dissipation, ε. Another portion goes into vertical mixing of thermocline water, thereby reducing the OBL temperature and increasing the potential energy. Additional turbulent and internal wave energy is generated at the mixed layer base through the shear of the mean inertial current. Energy from this process is also used to mix thermocline water and is dissipated through friction and internal wave propagation. Our objectives here are to understand in detail the physics of these processes and to quantify their relative importance in determining the OBL response to strong wind forcing.

Fig. 1.

Schematic showing main pathways for energy transferred to the upper ocean via the surface wind stress.

Fig. 1.

Schematic showing main pathways for energy transferred to the upper ocean via the surface wind stress.

Accurate prediction of kinetic and potential energy changes in the OBL is a critical step in modeling changes in the SST. For example, if surface waves cause increased transfer of mean current kinetic energy into turbulent kinetic energy through Langmuir circulation, then the sea state may cause more OBL cooling because of increased entrainment at the mixed layer base. This 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 fluxes 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 fluxes (e.g., Gargett and Moum 1995; Moum 1996; D’Asaro and Dairiki 1997). The majority of turbulence observations in the midlatitude ocean are limited to microstructure data collected during weak to moderate storm conditions. These data can provide estimates of the turbulence strength in a given region, but they cannot yield specific details on how turbulence was forced or what instability process created the turbulence.

Our approach for studying turbulence in the upper ocean is the application of large-eddy simulation (LES) models. LES models have been used extensively in the atmosphere to examine the structure and dynamics of the boundary layer with reasonable success (see, e.g., Khanna and Brasseur 1998). Recent oceanographic applications of LES models include studies of Langmuir circulation (Skyllingstad and Denbo 1995; McWilliams et al. 1997) diurnal mixing in the central tropical Pacific (Wang et al. 1998), and turbulence in the western Pacific warm pool (Skyllingstad et al. 1999). Here, we apply an ocean LES model to the problem of turbulence generation by wind-forced inertial currents. Our goal is to quantify each of the energy pathways shown in Fig. 1 and to link these pathways to specific turbulent processes through flow visualization and eddy statistics.

Results from LES provide a detailed four-dimensional dataset that can be used to reveal how mixing processes affect the average structure of the OBL and to determine if existing parameterizations represent mixing processes accurately. One-dimensional models of wind driven mixing (e.g., Large et al. 1994; Mellor and Yamada 1982; Price et al. 1986) do not explicitly model turbulent motions, but rely on empirical relationships between mixing parameters, such as entrainment rate or eddy viscosity, and mean ocean properties of buoyancy and shear. Typically, these mixing models determine shear-generated mixing rates using some form of the Richardson number,

 
formula

where

 
formula

is the square of the Brunt–Väisälä frequency, σθ is the potential density, g is gravity, ρO is a constant reference density, Sh2 = (∂u/∂z)2 + (∂υ/∂z)2, u and υ are the horizontal currents in the x and y directions, respectively, and z is the vertical coordinate. The condition Ri < 0.25 somewhere in the flow is necessary for linear, normal mode instability (Miles 1961; Howard 1961). However, boundary layer entrainment usually occurs at a somewhat higher measured average Ri, between 0.3 and 0.5, because the background shear is continuously forced.

Another aspect of the OBL that has not been addressed in mixed layer parameterizations concerns the role of surface wave effects and Langmuir circulation. The effects of Langmuir circulation are thought to dominate upper ocean turbulence in conditions of strong winds. However, only minimal observations exist that directly connect wind and waves with mixed layer growth (e.g., Smith 1998). Our simulations provide a means of separating the individual effects of shear driven OBL growth from entrainment caused by Langmuir cells. We find that for typical boundary layer depths (∼30 to 40 m) the Langmuir circulation has a relatively small effect on average boundary layer growth rate, but has a significant influence on when turbulence begins to erode the thermocline. As a result, boundary layer growth in cases with weak waves tends to lag behind the strong wave cases.

The paper is organized with the main goal of understanding the pathway wind energy takes in causing changes in the average OBL temperature structure. Following a brief discussion of the model and initial conditions in section 2, we begin our analysis by examining the evolution of horizontally averaged OBL properties in section 3. We relate changes in the average properties to specific terms in the turbulent and mean kinetic energy budgets, connecting the average OBL structure with turbulent processes. Section 4 presents a combination of flow visualization and statistics to examine how these organized turbulent circulations lead to mixed layer entrainment and strong vertical heat flux. Section 5 discusses the usefulness of mixing parameters derived from the LES results, such as gradient and bulk Richardson numbers, and how they might relate to similar indicators in a typical one-dimensional parameterization [specifically, the K-profile parameterization (KPP) of Large et al. (1994)]. Conclusions are presented in section 6.

2. The numerical model

We apply the LES model described in Skyllingstad et al. (1999) and Denbo and Skyllingstad (1996). This model is based on the time dependent Navier–Stokes equations with subgrid turbulence closure provided by the filtered structure function (FSF) approach of Ducros et al. (1996). Surface wave interaction is modeled using the Stokes vortex forcing term from Craik and Leibovich (1976). Energy is removed from the resolved turbulent flow field via the eddy viscosity in a manner consistent with the assumption that the spectrum of kinetic energy follows a −5/3 power law for scales below the model resolution (Ducros et al. 1996). In areas of the flow where the resolved spectra does not follow a −5/3 power law, the method may introduce errors by either removing too much energy or by not removing enough. In practice we find that the scheme yields an accurate solution for most of the turbulent boundary layer, but tends to underpredict turbulence kinetic energy in regions of strong stratification.

Eddy viscosity in the FSF approach is determined by first filtering the velocity fields using a three-pass Laplacian filter and then computing the velocity structure function from the filtered fields (see Ducros et al. 1996 for details). In the present application, we applied the four-neighbor formulation that uses only the horizontal differences in the horizontal velocity components to compute the structure function. This formulation is considered the preferred method by Ducros et al. (1996). We found that the FSF scheme avoided many of the problems encountered with methods that use the entire velocity field to infer eddy viscosity. For example, the original structure function model of Metais and Lesieur (1992) performed well in the mixed layer but produced excessive mixing in the upper thermocline because of increased diffusion associated with large-scale internal waves. Similar behavior was noted when using the Deardorff (1980) subgrid-scale parameterization. The FSF method, however, resulted in lower eddy viscosity values in the upper thermocline because the method is insensitive to the effects of eddy motions much larger than 4Δx (i.e., the internal wave velocity fields). In addition, we found that the FSF method develops both shear and turbulent kinetic energy profiles that are in good agreement with Monin–Obukhov similarity theory near the surface. This was an unexpected result that will be investigated in a separate paper on applying the FSF scheme in geophysical flow models.

