Wind-forced modulations of finite-depth gravity waves

Abstract

It has been shown recently that the modulations of deep-water gravity waves grow superexponentially under the effect of wind [S. Leblanc, “Amplification of nonlinear surface waves by wind,” Phys. Fluids 19, 101705 (2007)]. Finite depth is considered here. Based on Davey–Stewartson equations with forcing, stability analysis shows that longitudinal modulations are amplified when kh>1.363, where k is the wave number of the carrier wave and h is the water depth, but that oblique modulations dominate when 0<kh<2.199. In each case, amplification is characterized in the linear stage by superexponential growth. This latter is, however, so small when kh<0.380 that no modulation is expected in shallow water, although the carrier wave still grows, thanks to wind forcing.