## 1. Introduction

Ocean general circulation models (OGCMs) have long struggled to represent the Southern Ocean faithfully. Nevertheless, they typically have substantial biases in mixed layer depth (MLD) both when coupled to an atmosphere model (Sallée et al. 2013), or when uncoupled and forced by observation based surface fluxes (Downes et al. 2014). The magnitude of these shallow biases in late winter is truly remarkable, exceeding 400 m in a nearly circumpolar deep mixing band (DMB) between about 40° and 60°S (DuVivier et al. 2018), which includes key regions of water mass formation. Deficiencies in the ocean component have been implicated (Danabasoglu et al. 2012; Weijer et al. 2012). This view is consistent with DuVivier et al. (2018), who find that both uncoupled and coupled configurations of an OGCM have similarly shallow biases, despite weaker than observed stratification, and much stronger wind stress over the Southern Ocean when coupled. Furthermore, DMB biases are evident at eddy-resolving resolutions (Downes et al. 2014; DuVivier et al. 2018). Therefore, a significant portion of the biases may be attributable to inadequate representation ocean vertical mixing physics (Belcher et al. 2012). This possibility provides the overall motivation for this present work, which broadens the surface layer focus of Large et al. (2019, hereafter L19) to the boundary layer as a whole. It is one aspect of a larger Southern Ocean project, which includes DuVivier et al. (2018), to extend the attribution to surface forcing, the general circulation, the subsurface salinity structure, model resolution, and other factors.

Although OGCM resolution and numerics (e.g., vertical grid structure) can be the source of major differences between solutions, in a comprehensive review of ocean turbulence and mixing Burchard et al. (2008) assert that OGCM turbulent mixing terms are “the only terms about which there is debate.” Furthermore, they distinguish different strategies based on the resolved space scales and frequencies as direct numerical simulation (DNS) that solves the Navier–Stokes equations down to molecular scales, large-eddy simulation (LES) that solves the spatially filtered Navier–Stokes equations by resolving the most energetic large turbulence scales, statistical turbulence modeling (STM) that solves the Reynolds-averaged Navier–Stokes (RANS) equations with a large variety of closures, and empirical turbulence modeling (ETM) that typically predicts integral properties such as mixed layer depth. In principle, DNS and LES can be used to study turbulence itself (Lesieur 1997), while STM and ETM only represent the effects of turbulence.

As outlined by Burchard et al. (2008), the RANS equations contain unknown second moments, and equations for these moments contain even more unknown third moments, then continuing the process gives the Friedmann–Keller series (Keller and Friedmann 1924) of higher and higher order closures. Convergence of this series to more accurate solutions cannot be assured because of the growing number of unknowns. To be practical for geophysical flows the series must be truncated at an early stage (usually after the second) and there are typically further simplifications such as the neglect of horizontal mixing terms. Nonlocal transport refers to turbulent vertical fluxes that are independent of local gradients. It is of particular interest because OGCMs are not consistent in its treatment. It is represented in the Cheng et al. (2002) third-moment closure, and in the model for the fourth-order moments that is tested against LES and observations by Canuto et al. (2007). There is nonlocal transport in the level-3 second-order closure of Mellor and Yamada (1982), but not in the simpler lower levels, or in many other second-order closures where fluxes are formulated to be down-gradient, such as Kantha and Clayson (1994), Canuto et al. (1994), Bretherton and Park (2009), Harcourt (2013), and Harcourt (2015). However, nonlocal transport is fundamental to some low-order models, especially those that consider convective plumes, such as the unified convection scheme of Park (2014), as well as the combined eddy-diffusion, mass-flux model of Siebesma et al. (2007) and its recent extension (Tan et al. 2018).

Familiar examples of ETM are ocean mixed layer models of Kraus and Turner (1967) lineage and of Price et al. (1986), as well as the *K*-profile atmospheric boundary layer models of Troen and Mahrt (1986) and Holtslag and Boville (1993), plus the related *K*-profile parameterization (KPP) of the ocean boundary layer (Large et al. 1994). According to Smyth et al. (2002), ETM is more efficient than the STM, “forsaking theoretical development almost entirely in favor of simple empirical representations of specific processes.” For example, KPP does include nonlocal transport of buoyancy, but not of momentum because neither theory nor observations provided a suitable foundation. The Noh et al. (2003) improvements to Troen and Mahrt (1986) include both of these nonlocal transports, as well as an entrainment similar to that in KPP.

Since there are no direct measurements of turbulent fluxes in the open ocean, several recent advances in the understanding and hence modeling of upper-ocean mixing physics have utilized LES; for example, Wang et al. (1998), McWilliams et al. (1997), Grant and Belcher (2009), Harcourt and D’Asaro (2008), van Roekel et al. (2012), and Li and Fox-Kemper (2017). The latter discusses the others in some detail and evaluates schemes for incorporating the effects of Langmuir circulations driven by surface wave–induced Stokes drift in OGCMs. Additional relevant examples here include Brown and Grant (1997), Smyth et al. (2002), and Noh et al. (2004), because they focus on nonlocal transport, including momentum. In particular, Brown and Grant (1997) use LES to evaluate the two-scale mixing formulation of Frech and Mahrt (1995) for the atmospheric boundary layer and propose an improved scaling. Smyth et al. (2002) test the performance of KPP in an oceanic regime and suggest appropriate refinements based on LES comparisons. In addition to adding a nonlocal momentum transport with a form similar to those of buoyancy, they consider the effects of Stokes drift on the turbulent velocity scale and on the nonlocal momentum flux. Noh et al. (2004) go further and include the effects of wave breaking and Langmuir circulation. To be useful, LES need only mimic, but not necessarily reproduce nature, so most simulations are highly idealized.

The underlying physics of both local and nonlocal transport is contained in the LES equations and the relevant motions should be resolved by LES models. The present study investigates these transports in four particular LES of the Southern Ocean boundary layer forced by realistic winds, waves, and buoyancy. Specifically, the nonlocal transports of buoyancy and momentum are first diagnosed, then used to infer diffusivity and viscosity profiles through the boundary layer and finally to determine some basic relationships that provide a degree of order to the turbulent LES solutions. These LES results are independent of any STM or ETM assumptions, but in order to be more directly relevant to such models, the analysis utilizes some of their basic constructs, such as similarity theory from L19 and eddy diffusivity. However, neither evaluation of various boundary layer schemes and models against these results, nor examination of other familiar constructs, such as turbulent kinetic energy, is within the present scope.

## 2. Large-eddy simulations of the Southern Ocean boundary layer

This present study of the Southern Ocean boundary layer extends past idealized LES studies by examining ocean mixing in LES driven by highly variable winds, waves, and cooling at the Southern Ocean Flux Site (SOFS) near 47°S, 140°E (Schultz et al. 2012). In particular, the three idealized wind decay cases of L19 are not considered, because the weak buoyancy forcing (5 W m^{−2} cooling) may not be compatible with such a deep (≈100 m) imposed MLD, even though the surface layer turbulence down to 10 m is consistent with the realistic cases. Utilized are the four realistic cases; two from April 2010, AprN with no wave (calm) forcing and AprS with Stokes (wavy) forcing, and two June cases similarly denoted as JunN and JunS, and referred to as calm and wavy. These cases are described in detail in L19, so only the most relevant aspects are repeated, with the germane results summarized in section 3. First, the highly variable Southern Ocean forcing is shown for April and June in Fig. 1. The surface wind stress ** τ** and ocean density

*ρ*define the kinematic wind stress

^{−1}.

### a. The LES model with surface wave effects

The LES model used to study the turbulent ocean boundary layer is conventional (Moeng 1984; Sullivan et al. 1994; McWilliams et al. 1997) but is extensively modified to account for surface wave effects. Its dynamics integrate the wave-averaged, incompressible, and Boussinesq Craik–Leibovich equation set (Craik and Leibovich 1976; McWilliams et al. 1997). As such it solves the Navier–Stokes equations, including processes responsible for nonlocal momentum and buoyancy transport. It produces time evolving profiles of mean horizontal Eulerian flow **U** and buoyancy Θ, as well as the corresponding turbulent vertical fluxes ⟨*wu*⟩, ⟨*wυ*⟩, and ⟨*wθ*⟩, which are given by the correlations of fluctuations about these means in space and time plus subgrid-scale (SGS) contributions that are small everywhere except very near the surface. Similarly, the variance of velocity fluctuations plus SGS contributions give the turbulent kinetic energy (TKE).

