## 1. Introduction

Upper-ocean mixing models without explicit representations of surface waves may implicitly represent their impact when tuned to oceanic observations because of the natural correlation between wind and wave forcing. However, such models may be inaccurate if dimensional scales of surface waves do not scale simply with the wind, as is the case for variations in sea state or wave age at a given wind speed, or for variations in the relative strength of wave versus wind effects with the upper ocean mixed layer depth.

Mixed layer models have primarily sought to explicitly articulate surface wave effects in two generally distinct ways. One approach, after Craig and Banner (1994), accounts for the loss of energy from waves into the phase-averaged turbulent velocity fluctuations *k*–*ɛ*” models, where a second equation may predict dissipation *ɛ*, or in analogous “*k–kl*” models (a.k.a., q2–q2l) such as Mellor and Yamada (1982) and Kantha and Clayson (1994, hereafter KC94). Enhanced near-surface TKE and associated impacts on the turbulence length scale

Another way waves can alter mixed layer models is by accounting for the dynamical effects of surface wave Stokes drift *k* or *q*^{2} and infer an analogous modification in the prognostic equation for *ɛ* or *q*^{2}*l*.

Including the dynamics associated with surface waves in upper-ocean turbulence models appears warranted. Turbulence observations in a wide variety of ocean regimes find that vertical TKE below wave-bounded mixed layers is significantly elevated above

The model formulation presented here was initially motivated by the observation that the stability functions

## 2. Second-moment closure with Craik–Leibovich vortex forcing

### a. Reynolds equations

### b. Closure

Standard closure assumptions invoked in KC94 are used here with two minor changes, generalizing the deformation of turbulence by shear to formally include Stokes drift effects. The two generalizations are made ad hoc, and introduce two new model constants, *A*_{1}, *A*_{2}, *B*_{1}, *B*_{2}, *C*_{1}, *C*_{2}, *C*_{3}, and *S _{q}*.

### c. The ARSM, flux forms, and stability functions

*K*-profile parameterization (KPP; Large et al. 1994), making

Such expressions for stability functions [Eqs. (23)–(28)] are typically subject to “realizabilty constraints” when invoked in the context of a SMC model where dynamic conditions may at any time be far from the equilibrium state they represent. These limit the permitted ranges of some nondimensional forcing functions to avoid producing impossible states such as *G _{M}* < 0, or with improbably small levels of

*q*

^{2}by Eq. (12), and the prediction of the dissipation or “master” length scale

*l*to determine contributions to vertical fluxes down the local gradients of temperature, salinity, momentum and Stokes drift. The stability functions are restated in the appendix as ratios of polynomials, along with a summary of SMC model constants introduced in sections 3 and 4.

## 3. LES solutions for SMC comparison

### a. LES forcing case sets

To tune the new model, SMC predictions are compared here with steady-state solutions from LES in HD08, where the Craik–Leibovich vortex force models the interaction of waves and turbulence. Steady-state forcing cases in HD08 are specified from the 10-m wind speed *C _{D}*. The set of LES simulations is composed of several subsets of forcing cases labeled as “

**Σ**

_{i},” where

**Σ**

_{1}is a matrix of forcing cases varying wave age over four values in

_{2}repeats the eight fully developed cases

**Σ**

_{1}using a monochromatic approximation from Li and Garrett (1993) to specify

**Σ**

_{4}repeats the same eight

**Σ**

_{1}with

**Σ**

_{3a}and

**Σ**

_{3b}provide a continuation of the

**Σ**

_{1}to hurricane-strength winds

**Σ**

_{3a}, and one that remains constant for

**Σ**

_{3b}. Mixing dynamics in these high-wind, young-sea case sets are governed more strongly by entrainment zone shear. These case sets

**Σ**

_{1},

**Σ**

_{2},

**Σ**

_{3ab}, and

**Σ**

_{4}containing 55 LES steady-state solutions for wind and wave forcing that are described in greater detail in HD08 and are listed in correspondingly numbered tables therein. They are supplemented here by three additional wave-free forcing cases corresponding to

**Σ**

_{1}(

*U*

_{10}= 8.3 m s

^{−1}),

**Σ**

_{2}(

*U*

_{10}= 8.3 m s

^{−1}),

**Σ**

_{3}(

*U*

_{10}= 32.6 m s

^{−1}), and

**Σ**

_{4}(

*U*

_{10}= 8.3 m s

^{−1}), for a total of 59 cases. Figures 1–3 present dynamically scaled overviews of relevant mean LES profiles. Profiles from a subset of

**Σ**

_{1},

**Σ**

_{2}, and

**Σ**

_{4}are shown in Figs. 1 and 2, and profiles from high wind

**Σ**

_{3ab}are in Fig. 3.

