Wave-Current-Bathymetry Interaction Revisited: Modeling, Analysis and Asymptotics

Abstract

Starting from the free surface Euler equations, we derive a leading-order system in terms of surface variables, depending on the surface current and on the bathymetry through the depth-dependent Dirichlet-to-Neumann (DN) operator. The resulting system is shown to be well-posed using the theory of hyperbolic systems of pseudo-differential operators. We then consider wave propagation in slowly varying environments. As an explicit approximation to the DN operator, the semiclassical Weyl quantization of the symbol $g_b(X,\xi)=|\xi|\tanh(b(X)|\xi|)$ is shown to be both asymptotically accurate and consistent with the self-adjoint structure of the true operator, and to provide the natural framework for asymptotic analysis of the wave system. A central consequence of the resulting framework is that classical asymptotic models - including the wave action equation, the mild-slope equation, the Schr\"odinger equation, and the action balance equation - emerge systematically from a single formulation. By deriving these equations, we show how the simple leading order system with the Weyl quantization of the DN operator provides a unified and mathematically consistent framework for the asymptotic linear theory of wave-current-bathymetry interaction, hence providing a transparent, rigorous and accessible route from the primitive Euler equations to the mentioned asymptotic models. Throughout, numerical experiments are included to illustrate the analysis.