A new class of two-layer Green-Naghdi systems with improved frequency dispersion

TL;DR

This paper introduces an improved class of Green-Naghdi models for internal wave propagation that enhances frequency dispersion and stability, while preserving key physical properties and mathematical consistency.

Contribution

The paper develops a new class of Green-Naghdi models that improve frequency dispersion and stability without losing the original model's accuracy and Hamiltonian structure.

Findings

Models manage high-frequency Kelvin-Helmholtz instabilities.

Rigorous justification through consistency, well-posedness, and stability.

Applicable to original Green-Naghdi and Saint-Venant systems.

Abstract

We introduce a new class of Green-Naghdi type models for the propagation of internal waves between two (1+1)-dimensional layers of homogeneous, immiscible, ideal, incompressible, irrotational fluids, vertically delimited by a flat bottom and a rigid lid. These models are tailored to improve the frequency dispersion of the original bi-layer Green-Naghdi model, and in particular to manage high-frequency Kelvin-Helmholtz instabilities, while maintaining its precision in the sense of consistency. Our models preserve the Hamiltonian structure, symmetry groups and conserved quantities of the original model. We provide a rigorous justification of a class of our models thanks to consistency, well-posedness and stability results. These results apply in particular to the original Green-Naghdi model as well as to the Saint-Venant (hydrostatic shallow-water) system with surface tension.