## 1. Introduction

Diurnal warm layers (DWLs) form near the surface of the ocean on days with strong solar radiation, weak to moderate winds, and weak surface-wave activity. Reviewing existing literature, Kawai and Wada (2007) noted that DWLs are a widespread feature, found at all latitudes and characterized by typical sea surface temperature (SST) anomalies of

Recent field investigations with specialized instrumentation (Matthews et al. 2014; Sutherland et al. 2016; Moulin et al. 2018; Hughes et al. 2020) and numerical modeling studies (Sarkar and Pham 2019; Large and Caron 2015) have provided a consistent picture of the physical processes determining the evolution of DWLs in the ocean: strong surface buoyancy forcing tends to suppress turbulence below the DWL and induce a near-surface trapping of momentum, reflected in the evolution of a near-surface diurnal jet with speeds

This detailed understanding of the DWL dynamics was, however, almost exclusively gained based on investigations and long-term studies at tropical latitudes (e.g., Matthews et al. 2014; Bellenger and Duvel 2009), despite the observation that during the summer months diurnal SST anomalies at high latitudes may be as large as those found in tropical regions (Kawai and Wada 2007). The few available studies of high-latitude DWLs (e.g., Eastwood et al. 2011; Jia et al. 2023) found a widespread occurrence also in the Arctic Ocean. Jia et al. (2023) reported a repeated occurrence of DWLs with significant warming amplitudes > 2°C, including extreme events with amplitudes > 5°C, during the two summer months of their measurement campaign at latitudes of up to 80°N. A recent evaluation of turbulence models, including their performance under strong diurnal warming (Johnson et al. 2023), was for a low-latitude DWL. Due to the lack of detailed observations and numerical studies of high-latitude DWLs, our understanding of the energetics and parameterization of these features is limited at the moment.

A few recent studies focusing on the impact of surface-wave effects on DWLs (Kukulka et al. 2013; Pham et al. 2023; Wang et al. 2023) underlined the importance of Langmuir turbulence (LT) for the evolution of diurnal near-surface stratification, typically identifying a reduction of the diurnal SST amplitude and an increase of the DWL thickness due to stronger mixing. The ability of existing parameterizations (Price et al. 1986, hereafter PWP86; Fairall et al. 1996) and ocean turbulence models to reproduce these effects has not been systematically evaluated so far.

The goal of this paper is to investigate the relevance, implications, and parameterizations of different processes (in particular: LT and rotation effects in high-latitude DWLs) across the entire physically relevant parameter space. Section 2 introduces the models used in this study. Most of the analysis is based on a second-moment turbulence model that includes the effects of LT. To validate this model and evaluate its performance concerning the effects of LT, we use section 3 to compare it to our LES results for a typical DWL scenario. After that, in section 4, we use our validated second-moment model to investigate DWL energy budgets in a typical tropical versus a high-latitude DWL, thereby focusing especially on the effects of rotation and day length that have so far received only little attention. Last, in section 5, we attempt to provide a unified description of DWLs in the ocean by first identifying the key nondimensional parameters that govern their structure and evolution, and then evaluating the influence of these parameters across a large parameter space. Here, we also test the applicability of the frequently used bulk parameter model by PWP86 for high latitudes and DWLs influenced by LT.

## 2. Model formulation

### a. Momentum and buoyancy equations

*u*and

*υ*are the Reynolds-averaged velocities in the

*x*and

*y*directions,

*u*and

_{s}*υ*are the corresponding Stokes drift velocities,

_{s}*f*is the Coriolis parameter, and

*ν*and

*ν*are the molecular diffusivities of momentum and buoyancy (or heat), respectively. The vertical turbulent momentum fluxes (normalized here with a constant reference density

^{b}*ρ*

_{0}= 1027 kg m

^{−3}) are denoted by

*τ*and

_{x}*τ*. The evolution of the Reynolds-averaged buoyancy

_{y}*b*is determined by the vertical turbulent buoyancy flux

*G*and the radiative buoyancy flux

*I*due to penetrating shortwave radiation. We use the conventions that

_{b}*z*points vertically upward with

*z*= 0 at the surface, that all turbulent fluxes are positive upward, and that the radiative buoyancy flux

*I*is positive downward.

_{b}As boundary conditions for the momentum equations in (1) and (2), we describe the components of the (normalized) wind stress, *B*_{0} = *αgQ*_{ns}/(*ρ*_{0}*c _{p}*) at

*z*= 0, where

*Q*

_{ns}(positive downward) is the nonsolar heat flux, accounting for the longwave, latent, and sensible heat fluxes. Here,

*g*is the acceleration of gravity,

*c*the specific heat capacity, and

_{p}*α*the thermal expansion coefficient. Note that

*Q*

_{ns}and

*B*

_{0}will generally be negative (surface heat loss) in our study. Zero-flux conditions for the turbulent fluxes of momentum and buoyancy are applied at

*z*→ −∞ (practically, the lower boundary conditions are imposed at some finite value of

*z*that is sufficiently far below the surface to not affect the results).

### b. Surface forcing

*B*

_{0}< 0), and a periodic diurnal variability induced by the radiative heat flux according to

*T*is the period of the prescribed forcing (24 h),

_{p}*T*the duration of the daylight period with

_{d}*I*

_{0}> 0, and

*I*

_{max}the maximum radiative heat flux reached at

*T*/2 (midday). We performed numerical tests in which we compared this simplified solar radiation model with a more realistic radiation expression based on Stull (1988) and found only small differences in the DWL evolution that did not affect any of our conclusions. For our idealized study focusing on the basic mechanisms of DWL formation, the downward shortwave radiation

_{p}*I*will be computed from a simple absorption model of the form

*η*is the shortwave absorption scale. Note that in section 3 (model validation), and in some parts of the parameter space study in section 5, we will make the additional simplifying assumption that

*η*= 0, i.e., that all radiation is absorbed at the surface. The radiative buoyancy flux

*I*in (3) follows from

_{b}*B*

_{0}, the radiative buoyancy flux at the surface

*B*), are shown in Fig. 1.

