The RF stabilization of tearing modes with current condensation has the potential to increase stabilization efficiency and loosen power localization requirements. Such benefits stem from the cooperative feedback between the RF deposition and the resulting island temperature perturbation governed by diffusion. A self-consistent treatment of the damping of an rf ray as it traverses the island shows that low damping scenarios can require unfavorably high powers to overcome initial power leakage and effectively capitalize on the nonlinear effect. In this work, it is demonstrated that for such regimes, modulated stabilization schemes can achieve significant improvements in heating and current drive contributions to stabilization for the same average power as a continuous wave scheme. The impact of modulation frequency and duty cycle on the performance is explored, the results of which suggest modulation strategies in which the pulsing periods are kept on the order of a diffusive time.

1.
P. C.
de Vries
,
M. F.
Johnson
,
B.
Alper
,
P.
Buratti
,
T. C.
Hender
,
H. R.
Koslowski
,
V.
Riccardo
, and
JET-EFDA Contributors
, “
Survey of disruption causes at JET
,”
Nucl. Fusion
51
,
053018
(
2011
).
2.
P. C.
de Vries
,
M.
Baruzzo
,
G. M. D.
Hogeweij
,
S.
Jachmich
,
E.
Joffrin
,
P. J.
Lomas
,
G. F.
Matthews
,
A.
Murari
,
I.
Nunes
,
T.
Pütterich
,
C.
Reux
, and
J.
Vega
, “
The influence of an ITER-like wall on disruptions at JET
,”
Phys. Plasmas
21
,
056101
(
2014
).
3.
R. L.
Haye
,
R.
Prater
,
R.
Buttery
,
N.
Hayashi
,
A.
Isayama
,
M.
Maraschek
,
L.
Urso
, and
H.
Zohm
, “
Cross–machine benchmarking for ITER of neoclassical tearing mode stabilization by electron cyclotron current drive
,”
Nucl. Fusion
46
,
451
461
(
2006
).
4.
R. L.
Haye
,
A.
Isayama
, and
M.
Maraschek
, “
Prospects for stabilization of neoclassical tearing modes by electron cyclotron current drive in ITER
,”
Nucl. Fusion
49
,
045005
(
2009
).
5.
A. H.
Reiman
, “
Suppression of magnetic islands by rf driven currents
,”
Phys. Fluids
26
,
1338
1340
(
1983
).
6.
H.
Zohm
, “
Stabilization of neoclassical tearing modes by electron cyclotron current drive
,”
Phys. Plasmas
4
,
3433
3435
(
1997
).
7.
Q.
Yu
and
S.
Günter
, “
On the stabilization of neoclassical tearing modes by phased electron cyclotron waves
,”
Plasma Phys. Controlled Fusion
40
,
1977
1987
(
1998
).
8.
R.
Harvey
and
F.
Perkins
, “
Comparison of optimized ECCD for different launch locations in a next step tokamak reactor plasma
,”
Nucl. Fusion
41
,
1847
1856
(
2001
).
9.
S.
Bernabei
,
A.
Cardinali
,
G.
Giruzzi
, and
M.
Zabiégo
, “
Tearing mode stabilization in tokamaks with lower hybrid waves
,”
Nucl. Fusion
38
,
87
92
(
1998
).
10.
R.
Kamendje
,
S. V.
Kasilov
,
W.
Kernbichler
,
I. V.
Pavlenko
,
E.
Poli
, and
M. F.
Heyn
, “
Modeling of nonlinear electron cyclotron resonance heating and current drive in a tokamak
,”
Phys. Plasmas
12
,
012502
(
2005
).
11.
R. J.
La Haye
, “
Neoclassical tearing modes and their control
,”
Phys. Plasmas
13
,
055501
(
2006
).
12.
R. L.
Haye
,
J.
Ferron
,
D.
Humphreys
,
T.
Luce
,
C.
Petty
,
R.
Prater
,
E.
Strait
, and
A.
Welander
, “
Requirements for alignment of electron cyclotron current drive for neoclassical tearing mode stabilization in ITER
,”
Nucl. Fusion
48
,
054004
(
2008
).
13.
O.
Sauter
,
M. A.
Henderson
,
G.
Ramponi
,
H.
Zohm
, and
C.
Zucca
, “
On the requirements to control neoclassical tearing modes in burning plasmas
,”
Plasma Phys. Controlled Fusion
52
,
025002
(
2010
).
