On zonal steady solutions to the 2D Euler equations on the rotating unit sphere

TL;DR

This paper investigates the structure of stationary solutions to the 2D Euler equations on a rotating sphere, revealing the existence of non-zonal solutions near Rossby-Haurwitz flows and a rigidity result near rigid rotation, depending on the sphere's rotation.

Contribution

It constructs new non-zonal steady solutions near Rossby-Haurwitz flows and proves a rigidity result for solutions near rigid rotation, highlighting the influence of sphere rotation on solution structure.

Findings

Existence of non-zonal steady solutions near Rossby-Haurwitz flows.

Rigidity of solutions near rigid rotation under certain rotation conditions.

Bifurcation of solutions when rotation conditions are not met.

Abstract

The present paper studies the structure of the set of stationary solutions to the incompressible Euler equations on the rotating unit sphere that are near two basic zonal flows: the zonal Rossby-Haurwitz solution of degree 2 and the zonal rigid rotation $Y_1^0$ along the polar axis. We construct a new family of non-zonal steady solutions arbitrarily close in analytic regularity to the second degree zonal Rossby-Haurwitz stream function, for any given rotation of the sphere. This shows that any non-linear inviscid damping to a zonal flow cannot be expected for solutions near this Rossby-Haurwitz solution. On the other hand, we prove that, under suitable conditions on the rotation of the sphere, any stationary solution close enough to the rigid rotation zonal flow $Y_1^0$ must itself be zonal, witnessing some sort of rigidity inherited from the equation, the geometry of the sphere and the base flow. Nevertheless, when the conditions on the rotation of the sphere fail, the set of solutions is much richer and we are able to prove the existence of both explicit stationary and travelling wave non-zonal solutions bifurcating from $Y_1^0$, in the same spirit as those emanating from the zonal Rossby-Haurwitz solution of degree 2.