The Role of Higher Vorticity Moments in a Variational Formulation of Barotropic Flows on a Rotating Sphere

The effects of a higher vorticity moment on a variational problem for barotropic vorticity on a rotating sphere are examined rigorously in the framework of the Direct Method. This variational model differs from previous work on the Barotropic Vorticity Equation (BVE) in relaxing the angular momentum constraint, which then allows us to state and prove theorems that give necessary and sufficient conditions for the existence and stability of constrained energy extremals in the form of super and sub-rotating solid-body steady flows. Relaxation of angular momentum is a necessary step in the modeling of the important tilt instability where the rotational axis of the barotropic atmosphere tilts away from the fixed north-south axis of planetary spin. These conditions on a minimal set of parameters consisting of the planetary spin, relative enstrophy and the fourth vorticity moment, extend the results of previous work and clarify the role of the higher vorticity moments in models of geophysical flows.