## 2 Theory
### A Hydrodynamic Model of the Wind Wave Spectrum and Its Modulation
A simplified relation between the PCR wave number \(k\) and the basic cm-dm-scale GCW \(k_{g}\) can be written as follows:  
\[k_{g}\approx\frac{g}{\sigma k} \tag{6}\]  
In fact, the situation is more complicated, and the PCR wavelengths vary along the GCW profile [30], although here we ignore this fact and will suppose that PCR with a given wavelength is excited by GCW in a narrow band in the spectrum of GCW, centered around the wave number \(k_{g}\). Then, we insert into the kinetic equation some function describingPCR intensity as a function of the spectrum intensity of GCW with the wave number given by Equation (6). Despite PCR being bound to the cm-dm-scale GCW we, nevertheless, described them as if they were free ripples when using the kinetic equation for the wind wave spectrum action. Therefore, both free wind-generated capillary ripples and parasitic ripples generated by longer steep GCW were indistinguishable in our approach. However, because we focused mostly on the analysis of wind wave spectrum modulation in the presence of surface films, free capillary ripples were not practically excited (for them, usually \(\widetilde{\beta}^{eff}(k)<0\)) and all capillary waves were considered further as PCR. For parasitic waves, it follows from a number of wave tank experiments that their parameters are characterized by very fast growth when the slopes of GCW are large enough. We assume this dependence in the power form as  
\[k^{2}\widetilde{F}(k)\propto\left[k_{g}^{2}\widetilde{F}(k_{g})\right]^{m} \tag{7}\]  
Hence, the source of PCR \(\widetilde{I}(k_{g})\) associated with GCW reads  
\[\widetilde{I}(k_{g})=\alpha(k_{g})\widetilde{N}^{m}(k_{g}) \tag{8}\]  
Then, the r.h.s. of Equation (3) for capillary waves when neglecting the pressure pulsations compared to \(\widetilde{I}(k_{g})\) looks like  
\[Q(\widetilde{N}(k),u*,\gamma(\widetilde{\Gamma}))=\widetilde{\beta}^{eff}(k) \widetilde{N}(k)-\alpha(k)\widetilde{N}^{n+1}(k)+\widetilde{I}(k_{g}) \tag{9}\]  
Suppression of capillary waves due to films can be characterized by the hydrodynamic contrast \(K^{hydro}(k)=\frac{F_{nl}^{nl}(k)}{F_{0}^{eff}(k)}\). For mm-scale waves at low winds (of about 5 m/s and less), \(\beta_{0}^{eff}(k)<0\). In this case, when neglecting the second term in the r.h.s. of Equation (10), one can write the contrast \(K^{hydro}(k)\) for mm-scale ripples as follows:  
\[K^{hydro}(k)=\frac{N_{0}\left(k_{g}\right)\big{|}_{nsl}}{N_{0}\left(k_{g} \right)\big{|}_{sl}}=\frac{I_{0}\left(k_{g}\right)\big{|}_{nsl}}{I_{0}\left(k_ {g}\right)\big{|}_{sl}}\frac{\beta_{0}^{eff}(k)\Big{|}_{sl}}{\beta_{0}^{eff}( k)\big{|}_{nsl}}=\left[K^{hydro}\left(k_{g}\right)\right]^{m}\frac{\beta_{0}^{ eff}(k)\Big{|}_{sl}}{\beta_{0}^{eff}(k)\Big{|}_{nsl}} \tag{10}\]  
where \(K^{hydro}\left(k_{g}\right)\) denotes the contrast for basic cm-dm-scale GCW generating PCR, and \(nsl,sl\) mean non-slick (without film) and slick (with film) areas, respectively. Variations of the spectrum of mm-scale ripples under the action of IW contain several components. They are kinematic (straining) variations due to IW orbital velocities, variations of the damping coefficient due to the redistribution of surfactant concentration as well as variations of the wind wave growth (we neglect this effect in the following as insignificant), and variations of the PCR source due to kinematic modulation of cm-dm-scale waves. The hydrodynamic modulation transfer function is \(m_{h}=m_{h}(k,K,C,\ldots)\), which characterizes the wind wave spectrum variations \(m^{hydro}\frac{U_{0}}{\mathbb{C}}e^{ix(x-Ct)}=\frac{F}{F_{0}(k)}=\frac{\Delta F (k)}{F_{0}(k)}e^{ix(x-Ct)}=\frac{\widetilde{N}\left(k\right)-N_{0}(k)}{N_{0}( k)}=\frac{\Delta N(k)}{N_{0}(k)}e^{ix(x-Ct)}\) The definition of \(m^{hydro}\) usually implies that \(\Delta F(k)<<F_{0}(k)\) and can be written for capillary waves when using Equations (3)-(9), as follows  
\[