## 1. Introduction

The ocean surface boundary layer (OSBL) is mixed by turbulence, driven by a combination of surface waves, wind stress, and surface heat fluxes. Surface waves can result in mixing through wave breaking and through a “vortex force” interaction between the wave-induced Stokes drift (Stokes 1847) and the vorticity in the flow (Craik and Leibovich 1976). Instabilities created by this vortex force produce Langmuir turbulence, which is believed to be a significant contributor to upper-ocean mixing over the global ocean (Li et al. 2005; Belcher et al. 2012; D’Asaro 2014).

Heat fluxes into the ocean can be separated into shortwave radiative heating, where the radiation is absorbed over a depth of several meters (Denman 1973; Paulson and Simpson 1977), and surface heating (sensible and latent heating and longwave radiation). The OSBL often exhibits a strong diurnal cycle; during the day a shallow, weakly stratified (well mixed) layer is present, with a diurnal thermocline forming below this layer, and during the night a deeper convective well-mixed layer develops (Brainerd and Gregg 1993, 1995; Sutherland et al. 2013, 2014).

Large-eddy simulation (LES) has been used to study Langmuir turbulence, beginning with the work of Skyllingstad and Denbo (1995). Results from LES show that Langmuir turbulence can maintain weak stratification near the ocean surface in the presence of surface heating (Min and Noh 2004) and a diurnal thermocline in the presence of radiative and surface heating (Noh et al. 2009). Kukulka et al. (2013) showed that LES provides better agreement with observations of the diurnal cycle of upper-ocean temperature structure when the vortex force is included. In particular, in the daytime OSBL the stratification in an LES without Langmuir turbulence was greater than the observed stratification during periods of the Surface Waves Process Program (SWAPP) experiment, and including Langmuir turbulence in the LES reduced the stratification. The K-profile parameterization (KPP) model (Large et al. 1994) and the Price–Weller–Pinkel (PWP) model (Price et al. 1986), which are both popular OSBL models, produced stratification similar to the LES without Langmuir turbulence. Plueddemann and Weller (1999) showed that Langmuir circulations, which are structures associated with the presence of Langmuir turbulence, were present over the observation period simulated by Kukulka et al. (2013). They also showed that in periods of weak winds, where no Langmuir circulations were present, the observed stratification was similar to that obtained from the PWP model. Despite the previous LES studies, the structure of Langmuir turbulence in the presence of surface heating has not been investigated in detail and, as a result, the response of the wave-driven OSBL to a surface heat flux is not understood quantitatively.

In this study, LES is used to investigate the equilibrium structure of the wave-driven OSBL following the application of surface heating to an initially neutral OSBL. The study is confined to the OSBL driven by waves through the vortex force, rather than other wave effects such as wave breaking (Noh et al. 2004; Sullivan et al. 2007) and wave-induced nonbreaking turbulence (Babanin 2006). The goal of this work is to develop scalings for various OSBL properties, in particular the depth of the OSBL, and the profiles of dissipation and vertical velocity variance. For this reason, we principally investigate simulations with constant forcing that have reached equilibrium, rather than simulations of diurnal cycles or other transient scenarios. Focusing on idealized simulations also allows the effects of surface and shortwave radiative heating to be separated and compared. After a brief discussion of model details in section 2, the structure of the equilibrium OSBL is studied in section 3 using the budgets for the turbulence kinetic energy (TKE) and the turbulent heat flux. The variation in the depth of the OSBL with surface heating is investigated in section 4. The effects of shortwave radiative heating on the equilibrium mixed layer depth are investigated in section 5, and the resulting model is compared with an LES with diurnal radiative heating.

## 2. Model and simulations

The simulations use the Met Office Large Eddy Model (LEM). The LEM for atmospheric boundary layer studies is described in Shutts and Gray (1994). The modifications of the model to simulate the OSBL are described by Grant and Belcher (2009). These modifications include imposing a Stokes drift, which affects the momentum budgets through the vortex force, the Coriolis–Stokes force, and a modified pressure (Craik and Leibovich 1976; McWilliams et al. 1997). The Stokes drift profile is given by *z* is depth (negative); *δ* is the Stokes penetration length (positive), which is related to the wavelength of surface waves by *δ* = *λ*/(4*π*); and *x* axis. The turbulent Langmuir number _{t} = 0.3 (Li et al. 2005).

The model domain is 256 m × 256 m in the horizontal (*x*, *y*) and 90 m in the vertical (*z*), with resolutions of 2 and 0.6 m in the horizontal and vertical, respectively. The forcing for the following simulations was chosen so that the boundary layer turbulence was resolved, with the shallowest simulated boundary layers being just over 10 m deep. The resolution probably begins to have an effect in the shallowest mixed layers, but the resolved turbulent fluxes are much greater than the subgrid fluxes in all the simulations. The domain is horizontally doubly periodic. A damping layer is imposed below 65 m to damp gravity waves and prevent their reflection off the lower boundary.

Simulations are started from rest with a constant temperature layer, below which the fluid has constant stratification. All simulations are forced using a surface wind stress aligned with the *x* axis, with ^{−3} m s^{−1}, equivalent to a 10-m wind speed of approximately 5 m s^{−1}. The effects of rotation are included through the Coriolis parameter *f*. In all simulations, La_{t} = 0.3.