*T*and

_{p}*T*, the maximum total surface buoyancy flux at midday

_{d}*B*

_{max}, and the surface buoyancy loss

*B*

_{0}. Rather than

*T*, the more sensible parameter to describe the formation of DWLs is the heating period

_{d}*T*during which the total surface buoyancy flux

_{h}*B*is positive (see Fig. 1). From (4), it is clear that these two time scales are related according to

*x*direction,

*ρ*= 1.23 kg m

_{a}^{−3}is the air density,

*U*

_{10}is the 10-m wind speed, and

*C*1.7 × 10

_{d}=^{−3}is a constant drag coefficient. Introducing the friction velocity,

*V*are estimated from expressions of the form

_{s}*c*= 0.016 and

_{s}*C*= 2.67 × 10

_{s}^{−5}s

^{4}m

^{−2}are model constants. From these expressions, and the quadratic drag law mentioned above, the turbulent Langmuir number

*u*predicted by their model is a function of the surface Stokes velocity

_{s}### c. Second-moment turbulence modeling approach

*ν*and

_{t}*N*

^{2}=

*∂b*/

*∂z*.

*l*and a turbulent velocity scale

*k*

^{1/2}, where

*D*denotes the vertical transport of TKE and

_{k}*ε*the turbulence dissipation rate. The terms

*P*and

*P*are the Eulerian and Stokes shear production terms defined as (Harcourt 2013; H15):

_{s}*l*, required for the computation of the turbulent diffusivities, is inferred from the solution of a Mellor–Yamada-type transport equation for the product

*kl*in (A2) or, alternatively for comparison, from a transport equation for the inverse turbulence time scale

*ω*∝

*k*

^{1/2}

*l*

^{−1}in (A4). All relevant details of the turbulence closure models used in our study are summarized in appendix A.

### d. LES modeling approach

The LES approach is used to validate the second-moment models in section 3. The approach is based on the Craik–Leibovich equations to produce the Eulerian velocity, pressure, and buoyancy fields in a temporally evolving three-dimensional computational domain. Readers are referred to appendix B for the numerical implementation of the LES and the model setup.

From the LES, horizontally averaged profiles of velocities 〈*U _{i}*〉, buoyancy 〈

*B*〉, and turbulent fluxes

*u*,

*υ*, and

*b*) in (1)–(3), as elaborated in section 3. Here, we use angle brackets to denote the horizontal average of the LES fields and primes to denote the fluctuations.

## 3. Comparison of LES and second-moment models

In this section, we compare the second-moment turbulence closure models to our LES results for a typical tropical DWL scenario both with and without the effects of LT, focusing especially on the performance of the second-moment model for this newly included process. Both LES and second-moment models are driven with identical atmospheric and buoyancy forcing and use the same parametric surface-wave model by Li et al. (2017) to compute the Stokes velocities. Note that the effects of surface-wave breaking are not taken into account in our simulations.

The three different second-moment models that we want to test are described in detail in section 2c and appendix A. They include (i) the full model of H15, which represents LT effects in both the stability functions and the transport equations for *k* and *kl* in Eqs. (12) and (A2) through the additional Stokes production term *P _{s}*; (ii) the model of Kantha and Clayson (2004, hereafter KC04), which only considers the additional Stokes production terms in the transport equations but ignores the impact of LT on the stability functions; and (iii) the model of Kantha and Clayson (1994), which ignores LT effects entirely. Both the models of H15 and KC04 converge to the model of Kantha and Clayson (1994) for the special case of zero Stokes drift (

*u*= 0), which allows for a clear separation of LT effects from other modeling components. To compute the turbulent length scale

_{s}*l*, we used an extended version of the Mellor–Yamada equation for

*kl*in (A2) for all of the following simulations but we will also include a short comparison with a modified version of the

*k–ω*model in section 2 of the supplemental material.

All second-order moment model simulations were conducted with a modified version of the General Ocean Turbulence Model (GOTM), described in Umlauf et al. (2005). The time step for these simulations was set to 6 s, and the domain depth and grid size match those of the LES grid with a resolution of 0.05 m at the surface, gradually coarsening toward the bottom (see appendix B). These parameters were found to ensure numerical convergence and exclude any impact of the lower edge of the domain on the DWL properties.

For all the simulations in this section, we use a peak solar buoyancy flux of *I _{b}* = 2.3 × 10

^{−7}m

^{2}s

^{−3}at noon at the surface (see Fig. 1), which, for comparison, would correspond to a peak solar heat flux of

*I*

_{max}= 400 W m

^{−2}for a thermal expansion coefficient

*α*= 2.4 × 10

^{−4}K

^{−1}. To keep the setup for this model comparison as simple as possible, we also assume that the nonsolar surface buoyancy flux vanishes (

*B*

_{0}= 0) and that all short wave radiation is absorbed at the surface (

*η*= 0). The heating period is

*T*=

_{h}*T*= 12 h at a tropical latitude of 10°N (corresponding to

_{d}*f*= 2.53 × 10

^{−5}s

^{−1}and a local inertial period of

*T*= 69.1 h). A constant friction velocity of

_{f}*U*

_{10}= 3.1 m s

^{−1}. This results in a Monin–Obukhov length

*κ*= 0.4), which is more than an order of magnitude larger than the numerical grid spacing near the surface. For all GOTM runs in this section, the surface roughness length

*z*

_{0}that appears in the boundary condition (A9) for the turbulence length scale

*l*was set to

*z*

_{0}= 0.01 m. This model parameter is not well constrained. Our parameter studies in section 5 show, however, that the impact of

*z*

_{0}is negligible.

