Phase transitions of barotropic flow coupled to a massive rotating sphere - derivation of a fixed point equation by the Bragg method

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The kinetic energy of barotropic flow coupled to an infnitely massive rotating sphere by an unresolved complex torque mechanism is approximated by a discrete spin-lattice model of fluid vorticity on a rotating sphere, analogous to a one-step renormalized Ising model on a sphere with global interactions. The constrained energy functional is a function of spin-spin coupling and spin coupling with the rotation of the sphere. A mean field approximation similar to the Curie-Weiss theory, modeled after that used by Bragg and Williams to treat a two dimensional Ising model of ferromagnetism, is used to find the barotropic vorticity states at thermal equilibrium for given temperature and rotational frequency of the sphere. A fixed point equation for the most probable barotropic flow state is one of the main results.