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Equilibrium statistical theory for nearly parallel vortex filaments. (English) Zbl 1041.76038
In this outstanding paper the first mathematically rigorous equilibrium statistical theory for three-dimensional vortex filaments is developed 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. The authors consider 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. They 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.

MSC:
76B47 Vortex flows for incompressible inviscid fluids
76D17 Viscous vortex flows
82D15 Statistical mechanical studies of liquids
76F02 Fundamentals of turbulence
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76F55 Statistical turbulence modeling
76M35 Stochastic analysis applied to problems in fluid mechanics
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
86A05 Hydrology, hydrography, oceanography
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