The additional terms arising from phase averaging over the surface waves include Stokes–Coriolis, a vortex force, and a Bernoulli pressure head in the momentum equations, as well as horizontal advection by Stokes drift in the buoyancy equations, and additional production of both resolved and SGS energy by vertical gradients of Stokes drift (Sullivan et al. 2007). The model resolves neither the oscillatory, nor breaking wave motions, so wave quantities are unaffected by the currents and there are no explicit wave effects on pressure. The optional acceleration and energy generation due to nonconservative wave breaking (Sullivan et al. 2004) is not exercised, because effects are limited to the upper few meters (Noh et al. 2004).

Stokes advection and production of both SGS and resolved TKE are relatively well represented in the current simulations by accounting for a full wave spectrum. Also, in contrast to RANS models, the local SGS energy transfer from Stokes production in the LES can be either positive or negative since the Stokes shear is imposed in the turbulent flow field with a specified orientation and sign. In addition, the magnitude of the Stokes drift often exceeds the horizontal current speed near the surface, leading to substantially different advection of TKE.

*B*

_{0}, which can include both heat and freshwater forcing, and is negative in the more common situation of surface buoyancy loss. Therefore, it defines an equivalent surface heating (Fig. 1f) as

*g*= 9.81 m s

^{−2}, thermal expansion is constant at

*α*= 1.9 × 10

^{−4}K

^{−1},

*C*

_{p}is ocean heat capacity, and

*ρ*

_{o}an ocean reference density, such that (

*ρ*

_{o}

*C*

_{p}) = 4.1 MJ m

^{−3}K

^{−1}. Also, all the solar radiation is absorbed in the first model level, so these LES are more representative of nature at night than during the day.

The computational domains are 1280 points over 1250 m in each horizontal direction. By design, the vertical extents are about twice the deepest MLD, so there are 512 grid cells over depths of 300 m in the April cases and 500 m in the June. The vertical grid is stretched such that the cell thickness is about 0.375 m at the surface, increasing to 1.13 m (April) or 2.02 m (June) at the bottom. The respective time steps are about 0.9 and 1.5 s, and all the cases use the SOFS (47°S) Coriolis parameter, *f* = −1.064 × 10^{−4} s^{−1}, so the inertial period is 16.4 h. The grid choices are justified in L19, where the vertical velocity at 18 m from JunS shows downwelling features associated with Langmuir turbulence that are aligned with the wind for up to 500 m and separated across wind by hundreds of meters. The return upwelling is not aligned with the wind and dominated by horizontal scales up to about 200 m, associated with convection in a roughly 200-m-deep boundary layer.

### b. Meteorological and Stokes (wave) forcing

Time-varying wave effects are incorporated as an imposed vertical profile of the Stokes drift **U**_{S}(*z*) computed every 150 s from the directional wave height spectrum, with 72 directions and 55 frequencies from 0.031 to 2.3 Hz, as detailed in L19. These spectra are generated by forcing the wave prediction model WAVEWATCH III v2.22 (Tolman 2002) with the SOFS-observed winds of Fig. 1a and breaking, as described in Romero and Melville (2010). To better capture the low wavenumber/frequency components, the wave data at SOFS from ECMWF ERA-Interim global reanalysis (Berrisford et al. 2011) are used as initial and boundary conditions.

A Langmuir number is a measure of the relative strength of wind to wave effects. The turbulent Langmuir number (La) of McWilliams et al. (1997) is defined such that ^{−2} = 11 (McWilliams et al. 1997), as well as misalignment between the surface Stokes drift **U**_{S}(0) and the wind stress (Fig. 1c). Another wave parameter is the *e*-folding depth *d _{S}* of the Stokes drift speed (Fig. 1e). In general this depth varies only between 1 and 6 m, but deepens to 10 m during the weakest winds of April.

L19 identified six distinct April forcing regimes and four more in June. The time periods of each are shown at the top of Fig. 1. The first is denoted A1–11 for April hours 1–11. It is characterized by very large surface cooling (Fig. 1e; solid trace) due to strong winds from the southwest (toward 45° to 70° clockwise from north; Fig. 1b) that are aligned with the surface Stokes drift (Fig. 1c). There is much less cooling and more moderate winds over the next four daytime hours of regime A12–15. During the third regime (A16–23) the wind speed drops to about 6 m s^{−1} with the waves remaining stronger than wind-wave equilibrium. Over regime A23–31 the light winds maximize *d*_{S} and the highly variable direction leads to significant misalignment with the surface Stokes drift. Regime A32–40 is the only period of stable surface buoyancy forcing, *Q* > 0. It is followed by A44–50 when the wind rises and blows increasingly toward the south, while the buoyancy forcing remains near neutral (0 > *Q* > −20 W m^{−2}).

In June (Fig. 1; dashed), the first regime, J1–4, is characterized by weak and varying winds, so the major forcing is the moderate surface cooling and a dominant swell propagating from the west-southwest, as shown in L19. As the wind increases to more than 15 m s^{−1} over the next 23 h (J5–27), a local wave field develops and stays near equilibrium (La^{−2} ≈ 11), while the cooling is modulated by daytime solar heating. The solar heating reduces the net cooling to very near zero over the next regime, J28–33. During the last June regime, J34–41, the cooling increases while the wind weakens with highly variable direction and misalignment with the Stokes drift.

### c. Boundary layer depths

A fundamental characteristic of a boundary layer is its depth *h*, which is a measure of the depth over which surface forced turbulence is dominant. This depth provides a useful nondimensional vertical coordinate, namely, *σ* = *d*/*h*, where *d* is the distance from the surface. For present purposes, *h* is diagnosed from converged hourly LES statistics. As detailed in L19, there are three steps in finding *h* from profiles of buoyancy flux ⟨*wθ*⟩, such as those shown in Fig. 2 from both April cases, when the depth *h*_{i} of maximum ocean stratification reaches 180 m (Fig. 3). This depth is analogous to the inversion height in the atmosphere. The first step is to find the entrainment depth *h*_{e} (magenta trace), where ⟨*wθ*⟩_{e} is a minimum. In unstable forcing it is below the shallowest depth *d*_{F} (blue trace) of zero flux. Second, find the first depth, *h*_{max}, below *h*_{e}, where ⟨*wθ*⟩_{max} is a relative maximum. Finally, take *h* (red trace) as the first depth below *h*_{e} where the buoyancy flux equals the weighted average (1–0.95)⟨*wθ*⟩_{e} + 0.95⟨*wθ*⟩_{max}. As shown in Fig. 3, this depth is sometimes shallower than *h*_{i}, and sometimes deeper.

As noted in L19, a value for *h* is always found between *h*_{e} and *h*_{max}, significant turbulent fluxes are not found deeper, and there is little sensitivity to the somewhat arbitrary weight of 0.95. Usually the algorithm could be simplified by taking ⟨*wθ*⟩_{max} to be zero, but not when the boundary layer is also forced from the interior, for example by breaking internal waves, or equatorial undercurrent shear. Another example from early in regime A32–43 is shown in L19. It is associated with the rapid collapse of the boundary layer due to the onset of stable forcing and hence, *h*_{e} = 0. Such transitions and the subsequent day–night evolution of regime A32–43 are distinct and a challenge for the fixed LES vertical grid set, because of the rapid changes in the number of grid cells in the boundary layer.

In Fig. 2b, Stokes forcing reduces the time variability of ⟨*wθ*⟩, while enhancing the response of *d*_{F}, associated with the daytime solar heating of regime A12–15. It also deepens the boundary layer during stable forcing and throughout the following near-neutral regime A44–50.

In June the boundary layer depth and *h*_{i} are about 40 m deeper than in April. Otherwise, because unstable forcing persists throughout (*Q* < 0, Fig. 1e dashed), characteristics of the boundary layer and impact of Stokes forcing are very similar to the April cases prior to regime A32–43. In particular, the behavior associated with the diurnal cycle of heat flux approaching, but not quite reaching zero in regime J28–33 is very similar to A12–15, though the deeper boundary layer weakens the signal somewhat. For example, the rise and fall of *d*_{F} (blue trace) and the associated change in sign of the buoyancy flux at deeper depths are common. There is similar behavior during the daylight hours at the start of regime J5–27, but it is muted because of the greater cooling. Thus, this period of solar heating plus those of regimes A12–15, J28–32, and A32–44 form a progression of diurnal cycles where the maxima of surface heating are −42, −19, −13, and 228 W m^{−2}, respectively, and in the LES there is an increasing Stokes dependent response, modulated by *h*.

