Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-17T17:20:20.890Z Has data issue: false hasContentIssue false
  • English
  • Français

Relaxed magnetohydrodynamics with ideal Ohm's law constraint

Part of: Hamiltonian Methods in Plasma Physics

Published online by Cambridge University Press:  04 February 2022

R.L. Dewar Open the ORCID record for R.L. Dewar [Opens in a new window]  and
Z.S. Qu
R.L. Dewar*
Affiliation:
Mathematical Sciences Institute, The Australian National University, Canberra, ACT 2601, Australia
Z.S. Qu
Affiliation:
Mathematical Sciences Institute, The Australian National University, Canberra, ACT 2601, Australia
*
†Email address for correspondence: robert.dewar@anu.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zero-resistivity ‘ideal’ Ohm's law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywhere). This is illustrated by deriving dispersion relations for linear waves on an MHD equilibrium.

Keywords

plasma confinementplasma dynamicsplasma simulation
Type
Research Article
Information
Journal of Plasma Physics , Volume 88 , Issue 1 , February 2022 , 835880101
Check for updates
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bhattacharjee, A. & Dewar, R.L. 1982 Energy principle with global invariants. Phys. Fluids 25, 887–897.CrossRefGoogle Scholar
Burby, J.W. 2017 Magnetohydrodynamic motion of a two-fluid plasma. Phys. Plasmas 24, 082104–1–13.CrossRefGoogle Scholar
Burby, J.W., Kallinikos, N. & MacKay, R.S. 2020 Some mathematics for quasi-symmetry. J. Maths Phys. 61, 093503–1–22.CrossRefGoogle Scholar
Calkin, M.G. 1963 An action principle for magnetohydrodynamics. Can. J. Phys. 41, 2241–2251.CrossRefGoogle Scholar
Chandrasekhar, S. & Woltjer, L. 1958 On force-free magnetic fields. Proc. Natl Acad. Sci. USA 44, 285.CrossRefGoogle ScholarPubMed
Constantin, P., Drivas, T.D. & Ginsberg, D. 2021 On quasisymmetric plasma equilibria sustained by small force. J. Plasma Phys. 87, 905870111–1–30.CrossRefGoogle Scholar
Dennis, G.R., Hudson, S.R., Dewar, R.L. & Hole, M.J. 2014 a Multi-region relaxed magnetohydrodynamics with anisotropy and flow. Phys. Plasmas 21, 072512–1–10.Google Scholar
Dennis, G.R., Hudson, S.R., Dewar, R.L. & Hole, M.J. 2014 b Multi-region relaxed magnetohydrodynamics with flow. Phys. Plasmas 21, 042501–1–9.Google Scholar
Dewar, R.L. 1970 Interaction between hydromagnetic waves and a time-dependent, inhomogeneous medium. Phys. Fluids 13, 2710–2720.CrossRefGoogle Scholar
Dewar, R.L. 1977 Energy-momentum tensors for dispersive electromagnetic waves. Austral. J. Phys. 30, 533.CrossRefGoogle Scholar
Dewar, R.L., Burby, J.W., Qu, Z.S., Sato, N. & Hole, M.J. 2020 Time-dependent relaxed magnetohydrodynamics – inclusion of cross helicity constraint using phase-space action. Phys. Plasmas 27, 062504–1–22.CrossRefGoogle Scholar
Dewar, R.L., Yoshida, Z., Bhattacharjee, A. & Hudson, S.R. 2015 Variational formulation of relaxed and multi-region relaxed magnetohydrodynamics. J. Plasma Phys. 81, 515810604.CrossRefGoogle Scholar
Fathi, A. 2009 Weak KAM Theorem in Lagrangian Dynamics, Cambridge Studies in Advanced Mathematics, vol. 88. Cambridge University Press. 300 pp., ISBN: 0521822289.Google Scholar
Finn, J.M. & Antonsen, T.M. Jr. 1983 Turbulent relaxation of compressible plasmas with flow. Phys. Fluids 26, 3540.CrossRefGoogle 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.CrossRefGoogle Scholar
Hameiri, E. 1983 The equilibrium and stability of rotating plasmas. Phys. Fluids 26, 230–237.CrossRefGoogle Scholar
Hameiri, E. 1998 Variational principles for equilibrium states with plasma flow. Phys. Plasmas 5, 3270–3281.CrossRefGoogle Scholar
Hameiri, E. 2014 Some improvements in the theory of plasma relaxation. Phys. Plasmas 21, 044503–1–5.CrossRefGoogle Scholar
Hosking, R.J. & Dewar, R.L. 2015 Fundamental Fluid Mechanics and Magnetohydrodynamics. Springer Singapore, published online 2015. Book copyright 2016.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.CrossRefGoogle Scholar
Kanzow, C., Steck, D. & Wachsmuth, D. 2018 An augmented Lagrangian method for optimization problems in Banach spaces. SIAM J. Control Optim. 56, 272–291.CrossRefGoogle Scholar