In the surface heating simulations, the surface heat flux is initially zero, which allows Langmuir turbulence to develop a neutral boundary layer. After 50 000 s, the surface heat flux is increased linearly to a value *H*_{0} over 5000 s and is held constant for the remainder of the simulation. The simulations are then continued until the turbulence statistics and boundary layer depth reach an equilibrium, and the flow statistics are then averaged over 40 000 s. The depth of the neutral mixed layer at 50 000 s is defined as the initial mixed layer depth *h*_{i}. The introduction of heating to an initially neutral boundary layer in these simulations is intended to be analogous to the morning transition of the diurnal cycle.

This study uses seven sets of surface heating simulations, shown in Table 1. Simulations with different values of *f* were carried out representing strong (SS), moderate (SM), weak (SW), and no rotation effects (SN). Surface heating simulations with a larger Stokes penetration length (SD) and a shallower initial mixed layer (SH) were also carried out. A reference (neutral) simulation was carried out with no surface heat flux, and all other parameters were identical to the SM simulations. Also shown in Table 1 are the values of Langmuir stability length, *B*_{0} = −*αgH*_{0}/(*ρc*_{p}) is the surface buoyancy flux, *α* is the thermal expansion coefficient, *g* is gravitational acceleration, and *c*_{p} is the specific heat capacity of water (Belcher et al. 2012). The range of heat fluxes across the simulations is 0–64 W m^{−2}, which is significantly smaller than the typical daytime maximum heating. However, we are interested in the scaling of Langmuir turbulence, and scaling laws should be general for a given turbulent flow, providing the flows span the same nondimensional parameter space. Using weak surface forcing allows us to carry out a large number of simulations, spanning a wide range of parameter space. The nondimensional parameter space of these simulations includes reasonable heating (~500 W m^{−2}) with moderate to strong winds (10-m wind speed > 10 m s^{−1}) and wind waves.

Forcing parameters for the surface heating simulations. Shown are the simulation set abbreviation, surface heat flux *H*_{0}, Langmuir stability length *L*_{L}, Coriolis parameter *f*, Stokes drift penetration length *δ*, and the initial mixed layer depth *h*_{i}. Note that the highest values of *L _{L}* correspond to the lowest values of

*H*

_{0}.

## 3. Results

The following section begins with a discussion of the profiles of mean temperature and turbulent heat flux with surface heating and the definition of the mixed layer depth. Following this, the turbulence and mean properties of the OSBL are investigated using a nondimensional framework.

### a. Structure of the OSBL

Figure 1a shows the profiles of mean temperature *L*_{L} becomes smaller. Below the weakly stratified layer is a temperature jump, with a second weakly stratified layer below. The initial model stratification is below 53 m.

Figure 1b shows the turbulent heat flux *h*_{m} is then defined as the depth at which the line reaches zero. The layer of weak stratification near the surface is within the mixed layer (Fig. 1a). Below *h*_{m} the turbulent heat flux is still nonzero, indicating that the boundary layer depth *h*_{b}, where the flux goes to zero, is greater than *h*_{m}.

The temperature jump between the surface and 53 m does not exceed 0.1 K in any of the simulations, despite more than 20 h of heating, indicating that commonly used definitions of the mixed layer using temperature or density jumps (e.g., de Boyer Montégut et al. 2004) would not be sufficient for diagnosing the depth of the turbulent mixed layer in these simulations.

Figure 2 shows a schematic of the equilibrium structure of the upper ocean in LES with surface heating. The new thermocline is a layer containing turbulent, stratified flow below the mixed layer. Despite the simulations being run to equilibrium, we refer to this layer as the diurnal thermocline, as in these simulations it is analogous to the thermocline that is formed and eroded over the course of a diurnal cycle. The boundary layer consists of the mixed layer and the diurnal thermocline, and the boundary layer depth *h*_{b} is defined as the maximum depth to which TKE is transported by the turbulence (Fig. 5d). The residual layer is the layer of weak stratification below the boundary layer, and it contains decaying turbulence, present before the surface heat flux was introduced. When the heating is introduced, the turbulence below *h*_{b} becomes detached from the surface. The time scale of this detachment in the present simulations is approximately 30 min, as shown in the appendix. The initial stratification imposed in the LES remains at depths below *h*_{i}.

Figure 3 shows the mean profiles of along-wind current

Figure 4 shows the resolved turbulent fluxes of momentum *L*_{L} decreases, the

The *h*_{m}. Near the base of the mixed layer, the

### b. Turbulent kinetic energy budget

**u**′ and

*b*′ =

*αgθ*′ is the buoyancy;

*p*is pressure; and

*ε*is the rate of dissipation of energy through molecular viscosity. The terms on the right-hand side of Eq. (1) are, from left to right, Stokes production, shear production, buoyancy flux, TKE transport, and dissipation. Grant and Belcher (2009) used LES to show that the velocity and length scales of the TKE budget in Langmuir turbulence are

*h*

_{m}, respectively.

Figure 5 shows the first four terms on the rhs of Eq. (1) for two SM simulations and the neutral simulation. All the terms have been scaled by *L*_{L}. The region of significant Stokes drift gradient, and hence Stokes production, is confined to a near-surface layer with depth *O*(*δ*). As *L*_{L} decreases, *δ*/*h*_{m} increases, causing the Stokes production to be present deeper into the mixed layer.

