Self containment radius for rotating planar flows, single-signed vortex gas and electron plasma

A low temperature relation $R^2 = \Omega \beta / 4 \pi \mu$ between the radius $R$ of a compactly supported 2D vorticity (plasma density) field, the total circulation $\Omega$ (total electron charge) and the ratio $\mu / \beta$ (Larmor frequency), is rigorously derived from a variational Principle of Minimum Energy for 2D Euler dynamics. This relation and the predicted structure of the global minimizers or ground states are in agreement with the radii of the most probable vorticity distributions for a vortex gas of $N$ point vortices in the unbounded plane for a very wide range of temperatures, including $\beta = O(1)$. In view of the fact that the planar vortex gas is representative of many 2D and 2.5D statistical mechanics models for geophysical flows, the Principle of Minimum Energy is expected to provide a useful method for predicting the statistical properties of these models in a wide range of low to moderate temperatures.