## 1. Introduction

The Southern Ocean plays a disproportionately important role in the climate system by taking up more than 40% of the global ocean’s anthropogenic carbon inventory (Khatiwala et al. 2009) and by ventilating a significant fraction of recently warmed deep waters (Purkey and Johnson 2010; Kouketsu et al. 2011). Therefore, it is problematic that ocean general circulation models (OGCMs) have long struggled to represent the Southern Ocean faithfully. In coupled solutions of the Community Earth System Model, for example, the zonal-mean wind stress over the Southern Ocean is much stronger than observed. Nonetheless, ocean mixed layer depth (MLD) is still substantially too shallow in key regions of water mass formation, thus implicating deficiencies in the ocean component (Danabasoglu et al. 2012; Weijer et al. 2012). Similar biases remain at eddy-resolving resolutions, suggesting problems with representations of vertical physics (e.g., Fox-Kemper et al. 2011; Belcher et al. 2012). The magnitude of these shallow biases is truly remarkable, exceeding 400 m in midlatitudes (40°–60°S) during later winter (e.g., Downes et al. 2014; DuVivier et al. 2018), and is the motivation for the present work. It is one part of a larger Southern Ocean project to extend the bias attribution also to surface forcing, the general circulation, the subsurface salinity structure, model resolution and other factors (e.g., DuVivier et al. 2018).

The understanding and hence modeling of Southern Ocean mixing physics has long been plagued by a paucity of observations, but recent observational datasets provide an opportunity for progress. The Southern Ocean Flux Station (SOFS) is part of the Southern Ocean Time Series Observatory of the Australian Integrated Marine Observing System (http://imos.org.au/facilities/deepwatermoorings/sots). It is located in a region of extreme weather events and shallow winter MLD biases, 580 km southwest of Tasmania near 140°E, 47°S. In addition, the ARGO float program^{1} provides high vertical resolution temperature and salinity in the region and throughout the Southern Ocean.

Still missing, however, are direct measurements of turbulent fluxes in the open ocean. Instead, recent research has utilized idealized large-eddy simulation (LES; e.g., McWilliams et al. 1997; Grant and Belcher 2009; Harcourt and D’Asaro 2008; Roekel et al. 2012; 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. Furthermore, Belcher et al. (2012) used some of these LES results, available forcing data and simple energy scaling to argue that “wave-forcing and hence Langmuir turbulence could be important over wide areas of the ocean and in all seasons in the Southern Ocean.”

A solid foundation for understanding near surface turbulence is Monin–Obukhov similarity theory (Monin and Obukhov 1954; Wyngaard 2010). However, ocean applications (e.g., Large et al. 1994) rely on empiricism from the atmospheric surface layer, but not directly validated in the ocean. Therefore, similarity theory is reviewed and extended to Stokes forcing of the ocean in section 2. Our LES model, meteorological forcing, and calculations of Stokes drift and of boundary layer depth are described in section 3, where examples of the turbulent solutions provide qualitative validation of the simulations and justification of various configuration choices. In section 4, Southern Ocean LES are used to modify the empirical buoyancy similarity functions for the ocean and to formulate new Stokes similarity functions. A practical parameterization is developed and evaluated in section 5. A similar evaluation of alternatives follows in section 6, before the discussion and conclusion of section 7.

## 2. Monin–Obukhov similarity theory

Semiempirical Monin–Obukhov similarity theory is well established in the atmospheric surface layer (Foken 2006). This layer begins at a distance *d*_{o} from the surface and outside the direct influence of the surface and its roughness elements. It extends to a fraction *ε* of the planetary boundary layer depth *h*. The theory states that the only parameters (independent variables) governing the structure (dependent variables) of a neutral surface layer are the distance *d* from the boundary and the kinematic wind stress, *u**^{2} = *u** is the friction velocity, and *ρ* is fluid density. Throughout this layer the wind-driven Eulerian flow is generally aligned with the stress, so defining the mean components as *U*, aligned; *V*, orthogonal in the horizontal; and *W*, the upward vertical (*z* direction) gives a natural Cartesian coordinate system, where *W* is zero and *V* is usually much smaller than *U*. The velocity is then given by {*U* + *u*′, *V* + *υ*′, *w*′}, where *u*′, *υ*′, and *w*′ are fluctuations about the mean *U*, *V*, 0}.

*κ*= 0.40. The addition of buoyancy, with mean

*B*

_{0}, to the system adds one more independent variable,

*B*

_{0}/

*u**, but not a dimension. According to the Buckingham-pi theorem, the nondimensional shear and stratification then become functions of one dimensionless group (Wyngaard 2010), which is traditionally the stability parameter

*L*and the convective velocity scale

*B*

_{0}is defined to be positive when it stabilizes the surface layer, such that with unstable forcing

*ζ*and

*B*

_{0}

*h*) is positive. The nondimensional gradients are given by

*ζ*. Over land, the vertical shear

*ξ*:

*ξ*contains the important information about the Stokes forcing, such as the turbulent Langmuir number, La =

The search for Stokes similarity functions, *h*, as suggested by Harcourt and D’Asaro (2008), makes little difference.

