## 1. Introduction

Turbulence in the ocean surface boundary layer (OSBL) leads to vigorous mixing of properties from the atmosphere (or cryosphere) above and ocean interior below. The OSBL is therefore fundamental to net exchange of heat, gases, and other quantities between the atmosphere and the ocean interior. In large-scale ocean circulation models, such as for simulating Earth’s climate, the OSBL turbulent mixing is parameterized. These turbulent mixing parameterizations are formulated with insight about the characteristics of OSBL turbulence, which depend on the generation mechanism of the turbulence.

In this study we consider three mechanisms that drive OSBL turbulence: the Eulerian current shear (shear turbulence), gravitational instabilities (convective turbulence), and surface waves through the Stokes-drift shear (Langmuir turbulence). The primary focus of this study is to extend the energetics-based planetary boundary layer (ePBL) parameterization for shear and convective turbulence of Reichl and Hallberg (2018, hereafter RH18) to include the enhanced turbulent mixing due to Langmuir turbulence. This will be accomplished through analysis of the energetic implications of the interaction of the three types of turbulence in setting the vertically integrated entrainment buoyancy flux within the OSBL. Breaking waves are also critical to the generation and dissipation of OSBL turbulence, especially near the ocean surface, but are not considered for this study (see Sullivan et al. 2007; McWilliams et al. 2012).

### a. Langmuir turbulence in the OSBL

A property of ocean surface wave motion is a vertically sheared mean Lagrangian current, known as the Stokes drift ^{1} The surface magnitude of Stokes drift *λ* [*k* = 2*π*/*λ* is the wavenumber]. Superposition of waves of the full directional wavenumber wave spectra determines the net Stokes drift and its direction. The Stokes drift modifies the characteristics of turbulence relative to an environment without surface waves as its vertical shear plays a distinct role in turbulence generation from Eulerian current mean shear (e.g., Teixeira and Belcher 2002; Grant and Belcher 2009). The modified turbulence due to Stokes drift is known as Langmuir turbulence (following McWilliams et al. 1997), as the mechanism was proposed as a result of efforts to understand Langmuir circulations in the ocean (following Craik and Leibovich 1976).

Ocean climate simulations with and without parameterization of the effect of Langmuir turbulence suggest it plays a significant role in global OSBL processes (see Belcher et al. 2012; D’Asaro 2014; Fan and Griffies 2014; Li et al. 2016; Noh et al. 2016). Surface waves also play a critical role in OSBL near-surface mixing and air–sea fluxes through wave breaking (e.g., Melville 1996; Pizzo et al. 2016). Surface wave breaking in the OSBL primarily enhances the turbulence from traditional (law of the wall) scaling over a layer confined near to the surface (Agrawal et al. 1992; Anis and Moum 1995; Gerbi et al. 2009; Thomson et al. 2016). Langmuir turbulence, however, modifies turbulence properties and entrainment rates throughout the OSBL based on numerous descriptive studies based on field campaigns (e.g., Weller and Price 1988; Smith 1992; Kukulka et al. 2009; Hoecker-Martínez et al. 2016) and modeling studies (e.g., Skyllingstad and Denbo 1995; McWilliams et al. 1997; Noh et al. 2004; Li et al. 2005; Polton and Belcher 2007; Harcourt and D’Asaro 2008; Kukulka et al. 2010; Van Roekel et al. 2012). Langmuir turbulence is hypothesized to be the primary wave-driven mechanism modifying entrainment at the base of the OSBL (following Sullivan et al. 2007; McWilliams et al. 2012).

### b. Parameterizations for Langmuir turbulence

Parameterizations including the effect of Langmuir turbulence have included modifications to the *K*-profile parameterization (KPP; Large et al. 1994; McWilliams and Sullivan 2000; Smyth et al. 2002; Sinha et al. 2015; Reichl et al. 2016; Li et al. 2016; Li and Fox-Kemper 2017) and to turbulent kinetic energy (TKE) closure (D’Alessio et al. 1998; Axell 2002; Kantha and Clayson 2004; Harcourt 2013, 2015; Noh et al. 2016). In the KPP approach, the turbulent velocity scale is typically enhanced using LES-derived relationships between the turbulent vertical velocity variance^{2}

_{SL}) more accurately captures the effect of Langmuir turbulence on long-lasting, bulk turbulence properties (specifically the vertical average of vertical velocity variance over the boundary layer

*H*

_{bl}:

_{SLp}) was later introduced by Van Roekel et al. (2012) to include the effect of misalignment between Langmuir cells and the wind stress. Reichl et al. (2016) subsequently found that La

_{SLp}(in a slightly modified form) was preferred for parameterization of Langmuir turbulence in KPP for hurricane conditions. A KPP parameterization using La

_{SLp}also yielded the best prediction of global OSBL mixing in a comparative climate study by Li et al. (2016). More general definitions of the Langmuir number, such as based on the spectrally filtered surface Stokes drift (Kukulka and Harcourt 2017), have also been proposed. It is not known if a single form of the Langmuir number is preferred for parameterization of all aspects of Langmuir turbulence. We therefore consider possible forms of the Langmuir number in this study.

