TY - JOUR

T1 - Self-stretching of a perturbed vortex filament I. The asymptotic equation for deviations from a straight line

AU - Klein, Rupert

AU - Majda, Andrew J.

N1 - Funding Information:
~Supported through a post-doctoral grant by the Deutsche Forschungsgemeinschaft DFG. 2Present address: Institut fiir Technische Mechanik, Rhein.-Westf. Technische Hochschule Aachen, Templergraben 64, W-5100 Aachen, Germany. 3partially supported by grants A.R.O. DAAL 03-89-K-0013, O.N.R. N00014-89-J-1044 and NSF DMS 8702864.

PY - 1991/4/2

Y1 - 1991/4/2

N2 - 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.

AB - 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.

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U2 - 10.1016/0167-2789(91)90151-X

DO - 10.1016/0167-2789(91)90151-X

M3 - Article

AN - SCOPUS:0001612122

VL - 49

SP - 323

EP - 352

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 3

ER -