Turbulent air–sea interactions are known to play a central role in modulating heat and moisture exchanges at interannual to climatic scales. They also control the major part of the heat, moisture, and momentum exchanges in tropical cyclones (TCs) and, as a consequence, have a strong impact on cyclone intensity (e.g. ). Their accurate representation in climate or numerical weather prediction (NWP) models is thus a key step towards better modelling the climate evolution and extreme weather events.

Because the turbulent fluctuations in surface parameters cannot be represented explicitly in atmospheric models, turbulent fluxes are computed using “bulk” parameterisations as functions of mean atmospheric variables at the surface within the framework of the similarity theory proposed by MOST. For the wind stress τ, it reads as follows: 1τ=ρu*2=CdΔU2, with ρ the air density, u* the friction velocity, ΔU the difference between the wind speed at a reference level and the surface current Uc, and Cd the drag coefficient. Note that bulk algorithms generally include the effect of gustiness in the turbulent fluxes by including a term Ug in the ΔU difference, such that ΔU reads ΔU=((Ux-Ucx)2+(Uy-Ucy)2+Ug2)1/2, with Ux, Uy the components of the surface wind and Ucx, Ucy the components of the surface current. This gustiness term is not explicitly mentioned in the present work but is taken into account in the SURFEX v8.1 version of all bulk algorithms used here. Similarly, the heat fluxes are expressed as follows: 2H=ρcpChΔUΔθ,LE=ρLvCeΔUΔq, with cp the air heat capacity and Lv the latent heat of vaporisation. Δθ and Δq represent the vertical air–sea gradients of potential temperature and specific humidity, respectively. In neutral conditions and in the surface layer, where u* is supposed to be constant with height, U(z) may be represented as a logarithmic profile: 3U(z)=u*κlog⁡(z/z0), where κ is the von Karman's constant (≈0.4) and z0 the roughness length. Equivalently, one can write 4u*=Cd(z)U(z)=κU(z)log⁡(z/z0) in neutral conditions and in the absence of a surface current.The roughness length z0 is expressed as the sum of two terms representing the behaviour of the surface in (respectively) rough and viscous regimes : 5z0=αu*2g+0.11νu*, with ν the kinematic viscosity of dry air, g the gravitational acceleration, and α the Charnock coefficient. The Charnock coefficient was originally assumed to be constant, but its dependence on wave parameters allows the drag coefficient to vary more explicitly with the sea state. Defining the transfer coefficients Cd, Ch, and Ce with reasonable accuracy in various conditions of surface wind, stability, and sea state has been the subject of a considerable amount of work by many expert teams for at least the last 50 years and the motivation for many dedicated field campaigns.

Direct observations of the turbulent fluxes at sea on buoys, ships and platforms provide constraints on the mean value of the neutral drag coefficient and its growth with wind speed in the range of 10 m wind speed between 5 and 20 m s-1 (e.g. ). In this wind range, the momentum transferred from the wind to the sea surface is mainly used for the waves to grow up to a well-developed sea, in equilibrium with the wind (e.g. ). The part of the wind stress absorbed by the waves has been formulated to be dependent on the stage of development of the wind sea or wave age (defined as the ratio of the wave phase speed for the peak of the wave spectrum to the near-surface wind) by and . The wave development, in turn, impacts the Charnock parameter and roughness length through Eq. () and the friction velocity through Eq. (). Observations carried out with extreme care and in mainstream conditions (i.e. in the absence of swell or strong surface currents) do indeed show a large variability in the friction velocity and in the drag coefficient at a given wind speed (Fig. ). Several studies based on theoretical considerations or field observations attribute part of this variability to the effect of the wave growth on z0. Wave steepness (wave height divided by wavelength) is also a good proxy for the sea-state impact on the surface roughness . Several parameterisations of the wind stress with dependence on the wave age have been developed to be used in wind–wave coupled models (e.g. ). As a pioneer in the wind–wave coupling domain, the European Centre for Medium-Range Weather Forecasts (ECMWF) used coupled models for operational forecasts since 1998 and obtained improvement for surface pressure in medium-range NWP and for the 500 hPa geopotential at seasonal scales .

