Statistical properties of water waves. Part 1. Steady-state distribution of wind-driven gravity-capillary waves

Abstract

The Miles–Phillips model of the linear coupling between waves on the ocean surface and a fluctuating wind field is generalized to include the average effect of the nonlinear water-wave interactions in the dynamic equations for gravity–capillary waves. A statistical-linearization procedure is applied to the general problem and yields the optimum linear description of the nonlinear terms by linear terms. The linearized dynamic equations are stochastic with solutions that have stable moments, i.e. the average nonlinear interactions quench the linear instability generated by the coupling to the mean wind field. In particular, an asymptotic steady-state power-spectral density for the water-wave field is calculated exactly in the context of the model for various wind speeds.