### d. A qualitative evaluation of entrainment

In unstable forcing, the buoyancy flux (Fig. 2) is positive near the surface, vanishes at a depth *d*_{F} (blue trace), becomes most negative at *h*_{e}, then increases toward zero. This increase with depth is indicative of a time tendency for buoyancy to increase as heavier water from these depths is mixed with lighter water from above. This mixing is a buoyancy loss (cooling) to the column above *h*_{e}, as is the surface buoyancy loss, *B*_{0} < 0. In L19, the buoyancy gain below *h*_{e} over the first 9 h of AprS are compared to profiles from ARGO^{1} and both show buoyancy gain (entrainment) starting at about 168 m and peaking near 180 m. Thus, this LES does reproduce at least the overall character of the upper ocean at SOFS in April 2010. A quantitative assessment is precluded by the mismatch between the model and observations. In particular, the model is stationary with no advection, whereas the combined advection from float movement and ocean currents could be substantial. Also, even though the relatively frequent 7-h ARGO sampling can resolve most inertial variability, it is far from ideal because of an apparent semidiurnal internal tide.

## 3. The LES surface layer

*σ*=

*ε*of the boundary layer depth. It is based on dimensional analysis (Wyngaard 2010), which is extended to the ocean with surface wave (Stokes) effects in L19. The theory postulates that the important independent variables governing the structure of the surface layer are the distance

*d*from the surface and the forcing parameters

*B*

_{0}. The ocean extension adds a Stokes forcing parameter. With four independent variables and only two dimensions (length and time) the Buckingham–Pi theorem says that the nondimensional gradient of Θ depends on the product of a buoyancy similarity function

*ϕ*

_{s}of a parameter

*ζ*times a Stokes similarity function

*χ*

_{s}of another parameter

*ξ*:

*κ*= 0.40 is the von Kármán constant. Similarly the nondimensional shear becomes

**U**is preferred over the Lagrangian

**U**+

**U**

_{S}.

*w*

_{m,s}associated with surface layer viscosity

*K*

_{m}and diffusivity

*K*

_{s}and hence the turbulent Prandtl number, Pr = (

*K*

_{m}/

*K*

_{s}):

*m*,

*s*is used to refer to either momentum or scalars.

*P*

_{U}and

*P*

_{S}, is due to misalignment of the stress vector ⟨−

*w*

**u**⟩ with the Eulerian and Stokes shear, as captured by the dot products in (5) and (6).

*ζ*=

*d*/

*L*is the stability parameter, where the Monin–Obukhov depth

*L*is associated with the depth where the magnitude of the buoyant production, or suppression, of TKE equals the rate of shear production (Wyngaard 2010). To keep the same association, the Stokes production of TKE in an oceanic boundary layer is included in L19 with a

*ζ*parameter defined by

_{m,s}=

*ϕ*

_{m,s}from the AprN and JunN cases were computed across the surface layer. These calculations indicate that the surface layer extends to a fraction

*ε*= 0.1 of the boundary layer and that the functions

*A*

_{m}= 14 and

*A*

_{s}= 25, are consistent with

*κ*= 0.40. In the convective limit of neither wind (

^{−2}= 0;

*χ*

_{m,s}= 1;

*ζ*=

*d*/

*L*), the formulation (9) makes the velocity scales of (4) proportional to the convective velocity scale:

*A*

_{m}/

*A*

_{s})

^{1/3}= 0.86.

*ϕ*

_{m,s}are too limited. There are only 8 h of stable forcing during AprN and many fewer grid cells in the surface layer of the much thinner stable boundary layers of Fig. 2. Therefore, the traditional buoyancy similarity functions from Högström (1988) are retained:

*ζ*is that the depth dependencies of

*χ*

_{m}computed from (3) and

*χ*

_{s}from (2) are greatly reduced, such that

*χ*

_{m,s}may possibly be regarded as depth independent beyond breakpoints

*σ*

_{Bm}= 0.10 and

*σ*

_{Bs}= 0.07. However, these functions are best computed from the LES at shallower depths

*σ*, where the gradients of (3) and (2) are strongest, and where L19 finds that both functions increase at rate near unity. Therefore, every hour (2) and (3) are used to calculate a depth independent value:

*ζ*is evaluated at

*σ*. In practice, averaging over four consecutive LES grid cells greatly reduces computational noise.

*ζ*from (8). The choice for the

*ξ*parameter is the fraction of total surface layer production of TKE that is due to Stokes shear (6), because it gives the best fit for the hourly values from (12). It is given by

*ξ*> 0.35. Consistency with the calm cases requires

*χ*

_{m,s}(0) = 1, so linear interpolations are taken for

*ξ*between 0.0 and 0.35, along with constant extrapolation at the minimum values of quadratics at

*ξ*= 0.85 in (15) and

*ξ*= 0.72 in (14). Respectively, these constants are about 0.25 and 0.17 and their ratio, 1.5, becomes the upper limit of the turbulent Prandtl number.

## 4. The LES boundary layer

The LES data provide the information necessary to diagnose both the local and nonlocal turbulent vertical transports of both momentum and buoyancy throughout much of boundary layer. These transports are found to be connected to the surface layer through nondimensional shape functions of *σ*, and the nonlocal fluxes can exceed the local. However, the degree to which these LES results are reproduced in STM boundary layer models, or are incorporated in ETM, is beyond the present scope.

### a. Nonlocal diffusivity

*γ*

_{s}is referred to as a buoyancy gradient due to nonzero buoyancy variance. An alternative closure of this equation by Holtslag and Moeng (1991) produces a similar form, but

*γ*

_{s}arises from the divergence of third moments, that are neglected by Deardorff (1972). The LES model makes none of the assumptions of either of these closures. Its SGS closure is fundamentally different and well established (section 2). Therefore, the LES can be used to learn about the collective nonlocal buoyancy transport Γ

_{s}, its depth dependencies, and about expressing it as a product (

*K*

_{s}

*γ*

_{s}).

The LES show clear evidence of nonlocal buoyancy fluxes throughout periods of unstable forcing in two ways. In Fig. 2 there are depths *d*_{F} (blue trace), where the small positive buoyancy gradients evident in Fig. 3, give direct estimates of *γ*_{s} = ∂_{z}*θ*(*d*_{F}). Furthermore, the “noisy” zero contour of Fig. 3 indicates that typically there are a number depths *d*_{G} above the blue trace where the gradient is zero, such that the fluxes from Fig. 2 are direct measures of Γ_{s} = ⟨*wθ*⟩(*d*_{G}). These depths all lie well within the boundary layer and are usually beyond the surface layer, except during the near-neutral regime A41–50, and the diurnal cycle of A12–15 from AprS. An effect of Stokes forcing is a more extensive range of depths *d*_{G} (Fig. 3b). There are zero gradients and nonzero fluxes over the first few hours of the stable regime A33–40, but mostly below the boundary layer where the turbulence is likely a remnant of prior unstable forcing that takes a few hours to dissipate. In this stable regime the Stokes forcing reduces the positive buoyancy gradient throughout the boundary layer. Stokes effects also erode gradients during unstable regimes, as shown by the deeper 1 K km^{−1} and shallower −1 K km^{−1} contours from AprS compared to AprN in Fig. 3. Again, the gradients from JunN and JunS are not shown, because they behave in a similar fashion to the first 32 h of April.

*d*

_{G}, where there is zero gradient and a nonzero flux. Therefore, a potentially more useful nonlocal diffusivity is defined by

*d*

_{G}of zero gradient there is usually nonzero gradient below at

*d*

_{F}, even in regime A44–50. The factors in square brackets of (17) allow

*γ*

_{s}to be a function of depth, with the assumption that

*ϕ*

_{s}from (9) is an appropriate scaling. This assumption follows from the surface layer scaling of L19, but remains to be tested deeper in the boundary layer in section 4c. In particular, the LES solutions will be used to determine the exponent

*p*, as well as to make the connection to surface layer diffusivity

*K*

_{s}of (4).

### b. Nonlocal viscosity

*X*) aligned with the shear, the above partition between local and nonlocal momentum flux in the shear direction, becomes

_{z}

*V*= 0:

*X*′ in the stress direction and

*Y*′ perpendicular, with velocity components

*U*′ and

*V*′. Now a nonzero shear ∂

_{z}

*V*′ becomes a measure of

*wυ*′⟩ is zero by construction:

_{υ}and

*K*

_{u}(−∂

_{z}

*U*) and the nonlocal contribution Γ

_{u}in (18). Another is the behavior at large angles, especially |Ω| > 90°. Yet another, is the connection to the surface layer viscosity

*K*

_{m}in (4).