Shear production occurs over the entire mixed layer. In the neutral simulation, shear production has a local maximum at approximately 0.25*h*_{m}. This local maximum moves deeper into the mixed layer as *L*_{L} decreases and is below the region of significant Stokes production. The scaled *h*_{m}/*L*_{L} (Belcher et al. 2012) increases with decreasing *L*_{L}, indicating that the fraction of TKE produced through Stokes production, which is then used to transport heat through the mixed layer, increases with increasing surface heat flux.

The TKE transport takes energy from near the surface, where Stokes production is largest, and redistributes this energy deeper in the mixed layer. The layer of negative TKE transport covers an increasing fraction of *h*_{m} as *L*_{L} decreases. In the SM simulations, the positive region of TKE transport penetrates below the mixed layer into the diurnal thermocline and to the base of the boundary layer (Fig. 2).

Figure 6 shows the scaled dissipation *h*_{m} the scaled dissipation profiles collapse onto a single curve. This indicates that the effects of surface heating on dissipation profiles in the upper mixed layer are captured entirely by variations in *h*_{m}, and the scaled dissipation profiles are the same as for Langmuir turbulence with no surface heating. Near the base of the mixed layer, the scaled dissipation profiles show some variation as a result of changes in the TKE transport and shear production, although this spread is much less than the spread in the unscaled dissipation profiles shown by the shaded area. The dissipation below the mixed layer decreases less rapidly with depth in the simulations with surface heating than in the neutral simulation because of decaying turbulence in the residual layer.

### c. Vertical velocity variance

Figure 7 shows the profiles of vertical velocity variance *h*_{m}, where the resolution could be beginning to affect the turbulence. The shape of the *h*_{m}, neglecting the changes in magnitude, does not appear to be affected by the surface heat flux.

In the presence of surface heating, there is a significant increase in *h*_{m}, despite *h*_{m} in some simulations.

### d. Heat flux budget

There is greater stratification near the surface in shear-driven turbulence than in Langmuir turbulence (Noh et al. 2009; Kukulka et al. 2013). To understand why this is the case, we consider the budget for the turbulent heat flux.

*ε*

_{wθ}is the dissipation of

*H*

_{0}/(

*ρ*

_{0}

*c*

_{p}

*τ*), where

Figure 8 shows the scaled profiles of the first four terms on the right-hand side of Eq. (2) within the mixed layer for the SM simulation with *L*_{L} = 62 m. From the surface to 0.8*h*_{m}, the downward flux of heat is maintained by the Stokes and gradient terms in Eq. (2). The magnitude of the Stokes and gradient terms are comparable, with the Stokes term being the larger of the two. The pressure-scalar and buoyancy terms balance the Stokes and gradient terms in the upper half of the mixed layer. The flux transport and flux dissipation are not shown in the figure. In these simulations *ε*_{wθ} is small, because the fluxes are carried by the largest eddies rather than parameterized subgrid eddies, and the flux transport is small. Near the base of the mixed layer, where the heat flux is small, the downward flux tendency of the pressure-scalar and gradient terms balances the buoyancy term, which is consistent with the presence of gravity waves.

Kukulka et al. (2013) argued that parameterizations of Langmuir turbulence should include the effects of Stokes drift on TKE production. Our results show that parameterizations should also include the effects of Stokes drift on the production of

The budget for the turbulent heat flux is the basis for the nonlocal component of the KPP scheme in unstable conditions (Holtslag and Moeng 1991), and it explains the countergradient transport of heat observed in the convective atmospheric boundary layer. The nonlocal part of the scheme, which allows countergradient transport, arises from the source terms in the *κ*_{d} is a diffusivity). However, Fig. 8 shows that the Stokes term is an important source of *κ*_{d}, is larger in Langmuir turbulence than in shear-driven turbulence with the same surface friction velocity (McWilliams et al. 1997; Teixeira and Belcher 2002, 2010).

## 4. The equilibrium depth of the mixed layer and boundary layer

The depths of the mixed layer and the boundary layer are important for the development of OSBL parameterizations as they describe the length scales of turbulence. In addition, *h*_{b} controls the heat capacity of the OSBL and hence the variation of the sea surface temperature in response to a surface heat flux. The formation of an equilibrium OSBL in the presence of Langmuir turbulence and surface heating indicates that *h*_{m} and *h*_{b} could be diagnosed from the forcing used in the simulations.

The values of *h*_{m} and *h*_{b} for the equilibrium OSBL must be a function of the characteristic length scales of the turbulent flow. The length scales that are commonly used to characterize boundary layers under a stabilizing heat flux are the Ekman depth *κ* is the von Kármán constant. Diagnostic models for the depth of stable geophysical boundary layers are generally a combination of the Ekman depth and the Obukhov length (e.g., Zilitinkevich 1972; Garwood 1977; Zilitinkevich and Baklanov 2002; Zilitinkevich et al. 2002). As a result of the new velocity (*u*_{s0}) and length (*δ*) scales introduced by the presence of Stokes drift, additional length scales can be constructed for Langmuir turbulence such as the Langmuir–Ekman depth *L*_{L}.

