Fully nonlinear long-waves models in presence of vorticity

TL;DR

This paper extends Green-Naghdi equations to include general vorticity, enabling detailed modeling of large amplitude shallow water waves with complex surface-current interactions, and introduces a cascade approach for vorticity dynamics.

Contribution

It introduces a novel reduction of 2+1D vorticity dynamics to a finite cascade of 2D equations within fully nonlinear shallow water models.

Findings

Vorticity effects are incorporated via a Reynolds-like tensor.

Closure achieved at second order of the cascade.

Method to reconstruct 3D velocity fields from 2D equations.

Abstract

We study here Green-Naghdi type equations (also called fully nonlinear Boussinesq, or Serre equations) modeling the propagation of large amplitude waves in shallow water. The novelty here is that we allow for a general vorticity, hereby allowing complex interactions between surface waves and currents. We show that the a priori 2+1-dimensional dynamics of the vorticity can be reduced to a finite cascade of two-dimensional equations: with a mechanism reminiscent of turbulence theory, vorticity effects contribute to the averaged momentum equation through a Reynolds-like tensor that can be determined by a cascade of equations. Closure is obtained at the precision of the model at the second order of this cascade. We also show how to reconstruct the velocity field in the 2 + 1 dimensional fluid domain from this set of 2-dimensional equations and exhibit transfer mechanisms between the horizontal and vertical components of the vorticity, thus opening perspectives for the study of rip currents for instance.