14.
A. I.
Smolyakov
,
A.
Poye
,
O.
Agullo
,
S.
Benkadda
, and
X.
Garbet
, “
Higher order and asymmetry effects on saturation of magnetic islands
,”
Phys. Plasmas
20
,
062506
(
2013
).
15.
B.
Ayten
,
E.
Westerhof
, and
ASDEX Upgrade Team
, “
Non-linear effects in electron cyclotron current drive applied for the stabilization of neoclassical tearing modes
,”
Nucl. Fusion
54
,
073001
(
2014
).
16.
D.
Borgogno
,
L.
Comisso
,
D.
Grasso
, and
E.
Lazzaro
, “
Nonlinear response of magnetic islands to localized electron cyclotron current injection
,”
Phys. Plasmas
21
,
060704
(
2014
).
17.
O.
Février
,
P.
Maget
,
H.
Lütjens
,
J.-F.
Luciani
,
J.
Decker
,
G.
Giruzzi
,
M.
Reich
,
P.
Beyer
,
E.
Lazzaro
, and
S.
Nowak
, “
First principles fluid modelling of magnetic island stabilization by electron cyclotron current drive (ECCD)
,”
Plasma Phys. Controlled Fusion
58
,
045015
(
2016
).
18.
J. C.
Li
,
C. J.
Xiao
,
Z. H.
Lin
, and
K. J.
Wang
, “
Effects of electron cyclotron current drive on magnetic islands in tokamak plasmas
,”
Phys. Plasmas
24
,
082508
(
2017
).
19.
D.
Grasso
,
D.
Borgogno
,
L.
Comisso
, and
E.
Lazzaro
, “
Magnetic island suppression by electron cyclotron current drive as converse of a forced reconnection problem
,”
J. Plasma Phys.
84
,
745840302
(
2018
).
20.
D.
Grasso
,
E.
Lazzaro
,
D.
Borgogno
, and
L.
Comisso
, “
Open problems of magnetic island control by electron cyclotron current drive
,”
J. Plasma Phys.
82
,
595820603
(
2016
).
21.
F.
Poli
,
E.
Fredrickson
,
M.
Henderson
,
S.-H.
Kim
,
N.
Bertelli
,
E.
Poli
,
D.
Farina
, and
L.
Figini
, “
Electron cyclotron power management for control of neoclassical tearing modes in the ITER baseline scenario
,”
Nucl. Fusion
58
,
016007
(
2018
).
22.
Y.
Yoshioka
,
S.
Kinoshha
, and
T.
Kobayashi
, “
Numerical study of magnetic island suppression by RF waves in large tokamaks
,”
Nucl. Fusion
24
,
565
572
(
1984
).
23.
Q.
Yu
,
S.
Günter
,
G.
Giruzzi
,
K.
Lackner
, and
M.
Zabiego
, “
Modeling of the stabilization of neoclassical tearing modes by localized radio frequency current drive
,”
Phys. Plasmas
7
,
312
322
(
2000
).
24.
N.
Bertelli
,
D. D.
Lazzari
, and
E.
Westerhof
, “
Requirements on localized current drive for the suppression of neoclassical tearing modes
,”
Nucl. Fusion
51
,
103007
(
2011
).
25.
E.
Poli
,
C.
Angioni
,
F.
Casson
,
D.
Farina
,
L.
Figini
,
T.
Goodman
,
O.
Maj
,
O.
Sauter
,
H.
Weber
,
H.
Zohm
,
G.
Saibene
, and
M.
Henderson
, “
On recent results in the modelling of neoclassical-tearing-mode stabilization via electron cyclotron current drive and their impact on the design of the upper EC launcher for ITER
,”
Nucl. Fusion
55
,
013023
(
2015
).
26.
W.
Zhang
,
Z. W.
Ma
,
Y.
Zhang
, and
J.
Zhu
, “
Stabilization of tearing modes by modulated electron cyclotron current drive
,”
AIP Adv.
9
,
015020
(
2019
).
27.
Q.
Yu
,
X. D.
Zhang
, and
S.
Günter
, “
Numerical studies on the stabilization of neoclassical tearing modes by radio frequency current drive
,”
Phys. Plasmas
11
,
1960
1968
(
2004
).
28.
X.
Wang
,
X.
Zhang
,
B.
