## 1. Introduction

In this paper, we would like to present some new results on the generation of ocean waves by wind and the consequent feedback of the growing wind waves on the airflow [called the quasi-linear effect in previous works of Janssen (1982, 1989, 1991)]. In Janssen (1991), a parameterization of the effect of waves on the wind profile and the consequent slowing down of the wind was developed. This parameterization of the two-way interaction of wind and waves has been used in the operational wave model (WAM) since the end of 1991 and has produced good agreement with the most recent parameterizations of the surface drag coefficient *C _{D}* (see Edson et al. 2013). Nevertheless, in spite of the success of this approach, it is valid to ask questions regarding some of the assumptions that have been made.

*τ*

_{w}_{,lf}the wave-induced stress of the low-frequency gravity waves, by

*τ*

_{w}_{,hf}the wave-induced stress of the high-frequency gravity–capillary waves, and by

*τ*the viscous stress, the stress balance at the surface reads,

_{υ}^{−1}, the drag coefficient reduces for increasing winds. Since the publication of Powell et al. (2003) showing field observations of decreasing drag coefficients in high winds, attempts have been made to explain this phenomenon. For example, Makin (2005) suggests that in high winds, sea-spray droplets form a stable layer near the water surface, which damps the turbulence and reduces the drag coefficient. Others try to give a mechanical explanation for the reduction of drag for high winds. Examples are Kudryavtsev and Makin (2007) and Kukulka et al. (2007), who consider the sheltering of short waves due to airflow separation. Neither can explain a decreasing drag coefficient in high winds. This also applies for the quasi-linear theory of wind–wave generation put forward by Troitskaya et al. (2012). Here, a fairly simple explanation of the reduction of the drag coefficient in high winds is given. It is caused by the combination of a nonlinear wind-input source function and a sea state–dependent background roughness length as explained below.

*z*, which is similar to the Charnock relation but with a small value of the Charnock parameter. Thus,

_{B}*α*= 0.01. Later work by Jean Bidlot (ECMWF 2019) suggested a smaller value of

_{B}*α*= 0.0065.

_{B}This parameterization of the background roughness length has been used in the operational WAM model since the end of 1991 and has produced good agreement with the most recent parameterizations of the observed surface drag coefficient *C _{D}* (see Edson et al. 2013). Nevertheless, in spite of the success of this approach, it is valid to ask why the roughness length of the gravity–capillary waves would scale with the square of the friction velocity divided by acceleration of gravity

*g*and whether

*α*is a constant.

Here, it is suggested to calculate the background roughness length using a model for the gravity–capillary wave spectrum called Verification and Interpretation of *ERS-1* (VIERS-1). This model was proposed by Janssen et al. (1998), and it gives the spectrum of gravity–capillary waves as follows from the balance of a nonlinear wind-input source function, dissipation by wave breaking, and three- and four-wave nonlinear interactions. The surface stress then results from an iterative solution of the stress balance at the surface, as given in Eq. (1). An iterative solution is required because viscous stress and the low- and high-frequency stress all depend in a nonlinear way on the surface stress *τ _{a}*. If the surface stress is known, it is possible to obtain quantities such as the background roughness length and the drag coefficient, and we will study the dependence of these quantities on wind speed

*U*

_{10}in the range 1 ≤

*U*

_{10}≤ 80 m s

^{−1}and on the wave-age parameter

*c*is the phase speed of the peak of the wave spectrum. Regarding the validity of the background roughness length as proposed by Janssen (1991), it is found that for intermediate and older stages of development (

_{p}*χ*> 15) Eq. (2) gives order of magnitude agreement with the explicit calculation. However, there is a significant disagreement for extreme sea states being very young (

*χ*< 10). In that event, the parameterization of the background roughness [Eq. (2)] is seen to overestimate the roughness. In particular for hurricane cases, where young wind seas are prevalent, the difference in roughness length is important, because the new formulation of the stress over the oceans (including the nonlinear form of the wind-input source function) gives much smaller drag and therefore much larger wind speeds.

The program of this paper is as follows: In section 2, a brief summary is given of the extension of the coupling approach to the case of two-dimensional propagation. This work is based on the critical layer theory of Miles (1957), and the key result is that the growth rate of the waves by wind is proportional to the curvature in the mean wind profile at the critical height, while the slowing down of the wind, resulting in the drag, is found to depend explicitly on the wave spectrum. In 1D, there is only one contribution to the slowing down of the wind, which conserves mean vorticity, while in 2D, there is an additional contribution that conserves mean momentum. In the steady state, and including effects of turbulence on the mean flow, one may evaluate how the curvature of the mean wind depends on the wave spectrum. Following Miles (1965), this gives the nonlinear dependence of the growth rate of the waves by wind on the wave spectrum.

This is followed in section 3 by a brief presentation of the VIERS model (Janssen et al. 1998), which assumes that the gravity wave spectrum, obtained by a wave-prediction system, is given, while the gravity–capillary wave spectrum is obtained from the steady-state version of the energy balance equation using the “local” (in wavenumber space) approximation for the three- and four-wave interactions, while the wind-input source function is given in Janssen (1991), with additional nonlinear effects found in section 2. Then, from the gravity wave spectrum, one obtains the wave-induced stress of the gravity waves *τ _{w}*

_{,lf}, while the gravity–capillary spectrum gives the high-frequency stress

*τ*

_{w}_{,hf}, and the viscous stress at the surface may be evaluated as well. Then, as already mentioned, the total stress

*τ*is obtained by means of iteration. Note that in every step of the iteration process, one solves for the gravity–capillary spectrum, which depends on the surface stress (which varies during the iteration process) and the fixed energy flux from the gravity waves. In this fashion, the stress relation at the surface can be satisfied, because it can be shown empirically that the iteration process converges.

_{a}In section 4, a number of properties of the new model for surface stress are presented, and the key result is a “climatological” relation between the drag coefficient *C _{D}* and wind speed

*U*

_{10}. It is shown that up to a wind speed of about 23 m s

^{−1}, the drag coefficient increases with wind speed, which is then followed by a maximum in drag at about 30 m s

^{−1}, while for higher winds, a decrease of drag coefficient with wind speed is found. Below the wind speed of 23 m s

^{−1}, there is a good agreement with the observed wind speed dependence of the drag as obtained by Edson et al. (2013). The model results are also in qualitative agreement with observations of decreasing drag with wind speed for hurricane conditions. It is also shown how the wave-induced stress of the gravity waves and of the gravity–capillary waves depend on wind speed. Owing to the combination of the quasi-linear effect and nonlinearity, the momentum transfer to the high-frequency gravity–capillary waves is quenched for high winds when the waves are steep. As a consequence, for high winds, i.e.,

*U*

_{10}> 23 m s

^{−1}, the wave-induced stress to the gravity–capillary waves

*τ*

_{w}_{,hf}vanishes. This has important consequences for the dependence of the drag coefficient on wind speed. Also, when

*τ*

_{w}_{,hf}vanishes, it is possible to find an expression for the surface stress, as according to Eq. (1), ignoring the viscous stress, the surface stress equals the low-frequency wave-induced stress. Using a simple expression for the wave spectrum and the wind-input term, an approximate expression for the wave-induced stress is obtained, which helps explain under which circumstances

*C*decreases with increasing wind speed. At around a wind speed of 30 m s

_{D}^{−1}, the drag coefficient attains a maximum, and above that, wind speed nonlinearity becomes so strong that the wind input and the low-frequency wave-induced stress are reduced to a considerable extent. As a consequence, a reduction of the drag coefficient for increasing wind speed is found, in qualitative agreement with recent findings from observation campaigns (e.g., Powell et al. 2003; Powell 2007; Jarosz et al. 2007; Holthuijsen et al. 2012). Laboratory experiments also provide information on the drag for high wind speeds nowadays, but these data suggest that for strong winds, there is a saturation in the drag. However, one needs to be careful using these results in a geophysical context because gravity wave spectra in a tank have a much higher peak frequency than in the field. Nevertheless, the revised estimates of ocean surface drag by Curic and Haus (2020) show good agreement with the empirical relation suggested by Edson et al. (2013).

Donelan (2018) tries to explain the reduction in drag in terms of flow separation that occurs just after the crest of a steep surface gravity wave. Simulations of turbulent flow over steep, nonbreaking surface waves by, for example, Sullivan et al. (2000) also suggest the presence of flow separation at the crest and the sheltering of the troughs when the critical layer is close to the surface. Clearly, sheltering is a nonlinear effect. In the present paper, we concentrate on the critical-layer mechanism, and it will be seen that the nonlinear wind input also plays an important role in reducing the drag. Following Hara and Belcher (2002), it is tempting to interpret the reduction of the wind input to the short waves in terms of a sheltering of the short waves by the long waves since waves remove momentum from the wind, thus leaving a smaller stress to force the growth of the short waves.

This section is concluded with a brief discussion on the choice of a number of tuning parameters in the model. For example, results for mean square slope are compared with observations from Cox and Munk (1954) in order to determine the starting wavenumber of the model for the gravity–capillary wave spectrum.