For LES case sets as in Fig. 1, (left) simulated TKE dissipation profiles

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

For LES case sets as in Fig. 1, (left) simulated TKE dissipation profiles

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

For LES case sets as in Fig. 1, (left) simulated TKE dissipation profiles

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

For high-wind LES case set **Σ**_{3ab}, composed of subsets **Σ**_{3a} (surface drag saturates at 0.0023) and **Σ**_{3b} (surface drag continues to increase with wind), mean profiles of (a) crosswind

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

For high-wind LES case set **Σ**_{3ab}, composed of subsets **Σ**_{3a} (surface drag saturates at 0.0023) and **Σ**_{3b} (surface drag continues to increase with wind), mean profiles of (a) crosswind

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

For high-wind LES case set **Σ**_{3ab}, composed of subsets **Σ**_{3a} (surface drag saturates at 0.0023) and **Σ**_{3b} (surface drag continues to increase with wind), mean profiles of (a) crosswind

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

### b. Scaling for bulk and near-surface TKE components

This scaling effectively absorbs variations in the Stokes drift *e*-folding depth scale **Σ**_{2} (Figs. 1a–c) and **Σ**_{4} (Figs. 1j–l), and for the mature *C _{p}*/

*U*

_{10}= 1.2 (Figs. 1d–f) and young

*C*/

_{p}*U*

_{10}= 0.6 (Figs. 1g–i) sea components of

**Σ**

_{1}. These TKE component profiles are shown for high wind cases

**Σ**

_{3ab}in Figs. 3a and 3b. As noted in HD08,

_{t}varies only with changes in

*C*with

_{D}*U*

_{10}at constant

*C*/

_{p}*U*

_{10}.

This causes the scaling of TKE components on La_{t} to appear more effective within some Fig. 1 plots of LES case subsets at fixed *C _{p}*/

*U*

_{10}than it does between differing

*C*/

_{p}*U*

_{10}plots. The LES model used in HD08 advects

### c. Near-surface scaling of dissipation length scale l with depth

*e*

_{SG}, and that includes parameterized subgrid buoyancy fluxes and dissipation

Figure 2 shows the diagnosis of the dissipation length scale **Σ**_{2} and **Σ**_{4}, and for the young *C _{p}*/

*U*

_{10}= 0.6 and mature

*C*/

_{p}*U*

_{10}= 1.2 sea components of

**Σ**

_{1}, arrayed as in Fig. 1; Figs. 3d–f provide the corresponding profiles for high-wind cases

**Σ**

_{3ab}. Profiles of estimated

In Figs. 2 and 3 profiles are also shown for estimated

Figure 4 compares profiles of **Σ**_{1} (*U*_{10} = 14.8 m s^{−1}, *C _{p}*/

*U*

_{10}= 0.8) for a young sea, Σ

_{1}(

*U*

_{10}= 18.1 m s

^{−1},

*C*/

_{p}*U*

_{10}= 1.2) for a mature sea, and for a high-wind case with very young waves and strong pycnocline shear,

**Σ**

_{3b}(

*U*

_{10}= 44.5 m s

^{−1},

*C*/

_{p}*U*

_{10}= 0.6).

The length-scale profiles _{1}(*U*_{10} = 14.8 m s^{−1}, *C _{p}*/

*U*

_{10}= 0.8), (b) Σ

_{1}(

*U*

_{10}= 18.1 m s

^{−1},

*C*/

_{p}*U*

_{10}= 1.2), and (c) Σ

_{3b}(

*U*

_{10}= 44.5 m s

^{−1},

*C*/

_{p}*U*

_{10}= 0.6). Estimates of these profiles (est.

*l*

_{LES}) are computed from mean subgrid TKE profiles

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

The length-scale profiles _{1}(*U*_{10} = 14.8 m s^{−1}, *C _{p}*/

*U*

_{10}= 0.8), (b) Σ

_{1}(

*U*

_{10}= 18.1 m s

^{−1},

*C*/

_{p}*U*

_{10}= 1.2), and (c) Σ

_{3b}(

*U*

_{10}= 44.5 m s

^{−1},

*C*/

_{p}*U*

_{10}= 0.6). Estimates of these profiles (est.

