Stability of shear shallow water flows with free surface

TL;DR

This paper investigates the stability of inviscid shear shallow water flows with free surface using the Benney equations, establishing conditions for stability and hyperbolicity, and exploring effects of stratification and vorticity.

Contribution

It provides new criteria for stability and hyperbolicity of shear shallow water flows, including explicit Riemann invariants and Hamiltonian structure for certain flow classes.

Findings

Monotonic convex velocity profiles are stable.

Derived hydrodynamic approximations have Hamiltonian structure.

Vorticity has a stabilizing effect in stratified flows.

Abstract

Stability of inviscid shear shallow water flows with free surface is studied in the framework of the Benney equations. This is done by investigating the generalized hyperbolicity of the integrodifferential Benney system of equations. It is shown that all shear flows having monotonic convex velocity profiles are stable. The hydrodynamic approximations of the model corresponding to the classes of flows with piecewise linear continuous and discontinuous velocity profiles are derived and studied. It is shown that these approximations possess Hamiltonian structure and a complete system of Riemann invariants, which are found in an explicit form. Sufficient conditions for hyperbolicity of the governing equations for such multilayer flows are formulated. The generalization of the above results to the case of stratified fluid is less obvious, however, it is established that vorticity has a stabilizing effect.