Counterexamples to Moffatt's statements on vortex knots

Oleg Bogoyavlenskij
Phys. Rev. E 95, 043104 – Published 12 April 2017

Abstract

One of the well-known problems of hydrodynamics is studied: the problem of classification of vortex knots for ideal fluid flows. In the literature there are known Moffatt statements that all torus knots Km,n for all rational numbers m/n (0<m/n<) are realized as vortex knots for each one of the considered axisymmetric fluid flows. We prove that actually such a uniformity does not exist because it does not correspond to the facts. Namely, we derive a complete classification of all vortex knots realized for the fluid flows studied by Moffatt and demonstrate that the real structure of vortex knots is much more rich because the sets of mutually nonisotopic vortex knots realized for different axisymmetric fluid flows are all different. An exact formula for the limit of the hydrodynamic safety factor qh at a vortex axis is derived for arbitrary axisymmetric fluid equilibria. Another exact formula is obtained for the limit of the magnetohydrodynamics safety factor q at a magnetic axis for the general axisymmetric plasma equilibria.

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  • Received 1 August 2016
  • Revised 12 January 2017

DOI:https://doi.org/10.1103/PhysRevE.95.043104

©2017 American Physical Society

Physics Subject Headings (PhySH)

  1. Research Areas
Fluid Dynamics

Authors & Affiliations

Oleg Bogoyavlenskij

  • Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada K7L 3N6

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Issue

Vol. 95, Iss. 4 — April 2017

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