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 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 $\mathrm{\beta ̃}$ in the range ${\text{0=β̃}}_{*}{\mathrm{<\beta ̃<\beta ̃}}_c\text{=4π/3.}$ Phase transitions occur at the positive value ${\mathrm{\beta ̃}}_c$ and ${\mathrm{\beta ̃}}_{*}\text{=0.}$ Spin–spin correlations are calculated giving two-point vorticity correlations that are important to the study of turbulence. There are exactly three distinct phases in the energy-enstrophy theory for 2D flows, namely (a) an uncorrelated high positive temperature phase, (b) an antiferromagnetic checkerboard vorticity pattern at low positive temperature, and (c) a highly correlated physical domain scale vorticity pattern (for instance, a large positive vorticity region surrounded by a sea of negative vorticity) at negative temperatures. The boundary ${\mathrm{\beta ̃}}_{*}\text{=0}$ agrees with the known numerical and analytical results on the occurrence of coherent or ordered structures at negative temperatures. The critical temperature ${\mathrm{\beta ̃}}_c\text{>0}$ is new, as is the corresponding checkerboard low positive temperature phase. Physical interpretations of the results in this paper are obtained.