To save computational resources for the LES, all simulations in this section start at 0500 local time (1 h before the start of the radiative buoyancy forcing) rather than at midnight. Note, however, that in all the following sections, the beginning of the simulations is at midnight.

The horizontally averaged LES results are shown in Fig. 2, comparing simulations without (*u _{s}* = 0) and with LT. In both cases, the buoyancy structure (Figs. 2a,b) shows the evolution of DWLs with similar characteristics. LT effects are clearly noticeable especially in the reduced near-surface buoyancy in the simulation with wave forcing, which is consistent with the reduced near-surface stratification due to LT-enhanced mixing (Figs. 2c,d). The Eulerian shear (Figs. 2e,f) in the simulation with LT deviates from its counterpart with

*u*= 0 significantly in the upper 2 m, where the Stokes shear production

_{s}*P*becomes the dominant source of turbulence (

_{s}*u*decays to approximately 10% of its surface value within the uppermost 0.65 m). This effect is also clearly evident in the Richardson number, Ri =

_{s}*N*

^{2}

*S*

^{−2}, which does not account for Stokes shear (Figs. 2g,h).

Figures 3 and 4 compare the DWL evolution in the LES (with and without LT) and the second-moment models for four selected points in time (marked in Fig. 2). This comparison shows that the overall characteristics of the LES are well reproduced by all models: both the DWL thicknesses and the vertical structures of buoyancy, velocity, and the turbulent momentum flux closely correspond to those predicted by the LES. Significant differences are largely confined to the upper 1–2 m, where the LES suggest a strong reduction of stratification and shear due to the effects of LT. For the period between 1200 and 1500 local time, when DWL anomalies are most distinct, the inlay plots in Figs. 3d and 3f show that the inclusion of LT effects leads to a significant reduction of the near-surface velocity. Only the model of H15 is in close agreement with the LES, while the model of KC04 clearly underestimates the additional mixing of momentum due to LT effects, underlining the importance of the Stokes shear term in (10). For the near-surface buoyancy profiles (see inlay plots in Figs. 3c,e), differences between the second-moment models are less pronounced, and all tend to underestimate the reduction of near-surface stratification due to LT. Differences between the simulations with and without LT become especially clear in the gradient Richardson number shown in Figs. 4b, 4d, 4f, and 4h. The pronounced near-surface peak in Ri is captured only by the most advanced model of H15 as shown in Fig. 4.

It is worth noting that all second-moment models predict virtually identical profiles underneath the thin near-surface region directly affected by Stokes production. For the LES, the negligible effect of LT below the thin-surface layer is the case only for the late-stage DWLs (Figs. 3e–g), while the DWL evolution in the morning and around noon (Figs. 3a–d) shows weak but significant LT effects also below the Stokes layer. These LT effects on the mean fields are accompanied by inflectional shear, similar to observations of inflectional shear by Hughes et al. (2021) when convective cooling commenced at sundown, as well as by enhanced TKE transport from the Stokes layer toward the layer underneath (Li and Fox-Kemper 2020).

Overall, we conclude that the performance of the model of H15 is most satisfying, and we will therefore use this model for all of the following numerical investigations. As shown in section 2 of the supplemental material, simulations conducted with a modified version of the *k–ω* model (see appendix A), using the same stability functions of H15, yield very similar results, providing support for the robustness of our results.

## 4. DWL energy budgets at low and high latitudes

In this section, we derive energy budgets for DWLs and use these to investigate the effects of rotation and heating time on high-latitude DWLs.

### a. Theory

*h*based on expressions of the form

*b*

_{ref}at some reference level

*z*

_{ref}below the DWL, and

*φ*a shape factor that depends on the vertical structure of the buoyancy profile. For example, it can be shown that

*φ*= 1/2 and

*φ*= 1/3 correspond to the cases of well-mixed and linearly stratified DWLs, respectively. For comparison, it is worth noting that Fairall et al. (1996) assumed DWLs with linear stratification (

*φ*= 1/3), whereas applying (16) and (17) to the empirical DWL profiles in expression (17) of Gentemann et al. (2009) yields

*φ*≈ 0.2–0.4 with a transition from exponential to more well-mixed profiles depending on wind speed. In our model,

*φ*changes in time during the evolution of the DWL.

*z*

_{ref}and the surface, the time derivative of the relation in (16) can be expressed as

*b*

_{ref}has a negligible effect. Our idealized simulations show that the variability of

*b*

_{ref}indeed becomes negligible shortly after the DWL has formed, isolating the reference level from surface buoyancy forcing.

*E*

_{pot}induced by the presence of the DWL. Reformulating (3) in terms of

*z*, and integrating by parts, yields an equation for the evolution of the potential energy anomaly:

*b*

_{ref}is found to be negligible in our simulations and has therefore been neglected in (19).

*w*=

_{e}*dh*/

*dt*.