### a. The turbulent kinetic energy equation

*P*

_{U},

*P*

_{S}and

*P*

_{B}, respectively. Variations in these parameters arise from failures of surface forcing, as given by

*u**, La

^{−2}, and (−

*B*

_{0}

*h*), to account for all the variability in TKE production in the surface layer. Much of this variability is due to misalignment of the stress vector (

*P*

_{U}or

*P*

_{S}negative, for example when there are counter inertial currents, or wave components propagating into the wind.

*K*

_{m}and diffusivity

*K*

_{s}associated with these transfers, and hence the turbulent Prandtl number, Pr = (

*K*

_{m}/

*K*

_{s}). The theory says that they depend, respectively, on turbulent velocity scales

*w*

_{m}and

*w*

_{s}:

Thus, the velocities *w*_{m} and *w*_{s} become scales for vertical fluxes in the surface layer and perhaps beyond, and possibly for other turbulent statistics, such as TKE, with the LES providing a means of verification. This scaling is fundamentally different from solving (9) either diagnostically assuming steady state, or prognostically. A classic example of the latter is the Mellor and Yamada (1982) level-2.5 boundary layer, where the velocity scales become proportional to the independent variable, TKE^{1/2}. A recent diagnostic example is Chor et al. (2018), where the dissipation in (9) becomes the key independent variable, with

In section 4b, LES of the Southern Ocean boundary layer in April and June are used to extend similarity theory to include Stokes-forced surface layers by formulating *P*_{U}, *P*_{S}, and *P*_{B} accounting for misalignment and other sources of variability. This extension does not involve a complete new dimensional analysis with

## 3. The large-eddy simulations of the Southern Ocean

Our LES model, as adapted to simulate ocean boundary layers, is well tested and documented (e.g., Sullivan et al. 2012; Roekel et al. 2012; Kukulka et al. 2013). The model dynamics integrate the wave-averaged, incompressible, and Boussinesq Craik–Leibovich equation set (McWilliams et al. 1997). The subgrid-scale (SGS) fluxes are modeled with the eddy viscosity described in Moeng (1984), Sullivan et al. (1994), and McWilliams et al. (1997). The additional terms arising from phase averaging over the surface waves include: Stokes–Coriolis, vortex force, and a Bernoulli pressure head in the momentum equations, and horizontal advection by Stokes drift in the scalar equations, as well as additional production of subgrid-scale energy by vertical gradients of Stokes drift (Sullivan et al. 2007). Submesoscale turbulence structures do not develop, because neither mesoscale straining by ocean eddies, nor unstable frontal features are imposed, as in Hamlington et al. (2014), McWilliams (2016), and Sullivan and McWilliams (2018). Wave breaking (e.g., Sullivan et al. 2004, 2007) is not considered. The ocean surface is flat, with a *w*′ = 0 boundary condition, but in contrast to LES over land neither *u*′ = 0, nor *υ*′ = 0, are imposed. The distance *d*_{o} needs to be deeper than the influence of these boundary conditions, as well as other surface influences, so it is determined empirically (section 4).

*g*(1 −

*g*[

*α*(

*T*−

*T*

_{o}) −

*β*(

*S*−

*S*

_{o})], where

*g*= 9.81 m s

^{−2}is gravitational acceleration and

*T*,

*S*, and

^{−3}are, respectively, ocean potential temperature, salinity, and density at a typical SOFS autumn surface temperature

*T*

_{o}= 12°C and salinity

*S*

_{o}= 35 psu. Thermal expansion is a constant,

*α*= 1.9 × 10

^{−4}°C

^{−1}, and

*β*= 7.8 × 10

^{−4}is haline contraction. The surface buoyancy forcing

*B*

_{0}can include both heat and freshwater fluxes. It is related to an equivalent surface heat flux into the ocean

*Q*

_{0}by

*C*

_{p}is ocean heat capacity and (

*C*

_{p}) = 4.1 MJ m

^{−3}K

^{−1}.

Seven large-eddy simulations are documented in Table 1. Four of these mimic two specific time periods at SOFS in April and June of 2010. The two cases of Stokes forcing are denoted as AprS and JunS, and the No-Stokes as AprN and JunN. The SOFS air–sea flux mooring (Schultz et al. 2012) provides the meteorological forcing. These simulations are spun up by holding the forcing fixed for 12 h in order to generate turbulence over the depth of the boundary layer. In the other three cases, D24S, D12S, and D06S, winds are idealized and generate surface waves that are always perfectly aligned, there is a small constant surface buoyancy loss (^{−2}), and *h* is initially 96 m. The wind blows steadily to the east for 12 h, then drops from 20 to 4 m s^{−1} on 24-, 12-, and 6-h time scales, respectively, and afterward remains constant. All seven LES use the SOFS (47°S) Coriolis parameter, ^{−1}, so the inertial period is 16.4 h.

