An Euler Poincar\'e framework for the multilayer Green Nagdhi equations

TL;DR

This paper introduces a new Euler Poincaré framework for multilayer Green-Nagdhi equations, revealing natural derivation, extending to traveling waves, and demonstrating distinct wave behaviors through numerical solutions.

Contribution

It provides a novel geometric derivation of multilayer Green-Nagdhi equations and extends the framework to traveling wave solutions, highlighting differences from traditional models.

Findings

Multilayer waves show distinct behaviors compared to single-layer models.

The framework naturally derives multilayer Green-Nagdhi equations from Euler Poincaré theory.

Numerical solutions reveal intriguing differences in wave dynamics.

Abstract

The Green Nagdhi equations are frequently used as a model of the wave-like behaviour of the free surface of a fluid, or the interface between two homogeneous fluids of differing densities. Here we show that their multilayer extension arises naturally from a framework based on the Euler Poincare theory under an ansatz of columnar motion. The framework also extends to the travelling wave solutions of the equations. We present numerical solutions of the travelling wave problem in a number of flow regimes. We find that the free surface and multilayer waves can exhibit intriguing differences compared to the results of single layer or rigid lid models.