*u*and

*υ*, respectively, adding the results, and integrating from

*z*

_{ref}to the surface. Ignoring again the molecular flux terms for simplicity, this yields an energy budget of the form:

**k**is the upward unit vector and

**u**

^{0}the velocity at the surface. The terms on the right hand side of (21) can be interpreted as (i) the work performed by the wind stress on the DWL, (ii) the exchange of kinetic energy with the surface wave field due to the effect of rotation (see, e.g., Suzuki and Fox-Kemper 2016), and (iii) the loss of kinetic energy to TKE by turbulence shear production. Similar to the negligible effect of temporal variations of

*b*

_{ref}in (18) and (19), we also find that the temporal variability of the reference kinetic energy,

### b. Results

To investigate the DWL energy budgets derived above, we compare a typical tropical case at 10°N with a high-latitude DWL at 70°N. The results were obtained using GOTM with the turbulence closure model of H15 that was shown to compare favorably to the LES in the previous section. We used a time step of 6 s and a grid spacing of 0.015 m at the surface, gradually coarsening toward the lower end of the numerical domain at 50-m depth. The atmospheric forcing and model parameters are summarized in Table 1. For both cases, we assumed that the surface buoyancy loss *B*_{0} due to cooling exactly compensates for the radiative buoyancy supply over the course of a day. The buoyancy forcing parameters in Table 1 were chosen to yield realistic summertime values for the peak radiative heat flux *I*_{max} for the corresponding latitudes, water temperatures, and thermal expansion coefficients. For the tropical case, the values in Table 1 correspond to *I*_{max} = 1000 W m^{−2}, using *α* = 3.4 × 10^{−4} K^{−1} for tropical 30°C water temperatures. Analogously, the parameters for the high-latitude case yield *I*_{max} = 680 W m^{−2} with *α* = 1.6 × 10^{−4} K^{−1} for 10°C water temperatures. Since the buoyancy flux is linearly proportional to *α*, it can correspond to different heat fluxes, depending on water temperature. To make our model more generally applicable, we have therefore formulated it in terms of buoyancy rather than temperature.

Atmospheric forcing and model parameters used for the analysis of the DWL energetics.

For the tropical case, as before, the period with nonzero solar radiation was chosen as *T _{d}* = 12 h (between 0600 and 1800 local time), whereas we assume

*T*= 18 h (between 0300 and 2100 local time) for the high-latitude case. The resulting effective heating periods

_{d}*T*, computed from (7), can be found in Table 1, together with all other model parameters that were kept constant. To determine the shortwave absorption length

_{h}*η*in (5), we varied

*η*and compared our GOTM results for the tropical scenario against a plot of the parametric temperature profile from Eq. (17) of Gentemann et al. (2009), who used a more complex nine-band absorption model for clear tropical waters. We find that our simple one-band model results in a very similar vertical DWL structure for

*η*= 0.87 m, which is the value we used for all simulations in this section (Table 1).

The evolution of the near-surface buoyancy for the two cases is shown in Figs. 5a and 6a. The DWL thickness *h,* one of the most important bulk parameters, is defined here by a simple density threshold, identifying the lower edge of the DWL with the vertical position where the buoyancy has decayed to 5% of its maximum value. Figures 5a and 6a show that this definition provides a plausible representation of the vertical extent of the DWL for both cases.

The reference level *z*_{ref} is chosen to coincide with the location of the minimum buoyancy in the water column, and *b*_{ref} found at this depth. This definition guarantees that the entire near-surface region affected by radiative heating is included in our analysis.

Figure 5b shows the evolution of the kinetic energy budget in (21) for the tropical case. During the initial DWL formation phase until approximately early afternoon, the work performed by the wind,

For the high-latitude case shown in Fig. 6, the work performed by the surface stress, *P _{s}* exceeds Eulerian shear production

*P*in the TKE budget. The net effect of the reduced turbulence production due to rotation is a complete collapse of entrainment after the initial DWL shoaling in the morning (blue curve in Fig. 6a).

To investigate the extent to which the strong surface buoyancy forcing and the stratification inside the DWL affect the energetics of turbulence, we computed the bulk flux coefficient,

The different contributions to the potential energy budget in (20) for the tropical and high-latitude cases are compared in Figs. 7b and 7c, respectively. During midday, in both cases, the largest fraction of the work performed by turbulence against gravity is used to mix down near-surface buoyant fluid generated by solar heating. According to (20), the ratio *η* = 0), this ratio is zero. We find

Figures 7b and 7c also show that the first hours after the initial generation of the DWLs are characterized by “detrainment” (*w _{e}* < 0) due to the restratifying effect of the increasing solar radiation. For the tropical case,

*w*changes sign in the late morning, and around 1400 local time the work required for the entrainment of dense fluid at the DWL base finally becomes the dominating term in the potential energy budget. This is in strong contrast to the high-latitude case, in which entrainment never becomes an energetically relevant factor.

_{e}Beyond the work required for DWL deepening, turbulent mixing may also act to change the vertical DWL buoyancy structure. The energetic implications of this third type of energy conversion in (20) can be quantified by considering changes in the shape parameter *φ*, which is easily computed from (16) and (17) after determining the DWL thickness *h* from the 5% buoyancy threshold discussed above (see blue lines in Figs. 5a and 6a). Figure 7d shows that during the morning and early afternoon, this parameter starts close to *φ* = 1/3, but increases to larger values over the course of the day, reflecting the tendency toward a more well-mixed DWL especially in the evening due to the decreasing solar buoyancy forcing. For comparison, the parametric temperature profiles in Gentemann et al. (2009) for this wind speed yield a constant *φ* ≈ 0.25, whereas the model of Fairall et al. (1996) corresponds to a constant *φ* = 1/3. These differences in *φ* between our model and the models of Fairall et al. (1996) and Gentemann et al. (2009) can be largely attributed to the thin near-surface convective layer generated by penetrating shortwave radiation in our simulations, which is not represented in the models of Fairall et al. (1996) and Gentemann et al. (2009). Figures 7b and 7c show that the work required for this partial homogenization of late-stage DWLs becomes significant only for the high-latitude case, where it dominates the potential energy balance during the afternoon.