Large-eddy simulations of the Southern Ocean in April and June 2010. The respective start dates are 29 Apr and 7 Jun 2010. The start times, *L*_{x} × *L*_{y} in the horizontal and *L*_{z} in the vertical and covered by a mesh of *N*_{x} × *N*_{y} × *N*_{z} grid cells. The vertical grids stretch from a finest resolution of

The choice of computational domain {*L*_{x}, *L*_{y}, *L*_{z}}, and mesh size {*N*_{x}, *N*_{y}, *N*_{z}} is based on past experience and then confirmed (section 3d) by inspecting the turbulent flow. The horizontal domain needs to be sufficiently wide to permit multiple coherent largest scale structures to develop independent of the periodic sidewall boundary conditions, while the mesh resolution needs to be sufficiently fine to capture small scales near the surface. Based on atmospheric and oceanic LES (e.g., Moeng and Sullivan 1994; Sullivan and Patton 2011; Sullivan et al. 2012) the horizontal domain should be about 5 times the boundary layer depth to minimize the influence of periodic sidewalls. The vertical extent of the domain *L*_{z} is about twice *h* to permit a smooth transition between the turbulent and stably stratified layers. The vertical grid is stretched algebraically with the ratio between neighboring cell thicknesses always less than 1.0035. Its resolution is finest, *e*-folding depth. Experience shows that the subgrid-scale model in LES tolerates a small amount of anisotropy, *L*_{x}/*N*_{x} < 3Δ*Z*_{min} (Sullivan et al. 2003). All these considerations lead to the domain choices in Table 1, which then give the maximum cell thickness

### a. Varying Southern Ocean surface conditions

The highly variable LES forcing is highlighted by overlapping the time series in Fig. 1. In April (blue traces), high winds (>20 m s^{−1}) persisted for nearly a day prior to *t* = 0 (Table 1), so there are large-amplitude waves with significant wave height *u** > 0.02 m s^{−1}; Fig. 1a), steady (Fig. 1d), and aligned with the surface Stokes drift (Fig. 1c), and there is substantial buoyancy loss (Fig. 1e). The wind weakens over next 4 h (A12–15; Fig. 1a), and there is significant diurnal variation in *Q*_{0}. The wind speed drops to 4 m s^{−1} in the third regime, A16–22, while the rise of La^{−2} to 15 (Fig. 1d), indicates that the waves (*H*_{S} ~ 5 m) remain stronger than wind-wave equilibrium. Over the next 9 h, A23–31, the winds are light, highly variable in direction, and hence greatly misaligned with the waves. The next regime A32–43 is characterized by a night–day–night transition (−10 < *Q*_{0} < 250 W m^{−2}), and includes all the stable forcing. The winds of the final April regime, A44–50, rise from about 6 to 15 m s^{−1} and blow increasingly toward the south, while the buoyancy forcing remains near neutral (^{−2}).

Characteristics of the April, June, and idealized decay forcing regimes from *t*_{1} to *t*_{2} (h) from time *t* = 0 (Table 1), with ranges when there is significant variability. The right-most column gives the empirical shallow limit of the surface layer *d*_{o} (section 4), with both No-Stokes (N) and Stokes forcing (S).

In the June case (Fig. 1; magenta), the simulations begin just after a wind and wave lull, so the moderate buoyancy loss is a major surface forcing initially and over the first June forcing regime, J1–4. Three subsequent regimes are also characterized in Table 2. 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), with *Q*_{0} becomes very close to zero, because of the solar heating. During the last June regime, J34–41, the wind weakens with highly variable direction and misalignment.

The wind speed decay (e.g., Fig. 1a; red) gives three regimes in each of the idealized cases; D1–26 until the peak in La^{−2}, D41–100 when the weak winds permit significant inertial motions to persist, and a transition regime D27–40.

### b. Stokes drift profiles

*ϖ*between 0.0310 and 2.3 Hz with a logarithmic resolution

*φ*. The wave model integration time step is 150 s. The combined directional wave height spectra

The Stokes drift forcing fields for the idealized cases were similarly generated with WAVEWATCH III, but assuming an unlimited fetch (i.e., single grid point) forcing the wave spectrum with the idealized time varying wind.

Two examples of directional wave spectra and Stokes drift profiles are shown in Fig. 2 to illustrate how these calculations account not only for transient wind forcing, but also for remotely generated swell. In the first, at hour 19 of the June cases, the broad spectral peak is due to low-frequency swell propagating to east-northeast (Fig. 2a) that gives the Stokes drift below 20 m (Fig. 2c). Above, the local wind waves developing in response to increasing wind to the southeast, cause the Stokes drift to veer toward the wind direction, though at the surface there is still a misalignment of about 20°. In the second example, from 12 h later, the still increasing winds reduce this misalignment, deepen the veering (Fig. 2d), and broaden the spectral peak in both direction and frequency (Fig. 2b). At both times the veering of the Stokes shear is well resolved by 25 grid points in the upper 10 m and captured in

### c. Boundary layer depths

Beginning about 11 h prior to the start of the April LES, there are seven ARGO profiles to 300-m depth at intervals of about 8 h (roughly half an inertial period), and distances from 7 to 30 km from SOFS. These profiles provide the LES initial conditions and some validation (section 2d). For both purposes consecutive profiles are averaged to reduce adiabatic signals due to inertial motions.

