Exact solutions of a energy-enstrophy theory for the barotropic vorticity equation on a rotating sphere

The equilibrium statistical mechanics of the energy-enstrophy theory for the barotropic vorticity equation is solved exactly in the sense that a explicitly non-Gaussian configurational integral is calculated in closed form. A family of lattice vortex gas models for the barotropic vorticity equation (BVE) is derived and shown to have a well-defined nonextensive continuum limit as the coarse-graining is refined. This family of continuous-spin lattice Hamiltonians is shown to be nondegenerate under different point vortex discretizations of the BVE. Under the assumption that the energy and the enstrophy (mean squared absolute vorticity) are conserved, a long range version of Kac’s Spherical Model with logarithmic interaction is derived and solved exactly in the zero total circulation or neutral vortex gas case by the method of steepest descent. The spherical model formulation is based on the fundamental observation that the conservation of enstrophy is mathematically equivalent to Kac’s spherical constraint. Two new features of this spherical model are (i) it allows negative temperatures, and (ii) a nonextensive thermodynamic limit where the strength of