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Variational formulation of relaxed and multi-region relaxed magnetohydrodynamics

Part of: PLA Tribute to V D Shafranov: 1929-2014

Published online by Cambridge University Press:  27 November 2015

R. L. Dewar Open the ORCID record for R. L. Dewar [Opens in a new window] ,
Z. Yoshida ,
A. Bhattacharjee  and
S. R. Hudson
R. L. Dewar*
Affiliation:
Centre for Plasmas and Fluids, Research School of Physics and Engineering, The Australian National University, Canberra, ACT 2601, Australia
Z. Yoshida
Affiliation:
Graduate School of Frontier Sciences, University of Tokyo, Kashiwa, Chiba 277-8561, Japan
A. Bhattacharjee
Affiliation:
Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton, NJ 08543, USA
S. R. Hudson
Affiliation:
Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton, NJ 08543, USA
*
†Email address for correspondence: [email protected]
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Abstract

Ideal magnetohydrodynamics (IMHD) is strongly constrained by an infinite number of microscopic constraints expressing mass, entropy and magnetic flux conservation in each infinitesimal fluid element, the latter preventing magnetic reconnection. By contrast, in the Taylor relaxation model for formation of macroscopically self-organized plasma equilibrium states, all these constraints are relaxed save for the global magnetic fluxes and helicity. A Lagrangian variational principle is presented that leads to a new, fully dynamical, relaxed magnetohydrodynamics (RxMHD), such that all static solutions are Taylor states but also allows state with flow. By postulating that some long-lived macroscopic current sheets can act as barriers to relaxation, separating the plasma into multiple relaxation regions, a further generalization, multi-region relaxed magnetohydrodynamics (MRxMHD) is developed.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 6 , December 2015 , 515810604
Copyright
© Cambridge University Press 2015 

