A simple analytical model for surface water waves on a depth-varying current

A perturbation analysis is presented in which a series of small amplitude r egular waves co-exist with an arbitrarily sheared current, U(z). Assuming t hat the current velocity is weak, i.e. U(z)/c = O(epsilon), the solution is extended to O(epsilon (2)), where c is the phase velocity and epsilon = ak the wave steepness. This provides a first approximation to the non-linear wave-current interaction, and allows simple explicit solutions for both the modified dispersion relation and the water-particle kinematics to be deriv ed. These solutions differ from the existing irrotational models commonly u sed in design and, in particular, highlight the importance of the near-surf ace vorticity distribution. These results are shown to be in good agreement with laboratory data provided by Swan et. al. [J. Fluid Mech (2001, in pre ss)]. Perhaps more surprisingly, good agreement is also achieved in a numbe r of strongly non-linear wave-current combinations, where the results of th e present analytical solution are compared with a fully non-linear numerica l wave-current model. (C) 2001 Published by Elsevier Science Ltd.