We describe the construction of stepped-pressure equilibria as extrema of a multi-region, relaxed magnetohydrodynamic (MHD) energy functional that combines elements of ideal MHD and Taylor relaxation, and which we call MRXMHD. The model is compatible with Hamiltonian chaos theory and allows the three-dimensional MHD equilibrium problem to be formulated in a well-posed manner suitable for computation. The energy-functional is discretized using a mixed finite-element, Fourier representation for the magnetic vector potential and the equilibrium geometry; and numerical solutions are constructed using the stepped-pressure equilibrium code, SPEC. Convergence studies with respect to radial and Fourier resolution are presented.
REFERENCES
1.
M.
McGann
, S. R.
Hudson
, R. L.
Dewar
, and G.
von Nessi
, Phys. Lett. A
374
, 3308
–3314
(2010
).2.
H. L.
Berk
, J. P.
Freidberg
, X.
Llobet
, P. J.
Morrison
, and J. A.
Tataronis
, Phys. Fluids
29
, 3281
(1986
).3.
R. L.
Dewar
, M. J.
Hole
, M.
McGann
, R.
Mills
, and S. R.
Hudson
, Entropy
10
, 621
(2008
).4.
O. P.
Bruno
and P.
Laurence
, Commun. Pure Appl. Math.
49
, 717
(1996
).5.
J. B.
Taylor
, Rev. Mod. Phys.
58
, 741
(1986
).6.
Z.
Yoshida
and R.
Dewar
, J. Phys. A: Math. Theor.
45
, 365502
(2012
).7.
S. P.
Hirshman
and H. K.
Meier
, Phys. Fluids
28
, 1387
(1985
).8.
S. P.
Hirshman
and J.
Breslau
, Phys. Plasmas
5
, 2664
(1998
).9.
H.
Boozer
, Rev. Mod. Phys.
76
, 1071
(2005
).10.
S.
Wiggins
, Introduction to Applied Dynamical Systems and Chaos, Texts in Applied Mathematics
(Springer-Verlag
, New York
, 1990
), Vol. 2
.11.
A. J.
Lichtenberg
and M. A.
Lieberman
, Regular and Chaotic Dynamics
, 2nd ed. (Springer-Verlag
, New York
, 1992
).12.
J. D.
Meiss
, Rev. Mod. Phys.
64
, 795
(1992
).13.
M. D.
Kruskal
and R. M.
Kulsrud
, Phys. Fluids
1
, 265
(1958
).14.
R. Y.
Neches
, S. C.
Cowley
, P. A.
Gourdain
, and J. N.
Leboeuf
, Phys. Plasmas
15
, 122504
(2008
).15.
E. R.
Solano
, Plasma Phys. Controlled Fusion
46
, L7
(2004
).16.
T. E.
Evans
, R. A.
Moyer
, P. R.
Thomas
, J. G.
Watkins
, T. H.
Osborne
, J. A.
Boedo
, E. J.
Doyle
, M. E.
Fenstermacher
, K. H.
Finken
, R. J.
Groebner
et al, Phys. Rev. Lett.
92
, 235003
(2004
).17.
W. A.
Cooper
, J. P.
Graves
, A.
Pochelon
, O.
Sauter
, and L.
Villard
, Phys. Rev. Lett.
105
, 035003
(2010
).18.
19.
J.
Nuehrenberg
and R.
Zille
, in Proceedings of the Workshop on Theory of Fusion Plasmas
, edited by A.
Bondeson
, E.
Sindoni
, and F.
Troyon
(Societa Italiana di Fisica – Editrice Compositori
, Bologna
, 1988
), p. 3
.20.
21.
A. N.
Kolmogorov
, Dokl. Akad. Nauk. SSR
98
, 469
(1954
).22.
J.
Moser
, Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl
II
, 1
(1962
).23.
V. I.
Arnold
, Russ. Math. Surveys
18
, 9
(1963
).24.
J.
Moser
, Stable and Random Motions
(Princeton University Press, Princeton
, NJ
, 1973
).25.
V. I.
Arnold
, Mathematical Methods of Classical Mechanics
(Springer-Verlag
, New York
, 1978
).26.
I.
Niven
, Irrational Numbers
(The mathematical association of America
, USA
, 1956
).27.
28.
J. N.
Mather
, Topology
21
, 457
(1982
).29.
I. C.
Percival
, in AIP Conf. Proc. of Nonlinear Dynamics and the Beam-Beam interaction
, edited by M.
Month
and J. C.
Herra
(AIP
, New York
, 1979
), Vol. 57
.30.
W.
Li
and P.
Bak
, Phys. Rev. Lett.