The bulk of the experiments are performed on a 160 × 160 horizontal mesh with 64 vertical grid points and uniform spacing of 2.0 m. We also present results from a single experiment to examine small-scale circulations that uses a 256 × 256 horizontal mesh with 128 vertical grid points and 0.75-m resolution. The domain sizes considered with the LES model are by necessity much smaller than the Rossby radius of deformation. Therefore, mesoscale adjustments in the flow are not possible and are assumed to be less important than the locally accelerated flow. This assumption could introduce errors that affect the character of turbulence; for example, large-scale pressure gradients could cause variations in the vertical shear of the mean current leading to Kelvin–Helmholtz instability.

Boundary conditions are periodic in the horizontal direction with a rigid lid at the model top and an open boundary condition based on Klemp and Durran (1983) at the model bottom. Time step length for the integrations ranged from 0.5 to 2 seconds depending on the maximum mean current velocity and the Courant–Friedrichs–Lewy limit imposed by the third-order Adams–Bashforth method (see Durran 1991). Simulations typically required 2 hours of wall clock time per simulated hour on a Hewlett Packard Exemplar computer running in a 32-node subcomplex.

Experiments were designed following the idealized conditions given by CL that represent typical fall conditions in the North Pacific with an initial mixed layer depth of hm and a seasonal thermocline depth of ht. Following CL temperature profiles were initialized as

 
formula

using parameters representing the Ocean Storms dataset (Tm = 12°C, hm = 35 m, ht = 50 m, ΔTt = 4°C, Δht = 15 m). In all simulations, salinity was held constant at 35.2 psu. Wind forcing for the experiments was prescribed using

 
formula

where ω* = ωf represents constant angular rotation rate relative to an inertially rotating reference frame, ω is the angular rotation rate relative to the earth’s surface, A is the maximum wind stress (set to 1.4 N m−2), and ts = 24 h is the storm duration. We consider two specific cases of wind rotation rates: ω* = 0 s−1 [i.e., ω = f, or inertially rotating (resonant) winds], and ω* = −1.01 × 10−4 s−1 (i.e., a case of off-resonant winds; for a latitude of 45°N, this scenario corresponds to ω = 0, or a locally steady wind direction). The inertially resonant case is similar to the passage of an occluding midlatitude cyclone. Initially the winds are weak northeasterly. As the storm approaches they increase and veer to the east, southeast, and south. The storm passes causing the winds to shift to the west and weaken.

Cases with the Ocean Storms parameters were chosen because they represent the basic case presented in CL. However, because of the strong currents and shear generated in the resonant storm forcing case, numerical effects have a large impact in the simulation at the storm peak. In particular, rapid advection over the grid resulted in considerable smoothing and reduction of small-scale turbulent energy. In addition, rotation of the flow field with periodic boundaries caused disruption of large-scale coherent motions so that turbulence was concentrated in a middle range of scales. To overcome these problems in the resonant cases, we used two coordinate transformations of the model grid. First, we transformed the LES domain to a rotating reference frame so that Coriolis terms are essentially set to zero (except for the Stokes term, see the appendix). As pointed out by CL, perfectly resonant winds have an effect similar to winds with constant direction at the equator. We next performed a transformation by subtracting out the average surface value of the currents, thereby moving the domain along with the surface flow field as was done in Skyllingstad et al. (1999). The slight inaccuracy introduced by these coordinate transforms is justified by the reduction in advective smoothing and other numerical artifacts (see appendix). For the off-resonance cases, changing the grid reference does not reduce the effects of rotation because the flow has a different mean velocity direction at every depth. Therefore, off-resonance cases were only transformed by following the average surface current.

Two surface wave scenarios were applied in the simulations using a vertical Stokes drift profile for a monochromatic wave field,

 
formula

where k = 2π/λ and hs is the wave height or twice the wave amplitude. Conditions for the two scenarios, designated “strong” and “weak,” were defined with hs = 6, λ = 130 and hs = 2, λ = 30, respectively. Strong waves in our experiments denote a deeper profile of Stokes drift velocity in comparison with the weak wave scenario. We chose these two simple scenarios to make an initial assessment of surface wave forcing relative to inertially resonant currents, even though in actual ocean storms it is very unlikely that these conditions would exist for as long as the simulations (∼24 hours). For example, in the rotating, resonant wind case, the motion of the storm, waves and wind direction would have to combine such that the wind and wave motion is always in the same direction as the surface water velocity. In future work, the idealized results presented here will be extended and validated using more realistic surface forcing fields.

Experiments were performed as summarized in Table 1. Latitude for all experiments was set to 45°N, representing midlatitude conditions. The resonant wind cases correspond to winds that rotate in an inertial sense relative to a point located at 45°N, whereas the off-resonance cases represent a fixed wind direction. Simulations were performed for 24 hours, which encompassed the main peak of storm winds as defined by (3) and the major thermocline mixing event.

Table 1.

Experimental parameters governing the idealized wind and wave forcing.

Experimental parameters governing the idealized wind and wave forcing.
Experimental parameters governing the idealized wind and wave forcing.

3. Evolution of the mean profiles

Observations of strong mixing events in the open ocean often consist of time series of temperature as a function of depth at a fixed point, for example, as analyzed by Large and Crawford (1995). The importance of resonant wind interaction is inferred from these data by examining the end effects of turbulent mixing on the mean temperature and current structure of the upper ocean. Our analysis will focus first on this aspect of inertial resonance by examining the evolution of the horizontally averaged velocity and temperature for each of the experiments. We also examine the dependence of mean flow evolution on the strength of the surface wave field.

a. Horizontally averaged fields

Time-depth sections of the horizontally averaged velocity components, u and υ, for each of the four forcing scenarios demonstrate how resonance affects the upper-ocean momentum structure (Fig. 2). In both resonant cases (Figs. 2a,c,e,g), the u and υ increase dramatically in response to the surface wind stress. The faster moving average currents produced by resonant forcing create strong current shear that coincides with the mixed layer deepening. In contrast, the off-resonance velocity fields (Figs. 2b,d,f,h) form a weak, rotating current system, with momentum being removed by the wind stress every half inertial cycle as the currents oppose the surface wind. Current shear at the mixed layer base in these cases is greatly reduced in comparison to the resonant cases.

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.

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.

Horizontally averaged temperature structure for each of the four forcing scenarios are presented in Fig. 3. Inertially resonant wind forcing causes markedly stronger mixing in comparison to the constant wind forcing in the off-resonance cases. For all cases, the vertical temperature structure changes only slightly before about hour 5 as the wind stress increases. After this time, the resonant cases begin showing stronger vertical mixing of the thermocline that accelerates rapidly at about hour 8 with the thermocline expanding in vertical extent from 15 m to ∼50 m by the end of the simulation. In contrast, the off-resonance cases show only limited changes in the thermocline vertical extent, expanding from 15 m to ∼25 m. We note that the depth of the density mixed layer depth does not change significantly in the simulations (using Δσt = 0.01 as a measure), in agreement with CL and Large and Crawford (1995).

