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

TL;DR

This paper develops a unified, rigorous framework for modeling wave-current-bathymetry interactions starting from the Euler equations, deriving asymptotic models via Weyl quantization and asymptotic analysis.

Contribution

It introduces a new asymptotic approximation of the Dirichlet-to-Neumann operator using Weyl quantization, unifying various classical wave models within a single framework.

Findings

The Weyl quantization provides an accurate asymptotic approximation of the DN operator.

Classical models like the wave action and Schrödinger equations emerge systematically from the framework.

Numerical experiments validate the theoretical analysis.

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.