Euler equation on a fast rotating ellipsoid

TL;DR

This paper generalizes the analysis of fast rotating Euler equations from spheres to biaxial ellipsoids, demonstrating that zonal flows persist in this more realistic geometry and are independent of the rotation rate.

Contribution

It extends the theoretical framework of Euler flows on rotating spheres to biaxial ellipsoids, establishing the persistence of zonalization in this new setting.

Findings

Time-averaged velocity remains bounded in Sobolev norms.

Velocity converges to a latitude-dependent zonal flow.

Zonalization phenomenon persists on biaxial ellipsoids.

Abstract

This paper extends the analytical study of the incompressible Euler equations from the classical spherical setting to the more realistic geometry of a biaxial ellipsoid. Motivated by the work of Cheng and Mahalov on fast rotating spheres and Xu on Rossby-Haurwitz solutions on ellipsoids, we adapt their framework to establish a parallel result for Euler flows on a rotating ellipsoidal surface. In the regime of rapid rotation, we prove that the time-averaged velocity field remains uniformly bounded in Sobolev norms independent of the rotation rate and converges to a longitude-independent zonal flow depending only on latitude. This shows that the zonalization phenomenon discovered by Cheng and Mahalov on the sphere persists on biaxial ellipsoids, thereby bridging the gap between spherical and ellipsoidal theories of fast rotating Euler dynamics.