A unique feature of our LES is the deep depth of maximum stratification *h* > *h*_{i} shown for the A3 profile of Fig. 3c.

An important quantity derived from 1-h LES statistics is a boundary layer depth *h* to which surface-forced turbulence penetrates. It is calculated from the profiles of buoyancy flux and an unstable example from hour 3 of AprS, is shown by the blue profile (A3) of Fig. 3c. There are three steps: first, find the entrainment depth *h*_{e} (blue asterisk at 165 m), where *h*_{i}; second, find the first depth *h*_{max} below *h*_{e} where *h* as the first depth (182 m) below *h*_{e} and above *h*_{max}, where the buoyancy flux equals the weighted average (1–0.95)*h* is always found, there is never significant momentum or buoyancy flux below, and it is insensitive to the somewhat arbitrary weight, 0.95. Also, it is applied throughout the transition regime A32–43, as shown by the red profiles of Fig. 3c. When the buoyancy flux first increases with depth (e.g., A34 and A38), *h*_{e} is taken to be at the surface, and *h* shoals from 31 to about 21 m. Later in the afternoon, at A40 for example, the reduced solar heating makes the buoyancy flux near the surface less negative than found deeper, so there is a calculated entrainment depth at about 12 m and boundary layer depth again at about 31 m. Usually, the algorithm could be simplified by taking *h* is very abrupt.

### d. The LES solutions at SOFS

The wind, wave, and buoyancy forcing are all stronger and more highly variable than in most LES studies. This forcing combined with the deep boundary layer presents an unusual challenge for the model to capture all the turbulence structures as well as their interactions. Although the LES need only mimic, but not necessarily reproduce nature, there are three basic issues to address.

First, do the LES reproduce at least the overall character of the observed time evolution of the upper ocean at SOFS in the autumn of 2010? The ARGO profiles suggest that they do quite well, though even the relatively frequent April sampling is very far from ideal. The ARGO float moves about 36 km in 46 h, and the ocean responds to real not parameterized forcing and to 3D advection not in the LES. Furthermore, the ARGO profiles indicate that there is a very large adiabatic signal from the semidiurnal internal tide that is missing in the LES. Although undersampled, the phase is such that the signal is minimized by differencing the consecutive average profile ending at hour 3, from the consecutive average ending at hour 19, with the result shown in Fig. 3b (dashed magenta). The nonzero difference at 200 m continues past 275 m and is consistent with lateral advection, or float travel through lateral gradients, or a tidal remnant. Compared to the LES (blue trace), the greater ARGO buoyancy loss above 150 m could include any combination of the above possibilities, including the different time intervals, or too little LES entrainment, with no way to discriminate. Nonetheless, both differences show buoyancy gain (entrainment) starting at about 168-m depth and peaking near 180 m, where the ARGO change is only about 10% greater than that from AprS.

Second, are the scales of motion generated by this forcing and their interactions resolved by the computational domains and grids (Table 1)? Flow visualization of the turbulent solutions provides a means of observing the dominant features. Figure 4 is an example of vertical velocity in the lower surface layer at 18 m. At this time (hour 19 of JunS) the winds and buoyancy flux are moderate, but the waves are complicated (Fig. 2a). There are multiple spatially elongated coherent downwelling (blue) structures aligned roughly in the direction of the wind (black vector at the top left). Such features are familiar from Langmuir turbulence associated with surface wave and wind coupling, even though the Stokes drift at this depth is nearly orthogonal (Fig. 2b). The structures are typically about 100 m wide, up to 800 m long, and separated by more than 300 m. In contrast Roekel et al. (2012) show much smaller and closer upwelling features (10 m wide, 200 m long, and 20 m apart), and roughly symmetric downwelling from their idealized LES. Larger scales in Fig. 4 are expected because of the deeper boundary layer. However, the different character of the upwelling (red), in particular its greater area, and hence the asymmetry of the *w*′ distribution, is characteristic of convective cells. These cells are coherent over several hundred meters. Thus, the large domains of Table 1 appear to be necessary and sufficient. They also improve the convergence of the hourly LES statistics.

Third, is the range of LES forcing representative of the SOFS site in autumn? The combined wind and buoyancy forcing during regime A1–11 is the strongest in the SOFS record for April, while A23–43 is among the weakest. The frequency of occurrence distribution created by Belcher et al. (2012) for unstable conditions in Southern Ocean winter is a function of La and *h*/*L*_{L} = *w**^{3}/(La^{−2}*u**^{3}). It is distributed narrowly in the range 5 < La^{−2} < 20 and broadly across six orders of magnitude of *h*/*L*_{L} < 1000, with a peak near (La^{−2} = 11, *h*/*L*_{L} = 0.2). More specifically at SOFS, La^{−2} exceeds a value of 8 more than 80% of the time from June through August. Thus, the 7–16 range of La^{−2} in Fig. 1d shows that the relative strength of Stokes to wind forcing in Fig. 1 is representative of most of the Southern Ocean, particularly at SOFS in autumn. The No-Stokes cases are in the very different regime of La^{−2} = 0. The forcing of Fig. 1 also spans the range of Southern Ocean *h*/*L*_{L}. Unstable values nearly reach 1000 for a few hours of A23–31, with *h*/*L*_{L} < 1 throughout D1–26 of D24S, for example. Negative values come from the stable forcing of regime A32–43.