Figure 9 shows *h*_{m} and *h*_{b} against *L*_{L} for all simulations with surface heating. The mixed layer and boundary layer become deeper as *L*_{L} increases within each set of simulations. Kukulka et al. (2013) suggested that an alternative stability length, *L*_{K} = *δ* ln(*L*_{L}/*δ*), could provide an estimator for *h*_{m}, but the results from the present simulations show that *L*_{K} is not a useful estimator for *h*_{m}. Increasing *δ* by a factor of 3 does not significantly affect *h*_{m}, indicating that *h*_{m} is more sensitive to changes in *L*_{L} than to changes in *δ*.

Both *h*_{m} and *h*_{b} show some dependence on *f*; the mixed layer becomes deeper as *f* decreases and *h*_{m} (as in Fig. 2) for small values of *f*. Grant and Belcher (2011) showed that, in the presence of a stratified shear layer, Langmuir turbulence is confined to the mixed layer, defined as the depth over which TKE is transported by the turbulence. For the SN (*f* = 0) simulations the mixed layer depth has been calculated using the TKE transport, but these *h*_{m} have been plotted in Fig. 9b, alongside the other depths defined through TKE transport. In the SN simulations the combined depth of the mixed layer and the shear stratified layer is the boundary layer depth, which does not reach a steady state (Grant and Belcher 2011) and is not relevant to the effects of heating on Langmuir turbulence. As a result, the values of *h*_{b} from the SN simulations are not plotted. Figure 9b indicates that *L*_{L} is the main control on *h*_{b}.

Dimensional consistency requires that *h*_{m} and *h*_{b} must depend only on length scales within the simulations. The boundary layer depth does not vary significantly over a range of *δ* = [4.8 m, 14.4 m], which indicates that *L*_{L} is the only length scale with a significant effect on the value of *h*_{b} that can be constructed from the surface forcing parameters in these simulations. However, it is clear that a simple relationship of the form *h*_{b} = *γL*_{L}, where *γ* is a constant, does not describe the variation in *h*_{b}.

The only length scale that has not been considered is the initial mixed layer depth *h*_{i}. Both *h*_{m} and *h*_{b} must asymptote toward *h*_{i} as *L*_{L} tends toward infinity. This indicates that *h*_{i}, along with *L*_{L}, could be responsible for the nonlinear variation of *h*_{m} with *L*_{L} seen in Fig. 9. In addition, for all *L*_{L} investigated here, simulations using *h*_{i} = 36 m have a smaller boundary layer depth than simulations using *h*_{i} = 53 m (note that in Fig. 9 the stars and diamonds indicate simulations that differ only in their values of *h*_{i} and *L*_{L}).

*h*

_{m}and

*h*

_{b}through the turbulence present within the initial mixed layer. The introduction of surface heating adds a sink of TKE due to the buoyancy flux in Eq. (1). The largest, energy-containing, turbulent eddies are supplied with energy by Stokes production. The largest turbulent eddies are also responsible for the transport of heat and so lose energy through the buoyancy flux. As the mixed layer becomes shallower, the depth-averaged Stokes production increases as

*B*

_{0}/2. We assume thatwhere

*β*is a constant. Equation (3) can be rearranged to provide a relationship between the steady mixed layer depth and the initial mixed layer depth, as a function of the Langmuir stability lengthFigure 10a shows the nondimensional mixed layer depth

*h*

_{m}/

*h*

_{i}as a function of the initial stability parameter

*h*

_{i}/

*L*

_{L}for all surface heating simulations. The model of Eq. (4) shows good agreement with all of our simulations using (2

*β*)

^{−1}= 3.5. Figure 10b shows

*h*

_{b}/

*h*

_{i}against

*h*

_{i}/

*L*

_{L}. Also shown is Eq. (4), where

*h*

_{m}has been replaced by

*h*

_{b}, and (2

*β*)

^{−1}= 3.0, which provides a good fit to

*h*

_{b}for all the simulations. The boundary layer depths for simulations which differ only in

*h*

_{i}collapse better in Fig. 10b than in Fig. 9b.

The dependence of *h*_{m} and *h*_{b} on the initial mixed layer depth is an interesting result. As far as the authors are aware, the possibility that the depth of the initial boundary layer could affect the depth of an equilibrium stable boundary layer has not been recognized previously. This dependence on *h*_{i} means that it is not possible to create a diagnostic model for *h*_{m} and *h*_{b} that is based on only the instantaneous forcing parameters, although for *h*_{i}/*L*_{L} ≫ 1, Eq. (4) tends to *h*_{m} ∝ *L*_{L}. The relationship proposed in Eq. (4) is only valid for Langmuir turbulence La_{t} < 0.5 (Grant and Belcher 2009).

Min and Noh (2004) carried out LES of the mixed layer with surface heating and estimated the mixed layer depth through tracer dispersion. The results of Min and Noh (2004) fit smoothly with our data (Figs. 9, 10). Thus, the model of Eq. (4) shows good agreement with the simulations of Min and Noh (2004) as well as with the current simulations.

