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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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References

del Álamo, J., Jimenez, J., Zandonade, P. & Moser, R. D. 2006 Self-similar vortex clusters in the turbulent logarithmic region. J. Fluid Mech. 561, 329–358.Google Scholar
Alexandrova, O., Mangeney, A., Maksimovic, M., Cornilleau-Wehrlin, N., Bosqued, J.-M. & André, M. 2006 Alfvén vortex filaments observed in magnetosheath downstream of a quasi-perpendicular bow shock. J. Geophys. Res. 111 (A12), A12208.Google Scholar
de Andrade, L. C. G. 2006 Vortex filaments in MHD. Phys. Scr. 73, 484.Google Scholar
Arms, R. J. & Hama, F. R. 1965 Localized-induction concept on a curver vortex and motion of an elliptic vortex ring. Phys. Fluids 8, 553–559.Google Scholar
Bewley, G. P., Paoletti, M. S., Sreenivasan, K. R. & Lathrop, D. P. 2008 Characterization of reconnecting vortices in superfluid helium. Proc. Natl Acad. Sci. USA 105, 13707–13710.Google Scholar
Boersma, J. & Wood, D. H. 1999 On the self-induced motion of a helical vortex. J. Fluid Mech. 384, 263–279.Google Scholar
Da Rios, L. S. 1906 Sul moto d’un liquido indefinite con un filetto vorticoso di forma qualunque. Rend. Circ. Mat. Palermo 22, 117.Google Scholar
Das, C., Kida, S. & Goto, S. 2001 Overall self-similar decay of two-dimensional turbulence. J. Phys. Soc. Japan 70, 966–976.Google Scholar
Fernandez, V. M., Zabusky, N. J. & Gryanik, V. M. 1995 Vortex intensification and collapse of the Lissajous-elliptic ring: single-and multi-filament Biot-Savart simulations and visiometrics. J. Fluid Mech. 299, 289–331.Google Scholar
Fukumoto, Y. 1997 Stationary configurations of a vortex filament in background flows. Proc. R. Soc. Lond. A 453, 1205–1232.CrossRefGoogle Scholar
Gutiérrez, S., Rivas, J. & Vega, L. 2003 Formation of singularities and self-similar vortex motion under the localized induction approximation. Comm. Partial Differential Equations 28, 927–968.Google Scholar
Hasimoto, H. 1971 Motion of a vortex filament and its relation to elastica. J. Phys. Soc. Japan 31, 293–294.Google Scholar
Kida, S. 1981 A vortex filament moving without change of form. J. Fluid Mech. 112, 397–409.Google Scholar
Kida, S. 1982 Stability of a steady vortex filament. J. Phys. Soc. Japan 51, 1655–1662.Google Scholar
Kimura, Y. 1987 Similarity solutions of two-dimensional point vortices. J. Phys. Soc. Japan 56, 2024–2030.Google Scholar
Kimura, Y. 2009 Self-similar collapse of a 3D straight vortex filament model. Geophys. Astrophys. Fluid Dyn. 103, 135–142.Google Scholar
Kimura, Y. 2010 Self-similar collapse of 2D and 3D vortex filament models. Theor. Comput. Fluid Dyn. 24, 389–394.Google Scholar
Lipniacki, T. 2000 Evolution of quantum vortices following reconnection. Eur. J. Mech. (B/Fluids) 19, 361–378.Google Scholar
Lipniacki, T. 2003a Quasi-static solutions for quantum vortex motion under the localized induction approximation. J. Fluid Mech. 477, 321–337.Google Scholar
Lipniacki, T. 2003b Shape-preserving solutions for quantum vortex motion under localized induction approximation. Phys. Fluids 15, 1381–1395.Google Scholar
Miyamoto, K. 1980 Plasma Physics for Nuclear Fusion. MIT Press.Google Scholar
Moore, D. W. & Saffman, P. G. 1972 The motion of a vortex filament with axial flow. Phil. Trans. R. Soc. Lond. A 272 (1226), 403–429.Google Scholar
Mück, B., Günther, C., Müller, U. & Bühler, L. 2000 Three-dimensional MHD flows in rectangular ducts with internal obstacles. J. Fluid Mech. 418, 265–295.Google Scholar
Mukhovatov, V. S. & Shafranov, V. D. 1971 Plasma equilibrium in a tokamak. Nucl. Fusion 11 (6), 605–633.Google Scholar
Pelz, R. B. 1997 Locally self-similar, finite-time collapse in a high-symmetry vortex filament model. Phys. Rev. E 55, 1617.Google Scholar
Ricca, R. L. 1991 Rediscovery of Da Rios equations. Nature 352, 561–562.Google Scholar