## 4. Similarity functions in the LES surface layer

The nondimensional gradients *d*_{o} and

With only 8 h of stable forcing, (6) is assumed to hold for the ocean LES. However, the two No-Stokes cases are used in section 4a to adapt the unstable buoyancy similarity functions,

### a. The ${\varphi}_{m}$ and ${\varphi}_{s}$ similarity functions in the No-Stokes LES

Only two periods with *Q*_{0} < −40 W m^{−2} and relatively steady winds are analyzed; namely, the first 22 h of AprN (regimes A1–11, A12–15, A16–22) and the first 27 h of JunS (regimes J1–4 and J5–27). To span the surface layer, *σ* = *d*/*h* = 0.04, 0.06, 0.08, 0.10, 0.12, and 0.14. Thus, the region nearer the surface where SGS fluxes may become important is avoided.

*d*/

*L*) and

*d*/

*L*) are computed from (7) and (8), respectively, at all nine depths. The resulting nondimensional products (

*σ*in Figs. 5a and 5b, respectively. Although (2) and (3) say they should both equal unity, the former is clearly biased low, but the latter is biased high. Such opposing biases cannot be reconciled through the von Kármán constant. However, Figs. 5c and 5d show that they can be reconciled with

*κ*= 0.4 by adopting the unstable functions

*σ*in the range between 0.02 and 0.10, where there are 467 data points and the averages of

*w**, and the coefficients give Pr = 0.82.

The points, especially in Fig. 5d, become more scattered and smaller on average for *ε* = 0.1 as the farthest extent of the surface layer. At smaller values of *σ*, high biases begin to develop and this tendency is used to determine the nearer extent, *d*_{o}, empirically. The dependency of *d*_{o} on the forcing regime is shown Table 2, for both No-Stokes and Stokes cases. It is always found to be less than 4 m. It is most variable for AprS when it shallows from 3.8 m to less than a meter in the transition regime (A32–43), then deepens to 2.3 m in the near-neutral regime (A44–50). There is less variability without surface wave complications. For example, *d*_{o} is 3.0 m throughout JunN. It is the same for both shear and stratification, except during the initially strong winds of D1–26, when the turbulent flow may not still be equilibrated with the initial deep inversion and the respective values are 1.9 and 0.3 m.

### b. The ${\chi}_{m}$ and ${\chi}_{s}$ functions in the Stokes LES

Compatibility with No-Stokes empiricism constrains the choices of independent variables for the dimensional analysis of the wind-, wave-, and buoyancy-forced ocean boundary layer. Judicious choices include distance *d* and buoyancy *V*_{Stokes}. The velocity variable, say *μ*, associated with the wind forcing must become *u** for La^{−2} = 0, so a general form is *R* and the exponent, as well as *V*_{Stokes}, are to be determined empirically from the LES solutions.

The additional factor, (1 + *R*La^{−2})^{p} can be incorporated into the Stokes similarity functions, *d**V*_{Stokes}*ζ* by a factor *R* and *p* are chosen to minimize the depth dependencies of *ζ* from (1). Particularly useful for this purpose is regime D1–26 of D24S, because the waves and wind are always aligned, the buoyancy is only weakly unstable, while the wind is initially very strong before falling smoothly to about 6 m s^{−1} (Fig. 1). Also analyzed here are all the regimes of Fig. 5 plus A26–33 where Stokes forcing appears to overcome the highly variable wind direction.

In (1), *ζ* is proportional to the ratio (−*w**^{3}/*u**^{3}), so Stokes effects are included by modifying the denominator. The addition of *R* = 1, adding *R* = *P*_{S}, while replacing *R* = (*P*_{S}/*P*_{U}). A denominator of (*p* between *Z*_{min}/2). Over the range −1 < *p* < 1, the least depth dependency is found with *R* = (*P*_{S}/*P*_{U}) and *p* = 1/2.

*σ*bins with standard deviations are shown in Fig. 6. The bin averages in both Fig. 6a and Fig. 6b, increase with

*σ*at about unity slope to breakpoints at

*σ*= 0.20. A similar exercise with

*R*= 0 gives significantly steeper slopes of 1.8 and 1.5, respectively, as shown by the dashed lines. In addition to weakening depth dependencies, modifying

*ζ*reduces the standard deviations in the range 0.08

*ζ*will hereafter be defined by

*L*is identified with the depth where the buoyant production, or suppression, of TKE equals the rate of shear production (Wyngaard 2010). Adding Stokes production to that of shear would increase this depth by a factor that (20) says is the term in square brackets.