*l*

_{LES}) are computed from mean subgrid TKE profiles

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

The length-scale profiles _{1}(*U*_{10} = 14.8 m s^{−1}, *C _{p}*/

*U*

_{10}= 0.8), (b) Σ

_{1}(

*U*

_{10}= 18.1 m s

^{−1},

*C*/

_{p}*U*

_{10}= 1.2), and (c) Σ

_{3b}(

*U*

_{10}= 44.5 m s

^{−1},

*C*/

_{p}*U*

_{10}= 0.6). Estimates of these profiles (est.

*l*

_{LES}) are computed from mean subgrid TKE profiles

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

The estimates of

For comparison, profiles of a related Langmuir turbulence dissipation length scale

Further development and tuning of a SMC model assumes that in the near-surface region of strong vortex force TKE production, a corresponding growth in the dissipation length scale occurs, with the result that both it and the vertical TKE component may be about double their values in the nonwave case. This expectation is in line with the qualitative understanding that the presence of Langmuir circulation structures, embedded in the turbulent boundary layer flow, entails an increase in the energy injection rate and energy levels at the larger *O*(*H*_{ML}) scales characterizing the separation between jets. As a result there should be a corresponding near-surface increase in the turbulence decay time scale

## 4. The second-moment closure model

*q*

^{2}and the stability functions [Eqs. (25)–(28)] are extended into a SMC model with the prognostic determination of

*l*after KC04 through

Standard closure constants are retained as possible following KC94 and KC04, with unaltered values for {*A*_{1} = 0.92, *A*_{2} = 0.74, *B*_{1} = 16.6, *B*_{2} = 10.1, *C*_{1} = 0.08, *C*_{2} = 0.7, *C*_{3} = 0.2, *S*_{q2} = 0.41*S _{H}*,

*S*

_{l}= 0.41

*S*,

_{H}*E*

_{2}= 1.0}, and new constants for Stokes effects in third moment closures are taken here to be

*q*

^{2}(Fig. 2, middle column, dashed lines), and using the more appropriate Langmuir number scaling assumption it is set at

*S*

_{l}over its KC94 value has not been found necessary. Indeed

*S*

_{l}=

*S*

_{q2}appears necessary to produce broadly well-behaved

Near-surface LES comparisons motivated a modification of the wall damping function with a dependence on La_{t} in

The pattern of *l* values diagnosed from LES solutions in the lower boundary layer are better replicated by increasing the buoyancy forcing coefficient to *E*_{3} = 5.0, in line with the generally larger values suggested by Burchard (2001) as an alternative to restricting *l* values used to compute the stability function so that it not fall below the Ozmidov scale *l* values on the upper end in the stability functions to keep *l* values throughout its use in determining the stability functions, but not directly to the prediction of

Toward the bottom of the mixed layer a limitation is imposed on the relative vertical decay with depth of each eddy coefficient

Given these model features governing the prediction of *q ^{2}l* and eddy diffusivities, the coefficient

*E*

_{6}of the vortex TKE production is left to be determined by tuning SMC predictions to match LES results. Here, this is done for the ensemble of LES cases on the basis of

*E*

_{6}presents a dilemma between tuning to minimize error in predicting

*E*

_{6}= 4.0 versus predicting energy levels (Figs. 5a,c) using

*E*

_{6}= 7.0. To predict mean vertical TKE (Figs. 5e,f) by inference from the SMC equilibrium model [Eq. (17c)] for comparison with Lagrangian float measurements (D’Asaro 2001; Harcourt and D’Asaro 2010), an intermediate value of

*E*

_{6}= 5 would be better. Figure 6 shows SMC to LES comparisons on the (power) rate of energy conversion into work

*E*

_{6}= 7.0 when, as is usually the case, predicting vertical buoyancy flux is the primary consideration. Some outlying values are due to excessive entrainment in LES case set

**Σ**

_{4}owing to inadequate vertical domain size and excessive interaction with the radiative bottom boundary conditions. Still, even after discounting these, the tuning to entrainment is not strongly compelling; there is a large scatter (Figs. 6a,c) and the poor dissipation length-scale predictions for

*E*

_{6}= 7.0 (Fig. 5a) belie the model’s purportedly more accurate articulation of the underlying physics. The equivocation here on the value of

*E*

_{6}echoes different values reported by KC04 (

*E*

_{6}= 4.0) and Carniel et al. (2005, p. 36), which recharacterizes the KC04 value to

*E*

_{6}= 7.2. Kantha et al. (2010) reconfirm that the value in KC04 was reported incorrectly and that the larger value should be used. However, it is unclear if the ambiguity here stemming from tuning priorities is directly relevant to the similarly different choices for

*E*

_{6}reported for these prior studies.

