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. In not fixing the angular momentum of the fluid component, there is no Hamiltonian equations of motion of the fluid component of the coupled system. This family is used to formulate a statistical equilibrium model for the energy - relative enstrophy theory of the coupled barotropic fluid - rotating sphere system, known as the spherical model, which because of its microcanonical constraint on relative enstrophy, does not have the low temperature defect of the classical energy - enstrophy theory. This approach differs from previous works and through the quantum - classical mapping between quantum field theory in spatial dimension $d$ and classical statistical mechanics in dimension $d+1,$ provides a new example of Feynman's generalization of the Least Action Principle to problems that do not have a standard Lagrangian or Hamiltonian. A simple mean field theory for this statistical equlibrium 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 $T_{-}.$ 
Chjan Lim