Under the assumption of horizontal homogeneity generally retained in general circulation models, the contribution from Stokes drift to the turbulent kinetic energy (TKE) prognostic equation arises from the vortex force vertical term WVFz=ũs∂zu+ṽs∂zv in the hydrostatic relation (Eq. ). Mimicking the way the TKE equation is usually derived see, e.g., and using an averaging operator ⋅ satisfying the “Reynolds properties”, we find that the turbulent fluctuations, defined as ϕ′=ϕ-ϕ,(ϕ=p,ρ,u,v), associated with the WVFz term are (WVFz)′=ũs∂zu′+ṽs∂zv′. After multiplication by w′ and averaging, we obtain w′(WVFz)′=ũs∂zu′w′+ṽs∂zv′w′-ũsu′∂zw′-ṽsv′∂zw′, where the last two terms on the right-hand side cancel, with similar terms appearing when forming the equations for u′∂tu′ and v′∂tv′ (see Eqs. A.7 and A.8 in ). The extra terms associated with the Stokes drift in the horizontally homogeneous TKE equation are thus us∂zu′w′ and vs∂zv′w′, which can be further rewritten as ũs∂zu′w′=-u′w′∂zũs+∂zũsu′w′,ṽs∂zv′w′=-v′w′∂zṽs+∂zṽsv′w′. The first term will modify the shear production term; it can also be derived by taking the Lagrangian mean of the wave-resolved TKE equation . The second will enter the TKE transport term that is usually parameterized as -Ke∂ze. The prognostic equation for the turbulent kinetic energy e in NEMO under the assumption that Ke=Avm, with Avm the eddy viscosity, is thus 15∂te=Avme32(∂ku)2+(∂kv)2+(∂ku)(∂kũs)+(∂kv)(∂kṽs)-AvtN2+1e3∂kAvme3∂ke-cϵe3/2lε2, with Avt the turbulent diffusivity, N the local Brunt–Väisälä frequency, lε a dissipative length scale, and cε a constant parameter (generally such that cε≈1/2). Once the value of e is know, eddy diffusivity and viscosity are given by Avm=Cmlme,Avt=Avm/Prt, with Prt the Prandtl number (see Sect. 10.1.3 in , for the detailed computation of Prt), lm a mixing length scale, and Cm a constant.

In addition to the modification of the shear production term in the TKE equation, the wave will affect the surface boundary condition for e, lm, and lε. The Dirichlet boundary condition traditionally used in NEMO for the TKE variable is modified into a Neumann boundary condition, 16Avme3∂kez=z1=-ρ0g∫02π∫0∞Socedωdθ=Φoce, meaning that the injection of TKE at the surface is given by the dissipation of the wave field via the wave–ocean Soce term, which is a sink term in the wave model energy balance equation usually dominated by wave breaking, converted into an ocean turbulence source term. In practice, this sum of Soce is obtained as a residual of the source term integration; hence, it also includes unresolved fluxes of energy to the high-frequency tail of the wave model. Due to the placement at cell interfaces of the TKE variable on the computational grid, the TKE flux is not applied at the free surface but at the center of the topmost grid cell (i.e., at z=z1). This amounts to interpreting the half-grid cell at the top as a constant flux layer, which is consistent with the surface layer Monin–Obukhov theory.

The length scales lm and lε are computed via two intermediate length scales lup and ldwn, respectively estimating the maximum upward and downward displacement of a water parcel with a given initial kinetic energy. lup and ldwn are first initialized to the length scale proposed by : lup(z)=ldwn(z)=2e(z)/N2(z). The resulting length scales are then limited not only by the distance to the surface and to the bottom but also by the distance to a strongly stratified portion of the water column such as the thermocline. This limitation amounts to controlling the vertical gradients of lup(z) and ldwn(z) such that they are not larger that the variations of depth ∂kl⋅≤e3,l⋅=lup,ldwn Then, the dissipative and mixing length scales are given by lm=lupldwn and lε=min⁡lup,ldwn. Following (their Sect. 4.2.3), a boundary condition consistent with the Monin–Obukhov similarity theory for the length scale ldwn (while lup necessitates only a bottom boundary condition) is ldwn(z=η)=κ(Cmcε)1/4Cmz0, with κ the von Karman constant and Cm and cε the constant parameters in the TKE closure. The surface roughness length z0 can be directly estimated from the significant wave height provided by the wave model as z0=1.6Hs their Eq. 5, which provides a proxy for the scale of the breaking waves. Note that in our study, no explicit parameterization of the mixing induced by near-inertial waves has been added . As highlighted by , without activating this ad hoc parameterization in the standard NEMO TKE scheme, the model does not mix deeply enough. They also speculated that this ad hoc mixing could mask the effects of wave-related mixing processes such as Langmuir turbulence. For this reason, it is not used in the present simulations.

