Wind–Wave Interaction for Strong Winds

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.

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 U₁₀ in the range 1 ≤ U₁₀ ≤ 80 m s⁻¹ and on the wave-age parameter $\chi =c_p/u_{*}$, where c_p 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 (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_a 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.

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 U₁₀. 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., U₁₀ > 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_D decreases with increasing wind speed. At around a wind speed of 30 m s⁻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 $u_{*}$. For this reason, the scaling is sometimes referred to as friction velocity scaling. It is found that, while for young wind sea the classical scaling relations work, for old wind sea, deviations may be found.

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 U₀(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 ${\tau }_w^{(x)}=-⟨uw⟩$ consists of two contributions, namely, one contribution giving rise to diffusion of vorticity (also present in one-dimensional theory) and a second contribution that gives rise to diffusion of momentum. In one dimension, in the absence of turbulence, the coupled system evolves toward a steady state with a linear wind profile; hence, the wind–wave growth rates vanish for large times. However, in two dimensions, the equilibrium wind profile is not a linear profile because the momentum diffusion contribution tries to maintain the curvature in the wind profile. As a consequence, even in the presence of air turbulence, wave growth is in two dimensions larger than in the case of one-dimensional propagation. Here, we summarize the main results, while details of the calculations are given in the appendix.

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 ${\tau }_w^{(x)}=-⟨uw⟩$ on the mean flow, which depends on the wave spectrum 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 ${\tau }_w^{(y)}=-⟨\upsilon w⟩$, which leads to additional turning of the wind profile. However, in the present treatment, the contribution of the cross stress will be ignored. A reason for this is that the wave-induced stress is determined to a large extent by the high-frequency waves. The response time of these short waves is small so that in practice they are in equilibrium with the wind and have a symmetrical angular distribution with respect to the wind direction. Therefore, the cross stress is assumed to be small.

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.

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_w was to the surface stress ${\tau }_a={\rho }_a{u}_{*}^2$. Typically, accuracy of four digits was achieved.

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 $C_D=u_{*}^2/U_{10}^2$, depends on wave age, which is in qualitative agreement with observations by Donelan (1982) and Smith et al. (1992). Finally, it was shown that results for the drag depend in a sensitive manner on how the Phillips parameter depends on the sea state. If the Phillips parameter 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.

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 U₁₀ 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).

In this derivation, the assumption is that the most important factor that determines the growth rate is the curvature shear ratio $W_c^{\mathbf{″}}/\left|W_c^{\mathbf{\prime }}\right|$ at the critical height, while other parameters, such as the vertical velocity, depend only weakly on the wind profile. Now, from the stress balance [Eq. (12)], one obtains the curvature shear ratio at the critical height z_c, and the result is substituted in the expression for the growth rate given in Eq. (3). Denoting by γ₀ the growth rate for a logarithmic wind profile, one finds after some algebra the following simple expression for the renormalized growth rate,

$\gamma ={\gamma }_0\frac{1+N_1}{1+N_2},$(23)

where the renormalization factors N₁ and N₂ depend on the angular average of the product of linear growth rate γ₀ and the wavenumber spectrum F(k). They read,

$N_1=\frac{k^3}{ϵ\kappa u_{*}}\int d\theta {\gamma }_{0}F(k,\theta ){\sin }^2\theta ,N_2=\frac{k^3}{ϵ\kappa u_{*}}\int d\theta {\gamma }_{0}F(k,\theta ).$(24)

Note the similarities between the renormalization factors; the only difference is the additional factor of sin²θ in the angular average in N₁. For typical angular dependencies in growth rates and two-dimensional spectra, the difference in size between N₁ and N₂ is a factor of 6; hence, N₁ = N₂/6. The growth rate γ₀ is the one according to linear theory with a logarithmic wind profile. For practical applications, we will use the parameterization for γ₀ given in Eqs. (18), (20), and (21) with a first-guess value of the parameter β_max of 1.2.

