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

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.