Selfstretching of a perturbed vortex filament I. The asymptotic equation for deviations from a straight line
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 selfinduction equation in that it includes selfstretching 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 selfinduction 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 selfstretch 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 NavierStokes 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 smallamplitude but shortwavelength 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.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 April 1991
 DOI:
 10.1016/01672789(91)90151X
 Bibcode:
 1991PhyD...49..323K