To estimate the importance of the nonlinear corrections to the growth rate of the wind waves, we will replace the angular average by a factor times the product of wind input in the wind direction and the wavenumber spectrum F(k). For example, N₂ becomes

$N_2={\mathrm{\Delta }}_{\gamma }\frac{k^3}{ϵ\kappa u_{*}}{\gamma }_0(k,\theta =0)F(k),$(25)

where the choice for the magnitude of Δ_γ, measuring the size of angular effects, is guided by the following consideration. Assume that the wave spectrum has a cos² directional distribution. Hence,

$F(k,\theta )=\frac{1}{N}{\cos }^2\theta F(k),N=\pi /2.$

Since the growth rate according to Eq. (18) is also proportional to cos²θ, the directional terms give rise to a correction factor,

${\mathrm{\Delta }}_{\gamma }=\frac{1}{N}{\displaystyle {\int }_{-\pi /2}^{\pi /2}d}\theta {\cos }^4\theta =\frac{3}{4};$

hence, we take Δ_γ = 3/4. Estimating the magnitude of N₁ involves an additional factor sin²θ. The relevant integral equals 1/8 or N₁ = N₂/6. Since N₁ < N₂, nonlinear effects reduce the growth rate.

Using the approximations for N₁ and N₂, 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 γ₀/ω with the nonlinear normalized growth γ/ω, where γ is given by Eq. (23), and the renormalization factors N₂ are given in Eq. (24), while N₁ = N₂/6. The relevant solutions for the drag coefficient C_D and the friction velocity $u_{*}$ are obtained using an iteration scheme as described in section 4. From Fig. 1, it is striking to see that nonlinear effects are only relevant for short waves with 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.

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Dependence of linear and nonlinear growth rate on wavenumber for a low wind speed of 15 m s⁻¹ and a high wind speed of 50 m s⁻¹.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0293.1

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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).

For wavenumbers higher than k_3w, 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.

c. Exact solution of short-wave energy balance

Combining now the explicit expressions for the source terms, the energy balance equation [Eq. (19)] becomes

$\frac{\partial }{\partial k}\mathrm{\Phi }(k)=\Gamma \frac{{\omega }^2}{k^4}B,$(43)

where the parameter Γ is defined as

$\Gamma =ϵ\beta \omega {\left(\frac{u_{*}}{c}\right)}^2-{\gamma }_d,$(44)

and gives the net effect of wind input and dissipation. Note that the energy flux Φ(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

$B={\left(\frac{{\upsilon }_g}{{\alpha }_3}\right)}^{1/2}{c}^{-2}\left\{{\mathrm{\Phi }}_0^{1/2}+\frac{1}{2{\alpha }_3^{1/2}}{\displaystyle {\int }_{k_{3w}}^{k}d}k\frac{\Gamma }{k^2}{\upsilon }_g^{1/2}\right\},$(45)

where Φ₀ is the value of the energy flux at k = k_3w. 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

$B_{3w}={\left(\frac{{\mathrm{\Phi }}_0}{2{\alpha }_3}\right)}^{1/2}{c}_0^{-3/2}\frac{y{(1+3y^2)}^{1/2}}{(1+y^2){(y+y^3)}^{1/4}},$(46)

where y = k/k₀, k₀ = (g/T)^1/2 is the wavenumber that separates gravity waves and capillary waves, and c₀ = (gT)^1/4 (note that the minimum phase speed equals $c_0\sqrt{2}$). Therefore, in the gravity wave range (k < k₀), the degree of saturation increases with wavenumber like k^3/4, while in the capillary range, B_3w decreases with wavenumber like k^−3/4, and it attains its maximum value at k ≈ 1.32k₀.

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 $k=k_{3w}y(u_{*})k_0$, where the three-wave interactions are switched on, and therefore marks the start of the gravity–capillary wave spectrum. Clearly, as there is a pronounced increase in the degree of saturation in the gravity–capillary range, these short waves are expected to have an important contribution to the wave-induced stress.