In section 5, we will discuss the work done by introducing the new model for surface stress into a single gridpoint version of the EMWF version of the WAM model (ecWAM). To speed up the routine for the calculations of the background roughness, an approximation to the gravity–capillary spectrum is introduced, which ignores direct effects of wind input and dissipation. This so-called inertial subrange spectrum provides an accurate approximation of the wave-induced stress associated with the gravity–capillary waves. But quantities such as the mean square slope are less-accurately represented. Regarding the model results it is noted that, because of the introduction of the effects of surface tension and viscosity, the updated ecWAM model might deviate from the “classical” scaling relations proposed by Kitaigorodskii (1962). In these scaling relations, the relevant parameters such as wave energy, peak period, wave age, duration, fetch, etc. are made dimensionless by means of acceleration of gravity *g* and the friction velocity

In section 6, our conclusions are summarized. The present operational version of ecWAM (CY47R1) has an empirical sharp reduction of the background roughness parameter for winds above 33 m s^{−1} (ECMWF 2020), which we hope to replace with this new development. Also, since CY46R1, the input and dissipation source terms are based on those of Ardhuin et al. (2010), whereas the wind-input source function is based on Janssen (1991), except for the partial sheltering effect. Since the nonlinear wind-input term represents directly the effects of sheltering of the short waves by the long waves, it is hoped to replace the partial sheltering effect of Ardhuin et al. (2010) with the new nonlinear wind input.

A preliminary account of this work has been published as an ECMWF technical memorandum by Janssen and Bidlot (2021). Although the main threads are very similar, certain details have changed, in particular regarding the choice of the angular distribution and certain details of the wind input to the short waves.

## 2. Wind–wave interaction

Wind–wave interaction involves two aspects. First of all, waves are generated by wind, and we shall adopt for convenience a particular wind–wave generation mechanism, namely, Miles’ critical-layer theory, which was originally formulated as a theory for the case of one-dimensional propagation. In Miles (1957), it is found that in the so-called quasi-laminar approach, the critical height, the height where wind speed *U*_{0}(*z*) equals the phase speed *c* = *ω*/*k*, plays a very important role in the generation process. It turns out that the growth rate *γ* of the ocean waves by wind is proportional to the ratio of the curvature of the wind speed profile divided by its shear, both evaluated at the critical height. Hence, the momentum transfer from wind to waves is an example of the resonant interaction between wind and waves. The second aspect, the feedback of the growing waves on the wind, is discussed extensively by Janssen (1982, 1989, 2004). The resonant interaction process conserves momentum; while the waves are growing, they will extract considerable amounts of momentum from the air, resulting in a slowing down of the airflow. As a consequence, the drag of airflow over wind waves depends on the sea state because young, steep wind waves will extract much more momentum from the airflow than old, gentle wind waves.

We revisit the case of two-dimensional propagation that was only partially discussed in Janssen (1991). It will be seen that the expression for the growth of ocean waves by wind is very similar to the one-dimensional result, so that the growth rate is proportional to the curvature in the wind profile. However, the mean flow equation has an additional term because the wave-induced stress

*c*propagating in the direction

*θ*grow when the curvature in the wind, directed along the

*x*axis with profile

*U*

_{0}(

*z*), is negative at the critical height

*z*. Introducing the Doppler-shifted velocity

_{c}*W*=

*U*

_{0}(

*z*) cos

*θ*−

*c*, the critical height now follows from the condition

*W*= 0, or

*c*=

*U*

_{0}(

*z*) cos

_{c}*θ*. The growth rate of the waves by wind is essentially the same as in two dimensions. However, one needs to replace

*F*(

*k*,

*θ*) due to wind becomes

*ϵ*is the air–water density ratio, which is assumed to be small. Clearly, the growth rate

*γ*of ocean waves by wind is proportional to the curvature

*z*=

*z*(Miles 1957). Here, the wave-induced velocity

_{c}*χ*satisfies the Rayleigh equation,

According to Fabrikant (1976) and Janssen (1982), wave growth results in a slowing down of the airflow. For one-dimensional propagation, the ocean waves exert a wave-induced stress *F*(*k*, *θ*). This part of the wave-induced stress leads to vorticity diffusion. In the case of two-dimensional wave propagation, there is an additional contribution to the *x* component of the wave-induced stress, which is connected to momentum diffusion. This contribution influences the mean flow but still conserves total momentum so that it is not directly connected to wave growth. Furthermore, in the 2D case there is, in principle, also a cross stress

*x*and

*y*components of the wind by

*U*

_{0}and

*V*

_{0}and suppose that at the present time the mean flow is in the

*x*direction. The mean flow equations in the absence of turbulence and viscous effects are according to the appendix,

*υ*=

_{g}*c*/2) these coefficients assume the simple form

*θ*and not over wavenumber

*k*, because the integration over wavenumber can easily be performed, as the resonance condition

*W*= 0 is reflected by a Dirac

*δ*function with argument

*W*. As a consequence, wavenumber

*k*and angular frequency

*ω*are expressed in terms of the vertical coordinate

*z*through the resonance condition

*W*=

*U*

_{0}cos

*θ*−

*c*= 0 where the phase speed follows from the dispersion relation of free gravity waves. Assuming that the

*x*component of the wind velocity is positive, and for positive

*c*, the resonance condition implies a restriction on the values of the angle

*θ*, i.e., |

*θ*| ≤

*π*/2. This restriction reduces the

*θ*integration domain to those waves that have a projection onto the wind direction.

*y*component of the mean velocity becomes

*D*vanishes, and no

_{c}*y*component of the mean flow is generated; Eq. (7) is ignored.

Let us now discuss the consequences of the two-dimensional version of the wind–wave coupling a bit further. Compared to the one-dimensional version given in Janssen (1982), it is seen that the first term on the right-hand side of Eq. (5) is new. This term is connected to the *y* component of the wave-induced velocity and is therefore connected to vortex stretching as the mean flow vortex is in the *y* direction. This term will try to maintain the curvature in the wind profile and therefore it increases wave growth.

This has important consequences for the equilibrium conditions. In one dimension, in the absence of turbulence and viscosity, the mean flow would evolve toward a condition where the curvature of the wind profile vanishes (Janssen 1982); hence, wind–wave growth vanishes for large times. This is clearly not the case for the two-dimensional problem, as now the rate of change of the wind velocity is proportional to a linear combination of the shear and the curvature of the wind profile; therefore, wind–wave growth does not necessarily vanish for large times.

Now, the growth rate of the waves by wind is still proportional to the curvature of the wind profile. And therefore, the second term on the right-hand side of Eq. (5) is expected to be directly related to the process of the momentum transfer from wind to waves. This is indeed the case as the vertical integral from *z* = 0 to *z* = ∞ of the first term vanishes because the diffusion coefficient *D*_{⊥} vanishes at the boundaries; therefore, the first term conserves momentum. However, the vertical integral of the second term is finite and gives the total rate of change of airflow momentum. Finally, although the first term does not affect the momentum budget, it still is important because it may locally affect the curvature of the mean flow and the growth rate of a particular wave.

*x*direction,

*x*component of the wave momentum

*P*. Similarly, one finds for the

_{x}*y*momentum the conversation law

*y*direction does not change because the

*y*component of the wave momentum vanishes. Here, wave momentum

**P**, is defined as

*τ**equals the rate of change of wave momentum due to the wind, or,*

_{w}*γ*is the wind-induced growth rate given in Eq. (3). Therefore, knowing the wave momentum and knowing the growth rate of ocean waves by wind, one may immediately obtain the wave-induced stress exerted on the mean airflow.

*x*-momentum balance. In the steady state, the

*x*-momentum balance becomes

*κ*is the von Kármán constant and

*z*is a background roughness length that represents the slowing down of the airflow by additional effects such as the growth of gravity–capillary waves. This process will be discussed in more detail in the next section.

_{B}*τ*is given in Eq. (11). The above relation will play a key role in subsequent developments of the coupling between wind and waves.

_{w}The interaction of wind and waves is now described by the set of Eqs. (3), (4), (6), (11), and (12), while the surface stress is given by Eq. (13). The above set of equations shows that there is a strong two-way interaction between wind and waves. While the waves grow, they extract considerable amounts of momentum from the airflow, resulting in a slowing down of the wind as the wind profile is forced by the waves toward an almost-linear wind profile. This change in wind profile, which mainly occurs close to the water surface, corresponds to an increase in roughness length and a reduction in the high-frequency part of the wind input and the wave-induced stress.

The strong interaction of wind and waves was studied by Janssen (1989) for the one-dimensional version of the above quasi-linear set of equations (i.e., for vanishing *D*_{⊥}). The properties of the one-dimensional coupling were explored by searching for a given spectral shape for steady-state solutions of the airflow over wind waves by means of an iteration method. The rate of convergence of this procedure was judged using Eq. (13), i.e., by checking how close the total stress *τ _{υ}* +

*τ*

_{turb}+

*τ*was to the surface stress

_{w}*α*was assumed to depend in a sensitive manner on the wave age

_{p}*χ*= 5) are steep, while, on the other hand, old wind waves (

*χ*= 25) are gentle, corresponding to a smooth sea state.

