Modulation instability and rogue waves for shear flows with a free surface

Abstract

We study free surface gravity waves in the presence of a depth-dependent shear current with a nonzero vorticity gradient. The evolution of weakly nonlinear, narrow-band wave packets is governed by a nonlinear Schrödinger equation. When dispersion and nonlinearity are of the same (opposite) signs, modulation instability will be present (absent), and rogue waves represented by breathers can (cannot) occur, respectively. For irrotational flows, rogue waves only occur for sufficiently deep water, or more precisely, kh>1.363, where k is the wave number of the carrier envelope and h is the water depth. While the irrotational and linear shear current cases have been treated previously, the present study demonstrates the importance of a shear current with a nonzero vorticity gradient. For a concave current, the threshold for modulation instability of a wave packet moving with (or against) the shear current will reduce (or increase) the numerical bound of 1.363, respectively. The opposite will hold for the case of a convex current. The growth rate of a disturbance will also be larger for concave currents in comparison with convex and linear currents. The streamline patterns for the transient case of rogue waves will be illustrated for the simple case of a linear shear current. Shear currents near the sea surface are known to have a profound influence on the dynamics of wind-generated waves, and the present investigation bears on this.