57
, 655
(1986
).31.
R. S.
MacKay
, J. D.
Meiss
, and I. C.
Percival
, Phys. Rev. Lett.
52
, 697
(1984
).32.
S. R.
Hudson
and J.
Breslau
, Phys. Rev. Lett.
100
, 095001
(2008
).33.
R. S.
MacKay
, J. D.
Meiss
, and I. C.
Percival
, Physica D
13
, 55
(1984
).34.
J. D.
Meiss
, Phys. Rev. A.
34
, 2375
(1986
).35.
I.
Mezic
and S.
Wiggins
, Chaos
9
, 213
(1999
).36.
Z.
Levnajic
and I.
Mezic
, Chaos
20
, 033114
(2010
).37.
M. G.
Rusbridge
, Plasma Phys.
19
, 499
(1977
).38.
39.
40.
Z.
Yoshida
and Y.
Giga
, Math. Z.
204
, 235
(1990
).41.
42.
G.
Marsh
, Force-Free Magnetic Fields: Solutions, Topology and Applications
(World Scientific
, Singapore
, 1996
).43.
S. R.
Hudson
, M. J.
Hole
, and R. L.
Dewar
, Phys. Plasmas
14
, 052505
(2007
).44.
U.
Locatelli
, C.
Froeschlé
, E.
Lega
, and A.
Morbidelli
, Physica D
139
, 48
(2000
).45.
46.
47.
W.
Park
, D.
Monticello
, H.
Strauss
, and J.
Manickam
, Phys. Fluids
29
, 1171
(1986
).48.
S.
Cappello
and D.
Biskamp
, Nucl. Fusion
36
, 571
(1996
).49.
50.
S. C.
Jardin
, J.
Breslau
, and N.
Ferraro
, J. Comput. Phys.
226
, 2146
(2007
).51.
C. R.
Sovinec
, T. A.
Gianakon
, E. D.
Held
, S. E.
Kruger
, and D. D.
Schnack
, Phys. Plasmas
10
, 1727
(2003
).52.
D.
del Castillo-Negrete
and L.
Chacon
, Phys. Rev. Lett.
106
, 195004
(2011
).53.
F.
Bauer
, O.
Betancourt
, and P.
Garabedian
, Magnetohydrodynamic Equilibrium and Stability of Stellarators
(Springer-Verlag
, Berlin
, 1984
).54.
S. P.
Hirshman
and J. P.
Whitson
, Phys. Fluids
26
, 3553
(1983
).55.
O.
Betancourt
, Commun. Pure Appl. Math.
41
, 551
(1988
).56.
M.
Taylor
, J. Comput. Phys.
110
, 407
(1994
).57.
A.
Bhattacharjee
, T.
Hayashi
, C. C.
Hegna
, N.
Nakajima
, and T.
Sato
, Phys. Plasmas
2
, 883
(1995
).58.
59.
A. H.
Boozer
and N.
Pomphrey
, Phys. Plasmas
17
, 110707
(2010
).60.
J. B.
Taylor
, Phys. Rev. Lett.
33
, 1139
(1974
).61.
J. M.
Finn
and T. M.
Antonsen
, Jr., Comm. Plasma Phys. Controlled Fusion
9
, 111
(1985
).62.
M. A.
Berger
, Plasma Phys. Controlled Fusion
41
, B167
(1999
).63.
G.
Ciraolo
, F.
Briolle
, C.
Chandre
, E.
Floriani
, R.
Lima
, M.
Vittot
, M.
Pettini
, C.
Figarella
, and P.
Ghendrih
, Phys. Rev. E.
69
, 056213
(2004
).64.
J.-K.
Park
, A. H.
Boozer
, and A. H.
Glasser
, Phys. Plasmas
14
, 052110
(2007
).65.
C. A.
Nührenberg
and A. H.
Boozer
, Phys. Plasmas
10
, 2840
(2003
).66.
L.
Woltjer
, Proc. Natl. Acad. Sci. U.S.A.
44
, 489
(1958
).67.
R.
Kaiser
and A.
Salat
, Phys. Plasmas
1
, 281
(1994
).68.
D.
Lortz
and G. O.
Spies
, Phys. Plasmas
1
, 249
(1994
).69.
70.
G. O.
Spies
and J.
Li
, Phys. Plasmas
1
, 2901
(1994
).71.
G. O.
Spies
, D.
Lortz
, and R.
Kaiser
, Phys. Plasmas
8
, 3652
(2001
).72.
73.
R.
Kaiser
and H.
Uecker
, Quantum J. Mech. Appl. Math.
57
, 1
(2004
).74.