Langmuir mixing is parameterized following the approach of . This parameterization takes the form of an additional source term PLC in the TKE equation (). PLC is defined as PLC=wLC3dLC, where wLC represents the vertical velocity profile associated with Langmuir cells and dLC their expected depth. Following , wLC and dLC are given by wLC=cLC‖u^LCs‖sin⁡-πzdLC,if-z≤dLC0,otherwise,-∫-dLCηN2(z)zdz=‖u^LCs‖22, where ‖u^LCs‖ is the portion of the surface Stokes drift contributing to Langmuir cell intensity and cLC a constant parameter. In the absence of information about the wave field it is generally assumed that ‖u^LCs‖∝‖τ‖. As mentioned in the Introduction, and showed that the intensity of Langmuir cells is largely influenced by the angle between the Stokes drift and the wind direction. To reflect this dependency we account for this angle in our definition of ‖u^LCs‖ via 17‖u^LCs‖=max⁡us(η)⋅eτ,0, with eτ the unit vector in the wind stress direction. The difference between the surface Stokes drift ‖us(η)‖ and ‖u^LCs‖ given by Eq. () is shown in Fig.  and compared to the usual parameterization of ‖us(η)‖ as 0.377‖τoce‖/ρ0 in the uncoupled case see. The modulation of ‖u^LCs‖ depending on the wind stress orientation significantly reduces the input of the surface Stokes drift contributing to Langmuir cell intensity, especially in the Southern Ocean, while other regions are less affected. Finally, a value for the parameter cLC must be chosen. Based on single-column experiments detailed in Appendix , we find that parameter values in the range 0.15–0.3 provide satisfactory results compared to the LESs of and will be considered for the numerical experiments discussed later in Sect. .

While the parameterization was already implemented in NEMO, there are three major novelties in our implementation: (i) the online coupled strategy allows us to use the surface Stokes drift directly delivered by the wave model instead of the original value empirically estimated from the wind speed (e.g., 1.6 % of the 10 m wind). (ii) We only considered the component of the Stokes drift aligned with the wind, and (iii) based on a series of single-column simulations (see Appendix ) the coefficient cLC evaluated as 0.15 by is set to a 0.3 value. Those changes, together with the new surface boundary condition for the TKE equation, lead to a deeper penetration of the TKE inside the mixed layer and, as shown in Sect. , greatly improved the MLD distribution.

NEMO is a state-of-the-art primitive-equation, split–explicit, free-surface oceanic model whose equations are formulated both in the vector-invariant and flux forms (see Eq.  for the vector-invariant form). The equations are discretized using a generalized vertical coordinate featuring, among others, the z⋆ coordinate with partial-step bathymetry and the σ coordinate, as well as a mixture of both . For efficiency and accuracy in the representation of external gravity wave propagation, model equations are split between a barotropic mode and a baroclinic mode to allow the possibility to adopt specific numerical treatments in each mode. The NEMO equations are spatially discretized on an Arakawa C grid in the horizontal and a Lorenz grid in the vertical, and the time dimension is discretized using a leapfrog scheme with a modified Robert-Asselin filter to damp the spurious numerical mode associated with leapfrog . For the current study the NEMO equations have been modified to include wave effects as described in Eqs. () and (). Moreover, the modifications to the standard NEMO one-equation TKE closure scheme are given in Sect. .

