Green-Naghdi dynamics of surface wind waves in finite depth

The Miles' quasi laminar theory of waves generation by wind in finite depth h is presented. In this context, the fully nonlinear Green-Naghdi model equation is derived for the first time. This model equation is obtained by the non perturbative Green-Naghdi approach, coupling a nonlinear evolution of water waves with the atmospheric dynamics which works as in the classic Miles' theory. A depth-dependent and wind-dependent wave growth. is drawn from the dispersion relation of the coupled Green-Naghdi model with the atmospheric dynamics. Different values of the dimensionless water depth parameter delta = gh/U-1, with g the gravity and U-1 a characteristic wind velocity, produce two families of growth rate gamma in function of the dimensionless theoretical wave-age c(0): a family of gamma with h constant and U-1 variable and another family of gamma with U-1 constant and h variable. The allowed minimum and maximum values of gamma in this model are exhibited.