Equilibrium statistical theory for nearly parallel vortex filaments

The first mathematically rigorous equilibrium statistical theory for three-dimensional vortex filaments is developed here in the context of the simplified asymptotic equations for nearly parallel vortex filaments, which have been derived recently by Klein, Majda, and Damodaran. These simplified equations arise from a systematic asymptotic expansion of the Navier-Stokes equation and involve the motion of families of curves, representing the vortex filaments, under linearized self-induction and mutual potential vortex interaction. We consider here the equilibrium statistical mechanics of arbitrarily large numbers of nearly parallel filaments with equal circulations. First, the equilibrium Gibbs ensemble is written down exactly through function space integrals; then a suitably scaled mean field statistical theory is developed in the limit of infinitely many interacting filaments. The mean field equations involve a novel Hartree-like problem with a two-body logarithmic interaction potential and an inverse temperature given by the normalized length of the filaments. We analyze the mean field problem and show various equivalent variational formulations of it. The mean field statistical theory for nearly parallel vortex filaments is compared and contrasted with the well-known mean field statistical theory for two-dimensional point vortices. The main ideas are first introduced through heuristic reasoning and then are confirmed by a mathematically rigorous analysis. A potential application of this statistical theory to rapidly rotating convection in geophysical flows is also discussed briefly.