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References

Araki, K. 2015 Differential-geometrical approach to the dynamics of dissipationless incompressible Hall magnetohydrodynamics: I. Lagrangian mechanics on semidirect product of two volume preserving diffeomorphisms and conservation laws. J. Phys. A: Math. Theor. 48, 175501,1–16.CrossRefGoogle Scholar
Arnold, V. I. & Khesin, B. A. 1998 Topological methods in hydrodynamics. In Applied Mathematical Sciences, vol. 125. Springer.Google Scholar
Berger, M. A. 1999 Introduction to magnetic helicity. Plasma Phys. Control. Fusion 41, B167–B175.Google Scholar
Bernstein, I. B., Frieman, E. A., Kruskal, M. D. & Kulsrud, R. M. 1958 An energy principle for hydromagnetic stability problems. Proc. R. Soc. Lond. A 244, 17–40.Google Scholar
Bevir, M. K. & Gray, J. W. 1982 Relaxation, flux conservation and quasi steady state pinches. In Proceedings of the Reversed Field Pinch Theory Workshop, Los Alamos, NM, USA, 28 Apr. – 2 May 1980 (ed. Lewis, H. R.), pp. 176–180. Los Alamos National Laboratory.Google Scholar
Bhattacharjee, A. & Dewar, R. L. 1982 Energy principle with global invariants. Phys. Fluids 25, 887–897.CrossRefGoogle Scholar
Bhattacharjee, A., Hayashi, T., Hegna, C. C., Nakajima, N. & Sato, T. 1995 Theory of pressure-induced islands and self-healing in three-dimensional toroidal magnetohydrodynamic equilibria. Phys. Plasmas 2, 883–888.Google Scholar
Boozer, A. H. & Pomphrey, N. 2010 Current density and plasma displacement near perturbed rational surfaces. Phys. Plasmas 17, 110707,1–4.Google Scholar
Cary, J. R. & Kotschenreuther, M. 1985 Pressure induced islands in three-dimensional toroidal plasma. Phys. Fluids 28, 1392–1401.Google Scholar
Comisso, L., Grasso, D. & Waelbroeck, F. L. 2015a Extended theory of the Taylor problem in the plasmoid-unstable regime. Phys. Plasmas 22, 042109,1–12.Google Scholar
Comisso, L., Grasso, D. & Waelbroeck, F. L. 2015b Phase diagrams of forced magnetic reconnection in Taylor’s model. J. Plasma Phys. 81, 495810510,1–15. Part of a collection on ‘Complex plasma phenomena in the laboratory and in the universe’.Google Scholar
Cordoba, D. & Marliani, C. 2000 Evolution of current sheets and regularity of ideal incompressible magnetic fluids in 2d. Commun. Pure Appl. Maths LIII, 0512–0524.Google Scholar
Dennis, G. R., Hudson, S. R., Dewar, R. L. & Hole, M. J. 2013a The infinite interface limit of multiple-region relaxed MHD. Phys. Plasmas 20, 032509,1–6.CrossRefGoogle Scholar
Dennis, G. R., Hudson, S. R., Dewar, R. L. & Hole, M. J. 2014 Multi-region relaxed magnetohydrodynamics with flow. Phys. Plasmas 21, 042501,1–9.Google Scholar
Dennis, G. R., Hudson, S. R., Terranova, D., Franz, P., Dewar, R. L. & Hole, M. J. 2013b A minimally constrained model of self-organized helical states in reversed-field pinches. Phys. Rev. Lett. 111, 055003,1–5.CrossRefGoogle ScholarPubMed
Dewar, R. L. 1970 Interaction between hydromagnetic waves and a time-dependent, inhomogeneous medium. Phys. Fluids 13, 2710–2720.Google Scholar
Dewar, R. L. 1976 Renormalised canonical perturbation theory for stochastic propagators. J. Phys. A: Math. Gen. 9, 2043–2057.Google Scholar
Dewar, R. L. 1978 Hamilton’s principle for a hydromagnetic fluid with a free boundary. Nucl. Fusion 18, 1541–1553.CrossRefGoogle Scholar
Dewar, R. L., Bhattacharjee, A., Kulsrud, R. M. & Wright, A. M. 2013 Plasmoid solutions of the Hahm–Kulsrud–Taylor equilibrium model. Phys. Plasmas 20, 082103,1–7.CrossRefGoogle Scholar
Dewar, R. L., Hole, M. J., McGann, M., Mills, R. & Hudson, S. R. 2008 Relaxed plasma equilibria and entropy-related plasma self-organization principles. Entropy 10, 621–634.CrossRefGoogle Scholar
Dombre, T., Frisch, U., Greene, J. M., Hénon, M., Mehr, A. & Soward, A. M. 1986 Chaotic streamlines in the abc flows. J. Fluid Mech. 167, 353–391.CrossRefGoogle Scholar
Freidberg, J. P. 1982 Ideal magnetohydrodynamic theory of magnetic fusion systems. Rev. Mod. Phys. 54, 801–902.Google Scholar
Freidberg, J. P. 1987 Ideal Magnetohydrodynamics. Plenum.Google Scholar
Frieman, E. & Rotenberg, M. 1960 On hydromagnetic stability of stationary equilibria. Rev. Mod. Phys. 32, 898–902.CrossRefGoogle Scholar
Goldstein, H. 1980 Classical Mechanics, 2nd edn. Addison-Wesley.Google Scholar
Grad, H. 1967 Toroidal containment of a plasma. Phys. Fluids 10, 137–154.Google Scholar
Hahm, T. S. & Kulsrud, R. M. 1985 Forced magnetic reconnection. Phys. Fluids 28, 2412–2418.CrossRefGoogle Scholar
Hameiri, E. 2014 Some improvements in the theory of plasma relaxation. Phys. Plasmas 21, 044503,1–5.Google Scholar
Hegna, C. C. & Bhattacharjee, A. 1989 Magnetic island formation in three-dimensional plasma equilibria. Phys. Fluids B 1, 392–397.CrossRefGoogle Scholar
Helander, P. 2014 Theory of plasma confinement in non-axisymmetric magnetic fields. Rep. Prog. Phys. 77, 087001,1–35.Google Scholar
Hole, M. J., Hudson, S. R. & Dewar, R. L. 2007 Equilibria and stability in partially relaxed plasma–vacuum systems. Nucl. Fusion 47, 746–753.Google Scholar
Hosking, R. J. & Dewar, R. L. 2015 Fundamental Fluid Mechanics and Magnetohydrodynamics. Springer.Google Scholar
Hudson, S. R., Dewar, R. L., Dennis, G., Hole, M. J., McGann, M., von Nessi, G. & Lazerson, S. 2012 Computation of multi-region relaxed magnetohydrodynamic equilibria. Phys. Plasmas 19, 112502,1–18.Google Scholar