## 5. Key parameters and DWL parameterization

### a. Identification of dimensional and nondimensional parameters

The evolution and physical properties of the DWLs in our idealized simulations are affected by a number of independent dimensional parameters, imposed by the atmospheric forcing and the properties of the surface wave field. The former includes the constant wind stress, quantified here with the help of the friction velocity *U*_{10}), and the parameters describing the idealized buoyancy forcing shown in Fig. 1: the maximum total buoyancy flux at midday *B*_{max}, the (constant) buoyancy loss at the surface *B*_{0}, the heating period *T _{h}*, and the period of the periodic forcing

*T*. For penetrating shortwave radiation, the vertical absorption scale

_{p}*η*in (5) has to be considered as an additional parameter.

The surface wave field affects the problem through the surface Stokes velocity *k _{p}*. Note that the same two dimensional parameters would also appear for the simpler case of monochromatic waves (Kukulka et al. 2013). However, in the equilibrium wave model of Li et al. (2017) used in our study, both

*k*depend on the wind speed through (8) and (9), and therefore do not constitute independent dimensional parameters.

_{p}Finally, as all model parameters of the turbulence model are nondimensional, no additional dimensional parameters are introduced, with a single exception: the upper boundary condition for the turbulent length scale *l* in (A9) involves the surface roughness length *z*_{0} that we consider in the following as an additional independent parameter.

*L*, the heating period

*T*shown in Fig. 1 as the relevant time scale, we can nondimensionalize the key variables of our problem (Table 2), and derive nondimensional versions of the transport equations of momentum and buoyancy in (1)–(3). From these nondimensional transport equations, it is straightforward to identify two key nondimensional parameters of the problem. The first is the nondimensional Coriolis parameter,

_{h}*T*. The second parameter,

_{f}*T*

_{h}B_{max}). For simplicity, we ignore the molecular transport terms in (1)–(3) for our dimensional analysis, as their effect is only marginal in our simulations.

Definition of nondimensional variables denoted by the hat (

*B*

_{0}/

*B*

_{max}and the time scale ratio

*T*/

_{h}*T*appear as independent nondimensional parameters in our model for the buoyancy forcing in Fig. 1. To reduce the number of free parameters and allow for quasi-periodic solutions, we will assume in most of the parameter studies that the daily average of the total buoyancy flux is zero, i.e., that the incoming solar radiation is exactly compensated by the net surface buoyancy loss

_{p}*B*

_{0}

*T*. With this constraint,

_{p}*B*

_{0}and

*B*

_{max}are no longer independent:

*T*/

_{h}*T*and

_{p}*T*/

_{d}*T*. The final nondimensional parameter associated with the buoyancy forcing is the nondimensional absorption scale,

_{p}Finally, as pointed out in the context of (8) above, the wave model of Li et al. (2017) predicts a constant value of the Langmuir number

The surface roughness length, *z*_{0}, which represents the length scale of turbulence at the surface, transforms into the nondimensional roughness parameter

All nondimensional parameters present in this study are summarized in Table 3. We carefully checked that different numerical solutions indeed collapse if all nondimensional parameters are kept constant and all variables are nondimensionalized as in Table 2.

The nondimensional parameters. Note that the variability of some parameters appearing in parentheses is restricted on our model.

### b. Nondimensional *PWP86* model

*h*, buoyancy anomaly

*a*

_{1},

*a*

_{2}, and

*a*

_{3}denote nondimensional model constants, and

*F*is a nondimensional model function accounting for the effect of rotation:

*η*= 0, i.e., if all shortwave radiation is absorbed at the surface. In section 5d, we will suggest a possible generalization for the case of penetrating shortwave radiation.

### c. Parameter space studies

Before we tested the scaling relations by PWP86 over a wide parameter range, we performed parameter space studies for the nondimensional parameters *T _{h}*/

*T*,

_{p}*R*over the physically relevant range from

*R*= 10

^{−4}to 10

^{−2}, and individually tested the impact of the above nondimensional parameters. For this parameter space study, we again used the closure model of H15 with the same time step and the same number of grid cells as in section 4. However, the depth of the water column was now automatically adjusted to 10 times the DWL thickness at midday to ensure that the lower edge of the numerical domain had no significant impact on the results.

As shown in Fig. 8, we find that the nondimensional parameters *T _{h}*/

*T*,

_{p}*T*/

_{h}*T*and

_{p}*B*

_{0}/

*B*

_{max}may have a larger impact for longer simulation periods of several days, where they may affect the nighttime DWL reset and thus the quasi-periodic evolution of the surface layer structure. Similarly, the peak wavenumber

To test the scaling relations by PWP86, we performed a parameter space study using H15, consisting of 200 model runs, in which we varied *R* from 10^{−4} to 10^{−2} and *T _{h}*/

*T*from 0 to 0.79).

_{f}We especially focused on the model performance in high-latitude regions (*T _{h}*/

*T*> 0.5), which are not well explored at the moment and for which the model assumptions of PWP86 are uncertain. We again assume surface absorption (

_{f}*B*

_{0}/

*B*

_{max}= −0.466,

*T*/

_{h}*T*= 0.4). The roughness length is set to

_{p}In Fig. 9, we show simulation results for the nondimensional DWL thickness *t* = *T _{p}*/2, i.e., at midday. These quantities are normalized by the PWP86 scaling relations in (27), (28), and (29), respectively, to reveal the variability of the model parameters

*a*

_{1},

*a*

_{2}, and

*a*

_{3}. Figure 9 shows that the performance of the PWP86 scaling is generally excellent, except for a weakly forced regime with

A more detailed analysis showed that turbulent and molecular diffusivities become comparable in this regime, and that Ri at the DWL base becomes much larger than the critical value for shear instability, indicating an absence of turbulent entrainment. It is worth noting that Hughes et al. (2020) studied this regime in more detail, based on high-resolution observations and a 1D model with a simpler turbulence closure without LT but similar radiative and atmospheric forcing parameters. From their simulations, these authors identified a critical wind speed of *U*_{10} = 2 m s^{−1} below which turbulent mixing does not occur. This is equivalent to *R* = 6.6 × 10^{−4} for the buoyancy forcing used in their study, and therefore consistent with the more generally applicable nondimensional threshold suggested by our simulations with a more advanced turbulence model that also included Langmuir effects. Overall, this indicates that molecular effects become significant in this regime, suggesting that the Reynolds and Prandtl numbers are additional relevant nondimensional parameters that have to be considered. As the effects of these parameters are not accounted for in any of the high-Re models used in our study, we do not investigate this regime any further here.

