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 dimensionalapproximations to the rest frame kinetic energy of a barotropic fluid coupledto a massive rotating sphere. In not fixing the angular momentum of the fluidcomponent, there is no Hamiltonian equations of motion of the fluid componentof the coupled system. This family is used to formulate a statisticalequilibrium model for the energy - relative enstrophy theory of the coupledbarotropic fluid - rotating sphere system, known as the spherical model, whichbecause of its microcanonical constraint on relative enstrophy, does not havethe low temperature defect of the classical energy - enstrophy theory. Thisapproach differs from previous works and through the quantum - classicalmapping between quantum field theory in spatial dimension $d$ and classicalstatistical mechanics in dimension $d+1,$ provides a new example of Feynman'sgeneralization of the Least Action Principle to problems that do not have astandard Lagrangian or Hamiltonian. A simple mean field theory for thisstatistical equlibrium model is formulated and solved, providing preciseconditions on the planetary spin and relative enstrophy in order for phasetransitions to occur at positive and negative critical temperatures, $T_{+}$and $T_{-}.$