Instability growth rates of crossing sea states for surface gravity waves in the case of wind blowing over water

In this paper, the influence of uniform wind flow for nonlinearly interacting surface waves in deep water using two coupled (2+1)-dimensional fourth-order nonlinear Schrödinger equations is studied. Starting from fourth-order coupled equations, the modulational instability of nonlinearly interacting waves in a situation of crossing sea states on an infinite depth of water is discussed. The inclusion of fourth-order terms to the nonlinear Schrödinger equation contributes to an improvement on the results associated with the stability of finite amplitude waves. The key point of this paper is that the present fourth-order results give significant improvements in the stability properties from the third-order results and consistent with the previous results. The stability conditions from the quartic nonlinear dispersion relation are obtained and employing those conditions the stable-unstable regions have been drawn. The three-dimensional contour maps of instability growth rate are also plotted. The growth rate of instability is shown to be appreciably much higher when the wind velocity approaches toward the critical value.