Modulational instability and wave amplification in finite water depth

The modulational instability of a uniform wave train to side band perturbations is one of the most plausible mechanisms for the generation of rogue waves in deep water. In a condition of finite water depth, however, the interaction with the sea floor generates a wave-induced current that sub- tracts energy from the wave field and consequently attenuates the instability mechanism. As a result, a plane wave remains stable under the influence of collinear side bands for relative depths kh 1.36 (where k is the wavenumber of the plane wave andh is the water depth), but it can still destabilise due to oblique perturbations. Using direct numerical simulations of the Euler equations, it is here demonstrated that oblique side bands are capable of triggering modulational instabil- ity and eventually leading to the formation of rogue waves also forkh 1.36. Results, nonetheless, indicate that modu- lational instability cannot sustain a substantial wave growth forkh < 0.8.