The results of the iteration process are presented in Janssen (1989); for a summary see Janssen and Bidlot (2018). In the present context, the most important result is the impact of the sea state on the wind profile. Young waves have a large roughness, giving a considerable slowing down of the wind, and therefore the equilibrium wind is quite reduced compared to the case of old wind sea, for which the airflow is much smoother. However, the shape of the wind profile away from the surface is still logarithmic, but close to the surface there are deviations from the logarithmic wind profile, which are a reflection of the impact of growing waves on the wind. In particular, it was shown in Janssen (1989) that for young wind sea, the wave-induced stress dominates the total stress near the surface, giving an additional slowing down of the wind, and hence a rougher airflow. As a consequence, the drag coefficient, defined as *α _{p}* has a much less-sensitive dependence on the wave age, e.g.,

*α*∼

_{p}*χ*

^{−2/3}, the drag is virtually independent of the sea state.

### a. Parameterization of quasi-linear theory

*τ*

_{turb}+

*τ*=

_{w}*τ*, or explicitly with

_{a}*l*=

*κ*(

*z*+

*z*),

_{B}^{1}

*z*, but in two dimensions, it only holds for

*z*= 0, so we will only use the stress balance relation at the surface. In the next step, we use the fit of the wind profile to the numerical data of Janssen (1989),

*z*

_{0},

*α*the Charnock parameter. Here,

*τ*(0) at the surface is obtained from the wave model.

_{w}Another advantage of using the logarithmic wind profile is that it provides a simple parameterization of the wave growth by wind. To obtain the growth rate *γ*, one needs to solve the Rayleigh equation, which cannot be solved exactly. Instead, we use as a starting point an approximate expression for the growth rate that has been obtained by Miles (1993) by means of asymptotic matching, a result that also holds for two-dimensional propagation (see the appendix).

*θ*the wave propagation direction and

*ϕ*the wind direction, the main result for the growth rate in two spatial dimensions is

*β*is given by

*γ*= 0.5771, Euler’s constant. Here,

_{E}*y*=

_{c}*k*(

*z*

_{0}+

*z*) is the dimensionless critical height and

_{c}*ϵ*is the air–water density ratio. This expression is valid for slow waves only, so in order to have a reasonable approximation also for the long waves, parameters were rescaled by replacing

*λ*= 0.281 by

*λ*= 1 and by replacing

*π*by the factor 1.2. In addition, in the formula for the critical height, the parameter

*z*= 0.008. As a result, the following parameterization for the Miles’ parameter

_{α}*β*is used,

*β*

_{max}= 1.2, and

*z*

_{0}will be large, resulting in large dimensionless critical height

*y*, easily reaching values

_{c}*y*= 1 for the short waves with large wavenumber

_{c}*k*. In other words, the Miles’ parameter

*β*, and therefore the growth rate

*γ*, will vanish for these short waves, giving a finite wave-induced stress.

*C*of the coupled ocean–wave atmosphere system (IFS–ecWAM with the feedback of the waves on the airflow switched on) has been performed and reported in Janssen and Bidlot (2018). Two examples were discussed in particular. First, the modeled sea state–dependent drag at a height of half the peak wavelength was compared to a parameterization proposed by Hwang (2005), which is based on observations from a number of field campaigns, and a good agreement was found. It is of the form

_{D}*λ*is the peak wavelength, and the wave age

_{p}*χ*is defined as

*c*the phase speed of the peak of the spectrum. The constants

_{p}*A*and

*a*are given by

*A*= 1.220 × 10

^{−2}and

*a*= −0.704. Second, the drag coefficient at 10-m height as function of wind speed

*U*

_{10}was validated against an empirical fit of Edson et al. (2013), obtained from eddy correlation data for the COARE 4.0 parameterization of the drag. This empirical fit is valid up to a wind speed of about 23 m s

^{−1}, and on average there is a good agreement between modeled drag and observed drag.

As mentioned in the introduction, an important assumption in the present approach is that the effect of the short gravity–capillary waves is represented by a background roughness length as given by Eq. (2). The validity of this assumption needs to be tested, and for this reason, one needs to give an explicit calculation of the momentum transfer to the short gravity–capillary waves. An attempt to explicitly calculate the momentum transfer to the short waves is given in the next section, using a model for the short gravity–capillary waves, called the VIERS model (Janssen et al. 1998). Before we do this, we have to discuss the inclusion of nonlinear effects in the expression of the growth rate of waves by wind.

### b. Full nonlinear theory

In the previous section, it has been mentioned that the original approach results in drag coefficients that are in good agreement with well-known parameterizations of drag against wind speed or wave age that are obtained from observation campaigns. These observational fits are restricted to wind speeds *U*_{10} that are less than about 23 m s^{−1}. Despite this good agreement, one may question a number of assumptions underlying the original approach. Here, we discuss the validity of the assumption that the wind profile has a logarithmic shape.

The coupled equations, in particular the stress balance [Eq. (12)], show that there is a strong interaction between wind and waves. Clearly, the resulting slowing down of the wind is a nonlinear effect, because its impact depends on the angular average of the wave spectrum. Therefore, one would expect that the growth rate of the waves by wind depends on the wave spectrum as well, and as a consequence, there would be deviations from the logarithmic wind profile. In Eq. (A4) a brief, approximate derivation of the sea state dependence on the growth rate *γ* is given. A similar result was already obtained by Miles (1965), and it has been utilized in the VIERS model (Janssen et al. 1998; see also Caudal 2002).

*z*, and the result is substituted in the expression for the growth rate given in Eq. (3). Denoting by

_{c}*γ*

_{0}the growth rate for a logarithmic wind profile, one finds after some algebra the following simple expression for the renormalized growth rate,

*N*

_{1}and

*N*

_{2}depend on the angular average of the product of linear growth rate

*γ*

_{0}and the wavenumber spectrum

*F*(

*k*). They read,

^{2}

*θ*in the angular average in

*N*

_{1}. For typical angular dependencies in growth rates and two-dimensional spectra, the difference in size between

*N*

_{1}and

*N*

_{2}is a factor of 6; hence,

*N*

_{1}=

*N*

_{2}/6. The growth rate

*γ*

_{0}is the one according to linear theory with a logarithmic wind profile. For practical applications, we will use the parameterization for

*γ*

_{0}given in Eqs. (18), (20), and (21) with a first-guess value of the parameter

*β*

_{max}of 1.2.

*F*(

*k*). For example,

*N*

_{2}becomes

*, measuring the size of angular effects, is guided by the following consideration. Assume that the wave spectrum has a cos*

_{γ}^{2}directional distribution. Hence,

^{2}

*θ*, the directional terms give rise to a correction factor,

*= 3/4. Estimating the magnitude of*

_{γ}*N*

_{1}involves an additional factor sin

^{2}

*θ*. The relevant integral equals 1/8 or

*N*

_{1}=

*N*

_{2}/6. Since

*N*

_{1}<

*N*

_{2}, nonlinear effects reduce the growth rate.

Using the approximations for *N*_{1} and *N*_{2}, we have plotted in Fig. 1 for two wind speeds the impact of nonlinear corrections on the growth rate as function of wavenumber *k* by comparing the linear normalized growth rate *γ*_{0}/*ω* with the nonlinear normalized growth *γ*/*ω*, where *γ* is given by Eq. (23), and the renormalization factors *N*_{2} are given in Eq. (24), while *N*_{1} = *N*_{2}/6. The relevant solutions for the drag coefficient *C _{D}* and the friction velocity

*k*> 1. As according to critical-layer theory, there is a direct correspondence between height

*z*and wavenumber

*k*through the resonance condition, it immediately follows that only the wind profile close to the surface deviates from the logarithmic profile due to nonlinear effects. Furthermore, Fig. 1 also illustrates that the nonlinear effect may be quite important for strong winds on the order of 50 m s

^{−1}, but it only plays a small role for “moderate” wind speeds on the order of 10–20 m s

^{−1}. This suggests that for relatively low wind speeds, the use of the logarithmic wind profile [Eq. (16)] is justified. The original approach seems to be valid for wind speeds less than about 20–25 m s

^{−1}.

## 3. A model for the short waves

The model for the short waves is based on work done by the VIERS-1 group in the 1990s. The main objective of this group was to obtain a physics-based model for the radar backscatter, and one of the main tasks was to obtain a model of the short gravity–capillary waves, because these short waves give an important contribution to the radar backscatter through the Bragg scatter mechanism. A detailed description of the short wave model is given in Janssen et al. (1998), and therefore we will suffice with a brief explanation, discussing mainly deviations from the original model, which solves the one-dimensional energy balance equation in wavenumber space. For directional aspects we refer to Caudal (2002).

