Feedback effects between resonance surface and space harmonics of external perturbations in tokamak
Resonant magnetic surfaces in a tokamak can amplify the spatial harmonics of external perturbations, which may come from other resonant surfaces, from error fields, or from a feedback system. The behavior of this active resonant media can be roughly approximated with a system of coupled Van der Pole...
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irk-123456789-793102015-04-01T03:02:04Z Feedback effects between resonance surface and space harmonics of external perturbations in tokamak Semenov, I.B. Mirnov, S.V. Fredrickson, E.D. Voznesensky, V.A. Magnetic confinement Resonant magnetic surfaces in a tokamak can amplify the spatial harmonics of external perturbations, which may come from other resonant surfaces, from error fields, or from a feedback system. The behavior of this active resonant media can be roughly approximated with a system of coupled Van der Pole oscillators. The effect of frequency injection locking (or spatial harmonics injection locking in the plasma frame) is typical for these nonlinear systems. It happens when the amplitude of one modes increases and this mode becomes a dominant mode. Transition into synchronized condition can occur in a time scale of ~ 50 -100 µsec. For a tokamak it means that the stability of a large scale MHD perturbation can change jumpily, because frequency (phase) lock may create a positive feedback between resonant surfaces (or between resonant surfaces and the external feedback system). This effect probably determines the explosive dynamic of the disruptive instability. Резонансна магнітна поверхня в токамаці може підсилювати просторові гармоніки збурювань інших резонансних поверхонь, збурювання просторових гармонік обмоток полоідального і тороідального полів (Error field) або обмоток зворотних зв'язків (Feedback field). Поводження цього активного резонансного середовища грубо можна апроксимувати системою зв'язаних генераторів Ван дер Поля. Ефект захоплення частоти (або захоплення просторових гармонік збурювань у системі координат, зв'язаної з плазмою), є типовим для подібних нелінійних систем. Він відбувається в тому випадку, коли амплітуда однієї з мод збільшується і ця мода стає домінантною модою. Перехід у стан захоплення (синхронізації) частоти відбувається за часи ~ 50 -100 µsec. У цей момент стійкий стан великомасштабних МГД-збурювань може стрибком стати нестійким унаслідок появи позитивного зворотного зв'язку між резонансними поверхнями (або між резонансними поверхнями і системою зворотних зв'язків). Цей ефект можливо визначає вибуховий характер розвитку нестійкості зриву. Резонансная магнитная поверхность в токамаке может усиливать пространственные гармоники возмущений других резонансных поверхностей, возмущения пространственных гармоник обмоток полоидального и тороидального полей (Error field) или обмоток обратных связей (Feedback field). Поведение этой активной резонансной среды грубо можно аппроксимировать системой связанных генераторов Ван дер Поля. Эффект захвата частоты (или захвата пространственных гармоник возмущений в системе координат, связанной с плазмой), является типичным для подобных нелинейных систем. Он происходит в том случае, когда амплитуда одной из мод увеличивается и эта мода становится доминантной модой. Переход в состояние захвата (синхронизации) частоты происходит за времена ~ 50 -100 µsec. В этот момент устойчивое состояние крупномасштабных МГД-возмущений может скачком стать неустойчивым вследствие появления положительной обратной связи между резонансными поверхностями (или между резонансными поверхностями и системой обратных связей). Этот эффект возможно определяет взрывной характер развития неустойчивости срыва. 2005 Article Feedback effects between resonance surface and space harmonics of external perturbations in tokamak / I.B. Semenov, S.V. Mirnov, E.D. Fredrickson, V.A. Voznesensky // Вопросы атомной науки и техники. — 2005. — № 2. — С. 11-13. — Бібліогр.: 15 назв. — англ. 1562-6016 PACS: 52.55.Fa; 52.27.Gr; 52.35.Mw http://dspace.nbuv.gov.ua/handle/123456789/79310 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Magnetic confinement Magnetic confinement |
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Magnetic confinement Magnetic confinement Semenov, I.B. Mirnov, S.V. Fredrickson, E.D. Voznesensky, V.A. Feedback effects between resonance surface and space harmonics of external perturbations in tokamak Вопросы атомной науки и техники |
