Formal Stability of Tetrahedral Non-Zonal Flows on the Sphere

We investigate the formal stability of finite-amplitude non-zonal flows bifurcating from the trivial state in the unforced 2D Euler equations on the sphere. To bypass the degeneracy of the spherical Laplacian and filter out the low-frequency Fj{\o}rtoft instabilities, we restrict the functional space to the invariant subspace of the tetrahedral symmetry group. Using Arnold's Energy-Casimir method, we prove that the linearized elliptic operator derived via Liapunov-Schmidt reduction acts as the Hessian of the conserved functional. By tracking the critical eigenvalue along the bifurcating branches via the Crandall-Rabinowitz theorem, we establish a relation between the bifurcation topology and formal stability. Applying this framework to four distinct geophysical profile functions, we demonstrate that subcritical polynomial and supercritical sine-Gordon flows achieve a negative-definite second variation, that is, their formal stability. In contrast, subcritical sinh-Gordon and supercritical Liouville exponential flows generate saddle points, making them unstable. This classification identifies the specific nonlinear interactions required for the persistence of large-scale coherent waves in planetary atmospheres.