*S*

_{in}represents the input of wind to waves,

*S*

_{nonl}describes three- and four-wave interactions,

*S*

_{visc}describes viscous dissipation,

*S*

_{br}describes dissipation due to whitecapping, and

*S*

_{slicks}describes the resonant energy transfer between surface waves and slicks (Marangoni effect). The energy balance equation is solved as a boundary value problem in wavenumber space by providing the energy flux from the long to the short waves at a boundary

*k*=

*k*

_{3}

*, which is essentially the wavenumber where three-wave interactions start to become important.*

_{w}*k*=

*k*

_{3}

*, knowledge of the gravity part of the wave spectrum is required. Here in the model development phase, JONSWAP spectra in wavenumber space are used, while later in the paper, modeled gravity wave spectra will be used. Assuming that the boundary is in the tail of the spectrum because*

_{w}*k*

_{3}

*is at a sufficiently high wavenumber, the spectrum*

_{w}*F*(

*k*) at the boundary is given by the Phillips spectrum,

*α*, which in the modeling phase depends on wave age

_{p}*χ*according to the scaling relation,

*A*= 0.24 and

*B*= 1. This choice of parameters is in fair agreement with the reanalysis of JONSWAP data performed by Günther (1981). Note that JONSWAP observations were obtained at fairly modest wind speed observations on the order of 10 m s

^{−1}, and as a consequence, the sea state was usually fairly old. The above scaling law suggests that the Phillips parameter would continue to increase for decreasing wave age

*χ*, but since waves have a limiting steepness, it seems likely that also the Phillips parameter is limited. This saturation behavior of the Phillips parameter has been modeled by a tanh profile so that,

*α*is also used to limit the evolution of the wave spectrum when integrating the wave model in time (ECMWF 2020). The appropriateness of the choice of maximum value for

_{p}*α*has been extensively tested at ECMWF by Jean Bidlot and by Jean-Michel Léfèvre and Lotfi Aouf at Meteo-France during the MyWave project. It is remarked that the choice for a limiting steepness and Phillips’ parameter will have important consequences for the behavior of the surface stress for young wind seas.

For wavenumbers higher than *k*_{3}* _{w}*, a new regime is entered because three-wave interactions start to play a role in the steady-state energy balance equation. In the following, we shall only discuss a theory for the one-dimensional wavenumber spectrum, while, if needed, effects of the angular distribution of the short waves are provided in a fairly simple fashion.

*F*(

*k*), which is related to the Fourier transform of the autocorrelation of the surface elevation

*η*, is normalized in such a way that,

*η*

^{2}⟩ is the wave variance. The wave energy

*E*, apart from a factor

*ρ*, then follows from,

_{w}*g*is acceleration of gravity and

*T*is surface tension.

### a. Wind-input source function

*γ*is given by the one-dimensional version of Eq. (23). After some rearrangement, one finds,

*α*

_{1}=

*α*

_{2}/6. Furthermore,

*β*is given by Eqs. (20) and (21). The form for the growth rate is also assumed to be valid for gravity–capillary waves. We have seen that for relatively low wind speeds (

*U*

_{10}< 20) nonlinear effects are small so that

*γ*≈

*γ*

_{0}. Plant (1982) compared this expression for the growth rate

*γ*of the waves with empirical data, and he found that on average the coefficient

*ϵβ*has the value of 26. However, in the context of quasi-linear theory, the parameter

*β*is not a constant [see, e.g., Eq. (20)], and it even vanishes when the dimensionless critical height

*y*is equal to or larger than 1. This typically occurs for the short waves.

_{c}*P*=

*ρ*(

_{w}ω**k**)

*F*(

**k**), while the growth rate

*γ*is given by Eq. (33). The wave-induced stress is evaluated for the Phillips spectrum,

*k*

^{−4}spectral tail. In keeping with the vanishing of the growth rate at

*y*=

_{c}*k*(

_{c}*z*

_{0}+

*z*) = 1, a wavenumber cutoff

_{c}*k*=

*k*is introduced, and the growth parameter

_{c}*β*is assumed to be a finite constant in the range

*k*<

_{p}*k*<

*k*, while it vanishes outside that range. Note, however, that this crude approximation of the quasi-linear effect may affect the wave-induced stress value. For this reason, we therefore have chosen an average value of

_{c}*β*, which deviates somewhat from its typical value of 26, namely,

*β*= 27.5.

*τ*,

_{a}*ω*and

_{p}*ω*are the peak and cut-off angular frequencies corresponding to the peak wavenumber

_{c}*k*and the cut-off wavenumber

_{p}*k*. By studying the integrand of Eq. (36) for large

_{c}*ω*, it is clear that it is essential to have a wavenumber cut off, because otherwise there would be a logarithmic singularity for the Phillips spectrum. In other words, for gravity waves, the quasi-linear effect is essential for obtaining a finite answer for the strength of the coupling between wind and waves. This latter statement is, however, not true when capillary effects are taken into account.

*. Assuming that the wind blows in the*

_{ϕ}*x*direction, the wind-input term then involves an additional factor of cos

^{2}

*θ*, where

*θ*is the propagation direction. In addition, assuming that the wave spectrum is symmetrical with respect to the wind direction, only the

*x*component of the wave-induced stress is finite, while, because of symmetry, the cross component vanishes. The total stress involves an additional factor cos

*θ*. Finally, in order to get an idea about the importance of directional effects, a cos

^{2}directional distribution for the wave spectrum is assumed. Then, with

*N*=

*π*/2, the correction factor Δ

*becomes*

_{ϕ}*was used as a tuning parameter to ensure that for the lower wind speeds, i.e.,*

_{ϕ}*U*

_{10}< 23 m s

^{−1}, agreement with the Edson et al. (2013) empirical fit for the drag coefficient was obtained. For the long gravity waves, this resulted in an optimal value of Δ

*= 0.62, suggesting that either wind input or wave spectrum are broader than indicated above. In fact, using the updated version of the ecWAM model, to be discussed in section 5, it turns out that the optimal value of Δ*

_{ϕ}*is adequate. For the short waves, we used a correction that depended on the friction velocity,*

_{ϕ}*ω*, the normalized wave-induced stress assumes the simple form

*τ*/

_{w}*τ*is proportional to the product of the Phillips parameter and logarithms involving the cut-off angular frequency

_{a}*ω*and the angular frequency

_{c}*ω*, and hence the sea state dependence of the normalized wave-induced stress is determined by how

_{p}*α*and, say, the ratio

_{p}*ω*/

_{c}*ω*depend on the wave age

_{p}*χ*. Typically, for increasing wave age, waves become gentler. The Phillips parameter

*α*decreases while the range parameter

_{p}*ω*/

_{c}*ω*increases, since older wind sea implies longer waves and a decrease in angular peak frequency. Therefore, as already discussed in Janssen (1989), the wave-age dependence of the wave-induced stress depends on the competition of these two factors. It turns out that for the presently chosen sensitive wave-age dependence of

_{p}*α*[see Eq. (28)] the wave-induced stress will decrease with increasing wave age. The exception is for very young wave age,

_{p}*χ*<

*A*/

*α*

_{max}, when the Phillips parameter approaches the constant value

*α*

_{max}. Under these circumstances, the normalized wave-induced stress will increase with increasing wave age. In other words, the wave-induced stress will attain a maximum for wave ages on the order of 5–10. As will be evident in section 4, this special behavior of the wave-induced stress will have profound consequences for the dependence of the drag on wind speed in hurricane conditions.

In the original VIERS model, a slightly different input source function was used. It had a similar form as in Eq. (33) but with *α*_{1} vanishing, because at that time it was not realized that for two-dimensional propagation there was an extra contribution to maintain the curvature in the wind profile.

### b. Nonlinear interactions

*k*), one thus has

*k*) reads,

*υ*is the group velocity ∂

_{g}*ω*/∂

*k*,

*B*is the angular average of the degree of saturation (Phillips 1985),

*α*

_{3}and

*α*

_{4}give the strength of the three- and four-wave interactions, respectively. The coefficients

*α*

_{3}and

*α*

_{4}may still depend on the ratio

*c*/

*υ*. In particular,

_{g}*α*

_{3}should vanish in the gravity wave regime because three-wave interactions are not possible there. For this reason,

*k*

_{3}

*is chosen in such a way that it is connected to the minimum in the phase velocity*

_{w}*c*=

*ω*/

*k*. This minimum occurs at

*y*is typically less than one. A satisfactory choice that was tried is

*y*= 1/2, but a more refined choice was proposed by Janssen and Wallbrink (1997), who made improvements to the original VIERS model with the aim to obtain better agreement between observed and simulated radar backscatter

*σ*

_{0}. They found a better agreement when the starting wavenumber was chosen to depend on the friction velocity

*y*now depends on the friction velocity in such a way that in agreement with the observations of Jähne and Riemer (1990) [and of Donelan and Plant (2009)] the gravity–capillary spectrum extends over a wider wavenumber range for stronger winds.

In the VIERS model, three dissipative processes are assumed to play a role in the gravity–capillary regime, namely, viscous dissipation, wave breaking, and damping due to slicks. These processes have been described in some detail by Janssen et al. (1998) and will be denoted in this paper by *S*_{diss} = −*γ _{d}F*(

*k*). In the numerical experiments discussed here, effects of damping due to slicks will be ignored.

