The effects of a higher vorticity moment on a variational problemfor barotropic vorticity on a rotating sphere is examined rigorouslyin the framework of the Direct Method. This variational modeldiffers from previous work on the Barotropic Vorticity Equation(BVE) in relaxing the angular momentum constraint, which then allowsus to state and prove theorems that give necessary and sufficientconditions for the existence and stability of constrained energyextremals in the form of super and sub-rotating solid-body steadyflows. Relaxation of angular momentum is a necessary step in themodeling of the important tilt instability where the rotational axisof the barotropic atmosphere tilts away from the fixed north-southaxis of planetary spin. These conditions on a minimal set ofparameters consisting of the planetary spin, relative enstrophy andthe fourth vorticity moment, extend the results of previous work and clarify the role of the higher vorticity moments in models of geophysical flows.