| description |
Resonant magnetic surfaces in a tokamak can amplify the spatial harmonics of external perturbations, which may come from other resonant surfaces, from error fields, or from a feedback system. The behavior of this active resonant media can be roughly approximated with a system of coupled Van der Pole oscillators. The effect of frequency injection locking (or spatial harmonics injection locking in the plasma frame) is typical for these nonlinear systems. It happens when the amplitude of one modes increases and this mode becomes a dominant mode. Transition into synchronized condition can occur in a time scale of ~ 50 -100 µsec. For a tokamak it means that the stability of a large scale MHD perturbation can change jumpily, because frequency (phase) lock may create a positive feedback between resonant surfaces (or between resonant surfaces and the external feedback system). This effect probably determines the explosive dynamic of the disruptive instability. |
| format |
Article |
| author |
Semenov, I.B. Mirnov, S.V. Fredrickson, E.D. Voznesensky, V.A. |
| author_facet |
Semenov, I.B. Mirnov, S.V. Fredrickson, E.D. Voznesensky, V.A. |
| author_sort |
Semenov, I.B. |
| title |
Feedback effects between resonance surface and space harmonics of external perturbations in tokamak |
| title_short |
Feedback effects between resonance surface and space harmonics of external perturbations in tokamak |
| title_full |
Feedback effects between resonance surface and space harmonics of external perturbations in tokamak |
| title_fullStr |
Feedback effects between resonance surface and space harmonics of external perturbations in tokamak |
| title_full_unstemmed |
Feedback effects between resonance surface and space harmonics of external perturbations in tokamak |
| title_sort |
feedback effects between resonance surface and space harmonics of external perturbations in tokamak |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2005 |
| topic_facet |
Magnetic confinement |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/79310 |
| citation_txt |
Feedback effects between resonance surface and space harmonics of external perturbations in tokamak / I.B. Semenov, S.V. Mirnov, E.D. Fredrickson, V.A. Voznesensky // Вопросы атомной науки и техники. — 2005. — № 2. — С. 11-13. — Бібліогр.: 15 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT semenovib feedbackeffectsbetweenresonancesurfaceandspaceharmonicsofexternalperturbationsintokamak AT mirnovsv feedbackeffectsbetweenresonancesurfaceandspaceharmonicsofexternalperturbationsintokamak AT fredricksoned feedbackeffectsbetweenresonancesurfaceandspaceharmonicsofexternalperturbationsintokamak AT voznesenskyva feedbackeffectsbetweenresonancesurfaceandspaceharmonicsofexternalperturbationsintokamak |
| first_indexed |
2025-07-06T03:23:35Z |
| last_indexed |
2025-07-06T03:23:35Z |
| _version_ |
1836866303478988800 |
| fulltext |
FEEDBACK EFFECTS BETWEEN RESONANCE SURFACES AND SPACE
HARMONICS OF EXTERNAL PERTURBATIONS IN TOKAMAK
I.B. Semenov 1, S.V. Mirnov 2, E.D. Fredrickson 3, V.A. Voznesensky 1
1 RRC “Kurchatov Institute”, 123182, Moscow, Russia, isemenov@nfi.kiae.ru;
2 TRINITI, Troitsk, 142092, Moscow Reg., Russia, mirnov@triniti.ru;
3 Plasma Physics Laboratory, Princeton, NJ 08543, USA, efredrickson@pppl.gov
Resonant magnetic surfaces in a tokamak can amplify the spatial harmonics of external perturbations, which may
come from other resonant surfaces, from error fields, or from a feedback system. The behavior of this active resonant
media can be roughly approximated with a system of coupled Van der Pole oscillators. The effect of frequency injection
locking (or spatial harmonics injection locking in the plasma frame) is typical for these nonlinear systems. It happens
when the amplitude of one modes increases and this mode becomes a dominant mode. Transition into synchronized
condition can occur in a time scale of ~ 50 -100 µsec. For a tokamak it means that the stability of a large scale MHD
perturbation can change jumpily, because frequency (phase) lock may create a positive feedback between resonant
surfaces (or between resonant surfaces and the external feedback system). This effect probably determines the explosive
dynamic of the disruptive instability.
