Linear-shear-current modified Schrödinger equation for gravity waves in finite water depth.

Abstract

A nonlinear Schrödinger equation for the propagation of two-dimensional surface gravity waves on linear shear currents in finite water depth is derived. In the derivation, linear shear currents are assumed to be a linear combination of depth-uniform currents and constant vorticity. Therefore, the equation includes the combined effects of depth-uniform currents and constant vorticity. Next, using the equation, the properties of the modulational instability of gravity waves on linear shear currents are investigated. It is showed that shear currents significantly modify the modulational instability properties of weakly nonlinear waves. Furthermore, the influence of linear shear currents on Peregrine breather which can be seen as a prototype of freak waves is also studied. It is demonstrated that depth-uniform opposing currents can reduce the breather extension in both the time and spatial domain in intermediate water depth, but following currents has the adverse impact, indicating that a wave packets with freak waves formed on following currents contain more hazardous waves in finite water depth. However, the corresponding and coexisting vorticity can counteract the influence of currents. Additionally, if the water depth is deep enough, shear currents have negligible effect on the characteristics of Peregrine breathers.