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 defined in the text is plotted as a dark solid line. Also shown is the sea surface temperature for all four cases (e).

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 defined in the text is plotted as a dark solid line. Also shown is the sea surface temperature for all four cases (e).

Surface wave effects are of secondary importance compared with wind resonance in these cases. The effect of wave forcing is most noticeable in the near-surface temperature and momentum, and in the timing of thermocline changes. In both resonant and off-resonance scenarios, wind forcing creates near-uniform temperature and momentum fields down to ∼15 m with weak waves and ∼25 m for strong waves. In the off-resonance strong wave case, surface waves tend to reduce the peak velocities; however the depth of significant mixing appears greater as shown by the reduced sea surface temperature. In general, stronger waves cause relatively greater mixing near the surface, leading to decreased vertical gradients in momentum and temperature. The evolution of the average surface temperature for each of the cases (Fig. 3e) further demonstrates the relatively minor role that wave forcing has in controlling the thermal structure, especially when compared with changes forced by resonance. Overall, the behavior of the OBL as shown in Figs. 2, 3 is consistent with observations of wind forced mixing and the one-dimensional model results presented in CL.

b. Momentum, heat, and energy budgets

In this section, we analyze the horizontally averaged budget equations for energy and momentum to demonstrate how increased momentum produced by wind resonance acts to change the OBL temperature structure. Because wave effects were found to have a minimal effect in comparison to resonant wind forcing, the remainder of the analysis will concentrate on results from the weak wave scenario. Horizontally averaged momentum is governed by

 
formula

where U and V are the horizontally averaged velocity components in the zonal direction, x, and meridional direction, y, respectively; u and υ are the corresponding total velocity components; subscript s represents the Stokes drift velocity components; primes denote the deviation from the horizontal average; f is the Coriolis parameter; and Km is the momentum subgrid scale diffusion. In the analysis of (5), we focus on two dominant terms on the right-hand side of each equation. The first is the stress divergence, which is the sum of the resolved and subgrid-scale (SGS) terms in square brackets. The second is Coriolis force acting on the mean current, that is (fV, −fU). These are shown as functions of depth and time in Fig. 4.

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.

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.

As shown by Figs. 2 and 4, flux divergence in the resonant case is closely aligned with the mean momentum direction throughout the simulation time period. The Coriolis term in the resonant case (Figs. 4b,d) causes the mean currents to rotate at the same rate as the input wind stress and vertical momentum flux. In the off-resonance case, zonal flux divergence does not change direction, but increases and decreases in response to the surface wind forcing (Fig. 4e). As a consequence, rotation through the Coriolis term causes the mixed layer current system to oppose the wind stress, reducing the transfer of momentum from the wind into the ocean. The evolution of the meridional accelerations for the off-resonance case are consistent with this picture, showing the decrease and then increase in the meridional flow as the currents rotate inertially (Figs. 2d,h). Turbulent transport of meridional momentum tends to redistribute the momentum vertically, removing negative momentum from the near surface layer and depositing it near the OBL base (Fig. 4g).

The horizontally averaged heat budget is described through a turbulent transport equation,

 
formula

where w is the vertical velocity, T is the temperature, Kh is the subgrid diffusion of heat, overbars denote horizontal averages, and primes denote perturbations from the horizontal average. Plots of the resolved vertical heat flux, ρCpwT, shown in Fig. 5, demonstrate the impact resonant wind forcing can have on the OBL heat budget. Peak values are about −7000 W m−2 in the resonant case and −2000 W m−2 during off resonance and represent the downward transport of heat stored during the summer in the OBL. These values are similar to estimates reported by Large and Crawford (1995), Large et al. (1986), and Krauss (1981) for storm forced vertical heat fluxes, although the resonant case is at the upper end of the observed estimates. This is understandable given that perfect resonance, as assumed in the idealized simulations, represents an upper bound on the transport of momentum from the wind to the ocean, and therefore the maximum case for shear production of turbulence.

Fig. 5.

Horizontally averaged resolved heat flux from the (a) resonant and (b) off-resonance cases.

Fig. 5.

Horizontally averaged resolved heat flux from the (a) resonant and (b) off-resonance cases.

Analysis of the momentum and heat budgets shows that vertical turbulent transport is the dominant mechanism for changing the average properties of the OBL. To further understand how the average momentum fields affect turbulent production, we next examine the turbulent kinetic energy defined as

 
formula

where the index i denotes the x, y, and z directions, respectively. Turbulent kinetic energy is strongly affected by wind resonance as shown by plotting TKE as a function of time (Fig. 6). With off-resonance winds, TKE follows the wind stress forcing, which increases until about hour 12 and then declines through the remainder of the simulation. In contrast, the resonant case has a much different behavior with TKE increasing steadily over time and deepening with the growing mixed layer. TKE increases until about hour 13 (after the maximum winds) and then gradually falls off throughout the rest of the simulation. In the resonant case, TKE continues to increase after the peak wind because the inertial currents are still gaining strength from the wind stress. Eventually, the production of TKE decreases when the momentum input from the wind cannot maintain strong shear and corresponding shear production of turbulence, as is shown below in the TKE budget analysis.

Fig. 6.

Time–depth section of the specific turbulent kinetic energy for (a) resonant and (b) off-resonance cases.

Fig. 6.

Time–depth section of the specific turbulent kinetic energy for (a) resonant and (b) off-resonance cases.

The budget of TKE is defined using

 
formula

where

 
formula

ρ is the density, g is gravity, q is the subgrid scale turbulent kinetic energy, and P is the pressure (see Skyllingstad et al. 1999). The first term (I) is the shear production, which accounts for the generation of turbulence by instabilities in the mean shear flow. The next term (II) represents the transfer of energy between TKE and potential energy through the buoyant production term. This term represents energy lost or gained by turbulence in doing work against gravity, lifting cold water or forcing warm water downward. Following the buoyant production term is the subgrid dissipation term (III), which accounts for the loss of kinetic energy via friction. The turbulent and subgrid transport terms (IV) redistribute turbulent energy vertically, but do not change the integrated energy for a closed system. The budget is closed with the Stokes production and pressure transport terms (V). Individual terms in (7) are evaluated by multiplying the momentum budget terms by the appropriate perturbation velocity fields before time differencing. This procedure introduces small estimation errors because the third order Adams–Bashforth time differencing scheme smooths the physical mode of the equations as ui + Ui approaches Δxt. However, this error is minor; the budget residual is typically much smaller than the significant terms.

