Simplied Equations for the Interaction of Nearly Parallel 3-D Vortex Filaments,

Abstract

New simplified asymptotic equations for the interaction of nearly parallel vortex filaments are derived and analysed here. The simplified equations retain the important physical effects of linearized local self-induction and nonlinear potential vortex interaction among different vortices but neglect other non-local effects of self-stretching and mutual induction. These equations are derived systematically in a suitable distinguished asymptotic limit from the Navier–Stokes equations. The general Hamiltonian formalism and conserved quantities for the simplified equations are developed here. Properties of these asymptotic equations for the important special case involving nearly parallel pairs of interacting filaments are developed in detail. In particular, strong evidence is presented that for any filament pair with a negative circulation ratio, there is finite-time collapse in a self-similar fashion independent of the perturbation but with a structure depending on the circulation ratio. On the other hand, strong evidence is presented that no finite-time collapse is possible for perturbations of a filament pair with a positive circulation ratio. The present theory is also compared and contrasted with earlier linear and nonlinear stability analyses for pairs of filaments.