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

The equilibrium statistical mechanics of the energy-enstrophy theory for th e two-dimensional (2D) Euler equations is solved exactly. A family of latti ce 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 vor tex discretizations of the Euler equations. Under the assumptions that the energy, total circulation and the enstrophy (mean squared vorticity) are co nserved, this lattice vortex gas model is equivalent to a long range versio n of Kac's exactly solvable spherical model with logarithmic interaction. T he spherical model formulation is based on the fundamental observation that the conservation of enstrophy is mathematically equivalent to Kac's spheri cal constraint. This spherical model is shown to have a free energy that is analytic in the properly scaled inverse temperatures <(<beta>)over tilde> in the range 0=<(<beta>)over tilde>(*)<<(<beta>)over tilde><<(<beta>)over t ilde>(c)=4 pi /3. Phase transitions occur at the positive value <(<beta>)ov er tilde>(c) and <(<beta>)over tilde>(*)=0. Spin-spin correlations are calc ulated giving two-point vorticity correlations that are important to the st udy of turbulence. There are exactly three distinct phases in the energy-en strophy theory for 2D flows, namely (a) an uncorrelated high positive tempe rature phase, (b) an antiferromagnetic checkerboard vorticity pattern at lo w positive temperature, and (c) a highly correlated physical domain scale v orticity pattern (for instance, a large positive vorticity region surrounde d by a sea of negative vorticity) at negative temperatures. The boundary <( <beta>)over tilde>(*)=0 agrees with the known numerical and analytical resu lts on the occurrence of coherent or ordered structures at negative tempera tures. The critical temperature <(<beta>)over tilde>(c)>0 is new, as is the corresponding checkerboard low positive temperature phase. Physical interp retations of the results in this paper are obtained. (C) 2001 American Inst itute of Physics.