### c. Exact solution of short-wave energy balance

*k*), as given by Eq. (39), is a function of the degree of saturation

*B*(

*k*). Since in practice the degree of saturation

*B*is on the order 0.1 or less, it is a fair approximation to disregard four-wave interactions in the expression for the energy flux. Retaining therefore only three-wave interactions, the energy balance Eq. (43) may be solved exactly,

^{2}and the result for the degree of saturation becomes

_{0}is the value of the energy flux at

*k*=

*k*

_{3}

*. It is of interest to discuss the terms in Eq. (45) separately. The first term is related to the effect of three-wave interactions. In the absence of wind input and dissipation, it follows from the condition of a constant energy flux in wavenumber space. The resulting spectrum is called the inertial subrange spectrum. Using the dispersion relation for pure gravity–capillary waves, the degree of saturation according to the constant energy flux condition becomes*

_{w}*y*=

*k*/

*k*

_{0},

*k*

_{0}= (

*g*/

*T*)

^{1/2}is the wavenumber that separates gravity waves and capillary waves, and

*c*

_{0}= (

*gT*)

^{1/4}(note that the minimum phase speed equals

*k*<

*k*

_{0}), the degree of saturation increases with wavenumber like

*k*

^{3/4}, while in the capillary range,

*B*

_{3}

*decreases with wavenumber like*

_{w}*k*

^{−3/4}, and it attains its maximum value at

*k*≈ 1.32

*k*

_{0}.

Effects of wind input and dissipation (Γ term) are represented by the second term in Eq. (45) and result in a modification of the “inertial” subrange spectrum given in Eq. (46). The degree of saturation now becomes a function of the friction velocity while, for large wavenumbers, dissipation becomes important, giving a rapid decay in the high-wavenumber range. Examples of degree of saturation spectra for different wave ages (ranging from a wave age *χ* between 5 and 25 in steps of 5) at a constant wind speed of 15 m s^{−1} are shown in Fig. 2. The kink in these spectra is at the wavenumber

## 4. Determination of the surface stress

### a. Method

*τ*or the friction velocity

_{a}*P*=

*ρ*(

_{w}ω**k**)

*F*(

**k**), while the growth rate

*γ*is given by Eqs. (23) and (25). Since now we have an explicit model for the background roughness, the turbulent stress, as given in Eq. (6), simplifies as the mixing length does not contain the background roughness; hence,

*l*(

*z*) =

*κz*, which vanishes at the surface. As a consequence, the turbulent stress vanishes at the surface so that the surface stress becomes

*τ*, the stress balance is solved by iteration. In this manner, a consistent solution for the spectrum of short and long waves is obtained, and at the same time, a consistent estimate of the stress over growing wind waves is found.

_{a}The final part of the solution procedure concerns a method to generate realistic spectra, using the JONSWAP spectrum. In the first step, a first guess of the wave age *U*_{10} is chosen. Using a constant drag coefficient *C _{D}* = 1.5 × 10

^{−3}a first guess for the friction velocity

*c*is determined. The Phillips parameter

_{p}*α*then follows from Eq. (36). The spectral parameters

_{p}*α*and, using the deep-water dispersion relation, the peak frequency

_{p}*ω*are now known so that the wave spectrum is given. During the iteration process, the wave spectrum is fixed, while the surface stress or friction velocity is updated every step until convergence is obtained. The convergence criterion is that the relative error in

_{p}^{−5}. Convergence is always achieved but may require on the order of 100 iterations.

### b. First results and comparison with approximate C_{D}

First results of the degree of saturation spectrum for the case of a wind speed of 15 m s^{−1} and wave ages ranging from *χ* = 5 to *χ* = 25 are shown in Fig. 2. The short-wave spectra are in qualitative agreement with the observations obtained by Jähne and Riemer (1990) [see for a later reference Donelan (2018)]. For an illuminating discussion of this comparison, please consult Caudal (2002).

The kink in these spectra is at wavenumber *k* = 1.3*k*_{0}, after which it rapidly decays to zero. Note that if the degree of saturation spectrum is a constant, then, as pointed out in section 3a, the wave-induced stress has a logarithmic singularity, unless the quasi-linear effect is taken into account. However, for gravity–capillary waves, there is, as is clear from Fig. 2, not such a catastrophe, as the degree of saturation spectrum vanishes for large wavenumbers, so even in the linear approximation, a finite wave-induced stress results. Nevertheless, the quasi-linear effect will be seen to play a vital role in limiting the drag for high wind speeds.

In Fig. 3, the wave-age dependence of the drag coefficient *C _{D}* (

*z*= 10) is shown for a wind speed of 15 m s

^{−1}. The graph shows that for old wind sea (

*χ*≈ 25), the drag coefficient is low, and it increases with decreasing wave age until a maximum of 2.25 × 10

^{−3}is reached for

*χ*≈ 7. The reason that a maximum in drag coefficient occurs is, as already discussed below [Eq. (37)], connected to the assumption that surface gravity waves have a limiting steepness for extremely young wave ages. For completeness, in the right panel of Fig. 3, the dependence of the different stress components of the momentum balance on wave age is also shown. For the relative high wind speed of 15 m s

^{−1}, the viscous stress plays a minor role in the stress balance at the surface. The low-frequency part of the wave-induced stress

*τ*

_{w}_{,lf}/

*τ*always gives a substantial contribution to the total stress, but this is clearly not the case for its high-frequency part, which gives a small contribution in the wave-age range from 3 to 11. The high-frequency stress vanishes because the cut-off wavenumber

_{a}*k*is below

_{c}*k*

_{3}

*; hence, there is no wind input to the gravity–capillary waves. In other words, in that wave-age range, the stress is completely determined by the low-frequency waves. In that event, we have*

_{w}*τ*=

_{a}*τ*

_{w}_{,lf}so that it makes sense to compare the results from the iteration scheme with the simple expression for the low-frequency wave-induced stress given by Eq. (37). Indeed, a comparison of the analytical result for the drag coefficient

*τ*

_{w}_{,hf}is represented by the turbulent stress

*τ*

_{turb}at the surface, where the mixing length is given by

*l*(0) =

*κz*. For the logarithmic profile Eq. (16), one then finds the simple relation,

_{B}^{−1}a fair agreement with the results of the explicit model for the background roughness is found. However, for higher wind speeds, the reduction of wind–wave growth at high frequencies is so large that a considerable dependence of the background roughness length on wave age is noted. As a consequence, the drag of airflow over really young wind waves is much less than anticipated from the present operational system. This has important consequences in particular for extreme conditions such as experienced in hurricane cases, as will be discussed in some more detail below. Finally, for wind speeds lower than 10 m s

^{−1}, the background roughness length is considerably larger than the operational choice of 0.0065, and, as a consequence, the new formulation should have impact in particular in the tropics where low wind speeds prevail.

### c. Climatology

*χ*and wind speed

*U*

_{10}. In the wind speed range of 0 <

*U*

_{10}< 25, it is approximately given by

The surface stress according to the present approach is now obtained using the JONSWAP spectrum with wave age given by Eq. (49), and the climatological relation between drag coefficient and wind speed is presented in Fig. 5. It is seen that for low wind speeds, up to *U*_{10} = 25 m s^{−1}, the drag coefficient increases with wind speed in close agreement with a fit of mean drag coefficient versus wind speed found by Hersbach [cf. Edson et al. (2013)]. This “empirical” fit is only valid for wind speeds less than 23 m s^{−1}. For larger winds, there are not enough reliable data yet (cf. Powell et al. 2003; Jarosz et al. 2007; Powell 2007; Holthuijsen et al. 2012), but these observations do suggest that the drag coefficient saturates and starts to decrease from *U*_{10} = 30 m s^{−1} to *U*_{10} = −35 m s^{−1} onward. The present model calculations seem to confirm this picture, in agreement with the analytical result [Eq. (37)].

Note that the analytical formula is valid for the case that the low-frequency wave-induced stress is dominant in the stress balance. This is indeed the case in the wind speed range 20–40 m s^{−1} as follows from Fig. 6, which shows the average Charnock parameter, the background Charnock parameter, and the viscous dimensionless length *α*_{visc}^{3} as function of wind speed. Clearly, the background roughness length becomes vanishingly small above a wind speed of around 20 m s^{−1}; hence, the low-frequency wave-induced stress dominates the stress balance for larger wind speeds up to 40 m s^{−1}.

Returning to Fig. 5, a comparison of the results of some sensitivity experiments is also shown. First, we show the climatological drag wind speed relation for the operational system, which has a constant dimensionless background roughness length equal to 0.0065. These have the label OLD. The drag coefficient starts to saturate from a wind speed of about 30 m s^{−1}. But, clearly, there are significant differences between the operational results and the present approach. The main reason is that for large winds, when the waves become steep, the momentum transfer from the airflow to the high-frequency waves is quenched, giving a considerable reduction of the background roughness and owing to the nonlinear wind input a considerable reduction of the low-frequency wave-induced stress. Furthermore, the last two experiments show that both the combination of a quasi-linear wind input and gravity–capillary roughness and the combination of nonlinear wind input and constant background roughness are not sufficient to have considerable reductions of drag at large wind. Clearly, this requires both a nonlinear wind input and gravity–capillary roughness.