Figure 5 shows the distributions of Ω from AprN and AprS. Again, the June distributions are qualitatively similar to Fig. 5 prior to the transition regime A32–43 and not shown. Clearly, *h* (red trace). The near-neutral regime A44–50 shows a similar pattern, with 0 > Ω > −40°. In contrast, there are rapid changes in sign during the weak and variable winds of regime A23–31, as well as the transition and stable forcing of A32–43.

The effect of Stokes drift is most evident in the upper 100 m. In the more orderly regimes, A1–11, A12–15, and A16–22 when it is roughly aligned with a strong enough wind stress (Fig. 1c), the Stokes drift rotates the stress vector to be more its direction, such that Ω of the Eulerian shear (middle panel) becomes much more negative than in AprN, and even by 20 m it is about −60°. However, the stress does not rotate quite enough to align with the Lagrangian shear, so above about 50 m the associated Ω (bottom panel) is about 10°. In regime A44–50 the angle of the wind (Ω_{τ} in Fig. 4) is positive, and the Stokes drift rotates the stress past the Eulerian shear vector such that Ω becomes positive for the Eulerian shear (middle panel), but little changed for the Lagrangian.

The ratio of the across-shear to the along-shear momentum flux is tan(Ω). The along shear ⟨*wu*⟩ dominates in the upper boundary layer where |Ω| < 10°. At depth, the across shear ⟨*wυ*⟩ is often comparable, 30° < |Ω| < 60°, and occasionally significantly greater.

### c. Nondimensional shape functions

*w*

_{m,s}are the associated turbulent velocity scales and where

*ζ*is calculated from (8) at a reference depth

*σ*

_{r}. Equations (22) and (23) are general in that they define unknown nondimensional shape functions

*G*

_{m,s}(

*σ*) that may or may not provide a useful connection between the surface layer and the whole of the boundary layer. Particular issues for the LES to address are the choice of reference depth, the behaviors of

*G*

_{s}and

*G*

_{m}and their differences, as well as how they vary with depth, with time and from case to case.

*χ*

_{m,s}(

*ξ*) is calculated from surface forcing and surface layer gradients according to (12). Two options are explored:

*σ*

_{r}=

*ε*and

*σ*

_{r}=

*σ*.

First consider *G*_{m} from the calm cases, with *σ*_{r} options, because there are nearly continuous profiles of *χ*_{m} = 1. Calculations of _{z}*V*′| > 10^{−4} s^{−1} to avoid division by small numbers. However, this limit is reduced to 10^{−5} s^{−1} over hours 10–18 of AprN in order to capture Ω transitions from negative to positive at depth, and over the stable A33–40 regime, so that all forcing regimes are represented. Excluded are AprN times near the stable/unstable transitions at hours 33 and 41, and near hour 28 when the wind is weakest and very variable in direction, with |Ω| < 40° a further restriction that avoids instances when there appears to be insufficient equilibrium for present purposes. The latter is relaxed to |Ω| < 60° for JunN. For both *σ*_{r} options, the more than 7000 remaining values of *G*_{m} (calm) are fit to a quartic regression (not shown) and are also averaged over *σ* bins of width 0.1. The bin averages are shown in Fig. 6, with vertical bars extending ±1 standard deviation. The bilinear reference curve is provided only to facilitate comparison of these results to the wavy cases and *G*_{s} results to follow.

With the *σ*_{r} = *σ* option the standard deviations of Fig. 6b are smaller, though by less 0.01, and the overall correlation coefficient of the quartic regression is higher; 0.35 compared to 0.28, so henceforth it is the marginally preferred option over *σ*_{r} = *ε* (Fig. 6a). With either option there are large standard deviations, but the scatter is not random. Profiles from a specific time usually fall entirely above or below all the bin averages. This behavior, suggests that the highly variable forcing may be playing a role. Nevertheless, the bin averages of Fig. 6b vary smoothly with depth. Thus they provide one realization of an average nondimensional momentum shape function that relates momentum flux ⟨*wυ*⟩ and shear ∂_{z}*V*′ across the entire boundary layer, to the depth *h*, to the surface forcing *u**, and to the surface layer shear used to compute Ψ_{m} from (3).

The wavy cases are more relevant to the ocean and the strong and persistent Stokes forcing allows _{z}*V*′| and on April times near hours 28, 33, and 41 are again applied. Equilibrium issues are avoided by requiring ∂_{z}*V*′ > 10^{−4} s^{−1} during the low and variable winds of regime A23–31. The bin averages of *G*_{m} (wavy), with *σ*_{r} = *σ*, are shown in Fig. 7. They reveal significant differences between the two shears. The Eulerian shear gives the most consistency with the calm results of Fig. 6b, especially for *σ* < 0.4, and the standard deviations overlap at all *σ*. Thus, Fig. 7a can be regarded as a second realization of *G*_{m}(*σ*).

Using the Lagrangian shear to calculate *σ* = 0.4, where the Stokes shear is much smaller than the Eulerian. Above, more of the Lagrangian shear is aligned with the momentum flux and the smaller across-shear flux is not fully compensated by smaller shear in (21) such that *G*_{m} becomes systematically reduced, and hence even smaller than the calm, and the peak shifts. The two major issues are how this tendency diminishes as the calm limit is approached, and the inconsistency with using Eulerian shear to compute *χ*_{m} from (12). Using the Lagrangian shear for this calculation too gives very large values of *χ*_{m}, such that the resulting *G*_{m} averages need to be reduced by a factor of 2 (asterisks) to fall within the limits of Fig. 7b. The standard deviations are much greater than those shown in Fig. 7. They are roughly equal to the plotted scaled bin averages (asterisks).

There are fewer instances of *G*_{s} are shown in Fig. 8. To keep adequate precision in the buoyancy flux, [⟨*wθ*⟩(*d*_{G})/*g*] > 2 × 10^{−8} cm s^{−1} is always a requirement. The above April time restrictions are again applied. Division by small numbers is avoided by requiring (*gα*)^{−1}∂_{z}*θ*(*d*_{F}) to be greater than 0.02°C km^{−1}, in the units of Fig. 3, for AprN and JunN, and about an order of magnitude smaller for AprS and JunS, because of the small gradients under Stokes forcing. The calm cases with *p* = 1 (Fig. 8b; blue diamonds) give the most consistent results in terms of smallest standard deviations and best agreement with *G*_{m}, especially with the wavy of Fig. 7a. The sensitivity to *p* diminishes as *d*_{G} approaches *d*_{F}. At shallower *σ*, the calm cases with *p* = 0 (Fig. 8a; blue diamonds) increase more rapidly than *G*_{m}, and even more so for *p* < 0. For *p* > 1, the agreement with *G*_{m} progressively degrades.

Values of *G*_{s} from the wavy cases tend to be smaller and much more variable, such that the combined bin averages (black squares) are smaller than the calm for *σ* < 0.3 with bigger standard deviations everywhere. These impacts are more evident for *p* = 0 in Fig. 8a, where the agreement with *G*_{m} improves for *σ* < 0.4, but the standard deviations are everywhere greater than for *p* = 1 (Fig. 8b). With Stokes forcing, the effect of *P*_{S} > 0 in (8) is to make the stability parameter more negative and hence, *ϕ*_{s} a weaker function of *σ*, such that *p* = 1, which combined with the smaller scatter supports this choice for the exponent in (17).

### d. A composite shape function for buoyancy and momentum

The degree of consistency achieved with the choices of *σ*_{r} = *σ* (Fig. 6b), Eulerian shear (Fig. 7a), and *p* = 1 (Fig. 8b) suggests that a composite shape function *G*_{C}(*σ*) may be applicable for both momentum and buoyancy, with or without Stokes forcing. For this particular purpose, *G*_{s} values from JunS are excluded, because they scatter more than twice as much as AprS, even though the bin averages are nearly the same. This excessive JunS scatter is responsible for the bigger standard deviations about the black squares of (Fig. 8b). Its source can be traced to a precision issue in the calculation of the nonlocal buoyancy gradient *γ*_{s}(*d*_{F}) in the denominator of (17), when the gradient is near the precision of the processed LES data. As evident in Fig. 3 near hour 15 for example, the depths *d*_{F} (blue trace) can be at or just below a zero gradient contour. These contours indicate that the vertical buoyancy difference *δ*_{υ}Θ between neighboring grid cells is less than the precision of the LES space–time averages. In such situations, changes in *d*_{F} and in the convergence of these averages can combine to change small values of *δ*_{υ}Θ by as much as a factor of 10 from hour to hour, with associated variability in *G*_{s} insufficiently compensated by ⟨*wθ*⟩(*d*_{G}) in the numerator of (17). There are a number of factors that contribute to making this precision issue more prevalent in JunS. Relative to April cases, there is overall weaker stratification, deeper boundary layers and coarser vertical resolution. Relative to JunN, the Stokes forcing reduces gradients everywhere, as quantified in the surface layer by *χ*_{s} < 1 from (15).

