SELF-STRETCHING OF A PERTURBED VORTEX FILAMENT .1. THE ASYMPTOTIC EQUATION FOR DEVIATIONS FROM A STRAIGHT-LINE

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 Schrodinger equation, the new asymptotic propagation law becomes a cubic nonlinear Schrodinger 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 Schrodinger 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 delta = O(Re -1/2) of the filament. In a particular distinguished limit of wavelength, perturbation amplitude and filament core size the nonlocal induction integral has a simplified asymptotic representation and yields a contribution in the Schrodinger equation that directly competes with the cubic nonlinearity.