*d*

_{o}(Table 2) and

*σ*less than the breakpoints be shifted to a reference depth, such as

*ε*, according to the solid lines of Fig. 6:

A total of 273 hourly values of depth average

*V*

_{Stokes}, is chosen to minimize the spread and maximize the correlation coefficients of

*ξ*. In (20),

*ζ*is proportional to the ratio

*ξ*in Fig. 7. Quadratic regressions give the Stokes similarity functions:

*ξ*from (23). However, with the first option above, the correlations are only slightly smaller (0.88 and 0.89), and standard deviations marginally greater (0.037 and 0.032). Furthermore, the Eulerian shear

*ξ*, with an intercept at about 0.3.

A smooth transition to No-Stokes forcing requires the intercepts to be *ξ* between 0.0 and 0.35 and constant extrapolation at the minimum values of quadratics at *w*_{m} of (13) and *w*_{s} of (14), can be enhanced by Stokes forcing. The ratio of these factors, 1.46, becomes the upper limit of the Prandtl number.

## 5. Parameterizing the ocean surface layer

Use of the Stokes similarity functions (24) and (25) in OGCMs, requires knowledge of *u**^{3}, La^{−2}, and *w**^{3}, so the problem reduces to finding parameterizations for *P*_{U}, *P*_{S}, and *P*_{B}, as computed in section 4b. The regimes of Fig. 7 are used again for this purpose.

First consider *P*_{B}*w**^{3} from (12). Assuming a linear buoyancy flux profile from the surface to zero at a depth *P*_{B} = 0.095 for *P*_{B} may be well represented by a somewhat smaller constant, if the ratio

*P*

_{S}

*u**

^{3}La

^{−2}from (11) becomes the numerator of

*ξ*in (23) and a significant term in the denominator. The agreement between the above linear profile assumption and

*P*

_{B}, suggests making a similar assumption about the momentum flux profile. A linear profile in (11) gives an estimate:

*P*

_{S}to these estimates with comparable integration limits (first model level below

*Z*

_{min}/2) from section 4b. The linear regression (dashed line) gives a correlation coefficient higher than 0.99, but does not go through the origin. A compromise with the best fit through the origin gives the parameterization,

*P*

_{Se}, which lies within 5% of the linear regression over the range of Fig. 8.

*P*

_{U}

*u**

^{3}and its distinctly smaller values with Stokes forcing (AprS and JunS) than without. From (2) and (4), the shear in the integrand of (10) should scale with

*u**, but competing effects complicate the momentum flux. The success of (20) in reducing depth dependencies in section 4b, suggests scaling it with

*u**

^{2}. Additional scaling by

*P*

_{U}on a combined parameter, Λ =

*ϕ*

_{m}(1 + La

^{−2}

*P*

_{S}/

*P*

_{U}). The close fit of the LES data in Fig. 9a supports the above scaling and the cancellation of

*P*

_{U}, but a simple solution is to relate

*P*

_{U}directly to

*P*

_{U}, and the solution gives the parameterization

*ζ*from (20) requires

The benefit of incorporating buoyancy forcing and *P*_{S} = *P*_{U}, then plotting *P*_{U} against the resulting function of Langmuir number only, (1 + La^{−2})^{−1}, in Fig. 9c. Not only does the correlation fall to 0.67, but very few points fall near the linear regression (dashed). Also, the dark blue triangles from the stable forcing of regime A32–43, and the light blue diamonds from the near neutral regime A44–50 all cluster below the other points. Overall, the major effect of *P*_{U}, and to move points (e.g., the magenta) with high *P*_{U} to the right. The blue points remain clustered, because as the boundary layer deepens and during the day–night transition (Fig. 3c), the decrease in *P*_{S}/*P*_{U}) at about the same La^{−2}.

Multiple values of *P*_{U} are a problem for any attempt to parameterize *P*_{U} in terms of La^{−2} alone. For example, consider the red points from D24S in Fig. 9c. After La^{−2} rises after hour 12 (red time arrow) then falls to hour 39 (Fig. 1d), *P*_{U} is about 50% higher (2.4) at (1 + La^{−2})^{−1} = 0.72, than initially (1.6). However, this behavior is not evident Fig. 9b, because as La^{−2} increases, *P*_{S} falls, such that with *P*_{U} relatively steady, so is ^{−2} and *P*_{S} decrease, such that *P*_{U} increases nearly linearly with ^{−2}.

Large oscillations are evident in the points of Fig. 9c from D06S (magenta) as time advances after hour 41 (magenta arrow) and the amplitude of the inertial motions is about 12 cm s^{−1}. However, there are about seven cycles in 58 h (solid magenta curve), so the period is only about half the inertial, because the coupling of the inertial current to the wind stress weakens with depth. Twice every inertial period the dot product of (10), and hence *P*_{U}, are minima, because the current is nearly orthogonal to the wind stress. When the current then rotates either more downwind, or upwind, the wind stress accelerates the upper flow in the direction of the wind sooner than the deeper. Therefore, with either rotation a downwind shear develops that increases the dot product, and hence both *P*_{U} and *P*_{U}, such that the magenta points now scatter about the regression line. Another contributor to the narrow range and reduced scatter is inertial variability in *h* that affects both

### a. Evaluating the parameterization

A full parameterization has been developed for the velocity scales associated with turbulent viscosity

${P}_{B}^{\text{PAR}}$ = 0.090${P}_{S}^{\text{PAR}}$ = 0.94${P}_{Se}$ , with${P}_{Se}$ from (26)initial