For wind speed above 30 m s-1, the coupling regime controlling the stress transfer from the atmosphere to the waves is thought to be less dependent on the wave growth, as most waves are breaking. Direct measurements of wind stress are sparse but show no clear dependence on the wave age but a saturation or decrease for wind speeds above 30 to 35 m s-1 . This saturation itself is confirmed by other (more or less direct) observations (e.g. ), but the exact corresponding 10 m wind speed where it occurs, the maximum value of the drag coefficient, and its behaviour at higher wind speeds are still very uncertain. Indeed, all available estimates beyond 30 m s-1 are highly scattered (see Figs.  and ). Based on observations, there is no evidence that the scattering of the drag coefficient in wind speeds higher than 30 m s-1 may be due to wave age. According to previous work, the physical mechanisms likely to explain the observed saturation or decrease in the drag coefficient above 30 m s-1 are airflow separation due to wave breaking , changes in the wind profile close to the surface due to a high concentration of sea spray , or the inclusion of non-linear effects in the critical layer theory for wave growth with an explicit calculation of the momentum transferred to capillary–gravity waves . The saturation or decrease observed for cyclonic wind speeds must be reproduced in a parameterisation (using an analytical function or capping) to match the observations and enable a more realistic simulation of the tropical cyclone intensity .

Observations of the heat transfer coefficients show no clear dependence on the wind speed nor on the sea state. Estimations of sensible heat flux at sea from sonic anemometers are extremely noisy, resulting in a large dispersion between datasets (Figs.  and ). Measurements of the latent heat flux are done by gas analysers, which are very sensitive to rain, high humidity rates at sea, sea spray, and pollutants. All of this results in highly scattered values, even in the 5–20 m s-1 wind speed range (Fig. ). However, surface heat transfer plays a central role in TC intensification (e.g. ) and correctly representing it for strong winds in NWP models is a key step towards a better forecast of TC intensity. Besides, heat transfer plays a central role in modulating the climate-scale dynamics (in particular in the intertropical band) and can also control local processes even at low winds (e.g. ).

Several parameterisations of sea surface turbulent fluxes are available in the current SURFEX v8.1 surface model , the surface scheme embedded in the atmospheric models used at Météo-France. None of them, however, provides a match to observations for all wind speeds, including the cyclonic conditions, and the possibility of accounting for the wave growth effect on the roughness length and drag coefficient. The ECUME parameterisation (, updated from its initial version in ) is the default scheme used for operational NWP in the non-hydrostatic, limited-area model AROME and in the global model ARPEGE . ECUME is also used in ARPEGE within the CNRM-CM configurations for climate simulations . It is also commonly used for case studies with the research-oriented, non-hydrostatic, Meso-NH model . ECUME has been built by fitting scale parameters for wind, temperature, and humidity on observations and enables a close match of the transfer coefficients to observations (Figs.  and ). These transfer coefficients are expressed as polynomial functions of the 10 m wind speed only (the roughness length is a diagnostic parameter; Eqs. () and () are not part of the bulk algorithm). The COARE 3.0 parameterisation can also be used in SURFEX v8.1. It enables representing the impact of sea state on the roughness length through the use of the parameterisations of or (our Fig. b and c). It can also be used in coupled mode with a wave model, the Charnock coefficient (Eq. ) being computed within the wave model. Using SURFEX with the wave model WAVEWATCH III™ (WW3, ) has been made possible by the implementation of a surface coupling interface with the OASIS coupler in SURFEX by . COARE 3.0 has been fitted to observations of wind stress and heat fluxes in the tropics, for wind speeds up to 18 m s-1. It provides a good match with observations of wind stress up to 20 m s-1 (Fig. ) but does not reproduce the decrease in the drag coefficient for winds higher than 30 m s-1 (Figs. a and ). As a consequence, it is not suitable for representing the development of TCs or strong storms.

The new parameterisation presented here combines the two aspects of wind–wave coupling and reproducing the decrease in the drag by cyclonic winds. It is based on a very large set of field observations (see Sect.  for their selection) and ensures that their mean behaviour, in terms of drag and heat transfer coefficients, is well reproduced for wind speeds up to 60 m s-1. It is also based on the Charnock relationship with a dependency of the Charnock parameter on the wave age for wind speeds between 7 and 22 m s-1, corresponding to the growth of wind sea : 6α=AχB, where χ=cp/U10 is the wave age and A and B are polynomial functions of U10 (see Appendix  for more details). Note that, according to , the wave age should be computed as cp/U10 rather than cp/U10, but we use Eq. () for computing-cost reasons and we checked that the differences are negligible. For wind speeds of less than 22 m s-1, the WASP transfer coefficients closely follow those derived by , using a very large and carefully screened dataset. We do not pretend here to improve much the state of the art of turbulent fluxes at sea that can be used for wind–wave coupling but rather to design a tool that can be used with every atmospheric model coupled with SURFEX v8.1, producing realistic wind stress and heat fluxes at every wind speed. In addition, the drag coefficient varies as a function of the wave age for a given wind speed in the moderate- to strong-wind range where wave growth is the major process absorbing the wind energy. The next section presents the principle used for building the new parameterisation, the observations used to check the mean values of the transfer coefficients for a given wind speed, and the dependency of the drag coefficient on wage age. These options are discussed with respect to the literature and information from various datasets. Section 3 presents the four case studies that were used to validate and test this parameterisation. Some conclusions are given in Sect. 4.

