## 1. Introduction

A significant source of uncertainty in parameterizations of upper-ocean mixing stems from limitations in our ability to describe and predict the role of surface waves in boundary layer turbulence. The now classical generation of upper-ocean mixing parameterizations [e.g., Mellor and Yamada 1982, hereinafter MY2.5; Price et al. 1986; Kantha and Clayson 1994, hereinafter KC94; *K*-profile parameterization (KPP), Large et al. 1994] uses surface stress, heat, and salinity fluxes and the subsurface profiles of shear *O*(10–100) near surfaces with breaking waves (e.g., Agrawal et al. 1992; Terray et al. 1996); (obs 2) downwind upper-ocean mean shear *O*(2) in the above LES studies or in second-moment closure (SMC) models (e.g., Kantha and Clayson 2004) and does not account for the observed near-surface increase in

The combination of wave breaking and Langmuir turbulence has also been studied using LES techniques, combining CL vortex force with TKE injection into sundry combinations of the resolved and unresolved (subgrid) model scales (Noh et al. 2004, Sullivan et al. 2004, 2007; McWilliams et al. 2012), all to various effects depending on the relative scales of TKE injection by breaker forcing. The partition into resolved [typically > *O*(1) m] and unresolved components, and the distribution of resolved forcing scales, can have large impacts on mixed layer turbulent dynamics. Sullivan et al. (2007) shows that for young seas CL breaker–Stokes drift interactions can strongly impact mixed layer entrainment rates and increase the skewness of *O*(1) wave height of the surface.

*K*

_{H}and viscosity

*K*

_{M}to predict vertical fluxes of momentum and temperature

*θ*, that is,

As the dissipation of surface wave energy typically exceeds other upper-ocean TKE sources, initial modifications of SMC models after Craig and Banner (1994) and Craig (1996) addressed obs 1, representing energy lost from breaking waves as a surface TKE flux proportional to

Separately, several Langmuir turbulence modifications of KPP have followed LES-motivated suggestions (McWilliams and Sullivan 2000; Smyth et al. 2002) to adjust the empirical constants governing parameterized profiles of

SMC modifications for Langmuir turbulence in D’Alessio et al. (1998) and KC04 included the additional CL vortex force TKE production *all* of the second moments

*requires*that

*E*

_{i}and stability functions

*S*

_{q}and

*S*

_{l}for diffusive transport of

*q*

^{2}and

*q*

^{2}

*l*that are either constant (KC94, MY2.5) or fixed in relation to

*S*

_{M}or

*S*

_{H}(H13 and most other SMCs). The vertical fluxes in H13,

*H*consistent with both Lagrangian float measurements (obs 4 and D’Asaro et al. 2014) and with the HD08 LES-derived nondimensional scaling of

Section 2 introduces inhomogeneous near-surface pressure–strain

## 2. Near-surface closure for Langmuir turbulence in quasi-equilibrium ARSM

*z*axis is preserved in Eq. (23). In addition to the horizontal anisotropy of redirected TKE production, several other features of Eq. (23) distinguish it from the near-wall form of Eq. (19). Redirection of

While

Evaluation of the equilibrium model from LES results for the three example forcing cases as in Fig. 13 of H13. Using steady-state LES profiles of Reynolds buoyancy flux

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

Evaluation of the equilibrium model from LES results for the three example forcing cases as in Fig. 13 of H13. Using steady-state LES profiles of Reynolds buoyancy flux

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

Evaluation of the equilibrium model from LES results for the three example forcing cases as in Fig. 13 of H13. Using steady-state LES profiles of Reynolds buoyancy flux

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

Evaluation as in Fig. 2 of the new ARSM from LES results for the three example forcing cases, as in Fig. 14 of H13. 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 the new ARSM of Eqs. (26d)–(26f) (solid gray), the new ARSM with

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

Evaluation as in Fig. 2 of the new ARSM from LES results for the three example forcing cases, as in Fig. 14 of H13. 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 the new ARSM of Eqs. (26d)–(26f) (solid gray), the new ARSM with