Figure 9 shows the bin averages of *G*_{m} (calm) from AprN and JunN (red diamonds; Fig. 6b), of *G*_{m} (wavy) from AprS and JunS (red triangles; Fig. 7a), of *G*_{s} (calm) from AprN and JunN (blue diamonds; Fig. 8b), and of *G*_{s} (wavy) from AprS (blue triangles). The combined averages over narrower bins (black squares) are joined by the solid black line, with black vertical bars extending ±1 standard deviation. At all *σ* all four subset averages fall within these standard deviations, except for *G*_{m} (calm) for 0.6 < *σ* < 0.8. Thus, there a degree of order across the boundary layer in the LES solutions of the phase-averaged Navier–Stokes equations that justifies regarding the black trace as a composite shape function *G*_{C}(*σ*), at least for some purposes. However, the shape functions do tend to be smaller with Stokes forcing than without, and for buoyancy, especially for *σ* > 0.3. These distinctions may be important for some applications should they prove to be robust.

The dotted curve of Fig. 9 is the cubic function *σ*(1 − *σ*)^{2} proposed by O’Brien (1970), which is the basis of Troen and Mahrt (1986), Holtslag and Boville (1993), and KPP (Large et al. 1994). The fit from the surface to the composite peak at about *σ* = 0.25 is quite good. Deeper however, the cubic continues to rise to a peak at *σ* = 0.33. For 0.4 < *σ* < 0.8 it lies systematically more than a standard deviation above the composite, and it has much greater curvature than the approximately linear decrease of *G*_{c}(*σ*), such that the approaches to zero at *σ* = 1 are quite different.

## 5. Nonlocal transports

Hourly LES profiles are now used to determine functional dependencies of Γ_{m,}_{s} from across the boundary layer and relationships to *γ*_{m,s}. These are also compared to experience from the atmospheric boundary layer, where nonlocal fluxes of buoyancy are familiar.

### a. Nonlocal buoyancy flux

A buoyancy profile typically has zero gradient at between 10 and 60 discrete depths *d*_{G} (e.g., Fig. 3), where Γ_{s} is given by the buoyancy flux (e.g., Fig. 2), though some have more than 100 and others none. Figure 10a shows these Γ_{s} normalized by the surface buoyancy flux *B*_{0} but for clarity only from every tenth instance. Values are constrained to be unity at the surface and zero at (*d*/*d*_{F}) = 1, and overall a linear decrease between these limits is a reasonable fit to the data, with a linear regression giving a slope of −1.04 and intercept at 1.02. However, there is considerable spread. The blue trace joins the 10 plotted values of the 100 calculated from hour 30 of AprS, while the red joins the 10 from hour 3 of JunS. Although both profiles are nearly linear for *σ* > 0.3, the latter is steeper than −1, while the former is flatter. A linear decrease of buoyancy flux implies a constant cooling rate such that a well-mixed boundary layer would remain so, and the linearity of Γ_{s} in Fig. 10a suggests that nonlocal transport would produce such a tendency.

*G*

_{C}for 0.22 <

*σ*< 0.80 in Fig. 9 suggest the relationship

*R*

_{s}is determined from Fig. 10b. For this purpose all Γ

_{s}values from every unstable hour of all cases are used, except for those greater than

*B*

_{0}, because such behavior may be due to past conditions and indicative of insufficient equilibrium. The bin averages of Fig. 10b show that

*R*

_{s}= 4.7 ± 2.3 is approximately constant over the range 0.1 <

*σ*< 0.8 of available Γ

_{s}. However, the spread is considerable, with standard deviations typically about 50% of the mean, but perhaps it is less than expected given the spreads in Figs. 9 and 10a.

### b. Across-shear momentum flux

Both Γ_{υ} from (19) and _{υ} = ⟨*wυ*⟩ can be expressed as the product *K*_{m} = *w*_{m} *h G*_{C}. There is no clear separation between calm (blue) and wavy (red) points, though the former appear to have a high bias. Hence, the slope of the least squares fit through the origin (1.1 = ratio of the means) is greater than 1. The slope of the linear regression (0.96) is closer to 1, and the correlation coefficient is 0.84. Both Γ_{υ} and

Summary of various linear regressions, *y* = *R*_{0} + *R*_{1}*x*, where all velocities have units of cm s^{−1}. The correlation coefficient is *r*_{xy} and the *F* value is for the goodness-of-fit test. The ratio of the means *X–Y* coordinate system of Fig. 4, and hence to calculate the angle Ω. The Lagrangian shear is used in Fig. 12d, so Ω is different.

In addition to an across-shear momentum flux, Fig. 4 shows an opposing component of the wind stress *τ*_{Y}, where (−*τ*_{Y}/*ρ*) can be regarded as the kinematic across-shear wind stress. The two quantities Γ_{υ} and (−*τ*_{Y}/*ρ*) are related in Fig. 11b, with the statistics again given in Table 1. Although the correlation coefficient is 0.78, profiles of ⟨*wυ*⟩ tend to first increase, then decrease with depth. This dependency is similar to that of *G*_{C}(*σ*), so it is largely removed in Fig. 11c, with the correlation increasing to 0.91 and little systematic difference between the calm (blue) and wavy (red) cases.

Brown and Grant (1997) argue that because nonlocal transport depends on convection, it should depend on

*R*

_{m}equals 2.9 if it is taken to be the ratio of means, or 3.6 if the slope of the linear regression is used (Table 1) and the −0.13 cm

^{2}s

^{−2}intercept ignored. Equation (25) gives the same functional form for the dependencies of nonlocal fluxes Γ

_{m,s}on depth from the shape function and on surface flux.

### c. Nonlocal stratification and shear

_{m,s}=

*K*

_{m,s}

*γ*

_{m,s}and

*K*

_{m,s}=

*w*

_{m,s}

*hG*

_{C}(

*σ*) are useful constructs, then the nonlocal gradients are given by

*τ*

_{Y}/

*ρ*) for the across-shear momentum, or

*B*

_{0}for buoyancy. The only depth dependency is given by

*ϕ*

_{m,s}(

*σ*) within the square brackets, which for

*γ*

_{s}is consistent with an exponent

*p*= 1 in (17), as suggested by Fig. 8b.

*χ*equals 1 and the term in the square brackets of (27) becomes the reciprocal of

*R*

_{m}= 3.2, the range of

*σ*= 1 to 9.7 at

*ε*, while the corresponding range of

*R*

_{m}and

*R*

_{s}, largely compensated by the different stability functions with

*A*

_{m}= 14 and

*A*

_{s}= 25. There is no depth dependency in (28) and the

*σ*=

*ε*. However, the atmospheric results of Holtslag and Moeng (1991) suggest a value of

*σ*near 1, but outside the range of

*R*

_{s}= 4.7, but well within the 2.8–6.1 range given by

*R*

_{s}= 2.4 (4.7 minus 1 standard deviation).

### d. Countergradient momentum flux

*wu*⟩ of (18) and the across-shear ⟨

*wυ*⟩ = Γ

_{υ}from (19), but not Γ

_{u}. However, geometry gives ∂

_{z}

*V*′ = (−∂

_{z}

*U*)sin(Ω) and

*wu*⟩, such that the nonlocal flux contribution to ⟨

*wu*⟩ given by

*γ*

_{u}= (1 − cosΩ)∂

_{z}

*U*.

In Frech and Mahrt (1995) the nonlocal momentum flux is aligned with the bulk shear across the atmospheric boundary layer. This idea could be extended to the ocean LES, but the direction of the bulk shear would be constant with depth and usually closely aligned with the wind stress. Thus, Γ_{u} would vary with the kinematic wind stress aligned with the shear (−*τ*_{X}/*ρ*), in the same way Γ_{υ} varies with (−*τ*_{Y}/*ρ*) in Fig. 11c. However, Fig. 12c, shows that the LES do not behave this way. Not only do Γ_{u} and (−*τ*_{X}/*ρ*) usually have opposite signs, but they are otherwise unrelated and the functional form (27) does not hold for *γ*_{u}.

In Fig. 12b, the viscosity in the abscissa *K*_{m} uses the composite *G*_{C}(*σ*) in (22). With the (cosΩ) factor now included in the abscissa, departures from the 1:1 line are due to the scatter in *G*_{C} (Fig. 9) and to its biases when used to compute along-shear fluxes. These biases produce an apparent separation between the calm (blue) and wavy (red) results. For example, the high bias of *G*_{m} (calm) is reflected in the blue points falling above the 1:1 line at high values. An opposing bias in the wavy cases more than compensates, such that the regression line falls below for −⟨*wu*⟩ > 1.5 (cm s^{−1})^{2}. However, the opposite is found at smaller values where the majority of points fall, such that best fit through the origin has a slope 1.3 > 1 (Table 1).

