Energy extremals and Nonlinear Stability in a Variational theory of Barotropic Fluid - Rotating Sphere System

TL;DR

This paper introduces a variational principle for a barotropic flow on a rotating sphere, revealing conditions for energy extremals and stability, especially highlighting the stability of super-rotating states and the nature of counter-rotating states.

Contribution

It proposes a new variational framework for coupled fluid-sphere systems, analyzing energy extremals and stability in relation to angular momentum and enstrophy.

Findings

Unique global energy maximizer is a super-rotating solid-body flow.

Counter-rotating states are stable only as local minima, not globally.

Super-rotating states are always nonlinearly stable.

Abstract

A new variational principle - extremizing the fixed frame kinetic energy under constant relative enstrophy - for a coupled barotropic flow - rotating solid sphere system is introduced with the following consequences. In particular, angular momentum is transfered between the fluid and the solid sphere through a modelled torque mechanism. The fluid's angular momentum is therefore not fixed but only bounded by the relative enstrophy, as is required of any model that supports super-rotation. The main results are: At any rate of spin $\Omega $ and relative enstrophy, the unique global energy maximizer for fixed relative enstrophy corresponds to solid-body super-rotation; the counter-rotating solid-body flow state is a constrained energy minimum provided the relative enstrophy is small enough, otherwise, it is a saddle point. For all energy below a threshold value which depends on the relative enstrophy and solid spin $\Omega $, the constrained energy extremals consist of only minimizers and saddles in the form of counter-rotating states$.$ Only when the energy exceeds this threshold value can pro-rotating states arise as global maximizers. Unlike the standard barotropic vorticity model which conserves angular momentum of the fluid, the counter-rotating state is rigorously shown to be nonlinearly stable only when it is a local constrained minima. The global constrained maximizer corresponding to super-rotation is always nonlinearly stable.