Free surface waves on shear currents with non-uniform vorticity: third-order solutions

Free surface waves of moderate amplitude on a fluid endowed with vorticity are calculated by computer-assisted perturbation expansions. Solitary waves are generated by deriving the nonlinear evolution equation (NEE) for the free surface displacement. Another recursive iteration procedure is then performed on the NEE. Properties obtained from second-and third-order expansions are computed explicitly for the case of a linear shear profile, or uniform vorticity distribution. Comparisons with known results in the literature show excellent agreement for small amplitude waves. Applications to non-uniform vorticity distributions are feasible and valuable, as existing methods will generally fail for nonlinear shear currents. Algebraic shear profiles U(y) = ay(m) (a = a constant, m not necessarily an integer) are tested, and backward modes display peculiar properties. Examples include a non-monotonic trend in the half-width of solitary waves and a local maximum in the velocity of the wave.