Multilayer shallow-water model with stratification and shear

TL;DR

This paper introduces a multi-layer shallow-water model with stratification and shear, generalizing previous single-layer models to better capture thermodynamics, mixing, and instability phenomena in fluid dynamics.

Contribution

It extends Ripa's single-layer model to multiple layers with variable shear and stratification, improving accuracy in representing thermodynamics and instability processes.

Findings

Multi-layer model accurately captures internal modes and instabilities.

Two-layer configuration improves results over single-layer models.

Model effectively represents baroclinic instability phenomena.

Abstract

The purpose of this paper is to present a shallow-water-type model with multiple inhomogeneous layers featuring variable linear velocity vertical shear and startificaion in horizontal space and time. This is achieved by writing the layer velocity and buoyancy fields as linear functions of depth, with coefficients that depend arbitrarily on horizontal position and time. The model is a generalization of Ripa's (1995) single-layer model to an arbitrary number of layers. Unlike models with homogeneous layers the present model is able to represent thermodynamics processes driven by heat and freshwater fluxes through the surface or mixing processes resulting from fluid exchanges across contiguous layers. A model configuration with only one layer has been previously shown to provide: a very good representation of the exact vertical normal modes up to the first internal mode; an exact representation of long-perturbation (free boundary) baroclinic instability; and a very reasonable representation of short-perturbation (classical Eady) baroclinic instability. Here it is shown that substantially more accurate overall results with respect to single-layer calculations can be achieved by considering a stack of only a few layers. A similar behavior is found in ageostrophic (classical Stone) baroclinic instability by describing accurately the dependence of the solutions on the Richardson number with only two layers.