Factors governing the evolution of TKE are revealed by plotting significant terms from (7) as functions of time and depth as shown in Fig. 7. Each case displays the same basic pattern for the budget terms, with the Stokes drift and dissipation rate having maximum magnitude near the surface, and the buoyancy and shear terms peaking near the bottom of the OBL along with a secondary peak in the dissipation rate. The overwhelming difference between the resonant and off-resonance cases is the strength of the shear production term and corresponding dissipation rate, which are both about four times larger in the resonant case. The stronger turbulence produced in the resonant case directly affects the depth of the boundary layer by forcing entrainment at the boundary layer base as indicated by the buoyancy production term. This causes the maximum of shear production to descend as a function of time.

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.

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.

The shear production term represents the exchange of energy between the mean current and the turbulence. As such, it is an important link between the average momentum produced by the wind, and vertical mixing forced by turbulent fluxes. The dominance of the shear production term, particularly in the resonant case, shows that wind energy entering the system through the mean current is converted to turbulence, which leads to boundary layer growth and OBL cooling through the vertical transport of relatively cold thermocline water. A key component of both the shear production term and the average momentum budget is the resolved eddy momentum flux shown in Fig. 8. Although the momentum flux magnitude is comparable for both resonant and off-resonance cases, in the resonant case, which has greater mean shear (see Fig. 2), momentum transport leads to a stronger growth in turbulence through shear production. This physical regime is characteristic of entrainment via shear instability (e.g., Sun et al. 1998; Smyth and Moum 2000a,b; Werne and Fritts 1999; Cortesi et al. 1999). Shear and stratification are continually reduced by turbulent mixing, so the gradient Richardson number (1) tends to increase. In unforced flows, instability eventually ceases and turbulence decays. In this decaying state, Ri approaches an asymptotic value of approximately 0.5 (Smyth and Moum 2000a). In the present case, however, shear is continually reinforced by the inertial response to the surface forcing. Shear and stratification are thereby maintained in a quasi-equilibrium state characterized by Ri close to the critical value 0.25 (Miles 1961). Stability analyses of the mean currents reveal weakly unstable modes with wavelengths of a few hundred meters (methodological details are given in Sun et al. 1998). This confirms that our horizontal domain size is sufficient to accommodate the largest dynamically important scales of motion.

Fig. 8.

Time–depth cross section of average turbulent momentum fluxes: (a) uw for the resonant case, (b) uw for the off-resonant case, (c) υw for the resonant case, and (d) υw for the off-resonant case.

Fig. 8.

Time–depth cross section of average turbulent momentum fluxes: (a) uw for the resonant case, (b) uw for the off-resonant case, (c) υw for the resonant case, and (d) υw for the off-resonant case.

The final component in the energy budget of the OBL is the conversion of TKE into potential energy, PE, which is governed by the horizontally averaged PE budget,

 
formula

where PE = gρz/ρ0, ρ is the density, Km and Kh are the subgrid-scale eddy viscosity and scalar diffusion coefficients. The PE budget begins with a group of three transport terms, which redistribute potential energy vertically but do not change the integrated energy for a closed system. The next term represents the transfer of energy between TKE and PE through the buoyant production term and is of opposite sign from the buoyancy production term in the TKE budget. The final term in the PE budget accounts for changes in PE caused by subgrid scale mixing. Estimation of PE is performed by examining the change in PE after the advection step and then again after the subgrid diffusion. Thus, changes in PE resulting from numerical smoothing during advection are implicitly included in the advective transport term. The buoyant production term contribution is calculated explicitly.

Inspection of the vertically averaged PE budget terms for the resonant case shows that buoyant production is responsible for most of the change in PE (Fig. 9). The off-resonance case shows a similar behavior, but with a much weaker amplitude. In general, changes in the averaged PE mirror the strength of the TKE. Smaller contributions to the PE budget come from the subgrid mixing terms and the transport term. In a perfect numerical model, the transport term would be nearly equal to zero because of the periodic model domain and relatively small fluxes at the model bottom. However, large velocity components produced in shear-driven turbulence cause increased smoothing by the scalar advection scheme, which produces an appreciable change in the integrated PE. This term is considerably smaller than the buoyancy production and may be thought of as part of the subgrid-scale mixing since it tends to reduce the amplitude of small-scale flow features.

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.

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.

Vertical profiles of the PE budget terms during the peak of the wind stress (Fig. 10) provide a different view of how PE is redistributed by turbulence in the OBL. In particular, the advective transport term in the budget shows a much higher amplitude with increased PE in the upper portion of the OBL that is almost exactly balanced by decreased PE in the lower OBL and thermocline. This profile represents the turbulent exchange of relatively warm surface water with cold thermocline water, and is ultimately a measure of the change in sea surface temperature, since both the subgrid and buoyancy terms must equal zero at the surface. In simple terms, when cold water is forced upward, it must be replaced by some portion of the warmer water that is being displaced. For the northern Pacific scenario considered here, the relatively large local changes in PE represent the exchange of water warmed during the summer with cool thermocline water that is maintained by horizontal transport and cooling during the previous winter. Thus, wind forced mixing acts as a catalyst in moving heat stored near the surface to deeper water.

Fig. 10.

Time average vertical profiles of the PE budget terms taken from hours 11–12 from the (a) resonant and (b) off-resonance cases.

Fig. 10.

Time average vertical profiles of the PE budget terms taken from hours 11–12 from the (a) resonant and (b) off-resonance cases.

The distinction between the changes in the integrated PE and the local PE are important when considering the response of bulk mixed layer models (e.g., Kraus and Turner 1967; Davis et al. 1981). Bulk layer models typically predict the integrated energy budget so that changes in the potential energy can be related to surface input of wind energy and heat flux. Bulk layer models assume that entrainment adds a portion of thermocline water to the mixed layer without changing the interior thermocline structure. Local changes in the PE below the mixed layer, as shown in Fig. 10, cannot occur because of this assumption. Additional mixing through a Richardson number criterion, for example, as performed in the Price et al. (1986) model, can account for the local PE changes by modeling the wind-generated current shear and associated mixing beneath the mixed layer base.