### c. Outline

The primary goal of this study is to extend the RH18 parameterization to account for additional mixing due to Langmuir turbulence. We focus specifically on the effect of Langmuir turbulence on the bulk OSBL properties through the enhancement to the integrated entrainment buoyancy flux. In section 2, we introduce the primary datasets used for this study and the relevant numerical models. In section 3, we then describe relevant quantities and modifications needed to parameterize the turbulent mixing and Langmuir turbulence following the RH18 approach. In section 4, we demonstrate the application of these results in one-dimensional model simulations and compare our methods directly to results obtained using the approach of Li and Fox-Kemper (2017). The symbols and acronyms used within this study are listed for convenience in Tables 1 and 2.

List of symbols used for this study.

List of acronyms used for this study.

## 2. Methods

This study investigates the contribution of Langmuir turbulence to turbulent mixing in mixed convective and shear-turbulence regimes. There are numerous difficulties that prevent precise investigation of this process from in situ or laboratory settings. These difficulties include reproducing the necessary range of scales and phenomena in the laboratory and obtaining the accuracy, quantity, and type of measurements needed to carefully diagnose and isolate OSBL turbulent processes in the field. Therefore, we utilize idealized numerical simulations for the present investigation, though future validation against field data is critical.

### a. LES and one-dimensional MOM6 experiments

The primary numerical simulations used for this study are the large-eddy simulations (LES) previously presented in Li and Fox-Kemper (2017, hereafter LF17). We briefly review their LES experiments here and refer the reader to the original citation for details. LF17 simulates the wave-averaged Craik–Leibovich equations for a wide range of external forcings including wind (5–10 m s^{−1}), wave (broadband empirical spectra of varying wave age), and convective (from −5 to −500 W m^{−2}) conditions. To investigate the role of varying Langmuir turbulence we obtain simulations from LF17 employing Stokes drift calculated from the Donelan et al. (1985) empirical wave spectra with up to four different wave ages. This set of experiments covers typical wind, wave, and destabilizing buoyancy flux conditions in the global ocean in dimensionless space (see LF17).

We conduct three additional simulations using the identical model approach as LF17. The first simulation varies the Coriolis parameter *f*, since RH18 finds that ^{−2}), since a stabilizing flux restricts the turbulence and shoals the Obukhov depth *L*_{O}. The LES experiments considered here are grouped and labeled by ID numbers for quick reference and the full list is given in Table 3. Using these LES data, the effect of Langmuir turbulence within the ePBL/RH18 framework is assessed and ePBL is modified accordingly.

List of LES experiments. Note that for experiments 3–18 the latitude is set to 45°, while for experiment 19 (marked with an asterisk) the latitude was set to 22.5°.

A one-dimensional version of the Modular Ocean Model 6 (MOM6; Adcroft et al. 2018) is used here to drive the new parameterization and validate it against the LES. These one-dimensional simulations are initialized from the same temperature profiles and forced by similar surface forcing as the LES experiments. In addition, MOM6 can use KPP (Large et al. 1994; Van Roekel et al. 2018) via the Community Vertical Mixing (CVMix) package (see github.com/CVMix). This allows the modified ePBL parameterization to be compared directly to experiments with KPP-based Langmuir turbulence parameterization (such as LF17).

### b. The ePBL parameterization

*w*′ and buoyancy

*b*′. This flux describes the rate of conversion between turbulent kinetic energy (TKE) and potential energy (PE). The vertical integral of this flux (hereafter

*M*

_{e}) over the entrainment layer is given by

*Z*

_{eT}and

*Z*

_{eB}are the top and bottom depths of the entrainment zone. The entrainment zone, with thickness

*H*

_{e}=

*Z*

_{eT}−

*Z*

_{eB}, is defined to encompass the region in the OSBL where the buoyancy flux is negative,

*M*

_{e}.

*K*:

*M*

_{e}with the boundary layer depth,

*H*

_{bl}= −

*Z*

_{eB}, and

*K*via an implicit, iterative integral constraint:

*K*is parameterized from the product of a turbulent velocity

*w*

_{t}and turbulent length

*L*:

*C*

_{K}is a constant, empirical coefficient.

