Generalized Grad–Shafranov equation for non-axisymmetric MHD equilibria

Burby, J. W.; Kallinikos, N.; MacKay, R. S.

doi:10.1063/5.0015420

Generalized Grad–Shafranov equation for non-axisymmetric MHD equilibria

Journal Article · Mon Oct 05 04:00:00 EDT 2020 · Physics of Plasmas

DOI:https://doi.org/10.1063/5.0015420· OSTI ID:1735911

^[1]; ^[2]; ^[2]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Univ. of Warwick, Coventry (United Kingdom)

The structure of static MHD equilibria that admit continuous families of Euclidean symmetries is well understood. Such field configurations are governed by the classical Grad–Shafranov equation, which is a single elliptic partial differential equation in two space dimensions. By revealing a hidden symmetry, here we show that in fact all smooth solutions of the equilibrium equations with non-vanishing pressure gradients away from the magnetic axis satisfy a generalization of the Grad–Shafranov equation. In contrast to solutions of the classical Grad–Shafranov equation, solutions of the generalized equation are not automatically equilibria, but instead only satisfy force balance averaged over the one-parameter hidden symmetry. We then explain how the generalized Grad–Shafranov equation can be used to reformulate the problem of finding exact three-dimensional smooth solutions of the equilibrium equations as finding an optimal volume-preserving symmetry.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; Simons Foundation

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 1735911

Report Number(s):: LA-UR--20-24232

Journal Information:: Physics of Plasmas, Journal Name: Physics of Plasmas Journal Issue: 10 Vol. 27; ISSN 1070-664X

Publisher:: American Institute of Physics (AIP)Copyright Statement

Country of Publication:: United States

Language:: English

References (29)