Our discussion of the energy budget is concluded by examining the primary processes in the energy pathway from surface wind to changes in the PE (Fig. 11). Figure 11 shows the cumulative effect of each of the main sources and sinks of energy including the wind stress term defined as Usfcτ/ρ, where Usfc is the surface current. Focusing on the resonant case first, wind and wave energy ramps up following the prescribed wind stress, with a maximum increase at about hour 14 because of the stronger average surface currents at this time. About 40% of this input energy is lost via turbulence dissipation as momentum is transferred from the near surface throughout the OBL. Most of the shear production is directly cancelled by dissipation and buoyant production so that TKE (not shown) is much smaller than the energy budget terms shown in Fig. 11. At the end of the storm in the resonant cases, Fig. 11 shows that ∼7% of the input wind energy goes into changing PE. In the off-resonance case, the rotation of the currents causes the wind stress term to become an energy sink as the surface currents rotate to a direction opposing the wind stress. This greatly limits the mean current velocity and corresponding shear production of turbulence. Entrainment buoyancy flux at the bottom of the boundary layer (responsible 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.

Fig. 11.

Cumulative vertically integrated input wind energy, total kinetic energy (KE), PE, and turbulence dissipation rate for (a) resonant and (b) off-resonance cases.

Fig. 11.

Cumulative vertically integrated input wind energy, total kinetic energy (KE), PE, and turbulence dissipation rate for (a) resonant and (b) off-resonance cases.

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 flow 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 predicted 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 kinetic energy. For example, it is important to know whether turbulent eddies span the entire mixed layer depth or are small-scale features that only cause localized mixing. In this section, we present visualizations of the turbulent flow fields 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 ∼100 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 ∼5 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 than the budget simulations. Therefore, we initialized this case with horizontally averaged profiles of u, υ, and T from hour 8 of the budget calculation and limited the duration of the experiment to 3 hours. The results described in this section therefore correspond to a “snapshot” of the flow field taken at hour 11 from the budget experiments. Only the resonant, weak wave scenario was considered. Results are presented in an inertially rotated coordinate system where the x axis is aligned with the surface wind stress direction, so that u and υ represent the instantaneous downwind and cross-wind velocity components, respectively.

We begin the discussion with a look at the vertical profiles of horizontal, root-mean-square averages of the velocity components (Fig. 12). To facilitate interpretation, we conceptually divide the water column into three layers, with boundaries at depths 25 m and 60 m. Below 60 m, the flow consists of weak gravity waves. These motions play a minor role in the processes of interest, and will not be discussed in detail. Between 60 m and 25 m, the flow is dominated by downwind motions. This form of anisotropy in the energy-containing scales is typical of shear-driven turbulence (e.g., Smyth and Moum 2000a,b). In this layer, mixing of near-surface water with thermocline water is driven by the shear at the base of the wind-driven surface current. In the upper 25 m, the structure is more complex. Between 10 m and 25 m, we see anisotropy suggestive of Langmuir cells;that is, downwind motion is weak compared with crosswind and vertical motion. As the surface is approached, these motions become primarily crosswind. We also observe an increase in downwind velocity near the surface, indicating a coupling between the wind and the ocean eddy field via convergence of the stress component corresponding to vertical shear of the downwind current.

Fig. 12.

Horizontally averaged root mean square downwind u, cross wind υ, and vertical w velocities at hour 11.

Fig. 12.

Horizontally averaged root mean square downwind u, cross wind υ, and vertical w velocities at hour 11.

a. Flow visualizations

Cross-sections of the vertical velocity fields, horizontal perturbation velocity vectors, and horizontal perturbation temperature provide a closer view of the wind-driven OBL turbulent structure (Fig. 13). At this time, the depth of active mixing extends to ∼60 m, so the cross sections shown in Fig. 13 represent the main areas of turbulent production described in the TKE budget discussion. At 10-m depth, Langmuir circulation is indicated by the organized, linear structures in the flow field. The largest eddies behave much as in previous simulations of Langmuir circulation (Skyllingstad and Denbo 1995; McWilliams et al. 1997), with scales of ∼30 to 50 m, or roughly equal to the mixed layer depth. Regions of strong upwelling tend to correspond with upwind velocity perturbations (e.g., x = 110 m, y = 50 m), whereas regions of strong downwelling tend to correspond with downwind velocity perturbations (e.g., x = 125 m, y = 150 m). These circulations have a relatively long downwind coherent structure, although they are not oriented as in previous studies that show a consistent orientation to the right of the wind (e.g., Smith 1998). Here, the orientation of the circulations appears to be affected by a larger-scale circulation (roughly domain scale) that is noticeable in the near-surface perturbation temperature field (Fig. 13c). Where the large eddies force positive (negative) υ, the small scale vortices are aligned to the right (left) of the large eddies. We also noticed this behavior in the simulation used in the budget calculations and in an additional coarse resolution experiment with horizontal domain size of 720 m (not shown), indicating that the small domain size used in this case was not a determining factor in the Langmuir cell behavior. The large-eddy structure orientation in these simulations probably differs from past studies because of the intensity of the wind forcing and upper ocean currents, although the overall scale is similar to observations of 2–200 m Langmuir cells reported by Smith (1998) and Plueddemann et al. (1996).

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.

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.

Throughout most of the OBL, large-scale currents reflect the form of the Langmuir cells. However the coherence of these eddies deteriorates approaching the bottom of the boundary layer (Figs. 13b,d). Between 20 and 40 m, mean current shear starts to control turbulent processes, causing strong horizontal u perturbations at 40 m (cf. Fig. 12). These horizontal velocity perturbations are associated with large excursions in the temperature field and may define the exchange of water between the OBL and the thermocline. The role of the large eddies in transporting downwind momentum is shown clearly by the w component and horizontal vector plots; areas of downwelling have downwind velocities that are ∼0.4 m s−1 larger than the surrounding water (e.g., between y = 100 and 150 m). These eddies are also the primary mechanism for moving cold water from the bottom of the OBL upward to the surface as shown by the perturbation temperature. Because of the momentum transport, shear is enhanced beneath the downwelling circulations, causing a slight increase in turbulence intensity. Both the w and T′ fields show this effect at 40 m with regions of slightly enhanced small-scale structure centered beneath the downwelling center at y = ∼150 m.

A cross section at x = 120 m (Fig. 14) illustrates the complex form of the large-scale circulations. Dominant features include a cool upwelling zone near y = 60 m, and a more diffuse, warm downwelling region at y = 120 m. Turbulent processes associated with these downwelling and upwelling zones are visible in cross sections at y = 120 m and y = 60 m, respectively (Fig. 15). We focus first on the downwelling region. In the entrainment zone at z = ∼45 m, u velocity shear has a strong influence on the horizontal eddy circulation, for example, as shown by the injection of positive u momentum between x = 150 and 192 m and the ejection between x = 50 and 75 m. Coincident with the injections and ejections are regions of warmer and colder water, respectively, as indicated by the temperature perturbation field. Strong negative temperature perturbations in ejections extend upward from the thermocline to ∼35 m depth, while corresponding positive perturbations are stronger below ∼45 m.