*w*

_{t}and

*L*are given in RH18. For

*w*

_{t}, turbulent velocity sources are decomposed into convective

*z*via

*a*is a fixed vertical decay scale. The length scale is found from wall theory as

*z*

_{0}> 0 is an empirical surface roughness length. The exponent

*γ*in this length scale relation determines the vertical shape of

*L*(and hence

*K*), with

*γ*= 1 having a law-of-the-wall profile at both the surface and the boundary layer depth and

*γ*= 2 approximating the KPP cubic shape function. This simplified approach for parameterizing

*K*is motivated for implementing in present generation climate model configurations (see RH18) as it accommodates constraints in vertical grids (typically less than 100 vertical levels to resolve the full ocean) and long time steps (coupling time steps of order 1 h). Advancing computational capabilities may alleviate these constraints to an extent, though optimal resource allocation between model complexity and model ensemble size remains debated (see Hewitt et al. 2017). Additionally, efforts such as subcycling and using alternative hardware (specifically graphics processing units) to accommodate more complex

*K*parameterization approaches [e.g., TKE closure such as Harcourt (2015)] in climate models are a topic of present research.

The equations used to form the mixing coefficient by ePBL are closed via parameterizations for *M*_{e}. In section 3 we discuss this approach, including the existing parameterizations supplied by RH18 and the modifications needed for including Langmuir turbulence.

### c. KPP with LF17 Langmuir entrainment

We refer the reader to Large et al. (1994) and Van Roekel et al. (2018) for comprehensive descriptions of KPP and its algorithms. Here we briefly review the main points of KPP as pertaining to the LF17 Langmuir turbulence modification for comparison with the results of this study.

*G*(

*σ*) is a nondimensional shape function varying in the vertical between zero and unity based on

*σ*= |

*z*|/

*H*

_{bl}. Equation (11) is effectively equivalent to Eq. (6) if we assume

*L*≈

*G*(

*σ*)

*H*

_{bl}and the nondimensional coefficient is absorbed into the length and/or velocity scale. The definition of

*w*

_{KPP}and

*w*

_{t}can vary depending on the application, and these differences and their respective enhancement due to Langmuir turbulence are not discussed in detail here (see Van Roekel et al. 2012; Reichl et al. 2016; LF17; RH18).

*H*

_{bl}is diagnosed from the depth where vertical profile of the bulk Richardson number exceeds a critical value:

*K*profile) are determined iteratively so that the integrated energetic effect of the vertical mixing is equivalent to the energy available

*M*

_{e}[see Eq. (5) and RH18 for more detail]. The important difference between the KPP and ePBL approach is that

*M*

_{e}is directly parameterized for ePBL, but is only indirectly related to the KPP bulk Richardson number method.

## 3. *M*_{e} and its parameterization

We now discuss the parameterization of quantity *M*_{e} as it pertains to ePBL and Eq. (5). It is important to clarify the physical significance of this quantity. To aid this discussion, simplified profiles of *M*_{e}. In neutral and stabilizing regimes, *M*_{e} is equivalent to the total integral of *M*_{e} isolates only the vertical fraction of the boundary layer where

The peak entrainment buoyancy flux is defined as the value of *Z*_{e}, which is the depth where *κ* is the von Kármán coefficient, and *M*_{e} remains nonzero in such conditions due to the competition between mixing and the maintenance of the stable near-surface buoyancy gradient by the stabilizing surface flux. The bottom of the entrainment layer *Z*_{eB} is equivalent to the base of the surface boundary layer −*H*_{bl}, so these two parameters may be used interchangeably.

### a. M_{e} in shear and convective turbulence

RH18 presents a parameterization to predict *M*_{e} from nondimensional relationships between 1) the surface fluxes, 2) characteristic time scales (the rotation and/or buoyancy frequency), and 3) a characteristic length scale (*H*_{bl}). The effects of these factors on *M*_{e} is investigated by RH18 using a one-dimensional column model employing a form of the *k*–*ε* turbulent mixing parameterization. The primary focus of their study was to develop a framework to parameterize OSBL turbulent mixing by predicting *M*_{e} and subsequently using this quantity to constrain the turbulent mixing through the energetic effects on the mean state. This *M*_{e} parameterization by RH18 applies for shear-driven stabilizing, shear-driven neutral, convective, and mixed shear/convective turbulence, but does not consider Langmuir turbulence. We therefore will first compare the *k*–*ε* derived RH18 shear and convective turbulence parameterizations against the collection of non-Langmuir LES within this study (see section 2a for the LES description).

*M*

_{e}is parameterized by dividing into mechanical processes and convective processes:

*S*) and nonstabilizing (

*N*) regimes as

*c*

_{N1}= 0.275,

*c*

_{N2}= 8,

*c*

_{N3}= 5,

*c*

_{S1}= 0.2,

*c*

_{S2}= 0.4, and

*c*

_{Ψ}= 0.67). The proportionality

*M*

_{e}) through

*M*

_{e}occurs in this region). This reduction is represented by decreasing

The shear-turbulence quantity represented by

#### 1) Estimating ${n}_{*}$

*k*–

*ε*formulation and historical observations (e.g., Caughey and Palmer 1979, and studies cited therein). The LES experiments do not include any pure convection cases to provide a direct LES estimate of