The NEMO ocean model has been coupled to the WW3 wave model. In numerical models, waves are generally described using several phase and amplitude parameters. We provide only the details sufficient to understand the coupling of waves with the oceanic model here, and an exhaustive description of WW3 is given by . WW3 integrates the wave action equation with the spectral density of wave action Nw(kw,θw) discretized in wavenumber kw and wave propagation direction θw for the spectral space (the subscript w is used here to avoid confusion with previously introduced notations): 18∂tNw+∂ϕϕ˙Nw+∂λλ˙Nw+∂kwk˙wNw+∂θwθ˙wNw=Sσ, where λ is longitude, ϕ is latitude, and S is the net spectral source term that includes the sum of the rate of change of the surface elevation variance due to interactions with the atmosphere via wind–wave generation and swell dissipation (Satm), nonlinear wave–wave interaction (Snl), and interaction with the upper ocean that is generally dominated by wave breaking (Soce). Those parameterized source terms are important in wave–ocean coupling. Indeed, as shown earlier in Eq. (), the Soce term is used to compute the TKE flux transmitted to the ocean, and the Sin term enters the computation of the wave-supported stress. They are computed here following . In Eq. (), the dot variables correspond to a propagation speed given by the following. 19ϕ˙=cgcos⁡θw+vz=ηR-120λ˙=cgsin⁡θw+uz=ηRcos⁡ϕ-121θ˙w=cgsin⁡θwtan⁡ϕR-1+sin⁡θw∂ωw∂ϕ-cos⁡θwcos⁡ϕ∂ωw∂λkwR-122k˙w=-∂σ∂Hkkw⋅∇D-k⋅∇uh(z=η) Here, R is the Earth's radius, uh(z=η)=(uz=η,vz=η) represents the surface currents provided by the ocean model, cg is the group velocity, ωw the absolute radian frequency, and H the mean water depth. Equation () is solved for each spectral component (kw,θw) coupled by the advection and source terms. Equations ()–() show how the oceanic currents affect the advection of the wave action density; there are also indirect effects via the source term .

The practical coupling between NEMO and WW3 has been implemented using the OASIS-MCT software primarily developed for use in multicomponent climate models. This software provides the tools to couple various models at low implementation and performance overhead. In particular, thanks to MCT , it includes the parallelization of the coupling communications and runtime grid interpolations. For efficiency, interpolations are formulated in the form of a matrix–vector multiplication whereby the matrix containing the mapping weights is computed offline once for all. In practice, after compiling OASIS-MCT, the resulting library is linked to the component models so that they have access to the specific interpolation and data exchange subroutines. Now that we have described the different components involved in our coupled system, we go into the details of the nature of the data exchanged between both models.

The wave hindcasts presented here are all based on the WW3 model in its version 6.02 configured with a single grid at 0.5∘ resolution in longitude and latitude. A spectral grid with 24 directions and 31 frequencies is exponentially spaced over the interval [fmin⁡,fmax⁡] with fmin⁡=0.037 Hz and fmax⁡=0.7 Hz. A one-step monotonic third-order coupled space–time advection scheme (also called the ultimate quickest scheme) is used with a specific procedure to alleviate the so-called garden sprinkler effect . As suggested in , the dissipation induced by wave breaking is proportional to the local saturation spectrum see also. The wind input growth rate at high frequency is based on the formulation of with an additional “sheltering” term to reduce the effective winds for the shorter waves . For the computation of nonlinear wave–wave interactions, the discrete interaction approximation of is used. This last approximation is known to be inaccurate, but it is thought that the associated errors are usually compensated for by a proper adjustment of the dissipation source term . As mentioned earlier in Sect. , the model was run with 10 m winds, without any air–sea stability correction. No wave measurements were assimilated in the model, but the stand-alone wave model was developed based on spectral buoy and synthetic aperture radar (SAR) data and calibrated against altimeter data by adjusting the wind–wave coupling parameter . The WW3 time step for the global configurations is Δtww3=3600 s.

