Navier-Stokes equations on a rapidly rotating sphere

TL;DR

This paper proves that solutions to the Navier-Stokes equations on a rapidly rotating sphere tend to become zonal over time, with the flow simplifying to a stable steady state as rotation speed increases.

Contribution

It extends previous results from the eta-plane to a spherical setting, demonstrating the long-term zonal behavior and stability of solutions under rapid rotation.

Findings

Solutions become predominantly zonal in the long-term limit

Non-zonal energy is bounded by a small parameter 

Global attractor reduces to a single stable steady state with fast rotation

Abstract

We extend our earlier \beta-plane results [al-Jaboori and Wirosoetisno, 2011, DCDS-B 16:687--701] to a rotating sphere. Specifically, we show that the solution of the Navier--Stokes equations on a sphere rotating with angular velocity 1/\epsilon\ becomes zonal in the long time limit, in the sense that the non-zonal component of the energy becomes bounded by \epsilon M. We also show that the global attractor reduces to a single stable steady state when the rotation is fast enough.