Euler Equation on a Rotating Surface

TL;DR

This paper investigates the behavior of 2D Euler equations on rotating surfaces, focusing on conservation laws, long-term estimates, and the stability of zonal flows, with implications for planetary band structures.

Contribution

It introduces new analytical and numerical methods to analyze the stability and long-term behavior of solutions to Euler equations on rotating surfaces of revolution.

Findings

Long-time estimates for solutions to Euler equations on rotating surfaces

Stability analysis of stationary zonal fields

Numerical confirmation of theoretical stability results

Abstract

We study 2D Euler equations on a rotating surface, subject to the effect of the Coriolis force, with an emphasis on surfaces of revolution. We bring in conservation laws that yield long time estimates on solutions to the Euler equation, and examine ways in which the solutions behave like zonal fields, building on work of B.~Cheng and A.~Mahalov, examining how such 2D Euler equations can account for the observed band structure of rapidly rotating planets. Specific results include both an analysis of time averages of solutions and a study of stability of stationary zonal fields. The latter study includes both analytical and numerical work.