For the oceanic component, we use a global ORCA025 configuration at a 1/4∘ horizontal resolution . The vertical grid is designed with 75 vertical z levels with vertical spacing increasing with depth. Grid thickness is about 1 m near the surface and increases with depth to reach 200 m at the bottom. Partial steps are used to represent the bathymetry. The LIM3 sea ice model is used for the sea ice dynamics and thermodynamics . The vertical mixing coefficients are obtained from the one-equation TKE scheme described in Sect. , and the convective processes are mimicked using an enhanced vertical diffusion parameterization that increases vertical diffusivity to 10 m2 s-1 at which static instability occurs. Water density is computed from temperature and salinity through the use of a polynomial formulation of the nonlinear equation of state . The vector-invariant form of momentum advection uses for the vorticity and a specific formulation to control the Hollingsworth instability . Momentum lateral viscosity is biharmonic and acts along geopotential surfaces. It is set to a value of 1.5×1011 m4 s-1 at the Equator and varies proportionally to Δx3 away from the Equator. Advection of tracers is performed with a flux-corrected transport (FCT) scheme , and lateral diffusion of tracers is harmonic and acts along an iso-neutral surface. It is set to a value of 300 m2 s-1 at the Equator, which varies proportionally to Δx. The bottom friction is nonlinear and the lateral boundary condition is free-slip. In this setup, the baroclinic time step is set to Δtnemo=900 s and a barotropic time step 30 times smaller. Compared to the standard uncoupled ORCA025 configuration, the additional computational cost associated with WW3 and the exchanges through the coupler is about 20 %.

The atmospheric fields used to force both ocean and wave models are based on the ECMWF (European Centre for Medium-Range Weather Forecasts) ERA-Interim reanalysis . Corrections have been applied to guarantee that the ERA-Interim mean states for rainfall as well as shortwave and longwave radiative fluxes are consistent with satellite observations from the Remote Sensing Systems (RSS) Passive Microwave Water Cycle (PMWC) product and GEWEX SRB 3.1 data (https://gewex-srb.larc.nasa.gov/, last access: 2 July 2020). Momentum and heat turbulent surface fluxes are computed using the IFS bulk formulation from the AeroBulk package using air temperature and humidity at 2 m, mean sea level pressure, and 10 m winds.

Sensitivity experiments have been conducted to check the proper implementation of various components of the present coupled modeling system. For the sake of clarity, our developments are split into four components: (i) the modification of the wind stress by waves through the Charnock parameter and the inclusion of wave-supported stress, (ii) the modifications of the NEMO governing equations through the Stokes–Coriolis, vortex force, and wave-induced surface pressure terms, (iii) the addition of a Langmuir turbulence parameterization, and (iv) the modifications to the TKE scheme. As summarized in Table , sensitivity experiments are designed in such a way to incrementally increase the level of complexity and test the effect of each component. The No_CPL experiment corresponds to the classical NEMO setup in which the wave effect is parameterized through a wind-stress-dependent TKE surface boundary condition as suggested by . In this approach, a Dirichlet surface boundary condition is used and expressed as follows: e(z=η)=12(15.8αCB)2/3‖τatm‖ρ0 with αCB=100. Based on the results of we expect that in the uncoupled case the nature of the boundary condition (i.e., Dirichlet vs. Neumann) does not significantly impact numerical solutions

In the authors consider a Dirichlet boundary condition such that e(z=η)=12(15.8αCB)2/3u⋆2 and an equivalent Neumann condition Ke∂zez=η=2αCBu⋆3. The authors claim that numerical solutions using a Dirichlet condition instead of a Neumann condition are qualitatively similar.

The wave distribution being inhomogeneous on the globe, it is expected that with the wave-modified wind stress parameterization the stress should follow the wave patterns more closely. In Fig. , the seasonal average of the significant wave height and of the difference between the Charnock coefficient computed by the wave model and the default constant value used in the uncoupled case (αch0=0.018) are shown. As expected, the Charnock parameter tends to be stronger in the area where the waves are higher. Generally an increase in the Charnock parameter is observed in the northern and southern basin, while there is a net decrease in αch near the Equator. There is also a strong seasonality in the Northern Hemisphere, with a reduction in summer and a strong increase in winter. The differences between αch and αch0 are very latitudinal with very few longitudinal variations.

To isolate the effect of the Charnock parameter we compare the results obtained in the No_CPL and WS_CPL experiments. Those two experiments show relatively similar sea surface temperature patterns, meaning that the modification of the wind stress ‖τoce‖ between those two cases is primarily due to the use of different Charnock parameters and the inclusion of the wave-supported stress.