Finally, it is also of interest to test the present wind–wave model for other aspects of the sea state as it could be a useful model for the interpretation of satellite remote sensing observations. For example, an altimeter estimates the mean square slope (mss) of the sea surface, and it is of interest to see how well this model for the gravity and gravity–capillary waves is performing. In Fig. 7, a comparison for low wind speeds is made between the climatological mean square slope and observations from Cox and Munk (1954). It turns out that the mss is sensitive to a number of parameters in the VIERS model, such as the starting wavenumber *k*_{3}* _{w}* of the short-wave spectrum and the strength of the three-wave interactions

*α*

_{3}. By choosing

*k*

_{3}

*according to Eq. (41), with friction-velocity-dependent parameter*

_{w}*y*, and by choosing

*α*

_{3}= 6

*π*, a reasonable agreement with the Cox and Munk (1954) observations was achieved.

It is of great interest to further validate the high-frequency part of the wave spectrum and the related (low pass) mean square slope with the wide range of satellite instruments, including altimeters, scatterometers, radiometers, and reflectometers that are nowadays available (for a summary see, e.g., Hwang and Fan 2018). This work will commence in a systematic manner as soon as the new version of the wave model becomes operational. At present, we can already be optimistic considering the reasonably good agreement between radar backscatter *σ*_{0} from Ku-band radar altimeter and the model (Abdalla and Janssen 2019). Also, preliminary work shows good agreement between low-pass mss from solar reflection and the wave model. Note, finally, that in the past wave slopes from the operational ecWAM model have been used with some success in the calibration of the Synthetic Aperture Interferometric Radar Altimeter (SIRAL) on board *CryoSat 2* and across-track ocean slope measurements (Galin et al. 2013).

## 5. Toward an operational implementation

We have developed a procedure to obtain the surface stress from the sum of stresses determined by the growth of long gravity waves, short gravity–capillary waves, and viscous effects. In this procedure, the quasi-linear wind input was replaced by the nonlinear wind-input term given by Eq. (23) using the approximation [Eq. (25)]. Now, briefly, some aspects of the operational implementation of the present air–sea interaction procedure are discussed. First, a fast approximate expression for the short-wave spectrum is introduced, and it is shown that the approximation gives accurate results for the drag coefficient and surface stress. In the next step, this approach is implemented in a version of the ecWAM model, and some of the initial results are discussed.

### a. Approximate short-wave spectrum

*y*=

*k*/

*k*

_{0},

*k*

_{0}= (

*g*/

*T*)

^{1/2},

*c*

_{0}= (

*gT*)

^{1/4}and

*k*=

*k*

_{3}

*, which depends on the degree of saturation*

_{w}*B*

_{0}=

*α*/2. Here,

_{p}*α*is the Phillips parameter of the high-frequency part of the gravity wave spectrum. The starting wavenumber

_{p}*k*

_{3}

*is given by Eq. (42).*

_{w}The climatological results for the drag coefficient using the complete wave spectrum versus the inertial subrange spectrum have been compared in detail, and a very good agreement is found. To understand this better, consider as a typical example a wind speed of 15 m s^{−1}. In Fig. 8, a comparison between the wave spectrum based on the full energy balance equation and the inertial subrange spectrum is shown. Noting that the main interest is in an accurate representation of the wave-induced stress, in the right panel the growth rate of waves by wind is shown as well. Evidently, wind input vanishes for large wavenumber (in this case *k _{c}* is just below 1000), so that the very short waves do not contribute to the wave-induced stress. Realizing this, it is clear that regarding the surface stress, there is very good agreement between inertial subrange spectrum and the complete spectrum.

It is concluded that for operational purposes it is a fair approximation to use the inertial subrange spectrum in the calculation of the wave-induced stress of the gravity–capillary waves. However, for the accuracy of mss, this may be a somewhat different matter because there is no high-wavenumber cutoff for this parameter. As shown in Fig. 7 for low wind speeds (*U*_{10} < 20 m s^{−1}), the inertial subrange spectrum gives an accurate mss, but for larger wind speeds, there are differences, although these are believed to be relatively small.

### b. Introduction into the ecWAM model

To further test the present approach, we have implemented the new scheme into an older version of the ecWAM model, which is a single gridpoint version of the wave physics introduced by Janssen (1991). This implementation was preceded by reprogramming the software regarding the determination of the high-frequency stress *τ _{w}*

_{,hf}and the stress

*τ*in the surface layer. Previously, these quantities were determined using tabulated values of

_{a}*τ*

_{w}_{,hf}and

*τ*at regular values of input parameters such as wind speed, Charnock parameter, etc. Reading from a table in memory is apparently relatively slow these days, and Bidlot (ECMWF 2019) realized that one may as well do the actual calculation on the fly. So we upgraded the single gridpoint version of the ecWAM software to allow for both iteration (for the surface stress) and the explicit calculation of

_{a}*τ*

_{w}_{,hf}and

*τ*and no slowing down of the running of the wave model was found. Having done this upgrade, it was relatively straightforward to introduce the novel framework for calculation of the surface stress, including the actual calculation of the surface roughness connected with the growth of gravity–capillary waves.

_{a}The upgraded ecWAM model, which uses the nonlinear input term [Eq. (23)] with the exact renormalization factors [Eq. (24)], has a number of parameters that need to be fixed. For the nonlinear transfer, we are using discrete interaction approximation (DIA) as implemented in ecWAM and have tuned the whitecap dissipation source term of CY45R1 with a dissipation coefficient CDIS = 1.35. The parameters for the wind-input source function are *β*_{max} = 1.3, *z _{α}* = 0.008, while the parameter

*α*was renamed as

*α*and was determined by the unresolved roughness calculation presented in this paper. Furthermore, the parameters of the numerical semi-implicit scheme were chosen as XIMP = 2 and XDELF = 5. Finally, during numerical experimentation, it became evident that the parameters determining the wind-input term of the short waves and its directional distribution required further specification. The angular distribution factor was chosen as

_{B}*λ*in the Miles parameter

*β*[see Eq. (19)] that controls the high-wavenumber cutoff was chosen to become friction-velocity dependent as well. We took

### c. Discussion of results

To study the properties of the new wave modeling system, a number of duration-limited runs were made for wind speeds ranging from 1 to 25 m s^{−1}. The duration of the runs was one day. The discussion of results is started by noting that the introduction of the effects of gravity–capillary waves into the ecWAM model may cause a breakdown of the universality of the usual scaling relations for wave growth, Charnock parameter, etc. Normally, it is possible to obtain universal scaling relations between parameters such as wave variance, peak period, wave age, etc. by scaling the relevant dimensional parameters by means of acceleration of gravity *g* and the friction velocity

The first interesting parameter to study is the dimensionless wave variance *E* = ⟨*η*^{2}⟩ is the wave variance. In Fig. 9, we have plotted *ϵ* as function of the wave-age parameter *χ* for four values of the wind speed. A fit to the numerical data gives the relation *ϵ* = 0.026*χ*^{3.32}. This result is very close to results of a reanalysis of fetch limited observations by Kahma and Calkoen as reported in Komen et al. (1996). They found, using a sea state–dependent roughness length of Donelan (1990), that *ϵ* = 0.025*χ*^{3.31}, so there is only a small difference in fitting coefficients. It is remarkable to see that for *ϵ* there is over a large range of wind speeds “universal” scaling behavior with respect to the wave age *χ* although for “older” sea states scatter in the numerical data around the mean fit between *ϵ* and *χ* increases. A similar remark applies to the dependence of the Phillips parameter *α _{p}* on the wave-age parameter, shown in the right panel of Fig. 9. According to Eq. (28), a universal relation between these two parameters exists, and this relation is plotted in Fig. 9 together with the numerical results for

*α*versus

_{p}*χ*from the ecWAM model for wind speeds equal to or larger than 10 m s

^{−1}. Although the ecWAM model gives a small overestimation of the Phillips parameter, the agreement between numerical results and the parameterization [Eq. (28)] is good. Thus, it is fair to conclude that according to the new version of the ecWAM model, the relation between Phillips’ parameter and wave age is universal. However, as will be seen in a moment, this conclusion about universality does not hold for all parameters. An example is the relation between drag coefficient and wave age.

