Abstract
Restricted Lie point symmetries are derived for the axially symmetric steady solutions to the ideal magnetohydrodynamics equations. The symmetries transform vectors of magnetic field B and plasma velocity V linearly with coefficients depending on a function u(z, r). A reduction of the eight MHD equilibrium equations to a single second-order partial differential equation for the function u(z, r) is obtained. Analogous Lie point symmetries and reduction are derived for the translationally invariant MHD equilibria. Applications of the symmetry transforms are indicated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Kruskal MD, Kulsrud RM (1958) Equilibrium of a magnetically confined plasma in a toroid. Phys Fluids 1: 265–274
Spitzer L (1958) The stellarator concept. Phys Fluids 1: 253–264
Grad H, Rubin H (1958) Hydromagnetic equilibria and force-free fields, In: Proceedings of the second United Nations international conference on the peaceful uses of atomic energy, 31. United Nations, Geneva, pp 190–197
Shafranov VD (1958) On magnetohydrodynamical equilibrium configurations. Soviet Phys JETP 6: 545–554
Chandrasekhar S (1956) Axisymmetric magnetic fields and fluid motions. Astrophys J 124: 232–243
Chandrasekhar S, Prendergast KH (1956) The equilibrium of magnetic stars. Proc Natl Acad Sci USA 42: 5–9
Bogoyavlenskij OI (2000) Helically symmetric astrophysical jets. Phys Rev E 62: 8616–8627
Bogoyavlenskij OI (2002) Symmetry transforms for ideal magnetohydrodynamics equilibria. Phys Rev E 66(056410): 1–11
Bogoyavlenskij OI (2002) Method of symmetry transforms for ideal MHD equilibrium equations. Contemp Math 301: 195–218
Gardner CS, Greene JM, Kruskal MD, Miura RM (1967) Method for solving the Korteweg–de Vries equation. Phys Rev Lett 19: 1095–1097
Hirota R (1971) Exact N-soliton of the Korteweg - de Vries equation for multiple collision of solitons. Phys Rev Lett 27: 1192–1193
Ablowitz MJ, Kaup DJ, Newell AC, Segur H (1973) Method for solving the sine-Gordon equation. Phys Rev Lett 30: 1262–1264
Alfven H, Falthammar C-G (1963) Cosmical electrodynamics. Fundamental principles. Clarendon Press, Oxford
Bogoyavlenskij OI (2000) Astrophysical jets as exact plasma equilibria. Phys Rev Lett 84: 1914–1917
Bogoyavlenskij OI (2000) Counterexamples to Parker’s theorem. J Math Phys 41: 2043–2057
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bogoyavlenskij, O. Restricted Lie point symmetries and reductions for ideal magnetohydrodynamics equilibria. J Eng Math 66, 141–152 (2010). https://doi.org/10.1007/s10665-009-9326-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-009-9326-7