PACS: 52.55.Fa; 52.27.Gr; 52.35.Mw
We use the following definitions:
• Active mode is the tearing or kink mode which has
positive or close to 0 positive growth. The behavior of
this mode depends on main plasma parameters such as
shear, pressure profile, etc
• Passive mode is the mode which has negative growth or
close to 0 negative growth. The behavior of this mode
depends on amplitude and phase of the external
perturbations.
• Dominant mode is the mode, which determines the
global phase synchronization of the secondary
active or passive modes.
Perturbations in a torus are coupled so that the
development of an active mode at a resonance surface
may excite a cascade of secondary passive (slave) modes
at other appropriate m/n resonance surfaces q(r)=(m/n),
where r- m is the poloidal, and n is the toroidal number of
the helical perturbation (Fig.1a). Real excitation of these
modes depends on the spatial structure of the driving
perturbation (which comes from the active mode or from
an external error field) and depends on conditions of the
mode excitation, such as the geometry of the plasma, the
current and pressure density profile, the values of toroidal
magnetic field and plasma current, electromagnetic
conditions of the vacuum wall and also depends on the
relative velocity of the resonance surfaces and external
perturbations. In the laboratory frame these phenomena
can be described in frequency terms as resonance,
frequency capture or mode locking. Destruction of
resonances between different resonance surfaces and
between resonance surfaces and error fields may greatly
improve high-performance tokamak operation.
Experiments show the number of different m/n resonant
perturbations, measured in different discharges at q≤5, do not
exceed 16. This particular case is shown in Fig. 1b.
Usually, total amount of active and passive modes,
which are developing in one shot simultaneously do not
exceed 4-6 (for the case Fig. 1a). In Fig. 1b the circus
show possible modes excitation in the case, for example,
when the active primary mode m/n=1 excites the set of
passive secondary modes 2/1, 3/1, 4/1, 5/1.
7/
3
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
q(
ρ)
ρ
7/
2
m
/n
=1
/1
,2
/2
,3
/3
,4
/4
2/
1,
4/
2,
6/
3,
8/
4
3/
1,
6/
2,
9/
3,
12
/4
4/
1,
8/
2,
12
/3
5/
1,
10
/2
5/
4
4/
3 3/
2,
6/
4
7/
4
5/
3
5/
2,
10
/4
9/
4
10
/3
11
/3
9/
2
11
/4
8/
3
Theoretically 32 active or
passive modes could be
abserved at rational
surfaces in the case
q=m/n=(1-12)/(1-4)<5
Fig. 1a
Fig.1
Problems of Atomic Science and Technology. Series: Plasma Physics (11). 2005. № 2. P. 11-13 11
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
q(
ρ)
ρ
5/
1
m
/n
=1
/1
,2
/2
,
3/
3,
4/
4
4/
3
Experimentally ~16 Active or
Passive coupled modes could
be observed in tokamak at
q(a)=m/n<5
2/
1,
4/
2,
6/
3
3/
1,
6/
2,
9/
3
3/
2 5/
3
4/
1
5/
2
E
R
R
O
R
F
IE
LD
FE
E
D
B
A
C
K
F
IE
LD
One of the
possible
feedback channels
A
P
P
P
P
Fig. 1b injectedA
a
b
a
b
cThe secondary modes interact with each other on the
tertiary basis and interact with space harmonics of error
fields - Aerror
sum and feedback systems - A feedback
sum . In
such a way the complicated system of resonant
interactions is realized [1, 2, 3]. At each resonance
surface the sum of these external perturbations are
forming the resultant perturbation Ainjected
sum , which
determines the dynamics of this magnetic surface (for
example 2/1 at Fig. 1b).