Kumar, A., Qu, Z., Doak, J., Dewar, R.L., Hezaveh, H., Nührenberg, C., Aleynikova, K., Hole, M.J., Hudson, S.R., Loizu, J. & Baillod, A. 2021 a Ideal MHD instabilities of multi-region relaxed MHD. Plasma Phys. Control. Fusion submitted.Google Scholar
Kumar, A., Qu, Z., Hole, M.J., Wright, A.M., Loizu, J., Hudson, S.R., Baillod, A., Dewar, R.L. & Ferraro, N.M. 2021 b Computation of linear MHD instabilities with the multi-region relaxed MHD energy principle. Plasma Phys. Control. Fusion 63, 045006.CrossRefGoogle Scholar
Levnajić, Z. & Mezić, I. 2010 Ergodic theory and visualization. I. Mesochronic plots for visualization of ergodic partition and invariant sets. Chaos 20, 033114.CrossRefGoogle ScholarPubMed
Meiss, J.D. 1992 Symplectic maps, variational principles, and transport. Rev. Mod. Phys. 64, 795–848.CrossRefGoogle Scholar
Mezić, I. & Wiggins, S. 1999 A method for visualization of invariant sets of dynamical systems based on the ergodic partition. Chaos 9, 213.CrossRefGoogle ScholarPubMed
Moffatt, H.K. 2015 Magnetic relaxation and the Taylor conjecture. J. Plasma Phys. 81, 905810608.CrossRefGoogle Scholar
Morrison, P.J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467–521.CrossRefGoogle Scholar
Newcomb, W.A. 1958 Motion of magnetic lines of force. Ann. Phys. 3 (4), 347–385.CrossRefGoogle Scholar
Newcomb, W.A. 1962 Lagrangian and Hamiltonian methods in magnetohydrodynamics. Nucl. Fusion Suppl. Part 2, 451–463.Google Scholar
Nocedal, J. & Wright, S.J. 2006 Numerical Optimization, 2nd edn, Operation Research and Financial Engineering, vol. 25. Springer. 664 pp. ISBN-13: 978-0387-30303-1.Google Scholar
Nührenberg, J. & Zille, R. 1988 Quasi-helically symmetric toroidal stellarators. Phys. Lett. A 129, 113–117.CrossRefGoogle Scholar
Panofsky, W.K.H. & Phillips, M. 1962 Classical Electricity and Magnetism, 2nd edn. Addison-Wesley.Google Scholar
Qu, Z.S., Dewar, R.L., Ebrahimi, F., Anderson, J.K., Hudson, S.R. & Hole, M.J. 2020 Stepped pressure equilibrium with relaxed flow and applications in reversed-field pinch plasmas. Plasma Phys. Control. Fusion 62, 054002.CrossRefGoogle Scholar
Qu, Z.S., Hudson, S.R., Dewar, R.L., Loizu, J. & Hole, M.J. 2021 On the non-existence of stepped-pressure equilibria far from symmetry. Plasma Phys. Control. Fusion 63, 125007.CrossRefGoogle Scholar
Rodriguez, E., Helander, P. & Bhattacharjee, A. 2020 Necessary and sufficient conditions for quasisymmetry. Phys. Plasmas 27, 062501–1–5.CrossRefGoogle Scholar
Taylor, J.B. 1974 Relaxation of toroidal plasma and generation of reverse magnetic fields. Phys. Rev. Lett. 33, 1139–1141.CrossRefGoogle Scholar
Taylor, J.B. 1986 Relaxation and magnetic reconnection in plasmas. Rev. Mod. Phys. 58, 741–763.CrossRefGoogle Scholar
Vallis, G.K., Carnevale, G.F. & Young, W.R. 1989 Extremal energy properties and construction of stable solutions of the Euler equations. J. Fluid Mech. 207, 133–152.CrossRefGoogle Scholar
Vanneste, J. & Wirosoetisno, D. 2008 Two-dimensional Euler flows in slowly deforming domains. Physica D 237, 774–799.CrossRefGoogle Scholar
Vladimirov, V.A., Moffatt, H.K. & Ilin, K.I. 1999 On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. Part 4. Generalized isovorticity principle for three-dimensional flows. J. Fluid Mech. 390, 127–150.CrossRefGoogle Scholar
Webb, G.M. 2018 Magnetohydrodynamics and Fluid Dynamics: Action Principles and Conservation Laws, Lecture Notes in Physics, vol. 946. Springer International Publishing.CrossRefGoogle Scholar
Webb, G.M. & Anco, S.C. 2017 On magnetohydrodynamic gauge field theory. J. Phys. A: Math. Theor. 50, 255501–1–34.CrossRefGoogle Scholar
Webb, G.M. & Anco, S.C. 2019 Conservation laws in magnetohydrodynamics and fluid dynamics: Lagrangian approach. AIP Conf. Proc. 2153, 020024.CrossRefGoogle Scholar
Woltjer, L. 1958 a A theorem on force-free magnetic fields. Proc. Natl Acad. Sci. USA 44, 489–491.CrossRefGoogle ScholarPubMed
Woltjer, L. 1958 b On hydromagnetic equilibrium. Proc. Natl Acad. Sci. USA 44, 833.CrossRefGoogle ScholarPubMed
Yokoi, N. 2013 Cross helicity and related dynamo. Geophys. Astrophys. Fluid Dyn. 107, 114–184.CrossRefGoogle Scholar
Supplementary material: PDF

Dewar and Qu supplementary material

Dewar and Qu supplementary material

Download Dewar and Qu supplementary material(PDF)
PDF 1.5 MB