Figure a illustrates that the Charnock parameter mostly affects the drag coefficient CD, and hence the surface wind stress, for large winds. The ocean–wave coupling does not lead to appreciable differences in the drag coefficient CD for wind speeds lower than 8 m s-1. On the contrary, since large values of the Charnock parameter are observed for large wind speeds, the coupling significantly increases the drag (as well as its variance) at high winds. Figure b shows how the wind stress is modified by this increase in the drag coefficient jointly with the wave-supported stress, which tends to decrease the wind stress magnitude (Fig. ). At low wind speed the wind stress magnitude is not affected by the coupling with waves, while for strong winds the increase in wind stress associated with the increased drag coefficient is always larger than the decrease associated with the wave-supported stress. This latter effect reduces the wind stress by no more than 2 %; for the characteristic scales of our study, this correction is thus almost negligible. The wind stress changes due to the coupling with waves seen in our simulations are very localized in time and space and it is thus difficult to conclude on their overall effect on upper-ocean dynamics such as Ekman pumping and the surface currents.

As described in Sect. , in the ocean–wave coupled case, the surface boundary condition for the TKE equation is a Neumann condition whose value is directly given by the wave model, unlike the uncoupled case in which a Dirichlet condition is imposed. We aim here to assess the impact on the order of magnitude of the near-surface TKE. Since the Neumann boundary condition is applied at the center of the topmost grid box (i.e., approximately at 50 cm of depth), we compare in Fig.  the TKE value at 1 m of depth between the coupled (All_CPL2) and the uncoupled (No_CPL) case. Positive values mean that near-surface TKE is larger in the coupled simulation.

It shows an almost homogeneous increase in the TKE (up to more than 100 %) in the extratropical areas. While low seasonal variability in the extratropical areas is visible in Fig. , a spatial averaging by hemisphere (Fig. ) highlights seasonal variability with a strong increase in both the near-surface TKE value and the TKE difference between the two experiments during winter. In Figs.  and (and also in the remainder of the paper), the spatial averaging is made between 25 and 60∘ S in the Southern Hemisphere and between 25 and 60∘ N in the Northern Hemisphere to avoid any conflicts with sea ice and to remove the equatorial region from the comparison. The increase in the surface TKE injection associated with waves is expected to contribute to an overall increase in mixed layer depth provided that the mixing length diagnosed by the turbulent closure scheme allows for the effective propagation of this additional TKE deeper in the mixed layer.

In this section, we evaluate the wave effect on vertical mixing using the mixed layer depth (MLD) as a relevant metric. Figure  represents the seasonally averaged difference in MLD between the coupled (All_CPL2) and the uncoupled (No_CPL) case relative to the No_CPL case (i.e., (hMLDnoCPL-hMLDCPL)/hMLDnoCPL with hMLD considered negative downward). It shows a significant deepening of the mixed layer at high latitudes in the coupled case with only a very few localized mixed layer shallowing up to 60 %, mainly in the Southern Hemisphere.

To assess whether the overall deepening of the mixed layer is realistic, we make a comparison with available observations. Available observations for 2014 were extracted following an updated dataset from . The MLD has been computed as the depth at which the density is 3 % smaller that the density at 10 m as in . Figure  represents the spatially averaged MLD; the blue line is the spatially averaged MLD obtained from ARGO floats (available during the same period) in both hemispheres. In the Northern Hemisphere (Fig. a), there is only a slight improvement compared to data during winter and late summer when implementing the coupling with waves. In the Southern Hemisphere (Fig. b) the situation is rather different.

From January to July, the deepening of the MLD induced by the wave coupling significantly reduces the bias between the model and ARGO data. From July to December, results in the coupled case show an overestimation of MLDs, which were already too deep in the uncoupled case, thereby increasing the bias between the data and model. Since mesoscale activity makes direct comparisons to data unreliable for such a short period of time, we compare the normalized distribution of MLD between the different simulations and available ARGO data. Results are presented in Fig.  for the year 2014 (panel a) and during summer only (panel b). In both cases the improvement in the Northern Hemisphere is very modest. As far as the Southern Hemisphere is concerned the coupling with waves leads to a significant improvement compared to the MLD derived from ARGO floats despite the fact that there are still too many low MLD values in the range 50–100 m. In comparison with the uncoupled case there is a more realistic spreading toward deeper mixed layer depths. More particularly in summer (Fig. b), the probability density function (PDF) in the coupled case matches the one computed from ARGO data almost perfectly. Despite the fact that we did not activate the ad hoc extra mixing induced by near-inertial waves , our implementation of the wave–ocean interaction leads to a significant deepening of the MLD in a realistic way.

