# Equilibrium statistical theory for nearly parallel vortex filaments

Pierre Louis Lions, Andrew Majda

Research output: Contribution to journalArticle

### Abstract

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.

Original language English (US) 76-142 67 Communications on Pure and Applied Mathematics 53 1 Published - Jan 2000

### Fingerprint

Vortex Filament
Vortex flow
Mean Field
Filament
Geophysical Flows
Mean Field Equation
Point Vortex
Variational Formulation
Statistical mechanics
Interaction
Statistical Mechanics
Function Space
Convection
Asymptotic Expansion
Vortex
Proof by induction
Rotating
Navier-Stokes Equations
Logarithmic
Ensemble

### ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics

### Cite this

Equilibrium statistical theory for nearly parallel vortex filaments. / Lions, Pierre Louis; Majda, Andrew.

In: Communications on Pure and Applied Mathematics, Vol. 53, No. 1, 01.2000, p. 76-142.

Research output: Contribution to journalArticle

@article{4cd0ac810ccc4292bc4887d503617a5b,
title = "Equilibrium statistical theory for nearly parallel vortex filaments",
abstract = "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.",
author = "Lions, {Pierre Louis} and Andrew Majda",
year = "2000",
month = "1",
language = "English (US)",
volume = "53",
pages = "76--142",
journal = "Communications on Pure and Applied Mathematics",
issn = "0010-3640",
publisher = "Wiley-Liss Inc.",
number = "1",

}

TY - JOUR

T1 - Equilibrium statistical theory for nearly parallel vortex filaments

AU - Lions, Pierre Louis

AU - Majda, Andrew

PY - 2000/1

Y1 - 2000/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0034583022&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034583022&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034583022

VL - 53

SP - 76

EP - 142

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 1

ER -