Wu
,
S.
Zhu
, and
Y.
Hu
, “
Numerical study on the stabilization of neoclassical tearing modes by electron cyclotron current drive
,”
Phys. Plasmas
22
,
022512
(
2015
).
29.
L.
Chen
,
J.
Liu
,
G.
Sun
,
P.
Duan
, and
J.
Sun
, “
Modeling of the influences of electron cyclotron current drive on neoclassical tearing modes
,”
Phys. Plasmas
22
,
052120
(
2015
).
30.
F.
Widmer
,
P.
Maget
,
O.
Février
,
H.
Lütjens
, and
X.
Garbet
, “
Non-linear simulations of neoclassical tearing mode control by externally driven RF current and heating, with application to ITER
,”
Nucl. Fusion
59
,
106012
(
2019
).
31.
H.
Zohm
,
G.
Gantenbein
,
G.
Giruzzi
,
S.
Günter
,
F.
Leuterer
,
M.
Maraschek
,
J.
Meskat
,
A.
Peeters
,
W.
Suttrop
,
D.
Wagner
,
M.
Zabiégo
,
A. U.
Team
, and
E.
Group
, “
Experiments on neoclassical tearing mode stabilization by ECCD in ASDEX upgrade
,”
Nucl. Fusion
39
,
577
580
(
1999
).
32.
R.
Prater
,
R. L.
Haye
,
J.
Lohr
,
T.
Luce
,
C.
Petty
,
J.
Ferron
,
D.
Humphreys
,
E.
Strait
,
F.
Perkins
, and
R.
Harvey
, “
Discharge improvement through control of neoclassical tearing modes by localized ECCD in DIII-D
,”
Nucl. Fusion
43
,
1128
1134
(
2003
).
33.
C. D.
Warrick
,
R. J.
Buttery
,
G.
Cunningham
,
S. J.
Fielding
,
T. C.
Hender
,
B.
Lloyd
,
A. W.
Morris
,
M. R.
O'Brien
,
T.
Pinfold
,
K.
Stammers
,
M.
Valovic
,
M.
Walsh
,
H. R.
Wilson
, and
COMPASS-D and RF teams
, “
Complete stabilization of neoclassical tearing modes with lower hybrid current drive on compass-D
,”
Phys. Rev. Lett.
85
,
574
577
(
2000
).
34.
G.
Gantenbein
,
H.
Zohm
,
G.
Giruzzi
,
S.
Günter
,
F.
Leuterer
,
M.
Maraschek
,
J.
Meskat
, and
Q.
Yu
, “
Complete suppression of neoclassical tearing modes with current drive at the electron-cyclotron-resonance frequency in ASDEX Upgrade tokamak
,”
Phys. Rev. Lett.
85
,
1242
1245
(
2000
).
35.
H.
Zohm
,
G.
Gantenbein
,
A.
Gude
,
S.
Günter
,
F.
Leuterer
,
M.
Maraschek
,
J.
Meskat
,
W.
Suttrop
,
Q.
Yu
,
ASDEX Upgrade Team
, and
ECRH Group
, “
The physics of neoclassical tearing modes and their stabilization by ECCD in ASDEX Upgrade
,”
Nucl. Fusion
41
,
197
202
(
2001
).
36.
A.
Isayama
,
Y.
Kamada
,
S.
Ide
,
K.
Hamamatsu
,
T.
Oikawa
,
T.
Suzuki
,
Y.
Neyatani
,
T.
Ozeki
,
Y.
Ikeda
,
K.
Takahashi
,
K.
Kajiwara
, and
JT-60 Team
, “
Complete stabilization of a tearing mode in steady state high-pH-mode discharges by the first harmonic electron cyclotron heating/current drive on JT-60U
,”
Plasma Phys. Controlled Fusion
42
,
L37
L45
(
2000
).
37.
R. J.
La Haye
,
S.
Günter
,
D. A.
Humphreys
,
J.
Lohr
,
T. C.
Luce
,
M. E.
Maraschek
,
C. C.
Petty
,
R.
Prater
,
J. T.
Scoville
, and
E. J.
Strait
, “
Control of neoclassical tearing modes in DIII-D
,”
Phys. Plasmas
9
,
2051
2060
(
2002
).
38.
C.
Petty
,
R. L.
Haye
,
T.
Luce
,
D.
Humphreys
,
A.