M. J.
Hole
, S. R.
Hudson
, and R. L.
Dewar
, J. Plasma Phys.
72
, 1167
(2006
).75.
M. J.
Hole
, S. R.
Hudson
, and R. L.
Dewar
, Nucl. Fusion
47
, 746
(2007
).76.
R.
Mills
, M.
Hole
, and R.
Dewar
, J. Plasma Phys.
75
, 637
–659
(2009
).77.
M. J.
Hole
, R.
Mills
, S. R.
Hudson
, and R. L.
Dewar
, Nucl. Fusion
49
, 065019
(2009
).78.
S.
Hudson
, R.
Dewar
, M.
Hole
, and M.
McGann
, Plasma Phys. Controlled Fusion
54
, 014005
(2012
).79.
O.
Betancourt
and P.
Garabedian
, Proc. Natl. Acad. Sci. U.S.A.
72
, 926
(1975
).80.
F.
Bauer
, O.
Betancourt
, and P.
Garabedian
, A Computational Method in Plasma Physics
(Springer-Verlag
, New York
, 1978
).81.
F.
Bauer
, O.
Betancourt
, and P.
Garabedian
, A Computational Method in Plasma Physics
(Springer-Verlag
, New York
, 1982
).82.
O.
Betancourt
and P. R.
Garabedian
, Commun. Pure Appl. Math.
35
, 365
(1982
).83.
A.
Bhattacharjee
, J. C.
Wiley
, and R. L.
Dewar
, Comput. Phys. Commun.
31
, 213
(1984
).84.
S. P.
Hirshman
, W. I.
van Rij
, and P.
Merkel
, Comput. Phys. Commun.
43
, 143
(1986
).85.
A. H.
Reiman
and H. S.
Greenside
, Comput. Phys. Commun.
43
, 157
(1986
).86.
K.
Harafuji
, T.
Hayashi
, and T.
Sato
, J. Comput. Phys.
81
, 169
(1989
).87.
S. P.
Hirshman
and O.
Betancourt
, J. Comput. Phys.
96
, 99
(1991
).88.
Y.
Suzuki
, N.
Nakajima
, K.
Watanabe
, Y.
Nakamura
, and T.
Hayashi
, Nucl. Fusion
46
, L19
(2006
).89.
S. P.
Hirsman
, R.
Sanchez
, and C. R.
Cook
, Phys. Plasmas
18
, 062504
(2011
).90.
91.
92.
H. J.
Gardner
and D. B.
Blackwell
, Nucl. Fusion
32
, 2009
(1992
).93.
R. L.
Dewar
and S. R.
Hudson
, Physica D
112
, 275
(1998
).94.
J.
Nocedal
and S. J.
Wright
, Numerical Optimization
(Springer-Verlag
, New York
, 1999
).95.
R. B.
White
, The Theory of Toroidally Confined Plasmas
(Imperial College
, London
, 2001
).96.
S.
Lazerson
, E.
Lazarus
, S.
Hudson
, N.
Pablant
, and D.
Gates
, “3D Equilibrium Effects due to RMP application on DIII
,” in Conference proceedings: 39th European Physical Society Conference on Plasma Physics and 16th International Congress on Plasma Physics Stockholm, Sweden (2012
), p. 4077
, http://epsicpp2012.spp.ee.kth.se/.97.
98.
K.
Ichiguchi
, M.
Wakatani
, T.
Unemura
, T.
Tatsuno
, and B. A.
Carrreras
, Nucl. Fusion
41
, 181
(2001
).99.
K.
Ichiguchi
and B. A.
Carreras
, Nucl. Fusion
51
, 053021
(2011
).100.
C.
Hegna
, Nucl. Fusion
51
, 113017
(2011
).101.
102.
D.
Bonfiglio
, L.
Chacon
, and S.
Cappello
, Phys. Plasmas
17
, 082501
(2010
).103.
N. M.
Ferraro
, S. P.
Jardin
, and P. B.
Synder
, Phys. Plasmas
17
, 102508
(2010
).104.
M. G.
Schlutt
, C. C.
Hegna
, C. R.
Sovinec
, S. F.
Knowlton
, and J. D.
Hebert
, Nucl. Fusion
52
, 103023
(2012
).105.
Z.
Yoshida
and S. M.
Mahajan
, Phys. Rev. Lett.
88
, 095001
(2002
).106.
P. R.
Garabedian
, Commun. Pure Appl. Math.
51
, 1019
(1998
).107.
M.
Finn
and L.
Chacon
, Phys. Plasmas
12
, 054503
(2005
).© 2012 American Institute of Physics.
2012
American Institute of Physics
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