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

Chjan C. Lim
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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 $Ω$ 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 $Ω$, 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.

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Chjan C. Lim. 2006-08-09. Energy extremals and Nonlinear Stability in a Variational theory of Barotropic Fluid - Rotating Sphere System. https://doi.org/10.1063/1.2347900

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