Modified nonlinear Schrödinger equation for gravity waves with the influence of wind, currents, and dissipation

Abstract

A new modified nonlinear Schrödinger (MNLS) equation is derived for gravity waves with the presence of wind, dissipation, and shear currents in finite water depth. Horizontal surface currents are assumed stationary and slowly varying spatially. Using the MNLS equation, the modulational instability (MI) of deep-water gravity wave trains affected by wind and dissipation is considered. It was demonstrated that the modulational perturbation of waves is unstable or becomes unstable after several wave periods, whereas the dissipation will suppress the MI. Then, a new theoretical formula for predicting the maximum amplitude is derived to take into account the effect of vorticity, dissipation, and wind. The effect of dissipation becomes significant in strong currents, while wind can increase the height amplification. Furthermore, an explicit analytical Peregrine breather (PB) solution that considers the effect of vorticity, dissipation, and wind is presented. Opposing currents and winds will increase the height of PB. However, following currents and dissipation have opposite effects. The effects of the shear current, dissipation, and wind on nondimensional maximum amplitudes during the evolution of the Akhmediev breather are similar to PB solution.