Euler equations on a fast rotating sphere --- time-averages and zonal flows

TL;DR

This paper analyzes the behavior of inviscid flows on a rapidly rotating sphere, showing that time-averages tend to zonal flows due to Coriolis effects, using geometric methods.

Contribution

It provides a PDE-analytical proof that time-averages of solutions remain close to zonal flows, even from far initial conditions, leveraging Riemannian geometry and Hodge theory.

Findings

Time-averages of solutions are close to zonal flows.

Initial data can be arbitrarily far from zonal flows.

Coriolis variation drives the zonal flow tendency.

Abstract

Motivated by recent studies in geophysical and planetary sciences, we investigate the PDE-analytical aspects of time-averages for barotropic, inviscid flows on a fast rotating sphere $S^2$. Of particular interests are the incompressible Euler equations. We prove that the finite-time-average of the solution stays close to a subspace of \emph{longitude-independent zonal flows}. The intial data can be arbitrarily far away from this subspace. Meridional variation of the Coriolis parameter underlies this phenomenon. Our proofs use Riemannian geometric tools, in particular the Hodge Theory.