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, 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.
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Generalized Grad–Shafranov equation for non-axisymmetric MHD equilibria
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October 2020
Research Article|
October 05 2020
Generalized Grad–Shafranov equation for non-axisymmetric MHD equilibria
J. W. Burby
;
J. W. Burby
a)
1
Los Alamos National Laboratory
, Los Alamos, New Mexico 87545, USA
a)Author to whom correspondence should be addressed: [email protected]
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N. Kallinikos
;
N. Kallinikos
2
Mathematics Institute, University of Warwick
, Coventry CV4 7AL, United Kingdom
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R. S. MacKay
R. S. MacKay
2
Mathematics Institute, University of Warwick
, Coventry CV4 7AL, United Kingdom
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J. W. Burby
1,a)
N. Kallinikos
2
R. S. MacKay
2
1
Los Alamos National Laboratory
, Los Alamos, New Mexico 87545, USA
2
Mathematics Institute, University of Warwick
, Coventry CV4 7AL, United Kingdom
a)Author to whom correspondence should be addressed: [email protected]
Phys. Plasmas 27, 102504 (2020)
Article history
Received:
May 28 2020
Accepted:
September 02 2020
Citation
J. W. Burby, N. Kallinikos, R. S. MacKay; Generalized Grad–Shafranov equation for non-axisymmetric MHD equilibria. Phys. Plasmas 1 October 2020; 27 (10): 102504. https://doi.org/10.1063/5.0015420
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