We determined the model constants *a*_{1}, *a*_{2}, and *a*_{3} by calculating the mean of the PWP86-scaled model results shown in Fig. 9, excluding regions with *R* < 7 × 10^{−4}. Table 4 shows that the revised constants suggest a more than 30% increase in the DWL thickness (and a correspondingly smaller buoyancy/temperature anomaly) compared to the original values by PWP86, which is significant for many applications. Most importantly, differences between DWL bulk parameters from our GOTM simulations and those predicted by the (revised) PWP86 model rarely exceed 10% (with largest deviations observed at large *T _{h}*/

*T*) across the entire parameter range. Table 4 also shows that simulations without LT (not discussed in detail here) result in DWLs that are approximately 10% shallower and have a correspondingly larger buoyancy contrast.

_{f}Model constants *a*_{1}, *a*_{2}, and *a*_{3} of the PWP86 model appearing in (27)–(29). The original constants of PWP86 were converted to our notation according to: *a*_{1} = 0.45 × 2^{1/2} = 0.63, *a*_{2} = 1.5 × 2^{−3/2} = 0.53, and *a*_{3} = 1.5 × 2^{−1/2} = 1.06. The factor 1/2 arises from the relation *T _{h}* = 2

*P*, where

_{Q}*P*is the heating period in the notation of PWP86. The ranges given in the table correspond to the

_{Q}*maximum*deviations across the entire parameter range. Standard deviations (not shown) are considerably smaller. Here,

*t*

_{max}is the time of maximum buoyancy anomaly. All simulations were conducted for the case

The PWP86 scaling relations were originally proposed to predict DWL properties at the solar radiation peak (*t* = *T _{p}*/2). More relevant for many applications, including the interpretation of SST snapshots from satellite data, atmosphere–ocean coupling, and ecosystem applications, are, however, often the DWL properties at the peak of the DWL buoyancy or temperature anomaly in the afternoon. The timing of this peak cannot be determined from the PWP86 scaling. We therefore identified the (nondimensional) time

*t*

_{max}/

*T*of the maximum buoyancy anomaly numerically from our simulations.

_{p}Scaling our simulations at *t* = *t*_{max} with the expressions of PWP86 (see section 2 of the supplemental material) suggests that the PWP86 scaling also provides an excellent representation of the DWL bulk properties during the buoyancy peak in the afternoon, provided the model coefficients *a*_{1}, *a*_{2}, and *a*_{3} are appropriately adjusted. The values in Table 4 show that the DWL thickness and the buoyancy anomaly have increased by 29% and 47%, respectively, compared to midday, illustrating a strong modification of the DWL during the early afternoon. The small variability of the model coefficients in Table 4 supports the applicability of the PWP86 scaling also for this case, except for the diurnal jet, which shows a strong dependency on *T _{h}*/

*T*especially for large values of this parameter. We attribute this to the effect of the pronounced inertial oscillations at high latitudes that are not well represented by the scaling of PWP86. The good performance of the scaling of PWP86 for the depth and bulk buoyancy at this point in time is a surprising result, as we found that the model assumption of a constant bulk Richardson number

_{f}Figure 10a shows that the timing of the afternoon buoyancy peak is relatively robust, generally observed between 1500 and 1630 local time with a shift toward later times for larger *T _{h}*/

*T*. We attribute this shift to the suppression of entrainment of colder bottom waters due to stronger rotation effects at higher latitudes and/or a larger total buoyancy flux for larger

_{f}*T*. A similar shift toward later times is observed if the shortwave absorption scale

_{h}### d. Effect of penetrating shortwave radiation

To investigate the first-order impacts of penetrating shortwave radiation in the scaling of PWP86, we carried out a parameter space study similar to that shown in Fig. 9. Now, however, we varied *R* and *R* = 10^{−4}–10^{−2} and *T _{h}*/

*T*= 0.14 and 0.74, corresponding to our standard tropical and high-latitude cases from section 4, while keeping the other nondimensional parameters constant at

_{f}*T*/

_{h}*T*= 0.4,

_{p}*B*

_{0}/

*B*

_{max}= −0.466, and

*η*to the DWL thickness

*h*, which yields

*η*/

*h*(rather than

*η*/

*h*always stays well below 1 for the range of

*η*by multiplying the corresponding PWP86 expression in (27) with a function

*J*that depends on

*A*= 6.9 was obtained from fitting [the original prefactor of PWP86,

_{η}*a*remain unchanged for consistency with (27)–(29). As

_{i}*a*, and thus the model uncertainties, based on (32)–(34). The variability of the parameters in Table 5 suggests that the modified PWP86 scaling captures the effect of penetrating shortwave radiation with good accuracy for the tropical case. For the high-latitude case, however, the agreement is only moderate, suggesting the need for a more detailed analysis of the effect of penetrating radiation in high-latitude DWLs. For the according plots, please see section 2 of the supplemental material.