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The wave-age dependence (between 5 and 25 in steps of 5) of the degree of saturation spectrum as a function of wavenumber for a wind speed of 15 m s⁻¹.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0293.1

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4. Determination of the surface stress

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 $k=k_{3w}=y(u_{*})k_0$ where the three-wave interactions are switched on, and therefore this marks the start of the gravity–capillary wave spectrum. Thus, rather than being constant (as for the high-wavenumber part of the gravity range), the degree of saturation increases rapidly up to about k = 1.3k₀, 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_a 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 k_c is below k_3w; 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 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 $C_D={\tau }_w/U_{10}^2$ and the numerical result is shown in Fig. 3, and a reasonably good agreement is found for small wave ages.

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(left) The wave-age dependence of the drag coefficient for a wind speed of 15 m s⁻¹. The red line shows an approximate solution for the stress balance. (right) The wave-age dependence for the same wind speed of the normalized low- and high-frequency stress.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0293.1

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We are now in a position to determine the background roughness length, which is a quantity that originates from the theory of section 2. Here, the unresolved stress τ_w,hf is represented by the turbulent stress τ_turb at the surface, where the mixing length is given by l(0) = κz_B. For the logarithmic profile Eq. (16), one then finds the simple relation,

$z_B=z_0\sqrt{{\tau }_{w,\text{hf}}/{\tau }_a},$(48)

and in Fig. 4 the dimensionless background roughness length $g{z}_B/u_{*}^2$ is plotted as function of wave age for different wind speeds. In the present operational system, the dimensionless background roughness length has the constant value of 0.0065, and only in the low-wind-speed range of 10–15 m s⁻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.

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Comparison of dimensionless background roughness length $g{z}_B/u_{*}^2$ as function of wave age χ for different wind speeds.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0293.1

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c. Climatology

To conclude the discussion of the results of the new coupled wind/wave system, it would be of interest to be able to study “climatological” results for the drag coefficient so that a direct comparison with climatological results of the present ECMWF operational system can be made. However, when developing this new model of wind–wave interaction we had of course not yet introduced the new approach in a semioperational version of ecWAM. In addition, climatological results can be quite expensive, in particular when one is interested in extremes. Therefore, here we use a shortcut in order to obtain insight in the properties of the new approach. Using operational results obtained with the coupled IFS–ecWAM model one finds that on average there is a relation between wave age χ and wind speed U₁₀. In the wind speed range of 0 < U₁₀ < 25, it is approximately given by

$\chi =\frac{35}{1+0.005U_{10}^2},$(49)

and it is assumed that the relation also holds for larger wind speeds, although we have not been able to verify this because there were not sufficient extreme wind cases. Relation (49) expresses, in agreement with one’s expectations, that the stage of development of the sea state generated by low–wind speed events is on average much older than the sea state generated by extreme wind events.

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 U₁₀ = 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 U₁₀ = 30 m s⁻¹ to U₁₀ = −35 m s⁻¹ onward. The present model calculations seem to confirm this picture, in agreement with the analytical result [Eq. (37)].

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The climatological dependence of the drag coefficient C_D on wind speed U₁₀ in the range of 1–80 m s⁻¹ according to the old and the new approach. The analytical result (37) for the drag coefficient is shown for reference, and Hersbach’s climatological fit valid for low to moderate wind speeds is shown. In addition, the two final curves illustrate that the combination of nonlinear input and the gravity–capillary background roughness is required to give a substantial reduction in drag for steep, high wind speed waves.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0293.1

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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_visc³ 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.

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Climatological Charnock parameter α and background Charnock parameter α_B as a function of wind speed U₁₀.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0293.1

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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_3w of the short-wave spectrum and the strength of the three-wave interactions α₃. By choosing k_3w according to Eq. (41), with friction-velocity-dependent parameter y, and by choosing α₃ = 6π, a reasonable agreement with the Cox and Munk (1954) observations was achieved.

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Comparison of modeled and observed mss as function of wind speed. Also is shown impact on mss when inertial subrange spectrum is chosen.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0293.1

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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 σ₀ 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.