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

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 0 1 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 0 1 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, we are able to prove the existence of explicit stationary non-zonal solutions bifurcating from Y 0 1 , in the same spirit as those emanating from the zonal Rossby-Haurwitz solution of degree 2.

Traveling Waves Near Couette Flow for the 2D Euler Equation

“…The problem of finding steady states solutions for 2D Euler has been addressed by Nadirashvili in [77], where he studies the geometry and the stability of solutions, following the works of Arnold [3,4,5]. In recent years, a large number of results taking into account the geometry of the steady states and their flexibility/rigidity properties have emerged, see [19,20,22,24,52,53,54,67,74,78] and references therein.…”

“…Recently, there has been a growing interest in the study of existence or not of invariant structures, and their stability for 2D Euler near other shear flows and for related equations, see [22,36,70,72,73,78,80,81]. 1.5.…”

“…where L R i m,ε are given in ( 77) and (78) or equivalently by the right-hand side of ( 75) and (76). After that, the solution (λ ε,κ,m , h ε,κ,m ) will be given by…”

“…Uniqueness. So far we have shown that there exist λ and ( hR 1 n , hR 2 n ) given by 77), (78) for all n ∈ N and satisfying (146), (147), ( 148) and (149).…”

On maximally mixed equilibria of two-dimensional perfect fluids

“…Lin and Zeng [36] discovered non-shear Catseye steady states nearby (at low regularity) to the Couette shear flow, and there are also travelling waves with an order 1 velocity as showed by Castro and Lear [10]. Such steady structure have recently been identified nearby the Kolmogorov flow (in the analytic topology) and the Poiseuille flow by Coti Zelati, Elgindi and Widmayer [20] and by Nualart [48] on the rotating sphere nearby zonal flows. These results provide an obstruction to inviscid damping back to a shear flow for general perturbations nearby certain shear flows of a given structure.…”