_{i}Beyond its impact on the bulk DWL properties, our simulations also showed that penetrating shortwave radiation strongly affects the near-surface structure of the DWL buoyancy and velocity profiles. If *η* > 0, many of our simulations showed the evolution of a thin convective layer immediately below the surface. The overall effect of this additional mixing is a reduction of the surface buoyancy, similarly to the observed reduction caused by LT (see Fig. 3), suggesting that the two processes interact. In the following, we therefore investigate the combined effects of penetrating shortwave radiation and LT on the surface buoyancy *b*^{0} and surface velocity *V*^{0}, both nondimensionalized here by the corresponding bulk values

Figures 11a and 11b shows the variability of these variables as a function of the stability parameter *R* for a tropical *T _{h}*/

*T*= 0.14 and different combinations of simulations with and without LT and various values of

_{f}*T*/

_{h}*T*= 0.4,

_{p}*B*

_{0}/

*B*

_{max}= 0.466, and

*η*= 0 (Fig. 11a), while, surprisingly, LT

*increases*the surface buoyancy for the cases with

*η*> 0. In these cases, the near-surface buoyancy is characterized by unstable thermal stratification (convection) such that the additional mixing associated with LT now brings

*warmer*(less buoyant) fluid to the surface. For comparison, the linear DWL buoyancy profile assumed in Fairall et al. (1996) yields a constant

## 6. Discussion and conclusions

Based on state-of-the-art second-moment turbulence modeling, and supported by turbulence-resolving LES, we have shown that LT strongly impacts the DWL energetics, mainly by reducing the work performed by the surface stress and partly compensating this effect by Stokes shear production. Surface buoyancy and surface velocity are strongly reduced under LT, even under weak winds, which has important implications for air–sea exchange in coupled models. With an average increase in DWL thicknesses of only around 10%, the impact of LT on DWL bulk parameters, however, turned out to be moderate. We attribute this largely to the equilibrium wave model used in our study. Although equilibrium wave fields are typical in many situations, it is worth noting that previous LES studies with monochromatic nonequilibrium waves, focusing on swell effects (Kukulka et al. 2013), have shown a stronger impact of LT on DWL properties. Under nonequilibrium wind and wave conditions and deviations from a fixed La = 0.3, the scaling coefficients that were derived in section 5 may have to be adjusted.

Dimensional analysis and the parameter space studies in section 5 showed that the most relevant nondimensional parameters among those compiled in Table 3 are the following three: the stability parameter *T _{h}*/

*T*, and the shortwave radiation absorption scale

_{f}*R*,

*T*/

_{h}*T*, and

_{f}As shown in section 5, however, the three key parameters identified above do appear independently in the frequently used DWL scaling relations of PWP86. We showed that their model reliably predicts the most important DWL bulk parameters across a wide parameter range with our different sets of revised model coefficients that include the deepening of the DWL due to LT and other aspects of our more advanced turbulence model. We suggest a simple model extension to also account for the effects of penetrating shortwave radiation, which, however only yielded good agreement with our simulations for tropical DWLs, pointing at future work for a reliable description of high-latitude DWLs. One caveat of PWP86 applies to the low-energy regime with

The excellent performance of the simple PWP86 scaling relations was a somewhat unexpected result as our parameter space also included high-latitude DWLs for which the PWP86 modeling assumption of a constant bulk Richardson number formally breaks down. In view of increasing ice-free areas at high latitudes and strong DWL temperature anomalies already observed at high latitudes (Jia et al. 2023; Eastwood et al. 2011), it is likely that the physics of these DWLs (e.g., Sutherland et al. 2016) will receive increased attention in the future.

## Acknowledgments.

This paper is a contribution to the project L4 (Energy-Consistent Ocean-Atmosphere Coupling) of the Collaborative Research Centre TRR 181 “Energy Transfers in Atmosphere and Ocean,” funded by the German Research Foundation (DFG) under Grant 274762653 to L. Umlauf. H. T. Pham and S. Sarkar are pleased to acknowledge funding by NSF Grant OCE-1851390. We thank Kenneth Hughes and an anonymous reviewer for their valuable contributions. Qing Li and Ramsey Harcourt provided expert input on Langmuir turbulence in GOTM.

## Data availability statement.

Simulations in this manuscript were carried out with a modified version of the General Ocean Turbulence Model (GOTM). The used source code is archived at https://doi.org/10.5281/zenodo.8103884 (Klingbeil and Umlauf 2023). The LES data as well as the scripts that run GOTM and plot the figures shown in this article are archived at https://doi.org/10.5281/zenodo.10223915 (Schmitt 2023).

## APPENDIX A

### Second-Moment Turbulence Models

*ν*,

_{t}*k*and a turbulence length scale

*l*according to

*c*,

_{μ}Our analysis in sections 4 and 5 is based on the stability functions of H15 that constitute an improved version of an earlier model by Harcourt (2013) and are considered state of the art for the integration of LT effects in second-moment closure models. Note that the stability functions in (A1) are presented using the notation of the generic length scale (GLS) framework (Umlauf and Burchard 2003). They are related to their equivalents in Mellor–Yamada notation (see H15) as

H15 showed that if LT effects are included, *c _{μ}*,

*Nk*/

*ε*,

*Sk*/

*ε*,

*S*/

_{c}k*ε*, and

*S*/

_{s}k*ε*, where

*S*and

_{c}*S*defined in (15) represent the direct impact of Stokes shear on the stability functions that was ignored in earlier models of LT [the full expressions for the stability functions are shown in (33) of H15]. One example of these earlier models is the one of KC04 that is based on the original stability functions of Kantha and Clayson (1994), ignoring LT effects. In this case, we have

_{s}*P*and

*P*in (13) and (14) simplify accordingly.