Mixed layer turbulence properties from the second-moment closure (SMC) model using two tunings, one with (left) *E*_{6} = 7.0 and one with (right) *E*_{6} = 4.0 are compared against LES results for forcing case sets identified in HD08 as **Σ**_{1}, **Σ**_{2}, **Σ**_{3a}, **Σ**_{3b}, and **Σ**_{4}. Properties compared are (top) the maximum nondimensional dissipation length scale

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mixed layer turbulence properties from the second-moment closure (SMC) model using two tunings, one with (left) *E*_{6} = 7.0 and one with (right) *E*_{6} = 4.0 are compared against LES results for forcing case sets identified in HD08 as **Σ**_{1}, **Σ**_{2}, **Σ**_{3a}, **Σ**_{3b}, and **Σ**_{4}. Properties compared are (top) the maximum nondimensional dissipation length scale

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mixed layer turbulence properties from the second-moment closure (SMC) model using two tunings, one with (left) *E*_{6} = 7.0 and one with (right) *E*_{6} = 4.0 are compared against LES results for forcing case sets identified in HD08 as **Σ**_{1}, **Σ**_{2}, **Σ**_{3a}, **Σ**_{3b}, and **Σ**_{4}. Properties compared are (top) the maximum nondimensional dissipation length scale

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Metrics of mixed layer entrainment from the second-moment closure (SMC) model using two tunings, one with (left) *E*_{6} = 7.0 and one with (right) *E*_{6} = 4.0 are compared against LES results for forcing case sets identified in HD08 as **Σ**_{1}, **Σ**_{2}, **Σ**_{3a}, **Σ**_{3b}, and **Σ**_{4}. Metrics compared are (top) the

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Metrics of mixed layer entrainment from the second-moment closure (SMC) model using two tunings, one with (left) *E*_{6} = 7.0 and one with (right) *E*_{6} = 4.0 are compared against LES results for forcing case sets identified in HD08 as **Σ**_{1}, **Σ**_{2}, **Σ**_{3a}, **Σ**_{3b}, and **Σ**_{4}. Metrics compared are (top) the

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Metrics of mixed layer entrainment from the second-moment closure (SMC) model using two tunings, one with (left) *E*_{6} = 7.0 and one with (right) *E*_{6} = 4.0 are compared against LES results for forcing case sets identified in HD08 as **Σ**_{1}, **Σ**_{2}, **Σ**_{3a}, **Σ**_{3b}, and **Σ**_{4}. Metrics compared are (top) the

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

### SMC model profiles

Profiles of SMC eddy coefficients *E*_{6} and with the relative rate of vortex force TKE production, switching from

Mean profiles of SMC model eddy coefficients *E*_{6} = 4.0 or *E*_{6} = 7.0, for SMC simulations corresponding to the three LES case examples of Fig. 4, and for wave-free SMC results with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean profiles of SMC model eddy coefficients *E*_{6} = 4.0 or *E*_{6} = 7.0, for SMC simulations corresponding to the three LES case examples of Fig. 4, and for wave-free SMC results with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean profiles of SMC model eddy coefficients *E*_{6} = 4.0 or *E*_{6} = 7.0, for SMC simulations corresponding to the three LES case examples of Fig. 4, and for wave-free SMC results with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

The shape of the diffusivity profiles differs between the coefficients and varies with the relative penetration of Langmuir turbulence into the layer, with the order of buoyancy and momentum coefficients reverting to *K*-profile parameterization of Large et al. (1994) than they are to those in KC04 (their Fig. 1), due in large part to the higher value of *E*_{3} = 5.0 used for the effect of buoyancy flux on *E*_{6} = 7.0, but sometimes more consistent near the surface when the *E*_{6} = 4.0 SMC values are close to the LES profile. While LES and SMC **Σ**_{1} (*U*_{10} = 18.1 m s^{−1}, *C _{p}*/

*U*

_{10}= 1.2) of Fig. 9b with relatively deep penetration of Stokes shear, interior SMC TKE levels differ little at depths where Stokes shear is small. In these cases (Figs. 9a,c) where Stokes shear is either restricted within the surface layer or small compared to Eulerian shear, the most significant change in the strength of mixing then comes primarily from the larger new stability functions, and to a lesser extent from increases in