*M*

_{e}) as

*B*

_{Frac}is zero during purely mechanical driven turbulence and tends to one for purely convective driven turbulence. Extrapolating the value of

*B*

_{Frac}→ 1 can therefore provide a method to estimate

A linear extrapolation of this ratio provides an estimate as the upper bound (blue dashed line, Fig. 2) and yields *B*_{Frac} = 1. However, the ratio may level off at *B*_{Frac} > 0.75). Without additional experiments it is difficult to diagnose whether the departure from the trend is real or an artifact. Since the focus of this study is not convective turbulence, we will not investigate this value in detail, but instead test both values of

#### 2) Estimating ${m}_{*}$ and its components

*M*

_{e}can then be used to estimate

*B*

_{Frac}where convection is more important and hence likely dominates uncertainty significantly exceeds the RH18

*B*

_{Frac}fit well with the curve.

To evaluate the RH18 parameterization against all the cases from LF17 with no Langmuir turbulence we compare *M*_{e} from RH18 to that diagnosed from LES in Fig. 4a. The results predicted using the RH18 formulation are consistent with the results computed from the LES over the entire range of convective, stabilizing, and shear turbulence regimes. We therefore conclude that the RH18 parameterization for *M*_{e} compared to the LES in non-Langmuir conditions, even for the cases where

In Fig. 4 we also compare the RH18 parameterization against the full set of simulations from LF17 including Langmuir turbulence (Fig. 4b). We find that a significant fraction of *M*_{e} is underpredicted by RH18 in the simulations with Langmuir turbulence, particularly those in strongly mechanically forced conditions (darker markers). The RH18 parameterization is inadequate to capture the effect of Langmuir turbulence without modification. A proposed modification is presented in the following section.

### b. Enhancement to M_{e} in Langmuir turbulence

*M*

_{e}in the modified form:

*M*

_{e}beyond the level that would occur in shear-only turbulence without considering the fractional contribution of Langmuir turbulence and shear turbulence.

The value of _{t} as defined in Eq. (1) or La_{SL} as defined in Eq. (2). In general, La_{SL} has been the preferred parameter for scaling integrated turbulence metrics in previous studies (Harcourt and D’Asaro 2008; Van Roekel et al. 2012; Reichl et al. 2016; Li and Fox-Kemper 2017). However, different metrics can scale better with different forms and exponential relations of the Langmuir number (e.g., Reichl et al. 2016), while only the scaling of Stokes production is bounded by theory to scale as

*p*is an empirically found exponent. The value

*p*= 2 (Fig. 5a) is equivalent to the scaling of Stokes production (e.g., Belcher et al. 2012):

*p*= 1 (Fig. 5b), that is,

*B*

_{Frac}, indicated by the shading. To reduce this scatter we tested additional parameter dependency using a matrix of nondimensional parameter combinations from

*B*

_{f}, |

*f*|, and

*H*

_{bl}, yielding most significant reduction in RMS difference from LES with the ratio

*B*

_{f}and

*H*

_{bl}approaches the Ekman length,

*B*

_{f}is introduced as this scaling works to capture variability of

*M*

_{e}in both stabilizing and destabilizing

*B*

_{f}, though limited stabilizing cases are tested here. This ratio will cause the effect of Langmuir turbulence to be reduced (increasing the effective La

_{SL}) as |

*B*

_{f}| increases or

*f*| decreases, though future investigation to refine behavior as |

*f*| → 0 is needed. We express this ratio equivalently as the ratio of the Ekman depth and the absolute Obukhov depth,

_{t}and find

*c*

_{LT}is the slope in each respective panel of Fig. 5. This fit is intentionally conducted for the dimensional data to bias toward the stronger mechanical mixing events when the Langmuir turbulence signal is less susceptible to noise. To confirm the validity of the results across the parameter space, fits for the same relations is shown to nondimensional parameters in Fig. 6.

In Fig. 7, *M*_{e} estimated from Eq. (20) with _{t} as in Eq. (25) or La_{SL} as in Eq. (24) are compared with *M*_{e} diagnosed from LES. Both appear to reproduce *M*_{e} in the LES, though La_{SL} is preferred as it yields smaller RMS error. This is consistent with La_{SL} being preferred in previous studies based on scaling of enhanced *K* (Reichl et al. 2016), and

## 4. Results

In this section we will compare the detailed results of simulations with ePBL including the modification proposed in the previous section against a subset of the LES simulation. We will also include the LF17 model in this comparison and compare and contrast the two approaches in detail.

### a. Impact in one-dimensional simulations

Here we investigate the ability of ePBL with the new *K* throughout the water column in this study (see Reichl et al. 2016). We therefore introduce two versions of ePBL that are tested in detail in this section. The first version (ePBL-LT) uses ePBL as described in section 2b. The second version (ePBL-LT-Bulk) is similar to the first, but use arbitrarily large coefficients in Eqs. (8) and (9), thereby mimicking a well-mixed boundary layer parameterization [arbitrarily large *K* by Eq. (6)].

