Stability of degree-2 Rossby-Haurwitz waves

TL;DR

This paper proves the orbital stability of degree-2 Rossby-Haurwitz waves on a rotating sphere, confirming a recent conjecture, using a variational approach and analyzing rearrangements of functions.

Contribution

It establishes the orbital stability of degree-2 RH waves, providing a novel variational characterization and extending the analysis to related flows.

Findings

Confirmed orbital stability of degree-2 RH waves

Developed variational characterizations for solutions

Extended approach to degree-1 RH waves and zonal flows

Abstract

Rossby-Haurwitz (RH) waves are important explicit solutions of the incompressible Euler equation on a two-dimensional rotating sphere. In this paper, we prove the orbital stability of degree-2 RH waves, which confirms a conjecture proposed by A. Constantin and P. Germain in [Arch. Ration. Mech. Anal. 245, 587-644, 2022]. The proofs are based on a variational approach, with the main challenge being to establish suitable variational characterizations for the solutions under consideration. In this process, the set of rearrangements of a fixed function plays a vital role. We also apply our approach to the stability analysis of degree-1 RH waves, Arnold-type flows, and zonal flows with monotone absolute vorticity.