Hyatt
,
J.
Lohr
,
R.
Prater
,
E.
Strait
, and
M.
Wade
, “
Complete suppression of them = 2/n = 1 neoclassical tearing mode using electron cyclotron current drive in DIII-D
,”
Nucl. Fusion
44
,
243
251
(
2004
).
39.
F. A.
Volpe
,
A.
Hyatt
,
R. J.
La Haye
,
M. J.
Lanctot
,
J.
Lohr
,
R.
Prater
,
E. J.
Strait
, and
A.
Welander
, “
Avoiding tokamak disruptions by applying static magnetic fields that align locked modes with stabilizing wave-driven currents
,”
Phys. Rev. Lett.
115
,
175002
(
2015
).
40.
A.
Nelson
,
R. L.
Haye
,
M.
Austin
,
A.
Welander
, and
E.
Kolemen
, “
Simultaneous detection of neoclassical tearing mode and electron cyclotron current drive locations using electron cyclotron emission in DIII-D
,”
Fusion Eng. Des.
141
,
25
29
(
2019
).
41.
W.
Choi
,
R. L.
Haye
,
M.
Lanctot
,
K.
Olofsson
,
E.
Strait
,
R.
Sweeney
,
F. A.
Volpe
, and
DIII-D Team
, “
Feedforward and feedback control of locked mode phase and rotation in DIII-D with application to modulated ECCD experiments
,”
Nucl. Fusion
58
,
036022
(
2018
).
42.
A. H.
Reiman
and
N. J.
Fisch
, “
Suppression of tearing modes by radio frequency current condensation
,”
Phys. Rev. Lett.
121
,
225001
(
2018
).
43.
N. J.
Fisch
, “
Confining a tokamak plasma with rf-driven currents
,”
Phys. Rev. Lett.
41
,
873
876
(
1978
).
44.
N. J.
Fisch
and
A. H.
Boozer
, “
Creating an asymmetric plasma resistivity with waves
,”
Phys. Rev. Lett.
45
,
720
722
(
1980
).
45.
N. J.
Fisch
, “
Theory of current drive in plasmas
,”
Rev. Mod. Phys.
59
,
175
234
(
1987
).
46.
E.
Westerhof
,
A.
Lazaros
,
E.
Farshi
,
M.
de Baar
,
M.
de Bock
,
I.
Classen
,
R.
Jaspers
,
G.
Hogeweij
,
H.
Koslowski
,
A.
Krämer-Flecken
,
Y.
Liang
,
N. L.
Cardozo
, and
O.
Zimmermann
, “
Tearing mode stabilization by electron cyclotron resonance heating demonstrated in the TEXTOR tokamak and the implication for ITER
,”
Nucl. Fusion
47
,
85
90
(
2007
).
47.
E.
Rodríguez
,
A. H.
Reiman
, and
N. J.
Fisch
, “
Rf current condensation in magnetic islands and associated hysteresis phenomena
,”
Phys. Plasmas
26
,
092511
(
2019
).
48.
P. T.
Bonoli
and
R. C.
Englade
, “
Simulation model for lower hybrid current drive
,”
Phys. Fluids
29
,
2937
2950
(
1986
).
49.
R.
Prater
,
D.
Farina
,
Y.
Gribov
,
R.
Harvey
,
A.
Ram
,
Y.
Lin-Liu
,
E.
Poli
,
A.
Smirnov
,
F.
Volpe
,
E.
Westerhof
,
A.
Zvonkov
, and
ITPA Steady State Operation Topical Group
, “
Benchmarking of codes for electron cyclotron heating and electron cyclotron current drive under ITER conditions
,”
Nucl. Fusion
48
,
035006
(
2008
).
50.
C. F. F.
Karney
,
N. J.
Fisch
, and
F. C.
Jobes
, “
Comparison of the theory and the practice of lower-hybrid current drive
,”
Phys. Rev. A
32
,
2554
2556
(
1985
).
51.
C.
Tsironis
,
A. G.
Peeters
,
H.
Isliker
,
D.
Strintzi
,
I.
Chatziantonaki
, and
L.
Vlahos
, “
Electron-cyclotron wave scattering by edge density fluctuations in ITER
,”
Phys. Plasmas
16
,
112510
(
2009
).
52.
G.
Kurita
,
T.
Tuda
,
M.
Azumi
,
T.