These two approaches bound the possible modifications to *K*, since the first method will underestimate *K* in the presence of Langmuir turbulence, while the second method is the maximum possible *K* and therefore an overestimation. For comparison with the LF17 study we also employ the CVMix-KPP model using the LF17 modification for computing the boundary layer depth. We investigate results from experiments with 10 m s^{−1} wind speed in these results, since the net Langmuir turbulence effect tends to be most profound due to the strong wind and wave forcing (experiments 5, 9, and 12).

In Fig. 8, the temperature profile from two experiments with small (experiment 5, −5 W m^{−2}, left panels) and large (experiment 12, −50 W m^{−2}, right panels) convective contributions are shown. In each case the results for no wave (solid) and wave age = 1.2 (dashed) are plotted on the same axis at the similar time from each LES and the one-dimensional simulation. The length of the LES simulations in all cases are not the same, so therefore we present all results at the conclusion of the shortest duration LES run (1.07 simulation days). The LES and one-dimensional model turbulent spinup times are not comparable, meaning it may not be valid to compare the simulations at the same time step. However, we also investigated plotting LES/MOM6 results at the time of equivalent PE change (which is exactly comparable due to their identical vertical domain configuration) and found virtually identical results.

In general, the temperature profile comparisons suggest a few key conclusions. First, both the KPP and ePBL parameterizations are able to reasonably predict the entrainment without Langmuir turbulence in both experiments shown here (this is compared directly in Fig. 9). Second, the ePBL approach captures the enhanced cooling and entrainment due to Langmuir turbulence reasonably well for both simulations, while the KPP approach underestimates the total excess cooling in the Langmuir turbulence case. Third, the vertical structure of the mixing coefficient has a significant impact on the ability of the parameterization to predict a similar temperature profile at a similar time, indicating disagreement between the locality of where the mixing occurs in each experiment (e.g., note the different locations of relative vertical temperature homogenization between the green and red lines in both experiments). Finally, the excess homogenization of the temperature profile near the surface by Langmuir turbulence can only be captured by explicitly increasing *K* near the surface, a point particularly emphasized by the poor performance of the red-dashed profile compared to the solid red profile in Figs. 8c and 8d. We note that increasing *K* following Reichl et al. (2016) in ePBL-LT can improve this bias, but since their study only investigated strongly shear-driven mixing under hurricanes we do not present this result in detail but propose it as a motivation for future research.

There are two significant differences between the KPP method and the ePBL method that contribute to the differences seen in this set of one-dimensional simulations. First is that KPP also includes nongradient turbulent buoyancy flux (see, e.g., Van Roekel et al. 2018). The impact of the nongradient mixing is seen most clearly in the difference between the ePBL (red) and KPP (green) temperature profile in the stronger convection case (Figs. 9b,d). Second is that ePBL is numerically more robust to model time step and vertical resolution and better able to capture the enhancement of mixing due to Langmuir turbulence in these simulations where vertical resolution is much higher than typical applications in climate models. The second difference is entirely due to the sensitivity of KPP to implementation details, specifically the sensitivity of *M*_{e} relationships found here (this is explored in detail in section 4b). The disagreement between the simulations with ePBL and KPP here only emerge when the LF17

We next investigate bulk properties of the model simulation including the predicted rate of integrated potential energy change and the rate of temperature cooling at the surface. These two metrics indicate different aspects of the performance of the vertical mixing parameterizations, which when taken together indicate the ability of a parameterization to accurately simulate the entire ocean surface boundary layer. To mitigate the difference between the LES and one-dimensional model spinup process of turbulence here, we focus on these specific results averaged over the time period of the final inertial period in each LES model. The LES profiles are not held constant during the turbulence spinup, which would complicate comparison of the absolute change of surface temperature and potential energy relative to the initial condition. Therefore, we focus our investigation on the mean of the instantaneous rate of change of these quantities for a time period not including the spinup processes. We show the results from cases without waves (with squares, triangles, and diamonds indicating the three different experiments) and with the strongest wave forcing (wave age = 1.2, circles with like experiments connected with lines for clarity). We first show the surface temperature rate of change in the top grid cell (SST, with identical top gridcell thickness between LES and one-dimensional model), where we use identical vertical discretization in the one-dimensional model and the LES (Fig. 9a) for equivalence. We see that the ePBL-LT-Bulk and ePBL-LT simulations perform best for predicting consistent SST tendency both with and without Langmuir turbulence (cyan and red markers), with the LES result bounded between the ePBL-LT and ePBL-LT-Bulk result. These results indicate that further improved SST cooling rate simulation could be achieved through combining the ePBL-LT approach with a localized (in the vertical) mixing enhancement. The PE increase rate plots (Fig. 9b) show that all three experiments perform roughly equally well, with the ePBL-LT-Bulk simulations being between the KPP and ePBL-LT result.

