Large-amplitude solitary waves on a linear shear current (Workshop on Nonlinear Water Waves)

Abstract

This work considers two-dimensional steady motion of solitary waves progressing in permanent form with constant speed on a shear current of which the horizontal velocity varies linearly with depth. In particular, we focus on peaking of the wave crest with increase of amplitude. First, local flow analysis near the crest of the peaked wave using conformal mapping determines singularities of solutions at the wave crest. It is shown that the inner angle of the corner at the crest of the peaked wave does not depend on the magnitude of shear current. Next, solutions of the two long wave models, the Korteweg-de Vries (KdV) model and the strongly nonlinear model, for solitary waves on a linear shear current are numerically compared with those the full Euler system. Numerical examples demonstrate that these long wave models can qualitatively capture variation of peaking phenomenon with the magnitude of shear current. In addition, linear stability analysis of steady solutions of solitary waves is discussed using the two long wave models.