At the same time, one may wonder whether the discrepancy between modeled and observed Phillips’ parameter can be explained by a small underestimation of the surface stress, which may then give an overestimation of the wave-age parameter, thus shifting the *α _{p}*–

*χ*relation toward larger values of

*χ*. To check this, Fig. 10 gives a plot of drag coefficient at half the peak wavelength,

*C*(

_{D}*λ*/2), as function of wave age for four different wind speeds, and the numerical results are compared with Hwang’s parameterization [cf. Eq. (22)]. Clearly, simulated drag coefficients are systematically slightly lower than the ones according to Hwang, thus explaining why the simulated Phillips parameter in Fig. 9 is slightly higher compared to the parameterization [Eq. (36)]. However, for wave ages larger than 20, agreement between numerical results and Hwang’s relation is less convincing. At fixed wave age for smaller winds

_{p}*C*(

_{D}*λ*/2) is found to be systematically larger than for stronger winds. Near a wave age of 25, the relative difference amounts to about 20%.

_{p}The above finding is confirmed by studying the dependence of the Charnock parameter *α* on the wave-age parameter. This is shown in Fig. 11, where in the left panel numerical results for the Charnock parameter are shown as a function of wave age for different wind speeds. For young wind sea, an approximate scaling behavior is found; however, for old wind sea this is clearly not the case, suggesting a breakdown in universality of the “classical” scaling relations.

For completeness, we have also displayed in the right panel of Fig. 11 the dimensionless background roughness *α _{B}* as a function of wave age for different wind speeds. In the previous version of the ecWAM model, this parameter was assumed to be a constant, and it is of interest to see to what extent this choice was a valid assumption. Comparing this with Fig. 4, it is clear that there is a good agreement between the wave model calculations of this section and the simple model used in section 4. Obviously, for typical wind speeds on the order of 10 m s

^{−1}, the background roughness is fairly constant during the evolution of the sea state, thus for lower winds, the assumption of a constant background roughness seems a valid one, although it should be pointed out that the smaller the wind speed, the larger the background roughness

*α*. In other words, in low-wind-speed regions, e.g., in the deep tropics, the new system will give rise to a larger surface drag. On the other hand, for strong winds and young wind seas, the background roughness is seen to vanish, resulting in a smoother water surface and at the same time scaling behavior.

_{B}### d. First results with a coupled atmosphere–ocean wave system

We finally briefly comment on results obtained with a coupled atmosphere (IFS)–ocean wave system using the new formulation of the air–sea interaction approach. The atmospheric component is the IFS (CY47R1), which was run with the Tco 1279 grid (effectively about 9-km resolution) and 137 layers in the vertical. The ocean wave system is ecWAM, which was run with a spatial resolution of (1/8°), about 14 km, while the spectrum had 36 frequencies (on a logarithmic grid) and 36 angular directions. The empirical reduction of the background roughness was switched off. The forecast of 22 March 2019 was chosen because a number of tropical cyclones were present in the forecast. The results that are shown in the next figures are averages over the first 3 days of the forecast using output every hour.

The average drag coefficient at 10-m height as function of the 10-m wind speed is shown in Fig. 12. For relatively low wind speeds, both the old and the new scheme show good agreement with the empirical fit obtained by Edson et al. (2013). For large winds, considerable differences between the old and new scheme are noted. As argued before, these differences are caused by the combination of the explicit modeling of the background roughness length and the nonlinear effects introduced in the wind-input term.

To show that the short-wave model produces realistic spectra, we have plotted mean square slope, integrated over all wavenumbers, as function of the 10-m wind speed in Fig. 13. For comparison purposes, we have shown the measurements of mean square slope of Cox and Munk (1954) as well, and it is clear that there is good agreement between model results and observations. Finally, when we started the development of this new wind–wave interaction scheme, we made a simplifying assumption regarding the relation between average wave age and wind speed so that we could quickly generate results. For this reason, we have shown in the right panel of Fig. 13 the average wave age as function of wind speed, and it is clear that indeed, on average, the sea state is old for low wind speeds, while it is much younger for strong winds. However, the relation Eq. (49) used in the testing of this approach has considerably younger sea state compared to the present coupled system: at *U*_{10} = 50 m s^{−1} the present results have an average wave age of about 5, while Eq. (49) suggests lower values on the order of three. This may explain why for large winds the drag coefficients in the development phase are smaller than the ones found with the present coupled atmosphere–ocean wave system.

## 6. Conclusions

In this paper, we have discussed some new results on the generation of ocean waves by wind and the feedback of growing wind waves on the airflow given a strongly coupled air–water system. First, we have extended the wind-input term, which is based on the Miles (1957) critical-layer theory by including nonlinear effects, which for strong winds give rise to a considerable reduction of the growth of surface gravity waves and of the associated wave-induced stress. This nonlinear effect is in particular important for strong winds, i.e., *U*_{10} > 25–30 m s^{−1} where, together with an explicit representation of the background roughness length, it will give rise to relatively small drag. It is emphasized that for moderate wind speeds the nonlinear effect on the wind input may be neglected.

Second, we have scrutinized the assumption of a constant dimensionless background roughness length. Originally, an explicit calculation of the slowing down of the airflow by surface gravity waves was made while the effect of the gravity–capillary waves was parameterized using a constant dimensionless background roughness length [see Eq. (2)]. But this is a simplifying assumption that needs to be verified. For this reason, we have introduced an explicit calculation of the amount of momentum gravity–capillary waves receive using a model for the gravity–capillary waves that solves the one-dimensional energy balance equation for the short waves (the VIERS model). Again, the wind input is determined by a nonlinear version of the critical-layer theory of Miles (1957). As a consequence, for steep waves, wind input to the short waves is quenched. In practice, this means that when the sea state is nonlinear, the gravity–capillary waves will hardly receive momentum from the wind so that the low-frequency wave-induced stress dominates the stress balance at the surface. Hence, the simple expression for the drag [see Eq. (37)] plays a prominent role in the air–sea interaction problem.

The consequences of this approach have been incorporated in a single gridpoint version of the ecWAM model. By relaxing the accuracy requirement (relative accuracy reduced from 0.001% to 0.1%), the solution of the surface stress balance only requires typically a few iterations, thus making calculations of the surface stress on the fly practically feasible. The results from the new version of the ecWAM model are consistent with the simple model presented in section 3. For example, for large winds, the background roughness is small for young wind waves. As a consequence, it is emphasized that for young wind sea, the classical scaling relations for, for example, wave variance, peak frequency, and Phillips’ parameter still hold, but for old wind sea this is not quite the case, because the background roughness is finite and depends on additional parameters such as the surface tension and viscosity in air and water.

First trials with the Ardhuin et al. (2010) parameterization in the coupled IFS–ecWAM system, which involves both data assimilation and 10-day forecasting over a 30-day period, show that the new air–sea interaction approach produces realistic results. Here, the partial sheltering effect of Ardhuin et al. (2010) has been replaced by the nonlinear wind-input source function, which effectively models sheltering (see, e.g., Hara and Belcher 2002). Validation of forecast tropical cyclones shows that the new approach performs as well as the operational system (Magnusson et al. 2021).

## Acknowledgments.

We thank the reviewers for their helpful comments that improved the clarity of the manuscript and resulted in suggestions for future research.

## Data availability statement.

Figures 1–11 are a direct graphical representation using output of the wind–wave interaction model developed in this paper. Figures 12 and 13 were produced from results of large-scale simulations produced by an upgrade of ECMWF’s forecasting system with the new wind–wave interaction model in comparison with the operational model version, called control. These updated model data are available at https://doi.org/10.21957/ccsr-7e97, while the results of the control model are available at https://doi.org/10.21957/8ycd-y021.

In the original treatment of Janssen (1991), *l* = *κz* and the boundary condition of vanishing wind speed, *U*_{0}(*z* = *z _{B}*) = 0, was specified at

*z*=

*z*.

_{B}Although Eq. (45) is not an explicit expression for the degree of saturation *B*, since the growth rate of the waves depends through the roughness length and the friction velocity on *B*. Strictly speaking, an iteration of Eq. (45) is required.

The viscous dimensionless length is defined as the viscous roughness length made dimensionless with the factor

## APPENDIX

### Introducing 3D Effects

#### a. Basic equations

*U*

_{0}and

*V*

_{0}in the

*x*and

*y*direction, respectively. It is customary to rotate the coordinate system in such a way that the mean velocity is along the

*x*direction. In time-varying conditions the wind-velocity angle may be depending on time, so the rotation becomes time dependent, making the introduction of two wind-velocity components a necessity. Therefore,

**e**

*,*

_{x}**e**

*, and*

_{y}**e**

*are unit vectors in the*

_{z}*x*,

*y*, and

*z*direction. Thus, we deal with a plane parallel flow whose speed and density only depend on height

*z*. The equations for an adiabatic fluid do not explain the height dependence of wind speed and density because effects of small-scale turbulence are not taken into account. In addition, effects of turbulence on the wave-induced motion in the air are not considered.

*k*and

_{x}*k*the wavenumbers in the

_{y}*x*and

*y*direction, while

*ω*is the unknown angular frequency. Here, for given wavenumbers

*k*and

_{x}*k*, the task is to obtain

_{y}*ω*from the boundary value problem for the vertical velocity or displacement. To simplify things, we introduce polar coordinates with wavenumber

*k*and wave propagation direction

*θ*; hence,

*U*and wind direction

*ϕ*so that the wind speed components become

*U*is assumed to be given by the usual logarithmic wind profile, at least to lowest order in nonlinearity, so that

*z*

_{0}the roughness length. In principle, the wind direction

*ϕ*may be a function of time and height, but for simplicity, it is assumed that

*ϕ*is independent of height.

