Onsager's ensemble for point vortices with random circulations on the sphere
The problem of whether random sampling of circulations of point vortices in Onsager's stationary statistical \(N\)-point-vortex ensemble would, in the continuum \(N\rightarrow \infty \) limit, lead to the Euler continuum vorticities predicted by the Miller-Robert theory is rigorously addressed. The analysis is performed with a view towards two-dimensional smooth manifolds, while presently specialized to random point vortices that move on the 2-sphere \(S^2\). It is proved that typical point vortex distributions obtained from the Onsager ensemble in the \(N\rightarrow \infty \) limit are special solutions of the Euler equations of an incompressible inviscid fluid flow on \(S^2\). A close affinity with the outcomes of Miller-Robert theory is established and conditions stated for the correspondence to become exact. Not addressed in the paper is a connection of the point vortex methods with two-dimensional incompressible viscous Navier-Stokes flows.
incompressible Euler fluid flow on \(S^2\)
scientific article; zbMATH DE number 6135535
scientific article