Onsager's Ensemble for Point Vortices with Random Circulations on the Sphere

Onsager's ergodic point vortex (sub-)ensemble is studied for N vortices which move on the 2-sphere S ² with randomly assigned circulations, picked from an a-priori distribution. It is shown that the typical point vortex distributions obtained from the ensemble in the limit N→∞ are special solutions of the Euler equations of incompressible, inviscid fluid flow on S ². These typical point vortex distributions satisfy nonlinear mean-field equations which have a remarkable resemblance to those obtained from the Miller-Robert theory. Conditions for their perfect agreement are stated. Also the non-random limit, when all vortices have circulation 1, is discussed in some detail, in which case the ergodic and holodic ensembles are shown to be inequivalent.