  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 $T_{-}.$
