On the model formulations for the interaction of nonlinear waves and current

Abstract

We study the model formulations of wave–current interactions in the framework of Euler equations. This work is intrigued by a recent paper from Wang et al. (2018) (hereafter WMY), which proposes such a model for the evolution of nonlinear broadband surface waves under the influence of a prescribed steady and irrotational current without vertical shear. We show that WMY’s model can be derived from a more general model accounting for an arbitrary steady and irrotational current. Under further assumption of scale separation between waves and current (i.e. horizontally slowly-varying current), WMY’s model is equivalent to an earlier model, in contrast to WMY’s claim that their model includes additional higher-order effects in wave steepness. We demonstrate the usefulness of such models in a numerical study on wave blocking by opposing current, where the nonlinear effect on the caustic location and wave amplitude amplification is elucidated. We further show that the model formulation in the framework of Euler equations form a Hamiltonian system conserving the total energy of waves and current, justifying the theoretical significance of the model equations. Finally, we generalize the formulation to nonlinear wave evolution in the presence of a rotational current with constant vorticity, which overcomes a limitation of such models that has been overlooked in previous work.