The Effect of Random Wind Forcing in the Nonlinear Schrödinger Equation

Abstract

The influence of a strong and gusty wind field on ocean waves is investigated. How the random wind affects solitary waves is analyzed in order to obtain insights about wave generation by randomly time varying wind forcing. Using the Euler equations of fluid dynamics and the method of multiple scales, a random nonlinear Schrodinger equation and a random modified nonlinear Schrodinger equation are obtained for randomly wind forced nonlinear deep water waves. Miles theory is used for modeling the pressure variation at the wave surface resulting from the wind velocity field. The nonlinear Schrodinger equation and the modified nonlinear Schrodinger equation are computed using a relaxation pseudo spectral scheme. The results show that the influence of gusty wind on solitary waves leads to a randomly increasing ocean wave envelope. However, in a laboratory setup with much smaller wave amplitudes and higher wave frequencies, the influence of water viscosity is much higher. This leads to fluctuating solutions, which are sensitive to wind forcing.