Nonlinear stability bounds for inviscid, two-dimensional, parallel or circular flows with monotonic vorticity, and the analogous three-dimensional quasi-geostrophic flows

Abstract

Rigorous bounds are obtained on the mean normal displacement of vorticity or potential vorticity contours from their undisturbed parallel (or concentric) positions for incompressible planar flow, flow on the surface of a sphere, and three-dimensional quasi-geostrophic flow. It is required that the basic flows have monotonic distributions of vorticity, and it is this requirement that turns a particular linear combination of conserved quantities, a combination involving the linear or angular impulse and the areas enclosed by vorticity contours, into a norm when viewed in a certain hybrid Eulerian–Lagrangian set of coordinates. Liapunov stability theorems constraining the growth of finite-amplitude disturbances then follow merely from conservation of this norm. As a corollary, it is proved that arbitrarily steep, one-signed vorticity gradients are stable, including the limiting case of a circular patch of uniform vorticity.