Motion of a Vortex Filament in the Local Induction Approximation: Reformulation of the Da Rios-Betchov Equations in the Extrinsic Filament Coordinate Space

TL;DR

This paper reformulates the Da Rios-Betchov equations in extrinsic coordinate space, simplifying the analysis of vortex filament motion and aligning well with experimental observations of vortex dynamics.

Contribution

It introduces a new formulation of vortex filament equations in extrinsic space, facilitating easier computation and better understanding of vortex evolution and slip effects.

Findings

Reformulation aligns with the Betchov-Hasimoto solution in small-amplitude limit.

Qualitative agreement with laboratory experiments on vortex waves.

Provides insight into vortex filament slipping motion effects.

Abstract

In recognition of the highly non-trivial task of computation of the inverse Hasimoto transformation mapping the intrinsic geometric parameter space onto the extrinsic vortex filament coordinate space a reformulation of the Da Rios-Betchov equations in the latter space is given. The nonlinear localized vortex filament structure solution given by the present formulation is in detailed agreement with the Betchov-Hasimoto solution in the small-amplitude limit and is also in qualitative agreement with laboratory experiment observations of helical-twist solitary waves propagating on concentrated vortices in rotating fluids. The present formulation also provides for a discernible effect of the slipping motion of a vortex filament on the vortex evolution.