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

Evaluation as in Fig. 2 of the new ARSM from LES results for the three example forcing cases, as in Fig. 14 of H13. 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 the new ARSM of Eqs. (26d)–(26f) (solid gray), the new ARSM with

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

There are several remaining ARSM discrepancies and uncertainties in the near-surface closure. Errors in the shear layer adjacent the surface may call for additional inhomogeneous near-wall treatment using the form of Eq. (19) to represent the pressure “echo” effect, but an accurate parameterization of this would require higher LES resolution to reduce dependence on subgrid dynamics. Presumably,

## 3. SMC model improvements

Figure 4 shows the SMC based on the new ARSM and stability functions [Eq. (33)] replaces near-surface retrograde shear of H13 with a small prograde downwind shear layer in all three example LES cases, and for sufficiently strong CL forcing, the vertical TKE exhibits a subsurface maximum at a depth similar to LES predictions. Profile details in Fig. 4 occasionally correspond very closely, but because of the unaltered slow pressure–strain closure and the isotropic dissipation assumption, vertical TKE does not continue to decrease right next to the surface, limiting the SMC Eulerian shear to levels below LES predictions. The Figs. 2g–i vertical TKE comparison between LES and ARSM predictions at these depths suggests the need for a near-wall term for the slow pressure–strain closure. However, this model–model comparison is ambivalent over these top grid-layer depths, as LES predictions and their interpretation there depend strongly on the LES subgrid model that involves neither a momentum flux down the Stokes gradient nor any implications from the anisotropy of subgrid TKE (i.e., the LES subgrid is consistent with an SMC with constant

Mean profiles of (a)–(c) downwind horizontal velocity *E*_{6} = 6.0, the H13 SMC with *E*_{6} = 7.0, and for a SMC

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

Mean profiles of (a)–(c) downwind horizontal velocity *E*_{6} = 6.0, the H13 SMC with *E*_{6} = 7.0, and for a SMC

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

Mean profiles of (a)–(c) downwind horizontal velocity *E*_{6} = 6.0, the H13 SMC with *E*_{6} = 7.0, and for a SMC

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

Low SMC levels of midlayer vertical TKE in Fig. 4 correspond to an inadequate mixing of momentum below the depths of strong CL production. The quasi-equilibrium assumption of the ARSM, responsible for generating

Figures 5 and 6 reprise Figs. 5 and 6 of H13 to demonstrate the effect of changes to the SMC of Langmuir turbulence on bulk layer–averaged turbulence statistics for

Mixed layer turbulence properties from the new SMC with (a),(c),(e) *E*_{6} = 6 and from the H13 SMC with (b),(d),(f) *E*_{6} = 7 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 45, 1; 10.1175/JPO-D-14-0046.1

Mixed layer turbulence properties from the new SMC with (a),(c),(e) *E*_{6} = 6 and from the H13 SMC with (b),(d),(f) *E*_{6} = 7 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 45, 1; 10.1175/JPO-D-14-0046.1

Mixed layer turbulence properties from the new SMC with (a),(c),(e) *E*_{6} = 6 and from the H13 SMC with (b),(d),(f) *E*_{6} = 7 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 45, 1; 10.1175/JPO-D-14-0046.1

Metrics of mixed layer entrainment from new SMC with (a),(c) *E*_{6} = 6 and from the H13 SMC with (b),(d) *E*_{6} = 7; (a),(c) 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 45, 1; 10.1175/JPO-D-14-0046.1

Metrics of mixed layer entrainment from new SMC with (a),(c) *E*_{6} = 6 and from the H13 SMC with (b),(d) *E*_{6} = 7; (a),(c) 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 45, 1; 10.1175/JPO-D-14-0046.1

Metrics of mixed layer entrainment from new SMC with (a),(c) *E*_{6} = 6 and from the H13 SMC with (b),(d) *E*_{6} = 7; (a),(c) 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 45, 1; 10.1175/JPO-D-14-0046.1

