Maximum entropy states of quasi-geostrophic point vortices

Abstract

The statistical equilibrium state of quasi-geostrophic point vortices is investigated theoretically, based on the maximum entropy theory. We search for the state of maximum Shannon entropy under the constraints of the vertical vorticity distribution P(z), the angular momentum I, and the energy of the vortex system E. Solutions of the mean field equation are obtained by the numerical procedure proposed by Turkington and Whittaker. The most probable state in an infinite fluid domain is axisymmetric, whose radial distribution depends both on the vertical vortex distribution P(z) and the total energy of the vortex system E. At a certain critical energy value Ec, the number of microscopic state of fixed angular momentum becomes largest (zero-inverse temperature state), where the radial distribution is Gaussian at any vertical height. When the energy is smaller (E Ec: negative temper...