Rogue waves in shallow water in the presence of a vertically sheared current
Résumé
Two-dimensional rogue wave occurrence in shallow water on a vertically sheared current of constant vorticity is considered. Using Euler equations and Riemann invariants in the shallow water approximation, hyperbolic equations for the surface elevation and the horizontal velocity are derived and closed-form nonlinear evolution equation for the surface elevation is obtained. Following Whitham (1974), a dispersive term is added to this equation using the fully linear dispersion relation. With this new single first-order partial differential equation, vorticity effects on rogue wave properties are studied numerically. Besides, the Boundary Integral Element Method (BIEM) and the KdV equation both with vorticity are used for this numerical investigation, too. It is shown that results from the generalised Whitham equation agree quite well with those from BIEM whereas those from the KdV model are quite different. The numerical simulations carried out with the generalised Whitham equation and BIEM show that the presence of an underlying vertically sheared current modifies rogue wave properties significantly. For negative vorticity the amplification factor and duration of extreme wave events are increased whereas it is the opposite for positive vorticity.