An extended nonlinear Schrödinger equation for water waves with linear shear flow, wind, and dissipation

Abstract

Based on potential flow theory, governing equations are developed for surface gravity waves affected by wind, dissipation, and a linear shear flow (LSF). The LSF is composed of a uniform flow and a shear flow with constant vorticity. Using the multiple-scale analysis method, a two-dimensional nonlinear Schrödinger equation (NLSE) describing the evolution of freak waves in water of finite depth is derived by solving the governing equations. The modulational instability (MI) of the NLSE is analyzed, and it is shown that uniform up-flow and positive vorticity require a lower angular frequency to sustain the MI than uniform down-flow and negative vorticity. Moreover, the low-frequency section requires stronger wind velocity to sustain the MI than the high-frequency section. In other words, young waves are more sensitive to the MI than old waves. In addition, the free surface elevation of freak waves as a function of time is examined for different uniform flows, vorticities, and wind forcing, and the results are compared with a measured freak-wave time series from the North Sea. It is found that the theory agrees with the observations. Furthermore, the LSF affects the height and steepness of freak waves, while wind forcing affects their symmetry. Hence, the MI, wave–current interactions, and wind–wave interactions may be responsible for generating freak waves in realistic ocean scenarios.