A high-order spectral method for nonlinear water waves in the presence of a linear shear current

A direct numerical method is proposed to simulate nonlinear water waves with nonzero constant vorticity in a two-dimensional channel of finite or infinite depth. Such a vortical distribution represents a linearly varying shear current in the background flow. Our method is based on the reduction of this problem to a lower-dimensional Hamiltonian system involving surface variables alone. This is made possible by introducing the Dirichlet-Neumann operator and associated Hilbert transform which are described via a Taylor series expansion about the still water level. Each Taylor term is a sum of concatenations of Fourier multipliers with powers of the surface elevation, and thus is efficiently computed by a pseudo-spectral method using the fast Fourier transform. The performance of this numerical model is illustrated by examining the long-time evolution of Stokes waves on deep water and of solitary waves on shallow water. It is observed that a co-propagating current has a stabilizing effect on surface wave dynamics while a counter-propagating current promotes wave growth. In particular, the Benjamin-Feir instability of Stokes waves can be significantly reduced or enhanced. Our simulations also suggest the existence of stable rotational solitary waves if the vorticity is not too large in magnitude.