A long range spherical model and exact solutions of an energy enstrophy theory for two-dimensional turbulence

Abstract

The equilibrium statistical mechanics of the energy-enstrophy theory for the two-dimensional (2D) Euler equations is solved exactly. A family of lattice vortex gas models for the Euler equations is derived and shown to have a well-defined nonextensive continuum limit. This family of continuous-spin lattice Hamiltonians is shown to be nondegenerate under different point vortex discretizations of the Euler equations. Under the assumptions that the energy, total circulation and the enstrophy (mean squared vorticity) are conserved, this lattice vortex gas model is equivalent to a long range version of Kac's exactly solvable spherical model with logarithmic interaction. The spherical model formulation is based on the fundamental observation that the conservation of enstrophy is mathematically equivalent to Kac's spherical constraint. This spherical model is shown to have a free energy that is analytic in the properly scaled inverse temperatures ̃β in the range O = ̃β* < ̃β0 is new, as is the corresponding checkerboard low positive temperature phase. Physical interpretations of the results in this paper are obtained. © 2001 American Institute of Physics.