Existence of three-dimensional toroidal MHD equilibria with nonconstant pressure Bruno, Oscar P.; Laurence, Peter Communications on Pure and Applied Mathematics, Vol. 49, Issue 7 https://doi.org/10.1002/(SICI)1097-0312(199607)49:7<717::AID-CPA3>3.0.CO;2-C	journal	July 1996
Über die Existenz toroidaler magnetohydrostatischer Gleichgewichte ohne Rotationstransformation Lortz, Dietrich Zeitschrift für angewandte Mathematik und Physik ZAMP, Vol. 21, Issue 2 https://doi.org/10.1007/BF01590644	journal	March 1970
Variational method for three-dimensional toroidal equilibria Bhattacharjee, A.; Wiley, J. C.; Dewar, R. L. Computer Physics Communications, Vol. 31, Issue 2-3 https://doi.org/10.1016/0010-4655(84)90046-8	journal	February 1984
The geometry of cross sections to flows Fried, David Topology, Vol. 21, Issue 4 https://doi.org/10.1016/0040-9383(82)90017-9	journal	January 1982
Three-dimensional magnetohydrodynamic equilibria with continuous magnetic fields Hudson, S. R.; Kraus, B. F. Journal of Plasma Physics, Vol. 83, Issue 4 https://doi.org/10.1017/S0022377817000538	journal	July 2017
Low-shear three-dimensional equilibria and vacuum magnetic fields with flux surfaces Sengupta, Wrick; Weitzner, Harold Journal of Plasma Physics, Vol. 85, Issue 2 https://doi.org/10.1017/S0022377819000230	journal	April 2019
Differential forms for plasma physics MacKay, R. S. Journal of Plasma Physics, Vol. 86, Issue 1 https://doi.org/10.1017/S0022377819000928	journal	January 2020
Some New Variational Properties of Hydromagnetic Equilibria Grad, Harold Physics of Fluids, Vol. 7, Issue 8 https://doi.org/10.1063/1.1711373	journal	January 1964
Asymptotic Theory of Hamiltonian and other Systems with all Solutions Nearly Periodic Kruskal, Martin Journal of Mathematical Physics, Vol. 3, Issue 4 https://doi.org/10.1063/1.1724285	journal	July 1962
Toroidal Containment of a Plasma Grad, Harold Physics of Fluids, Vol. 10, Issue 1 https://doi.org/10.1063/1.1761965	journal	January 1967
Analytic, high-β solutions of the helical Grad–Shafranov equation Smith, D. R.; Reiman, A. H. Physics of Plasmas, Vol. 11, Issue 8 https://doi.org/10.1063/1.1763576	journal	August 2004
Computation of multi-region relaxed magnetohydrodynamic equilibria Hudson, S. R.; Dewar, R. L.; Dennis, G. Physics of Plasmas, Vol. 19, Issue 11 https://doi.org/10.1063/1.4765691	journal	November 2012
Existence of three-dimensional ideal-magnetohydrodynamic equilibria with current sheets Loizu, J.; Hudson, S. R.; Bhattacharjee, A. Physics of Plasmas, Vol. 22, Issue 9 https://doi.org/10.1063/1.4931094	journal	September 2015
Expansions of non-symmetric toroidal magnetohydrodynamic equilibria Weitzner, Harold Physics of Plasmas, Vol. 23, Issue 6 https://doi.org/10.1063/1.4954048	journal	June 2016
Verification of the SPEC code in stellarator geometries Loizu, J.; Hudson, S. R.; Nührenberg, C. Physics of Plasmas, Vol. 23, Issue 11 https://doi.org/10.1063/1.4967709	journal	November 2016
Theory and discretization of ideal magnetohydrodynamic equilibria with fractal pressure profiles Kraus, B. F.; Hudson, S. R. Physics of Plasmas, Vol. 24, Issue 9 https://doi.org/10.1063/1.4986493	journal	September 2017
Some mathematics for quasi-symmetry Burby, J. W.; Kallinikos, N.; MacKay, R. S. Journal of Mathematical Physics, Vol. 61, Issue 9 https://doi.org/10.1063/1.5142487	journal	September 2020
Geometric phase, rotational transforms, and adiabatic invariants in toroidal magnetic fields Bhattacharjee, A.; Schreiber, G. M.; Taylor, J. B. Physics of Fluids B: Plasma Physics, Vol. 4, Issue 9 https://doi.org/10.1063/1.860145	journal	September 1992
Plasma equilibrium with rational magnetic surfaces Boozer, Allen H. Physics of Fluids, Vol. 24, Issue 11 https://doi.org/10.1063/1.863297	journal	January 1981
Variational moment solutions to the Grad–Shafranov equation Lao, L. L. Physics of Fluids, Vol. 24, Issue 8 https://doi.org/10.1063/1.863562	journal	January 1981
Hydromagnetic equilibria and their proper coordinates Hamada, Shigeo Nuclear Fusion, Vol. 2, Issue 1-2 https://doi.org/10.1088/0029-5515/2/1-2/005	journal	January 1962
Non-axisymmetric, multi-region relaxed magnetohydrodynamic equilibrium solutions Hudson, S. R.; Dewar, R. L.; Hole, M. J. Plasma Physics and Controlled Fusion, Vol. 54, Issue 1 https://doi.org/10.1088/0741-3335/54/1/014005	journal	December 2011
Elimination of MHD current sheets by modifications to the plasma wall in a fixed boundary model Kim, E.; McFadden, G. B.; Cerfon, A. J. Plasma Physics and Controlled Fusion, Vol. 62, Issue 4 https://doi.org/10.1088/1361-6587/ab6d48	journal	February 2020
Stepped pressure equilibrium with relaxed flow and applications in reversed-field pinch plasmas Qu, Z. S.; Dewar, R. L.; Ebrahimi, F. Plasma Physics and Controlled Fusion, Vol. 62, Issue 5 https://doi.org/10.1088/1361-6587/ab7fc5	journal	April 2020
Elimination of current sheets at resonances in three-dimensional toroidal ideal-magnetohydrodynamic equilibria Mikhailov, M. I.; Nührenberg, J.; Zille, R. Nuclear Fusion, Vol. 59, Issue 6 https://doi.org/10.1088/1741-4326/ab0f50	journal	April 2019
Why carbon dioxide makes stellarators so important Boozer, Allen H. Nuclear Fusion, Vol. 60, Issue 6 https://doi.org/10.1088/1741-4326/ab87af	journal	May 2020
On the volume elements on a manifold Moser, J{ürgen Transactions of the American Mathematical Society, Vol. 120, Issue 2 https://doi.org/10.1090/S0002-9947-1965-0182927-5	journal	February 1965
The Theory of Toroidally Confined Plasmas White, Roscoe B. World Scientific https://doi.org/10.1142/p916	book	January 2014
Torusartige Plasmakonfigurationen ohne Gesamtstrom durch ihren Querschnitt im Gleichgewicht mit einem Magnetfeld Meyer, F.; Schmidt, H. U. Zeitschrift für Naturforschung A, Vol. 13, Issue 12 https://doi.org/10.1515/zna-1958-1201	journal	December 1958

Similar Records

A similarity reduction of the Grad-Shafranov equation

Journal Article · Thu Jul 15 00:00:00 EDT 2010 · Physics of Plasmas · OSTI ID:21389156

Generalized conditional symmetries and related solutions of the Grad-Shafranov equation

Journal Article · Tue Apr 15 00:00:00 EDT 2014 · Physics of Plasmas · OSTI ID:22253067

Symmetries, weak symmetries, and related solutions of the Grad-Shafranov equation

Journal Article · Fri Oct 15 00:00:00 EDT 2010 · Physics of Plasmas · OSTI ID:21421262

Related Subjects

70 PLASMA PHYSICS AND FUSION TECHNOLOGY
algebraic topology
calculus of variations
coordinate transformations
differentiable manifold
differential geometry
magnetohydrodynamics
quasisymmetric
stellarators
vector calculus
vector fields

Generalized Grad–Shafranov equation for non-axisymmetric MHD equilibria

Citation Formats

References (29)

Similar Records

Related Subjects