Abstract
This chapter is concerned with the magnetic relaxation problem, in which an electrically conducting fluid is initialised in some non-trivial state, and is subsequently allowed to relax to some minimum-energy state, subject to the magnetohydrodynamic (MHD) equations. No driving or forcing is applied during this relaxation process and some form of dissipation allows energy to decrease until the system reaches a relaxed state. Our problem is simple: can we understand or predict this relaxed state?
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
While you can make A ⋅B into a material scalar by choosing an appropriate gauge (Webb et al. 2010), this would remove the utility of field-line helicity as a measure of changes in magnetic topology. Instead, a gauge of A should be chosen that is fixed in time (at least on the boundaries where it affects the h(V t)).
References
K. Bajer, H.K. Moffatt, Magnetic relaxation, current sheets, and structure formation in an extremely tenuous fluid medium. Astrophys. J. 779, 169 (2013)
M.R. Bareford, A.W. Hood, P.K. Browning, Coronal heating by the partial relaxation of twisted loops. Astron. Astrophys. 550, A40 (2013)
M.A. Berger, Rigorous new limits on magnetic helicity dissipation in the solar corona. Geophys. Astrophys. Fluid Dyn. 30, 79–104 (1984)
M.A. Berger, An energy formula for nonlinear force-free magnetic fields. Astron. Astrophys. 201, 355–361 (1988)
A. Bhattacharjee, R.L. Dewar, Energy principle with global invariants. Phys. Fluids 25, 887–897 (1982)
D. Biskamp, Nonlinear Magnetohydrodynamics (Cambridge University, Cambridge, 1997)
P.K. Browning, Helicity injection and relaxation in a solar-coronal magnetic loop with a free surface. J. Plasma Phys. 40, 263–280 (1988)
S. Candelaresi, D.I. Pontin, G. Hornig, Magnetic field relaxation and current sheets in an ideal plasma. Astrophys. J. 808, 134 (2015)
J. Cantarella, D. DeTurck, H. Gluck, M. Teytel, The spectrum of the curl operator on spherically symmetric domains. Phys. Plasmas 7, 2766–2775 (2000)
A.R. Choudhuri, The Physics of Fluids and Plasmas: An Introduction for Astrophysicists (Cambridge University, Cambridge, 1998)
C.G. Gimblett, R.J. Hastie, P. Helander, Model for current-driven edge-localized modes. Phys. Rev. Lett. 96(3), 035006 (2006)
J. Heyvaerts, E.R. Priest, Coronal heating by reconnection in DC current systems - a theory based on Taylor’s hypothesis. Astron. Astrophys. 137, 63–78 (1984)
S.P. Hirshman, J.C. Whitson, Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria. Phys. Fluids 26, 3553–3568 (1983)
S.R. Hudson, R.L. Dewar, G. Dennis, M.J. Hole, M. McGann, G. von Nessi, S. Lazerson, Computation of multi-region relaxed magnetohydrodynamic equilibria. Phys. Plasmas 19(11), 112502 (2012)
A.S. Hussain, P.K. Browning, A.W. Hood, A relaxation model of coronal heating in multiple interacting flux ropes. Astron. Astrophys. 600, A5 (2017)
P. Laurence, M. Avellaneda, On Woltjer’s variational principle for force-free fields. J. Math. Phys. 32, 1240–1253 (1991)
H.K. Moffatt, The energy spectrum of knots and links. Nature 347, 367–369 (1990)
H.K. Moffatt, Relaxation under topological constraints, in Topological Aspects of the Dynamics of Fluids and Plasmas, ed. by H.K. Moffatt, G.M. Zaslavsky, P. Comte, M. Tabor (Springer Netherlands, Dordrecht, 1992), pp. 3–28
H.K. Moffatt, Magnetic relaxation and the Taylor conjecture. J. Plasma Phys. 81(6), 905810608 (2015)
R. Paccagnella, Relaxation models for single helical reversed field pinch plasmas. Phys. Plasmas 23(9), 092512 (2016)
D.I. Pontin, G. Hornig, The structure of current layers and degree of field-line braiding in coronal loops. Astrophys. J. 805, 47 (2015)
D.I. Pontin, S. Candelaresi, A.J.B. Russell, G. Hornig, Braided magnetic fields: equilibria, relaxation and heating. Plasma Phys. Controlled Fusion 58(5), 054008 (2016)
A. Reiman, Minimum energy state of a toroidal discharge. Phys. Fluids 23, 230–231 (1980)
R.L. Ricca, F. Maggioni, On the groundstate energy spectrum of magnetic knots and links. J. Phys. A Math. Gen. 47(20), 205501 (2014)
A.J.B. Russell, A.R. Yeates, G. Hornig, A.L. Wilmot-Smith, Evolution of field line helicity during magnetic reconnection. Phys. Plasmas 22(3), 032106 (2015)
C.B. Smiet, S. Candelaresi, D. Bouwmeester, Ideal relaxation of the Hopf fibration. Phys. Plasmas 24(7), 072110 (2017)
J.B. Taylor, Relaxation of toroidal plasma and generation of reverse magnetic fields. Phys. Rev. Lett. 33, 1139–1141 (1974)
J.B. Taylor, Relaxation and magnetic reconnection in plasmas. Rev. Mod. Phys. 58, 741–763 (1986)
G. Valori, B. Kliem, T. Török, V.S. Titov, Testing magnetofrictional extrapolation with the Titov-Démoulin model of solar active regions. Astron. Astrophys. 519, A44 (2010)
A.A. van Ballegooijen, E.R. Priest, D.H. Mackay, Mean field model for the formation of filament channels on the sun. Astrophys. J. 539, 983–994 (2000)
G.M. Webb, Q. Hu, B. Dasgupta, G.P. Zank, Homotopy formulas for the magnetic vector potential and magnetic helicity: the Parker spiral interplanetary magnetic field and magnetic flux ropes. J. Geophys. Res. Space Phys. 115, A10112 (2010)
L. Woltjer, A theorem on force-free magnetic fields. Proc. Natl. Acad. Sci. 44, 489–491 (1958)
A.R. Yeates, G. Hornig, The global distribution of magnetic helicity in the solar corona. Astron. Astrophys. 594, A98 (2016)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 CISM International Centre for Mechanical Sciences
About this chapter
Cite this chapter
Yeates, A.R. (2020). Magnetohydrodynamic Relaxation Theory. In: MacTaggart, D., Hillier, A. (eds) Topics in Magnetohydrodynamic Topology, Reconnection and Stability Theory. CISM International Centre for Mechanical Sciences, vol 591. Springer, Cham. https://doi.org/10.1007/978-3-030-16343-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-16343-3_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-16342-6
Online ISBN: 978-3-030-16343-3
eBook Packages: EngineeringEngineering (R0)