*W*=

*U*

_{0}cos

*θ*+

*V*

_{0}sin

*θ*−

*c*=

*U*cos(

*θ*−

*ϕ*) −

*c*and

*c*=

*ω*/

*k*is the magnitude of the phase speed of the surface gravity wave. Introducing the perturbation of the streamlines

*ψ*=

*w*/

*W*, one obtains after some algebra the following differential equation for

*ψ*,

*z*where the Doppler-shifted velocity

_{c}*W*vanishes,

*U*

_{0}is replaced by the effective wind

*U*

_{0}cos(

*θ*−

*ϕ*), and as a consequence, the friction velocity

*z*

_{0}). Therefore, results for the 2D problem can immediately be used to write down the solution for the 3D problem.

#### b. Growth rate in 3D

*ψ*at |

*z*| → ∞. In Janssen (2004), the boundary value problem for the 2D case has been solved for the case of a jump in the density profile, representing wind blowing over a water surface at

*z*= 0. The 3D solution now follows immediately by replacing the friction velocity

*ϵ*=

*ρ*/

_{a}*ρ*is the air–water density ratio, while

_{w}*ψ*′(0)′ follows from the solution of the boundary value problem in air, i.e.,

*W*on

*U*cos(

*θ*−

*ϕ*) instead of

*U*

_{0}.

*ϵ*is small. Using now the vertical component of the wave-induced velocity instead of the displacement of the streamlines

*ψ*=

*w*/

*W*, one finds from Eq. (A4) in terms of the normalized velocity

*χ*=

*w*/

*w*(0) the Rayleigh equation,

*W*

_{0}=

*U*

_{0}−

*c*

_{0}and

*γ*of the energy of the waves then follows from solving Eq. (A6) to first order in

_{E}*ϵ*. As a result,

*β*is given by Eq. (A12) and

*y*involves a cos(

_{c}*θ*−

*ϕ*) factor as well,

#### c. Wave-induced stress in 3D

The principal goal of this section is to determine the wave-induced stress in terms of the vertical velocity profile. Note that in 3D, the wave-induced stress has two components, one along the wind direction and one across. The along and cross components are connected on one hand to wind–wave growth and on the other hand to stretching of the vortex. First, the stress for a single wave is obtained, and then this is followed by an expression for many waves that are characterized by the wave spectrum.

*x*direction, is given by

*θ*=

*k*+

_{x}x*k*−

_{y}y*ωt*, and dropping the hats, the

*x*component of the wave stress becomes

*u*in terms of

*w*. From zero divergence we find

*u*in terms of

*υ*and

*w*,

*υ*can be expressed in terms of

*u*and

*w*using the

*x*and

*y*component of the Euler equations. This gives,

*W*

_{0}=

*U*cos(

*θ*−

*ϕ*) −

*c*. Substitution of Eq. (A18) in Eq. (A17) then gives,

*u*=

*iw*′/

*k*as

_{x}*k*vanishes. Then, using Eq. (A16), the wave-induced stress in the

_{y}*x*direction becomes

*η*through ∂

*η*/∂

*t*=

*w*at

*z*= 0. Write

*w*=

*w*

_{0}

*χ*where

*χ*satisfies the Rayleigh equation with boundary condition

*χ*(0) = 1. One finds −

*iωη*=

*w*

_{0}; hence,

*υ*component of the velocity) and gives rise to diffusion of momentum. The second term is the one that occurs also for one-dimensional propagation, and this term will give rise to diffusion of vorticity.

*F*(

**k**) is the wavenumber spectrum. After some algebra, the

*x*component of the wave-induced stress becomes

*x*component of the wind velocity, i.e.,

*W*

_{0}=

*U*cos(

*θ*−

*ϕ*) −

*c*. The parallel diffusion coefficient has, apart from the

*θ*dependence, a very similar form as the perpendicular and the cross-diffusion coefficient. In addition, there is, because of the critical layer, a simplification possible of the expression of the diffusion coefficients. Evaluating the integrals over

*k*and using the linear dispersion relation for surface gravity waves so that

*υ*=

_{g}*c*/2, one finds,

*k*is expressed in terms of the vertical coordinate

*z*through the resonance condition

*W*

_{0}=

*U*

_{0}cos(

*θ*−

*ϕ*) −

*c*= 0, where here the phase speed follows from the dispersion relation of free gravity waves. The resonance condition also implies a restriction for the integration in the

*θ*domain: only those waves are included that have a (positive) projection onto the wind direction, i.e., |

*θ*| ≤

*π*/2.

*y*component of the mean velocity then becomes

*x*and

*y*components of the wind velocity are very similar indeed.

It would be of interest to study the consequences of this set of coupled equations for the evolution of wind waves and surface winds, namely, Eqs. (A27) and (A33). In particular, it would be interesting to study whether, for example, in the case of an asymmetric distribution of surface gravity waves around the wind direction there will be a turning of the wind by the waves and to what extent this matters. So far this has not been done. Instead, in the main part of the paper we will make the assumption that the cross-diffusion coefficient vanishes so that no appreciable cross component of the wind is generated. This is a reasonable assumption, since most of the stress is determined by the short waves that are very quickly in equilibrium with the wind.

#### d. Nonlinear effects

The coupled equations, in particular the stress balance Eq. (12), show that there is a strong interaction between wind and waves. Clearly, the resulting slowing down of the wind is a nonlinear effect because its impact depends on the angular average of the wave spectrum. Therefore, one would expect that the growth rate of the waves by wind depends on the wave spectrum as well, and as a consequence, there would be deviations from the logarithmic wind profile. Here, a brief, approximate derivation of the sea state dependence on the growth rate is given. A similar result was already obtained in Miles (1965), and it has been utilized in the VIERS model (Janssen et al. 1998; see also Caudal 2002). This relatively simple proof is possible because of the nature of the resonant wave, mean-flow interaction, which allows a direct correspondence between wavenumber space and the vertical through the resonance condition *c* = *U*_{0}(*z _{c}*) cos

*θ*.

*z*. For a logarithmic wind profile, this ratio is given by

_{c}*z*, and it is of interest to learn to what extent nonlinearity will affect the curvature shear ratio. Now, from the stress balance [Eq. (12)] one finds for this ratio at the critical height

_{c}*z*,

_{c}*D*=

_{W}*D*

_{⊥}+

*D*

_{‖}, and a prime denotes differentiation with respect to

*z*. For a linear system with a logarithmic wind profile, one would ignore the wave diffusion coefficients and the curvature shear ratio would, as expected, reduce to

_{c}*z*. Eliminating the ratio of curvature to shear in the expression for the growth rate in Eq. (3), one finds,

_{c}*γ*

_{0}according to linear theory as the growth rate at

*D*=

_{W}*D*

_{⊥}= 0, i.e.,

*χ*|

_{c}^{2}in terms of the growth rate according to linear theory, i.e.,

*χ*|

_{c}^{2}, one can evaluate the wave diffusion coefficients. For example,

*D*

_{⊥}one finds,

^{2}factor, the expressions for the wave diffusion coefficients are identical.

*W*(

*z*) = 0, which implies the simple relation

_{c}*c*=

*U*

_{0}(

*z*) cos

_{c}*θ*. Here, in first order of approximation the wind profile assumes the logarithmic profile with roughness length

*z*

_{0}, as given by Eq. (16). Therefore, each wave-related factor in the expression for

*D*

_{⊥}involves a logarithmic function of height. Now, evaluating

*D*

_{⊥}using the mixing length

*l*(

*z*) =

*κ*(

*z*+

*z*) one finds for large

_{B}*z*,

*γ*

_{0}and the wave spectrum can all be expressed in terms of the phase speed; hence, through the resonance condition, they contain a logarithmic dependence on

*z*, which is slowly varying compared to a linear dependence on

_{c}*z*. Only the front factor, which is linearly dependent on

_{c}*z*, and the

_{c}*z*, are rapidly varying functions of height. To a good approximation, the

_{c}*z*derivative of

_{c}*D*

_{⊥}

*χ*|

_{c}^{2},

*D*, and

_{W}*N*

_{1}and

*N*

_{2}depend on the angular average of the product of linear growth rate and the wavenumber spectrum. They read,

^{2}

*θ*in the angular average in

*N*

_{1}. For typical angular dependencies in growth rates and two-dimensional spectra, the difference in size between

*N*

_{1}and

*N*

_{2}is a factor of 6; hence,

*N*

_{1}=

*N*

_{2}/6. The growth rate

*γ*

_{0}is the one according to linear theory with a logarithmic wind profile. For practical applications, we will use the parameterization for

*γ*

_{0}given in Eqs. (18), (20), and (21) with a first-guess value of the parameter

*β*

_{max}of 1.2.

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