Dynamics of helical vortices and helical-vortex rings

E. B. Sonin^a

Racah Institute of Physics, Hebrew University of Jerusalem - Jerusalem 91904, Israel

^a sonin@cc.huji.ac.il

Received: 3 December 2011
Accepted: 9 January 2012

Abstract

The letter considers dynamics of helical vortices and helical-vortex rings either solving directly the equations of motions of a vortex line or using canonical relations following from the Hamiltonian equations of motion. An analytical solution in elliptical integrals was found for helical-vortex rings in the local-induction approximation. The analysis based on the canonical Hamilton relation provides a clear physical explanation for anomalous velocities of helical-vortex rings, i.e., for suppression of the velocity and even inversion of its direction at sufficiently large amplitude of the helical distortion. The extended local-induction approximation is suggested, which provides an exact solution for the equations of motion of helical vortices and helical-vortex rings in the limit when the small-pitch helical vortex reduces to a cylindric sheet of uniform vorticity.