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Motion of open vortex-current filaments under the Biot–Savart model

Published online by Cambridge University Press:  12 December 2017

Daniel T. Kennedy  and
Robert A. Van Gorder Open the ORCID record for Robert A. Van Gorder [Opens in a new window]
Daniel T. Kennedy
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
Robert A. Van Gorder*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
*
†Email address for correspondence: Robert.VanGorder@maths.ox.ac.uk
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Abstract

Vortex-current filaments have been used to study phenomena such as coronal loops and solar flares as well as tokamaks, and recent experimental work has demonstrated dynamics akin to vortex-current filaments on a table-top plasma focus device. While MHD vortex dynamics and related applications to turbulence have attracted consideration in the literature due to a wide variety of applications, not much analytical progress has been made in this area, and the analysis of such vortex-current filament solutions under various geometries may motivate further experimental efforts. To this end, we consider the motion of open, isolated vortex-current filaments in the presence of magnetohydrodynamic (MHD) as well as the standard hydrodynamic effects. We begin with the vortex-current model of Yatsuyanagi, Hatori & Kato (J. Phys. Soc. Japan, vol. 65, 1996, pp. 745–759) giving the self-induced motion of a vortex-current filament. We give the ‘cutoff’ formulation of the Biot–Savart integrals used in this model, to avoid the singularity at the vortex core. We then study the motion of a variety of vortex-current filaments, including helical, planar and self-similar filament structures. In the case where MHD effects are weak relative to hydrodynamic effects, the filaments behave as expected from the pure hydrodynamic theory. However, when MHD effects are strong enough to dominate, then we observe structural changes to the filaments in all cases considered. The most common finding is reversal of vortex-current filament orientation for strong enough MHD effects. Kelvin waves along a vortex filament (as seen for helical and self-similar structures) will reverse their translational and rotational motion under strong MHD effects. Our findings support the view that vortex-current filaments can be studied in a manner similar to classical hydrodynamic vortex filaments, with the primary role of MHD effects being to change the filament motion, while preserving the overall geometric structure of such filaments.

JFM classification

MHD and Electrohydrodynamics: MHD and Electrohydrodynamics Vortex Flows: Vortex dynamics Vortex Flows: Vortex Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 836 , 10 February 2018 , pp. 532 - 559
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Copyright
© 2017 Cambridge University Press 

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