To more carefully examine the effect of Langmuir turbulence in the models we then take the difference between the results with and without Stokes drift (Figs. 9c,d). In this case we find that ePBL-LT performs better for predicting the change in both SST-cooling and PE increase rate due to Langmuir turbulence. The offset between the LES and column model result grows for both KPP and ePBL-LT as the contribution due to Langmuir turbulence grows (Figs. 9c,d). However, the ePBL-LT-Bulk does not show a similar increase in error, again suggesting that the localized mixing enhancement can further improve these results.

### b. The relationship between M_{e} and $\overline{{w}^{\prime}{b}_{e}^{\prime}}$

*M*

_{e}, the integral of the buoyancy flux over the entrainment region (see Fig. 1). To compare our results with LF17, we investigate the relationship between

*M*

_{e}and

*M*

_{e}[Eq. (3)] against

*Z*

_{eT}, which can be diagnosed within the LES result. The region between

*Z*

_{eT}and

*Z*

_{e}makes up the upper region of the entrainment layer. There exists a secondary region below

*Z*

_{e}, where the buoyancy flux remains negative (indicating active work against gravity) but increases (becomes less negative) moving downward to the base of the entrainment layer

*Z*

_{eB}where the buoyancy flux becomes zero. The secondary region contributes a smaller fraction of

*M*

_{e}in the shear-driven case (Fig. 1, left panel), but can make up roughly half of

*M*

_{e}in the convective case (Fig. 1, right panel). The buoyancy flux profile in the entrainment layer approximately forms a pair of triangular shaped regions, approximately equating the mean buoyancy flux within the entrainment layer with

*M*

_{e}is found from this mean buoyancy flux multiplied by the thickness of this layer (

*H*

_{e}=

*Z*

_{eT}−

*Z*

_{eB}):

*M*

_{e}and

*H*

_{e}between shear and convective regimes (as demonstrated in Fig. 1).

*Z*

_{eB}, from

*Z*

_{e}and

*Z*

_{eB}using our definition of the base of the OSBL (internal wave-breaking or interior-driven turbulence would complicate this scenario, but we neglect internal processes for this discussion). For simplicity, this depth is assumed to scale linearly by a coefficient

*A*with the entrainment depth:

*A*from the LES results by

*A*varies based on the buoyancy TKE production fraction

*B*

_{Frac}[Eq. (18)]. Through a series of least-mean square adjustments based on the total integrated mixing (the final version appearing in Fig. 10c), we derive a relationship between

*A*and

*B*

_{Frac}:

*α*

_{1}= 0.12,

*α*

_{2}= 0.16, and

*α*

_{3}= 10 are empirical coefficients. The coefficients are fit in this method to bias the curve not to best fit the data weighted evenly, but to weight the curve toward data points representing simulations with more significant total mixing (more negative

*M*

_{e}). The inclusion of this

*B*

_{Frac}dependence in

*A*is justified by a significantly reduced RMS error between the empirical

*M*

_{e}and the LES

*M*

_{e}versus using (for example) a constant mean value of

*A*. The final relationship to estimate

*M*

_{e}is given by

*M*

_{e}from

Thus, we find that by accounting for the variability of the entrainment layer thickness *H*_{e} through the diagnosed peak entrainment buoyancy flux *Z*_{e}, and the diagnosed buoyancy production fraction *B*_{Frac}, we can infer the integrated entrainment buoyancy flux *M*_{e} with a high degree of accuracy. Therefore, the LF17 entrainment buoyancy flux parameterization can be used to estimate *M*_{e}, which is investigated in the following section. In the appendix we discuss the implications of Eq. (32) for estimating the OSBL thickness.

*M*

_{e}(section 3) to predict

*M*

_{e}from Eq. (20) is compared with the LES results. The consistency between Eq. (37) and LES suggests that our new parameterization for

*M*

_{e}modified to include Langmuir turbulence [Eq. (20)] is valid for predicting both

*M*

_{e}and

*M*

_{e}and

### c. Estimating M_{e} following LF17

Using Eq. (36) we can estimate *M*_{e} from LF17 by substituting Eq. (27) for *M*_{e} diagnosed from LES with those predicted by the LF17 parameterization of *M*_{e} based on their parameterization. This result is expected because this LES data is the same that was used to derive their parameterization.

*M*

_{e}. The latter tuning strategy gives

*M*

_{e}by approximately 25% (Figs. 12b,d). Note that we do not suggest use of Eq. (38) in place of the LF17 relationship in their modifications of KPP. Rather, we present this alternate fit here to compare

*M*

_{e}predicted by their method in a more representative manner, since this metric was not used as the basis for determining the coefficients in their study. The different coefficients in Eqs. (27) and (38) result from different weights put on the Langmuir regime versus convection regime.

