Extremal Free Energy in a Simple Mean Field Theory for a Coupled Barotropic Fluid - Rotating Sphere System

A family of spin-lattice models are derived as convergent finite dimensional approximations to the rest frame kinetic energy of a barotropic fluid coupled to a massive rotating sphere. A simple mean field theory for this statistical equilibrium model is formulated and solved, providing precise conditions on the planetary spin and relative enstrophy in order for phase transitions to occur at positive and negative critical temperatures, T+ and T−. When the planetary spin is relatively small, there is a single phase transition at T − < 0, from a preferred mixed vorticity state v = m for all positive temperatures and T < T − to an ordered pro-rotating (west to east) flow state v = nu for T− < T < 0. When the planetary spin is relatively large, there is an additional phase transition at T+ > 0 from a preferred mixed state v = m above T+ to an ordered counter-rotating flow state v = nd for T < T+.