  Exact solutions of a family of Heisenberg-Ising spin-lattice models for a
coupled barotropic flow - massive rotating sphere system under microcanonical
constraint on relative enstrophy is obtained by the method of spherical
constraint. Phase transitions representative of Bose-Einstein condensation in
which highly ordered super and sub-rotating states self-organize from random
initial vorticity states are calculated exactly and related to three key
parameters - spin of sphere, kinetic energy of the barotropic flow which is
specified by the inverse temperature and amount of relative enstrophy which is
held fixed. Angular momentum of the barotropic fluid relative to the rotating
frame of the infinitely massive sphere is the main order parameter in this
statistical mechanics problem $-$ it is not constrained either canonically nor
microcanonically as coupling between the fluid and the rotating sphere by a
complex torque is responsible for its change. This coupling and exchange of
angular momentum is a necessary condition for condensation in this spin-lattice
system. There is no low temperature defects in this model - the partition
function is calculated in closed form for all positive and negative
temperatures. Also note-worthy is the fact that this statistical equilibrium
model is not a mean field model and can be extended to treat fluctuations if
required in more complex coupled flows.
