Relaxing the sharp density stratification and columnar motion assumptions in layered shallow water systems

TL;DR

This paper provides a rigorous justification for bilayer shallow-water models as approximations to hydrostatic Euler equations in density-stratified flows, accommodating continuous stratification and refining solutions near the pycnocline.

Contribution

It introduces a new theoretical framework that relaxes previous assumptions of sharp density and velocity profiles, extending the validity of layered shallow water models to more realistic stratification.

Findings

Validates bilayer models for density-stratified flows

Accommodates continuous stratification in the theory

Extends applicability to multilayer systems

Abstract

We rigorously justify the bilayer shallow-water system as an approximation to the hydrostatic Euler equations in situations where the flow is density-stratified with close-to-piecewise constant density profiles, and close-to-columnar velocity profiles. Our theory accommodates with continuous stratification, so that admissible deviations from bilayer profiles are not pointwise small. This leads us to define refined approximate solutions that are able to describe at first order the flow in the pycnocline. Because the hydrostatic Euler equations are not known to enjoy suitable stability estimates, we rely on thickness-diffusivity contributions proposed by Gent and McWilliams. Our strategy also applies to one-layer and multilayer frameworks.