Phase transitions of barotropic flow on the sphere by the Bragg method

The kinetic energy of barotropic flow is approximated by a Discrete Model of Fluid Vorticity on a rotating sphere, analogous to a one-step renormalized Ising model on a sphere with global interactions. The constrained Hamiltonian is a function of spin-spin coupling and spin coupling with the rotation of the sphere. A mean field approximation, modeled after that used by Bragg and Williams to treat a two dimensional Ising model of ferromagnetism, is used to find the fluid vorticity states at thermal equalibrium for given temperature and rotational frequency of the sphere. In positive temperatures for fixed planetary spin there is a phase transition $T_c>T>0$ from a disordered state to a counter-rotating solid-body state. For fixed planetary spin at negative temperatures there is is a phase transition $T_c