## 5. Effects of radiative heating on mixed layer depth

In addition to the effects of surface heating, the ocean can also be stabilized by the absorption of shortwave radiation, which can penetrate many meters into the water column. This shortwave radiative heating is an important source of ocean heating during the daytime. To investigate the effects of radiative heating on mixed layer depth, simulations are carried out that are identical to the SM simulations but replace the surface heat flux *H*_{0} by an irradiance profile *I*(*z*). The rate of heating due to the divergence of shortwave radiation is then given by (*ρc*_{p})^{−1}∂*I*(*z*)/∂*z*. The irradiance profile is chosen to be a simple exponential decay *I*(*z*) = *I*_{0} exp(*z*/*ξ*), where *I*_{0} is the surface irradiance and *ξ* is the radiative penetration length (Denman 1973). For the radiative heating simulations, we define a buoyancy flux, *B*_{0} = −*αgI*_{0}/(*ρc*_{p}), and use *L*_{L} = 62, 93, and 124 m and *ξ* = 5, 7.5, and 10 m.

Figure 11 shows profiles of the turbulent buoyancy flux for the radiative heating and SM simulations with *L*_{L} = 124 m. In the radiative heating simulations, the *h*_{rad} increases with increasing *ξ*. In addition to the changes in the shape of the *ξ* becomes larger.

Figure 12a shows the mixed layer depth as a function of *L*_{L} for radiative heating simulations compared with the results for the SM simulations [see Eq. (7) for definition of *h*_{m} in radiative heating simulations]. The mixed layer depth in the radiative heating simulations is up to 70% larger than the surface heating simulations with the same *L*_{L}. Similar variation was observed in *h*_{b} across the simulations (not shown). This variation in *h*_{m} indicates that while Eqs. (3) and (4) are applicable to surface heating, they do not apply for radiative heating. Ultimately, it is the turbulence that drives the dynamics of the OSBL, and the most obvious difference between OSBL turbulence for surface heating and for shortwave radiation is the *B*_{0}, with the largest value *I*(−*h*_{m})/*I*_{0} = 0.1 for the present simulations. Second, the magnitude of the *ξ* increases. Finally, radiative heating introduces a region where *h*_{rad}.

*L*

_{L}, withwhere

*h*

_{m}as a function of

*L*

_{L}. However, we found a better fit to the simulations with the largest

*ζ*was achieved if (2

*β*)

^{−1}= 3. This indicates that while using

*L*

_{L}improves the model fit to radiative heating simulations, there are further, smaller, radiative heating profile effects that could be investigated in the future. The collapse of boundary layer depths for all radiative heating simulations (not shown) when plotted against

*ξ*tending to zero,

*L*

_{L}, which is equivalent to heating confined to a layer close to the surface, for example, because of radiation-absorbing material in the water column.

*h*

_{m}that depends on

*h*

_{rad}, a property of the turbulent flow. The value of

*h*

_{rad}can be approximated as a function of forcing parameters by considering the temperature budget. In the presence of radiative heating, the condition for the temperature gradient within the mixed layer to stay constant with time isFigure 13 shows that the sum of radiative and turbulent buoyancy fluxes is linear over most of the mixed layer. In this simulation approximately 10% of the radiation penetrates deeper than the mixed layer and the radiative heat flux is comparable to the turbulent buoyancy flux over the mixed layer. By noting that the irradiance profile is known, and assuming that the turbulent heat flux tends to zero at the surface and base of the mixed layer, Eq. (6) can be integrated over the mixed layer to giveThe value of

*h*

_{m}in the radiative heating simulations is found by fitting Eq. (7) to the

*ξ*= 10 m and

*L*

_{L}= 124 m. The turbulent buoyancy flux deviates from Eq. (7) near the base of the mixed layer and penetrates below

*h*

_{m}, although the majority of the total buoyancy flux through the mixed layer base is a result of the radiation penetrating beyond the mixed layer.

*L*

_{L}by

*B*

_{0},

*h*

_{i},

*ξ*. These parameters depend only on the surface forcing and the bulk OSBL properties, allowing

*h*

_{m}to be diagnosed.

In the real world the divergence of shortwave radiation contains several distinct decay depths (Paulson and Simpson 1977). The results of this idealized study indicate that the equilibrium mixed layer depth in the presence of a more realistic heating profile will be a weighted function of the individual decay depths and the surface forcing conditions.

### The diurnal cycle

The results in Fig. 12, and the parameterization for mixed layer depth given by Eqs. (4) and (5), were obtained for steady-state conditions. However, during the daytime the solar radiation varies with time, and it is of interest to see how well our diagnostic parameterization represents the daytime variation in mixed layer depth.

To investigate this, an LES was carried out using time-varying radiation. A neutral boundary layer was allowed to develop and then solar radiation, varying sinusoidally with a period of 24 h, was applied for 12 h (the heating phase of the sinusoid). The maximum irradiance was 64 W m^{−2}, the radiative decay depth was *ξ* = 10 m, and *h*_{i} = 50 m. The mixed layer depth was determined from the heat flux profiles averaged over 1800 s.

Figure 14 compares the mixed layer depth from the LES with that calculated using the parameterization given in Eq. (4), with *L*_{L} replaced by *β*)^{−1} = 3, as suggested by Fig. 12. The parameterized *h*_{m} starts to decrease when the irradiance starts to increase and reaches a minimum when the irradiance is greatest. For the first 3 h after the irradiance becomes positive, the mixed layer depth in the LES remains constant at the initial depth. After 3 h, *h*_{m} in the LES decreases with time until it is similar to the parameterized mixed layer depth.