Repeating this *G*_{C}(*σ*) exercise, but with everything else computed using the Lagrangian shear yields the results shown in Fig. 12d. The calm (blue) points are unchanged. With Stokes forcing, however, the much larger values of Lagrangian ∂_{z}|**U** + **U**_{S}| in (3) make *χ*_{m}(*ξ*) much greater than given by (14), such that *w*_{m} in (23) is very much reduced. The resulting reduction in *K*_{m} of Fig. 12d is much greater than increases in the shear at deeper depths, which leads to the pronounced low bias relative both to −⟨*wu*⟩ and to the calm values in Fig. 12d. This negative result is quantified in Table 1. It is further evidence that deeper shears are decoupled from the Stokes shear in the surface layer where it can dominate the Lagrangian.

## 6. Discussion and conclusions

Large-eddy simulations have been used to study the turbulent boundary layer in conditions representative of the Southern Ocean Flux Site during fall mixed layer deepening in 2010. These LES are intended only to mimic nature, which lessens requirements on surface forcing, for example. Although, such simplifications compromise verification, they are acceptable here, because the observational requirements are very far from being satisfied at SOFS. Nonetheless, the ARGO float profiles do provide realistic initial conditions and the large variations in the buoyancy forcing and in both the wind speed and direction were observed at SOFS. Also, the wavy LES with Stokes forcing, and the calm without, span nearly the full range of expected wave regimes.

### a. Basic results

Insofar, as the LES are faithful solutions of the Navier–Stokes equations, they provide some basic empirical results that should be represented by simpler models and parameterizations of turbulent ocean boundary layers, both statistical (STM) and empirical (ETM). The underlying construct of the analysis is general. It partitions the vertical fluxes into two terms: one proportional to the local gradients, and nonlocal terms that account for everything else, as expressed by (16) and (18), but makes no assumptions.

Evidence for nonlocal buoyancy flux in the unstable atmosphere is plentiful from observations (Deardorff 1966) and from LES (Moeng and Wyngaard 1989). In our unstable ocean LES, it is directly given by the positive flux Γ_{s} from Fig. 2, at each depth where the buoyancy gradient of Fig. 3 is zero. The nonlocal buoyancy gradient of Deardorff (1972) is directly given by the gradient*γ*_{s} in Fig. 3 at the depth where the flux of Fig. 2 equals zero.

The coordinate rotations of Fig. 4 are general, so nonlocal momentum transport is unequivocal in the ocean LES, because of the nonzero angles Ω between the LES stress and shear vectors (e.g., Fig. 5). The angle Ω is rarely zero and depends on Stokes drift and on the choice of shear between Eulerian and Lagrangian. The across-shear flux is proportional to tan(Ω), so it has similar dependencies, and is seldom negligible (|Ω| < 5°). It is entirely nonlocal, and is a direct measure of Γ_{υ}. Furthermore, Figs. 11b–d all show that this flux is related to (−*τ*_{Y}/*ρ*), as defined in Fig. 4. It can be of either sign, whereas the nonlocal contribution to the along-shear momentum flux is always positive (countergradient). Also where Ω is nonzero, there is a shear ∂_{z}*V*′ perpendicular to the momentum flux vector that gives a nonlocal shear

Nonlocal momentum transport can be similarly inferred from the lidar observations in the atmospheric boundary layer reported by Berg et al. (2013). Specifically, they also find a systematic nonzero angle between the shear and momentum flux vectors. Although they attribute this behavior to mesoscale effects, it is very similar to that of Ω, especially in Fig. 5a within 100 m of the surface and before hour 20. By design (L19) there is no mesoscale activity in the ocean LES.

### b. Similarity theory

Further progress is enabled by invoking specific constructs, beginning with Monin–Obukhov similarity theory in the surface layer, as extended in L19 for surface wave (Stokes) forcing. This theory says that the nondimensional stratification Ψ_{s} from (2) and shear Ψ_{m} from (3) should depend on the product of buoyancy similarity functions *ϕ*_{m,s} of a parameter such as *ζ* from (8) times Stokes similarity functions *χ*_{m,s} of a parameter such as *ξ* from (13). In L19, surface wave effects are found to greatly reduce the stratification and the Eulerian shear in the surface layer, with vertical fluxes maintained by viscosity and diffusivity enhancements given by the reciprocals of *χ*_{m,s}(*ξ*) given by (14) and (15). The depth dependencies of these reduced gradients in the surface layer are mostly captured by the dependency of *ζ* on (*d*/*L*), with the ratio of Stokes production of TKE from (6) to shear production from (5) providing a refinement according to (8). To the extent similarity theory is valid, these functions and dependencies should be universal and reproduced by LES in different regimes, as well as by simpler models and parameterizations.

In contrast, analogous functions are not found using the Lagrangian shear to give Ψ_{m} from (3). The Lagrangian shear in the surface layer is dominated by the prescribed Stokes shear, which is a property of the surface wave field. The dimensional analysis of similarity theory, however, should only apply to a single closed system, namely, the turbulent surface layer. It should not be expected to span two systems.

### c. Eddy diffusivity and viscosity

Eddy diffusivity and viscosity are traditional turbulence concepts, defined as the ratio of the turbulent flux (buoyancy or momentum), to the local mean gradient (stratification or shear). For surface fluxes passing through the surface layer to the boundary layer interior, similarity theory gives the associated vertical viscosity *K*_{m} and diffusivity *K*_{s} defined by (4). This concept fails deeper in the boundary layer, because nonzero fluxes are found where the local gradients are zero. However, it can be modified by defining a nonlocal viscosity _{z}*U*, such that the nonlocal contribution Γ_{u} is always countergradient.

The extension of the surface layer viscosity and diffusivity across the boundary layer according to (22) is a specific construct of some ETM, such as those based on Troen and Mahrt (1986). However, no assumptions are made about the shape functions *G*_{m,s}, which are found to scatter a great deal about a composite curve that is distinct from the cubic of O’Brien (1970). The scatter may possibly be produced by some STM, but is missing from ETM, though it could be generated stochastically. It is not random in the sense that LES values from any particular hour all tend to fall above or below the average, but could be more random in time. It is particularly big for *G*_{s} from JunS, because of a precision issue.

There is a very close match between the composite and the average profiles of both *G*_{m} (wavy) and *G*_{s} (calm). However, it is not clear if the higher values of *G*_{m} (calm) and lower values of *G*_{s} (wavy) are robust. These differences suggest that there may be missing, or suboptimal dependencies, or possibly that averaging over an even wider range of forcing conditions would give greater consistency. Also, buoyancy differences *δ*_{υ}Θ are required to be positive, so negative fluctuations about a small values are limited, but positive fluctuations are unbounded, and the resulting rectification could bias the gradient high, and hence *G*_{s} low. Further investigation of these issues could lead to shape functions that are more representative of eddy diffusivity and viscosity. However, to be broadly applicable refinements should be based on a wider range of ocean regimes, including more moderate wave conditions and shallower boundary layers, as well as on solutions with improved precision from more than one LES model.

### d. Boundary layer physics

*G*

_{m}and

*G*

_{s}in Fig. 9. This agreement relates surface fluxes, surface layer gradients, and deeper boundary layer fluxes and gradients. Rearrangement of terms in these calculations, with some cancellations yields

*wθ*⟩, ∂

_{z}Θ, ⟨

*wυ*⟩, and ∂

_{z}

*V*′ on the left hand side are from deeper in the boundary layer. In particular, ∂

_{z}Θ must come from

*d*

_{F}, where the stratification is nonzero, and so it is adjusted to the depth of the other terms with ratio of

*ϕ*

_{s}at the different depths. The terms in (32) are arranged such that the right- and left-hand side of the ~ can be interpreted as turbulent Prandtl numbers in the surface layer and deeper in the boundary layer, respectively.

