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 simpli fied asymptotic equations for nearly parallel vortex filaments, which have been derived recently by Klein, Majda, and Damodaran. These simplified equa tions arise from a systematic asymptotic expansion of the Navier-Stokes equ ation and involve the motion of families of curves, representing the vortex filaments, under linearized self-induction and mutual potential vortex int eraction. We consider here the equilibrium statistical mechanics of arbitra rily large numbers of nearly parallel filaments with equal circulations. Fi rst, the equilibrium Gibbs ensemble is written down exactly through functio n space integrals; then a suitably scaled mean held statistical theory is d eveloped in the limit of infinitely many interacting filaments. The mean fi eld equations involve a novel Hartree-like problem with a two-body logarith mic interaction potential and an inverse temperature given by the normalize d length of the filaments. We analyze the mean field problem and show vario us equivalent variational formulations of it. The mean field statistical th eory for nearly parallel vortex filaments is compared and contrasted with t he well-known mean field statistical theory for two-dimensional point vorti ces. The main ideas are first introduced through heuristic reasoning and th en are confirmed by a mathematically rigorous analysis. A potential applica tion of this statistical theory to rapidly rotating convection in geophysic al flows is also discussed briefly. (C) 2000 John Wiley & Sons, Inc.