Ricca, R. L. 1994 The effect of torsion on the motion of a helical vortex filament. J. Fluid Mech. 273, 241–259.Google Scholar
Ricca, R. L. 1996 The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics. Fluid Dyn. Res. 18, 245–268.Google Scholar
Sheel, T. K., Yasuoka, K. & Obi, S. 2007 Fast vortex method calculation using a special-purpose computer. Comput. Fluids 36, 1319–1326.Google Scholar
Shibata, K. 1996 New observational facts about solar flares from Yohkoh studies: evidence of magnetic reconnection and a unified model of flares. Adv. Space Res. 17 (4), 9–18.Google Scholar
Siggia, E. D. 1985 Collapse and amplification of a vortex filament. Phys. Fluids 28, 794–805.Google Scholar
Siggia, E. D. & Pumir, A. 1985 Incipient singularities in the Navier–Stokes equations. Phys. Rev. Lett. 55, 1749.Google Scholar
Sonin, E. B. 2012 Dynamics of helical vortices and helical-vortex rings. Europhys. Lett. 97, 46002.Google Scholar
Soto, L., Pavez, C., Castillo, F., Veloso, F., Moreno, J. & Auluck, S. K. H. 2014 Filamentary structures in dense plasma focus: current filaments or vortex filaments? Phys. Plasmas 21 (7), 072702.Google Scholar
Suzuki, T., Ito, A. & Yoshida, Z. 2003 Statistical model of current filaments in a turbulent plasma. Fluid Dyn. Res. 32, 247–260.Google Scholar
Umeki, M. 2010 A locally induced homoclinic motion of a vortex filament. Theor. Comput. Fluid Dyn. 24, 383–387.Google Scholar
Van Gorder, R. A. 2012a Exact solution for the self-induced motion of a vortex filament in the arclength representation of the local induction approximation. Phys. Rev. E 86, 057301.Google Scholar
Van Gorder, R. A. 2012b Integrable stationary solution for the fully nonlinear local induction equation describing the motion of a vortex filament. Theor. Comput. Fluid Dyn. 26, 591–594.Google Scholar
Van Gorder, R. A. 2013a Orbital stability for rotating planar vortex filaments in the Cartesian and arclength forms of the local induction approximation. J. Phys. Soc. Japan 82, 094005.Google Scholar
Van Gorder, R. A. 2013b Scaling laws and accurate small-amplitude stationary solution for the motion of a planar vortex filament in the Cartesian form of the local induction approximation. Phys. Rev. E 87, 043203.Google Scholar
Van Gorder, R. A. 2013c Self-similar vortex dynamics in superfluid 4He under the Cartesian representation of the Hall–Vinen model including superfluid friction. Phys. Fluids 25, 095105.Google Scholar
Van Gorder, R. A. 2015a The Biot–Savart description of Kelvin waves on a quantum vortex filament in the prescence of mutual friction and a driving fluid. Proc. R. Soc. A 471, 20150149.Google Scholar
Van Gorder, R. A. 2015b Helical vortex filament motion under the non-local Biot–Savart model. J. Fluid Mech. 762, 141–155.Google Scholar
Van Gorder, R. A. 2015c Non-local dynamics governing the self-induced motion of a planar vortex filament. Phys. Fluids 27, 065105.Google Scholar
Van Gorder, R. A. 2016 Self-similar vortex filament motion under the non-local Biot–Savart model. J. Fluid Mech. 802, 760–774.Google Scholar
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54, 641–663.Google Scholar
Yatsuyanagi, Y., Ebisuzaki, T., Hatori, T. & Kato, T. 2003 Filamentary magnetohydrodynamic simulation model, current-vortex method. Phys. Plasmas 10, 3181–3187.Google Scholar
Yatsuyanagi, Y., Hatori, T. & Kato, T. 1996 The equations of motion of a vortex-current filaments. J. Phys. Soc. Japan 65, 745–759.Google Scholar
Yatsuyanagi, Y., Hatori, T. & Kato, T. 1998 Numerical simulations of the vortex-current filaments motion. J. Phys. Soc. Japan 67 (1), 166–175.Google Scholar
Yatsuyanagi, Y., Hatori, T. & Kato, T. 2001 Chaotic reconnection due to fast mixing of vortex-current filaments. Earth Planets Space 53, 615–618.Google Scholar
Yoshimoto, H. & Goto, S. 2007 Self-similar clustering of inertial particles in homogeneous turbulence. J. Fluid Mech. 577, 275–286.Google Scholar
Zhou, H. 1997 On the motion of slender vortex filaments. Phys. Fluids 9, 970–981.Google Scholar