Experimentally it is possible to observe 3 forms of the
mode excitation: First, the passive eigenmode excitation
by corresponding harmonic of the external perturbation.
Second, the triggering of the active eigenmode (this
excitation is similar to hard excitation of the generator by
external perturbation or, for example, an excitation of
pendulum clock). In this case at the resonant surface, in
laboratory frame, it is possible to observe two identical
m/n modes with different frequencies. For example at q=2
(Fig. 1b), we can measure the free running 2/1eigenmode
and injected 2/1 mode with the frequency equals to 1/1 at
q=1. Third, it is possible to observe all modes
synchronization or frequency locking under the influence
of dominant mode. It happens when amplitude one of the
modes reaches the level of ~ 5 Gauss (data were corrected
according [4]). Synchronization (mode-locked mode) is
accompanied by sharp increase in amplitude. Mode-locked
mode, as a rule is observing directly before disruptive
instability. To all appearance, the sharp modes amplitude
growth is possible to explain by onset of positive feedback
between active modes. One possible feedback channels is
shown by dashed curve in Fig. 1b.Previously described
MHD-perturbation dynamics in coupled system of resonant
magnetic surfaces is similar to dynamics of coupled
nonlinear oscillators. For the first time it was mentioned by
Huygens [5] for the coupled pendulum system. This
phenomenon is well known in modern engineering and
describes by system of coupled quadrupoles (see Fig. 2)
based on Van der Pole generators [6, 7, 8].
Fig.2
Van der Pole equations set for coupled magnetic
surfaces is possible to write as:
where
– the sum amplitude of the injected perturbations at
resonant surface q;
ω0
q= 1
LqC q - free running frequency at
q=k=m /n ; k , Φk , μk - injected frequency, phase
and coupling coefficient between other resonant surfaces
at q≠k=m /n ;
α , β , Rg
q , RL
q - Van der Pole generator parameters
As is known, the locking amplitude
Ainjected
sum =Ainjected
locking describes for this system by Adler
ratio[9]:
where Q is the resonant quality of the system.
Application of the Adler ratio to resonant magnetic
surfaces is the subject of the next article. Note if the
coupled system has high Q and small frequency
difference, the locking maybe happen under very small
Ainjected
locking amplitude.
The fact of modes slippage and sharp transformation
into condition when dominant mode guides all other
modes is very important for feedback control studies
because the perturbation from feedback system in some
cases could play the role of dominant mode and excite
local positive feedback between internal modes, which
could lead to disruptive instability. Analytical model of
resonant field amplification in tokamaks is considered by
V. Pustovitov [10] and A. Boozer [11].
Simulation of the frequency locking dynamics as a
function of injected perturbation amplitude Ainjected
sum
was carried out by Dr. Yu. Mitrishkin. Fig. 3 shows the
sharp suppression of the free running perturbation and
transition into locking condition for the case of high
frequency difference ωinjected=2ω0 .
Fig.3
Note that the external error field can play an important
role as dominant mode with ω=0. Fig. 4 shows that case
(TFTR). As amplitude of stationary perturbation n=1
increases (Fig. 4a) the abrupt changes in mode structure
are happened (Fig. 4c).
12
injected
q
out
q
out
q
out
q
out AAAAA =+−− )1( 2
0
2 ωβα sum
Ainjected
sum =∑
k≠q
μk Aout
k Cos k tΦk t Aerror
sum A feedback
sum
qL
qC
injectedA q
outAq
LR
q
gR−
Nonlinear circuit
element
Quadrupole cellFig. 2
0 2 4 6 8 100
4
8
12
16
O
ut
pu
t S
pe
ct
ra
l D
en
si
ty
(A
.U
.)