${P}_{U}^{\text{PAR}}$ = 2.5${\varphi}_{m}$ (*ε*) from (18), with*ζ*at*d*=*εh*, from (20) using${P}_{S}^{\text{PAR}}$ and${P}_{U}^{\text{PAR}}$ update

${P}_{U}^{\text{PAR}}$ from (27), using${P}_{S}^{\text{PAR}}$ and${\varphi}_{m}$ (*ε*)*ξ*from (23), using${P}_{U}^{\text{PAR}}$ in (10),${P}_{S}^{\text{PAR}}$ in (11) and${P}_{B}^{\text{PAR}}$ in (12)${\chi}_{m}\left(\xi \right)$ from (24) and${\chi}_{s}\left(\xi \right)$ from (25);${\varphi}_{m}$ (*ε*) from (18), and${\varphi}_{s}$ (*ε*) from (19), using${P}_{S}^{\text{PAR}}$ and updated${P}_{U}^{\text{PAR}}$ in (20)${w}_{m}^{\text{PAR}}$ from (13), using${\chi}_{m}\left(\xi \right)$ and${\varphi}_{m}$ (*ε*)${w}_{s}^{\text{PAR}}$ from (14), using${\chi}_{s}\left(\xi \right)$ and${\varphi}_{s}$ (*ε*).

^{−2}reaches its maximum. All regimes from D24S and D12S are excluded because they are similar to D06S.

The calculations of *d*_{o} and *εh*), as discussed in section 4. Since the linear depth dependence of (13) and (14) is more valid nearer the surface the averaging stays as near as possible to

Parameterized turbulent velocity scales are compared to the LES in Fig. 10. For both *w*_{m} and *w*_{s} the correlation coefficient is 0.97, but there is a small high bias, especially for *d*” in the denominators of (28) and (29) and increase the LES values, especially the smaller when the forcing is weaker. There is relatively little scatter in AprN and JunN (blue points), which are confined to values less than 1.5 cm s^{−1}. Also in this region are the D06S points (red) from regime D41–100. The clusters of red D06S points near ^{−1}, and ^{−1} come from regime D1–26, then during D27–40 there is a relatively smooth transition of the red D06S points while La^{−2} falls from its peak. Wind speed and direction and the Stokes drift all vary only in AprS and JunS (black points). This combined variability tends to produce more spread; more so for *w*_{m} (Fig. 10a) than for *w*_{s} (Fig. 10b).

This qualitative comparison of Fig. 10 and Table 3 indicates that overall the LES turbulent velocity scales at

Summary statistics from comparisons of turbulent velocity scales with Stokes forcing from 188 h of AprS, JunS, and D06S. These scales (*X* = {*Y* = {*Y* = {*Y* = {*Y* = {*S*_{y} and *S*_{x}, rms is the root-mean-square difference, the correlation coefficient is *r*_{xy}, and the mean bias

## 6. Exploring simpler alternatives

The parameterization of section 5 assumes given Stokes profiles, but these are not always available. To demonstrate the benefits of a detailed calculation, for example (17), two simpler possibilities are considered. In the first the Langmuir number is the only wave information available, and in the second there is no knowledge of the wave field. Comparisons with *ε* from a total of 188 h from AprS, JunS, and D06S are shown in Fig. 11. The statistics are summarized Table 3, along those of

^{−1}are not reproduced and the rms difference is 0.91 compared to 0.58 cm s

^{−1}for

^{−1}, which leads to an rms difference about thrice that of

*w**

^{3}, and hence

*w** = 0 in stable conditions, are denoted as

^{−1}), such that the mean bias (0.70) becomes the lowest in Table 3.

Thus, by these measures, the formulation of (30) with either (31) or (32) is not as representative of the LES velocity scales as the parameterization of section 5. In particular, because the viscosity and diffusivity only differ by the ratio of

^{−2}= 11, as well as a monochromatic wave where the profile is exponential:

*η*=(2

*kh*)

^{−1}, for a horizontal wavenumber

*k*. Then, following Alves et al. (2003) 1.2 approximates the wave age (the ratio of phase speed

*c*

_{p}to wind speed), so that (

*c*

_{p}/1.2)

^{2}= (855

*u**

^{2}), where 855 is the ratio of ocean to air density, and

*C*

_{D}is the drag coefficient. Finally, assume linear dispersion to give

*k*=

*g*

*η*proportional to

*u**

^{2}/(

*gh*). The resulting turbulent velocity scales,

*z*= −

*εh*to the surface:

*ε*at large

*η*and by 1 at

*η*the ordinate and

*P*

_{S}from the LES versus

At hour 19 of JunS (magenta diamond), for example, *η* is about 0.04, such that the black curve of 12 gives *P*_{S} = 0.60, because *η* fails to capture the rotation with depth of

The circumstances are different at hour 1 of AprS (blue diamond). The severe underestimate of *η* = 0.16 from the black curve of Fig. 12), which with *h* = 180 m is still 60% of the surface drift at *P*_{S} stays relatively constant, but by hour 13 (blue triangle) the lower winds halve the decay scale to about 0.08, which increases

Despite aligned wind and waves, *η* decreases, initially because *h* deepens, then because the wind decreases. In contrast,

The very poor representation of *P*_{S}, by

By these measures *ξ* from *ξ*, as shown by Fig. 7.

