Statistical Point Vortex Theories

Abstract

In a paper that initiated the modern treatment of “statistical hydrodynamics,” Onsager (1949) described a point vortex-based theory of turbulence that has proven to be remarkably fruitful, if not all together perfect, and which continues to produce a lively and growing literature. Motivated largely by an attempt to explain the presence of large, isolated vortices in a wide range of turbulent flows (see the descriptive paper of McWilliams (1984))¹, Onsager’s theory adopts three main assumptions:

• The theory is based on the inviscid Euler equations in two dimensions, thereby ignoring viscous dissipation (considered to be important at small scales) and vortex-stretching (considered to be important at all scales).

• The theory is based on a point vortex discretization of the Euler equations thereby producing a “vortex gas,” which, while derived from the incompressible equations of motion, behaves in many respects like a compressible system with the ability of point vortices to cluster or expand.

• The theory makes use of equilibrium statistical mechanics to explain asymptotic properties of turbulent flows, typically thought to be more amenable to nonequilibrium techniques able to handle large flucuations far from equilibrium.