The parameterisation presented here is meant to be used for atmospheric numerical modelling, either operationally with the models of Météo-France of for a large variety of case studies with Meso-NH, with typically the first level at 5 to 20 m above sea level (a.s.l.). Whereas it may be tempting to use much finer sampling close to the surface to better represent its influence on the surface-layer or boundary-layer processes, we believe that doing so within the MOST framework leads to inconsistency (see , for a discussion). The mean values of the transfer coefficients should be representative of a large number of neutral conditions, and the only variability introduced is the impact of the wave age for wind speeds between 7 and 23 m s-1 (see Sect. ). Turbulent fluxes and transfer coefficients are usually derived from in situ measurements recorded using high-frequency sensors (sonic anemometers and gas analysers) with either the eddy covariance (EC) or the inertial–dissipative (ID) methods. While obtaining reliable estimates using the ID method is easier and more straightforward, it implies strong assumptions on the surface-layer structure, which restrict its use. In this study and for wind conditions up to 25 m s-1, we use only carefully checked datasets from measurements at 5 m a.s.l. or above computed using the EC method. Thanks to the effort of the observing community, a large number of such datasets exist and many of them were already used by for deriving the wind stress parameterisation COARE 3.5 (see Table  for a list). This results in more than 27 000 individual data (representing 10 to 30 min of measurements each) for Cd, 21 000 for Ch and 24 000 for Ce. This covers the wind speed up to 22 m s-1 for Cd, and 20 m s-1 for Ch and Ce. These observations were binned in intervals of 1 m s-1 of wind speed, screened and quality checked. The screening consists of evaluating the symmetry of the binned distributions and whether they correspond rather to normal or log-normal laws. Depending on the results, outliers more than 4 standard deviations from the mean values were removed.

Other historical datasets available in the literature (see Table ) have been used for the range of wind speed up to 30 m s-1. Direct EC measurements in strong winds are scarce and usually made airborne at height between 30 and 500 m a.s.l. . For extreme winds between 30 and 60 m s-1, only very few observations are available, especially for the heat transfer coefficients. Some of them are derived from profiles of dropsondes , mostly computed indirectly from the effect of the wind stress on the oceanic surface layer, which is more easily sampled than the atmospheric boundary layer in extreme conditions . The observations used in this study that correspond to extreme conditions are listed in Table . Among them, some are derived from the oceanic response to tropical cyclones using inversion techniques and come with uncertainties higher than more direct estimates (dropsondes). All these data were used as constraints to derive the transfer coefficients, with different principles for the drag or the heat transfer coefficients, as detailed below.

The neutral drag coefficient is first constructed as a mean value, depending on the wind speed only, and fitted to available observations in the wind range from 5 to 60 m s-1. Then, a variability depending on the wave age in the wind range of 7 to 25 m s-1 is added to the mean value.

The principle retained here for building the heat flux transfer coefficient is very similar to the one of the COARE 3.0 parameterisation. It is clear from Eq. () that the neutral transfer coefficients both for sensible and latent heat fluxes depend only on the roughness lengths z0x and z0. The values of the neutral transfer coefficients for turbulent heat ChN and CeN correspond to those of the COARE 3.0 parameterisation. Then, Eq. () is inverted to obtain the value of z0x, z0 being obtained in WASP as explained in Sect. . In the following, we use datasets of available observations to evaluate these parameters for wind speed in the range 0 to 60 m s-1. These observations are grouped in direct, EC measurements between 0 and 21 m s-1 for ChN and 0 and 19 m s-1 for CeN and less direct measurements for higher wind speed, available as mean values with estimates of uncertainties for a given wind range or in binned form (see Fig. ).