To better understand which components of the wave–ocean coupling are responsible for this improvement, the summer PDF in the Southern Hemisphere has been computed for each of the experiments described in Table . Results are shown in Fig. . First of all, it can be seen that all the wave–ocean interactions described in previous sections lead to an improvement in terms of mixed layer depth distribution compared to the uncoupled case. Indeed, the modification of the wind stress by the wave field introduced in WS_CPL increases both surface currents and near-surface TKE values, resulting in a slight deepening of the MLD. Adding the Stokes-drift-related terms in the primitive equations contributes only modestly to the deepening of the MLD, while most of the improvement results from the modified TKE scheme, with some slight improvement when the Langmuir parameterization is activated. It is somewhat reassuring to see that the better agreement with ARGO data is obtained when all components of the coupling are activated.

Since the near-surface mixing is strengthened by the coupling, we can expect an impact on sea surface temperature (SST). Figure  represents the time series of SST for each hemisphere. The Northern Hemisphere is characterized by a warm bias during summer with a very slight improvement when coupling with waves. In the Southern Hemisphere (Fig. b) the summer warm bias is reduced by half in the coupled simulation and a slight warming occurs during the winter. While the summer surface cooling might be linked to the mixed layer deepening, the winter warming might be rather linked to advection as observed by for the Baltic sea. It could also result from an increased heat content during summer, leading to higher SST during winter.

To better characterize the wave impact on the SST, we show in Fig. a the difference in terms of annual mean between the No_CPL experiment and OSTIA analysis, exhibiting a cold bias in the No_CPL simulation in equatorial and tropical regions and a warm bias in the northern part of the Pacific Ocean. The coupling with waves tends to diminish the cold bias (see Fig. b), especially in the Pacific Ocean, and the warm bias in the North Pacific is significantly reduced.

As already noticed by the warming in the equatorial and tropical regions mainly results from a lower wind stress caused by a value of the Charnock parameter lower than the value used in the uncoupled case (see Fig. b, d). A consequence is a decrease in the drag coefficient, leading to smaller turbulent exchange coefficients and reducing the heat flux. As mentioned above, in extratropical regions, some warm bias tends to be partially reduced by the extra mixing induced by the waves at high latitude and/or by the increased turbulent transfer coefficient. The tendency of the wave coupling to improve the near-surface temperature distribution can also be verified on a time–latitude Hovmöller diagram like the ones shown in Fig. . For instance, it can be seen that the summer warm bias in the Northern Hemisphere (Fig. a) coincides well with the cooling induced by the coupling with waves (Fig. b). Similarly we can also observe a warming in the tropical and equatorial regions (Fig. b) corresponding to the cold bias seen in Fig. a. In the southern extratropical region, a summer cooling is observed. It is induced by the wave coupling, whereas Fig. a shows a slight warm bias. During winter we can observe a warming in Fig. b north of 60∘ S, which again partially corresponds to a cold bias in Fig. a.

The last aspect of our solutions we would like to evaluate is the impact of the surface waves on surface currents and kinetic energy (KE). To do so, we show in Fig.  time series of the spatially averaged surface kinetic energy for both hemispheres. Whatever the hemisphere there is a net decrease in surface KE (up to 20 % in the south) when a coupling with the waves is included. This decrease in surface kinetic energy reflects a decrease in surface current magnitudes. Indeed, as detailed in Fig. , which represents the vertical profile of the horizontal components of the current in the oceanic surface boundary layer, the coupling with waves decreases both the surface current magnitudes and the shear. While currents from the WS_CPL are increased due to increased wind stress, the Stokes–Coriolis force when included in momentum equations leads to a decrease in velocities in the whole boundary layer as previously shown by (orange lines in Fig. ). The inclusion of vertical mixing due to waves and Langmuir circulation attenuates the currents in the surface layer, resulting in further reduced surface currents and stronger currents at the bottom of the boundary layer (purple lines in Fig. ). This concludes our checking of the proper functioning of the coupling with waves as described in the present paper.