The turbulence time scale for the initial neutral boundary layer, *τ* (see appendix). The diagnostic parameterization gives a reasonable prediction of the minimum mixed layer depth, but further work is needed to understand how the mixed layer depth varies in time.

Starting around 9 h, the parameterized mixed layer depth begins to increase as the radiation decreases, while the LES shows *h*_{m} remaining around 30–35 m. Since the parameterization depends on the instantaneous irradiance, it will give the same depth for increasing or decreasing heat flux. This means that the variation in the parameterized *h*_{m} is symmetric about the time of maximum irradiance. However, the boundary layer warms through the day and this will affect the evolution of mixed layer depth as the irradiance decreases with time.

In this study, the effects of surface fluxes and radiation have been considered separately. However, to simulate a realistic diurnal cycle, it is necessary to represent situations in which surface fluxes and radiation are present together. Typically, the surface flux will represent a loss of heat from the ocean and have the opposite sign to the solar radiative flux. In addition, the present study has only considered simple exponential radiation profiles, but the profile of absorption of radiation in the ocean is more complex. Further work is needed to understand the behavior of the mixed layer depth in more realistic situations.

## 6. Conclusions

This study has investigated the effects of surface heating on the ocean surface boundary layer (OSBL) driven by Langmuir turbulence. Large-eddy simulations were used to study the structure and equilibrium depth of the OSBL and mixed layer following the introduction of a constant surface heat flux to a neutral OSBL. The simulations cover a range of nondimensional parameter space that includes moderate to strong winds (10-m wind speed ≥ 10 m s^{−1}) with wind waves and moderate surface heating (~500 W m^{−2}) or weak winds and waves under weak surface heating.

In the presence of surface heating, the LES produces an OSBL consisting of a weakly stratified (mixed) layer near the surface, above a stratified thermocline, consistent with observations. The mixed layer is maintained by Langmuir turbulence, which produces a turbulent heat flux through the layer for two reasons. First, the Stokes drift transports heat down the Stokes drift gradient (away from the surface), regardless of the stratification. Second, the large vertical velocity variances

The dissipation *ε* and *h*_{m}, the same scaling as Langmuir turbulence without surface heating (Grant and Belcher 2009). This means that surface heating only affects equilibrium *ε* and

The equilibrium depths of the mixed layer *h*_{m} and boundary layer *h*_{b} were found to be a function of the Langmuir stability length *L*_{L} and the initial mixed layer depth *h*_{i} before the heat flux is introduced. The models for *h*_{m} and *h*_{b} developed here [Eq. (4)] are based on a depth-averaged energy balance for the largest scales of turbulent motion. Part of the average Stokes production, before heating is introduced, balances the sum of the increased average Stokes production and the energy used to redistribute heat after a surface heat flux is imposed. Equation (4) is successful for a range of rotation rates and Stokes penetration lengths, implying that *h*_{i} and *L*_{L} are the most significant length scales for the prediction of *h*_{m} and *h*_{b}. The values of *h*_{m}, from the simulations and Eq. (4), are consistent with previous LES studies (Min and Noh 2004).

Simulations were also carried out to investigate the effects of radiative heating on *h*_{m} and *h*_{b}. As radiation penetrates more deeply into the water column, the equilibrium mixed layer depth increases for the same surface irradiance. The effect of exponential heating profiles, with varying decay lengths, on *h*_{m} can be accounted for through a radiative Langmuir stability length

Most of the simulations used in this paper focus on the collapse of an initial turbulent mixed layer following the introduction of a constant surface heat flux and the equilibrium structure of this layer. These simulations are a simplified version of the morning transition of the diurnal cycle. Although the constant forcing is unrealistic in a diurnal cycle, it allows for reasonable averaging times and a simple diagnostic parameterization for the mixed layer depth. The diagnostic parameterization for mixed layer depth was compared to an LES using diurnally varying radiative heating. The minimum *h*_{m} in the LES was similar to that of the diagnostic parameterization. However, there was a lag between the radiative heat flux beginning and the LES mixed layer shoaling, which was not captured by the model.

The above results suggest that turbulence closure schemes should include Langmuir turbulence if they are to capture the structure and the depth of the OSBL over the diurnal cycle. Notably, Langmuir turbulence is necessary to explain the weakly stratified layer near the surface and the depths of the mixed and boundary layers. Further investigation is required to see whether the arguments that build Eq. (4) also apply to the surface-heated OSBL when wind and/or waves decrease, causing *L*_{L} and, presumably, *h*_{m} to decrease. In the current paper, all simulations have *δ* < *h*_{m}. Further studies could investigate the effects of strong heating with very weak waves, where *δ*/*h*_{m} > 1, as this nondimensional parameter could affect the scaling of Langmuir turbulence for very shallow mixed layers (Harcourt and D’Asaro 2008).

## Acknowledgments

This work was supported by the NERC under the OSMOSIS project, Grant NE/I020083/1. In addition, J.A.P. was supported by NOC Grant NE/I01993X/1, and B.C.P. was partially supported by NSF OCE 1350795. Comments from two anonymous reviewers helped improve this paper.

