Exact solutions of an energy-enstrophy theory for the barotropic vorticityequation on a rotating sphere

The equilibrium statistical mechanics of the energy-enstrophy theory for th e barotropic vorticity equation is solved exactly in the sense that a expli citly 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 lim it as the coarse graining is refined. This family of continuous-spin lattic e Hamiltonians is shown to be nondegenerate under different point vortex di scretizations of the EVE. Under the assumption that the energy and the enst rophy (mean-squared absolute vorticity) are conserved, a long-range version of Kac's spherical model with logarithmic interaction is derived and solve d exactly in the zero total circulation or neutral vortex gas case by the m ethod of steepest descent. The spherical model formulation is based on the fundamental observation that the conservation of enstrophy is mathematicall y equivalent to Kac's spherical constraint. Two new features of this spheri cal model are (i) it allows negative temperatures, and (ii) a nonextensive thermodynamic limit where the strength of the interaction scales with the n umber of lattice sites but where the size of the physical domain remains fi xed; novel interpretations of the saddle point criterion for negative tempe ratures will be formulated. This spherical model is shown to have a free en ergy that is analytic in the properly scaled inverse temperatures <(<beta>) over tilde> in the range 0 = <(<beta>)over tilde>(*) < <(beta )over tilde> < <(beta )over tilde>(c) = N(*)(2)pi (2)/2K in the nonextensive continuum l imit, with K being the fixed value of the enstrophy. The boundary <(<beta>) over tilde>(*) = 0 agrees with the known numerical and analytical results o n the occurrence of coherent or ordered structures at negative temperatures . Spin-spin correlations are calculated giving two-point vorticity correlat ions that are important to the study of turbulence. Physical interpretation s of the results in this paper are obtained and applied to planetary atmosp heres. (C) 2001 Elsevier Science B.V. All rights reserved.