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Variational formulation of relaxed and multi-region relaxed magnetohydrodynamics

  • R. L. Dewar (a1), Z. Yoshida (a2), A. Bhattacharjee (a3) and S. R. Hudson (a3)


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.


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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.
Arnold, V. I. & Khesin, B. A. 1998 Topological methods in hydrodynamics. In Applied Mathematical Sciences, vol. 125. Springer.
Berger, M. A. 1999 Introduction to magnetic helicity. Plasma Phys. Control. Fusion 41, B167B175.
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, 1740.
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. 176180. Los Alamos National Laboratory.
Bhattacharjee, A. & Dewar, R. L. 1982 Energy principle with global invariants. Phys. Fluids 25, 887897.
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, 883888.
Boozer, A. H. & Pomphrey, N. 2010 Current density and plasma displacement near perturbed rational surfaces. Phys. Plasmas 17, 110707,1–4.
Cary, J. R. & Kotschenreuther, M. 1985 Pressure induced islands in three-dimensional toroidal plasma. Phys. Fluids 28, 13921401.
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.
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’.
Cordoba, D. & Marliani, C. 2000 Evolution of current sheets and regularity of ideal incompressible magnetic fluids in 2d. Commun. Pure Appl. Maths LIII, 05120524.
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.
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.
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.
Dewar, R. L. 1970 Interaction between hydromagnetic waves and a time-dependent, inhomogeneous medium. Phys. Fluids 13, 27102720.
Dewar, R. L. 1976 Renormalised canonical perturbation theory for stochastic propagators. J. Phys. A: Math. Gen. 9, 20432057.
Dewar, R. L. 1978 Hamilton’s principle for a hydromagnetic fluid with a free boundary. Nucl. Fusion 18, 15411553.
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.
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, 621634.
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, 353391.
Freidberg, J. P. 1982 Ideal magnetohydrodynamic theory of magnetic fusion systems. Rev. Mod. Phys. 54, 801902.
Freidberg, J. P. 1987 Ideal Magnetohydrodynamics. Plenum.
Frieman, E. & Rotenberg, M. 1960 On hydromagnetic stability of stationary equilibria. Rev. Mod. Phys. 32, 898902.
Goldstein, H. 1980 Classical Mechanics, 2nd edn. Addison-Wesley.
Grad, H. 1967 Toroidal containment of a plasma. Phys. Fluids 10, 137154.
Hahm, T. S. & Kulsrud, R. M. 1985 Forced magnetic reconnection. Phys. Fluids 28, 24122418.
Hameiri, E. 2014 Some improvements in the theory of plasma relaxation. Phys. Plasmas 21, 044503,1–5.
Hegna, C. C. & Bhattacharjee, A. 1989 Magnetic island formation in three-dimensional plasma equilibria. Phys. Fluids B 1, 392397.
Helander, P. 2014 Theory of plasma confinement in non-axisymmetric magnetic fields. Rep. Prog. Phys. 77, 087001,1–35.
Hole, M. J., Hudson, S. R. & Dewar, R. L. 2007 Equilibria and stability in partially relaxed plasma–vacuum systems. Nucl. Fusion 47, 746753.
Hosking, R. J. & Dewar, R. L. 2015 Fundamental Fluid Mechanics and Magnetohydrodynamics. Springer.
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.
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.
Jensen, T. H. & Chu, M. S. 1984 Current drive and helicity injection. Phys. Fluids 27, 28812885.
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.
Kruskal, M. D. & Kulsrud, R. M. 1958 Equilibrium of a magnetically confined plasma in a toroid. Phys. Fluids 1, 265274.
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.
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.
Longcope, D. W. & Strauss, H. R. 1993 The coalescence instability and the development of current sheets in two-dimensional magnetohydrodynamics. Phys. Fluids B 5, 28582869.
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.
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, 33083314.
Mills, R., Hole, M. J. & Dewar, R. L. 2009 Magnetohydrodynamic stability of plasmas with ideal and relaxed regions. J. Plasma Phys. 75, 637659.
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467521.
Newcomb, W. A. 1962 Lagrangian and Hamiltonian methods in magnetohydrodynamics. Nucl. Fusion Suppl. Part 2, 451463.
Padhye, N. & Morrison, P. J. 1996a Fluid element relabeling symmetry. Phys. Lett. A 219, 287292.
Padhye, N. & Morrison, P. J. 1996b Relabeling symmetries in hydrodynamics and magnetohydrodynamics. Plasma Phys. Rep. 22, 869877.
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.
Potter, D. 1976 Waterbag methods in magnetohydrodynamics. In Methods in Computational Physics, vol. 16, pp. 4383. Academic.
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.
Rusbridge, M. G. 1991 The relationship between the ‘tangled discharge’ and ‘dynamo’ models of the magnetic relaxation process. Plasma Phys. Controll. Fusion 33, 13811389.
Salmon, R. 1988 Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech. 20, 225256.
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.
Stott, P. E., Wilson, C. M. & Gibson, A. 1977 The bundle divertor – part I: Magnetic configuration. Nucl. Fusion 17, 481496.
Taylor, J. B. 1974 Relaxation of toroidal plasma and generation of reverse magnetic fields. Phys. Rev. Lett. 33, 11391141.
Taylor, J. B. 1986 Relaxation and magnetic reconnection in plasmas. Rev. Mod. Phys. 58, 741763.
Waelbroeck, F. L. 1989 Current sheets and nonlinear growth of the $m=1$ kink-tearing mode. Phys. Plasmas B 1, 23722380.
Wang, X. & Bhattacharjee, A. 1995 Nonlinear dynamics of the $m=1$ kink-tearing instability in a modified magnetohydrodynamic model. Phys. Plasmas 2, 171181.
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.
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.
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, 545579.
White, R. B. 2013 Representation of ideal magnetohydrodynamic modes. Phys. Plasmas 20, 022105,1–4.
Wolfram Research, Inc. 2015 Mathematica, Version 10.1. Champaign, Il, USA: Wolfram Research.
Woltjer, L. 1958 A theorem on force-free magnetic fields. Proc. Natl Acad. Sci. USA 44, 489491.
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.
Yoshida, Z. & Giga, Y. 1990 Remarks on spectra of operator rot. Math. Z. 204, 235245.
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Variational formulation of relaxed and multi-region relaxed magnetohydrodynamics

  • R. L. Dewar (a1), Z. Yoshida (a2), A. Bhattacharjee (a3) and S. R. Hudson (a3)


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