Modified three-dimensional nonlinear Schrödinger equation in finite water depth for gravity waves with influence of slowly varying currents

A new modified three-dimensional (3D) nonlinear Schrödinger equation (3DMNLSE) is derived for gravity waves in the presence of wind, dissipation, and two-dimensional slowly varying currents, which include transverse and longitudinal currents in finite water depth. The effect of currents on modulational instability (MI) is investigated. The divergence (convergence) effect of longitudinal favorable (adverse) currents decreases (increases) the MI growth rate and region of instability while suppressing (enhancing) the formation of freak waves. Meanwhile, the transverse currents hardly affect the occurrence of MI. Furthermore, some results from the simulations with the space evolution of 3DMNLSE are presented. The results show the ubiquitous occurrence of freak waves in 3D wave fields under certain sets of initial conditions. We demonstrate that larger waves can be triggered when a weakly modulated wave train enters a region of adverse currents. The maximum amplitude of a freak wave depends on the ratio of the current velocity to the wave phase velocity. © 2024 Author(s).