An offline test was performed to assess the differences between the current version of ECUME used in the Météo-France NWP and climate runs on the one hand and WASP on the other. The SURFEX v8.1 model was used to compute the friction velocity and turbulent heat fluxes with either the ECUME or WASP scheme on the same dataset corresponding to observed atmospheric parameters, SST, and wave parameters Hs and Tp. This dataset consists of more than 53 000 hourly in situ measurements at the Lion buoy, located in the Gulf of Lion, between December 2001 and February 2014. They represent a large range of atmospheric conditions (Fig. ) with wind up to 25 m s-1, air temperature between 5 and 28 ∘C, relative humidity down to 40 %, and wave age (Cp/U10) as low as 0.4 due to strong wind and short fetch in mistral conditions. Strong winds in the Gulf of Lion correspond overall to the mistral and tramontane winds blowing offshore, resulting in strongly unstable conditions with dry air, young waves, and significant wave height up to 6 m. Figure  shows the difference obtained using WASP rather than ECUME on the fluxes of momentum and sensible and latent heat, as a function of the different surface conditions at the buoy. Warm colours indicate positive differences (the fluxes obtained using WASP are higher than those obtained using ECUME) and blue shades indicate negative differences. The comparison of friction velocities obtained using WASP and ECUME (Fig. a) shows that the difference does not depend at first order on the wind speed but on the wave age. As expected, young waves give higher friction velocities than older waves. The larger scattering of the difference which is obtained for the lowest and highest wave ages is an artefact due to the smaller size of the sample. For more common conditions, i.e. between 7 and 20 m s-1 and wave ages below 1, WASP gives consistently higher friction velocities than ECUME (8 %). In weaker wind conditions, the difference is not significant.

Both sensible and latent heat fluxes are generally lower with WASP than with ECUME (Fig. b, c). The difference in sensible heat flux is very dependent on the air–sea temperature gradient, especially for winds above 10 m s-1. In very unstable conditions, which are rather common in the Gulf of Lion, the difference reaches -150 W m-2, for only 30 to 40 W m-2 in stable conditions. The latent heat flux is lower whatever the conditions, except for weak wind and warm and moist conditions that are rarely met in the Gulf of Lion. It can be expected therefore that heat fluxes will be lower with WASP than with ECUME when simulating tropical cyclones, at least in the intensification phase with wind speed up to 25 m s-1.

The case study of a weak low-level flow across a sharp SST front in the Iroise Sea (, R2019 hereafter) is used to assess WASP in calm atmospheric conditions, with a strong change in atmospheric stratification over a few kilometres. The configuration used here is the same as in R19, and the reader can refer to this paper for a full description of the case study and modelling configuration.

The western Mediterranean region is regularly affected by heavy precipitation events (HPEs) that are characterised by a large amount of rainfall over a small area in a very short time (typically more than 100 mm in less than 1 d; ). These events regularly lead to flash flooding that is a major threat in the area, as it often causes severe damage and in some cases casualties (e.g. ). At a low level, strong winds with high SST as generally encountered in autumn govern heat transfer, which moistens and warms the air parcel, thus increases the instability, and finally intensifies the convection (e.g. ). The SST fine-scale structures and fronts in the Mediterranean are also known to play a role in low-level wind convergence , which is a key triggering mechanism for deep convection and HPEs at sea. Several case studies using kilometre-scale atmospheric models also showed that taking into account the modulation of the surface roughness by the waves can slow down the low-level flow, shifting the convergence lines and/or modifying the spreading of the cold pool formed below the convective system by precipitation evaporation (e.g. ).

The WASP parameterisation has already been tested with and without wave effect by hereafter S2020 on a HPE occurring in mid-October 2016 in south-eastern France. The wave parameters used as input of the parameterisation came from the wave model WW3 v5.16 in forced or coupled mode. In this case, the wave impact on the surface roughness reduces the low-level wind speed of more than 1 m s-1 over a large area and leads to a displacement of the HPE of 40 km towards sea. Since the sensitivity of the wave impact within WASP was already investigated in S2020, we use here the same case study and configuration to test the effect of using WASP in the AROME model with respect to the parameterisation ECUME currently used for operational forecasts. We first give a short summary of the configuration used and then present the results of the comparison.

WASP is designed to ensure the representation of the variability due to the wave growth and the saturation of the drag coefficient in case of cyclonic winds. The values of the transfer coefficient for heat are reasonably constrained by the observations for winds up to 20 m s-1, but between 20 and 60 m s-1 observations are too sparse for a robust fit. Case studies of tropical cyclones can help to validate indirectly the values chosen for the drag and heat transfer coefficients in the wind speed range with no observations or observations with large uncertainties.

The sensitivity of climate-scale runs to the turbulent flux parameterisation was tested in climate mode using the CNRM-CM model .