Fig. 14.

Vertical cross section along x = 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.

Fig. 14.

Vertical cross section along x = 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.

Fig. 15.

Vertical cross sections along (a) y = 120 m and (b) y = 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.

Fig. 15.

Vertical cross sections along (a) y = 120 m and (b) y = 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.

Similar weaker horizontal current perturbations at ∼30 to 40 m depth appear to be more linked with eddies driven from the surface. This region has a number of small scale (∼10 to 25 m) overturns that are actively mixing thermocline water into the mixed layer. The Langmuir region is characterized by strong downwelling pulses linked to the surface forcing, for example, at x = ∼25 m and ∼110 m. These vertical jets extend downward from about 5 m to 25 m and occasionally combine with circulations in the entrainment zone, indicating a possible enhancement of entrainment when small-scale Langmuir cells are particularly strong. Vectors in the Langmuir cells show a tendency to point more toward positive x, indicating the vertical transport of positive u momentum.

In comparison to the downwelling cross section, the flow field in the upwelling branch of the large eddy circulation has a more subdued turbulent structure. Eddy velocities in the entrainment zone are also fairly weak in comparison to the downwelling case, with less vertical excursions in the perturbation temperature field. Near the surface in the Langmuir zone, vertical velocities are dominated by upwelling, with vertical transport of negative u momentum indicated by the backward pointing vectors.

The flow visualization presented in Figs. 13–15 provides a snapshot of the processes responsible for transporting horizontal momentum and entraining thermocline water into the OBL. Large-scale eddies, forced near the surface by Stokes drift interaction with the mean current and shear production, produce regions of active downward transport of zonal momentum. Momentum flux convergence near the base of the mixed layer leads to intermittent injections and ejections of momentum that extend into the upper thermocline, enhancing the local exchange of thermocline water with the colder OBL. Because of the coherent large-eddy turbulence, areas of strong momentum and heat flux are localized beneath the downwelling portion of the eddies.

b. Quadrant analyses of vertical fluxes

Flow visualizations suggest that entrainment is produced by a wide range of flow scales. One method for identifying the importance of specific flow features, such as downward directed injections, is the application of quadrant analysis (Lin et al. 1996; Sullivan et al. 1998). Quadrant analysis divides the heat flux using conditional averages, separating the heat flux into four possible groups, w+T+, wT+, w+T, wT, where T is the perturbation from the horizontally averaged temperature and the superscript + and − refer to positive and negative perturbations, respectively. This method provides a way to discern systematic patterns in the entrainment flux, for example more cold water moving upward on average.

Application of quadrant analysis to hour 11 (Fig. 16) shows that heat flux in the shear-driven OBL has a complex structure for the negative fluxes, while positive fluxes are nearly equal. Because the flow is stably stratified, the negative heat flux dominates, giving a total flux with a maximum at ∼40 to 50 m depth. The peak values for the negative and positive fluxes are not located at the same depth; negative flux has a maximum value at ∼50 m, whereas the positive flux peaks at ∼55 m. One interpretation of this is that, on average, warm parcels are forced downward against gravity at ∼50 m (wT+), and tend to rebound upward at ∼55 m (w+T+). However, horizontal mixing and diffusion prevent a completely adiabatic process so that heat contained in transient eddies is diffused laterally, resulting in a net vertical heat flux. Additional cancellation of the positive and negative heat flux quadrants is produced by internal gravity waves, as identified above in the lower portion of the OBL. Internal waves act to lower the depth of the peak positive and negative heat fluxes relative to the depth of the total heat flux.

Fig. 16.

Quadrant analysis from hour 11. Superscript + and − denote conditional averaging with positive and negative perturbations, respectively. Fluxes are multiplied by ρCp.

Fig. 16.

Quadrant analysis from hour 11. Superscript + and − denote conditional averaging with positive and negative perturbations, respectively. Fluxes are multiplied by ρCp.

Focusing on just the negative flux quadrants, we note that the larger fraction of the negative flux in the entrainment layer is generated by relatively warm water being pushed downward. This corresponds to injections of momentum as shown in Fig. 13d. In contrast, above ∼30 m the negative heat flux is dominated by cold water being forced upward. This separation of the heat flux is consistent with the PE profiles shown in Fig. 10, namely, that wind driven turbulence tends to force cold 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 Sullivan et al. (1998) shows a significant difference between entrainment produced by shear and that produced by thermal convection. Sullivan et al. identified relatively large differences between each of the conditionally averaged heat fluxes so that the net result was that most of the entrainment flux 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 flux and the individual quadrant fluxes. Sullivan et al. found that heat flux quadrants in the convective boundary layer tended to cancel each other so that the net heat flux was smaller than all of the individual quadrant fluxes. Here, the largest flux is typically the total flux, indicating that in shear-dominated mixing less water recirculates adiabatically than in convective mixing.

Quadrant analysis of the zonal momentum flux, uw (Fig. 17), yields profiles consistent with the roughly linear growth in mean momentum as a function of time shown in Fig. 2. Because the wind stress is positive, almost all of the vertical momentum transport occurs through u+w and uw+. In the entrainment zone, u+w is slightly larger that uw+, indicating that downward injections of positive momentum have a slightly larger role in momentum transport in comparison to ejections of negative zonal velocity. A somewhat different momentum flux behavior was noted in Lin et al. (1996) for a neutral atmospheric boundary layer. Their average profile was similar to Fig. 17, with a peak negative momentum flux near the boundary but with reversed transport since the atmospheric system is a mirror image of the OBL. Momentum transport in the ABL was dominated by ejections (uw+) from the slow moving surface layer, whereas our results did not show a strong preference between u+w and uw+ near the upper boundary.

Fig. 17.

Quadrant analysis of the u component of momentum.

Fig. 17.

Quadrant analysis of the u component of momentum.

c. Turbulence spectra

The scales of turbulence produced by inertial current shear vary significantly as a function of depth in the boundary layer. This variation is quantified by examining the horizontally averaged velocity and temperature spectra from three model depths representing the Langmuir zone, middle of the mixed layer, and from the top of the entrainment zone (Fig. 18). Also shown are lines representing the −1 and −5/3 power laws, which can be thought of as describing the “production” and “inertial” subranges of turbulence for shear flow (Kader and Yaglom 1991; Katul and Parlange 1995). The generally accepted theory, based on wall-layer flow, is that energy injected at small wavenumbers through eddy asymmetries follows a −1 power law. At low wavenumbers, the flow is asymmetric and maintains a “memory” of the process that initially formed the eddy, for example like the horizontal jet shown in Fig. 13d at x = 180 m, y = 125 m that was produced by vertical advection of the mean shear. As energy cascades to small scales, the memory of injection events is lost and the flow becomes isotropic. The transition to isotropy is accompanied by the transition from a −1 to −5/3 power law and the establishment of an inertial subrange as presented in Kolmogorov’s (1941) theory.