In this study we have adapted the RH18 parameterization by adding the term *M*_{e} for the effect of Langmuir turbulence. We compare our result to LF17, who use a different approach to parameterize turbulent mixing modification due to Langmuir turbulence. In general, the approach given here and that of LF17 are compatible and provide a similar result, which is demonstrated here through comparisons of both methods with the LES results including *M*_{e} (Figs. 7 and 12), and the one-dimensional simulations (Figs. 8 and 9). The result presented here obtains a lower RMS error for predicting *M*_{e}, but this result is anticipated since LF17 did not target prediction of this quantity.

*H*

_{e}/

*H*

_{bl}in LF17. This is not a perfect interpretation of RH18, as the value 0.085 would also be variable in RH18, however, the dominant variability in

*M*

_{e}in LF17, RH18, and this study is controlled by this ratio of the entrainment layer thickness to the boundary layer thickness. This is why both parameterizations provide similar responses in the one-dimensional simulations here, particularly as both sets of empirical coefficients are determined using the same LES. One advantage of the approach taken here for implementing within the ePBL approach is that it provides an estimate for

*M*

_{e}from external forcing parameters without considering

*H*

_{e}. Note that, same as in Eq. (36), we are assuming simplified buoyancy flux profiles as shown in Fig. 1, which is valid based on the results of LES in quasi-equilibrium state. This simplification, for example, helps alleviate sensitivity of ePBL to vertical grid discretization, such as in section 4a.

## 5. Discussion

In this study we found that the new parameterization for *M*_{e} is capable of reproducing the simulated potential energy of the LES in a one-dimensional model that employs the ePBL mixing parameterization. We also found that this model is able to reproduce the effect of enhanced SST cooling by Langmuir turbulence across a range of wind and both stabilizing and destabilizing buoyancy flux conditions. Future research is needed to understand if these (and other) parameterizations remain valid over realistic conditions with diurnal cycles and penetrative solar radiation. A comparison between ePBL-LT, KPP with LF17 modifications, and many other parameterizations over realistic conditions in the global ocean is underway (Li et al. 2019, manuscript submitted to *J. Adv. Model. Earth Syst.*).

As observed through comparing ePBL-LT with ePBL-LT-bulk, another approach that may further improve agreement of the one-dimensional model with LES is through addition of an enhancement to *K*, such as that presented by Reichl et al. (2016). However, the Reichl et al. (2016) study focused only on strongly shear-driven hurricane conditions, motivating further investigations to better parameterize all effects of Langmuir turbulence in all forcing conditions for ocean models. We have also demonstrated how the integrated buoyancy flux (*M*_{e}) and the peak entrainment flux (

One specific future research topic is the role of nongradient mixing in convection and Langmuir turbulence and its parameterization for one-dimensional modeling. Nongradient mixing is a critical component of the KPP model in convective turbulence, which enables it to better reproduce the effect of large eddies in the OSBL. However, the one-dimensional model with KPP employed here does not reproduce the LES temperature profile, causing the inflection in the temperature profile to shoal significantly within the boundary layer, an effect of the nongradient flux representation in the model (see Fig. 8). Future research is needed to understand the vertical structure of *K*, nongradient buoyancy fluxes, and its modification by Langmuir turbulence (following Smyth et al. 2002; Noh et al. 2003; Sinha et al. 2015).

## 6. Conclusions

The present study investigates the role of Langmuir turbulence in enhancing OSBL turbulent mixing. Specifically, its effects on modifying the rate of change in potential energy through turbulent entrainment are compared to shear and convective turbulence. We first investigate parameterization of *M*_{e} in LES cases with and without Langmuir turbulence and find relations capable of accurately predicting this quantity. Using the LES results of LF17 with Langmuir turbulence, we further derive a parameterization to predict *M*_{e} in the RH18 framework in the presence of Langmuir turbulence [Eq. (20)]. This parameterization is optimized using the surface-layer averaged Langmuir number and is adjusted for the effects of convective turbulence on *M*_{e}. We then implement this new parameterization for *M*_{e} into the ePBL parameterization framework of RH18 and demonstrate its utility for one-dimensional vertical mixing parameterization.

We also demonstrate a method to estimate the entrainment buoyancy flux *M*_{e} (this study). The relationships used to derive these quantities also yields a new method to estimate the surface boundary layer depth, *H*_{bl}, that has potential utility for LES and TKE-based closure schemes (see the appendix). We conclude that the LF17 approach and the approach presented here yield generally similar results, which is not a surprising result since both are based on the same set of LES. However, we also find that implementing these relationships in time-stepping models that use ePBL and KPP can result in different behavior. This sensitivity suggests that careful consideration of potential effects of factors such as model discretization on parameterization physics is required when implementing these relationships.