_{s}*l*, we compute this quantity from a modified Mellor–Yamada-type transport equation for the variable

*kl*. These authors suggested to include an extra Stokes production term, analogous to the TKE budget in (12), in the original

*k–l*equation of Mellor and Yamada (1982), leading to an expression of the form:

*κ*is the von Kármán constant and

*L*the distance from the surface) is required to reproduce the logarithmic wall layer distribution close to the surface.

_{z}*D*summarizes the vertical transport terms, and

_{l}*c*

_{l}_{1}–

*c*

_{l}_{4}and

*c*denote nondimensional model constants (or functions) discussed in more detail below. The conversion relations between our notation and that originally used by KC04 and H15 are summarized in Table A1.

_{F}*ε*follows from the cascading relation

*c*in the logarithmic wall layer (Umlauf and Burchard 2005).

_{μ}*kl*in (A2), we also computed some of the solutions based on the

*k–ω*model by Umlauf et al. (2003), solving (12) combined with an equation of the form

*ω*denotes an inverse turbulence time scale defined as

*ω*in (A4) includes a Stokes production term recently suggested by Yu et al. (2022) to account for LT effects. The term

*D*denotes again the turbulent transport terms, and

_{ω}*c*

_{ω}_{1}–

*c*

_{ω}_{4}are nondimensional model constants (see Table A1).

*D*,

_{k}*D*, and

_{l}*D*appearing in (12), (A2), and (A4), respectively, are modeled by downgradient expressions:

_{ω}The model parameters *c _{l}*

_{1}and

*c*

_{l}_{2}, and similarly

*c*

_{ω}_{1}and

*c*

_{ω}_{2}for the

*k–ω*model (all compiled in Table A1) are well constrained by classical data for unstratified shear layers and decaying turbulence (e.g., Umlauf and Burchard 2003). The parameters

*c*

_{l}_{3}and

*c*

_{ω}_{3}determine the entrainment rate in stratified turbulent boundary layers. Their values follow from a condition on the so-called steady-state Richardson number, Ri

_{st}= 0.23, corresponding to the value of the Richardson number Ri in the entrainment layer at the base of the turbulent surface layer (Umlauf and Burchard 2005). Note that

*c*

_{l}_{3}= 2.4 is close to the value

*c*

_{l}_{3}= 2.5 used by H15. We also follow the suggestion by H15 to limit the vertical length scale by the Ozmidov scale,

*E*

_{4}= 1.33.

The most important model parameters in (A2) and (A4) in the context of LT are those multiplying the Stokes shear production terms, respectively. For the *k–l* equation, we adopt H15’s value *E*_{6} = 6, corresponding to *c _{l}*

_{4}= 3 in GLS notation. Note that this value is close to the revised

*E*

_{6}= 7.2 obtained from comparison to field measurements (see Kantha et al. 2010) of KC04. For the

*k–ω*model, we follow Yu et al. (2022) and choose

*c*

_{ω}_{4}= 0.15.

*S*is the stability function for the turbulent diffusivity of heat,

_{H}*S*/

_{l}*S*= 3.7 but, similar to Harcourt (2013), we find that

_{q}*S*=

_{l}*S*= 0.2 is more in line with the LES results. For the

_{q}*k–ω*model, the stability functions

*c*(see Umlauf et al. 2003) with constant proportionality factors expressed in terms of the turbulent Schmidt numbers

_{μ}*σ*and

_{k}*σ*(see Table A1).

_{ω}*z*

_{0}is the surface roughness length. For the upper boundary, these boundary conditions follow from the classical law-of-the-wall relations (see Umlauf and Burchard 2005). Please note that we do not consider the injection of TKE by breaking surface waves. A more detailed discussion of how

*z*

_{0}affects the class of models used in our study with and without wave breaking can be found in Umlauf and Burchard (2003).

## APPENDIX B

### Large-Eddy Simulations

*U*and buoyancy

_{i}*B*are numerically solved in the LES as follows:

*ω*is the vorticity and

_{k}*D*/

*Dt*=

*∂*/

*∂t*+

*U*/

_{j}∂*∂x*. The generalized pressure (Π) is defined as

_{j}*p*is the dynamic pressure and

*e*is the subgrid turbulent kinetic energy. A Poisson equation derived by taking the divergence of the momentum equation in (B1) is solved to obtain the modified pressure (

*p*/ρ

_{0}+ 2

*e*/3) using a multigrid method.

*ν*

_{sgs}:

*C*is set to be 0.5 and Δ

_{K}*is the grid filter width. Here,*

_{f}*i*,

*j*, and

*k*in the equation above indicate the grid indices in the

*x*,

*y*, and

*z*, directions, respectively. A unity subgrid Prandtl number is used to calculate the subgrid buoyancy flux

The computational domain is a rectangular box with dimensions of 64 × 64 × 72 m in the *x*, *y*, and *z* directions, respectively, using a grid size of 256^{3}. The grid is uniform in the horizontal directions with a spacing of 0.25 m. We use a fine vertical grid spacing of 0.05 m at the surface, and mildly stretch the grid at a rate of 3% in the region below.

The LES is initialized with zero velocity and a fixed buoyancy value throughout the domain. Periodicity is enforced at the horizontal boundaries. The wind stress components, *B*_{0} are applied at the top surface as implemented in the second-moment turbulence modeling approach. Homogeneous Neumann boundary conditions (zero gradients) are used at the bottom boundary for the horizontal velocity components and buoyancy while the vertical velocity component is set to zero at the bottom. A sponge layer is set up in the bottom 20 m to absorb possible fluctuations excited by turbulence in the surface layer.

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