Bifurcation of Tetrahedral Non-Zonal Flows in the 2D Euler Equations on a Rotating Sphere

TL;DR

This paper studies how non-zonal flows bifurcate from stationary solutions of the 2D Euler equations on a rotating sphere, using symmetry reduction and analyzing different nonlinear models.

Contribution

It introduces a symmetry-based reduction method to analyze bifurcations in non-zonal flows on the sphere, revealing how bifurcation topology depends on nonlinearity parity and mass conservation.

Findings

Bifurcation topology depends on nonlinearity parity and mass conservation.

Explicit bifurcation parameter derived via spectral projections.

Symmetry reduction bypasses kernel degeneracy in bifurcation analysis.

Abstract

We investigate the emergence of finite-amplitude non-zonal flows on the sphere $\mathbb{S}^2$ arising from stationary solutions to the 2D Euler equations. By restricting the Laplace-Beltrami eigenspace to the invariant subspace of the tetrahedral symmetry group $\mathbf{T}$, we bypass the $(2l+1)$-dimensional kernel degeneracy, obtaining a scalar Liapunov-Schmidt reduction. We analyze four distinct physical non-linearities: a polynomial model, the sine-Gordon and sinh-Gordon models, and the exponential (Liouville) model. We explicitly derive the bifurcation parameter via spectral projections, proving that the bifurcation topology (subcritical or supercritical) is not a geometric invariant, but is governed by the parity of the nonlinearity and the mass conservation.