Fig. 18.

Horizontal spectra of (a) T, (b) w, (c) u, and (d) υ at 10 m, 25 m, and 40 m at hour 11 as a function of wavenumber, ky = 2π/λ where λ is the wavelength. Spectra for u, υ, and w at 25 and 10 m are offset by 10 and 100, respectively.

Fig. 18.

Horizontal spectra of (a) T, (b) w, (c) u, and (d) υ at 10 m, 25 m, and 40 m at hour 11 as a function of wavenumber, ky = 2π/λ where λ is the wavelength. Spectra for u, υ, and w at 25 and 10 m are offset by 10 and 100, respectively.

Temperature spectra exhibit a −5/3 slope over at least a decade of wavenumbers. Velocity spectra tend to follow a −1 to −5/3 power law, although it appears that the inertial subrange in the entrainment zone is not well established (e.g., at 40-m depth). In the Langmuir zone at 10 m, spectra for all velocity components follow an inertial subrange starting from the largest scales resolved 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 field 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 −1 power law for u, but not for w and υ, implying that turbulence at these depths is gaining energy from the mean flow 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 −1 power law for the streamwise velocity, although they also noted a −1 power law for the υ component.

Below ∼40 m, the model u and w spectra do not display a significant range with a −5/3 dropoff, suggesting that the model has insufficient resolution in part of the entrainment zone. Following Skyllingstad et al. (1999), we apply two commonly used length scales, the Ozmidov scale Lo = (ɛ/N3)1/2 and the buoyancy length scale Lb = w/N, to see where buoyant stratification is acting to suppress vertical eddy dimensions (Fig. 19). These two length scales estimate the distance that turbulent motions can travel before being strongly affected by stratification. As Fig. 19 shows, stratification begins to control vertical eddy scales at depths below ∼45 m or in the lower half of the entrainment zone. For the coarse resolution budget calculation, the effects of buoyancy become important at an even shallower depth of about 25 m. Lack of a well-resolved inertial subrange does not necessarily mean that the energy containing scales are inaccurate. As Skyllingstad et al. (1999) demonstrated, even though LES produced dissipation rates were much smaller than observed values, the resolved heat flux was still quite accurate. This is because dissipation occurs at much smaller scales of motion than fluxes. Wang et al. (1998) also found that LES simulations of equatorial turbulence gave good results in comparison to observations, even though their simulations were also under-resolved in the stratified region of the flow. Nonetheless, comparisons of velocity spectra from the coarse resolution budget calculation (not shown) and the high resolution case presented in Fig. 18, indicate that turbulence energy levels in the low resolution simulations were reduced. The net effect of the stronger turbulence in the high resolution case is increased turbulent mixing, however, the relative importance of the TKE and PE budget terms was not significantly altered by the increased resolution.

Fig. 19.

Horizontally averaged Ozmidov length scale (Lo) and buoyancy length scale (Lb) from model hour 11.

Fig. 19.

Horizontally averaged Ozmidov length scale (Lo) and buoyancy length scale (Lb) from model hour 11.

Poor resolution in the thermocline as indicated by Lo and Lb may also help explain the relatively weak internal wave fields shown in our results. The model is unable to produce small-scale eddies in the strongly stratified regions of the flow. These eddies would likely generate small-scale internal waves that could transport momentum away from the mixed layer. Simulating these waves would require much higher resolution than is currently feasible. We also note that the simulations do not show significant internal wave growth via Kelvin–Helmholtz instability. Inclusion of a domain-scale pressure gradient (as would be forced by geostrophic adjustment) could cause a more unstable velocity profile to evolve. We have not attempted to examine this problem here, but hope to investigate the effects of large-scale pressure variations in future work.

5. Mixed layer parameters: Comparison to KPP model

Martin (1985), Large and Crawford (1995), and Large and Gent (1999) have shown that simplified one-dimensional models are capable of reproducing many of the average changes that occur in the wind-forced OBL. These models typically represent mixing rates through some combination of bulk stability parameters and empirically derived mixing coefficients. Here, as an example we examine the Large et al. (1994) KPP model. Eddy diffusivities in KPP are modeled using empirical profiles derived from atmospheric boundary layer measurements. Accurate boundary layer prediction using these profiles requires a robust estimate of the boundary layer depth. For KPP, boundary layer depth is estimated through a bulk Richardson number defined as

 
formula

where B(z) = ′/ρ, 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 = 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 fields (Fig. 20), are in general agreement with the range of Ric suggested by Large et al. (1994), although the depth where Rib = 0.3 appears to yield a boundary layer depth somewhat shallow in comparison to the depth of the peak heat flux. 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.

Fig. 20.

Contours of bulk Richardson number values 0.4 and 0.3 overlaid on the resolved turbulent heat flux, ρCpwT, for the resonant case.

Fig. 20.

Contours of bulk Richardson number values 0.4 and 0.3 overlaid on the resolved turbulent heat flux, ρCpwT, for the resonant case.

A bulk mixing coefficient estimate can be derived from the LES results by assuming a simple eddy viscosity budget equation for temperature and solving for Kh,

 
formula

This calculation is consistent as long as the vertical temperature profile is not completely mixed. Figure 21 shows Kh from the LES model along with Kh calculated using KPP as presented in CL. Comparison below ∼15 m shows good correspondence, with KPP values remaining within ∼30% of the LES prediction. The shape of the profile is somewhat different in the LES results, with slightly larger values below ∼60 m depth and smaller Kh in the main entrainment zone between 30 and 50 m. In general, however, the curves both have roughly linear profiles in the entrainment zone, and the differences between the two estimates may have more to do with the timing of mixing than with errors in the profile shape. We note, however, that qualitative comparison between the LES temperature time depth plots shown in Fig. 3 and KPP results reported in CL show that KPP is less diffusive and tends to maintain a sharper gradient of temperature in the thermocline than the LES. The differences shown in Fig. 21 are consistent with these qualitative differences and may indicate that the parameterized K profile or Ric used in determining OBL depth need adjustment for shear-driven entrainment mixing.

Fig. 21.

Horizontally averaged vertical mixing coefficient derived from the coarse resolution, resonant weak wave case using the resolved heat flux and average temperature gradient between hours 14–18, along with the equivalent mixing coefficient from the KPP model.

Fig. 21.

Horizontally averaged vertical mixing coefficient derived from the coarse resolution, resonant weak wave case using the resolved heat flux and average temperature gradient between hours 14–18, along with the equivalent mixing coefficient from the KPP model.

