Coherent structures in an energy-enstrophy theory for axisymmetric flows

The equilibrium statistical mechanics of a version of an energy-enstrophy theory for the axisymmetric Euler equations is solved exactly in the sense that a configurational integral is calculated in closed form. Under the assumption that the energy and the enstrophy (mean squared azimuthal 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 case in the standard thermodynamic limit. The spherical model formulation is based on the fundamental observation that the conservation of enstrophy expressed microcanonically is mathematically equivalent to Kac’s spherical constraint. Two-point vorticity correlations are calculated exactly in two qualitatively different phases separated by ${\mathrm{\beta ̃}}_{*}\text{=0.}$ Physical interpretations of the results in this paper are obtained and applied to the relaxed end-states of turbulent high Reynolds number round jets. The negative temperature phase is a very high energy (flow rate) and relatively low enstrophy turbulent flow state which has the form of an axially uniform axial round jet plus a long wavelength axial perturbation. The positive temperature phase corresponds to a low energy and high enstrophy (shear) flow state which consists of a disordered vorticity distribution.