Mean dissipation length scale *E*_{6} = 4.0 to *E*_{6} = 7.0, and for wave-free SMC results with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean dissipation length scale *E*_{6} = 4.0 to *E*_{6} = 7.0, and for wave-free SMC results with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean dissipation length scale *E*_{6} = 4.0 to *E*_{6} = 7.0, and for wave-free SMC results with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean energy profiles *E*_{6} = 4.0 or *E*_{6} = 7.0, and with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean energy profiles *E*_{6} = 4.0 or *E*_{6} = 7.0, and with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean energy profiles *E*_{6} = 4.0 or *E*_{6} = 7.0, and with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean vertical buoyancy flux profiles *E*_{6} = 4.0 or *E*_{6} = 7.0, and with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean vertical buoyancy flux profiles *E*_{6} = 4.0 or *E*_{6} = 7.0, and with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean vertical buoyancy flux profiles *E*_{6} = 4.0 or *E*_{6} = 7.0, and with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Figure 11 demonstrates one of the major outcomes of modifications introduced into the model by the component of momentum flux down the Stokes drift gradient, most markedly by comparison with SMC results where *E*_{6} = 7.0 for the case in Fig. 11b. For most comparison cases, however, the retrograde SCM shear continues to strengthen toward the surface and does not revert into a downwind shear in the approach to the surface boundary. Another significant difference between LES and SMC predictions is apparent in the comparison of vertical TKE profiles in Fig. 12, where the SMC profile is inferred from the equilibrium model Eq. (17c) for

Mean profiles of horizontal velocity components relative to midlayer values, with (a)–(c) downwind *E*_{6} = 4.0 or *E*_{6} = 7.0, and with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean profiles of horizontal velocity components relative to midlayer values, with (a)–(c) downwind *E*_{6} = 4.0 or *E*_{6} = 7.0, and with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean profiles of horizontal velocity components relative to midlayer values, with (a)–(c) downwind *E*_{6} = 4.0 or *E*_{6} = 7.0, and with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean vertical kinetic energy (VKE) profiles *E*_{6} = 4.0 or *E*_{6} = 7.0, and for wave-free SMC results with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean vertical kinetic energy (VKE) profiles *E*_{6} = 4.0 or *E*_{6} = 7.0, and for wave-free SMC results with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Mean vertical kinetic energy (VKE) profiles *E*_{6} = 4.0 or *E*_{6} = 7.0, and for wave-free SMC results with

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

In Figs. 13 and 14 the equilibrium model [Eq. (17)] for the Reynolds stress tensor is evaluated to examine the self-consistency of its predictions in the context of the three example LES case steady-state solutions. The comparison in Fig. 13 of the resolved TKE components with their corresponding right side equilibrium model expressions shows that while differences for the downwind TKE component

Evaluation of the equilibrium model from LES results for the three example forcing cases (Fig. 4). Given steady-state LES profiles of the Reynolds buoyancy flux (

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Evaluation of the equilibrium model from LES results for the three example forcing cases (Fig. 4). Given steady-state LES profiles of the Reynolds buoyancy flux (

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Evaluation of the equilibrium model from LES results for the three example forcing cases (Fig. 4). Given steady-state LES profiles of the Reynolds buoyancy flux (

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Evaluation as in Fig. 13 of equilibrium model from LES results for the three example forcing cases (Fig. 4). Given steady-state LES profiles the equilibrium model predictions of momentum flux profiles (i.e., the off-diagonal Reynolds stress tensor elements) are evaluated using Eqs. (17d)–(17f) (solid) and excluding Stokes drift contributions (dot-dashed) for self-consistency in the LES solutions (dotted). (d)–(f) Vertical flux of downwind momentum also shows profile from right of Eq. (17e) with new closure constant set to

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Evaluation as in Fig. 13 of equilibrium model from LES results for the three example forcing cases (Fig. 4). Given steady-state LES profiles the equilibrium model predictions of momentum flux profiles (i.e., the off-diagonal Reynolds stress tensor elements) are evaluated using Eqs. (17d)–(17f) (solid) and excluding Stokes drift contributions (dot-dashed) for self-consistency in the LES solutions (dotted). (d)–(f) Vertical flux of downwind momentum also shows profile from right of Eq. (17e) with new closure constant set to