## 7. Discussion and conclusions

The Stokes similarity functions (24) and (25) are the key new LES results that should be represented in simpler models and parameterizations. They account for surface wave effects in the velocity scales for turbulent viscosity and diffusivity in the surface layer, across a wide range of highly variable forcing, with misaligned wind and waves. However, it is important to acknowledge the caveat that the results presented are valid only for the surface layer of the seven LES of our study, with *ξ* equal zero and between about 0.35 and 0.86. Unfortunately, available observations are insufficient to determine qualitatively how well these LES results apply to the real ocean, where other effects, such as wave breaking, may provide additional production of TKE that needs to be included in (23). However, the ARGO comparison of Fig. 3b is encouraging.

The Wyngaard (1982) view of boundary layer physics is that they are distinct, and hence governed by their own mixing rules. An established basis for these rules is similarity theory, where the fundamental premise is that in a turbulent surface layer, dependent variables only depend on the distance from the surface and the forcing. Thus, Stokes drift is considered a forcing and the dot products in (10) and (11) accounted for. Hence, the mixing rules can now include the Stokes similarity functions in the surface layer and a challenge going forward is to discover if these functions play a role deeper in the boundary layer. An implication of the Wyngaard (1982) view and similarity theory is that in a boundary layer turbulent correlations such as the vertical fluxes *w*_{m} and *w*_{s} have been parameterized in section 5, in terms of the surface forcing plus the surface layer Stokes drift.

Adapting Monin–Obukhov similarity theory to the surface layer of the Southern Ocean LES confirms empirical results from the atmosphere, namely, *κ* = 0.40, *ε* = 0.1. It is necessary to determine the near surface limit of this layer *d*_{o} empirically for each forcing regime. With a judicious choice of independent variables, the extended dimensional analysis of section 4b remains compatible with this and other No-Stokes empiricism. Misalignment of the stress and shear vectors is accommodated by integrating the TKE production terms across the surface layer; given the Stokes drift profile as a forcing. The additional dimensionless group becomes the Stokes parameter *ξ*, the ratio of Stokes to total surface layer production of TKE. This ratio has a similar basis as the stability parameter *d*/*L*, because the Monin–Obukhov depth *L* is where the rate of shear production equals buoyant production, or suppression (Wyngaard 2010).

The unequal and variable values of *P*_{U}, *P*_{S}, and *P*_{B} are key to formulating the Stokes similarity functions *ζ* from (20) rather than from (1), depth averaging, and the Stokes similarity functions (24) and (25) of *ξ*. Each step reduces the spread, as quantified by comparable standard deviations. The overall reductions are from 0.33 to 0.036 for

The frequency of occurrence distribution of Belcher et al. (2012), is mentioned in section 3. It is divided into regimes dominated by production due to Eulerian shear, to Stokes shear, and to buoyancy, that they term wind, Langmuir, and convection, respectively. These regimes are based LES calculations of TKE dissipation at *h*/*L*_{L} < 0.08), which would not be unusual, whereas the Eulerian shear would seldom be important. To reach a similar conclusion using the average TKE production over the surface layer would require *P*_{U} from (10), *P*_{S} from (11), and *P*_{B} from (12) to be approximately equal, but they are not. Disregarding variability, but taking *P*_{B} = 0.09, *P*_{S} = 0.5 (Fig. 8), and *P*_{U} = 2.5 (Fig. 9) shifts the Stokes (Langmuir) regime to (La < 0.13; *h*/*L*_{L} < 0.4), which would be a rare occurrence. Although this shift leaves more of a role for convection and Eulerian shear, the Stokes shear production is still the biggest contributor (about 60%) near the peak of the distribution. This result is consistent with Fig. 7 where *ξ* is the Stokes contribution to the LES production of TKE in the surface layer, and falls between one-third and 85%.

The parameterized

From a practical modeling point of view, a Stokes drift profile requirement is a complication. The offline procedure of section 3 requires considerable expertise that is not commonly available. Unfortunately, the LES solutions are not well represented by functions of Langmuir number alone, such as (30). However, a conclusion is that the Stokes drift profile does provide sufficient information about the Stokes forcing to be an effective independent variable in the dimensional analysis extending similarity theory. A simple approximation such as the bulk Stokes drift (BSD) of section 6 is an attractive alternative. However, the biases and scatter may be unacceptable for some applications, especially when the Stokes forcing is less prominent (

## 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, and the patience of Dorothy Koch is gratefully acknowledged, as are the contributions of other Principal Investigators; Todd Ringler, Gokhan Danabasoglu, and Matt Long, and constructive discussions with Dan Whitt. The National Center for Atmospheric Research (NCAR) is sponsored by the National Science Foundation under Cooperative Agreement 1852977. LR was partially supported by the Office of Naval Research (N00014-16-1-2936). 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.

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^{1}

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.