Hudson, S. R., Hole, M. J. & Dewar, R. L. 2007 Eigenvalue problems for Beltrami fields arising in a three-dimensional toroidal magnetohydrodynamic equilibrium problem. Phys. Plasmas 14, 052505,1–12.Google Scholar
Jensen, T. H. & Chu, M. S. 1984 Current drive and helicity injection. Phys. Fluids 27, 2881–2885.CrossRefGoogle Scholar
K. Charidakos, I., Lingam, M., Morrison, P. J., White, R. L. & Wurm, A. 2014 Action principles for extended magnetohydrodynamic models. Phys. Plasmas 21, 092118,1–12.Google Scholar
Kruskal, M. D. & Kulsrud, R. M. 1958 Equilibrium of a magnetically confined plasma in a toroid. Phys. Fluids 1, 265–274.Google Scholar
Loizu, J., Hudson, S., Bhattacharjee, A. & Helander, P. 2015a Magnetic islands and singular currents at rational surfaces in three-dimensional magnetohydrodynamic equilibria. Phys. Plasmas 22, 022501,1–12.CrossRefGoogle Scholar
Loizu, J., Hudson, S. R., Bhattacharjee, A., Lazerson, S. & Helander, P. 2015b Existence of three-dimensional ideal-MHD equilibria with current sheets. Phys. Plasmas 22, 090704,1–5.Google Scholar
Longcope, D. W. & Strauss, H. R. 1993 The coalescence instability and the development of current sheets in two-dimensional magnetohydrodynamics. Phys. Fluids B 5, 2858–2869.Google Scholar
McGann, M.2013 Hamilton–Jacobi theory for connecting equilibrium magnetic fields across a toroidal surface supporting a plasma pressure discontinuity. PhD Thesis, Australian National University, Canberra ACT 0200, Australia.Google Scholar
McGann, M., Hudson, S. R., Dewar, R. L. & von Nessi, G. 2010 Hamilton–Jacobi theory for continuation of magnetic field across a toroidal surface supporting a plasma pressure discontinuity. Phys. Lett. A 374, 3308–3314.CrossRefGoogle Scholar
Mills, R., Hole, M. J. & Dewar, R. L. 2009 Magnetohydrodynamic stability of plasmas with ideal and relaxed regions. J. Plasma Phys. 75, 637–659.Google Scholar
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467–521.Google Scholar
Newcomb, W. A. 1962 Lagrangian and Hamiltonian methods in magnetohydrodynamics. Nucl. Fusion Suppl. Part 2, 451–463.Google Scholar
Padhye, N. & Morrison, P. J. 1996a Fluid element relabeling symmetry. Phys. Lett. A 219, 287–292.CrossRefGoogle Scholar
Padhye, N. & Morrison, P. J. 1996b Relabeling symmetries in hydrodynamics and magnetohydrodynamics. Plasma Phys. Rep. 22, 869–877.Google Scholar
Parker, E. N. 1994 Spontaneous current sheets in magnetic fields with applications to stellar x-rays. In International Series in Astronomy and Astrophysics, vol. 1. Oxford University Press.Google Scholar
Potter, D. 1976 Waterbag methods in magnetohydrodynamics. In Methods in Computational Physics, vol. 16, pp. 43–83. Academic.Google Scholar
Qin, H., Liu, W., Li, H. & Squire, J. 2012 Woltjer–Taylor state without Taylor’s conjecture: plasma relaxation at all wavelengths. Phys. Rev. Lett. 109, 235001,1–5.Google Scholar
Rusbridge, M. G. 1991 The relationship between the ‘tangled discharge’ and ‘dynamo’ models of the magnetic relaxation process. Plasma Phys. Controll. Fusion 33, 1381–1389.Google Scholar
Salmon, R. 1988 Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech. 20, 225–256.Google Scholar
Smiet, C. B., Candelaresi, S., Thompson, A., Swearngin, J., Dalhuizen, J. W. & Bouwmeester, D. 2015 Self-organizing knotted magnetic structures in plasma. Phys. Rev. Lett. 115, 095001,1–5.Google Scholar
Stott, P. E., Wilson, C. M. & Gibson, A. 1977 The bundle divertor – part I: Magnetic configuration. Nucl. Fusion 17, 481–496.Google Scholar
Taylor, J. B. 1974 Relaxation of toroidal plasma and generation of reverse magnetic fields. Phys. Rev. Lett. 33, 1139–1141.Google Scholar
Taylor, J. B. 1986 Relaxation and magnetic reconnection in plasmas. Rev. Mod. Phys. 58, 741–763.Google Scholar
Waelbroeck, F. L. 1989 Current sheets and nonlinear growth of the $m=1$ kink-tearing mode. Phys. Plasmas B 1, 2372–2380.Google Scholar
Wang, X. & Bhattacharjee, A. 1995 Nonlinear dynamics of the $m=1$ kink-tearing instability in a modified magnetohydrodynamic model. Phys. Plasmas 2, 171–181.Google Scholar
Webb, G. M., Dasgupta, B., McKenzie, J. F., Hu, Q. & Zank, G. P. 2014a Local and nonlocal advected invariants and helicities in magnetohydrodynamics and gas dynamics i: Lie dragging approach. J. Phys. A: Math. Theor. 47, 095501,1–33.Google Scholar
Webb, G. M., Dasgupta, B., McKenzie, J. F., Hu, Q. & Zank, G. P. 2014b Local and nonlocal advected invariants and helicities in magnetohydrodynamics and gas dynamics ii: Noether’s theorems and Casimirs. J. Phys. A: Math. Theor. 47, 095502,1–31.Google Scholar
Webb, G. M. & Zank, G. P. 2007 Fluid relabelling symmetries, Lie point symmetries and the Lagrangian map in magnetohydrodynamics and gas dynamics. J. Phys. A: Math. Theor. 40, 545–579.Google Scholar
White, R. B. 2013 Representation of ideal magnetohydrodynamic modes. Phys. Plasmas 20, 022105,1–4.Google Scholar
Wolfram Research, Inc. 2015 Mathematica, Version 10.1. Champaign, Il, USA: Wolfram Research.Google Scholar
Woltjer, L. 1958 A theorem on force-free magnetic fields. Proc. Natl Acad. Sci. USA 44, 489–491.Google Scholar
Yoshida, Z. & Dewar, R. L. 2012 Helical bifurcation and tearing mode in a plasma – a description based on Casimir foliation. J. Phys. A: Math. Gen. 45, 365502,1–36.Google Scholar
Yoshida, Z. & Giga, Y. 1990 Remarks on spectra of operator rot. Math. Z. 204, 235–245.Google Scholar