Mean field equilibria of single coherent vortices

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Abstract
We present the calculation of single-vortex statistical equilibria in a disk using the mean field theory respecting all the conservation laws of the two-dimensional Euler equations. The equilibrium may help in understanding the formation of coherent structures in experiments, numerical simulations, and planetary atmospheres. We calculate two-dimensional single-vortex solutions in the disk and confirm the bifurcation from symmetric to off-center vortices predicted by a linear perturbation analysis. With a second-order perturbation analysis and a calculation of the thermodynamic stability we find that both supercritical and subcritical bifurcations can occur, depending on the parameters. The shapes of the off-center vortices also are in good agreement with measurements on an electron plasma.