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Evaluation as in Fig. 13 of equilibrium model from LES results for the three example forcing cases (Fig. 4). Given steady-state LES profiles the equilibrium model predictions of momentum flux profiles (i.e., the off-diagonal Reynolds stress tensor elements) are evaluated using Eqs. (17d)–(17f) (solid) and excluding Stokes drift contributions (dot-dashed) for self-consistency in the LES solutions (dotted). (d)–(f) Vertical flux of downwind momentum also shows profile from right of Eq. (17e) with new closure constant set to

Citation: Journal of Physical Oceanography 43, 4; 10.1175/JPO-D-12-0105.1

Applying the same self-consistency test to the Reynolds stress tensor cross terms in Fig. 14 shows that for

On the other hand, the LES subgrid closure accounts (as in KC04) for only the additional vortex force production of subgrid TKE and does not include a subgrid momentum flux component down the Stokes gradient. The significance of this omission would increase as the ratio of subgrid to resolved TKE increases toward the surface, coincident with the discrepancy between LES and SMC Eulerian shear. As the excessive vertical momentum flux in the surface layer is similar to that component in the LES (dot-dashed in Figs. 14d–f) down the Eulerian shear, it is therefore also possible that this shear develops erroneously or excessively in the LES solutions in response to the lack of a correct subgrid momentum flux component down the Stokes drift gradient.

Other differences in the momentum fluxes suggest that several other smaller closure contributions are missing. The covariance (Figs. 14g–i) of horizontal momentum

## 5. Summary

A new level 2¼ second-moment closure (SMC) model was developed that extends the model of KC04 to include Langmuir turbulence effects in the algebraic Reynolds stress model (ARSM) and in the resulting stability functions and turbulent flux closure. This required adding vortex force TKE production in the ARSM, as well as the introduction of a new momentum flux component that is directed down the gradient of the Stokes drift, in addition to the conventional term down the gradient of the Eulerian momentum. Relative to KC04, the new model includes changes in the momentum flux closure [Eq. (18)] and in the response of stability functions to Stokes shear [Eq. (23)–(28)] that result directly and unequivocally from the inclusion of the CL vortex force in all components of the Reynolds stress tensor equation [Eq. (5)] that is used to derive the ARSM. Additional changes in the stability functions stem from the generalization of KC94 closure assumptions for pressure–strain and pressure–scalar correlations [Eqs. (8) and (9)] and are subject to corresponding choices in two new closure constants. Several other SMC model components were modified to conform to a suite of LES simulations for mixed layers with varying degrees of Langmuir forcing. Tuning the SMC model presents a dilemma between skill at predicting the dissipation length scale versus predicting mixed layer TKE and the entrainment rate. In general, the eddy coefficients for momentum flux due to Eulerian and Stokes shear vary independently with the relative strength of Langmuir and shear-driven turbulence because of the corresponding dependence of leading closure terms on different components of the TKE in the ARSM. Analysis of the equilibrium model using LES results suggests several closure terms are still missing, notably a pressure–strain contribution responsible for transferring vertical into crosswind TKE. The new SMC model improves the prediction of momentum profiles, reproducing a retrograde Eulerian shear in mixed layer interiors, and suggesting that downwind near-surface LES shear profiles not replicated by the SMC model may be at least partly due to missing LES subgrid flux components directed down the Stokes drift gradient.

Modifications made to the wall-damping function, to the decay of eddy coefficients below the mixed layer, and the introduction of functional dependence for

## Acknowledgments

This work was supported by the National Science Foundation (OCE0850551 and OCE0934580), the Office of Naval Research (N00014-08-1-0575), and by a grant of HPC resources from the Department of Defense High Performance Computing Modernization Program.

## APPENDIX

### Summary of Stability Functions and SMC Model Constants

*A*

_{1}= 0.92,

*A*

_{2}= 0.74,

*B*

_{1}= 16.6,

*B*

_{2}= 10.1,

*C*

_{1}= 0.08,

*C*

_{2}= 0.7, and

*C*

_{3}= 0.2. New ARSM closure constants are assumed to have values

*S*

_{q2}= 0.41

*S*,

_{H}*S*

_{l}= 0.41

*S*,

_{H}*E*

_{2}= 1.0,

*E*

_{3}= 5.0,

*E*

_{5}= 0), and

*E*

_{6}= 4.0 or

*E*

_{6}= 7.0.

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