6. Discussion and conclusions

The effects of resonant wind and wave forcing on the ocean boundary layer were examined using a large-eddy simulation turbulence model. Two scenarios were examined, one with constant or off-resonance winds and one with inertially rotating or resonant winds. As a secondary issue, the distinction between weak and strong surface wave forcing was also investigated. The evolution of average temperature and momentum profiles was consistent with previous observations and one-dimensional model predictions showing that resonant winds cause strong acceleration of the velocity field, 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 Langmuir cells were mostly confined 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 representation 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 significantly so that most of the integrated growth in turbulence energy is through vertical expansion of the OBL. After ∼6 h, mixing in both the resonant and off-resonance cases reaches the top of the seasonal thermocline and begins entraining cooler thermocline water into the OBL. In the resonant case, the buoyancy flux associated with entrainment is significant, amounting to about 7% of the input wind energy (this percentage should not be confused with previous mixing efficiency estimates that considered the wind energy at 10 m; here, input wind energy is that imparted by the wind stress and waves). The off-resonance case develops a much weaker current system as the wind acts to remove energy from the flow after ∼¼ to ½ inertial cycle, which limits the OBL deepening to about 50 m. In contrast, turbulent mixing in the resonant case extends to ∼100 m depth.

During the initial period of OBL growth, three distinctive layers can be identified in the simulation as shown in Fig. 12. In the upper ∼25 m, turbulence is dominated by vertical motions associated with wave induced Langmuir cells. Langmuir cells with scales up to several hundred meters are produced in this zone, with 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 in the schematic). The effects of shear in the Langmuir layer are minimized in comparison to the surface wave Stokes drift forcing and the vertical advection term. In the entrainment zone, shear begins to dominate in the production of turbulence. Here, the eddy flow field is punctuated by strong horizontal injections of energy produced by overturning eddies. The injection of energy from the mean flow into turbulence causes a maximum in the turbulent heat flux in this layer, which is responsible for cooling the surface waters and warming the upper thermocline. Below the turbulent boundary layer lies the internal wave zone. In this region, motions are produced by propagating internal waves and do not lead to significant mixing.

Analysis of the PE budget demonstrates how local changes in the PE are affected by eddy transport, with most of the local changes in PE attributed to water exchange that has no net impact on the integrated PE or TKE. Locally, however, the change in PE is much greater and represents the vertical exchange of water heated during the summer season with cold thermocline water established during the previous winter. Overall, our results are consistent with the one-dimensional modeling experiments performed by Crawford and Large (1996) using identical forcing and initial conditions. We found that bulk Richardson number criteria were consistent with values used in the KPP one-dimensional model, demonstrating that boundary layer depth is well-represented by nonlocal measures such as the bulk Richardson number. The wind-driven OBL is only slightly stable, so it behaves much like a neutrally forced boundary layer, although stratification leads to a somewhat lower mixing coefficient. As Fig. 21 shows, the effective Kh from the LES results is smaller than the KPP value throughout the depth of the entrainment layer. However, the depth of the K profile in KPP is limited by the bulk Richardson number, which defines a shallower mixed layer depth in comparison to the LES results. One possible modification to the KPP profile would be to use the local gradient Ri to adjust the value of Kh and increase the critical Rib to allow for a deeper boundary layer. In future research, we plan to investigate this possibility by performing a series of experiments with KPP and the LES model so that the relationship between Kh and local stability can be better determined (e.g., Wyngaard and Brost 1984).

Without a surface heat flux, the wind driven OBL is essentially a stratified, entraining boundary layer, which has few counterparts in other geophysical flow problems. Probably the closest analogue turbulent flow is encountered in the nocturnal atmospheric boundary layer (ABL) when surface stratification decouples the free atmosphere from the surface. Winds increase above the stable boundary layer in response to synoptic pressure gradient and can eventually generate turbulence that entrains downward toward the surface (this is also one of the few cases where the ABL does not need to be flipped over for comparison to the OBL!). Stable boundary layers in the atmosphere can be divided into two classes depending on the intermittancy of turbulence (Mahrt 1999). With strongly stable boundary layers, turbulence is very intermittent with long periods of quiescent flow interrupted by bursts of turbulence. Weakly stable boundary layers are characterized by almost continuous turbulence that acts to decrease the background stratification through engulfment of cooled surface air. We find that the latter category is most applicable to the storm-driven OBL, with stratification being maintained by entrainment of thermocline water rather than surface cooling in the ABL.

Our results again emphasize the importance of using wind forcing with high temporal resolution as originally suggested by Large et al. (1986). Wind variations on the scale of hours can make huge differences in the strength of upper ocean currents and subsequent mixing from shear production of turbulence. Also critical is the vertical resolution in one-dimensional representations of OBL mixing. Estimating the shear is crucial for determining the strength of turbulence production, and is directly related to the amount of vertical smoothing implicit in coarse vertical resolution models.

Acknowledgments

We are pleased to acknowledge the supercomputer time provided by the National Center for Atmospheric Research, which is funded by the National Science Foundation. We also thank the two anonymous reviewers for very useful comments and suggestions. This research was supported by the National Science Foundation through Grant OCE-9711862.

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APPENDIX

Model Domain Rotation

There are two main reasons rotating fluids are difficult to model in Cartesian coordinates. First, rotating currents can cause flow oscillations that result from the model resolution effectively changing as the mean current direction passes from grid parallel to grid diagonal directions. This causes variable numerical smoothing depending on the flow direction. The second effect of rotating currents is produced by misalignment of coherent vortices such as large scale convective rolls. Vortices are forced to have orientations and scales that fit the periodic domain rather than a natural scaling.

For resonant flows, these problems can be avoided to some degree by rotating the domain with the mean flow, which for perfect resonance is equivalent to setting f = 0.0 s−1. However, setting f = 0.0 s−1 is not an appropriate assumption for cases with a surface wave Stokes drift because of the Stokes drift Coriolis term. The solution is to retain the Stokes drift Coriolis term, as is shown by a straightforward analysis. We assume a simplified set of equation for a layer averaged velocity field,

 
formula

where u and υ are the average horizontal velocities, us and υs are the Stokes drift components, and τx and τy are the surface wind stress components. These equations can be combined using complex notation, U = u + iυ, Us = us + s, and τ = τx + y, yielding

 
Ut + if(U + Us) = τ.
(A2)

Applying a wind stress with rotation, τ = τ̃ exp(−ift), Us = Ũs(t) exp(−ift), and assuming a solution U = Ũ(t) exp(−ift) gives

 
formula

showing that the Stokes Coriolis term must be retained when rotating the model domain to accommodate inertial resonance.

Footnotes

Corresponding author address: Dr. Eric Skyllingstad, College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, OR 97331.