Zonal Shear Flows with a Free Surface: Hamiltonian Formulation and Linear and Nonlinear Stability

TL;DR

This paper develops a Hamiltonian framework for zonal shear flows with free surfaces, extending stability theorems and providing conditions for linear and nonlinear stability, including effects of vortex stretching.

Contribution

It introduces a Hamiltonian formulation that incorporates vortex stretching effects and generalizes existing stability theorems for zonal shear flows.

Findings

Generalized Flierl-Stern-Whitehead theorem for nonlinear structures

Derived stability conditions based on potential vorticity gradient

Extended theoretical understanding of free surface effects on flow stability

Abstract

General theoretical results via a Hamiltonian formulation are developed for zonal shear flows with the inclusion of the vortex stretching effect of the deformed free surface. These results include a generalization of the Flierl-Stern-Whitehead zero angular momentum theorem} for localized nonlinear structures on or off the beta-plane, and sufficient conditions for linear and nonlinear stability in the Liapunov sense - the latter are derived via bounds on the equilibrium potential vorticity gradient.