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 _ { - } . 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 = n _ { u } 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 = n _ { d } for T < T _ { + } . A detailed comparison is made between the results of the mean field theory and the results of Monte-Carlo simulations , dynamic numerical simulations and variational theory .