To further the discussion, these Prandtl numbers are computed wherever all the terms of (32) are available from the calculations of Fig. 9 plus JunS. For 174 instances of *σ* less than about 0.3 the mean ± 1 standard deviation of the ratio (*G*_{m}:*G*_{s}) is about 1 ± 0.5, and the ratio of deep to surface layer Prandtl numbers is higher, 1.1 ± 0.6. Figure 9 indicates that deeper these numbers should increase, and over all *σ* < 1 they both become about 2.0 ± 1.0. The Stokes similarity functions say that the addition of Stokes forcing reduces the surface layer shear more than the stratification (*χ*_{s} > *χ*_{m}), such that the term in the square brackets of (32) increases. Compensating changes in the fluxes, gradients, and ratios on the left-hand side are expected, but with realistic forcing the different ratios show that the compensation is not exact. Nevertheless, these relationships quantify that to about *σ* = 0.3 the boundary layer is strongly coupled to the surface layer, and that the deeper boundary layer does not become independent of the surface layer.

All the terms of (32) come directly from the LES, so to some degree such relationships appear to be a basic property of the turbulent LES solution of the Navier–Stokes equations in a boundary layer. Hence, they should be reproduced by simpler models and parameterizations.

No such relationships are evident when the Lagrangian shear is used in the surface layer (L19), and/or the boundary layer (Fig. 7). Therefore, the Stokes shear in the surface layer appears to be much more decoupled from deeper, mostly Eulerian, shears. This behavior supports the L19 treatment of Stokes drift has a forcing through the contribution of its shear to the production of TKE (6), as expressed by *P*_{S} in the stability parameter (8), and Stokes parameter *ξ* (13).

Additional support for the Wyngaard (1982) view is provided by (27) where *γ*_{s} and *γ*_{u} have very similar functional forms, provided the surface stress is taken to be the across-shear component of the wind stress. Furthermore, extrapolation to the convective limit shows that the coefficients are essentially the same; ranging from about 5 at *σ* = 1 to 11 at *ε*. Although the equivalent coefficients from studies of atmospheric convection are not depth dependent, their range is similar (2.5–10). Absent distinct boundary layer physics, such a close resemblance in the nonlocal behavior of ocean momentum, ocean buoyancy, and atmospheric convection would not be expected.

## Acknowledgments

This work was made possible by support from the U.S. Department of Energy (DOE) under solicitation DE-FOA-0001036, Climate and Earth System Modeling: SciDAC and Climate Variability and Change, Grant SC-00126005. The patience of Dorothy Koch is gratefully acknowledged, as are the contributions of other principal investigators; Todd Ringler, Gokhan Danabasoglu, and Matt Long and contributors; Alice DuVivier and Justin Small. A special thanks to Leonel Romero for providing the essential Stokes wave driving. The National Center for Atmospheric Research (NCAR) is sponsored by the National Science Foundation. The four SOFS simulations utilized resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract DE-AC02-05CH11231. The idealized simulations utilized high-performance computing on Yellowstone (ark:/85065/d7wd3xhc) provided by NCAR’s Computational and Information Systems Laboratory. The simulations can be made available upon request to the corresponding author, in accordance with the data policies of DOE, NSF, and NCAR.

## REFERENCES

Belcher, S., and et al. , 2012: A global perspective on mixing in the ocean surface boundary layer.

,*J. Geophys. Res.***39**, L18605, https://doi.org/10.1029/2012GL052932.Berg, J., J. Mann, and E. Patton, 2013: Lidar-observed stress vectors and veer in the atmospheric boundary layer.

,*J. Atmos. Oceanic Technol.***30**, 1961–1969, https://doi.org/10.1175/JTECH-D-12-00266.1.Berrisford, P., and et al. , 2011: The ERA-Interim Archive: Version 2.0. ERA Rep. Series 1, 23 pp., https://www.ecmwf.int/en/elibrary/8174-era-interim-archive-version-20.

Bretherton, C., and S. Park, 2009: A new moist turbulence parameterization in the Community Atmosphere Model.

,*J. Climate***84**, 3422–3448, https://doi.org/10.1175/2008JCLI2556.1.Brown, A., and A. Grant, 1997: Non-local mixing of momentum in the convective boundary layer.

,*Bound.-Layer Meteor.***84**, 1–22, https://doi.org/10.1023/A:1000388830859.Burchard, H., and et al. , 2008: Observational and numerical modeling methods for quantifying coastal ocean turbulence and mixing.

,*Prog. Oceanogr.***76**, 399–442, https://doi.org/10.1016/j.pocean.2007.09.005.Canuto, V., F. Minotti, C. Ronchi, M. Ypma, and O. Zeman, 1994: Second-order closure PBL model with new third-order moments: Comparison with LES data.

,*J. Atmos. Sci.***51**, 1605–1618, https://doi.org/10.1175/1520-0469(1994)051<1605:SOCPMW>2.0.CO;2.Canuto, V., Y. Cheng, and A. Howard, 2007: Non-local ocean mixing model and a new plume model for deep convection.

,*Ocean Modell.***16**, 28–46, https://doi.org/10.1016/j.ocemod.2006.07.003.Cheng, Y., V. Canuto, and A. Howard, 2002: An improved model for the turbulent PBL.

,*J. Atmos. Sci.***59**, 1550–1565, https://doi.org/10.1175/1520-0469(2002)059<1550:AIMFTT>2.0.CO;2.Craik, A., and S. Leibovich, 1976: A rational model for Langmuir circulations.

,*J. Fluid Mech.***73**, 401–426, https://doi.org/10.1017/S0022112076001420.Danabasoglu, G., S. Bates, B. Briegleb, S. Jayne, M. Jochum, W. Large, S. Peacock, and S. Yeager, 2012: The CCSM4 ocean component.

,*J. Climate***25**, 1361–1389, https://doi.org/10.1175/JCLI-D-11-00091.1.Deardorff, J., 1966: The counter-gradient heat flux in the atmosphere and in the laboratory.

,*J. Atmos. Sci.***23**, 503–506, https://doi.org/10.1175/1520-0469(1966)023<0503:TCGHFI>2.0.CO;2.Deardorff, J., 1972: Theoretical expression for the countergradient vertical heat flux.

,*J. Geophys. Res.***77**, 5900–5904, https://doi.org/10.1029/JC077i030p05900.Downes, S., and et al. , 2014: An assessment of Southern Ocean water masses and sea ice during 1988-2007 in a suite of interannual CORE-II simulations.

,*Ocean Modell.***18**, 67–94, https://doi.org/10.1016/j.ocemod.2015.07.022.DuVivier, A., W. Large, and R. Small, 2018: Argo observations of the deep mixing band in the Southern Ocean: A salinity modeling challenge.

,*J. Geophys. Res. Oceans***123**, 7599–7617, https://doi.org/10.1029/2018JC014275.Frech, M., and L. Mahrt, 1995: A two-scale mixing formulation for the atmospheric boundary layer.

,*Bound.-Layer Meteor.***73**, 91–104, https://doi.org/10.1007/BF00708931.Grant, A., and S. Belcher, 2009: Characteristics of Langmuir turbulence in the ocean mixed layer.

,*J. Phys. Oceanogr.***39**, 1871–1887, https://doi.org/10.1175/2009JPO4119.1.Harcourt, R., 2013: A second-moment closure model of Langmuir turbulence.

,*J. Phys. Oceanogr.***43**, 673–697, https://doi.org/10.1175/JPO-D-12-0105.1.Harcourt, R., 2015: An improved second-moment closure model of Langmuir turbulence.

,*J. Phys. Oceanogr.***45**, 84–103, https://doi.org/10.1175/JPO-D-14-0046.1Harcourt, R., and E. D’Asaro, 2008: Large eddy simulation of Langmuir turbulence in pure wind seas.

,*J. Phys. Oceanogr.***38**, 1542–1562, https://doi.org/10.1175/2007JPO3842.1.Högström, U., 1988: Non-dimensional wind and temperature profiles in the atmospheric surface layer.

,*Bound.-Layer Meteor.***42**, 55–78, https://doi.org/10.1007/BF00119875.Holtslag, A., and C.-H. Moeng, 1991: Eddy diffusivity and counter gradient transport in the convective atmospheric boundary layer.

,*J. Atmos. Oceanic Technol.***48**, 1690–1698, https://doi.org/10.1175/1520-0469(1991)048<1690:EDACTI>2.0.CO;2.Holtslag, A., and B. Boville, 1993: Local versus nonlocal boundary-layer diffusion in a global climate model.

,*J. Climate***6**, 1825–1842, https://doi.org/10.1175/1520-0442(1993)006<1825:LVNBLD>2.0.CO;2.Kantha, L., and C. Clayson, 1994: An improved mixed layer model for geophysical applications.

,*J. Geophys. Res.***99**, 25 235–25 266, https://doi.org/10.1029/94JC02257.Keller, L., and A. Friedmann, 1924: Differentialgleichungen fur die turbulente bewegung einer kompressiblen flussigkeit.