## Appendix

### Turbulence Decay in the Residual Layer

When surface heating is introduced in the simulations, the residual layer, which is within the initial neutral boundary layer, becomes detached from the surface (Fig. 2). The time scale of this detachment should be representative of the formation time scale of the stable mixed layer. Previous studies of the convective atmospheric boundary layer have investigated the decay of turbulence kinetic energy components after forcing is removed (Nieuwstadt and Brost 1986; Pino et al. 2006).

Figure A1 shows the evolution of *t* following the onset of a surface heat flux. The values are taken at a depth of 45 m, within the initial mixed layer but deeper than the equilibrium stable boundary layer. The symbols denote the Langmuir number for the simulation. Three simulations were used with La_{t} = 0.25, 0.3, and 0.4. These simulations are identical to the SM simulation with *L*_{L} = 124 m, except that the surface heat flux is introduced instantaneously at 50 000 s and the magnitude of *u*_{s0} varies between simulations to change La_{t}. The range of La_{t} can alternatively be expressed as ^{−1}. In all simulations

The response of

## REFERENCES

Babanin, A., 2006: On a wave-induced turbulence and a wave-mixed upper ocean layer.

,*Geophys. Res. Lett.***33**, L20605, doi:10.1029/2006GL027308.Belcher, S. E., and et al. , 2012: A global perspective on Langmuir turbulence in the ocean surface boundary layer.

,*Geophys. Res. Lett.***39**, L18605, doi:10.1029/2012GL052932.Brainerd, K., , and M. Gregg, 1993: Diurnal restratification and turbulence in the oceanic surface mixed-layer. 1. Observations.

,*J. Geophys. Res.***98**, 22 645–22 656, doi:10.1029/93JC02297.Brainerd, K., , and M. Gregg, 1995: Surface mixed and mixing layer depths.

,*Deep-Sea Res. I***42**, 1521–1543, doi:10.1016/0967-0637(95)00068-H.Craik, A. D. D., , and S. Leibovich, 1976: Rational model for Langmuir circulations.

,*J. Fluid Mech.***73**, 401–426, doi:10.1017/S0022112076001420.D’Asaro, E. A., 2014: Turbulence in the upper-ocean mixed layer.

,*Annu. Rev. Mar. Sci.***6**, 101–115, doi:10.1146/annurev-marine-010213-135138.Deardorff, J. W., 1972: Theoretical expression for the countergradient vertical heat flux.

,*J. Geophys. Res.***77**, 5900–5904, doi:10.1029/JC077i030p05900.de Boyer Montégut, C., , G. Madec, , A. Fischer, , A. Lazar, , and D. Iudicone, 2004: Mixed layer depth over the global ocean: An examination of profile data and a profile-based climatology.

,*J. Geophys. Res.***109**, C12003, doi:10.1029/2004JC002378.Denman, K. L., 1973: A time-dependent model of the upper ocean.

,*J. Phys. Oceanogr.***3**, 173–184, doi:10.1175/1520-0485(1973)003<0173:ATDMOT>2.0.CO;2.Garwood, R. W., Jr., 1977: An oceanic mixed layer model capable of simulating cyclic states.

,*J. Phys. Oceanogr.***7**, 455–468, doi:10.1175/1520-0485(1977)007<0455:AOMLMC>2.0.CO;2.Grant, A. L. M., , and S. E. Belcher, 2009: Characteristics of Langmuir turbulence in the ocean mixed layer.

,*J. Phys. Oceanogr.***39**, 1871–1887, doi:10.1175/2009JPO4119.1.Grant, A. L. M., , and S. E. Belcher, 2011: Wind-driven mixing below the oceanic mixed layer.

,*J. Phys. Oceanogr.***41**, 1556–1575, doi:10.1175/JPO-D-10-05020.1.Harcourt, R. R., , and E. A. D’Asaro, 2008: Large-eddy simulation of Langmuir turbulence in pure wind seas.

,*J. Phys. Oceanogr.***38**, 1542–1562, doi:10.1175/2007JPO3842.1.Holtslag, A. A. M., , and C.-H. Moeng, 1991: Eddy diffusivity and countergradient transport in the convective atmospheric boundary layer.

,*J. Atmos. Sci.***48**, 1690–1698, doi:10.1175/1520-0469(1991)048<1690:EDACTI>2.0.CO;2.Kukulka, T., , A. J. Plueddemann, , and P. P. Sullivan, 2013: Inhibited upper ocean restratification in nonequilibrium swell conditions.

,*Geophys. Res. Lett.***40**, 3672–3676, doi:10.1002/grl.50708.Large, W. G., , J. C. McWilliams, , and S. C. Doney, 1994: Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization.

,*Rev. Geophys.***32**, 363–403, doi:10.1029/94RG01872.Li, M., , C. Garrett, , and E. Skyllingstad, 2005: A regime diagram for classifying turbulent large eddies in the upper ocean.

,*Deep-Sea Res. I***52**, 259–278, doi:10.1016/j.dsr.2004.09.004.McWilliams, J. C., , P. P. Sullivan, , and C. H. Moeng, 1997: Langmuir turbulence in the ocean.

,*J. Fluid Mech.***334**, 1–30, doi:10.1017/S0022112096004375.Min, H., , and Y. Noh, 2004: Influence of the surface heating on Langmuir circulation.

