An evolutionary equation for weakly nonlinear wind waves on the surface of a finite-depth viscous fluid

A model is constructed for the generation of gravity capillary waves on the surface of finite-depth water by a turbulent wind near the stability threshold. On the basis of the numerical model of wave generation by a turbulent wind, the threshold values of the wind friction velocity u(*c) and the wave number of the most unstable disturbance k(c) are determined as functions of depth. The nonlinear viscous decrease rate and wind growth rate of waves are calculated near the stability threshold. For this purpose, a technique is used that is based on the use of curvilinear coordinates with lines close to streamlines beyond viscous boundary layers near the surface and the bottom. The coefficients of the Ginzburg-Landau equation are calculated, which is valid in the case when the wave number of the most unstable disturbance is different from the values at which resonance with the second harmonic and the mean flow is observed.