Nonlinear modulation of capillary-gravity waves on shear flows including the effects of wind input and dissipation

This study has provided an analytical investigation of the modulational instability (MI) of capillary-gravity (CG) wavetrains acted upon by wind forcing and dissipation starting from the nonlinear Schrödinger equation. The wind regime is that of the validity of Miles theory, and the viscosity is small. It is observed that there are several wavenumber bands within which these wavetrains are unstable to sideband perturbations. The outcomes shed some light on the influence of wind input and dissipation on the MI. In the wind-driven wave regime, the Miles theory provides an estimate of the growth rate of the wavetrain under the action of wind. It is observed that the MI is increased/reduced due to the effect of wind forcing/dissipation terms. The expression for the growth rate of instability for CG waves propagating at the free surface of a vertically sheared current is obtained. We have also made the stability analysis for pure capillary waves. Furthermore, an explicit analysis of the Peregrine breather solution for considering the effects of shear current, wind, and dissipation in the case of CG waves is presented.