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.

1.
A. H.
Boozer
,
Nucl. Fusion
60
,
065001
(
2020
).
2.
A.
Bhattacharjee
,
G. M.
Schreiber
, and
J. B.
Taylor
,
Phys. Plasmas
4
,
2737
(
1992
).
3.
R. B.
White
,
The Theory of Toroidally Confined Plasmas
, 3rd ed. (
Imperial College Press
,
London
,
2014
).
4.
D. R.
Smith
and
A. H.
Reiman
,
Phys. Plasmas
11
,
3752
(
2004
).
5.
H.
Grad
,
Phys. Fluids
10
,
137
(
1967
).
6.
D.
Lortz
,
Z. Angew. Math. Phys.
21
,
196
(
1970
).
7.
F.
Meyer
and
H. U.
Schmidt
,
Z. Naturforsch. A
13
,
1005
(
1958
).
8.
M. I.
Mikhailov
,
J.
Nürenberg
, and
R.
Zille
,
Nucl. Fusion
59
,
066002
(
2019
).
9.
E.
Kim
,
G. B.
McFadden
, and
A. J.
Cerfon
,
Plasma Phys. Controlled Fusion
62
,
044002
(
2020
).
10.
H.
Weitzner
,
Phys. Plasmas
23
,
062512
(
2016
).
11.
W.
Sengputa
and
H.
Weitzner
,
J. Plasma Phys.
85
,
905850209
(
2019
).
12.
O. P.
Bruno
and
P.
Laurence
,
Commun. Pure Appl. Math.
49
,
717
(
1996
).
13.
S. R.
Hudson
,
R. L.
Dewar
,
G.
Dennis
,
M. J.
Hole
,
M.
McGann
,
G.
von Nessi
, and
S.
Lazerson
,
Phys. Plasmas
19
,
112502
(
2012
).
14.
S. R.
Hudson
,
R. L.
Dewar
,
M. J.
Hole
, and
M.
McGann
,
Plasma Phys. Controlled Fusion
54
,
014005
(
2012
).
15.
J.
Loizu
,
S. R.
Hudson
,
A.
Bhattacharjee
,
S.
Lazerson
, and
P.
Helander
,
Phys. Plasmas
22
,
090704
(
2015
).
16.
J.
Loizu
,
S. R.
Hudson
, and
C.
Nührenberg
,
Phys. Plasmas
23
,
112505
(
2016
).
17.
Z. S.
Qu
,
R. L.
Dewar
,
F.
Ebrahimi
,
J. K.
Anderson
,
S. R.
Hudson
, and
M. J.
Hole
,
Plasma Phys. Controlled Fusion
62
,
054002
(
2020
).
18.
B. F.
Kraus
and
S. R.
Hudson
,
Phys. Plasmas
24
,
092519
(
2017
).
19.
S. R.
Hudson
and
B. F.
Kraus
,
J. Plasma Phys.
83
,
715830403
(
2017
).
20.
M.
Kruskal
,
J. Math. Phys.
3
,
806
(
1962
).
21.
R. S.
MacKay
,
J. Plasma Phys.
86
,
925860101
(
2020
).
22.
R.
Abraham
and
J. E.
Marsden
,
Foundations of Mechanics
(
AMS Chelsea Publishing, American Mathematical Society
,
2008
).
24.
J. W.
Burby
,
N.
Duignan
, and
J. D.
Meiss
, “
Integrability, normal forms, and magnetic axis coordinates
” (unpublished).
25.
M. E.
Taylor
,
Partial Differential Equations III: Nonlinear Equations
, Applied Mathematical Sciences Series, 2nd ed. (
Springer
,
2010
), Vol.
117
.
26.
P.
Jin
,
A.
Zhu
,
G. E.
Karniadakis
, and
Y.
Tang
, arXiv:2001.03750 (
2020
).
27.
J. W.
Burby
,
Q.
Tang
, and
R.
Maulik
, arXiv:2007.04496 (
2020
).
28.
A. H.
Boozer
,
Phys. Fluids
24
,
1999
(
1981
).
29.
L. L.
Lao
,
S. P.
Hirshman
, and
R. M.
Wieland
,
Phys. Fluids
24
,
1431
(
1981
).
30.
A.
Bhattacharjee
,
J. C.
Wiley
, and
R. L.
Dewar
,
Comput. Phys. Commun.
31
,
213
(
1984
).
31.
H.
Grad
,
Phys. Fluids
7
,
1283
(
1964
).
32.
J. W.
Burby
,
N.
Kallinikos
, and
R. S.
MacKay
,
J. Math. Phys.
61
,
093503
(
2020
).
33.
P.
Constantin
,
T. D.
Drivas
, and
D.
Ginsberg
, arXiv:2009.08860 (
2020
).
35.
J.
Moser
,
Trans. Am. Math. Soc.
120
,
286
(
1965
).
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