, Waltman, 395–405.*Proceedings of the First International Congress on Applied Mechanics*Kraus, E., and J. Turner, 1967: A one-dimensional model of the seasonal thermocline, II. The general theory and its consequences.

,*Tellus***19**, 98–105, https://doi.org/10.3402/tellusa.v19i1.9753.Large, W., J. McWilliams, and S. Doney, 1994: Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization.

,*J. Atmos. Sci.***32**, 363–403, https://doi.org/10.1029/94RG01872.Large, W., E. Patton, A. DuVivier, and P. Sullivan, 2019: Similarity theory in the surface layer of large-eddy simulations of the southern ocean with waves.

,*J. Phys. Oceanogr.***49**, 2165–2187, https://doi.org/10.1175/JPO-D-18-0066.1.Lesieur, M., 1997:

. 3rd ed. Fluid Mechanics and Its Applications, Vol. 84, Kluwer Academic, 563 pp.*Turbulence in Fluids*Li, Q., and B. Fox-Kemper, 2017: Assessing the effects of Langmuir turbulence on the entrainment buoyancy flux in the ocean surface boundary layer.

,*J. Phys. Oceanogr.***47**, 2863–2886, https://doi.org/10.1175/JPO-D-17-0085.1.Mailhot, J., and R. Benoit, 1982: A finite-element model of the atmospheric boundary layer suitable for use with numerical weather prediction models.

,*J. Atmos. Sci.***39**, 2249–2266, https://doi.org/10.1175/1520-0469(1982)039<2249:AFEMOT>2.0.CO;2.McWilliams, J., P. Sullivan, and C.-H. Moeng, 1997: Langmuir turbulence in the ocean.

,*J. Fluid Mech.***334**, 1–30, https://doi.org/10.1017/S0022112096004375.Mellor, G., and T. Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems.

,*Rev. Geophys.***20**, 851–875, https://doi.org/10.1029/RG020i004p00851.Moeng, C.-H., 1984: A large-eddy simulation model for the study of planetary boundary-layer turbulence.

,*J. Atmos. Sci.***41**, 2052–2062, https://doi.org/10.1175/1520-0469(1984)041<2052:ALESMF>2.0.CO;2.Moeng, C.-H., and J. C. Wyngaard, 1989: Evaluation of turbulent transport and dissipation closures in second-order modeling.

,*J. Atmos. Sci.***46**, 2311–2330, https://doi.org/10.1175/1520-0469(1989)046<2311:EOTTAD>2.0.CO;2.Monin, A., and A. Obukhov, 1954: Basic laws of turbulent mixing in the surface layer of the atmosphere.

,*Tr. Geofiz. Inst., Akad. Nauk SSSR***24**(151), 163–187.Noh, Y., W. Cheon, S. Hong, and S. Raasch, 2003: Improvement of the K-Profile Model for the planetary boundary layer based on large eddy simulation data.

,*Bound.-Layer Meteor.***107**, 401–427, https://doi.org/10.1023/A:1022146015946.Noh, Y., H. Min, and S. Raasch, 2004: Large eddy simulation of the ocean mixed layer: The effects of wave breaking and Langmuir circulation.

,*J. Phys. Oceanogr.***34**, 720–735, https://doi.org/10.1175/1520-0485(2004)034<0720:LESOTO>2.0.CO;2.O’Brien, J., 1970: A note on the vertical structure of the eddy exchange coefficient in the planetary boundary layer.

,*Rev. Geophys.***27**, 1213–1215, https://doi.org/10.1175/1520-0469(1970)027<1213:ANOTVS>2.0.CO;2.Park, S., 2014: A unified convection scheme (UNICON). Part I: Formulation.

,*J. Atmos. Sci.***71**, 3902–3930, https://doi.org/10.1175/JAS-D-13-0233.1.Price, J., R. Weller, and R. Pinkel, 1986: Diurnal cycling: Observations and models of the upper ocean’s response to diurnal heating, cooling and wind mixing.

,*J. Geophys. Res.***91**, 8411–8427, https://doi.org/10.1029/JC091iC07p08411.Romero, L., and W. K. Melville, 2010: Numerical modeling of fetch-limited waves in the Gulf of Tehuantepec.

,*J. Phys. Oceanogr.***40**, 466–486, https://doi.org/10.1175/2009JPO4128.1.Sallée, J., E. Shuckburgh, N. Bruneau, A. Meijers, T. Bracegirdle, and Z. Wang, 2013: Assessment of southern ocean mixed-layer depths in CMIP5 models: Historical bias and forcing response.

,*J. Geophys. Res. Oceans***110**, 1845–1862, https://doi.org/10.1002/jgrc.20157.Schultz, E., S. Josey, and R. Verein, 2012: First air-sea flux mooring measurements in the Southern Ocean.

,*Geophys. Res. Lett.***39**, L16606, https://doi.org/10.1029/2012GL052290.Siebesma, A., P. Soares, and J. Teixeira, 2007: A combined eddy-diffusivity mass-flux approach for the convective boundary layer.

,*J. Atmos. Sci.***64**, 1230–1248, https://doi.org/10.1175/JAS3888.1Smyth, W., E. Skylingstad, G. Crawford, and H. Wijesekera, 2002: Nonlocal fluxes and stokes drift effects in the K-profile parameterization.

,*Ocean Dyn.***52**, 104–115, https://doi.org/10.1007/s10236-002-0012-9.Sullivan, P., J. McWilliams, and C. Moeng, 1994: A subgrid-scale model for large-eddy simulation of planetary boundary-layer flows.

,*Bound.-Layer Meteor.***71**, 247–276, https://doi.org/10.1007/BF00713741.Sullivan, P., J. McWilliams, and W. Melville, 2004: The oceanic boundary layer driven by wave breaking with stochastic variability. Part 1. Direct numerical simulations.

,*J. Fluid Mech.***507**, 143–174, https://doi.org/10.1017/S002211200700897XSullivan, P., J. McWilliams, and W. Melville, 2007: Surface gravity wave effects in the oceanic boundary layer: Large-eddy simulation with vortex force and stochastic breakers.

,*J. Fluid Mech.***593**, 405–452, https://doi.org/10.1017/S002211200700897X.Tan, Z., C. Kaul, K. Pressel, Y. Cohen, T. Schneider, and J. Teixeira, 2018: An extended eddy-diffusivity mass-flux scheme for unified representation of subgrid-scale turbulence and convection.

,*J. Adv. Model. Earth Syst.***10**, 770–800, https://doi.org/10.1002/2017MS001162.Tolman, H., 2002: User manual and system documentation of WAVEWATCH-III version 2.22. Tech. rep., NOAA/NWS/NCEP/MMAB Tech. Note 222, 133 pp., https://polar.ncep.noaa.gov/mmab/papers/tn222/MMAB_222.pdf.

Troen, I., and L. Mahrt, 1986: A simple model of the atmospheric boundary layer: Sensitivity to surface evaporation.

,*Bound.-Layer Meteor.***37**, 129–148, https://doi.org/10.1007/BF00122760.van Roekel, L., B. Fox-Kemper, P. Sullivan, P. Hamlington, and S. Haney, 2012: The form and orientation of Langmuir cells for misaligned winds and waves.

,*J. Geophys. Res.***117**, C05001, https://doi.org/10.1029/2011JC007516.van Roekel, L., and et al. , 2018: The KPP boundary layer scheme: Revisiting its formulation and benchmarking one-dimensional ocean simulations relative to LES.

,*J. Adv. Model. Earth Syst.***10**, 2647–2685, https://doi.org/10.1029/2018MS001336.Wang, D., J. McWilliams, and W. Large, 1998: Large eddy simulation of the diurnal cycle of deep equatorial turbulence.

,*J. Phys. Oceanogr.***28**, 129–148, https://doi.org/10.1175/1520-0485(1998)028<0129:LESOTD>2.0.CO;2.Weijer, W., and et al. , 2012: The Southern Ocean and its climate in CCSM4.

,*J. Climate***25**, 2652–2675, https://doi.org/10.1175/JCLI-D-11-00302.1.Wyngaard, J., 1982: Lectures on the planetary boundary layer.

, D. Lilly and T. Gal-Chen, Eds., NATO ASI Series C, Vol. 114, D. Reidel, 603–650.*Mesoscale Meteorology—Theories, Observations and Models*Wyngaard, J. C., 2010:

. Cambridge University Press, 406 pp.*Turbulence in the Atmosphere*

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ARGO data are collected and made freely available by the International Argo Program and the national programs that contribute to it (http://www.argo.ucsd.edu, http://argo.jcommops.org). The ARGO Program is part of the Global Ocean Observing System.