## 4. Higher-order quasi-homogeneous closures and near-surface effects

*S*superscripts and where the first-order Eulerian term from KC94 in Eq. (15) would correspond to setting

The structure of Eq. (38) may be compared with the defects of the H13 ARSM (Figs. 2–3), though details may differ in the context of other SMCs that use the second-order Eulerian counterparts to Eq. (38) in the pressure–strain closure or that do not make the quasi-equilibrium assumption. The diagonal terms [Eq. (38a)] can indeed rotate the CL vortex TKE production entirely from *i* = *j* diagonal of *i* ≠ *j* off-diagonal elements using a different set

## 5. A numerical thought experiment in “free-range” Langmuir turbulence

*L*

_{z}= 128 m and width

*L*

_{x}=

*L*

_{y}= 256 m centered on

*z*= 0 m, and resolved isotropically with

*dx*=

*dy*=

*dz*= 1 m, an initially unstratified layer spanning the |

*z|*<

*L*

_{z}/4 central half is separated from free-slip upper and lower closed boundaries by stable layers spanning the upper and lower

*L*

_{z}/4, where stratification

^{−1}, and by sharp temperature jumps of 2°C each at |

*z|*=

*L*

_{z}/4 (Fig. 7a). Velocity

*z|*<

*L*

_{z}/6 central third of the domain is initially set to

Simulation of free-range Langmuir turbulence, where CL vortex production of TKE is removed from closed boundaries: (a) profiles of the initial

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

Simulation of free-range Langmuir turbulence, where CL vortex production of TKE is removed from closed boundaries: (a) profiles of the initial

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

Simulation of free-range Langmuir turbulence, where CL vortex production of TKE is removed from closed boundaries: (a) profiles of the initial

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

Figure 8 compares the LES-predicted

Comparison of second moments (a)

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

Comparison of second moments (a)

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

Comparison of second moments (a)

Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0046.1

## 6. Conclusions

Further modifications of pressure correlation closures in the H13 SMC of Langmuir turbulence have been developed from near-wall inhomogeneous closures and applied in an improved model. The anisotropy and vertical extent of the resulting changes to the ARSM for Langmuir turbulence is relatively successful in improving the near-surface profiles of mean momentum and TKE components predicted by the model. Improvements in SMC model entrainment, relative to HD08 LES predictions, were more significantly affected by partial reversion to the constant stability function for turbulence transport used in KC94 and KC04 than they were by the modification of stability functions. Indeed the question posed recently by Kantha et al. (2014) on the benefits of increased complexity from the inclusion of Langmuir physics in H13 is relevant. On the other hand, progress in predicting upper mixed layer dynamics below ocean surface waves has a much broader range of modeling goals, such as in the dispersion of pollutants or the role of bubble clouds in gas transfer. One virtue of the new SMC of Langmuir turbulence is that the stability functions [Eq. (33)] are only marginally more complex than in KC94, even though they depend upon a new parameter from a surface-proximity function. Dependence on a direction related to the horizontal orientation of near-surface Stokes shear and stress does not enter into the stability functions or the vertical flux eddy coefficients, but may be relevant to predicting horizontal TKE and fluxes of neutral or near-neutral scalars. The relationship between Langmuir cell orientation and features of the near-surface pressure–strain closure has received limited development here and requires further evaluation over a broader range of LES upper-ocean forcing regimes. Additional near-surface treatments for shear production may be called for, but these cannot be accurately evaluated where LES subgrid dynamics are significant. Also, integration of a TKE flux from wave attenuation into the new SMC of Langmuir turbulence remains an open question, as the interaction of injected TKE, and with the parameterized CL vortex force, introduces a new source of vertical momentum flux to near-surface depths and raises questions of how best to represent the anisotropy of the injected TKE in the ARSM.

## Acknowledgments

This work has benefited greatly from discussions with Eric D’Asaro, Andrey Shcherbina, Baylor Fox-Kemper, Brodie Pearson, Alan Grant, Stephen Belcher, Jeff Polton, and, at an early stage, from considerate comments pointing to near-surface closures offered by David T. Walker. 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.

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