Takizuka
, and
T.
Takeda
, “
Effect of local heating on the m = 2 tearing mode in a tokamak
,”
Nucl. Fusion
34
,
1497
1515
(
1994
).
53.
C. C.
Hegna
and
J. D.
Callen
, “
On the stabilization of neoclassical magnetohydrodynamic tearing modes using localized current drive or heating
,”
Phys. Plasmas
4
,
2940
2946
(
1997
).
54.
D. D.
Lazzari
and
E.
Westerhof
, “
On the merits of heating and current drive for tearing mode stabilization
,”
Nucl. Fusion
49
,
075002
(
2009
).
55.
E.
Rodríguez
,
A. H.
Reiman
, and
N. J.
Fisch
, “
Rf current condensation in the presence of turbulent enhanced transport
,”
Phys. Plasmas
27
,
042306
(
2020
).
56.
M.
Maraschek
,
G.
Gantenbein
,
Q.
Yu
,
H.
Zohm
,
S.
Günter
,
F.
Leuterer
,
A.
Manini
,
ECRH Group
, and
ASDEX Upgrade Team
, “
Enhancement of the stabilization efficiency of a neoclassical magnetic island by modulated electron cyclotron current drive in the ASDEX Upgrade tokamak
,”
Phys. Rev. Lett.
98
,
025005
(
2007
).
57.
F. A. G.
Volpe
,
M. E.
Austin
,
R. J.
La Haye
,
J.
Lohr
,
R.
Prater
,
E. J.
Strait
, and
A. S.
Welander
, “
Advanced techniques for neoclassical tearing mode control in DIII-D
,”
Phys. Plasmas
16
,
102502
(
2009
).
58.
A.
Isayama
,
G.
Matsunaga
,
T.
Kobayashi
,
S.
Moriyama
,
N.
Oyama
,
Y.
Sakamoto
,
T.
Suzuki
,
H.
Urano
,
N.
Hayashi
,
Y.
Kamada
,
T.
Ozeki
,
Y.
Hirano
,
L.
Urso
,
H.
Zohm
,
M.
Maraschek
,
J.
Hobirk
,
K.
Nagasaki
, and
JT-60 Team
, “
Neoclassical tearing mode control using electron cyclotron current drive and magnetic island evolution in JT-60U
,”
Nucl. Fusion
49
,
055006
(
2009
).
59.
W.
Kasparek
,
N.
Doelman
,
J.
Stober
,
M.
Maraschek
,
H.
Zohm
,
F.
Monaco
,
H.
Eixenberger
,
W.
Klop
,
D.
Wagner
,
M.
Schubert
,
H.
Schütz
,
G.
Grünwald
,
B.
Plaum
,
R.
Munk
,
K. H.
Schlüter
, and
ASDEX Upgrade Team
, “
NTM stabilization by alternating O-point EC current drive using a high-power diplexer
,”
Nucl. Fusion
56
,
126001
(
2016
).
60.
H.
Zohm
,
G.
Gantenbein
,
F.
Leuterer
,
A.
Manini
,
M.
Maraschek
,
Q.
Yu
, and
ASDEX Upgrade Team
, “
Control of MHD instabilities by ECCD: ASDEX Upgrade results and implications for ITER
,”
Nucl. Fusion
47
,
228
232
(
2007
).
61.
O.
Sauter
, “
On the contribution of local current density to neoclassical tearing mode stabilization
,”
Phys. Plasmas
11
,
4808
4813
(
2004
).
62.
M.
Erba
,
T.
Aniel
,
V.
Basiuk
,
A.
Becoulet
, and
X.
Litaudon
, “
Validation of a new mixed Bohm/gyro-Bohm model for electron and ion heat transport against the ITER, tore supra and START database discharges
,”
Nucl. Fusion
38
,
1013
1028
(
1998
).
63.
P.
Helander
, “
Classical and neoclassical transport in tokamaks
,”
Fusion Sci. Technol.
61
,
133
141
(
2012
).
64.
S. V.
Konovalov
,
A. B.
Mikhailovskii
,
T.
Ozeki
,
T.
Takizuka
,
M. S.
Shirokov
, and
N.
Hayashi
, “
Role of anomalous transport in onset and evolution of neoclassical tearing modes
,”
Plasma Phys. Controlled Fusion
47
,
B223
B236
(
2005
).
You do not currently have access to this content.