Gradient catastrophe and flutter in vortex filament dynamics

TL;DR

This paper investigates the gradient catastrophe and flutter instability in vortex filament dynamics, linking geometric and analytical aspects, and demonstrating the role of Painlevé-I in regularization.

Contribution

It reveals the geometric nature of flutter as an elliptic umbilic singularity and connects it to the Painlevé-I equation for regularization.

Findings

Flutter manifests as rapid oscillations in filament curves.

The phenomenon is linked to the gradient catastrophe in the dispersionless Da Rios system.

Painlevé-I equation governs the double scaling regularization.

Abstract

Gradient catastrophe and flutter instability in the motion of vortex filament within the localized induction approximation are analyzed. It is shown that the origin if this phenomenon is in the gradient catastrophe for the dispersionless Da Rios system which describes motion of filament with slow varying curvature and torsion. Geometrically this catastrophe manifests as a rapid oscillation of a filament curve in a point that resembles the flutter of airfoils. Analytically it is the elliptic umbilic singularity in the terminology of the catastrophe theory. It is demonstrated that its double scaling regularization is governed by the Painlev\'e-I equation.