Self-stretching of a perturbed vortex filament I. The asymptotic equation for deviations from a straight line

Rupert Klein, Andrew J. Majda

Research output: Contribution to journalArticle

Abstract

A new asymptotic equation is derived for the motion of thin vortex filaments in an incompressible fluid at high Reynolds numbers. This equation differs significantly from the familiar local self-induction equation in that it includes self-stretching of the filament in a nontrivial, but to some extent analytically tractable, fashion. Under the same change of variables as employed by Hasimoto (1972) to convert the local self-induction equation to the cubic nonlinear Schrödinger equation, the new asymptotic propagation law becomes a cubic nonlinear Schrödinger equation perturbed by an explicit nonlocal, linear operator. Explicit formulae are developed which relate the rate of local self-stretch along the vortex filament to a particular quadratic functional of the solution of the perturbed Schrödinger equation. The asymptotic equation is derived systematically from suitable solutions of the Navier-Stokes equations by the method of matched asymptotic expansions based on the limit of high Reynolds numbers. The key idea in the derivation is to consider a filament whose core deviates initially from a given smooth curve only by small-amplitude but short-wavelength displacements balanced so that the axial length scale of these perturbations is small compared to an integral length of the background curve but much larger than a typical core size δ=O(Re -1 2) of the filament. In a particular distinguished limit of wavelength, preturbation amplitude and filament core size the nonlocal induction integral has a simplified asymptotic representation and yields a contribution in the Schrödinger equation that directly competes with the cubic nonlinearity.

Original languageEnglish (US)
Pages (from-to)323-352
Number of pages30
JournalPhysica D: Nonlinear Phenomena
Volume49
Issue number3
DOIs
StatePublished - Apr 2 1991

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vortex filaments
Vortex Filament
Nonlinear equations
Straight Line
Stretching
Vortex flow
Reynolds number
Deviation
deviation
Wavelength
Filament
filaments
Navier Stokes equations
Proof by induction
induction
Cubic equation
high Reynolds number
Fluids
nonlinear equations
Nonlinear Equations

ASJC Scopus subject areas

  • Applied Mathematics
  • Statistical and Nonlinear Physics

Cite this

Self-stretching of a perturbed vortex filament I. The asymptotic equation for deviations from a straight line. / Klein, Rupert; Majda, Andrew J.

In: Physica D: Nonlinear Phenomena, Vol. 49, No. 3, 02.04.1991, p. 323-352.

Research output: Contribution to journalArticle

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