Effects of slowly varying depth and current on the evolution of a Stokes wavepacket

Abstract

We consider the effects of slowly varying depth and current on the evolution of a packet of Stokes waves. The lengthscale of one-dimensional depth variation is assumed to be much greater than that of the wave envelope, and the direction of the current is perpendicular to the depth contours. By the method of multiple scales, the wave envelope is found to satisfy a cubic Schrödinger equation with slowly varying coefficients The criterion of spatial instability to small sidebands is extended. Numerical integration shows that the nonlinear evolution of a wavepacket is directly related to the instability parameter, which depends strongly on the current and depth variation. The heuristic hypothesis of Djordjevic & Redekopp on the soliton evolution over a slope in the absence of current is assessed.