,*J. Phys. Oceanogr.***34**, 2630–2641, doi:10.1175/JPOJPO-2654.1.Monin, A. S., , and A. M. Obukhov, 1954: Basic laws of turbulent mixing in the surface layer of the atmosphere.

,*Tr. Geofiz. Inst., Akad. Nauk SSSR***24**, 163–187.Nieuwstadt, F., , and R. Brost, 1986: The decay of convective turbulence.

,*J. Atmos. Sci.***43**, 532–546, doi:10.1175/1520-0469(1986)043<0532:TDOCT>2.0.CO;2.Noh, Y., , H. Min, , and S. Raasch, 2004: Large eddy simulation of the ocean mixed layer: The effects of wave breaking and Langmuir circulation.

,*J. Phys. Oceanogr.***34**, 720–735, doi:10.1175/1520-0485(2004)034<0720:LESOTO>2.0.CO;2.Noh, Y., , G. Goh, , S. Raasch, , and M. Gryschka, 2009: Formation of a diurnal thermocline in the ocean mixed layer simulated by LES.

,*J. Phys. Oceanogr.***39**, 1244–1257, doi:10.1175/2008JPO4032.1.Paulson, C. A., , and J. J. Simpson, 1977: Irradiance measurements in the upper ocean.

,*J. Phys. Oceanogr.***7**, 952–956, doi:10.1175/1520-0485(1977)007<0952:IMITUO>2.0.CO;2.Pino, D., , H. J. Jonker, , J. V.-G. De Arellano, , and A. Dosio, 2006: Role of shear and the inversion strength during sunset turbulence over land: Characteristic length scales.

,*Bound.-Layer Meteor.***121**, 537–556, doi:10.1007/s10546-006-9080-6.Plueddemann, A. J., , and R. A. Weller, 1999: Structure and evolution of the oceanic surface boundary layer during the surface waves processes program.

,*J. Mar. Syst.***21**, 85–102, doi:10.1016/S0924-7963(99)00007-X.Polton, J. A., , and S. E. Belcher, 2007: Langmuir turbulence and deeply penetrating jets in an unstratified mixed layer.

,*J. Geophys. Res.***112**, C09020, doi:10.1029/2007JC004205.Price, J., , R. Weller, , and R. Pinkel, 1986: Diurnal cycling - Observations and models of the upper ocean response to diurnal heating, cooling, and wind mixing.

,*J. Geophys. Res.***91**, 8411–8427, doi:10.1029/JC091iC07p08411.Rossby, C.-G., , and R. B. Montgomery, 1935: The layer of frictional influence in wind and ocean currents.

*Papers in Physical Oceanography and Meteorology*, Vol. 3, No. 3, 101 pp., doi:10.1575/1912/1157.Shutts, G. J., , and M. E. B. Gray, 1994: A numerical modelling study of the geostrophic adjustment process following deep convection.

,*Quart. J. Roy. Meteor. Soc.***120**, 1145–1178, doi:10.1002/qj.49712051903.Skyllingstad, E., , and D. Denbo, 1995: An ocean large-eddy simulation of Langmuir circulations and convection in the surface mixed layer.

,*J. Geophys. Res.***100**, 8501–8522, doi:10.1029/94JC03202.Stokes, G. G., 1847: On the theory of oscillatory waves.

,*Trans. Cambridge Philos. Soc.***8**, 441–463.Sullivan, P. P., , J. C. McWilliams, , and W. K. Melville, 2007: Surface gravity wave effects in the oceanic boundary layer: Large-eddy simulation with vortex force and stochastic breakers.

,*J. Fluid Mech.***593**, 405–452, doi:10.1017/S002211200700897X.Sutherland, G., , B. Ward, , and K. Christensen, 2013: Wave-turbulence scaling in the ocean mixed layer.

,*Ocean Sci.***9**, 597–608, doi:10.5194/os-9-597-2013.Sutherland, G., , K. Christensen, , and B. Ward, 2014: Evaluating Langmuir turbulence parameterizations in the ocean surface boundary layer.

,*J. Geophys. Res.***119**, 1899–1910, doi:10.1002/2013JC009537.Teixeira, M. A. C., , and S. E. Belcher, 2002: On the distortion of turbulence by a progressive surface wave.

,*J. Fluid Mech.***458**, 229–267, doi:10.1017/S0022112002007838.Teixeira, M. A. C., , and S. E. Belcher, 2010: On the structure of langmuir turbulence.

,*Ocean Modell.***31**, 105–119, doi:10.1016/j.ocemod.2009.10.007.Zilitinkevich, S. S., 1972: On the determination of the height of the Ekman boundary layer.

,*Bound.-Layer Meteor.***3**, 141–145, doi:10.1007/BF02033914.Zilitinkevich, S. S., , and A. Baklanov, 2002: Calculation of the height of the stable boundary layer in practical applications.

,*Bound.-Layer Meteor.***105**, 389–409, doi:10.1023/A:1020376832738.Zilitinkevich, S. S., , A. Baklanov, , J. Rost, , A.-S. Smedman, , V. Lykosov, , and P. Calanca, 2002: Diagnostic and prognostic equations for the depth of the stably stratified Ekman boundary layer.

,*Quart. J. Roy. Meteor. Soc.***128**, 25–46, doi:10.1256/00359000260498770.