Exact solutions of an 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 nonGaussian 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 the interaction scales with the number of lattice sites but where the size of the physical domain remains fixed; novel interpretations of the saddle point criterion for negative temperatures will be formulated. This spherical model is shown to have a free energy that is analytic in the properly scaled inverse temperatures β̃ in the range 0=β̃∗<β̃<β̃c=N∗2π2/2K in the nonextensive continuum limit, with K being the fixed value of the enstrophy. The boundary β̃∗=0 agrees with the known numerical and analytical results on the occurrence of coherent or ordered structures at negative temperatures. Spin–spin correlations are calculated giving two-point vorticity correlations that are important to the study of turbulence. Physical interpretations of the results in this paper are obtained and applied to planetary atmospheres.