## 2 Theory
### A Hydrodynamic Model of the Wind Wave Spectrum and Its Modulation  
Wind waves can be described by the kinetic equation for the spectral density of the wave action \(\widetilde{N}(k)\equiv N(k,x,t)\) of GCW in the following form (see, e.g., [6]):  
\[ \begin{split}&\frac{\partial\widetilde{N}(k)}{\partial t}+[U(x,t)+c_ {gr}(k)]\frac{\partial\widetilde{N}(k)}{\partial x}-k\frac{\partial U(x,t)}{ \partial x}\frac{\partial\widetilde{N}(k)}{\partial k}=\\ &=Q(\widetilde{N}(k),u*,\gamma(k,E(\widetilde{\Gamma}),\sigma( \widetilde{\Gamma}))\end{split}  \tag{3}\]  
The wave action spectrum is related to the spectrum of wind wave heights as \(\widetilde{F}\)\((k)=\widetilde{N}\)\((k)\cdot k/\omega(k)\). In Equation (3), \(k\), \(c_{gr}(k)=\partial\omega(k)/\partial k\) and \(\omega(k)=(gk+\sigma k^{3})\) are the wave number, group velocity and frequency of GCW, respectively; \(g\) is the acceleration of gravity, \(\sigma(\Gamma)\) is the surface tension. \(Q\equiv Q(\widetilde{N}(k),u*,\gamma(k,E(\widetilde{\Gamma}),\sigma( \widetilde{\Gamma}))\) describes wind wave energy sources, sinks and nonlinear wave-wave interactions, \(u*\) is the wind friction velocity, \(\widetilde{\gamma}(k)\equiv\gamma(k,E(\widetilde{\Gamma}),\sigma( \widetilde{\Gamma}))\) is the wave damping coefficient [27], which, in turn, depends on the film elasticity \(E(\widetilde{\Gamma})\) and surface tension and, in general, on some other surfactant film characteristics, e.g., diffusion coefficient, viscosity, etc. Hereafter, an upper tilde denotes the functions varying due to non-uniform currents; those with a subscript \"0\" correspond to undisturbed, steady-state values. The r.h.s. of Equation (3) can be written as:  
\[Q=\widetilde{\beta}^{eff}(k)\widetilde{N}(k)+P_{a}+I_{nl}\Big{[}\widetilde{N} (k)\Big{]} \tag{4}\]  
In Equation (4), \(\widetilde{\beta}^{eff}(k)=\beta(k,u*)-2\gamma(k,E(\widetilde{\Gamma}),\sigma (\widetilde{\Gamma}))\), \(\beta(k,u*)\) is the wind growth rate due to the Miles wave generation mechanism--the growth rate dependence on wind velocity and wind wave number can be found, e.g., in [28]--\(P_{a}\) describes wind wave excitation due to atmospheric pressure pulsations--the so-called Phillips mechanism. Notably, that in further analysis, we neglect the variations of the wind growth rate over IW. One of the most difficult problems is to describe the nonlinear terms \(I_{nl}(\widetilde{N}(k))\) in the kinetic equation. The nonlinear term includes weakly nonlinear wave-wave interactions, and strongly nonlinear processes, such as wave breaking, responsible among others for the wind wave spectrum limitation. Wave-wave interactions are described by a collision integral which is extremely difficult to analyze. Wave breaking processes are also studied insufficiently and no analytical approach is developed to their description. Therefore, these complicated nonlinear processes are approximated by some empirical, usually algebraic expressions, similarly to [29], namely  
\[I_{nl}\Big{[}\widetilde{N}(k)\Big{]}=-\alpha(k)\widetilde{N}^{n+1}(k) \tag{5}\]  
Here, \(\alpha(k)\) and \(n\) are some empirical values. In the case when \(\widetilde{\beta}^{eff}(k)<0\), wind waves are not generated due to the Miles instability mechanism; thus, the Phillips excitation can generate wind waves, but of very small amplitudes. Therefore, neglecting the Phillips mechanism, one can consider that no waves are excited, in particular, with mm-scale wavelengths. Nevertheless, mm-scale waves appear, even \(\widetilde{\beta}^{eff}(k)<0\), because they can be generated by longer, cm-dm-scale waves if the Miles excitation condition is fulfilled for the latter. Thus, generated waves are known as bound waves, in particular, as parasitic capillary ripples (PCR) [15; 16; 17; 18; 19; 20; 26; 30].