An extension of this work is to explore the ePBL parameterization of RH18 with and without Langmuir turbulence in coupled climate simulations to better understand the effects of Langmuir turbulence on global mixed layer depth distributions, and to compare with the results of previous studies (Belcher et al. 2012; Fan and Griffies 2014; Noh et al. 2016; Li et al. 2016; Li and Fox-Kemper 2017). Preliminary results utilizing NOAA/GFDL’s Modular Ocean Model 6 configuration for climate simulations suggest significant improvement to the OSBL depth by accounting for Langmuir turbulence in ePBL, particularly at high latitudes including the Southern Ocean. This is generally in agreement with the previous studies on the subject. Detailed experiments and analysis of such result will be presented in a follow-up manuscript.

## Acknowledgments

We acknowledge many important discussions related to this work with Dr. Alistair Adcroft, Dr. Stephen Griffies, Dr. Robert Hallberg, and Dr. Baylor Fox-Kemper. We thank Dr. Andrew Shao for helpful comments on a draft of this manuscript. We also thank Dr. Luke Van Roekel and an anonymous reviewer for helpful comments that improved the quality of this manuscript. BR acknowledges support from the Cooperative Institute for Climate Science and the Cooperative Institute for Modeling the Earth’s System at Princeton University and the Carbon Mitigation Initiative through Princeton’s Environmental Institute at Princeton University. QL acknowledges support from NSF Grants 1258907 and 1350795 under the supervision of Dr. Baylor Fox-Kemper. This research was supported in part by the National Science Foundation under Grant NSF PHY-1748958. All the LES runs were conducted using computational resources and services at the Center for Computation and Visualization, Brown University, supported in part by the National Science Foundation EPSCoR Cooperative Agreement EPS-1004057. We are grateful to Peter Sullivan for making the LES code available.

## APPENDIX

### Estimating the Boundary Layer Depth

There are several methods to estimate the boundary layer depth *H*_{bl} in large-eddy simulations and two-equation closure models. A reason for this ambiguity is that there is often no discrete interface between the boundary layer and the interior, and attempting to diagnose a single value within the transition layer is uncertain. Some common methods include diagnosing from maximum *N*^{2} or a threshold criteria for a turbulent property such as the buoyancy flux, turbulent kinetic energy, or the dissipation. In this study the boundary layer depth *H*_{bl} can be estimated from the quantity *A* [Eq. (35)] and *Z*_{e} through relation (32). There are some potential benefits of using this method versus the other methods (e.g., arbitrariness of threshold values, *N*^{2} peaks due to interior/hysteresis), so we will investigate the usefulness of this relation here.

*A*, which in turn requires knowing

*B*

_{Frac}, or the buoyancy production fraction in the boundary layer. Care is required while estimating the integrated buoyancy production in the OSBL (

*B*

_{Frac}therefore requires knowing the boundary layer thickness, which adds undesirable complexity to the algorithm required to find

*H*

_{bl}. To eliminate this recursion, we approximate the buoyancy production term from

*G*

_{bl}computed from the LES result as demonstrated in both dimensional (Fig. A1a) and nondimensional (Fig. A1b) form, where

*B*

_{Frac}are estimated directly from the LES result without differentiating the OSBL contribution, since nearly all shear production occurs within the OSBL in the experiments conducted here. We thus use the quantity

*B*

_{Frac}without prior knowledge of

*H*

_{bl}(where

*Z*

_{B}is the bottom of the LES domain).

We compare the depth estimated from this new method and other simplified methods to that estimated using a threshold *Z*_{e}. Unsurprisingly, since we have previously shown the importance of the lower-entrainment region, this estimate underestimates the OSBL in all cases. We then (Fig. A2b) estimate the OSBL depth from the depth where *N*^{2} reaches its maximum, another common quantity used to evaluate OSBL depth in LES. This quantity reproduces the OSBL depth compared to the *A* = 0.2 to enhance *Z*_{e} by a factor of 1.2 in Fig. A2c. This result has a reasonable fit to the OSBL, but shows a clear departure between 1:1 scaling varying between shear and convective cases. Finally, we use the value of *A* predicted in this study [Eq. (35)] and estimate *H*_{bl} ≡ *Z*_{eB} in Fig. A2d. This result shows skill for predicting the OSBL depth.

The new approach for estimating *H*_{bl} from *Z*_{e} and *A* is advantageous over the *N*^{2} in transient conditions, as *N*^{2} may have more sensitivity to internal processes or hysteresis effects. The *Z*_{e} may also be hard to define in transient conditions, as the *N*^{2}.

This result requires further investigation in future studies to ensure robustness and to explore how applicable this method is for isolating OSBL from internally generated turbulence sources in nonidealized one-dimensional simulations. Such investigations are specifically needed to adapt this method for use in global ocean simulations that utilize TKE closure type mixing parameterizations, particularly when internal turbulence generation regions interact with the entrainment layer and the entrainment layer is therefore no longer entirely surface driven. Such scenarios would complicate evaluation of the shear-production contribution, since any internal shear layers should be neglected. Furthermore, the diagnosed quantity

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