ω 0 ω 02
Injected perturbation amplitude at ω 02ω=
Simulation
Fig. 3
3.953.943.93 3.96 3.97
TFTR #76778
Stationary Magnetic
Perturbation n=1
Ip
(M
A
)
G
au
ss
0
-5
5
0
Phase lock level
M
in
or
di
sr
up
tio
n
3.9512 3.9515 3.9518
0
180
360
B
(t
)a
s
br
ig
ht
ne
ss
P
lo
id
al
an
gl
e
(d
eg
re
es
)
Time (sec)
Locked mode
Fig. 4
(a)
(b)
(c)
Ip=2.43 MA
dΦk t
dt
=ω0
q 1/2Q Im Ainjected
sum /Aout
q
Fig.4
In the moment (t=3.9516 s) the stationary dominant
mode m=3/n=1 changes the rotational dominant mode
m=4/n=1 or “rotational Locked mode” m=4/n=1
transforms into stationary locked mode m=3/n=1.
Transition is accompanied by minor disruption and lasts
150-200 µs. (Toroidal rotation of the plasma column
apparently lasts out.). The dynamics of minor disruption
was described in details in [12] but it was not clear the
reason of stationary Locked mode. Above discussed
mechanism of phase velocities alignment (similarly to
Adler ratio) by dominant mode fills this gap.
Note, in the case of a stationary error field it is
difficult to observe the amplification of error field directly
using magnetic probes which produces a false impression
of precursor absence in some disruptions. Amplification
of a pulsed error field is easily observed in RWM
stabilization experiments [13]. The modes
synchronization with externally applied active
perturbation and amplification of this perturbation was
observed in T-10 [14], [15].
We thank to Ken Young (PPPL), Martin Peng
(PPPL), Ted Strait (GA), for help, fruitful discussions and
TFTR, NSTX and DIIID data.
This work was done under “US-RF program of
cooperation in fusion”.
REFERENCES
1.I. Semenov et al.//Proc. of 30th EPS Conference on
Controlled Fusion and Plasma Physics, St. Petersburg,
Russia. 2003, P-3.158.
2.I. Semenov et al.//MHD workshop “Control of MHD
stability by rotation”, San Diego, 2001.
3.S. Mirnov et al.//Phys. of Plasmas, 1998, v 5(11),
p.3950.
4.V.D. Pustovitov//Pis’ma v ZhETPh (78). 2003, N 5, p.
727-730.
5.H. Gujgens. Tree memuara po mekhanike. Мoscow:
AN SSSR, 1951.
6.M.I. Rabinovich, D.I. Trubetskov. Vvedenie v teoriyu
kolebanij I voln. М.: “Nauka”, 1992 (in Russian).
7.Ye.I. Minaev. Osnovy radiophisiki. М.: "Radio I
Svyaz’", 1990 (in Russian).
8.J. L. Summers, J. Brindley, P. H. Gaskell, et al. The
Role of Poincare-Andronov-Hopf Bifurcations in the
Application of Variable-Coefficient Harmonic Balance to
Periodically Forced Nonlinear Oscillators // Savage
Philosophical Transactions: Mathematical, Physical and
Engineering Sciences. Vol. 354, No. 1704, Jan. 15, 1996,
Jstor, pp. 143-168.
9.R. Adler // Proc. IRE 34, 351 (1946), [reprinted in Proc
IEEE 61, 1380 (1973)].
10.V. Pustovitov. Analytical Model of the Resonant
Field Amplification in Tokamak. // Proc. of 30th EPS
Conference on Controlled Fusion and Plasma Physics,
St. Petersburg, Russia, 2003.
11.A. Boozer // Phys. Plasmas (10). 2003, p.1458.
12.I. Semenov, et al.// Phys. of Plasmas (10). 2003, n. 3,
p. 664.
13.E.J. Strait, et al. // Proc. of 19th IAEA Fusion Energy
Conference, Lyon, France, 2002. EA-CN-94/EX/S2-1
14.Chudnovskiy et al. // Proc. of 30th EPS Conference
on Controlled Fusion and Plasma Physics, St.
Petersburg, Russia, 2003. P-3.11.
15.N. Ivanov, et al. // Proc. of 19th IAEA Fusion Energy
Conference, Lyon, France, 2002. EX/S2-3.
ЭФФЕКТЫ ОБРАТНОЙ СВЯЗИ МЕЖДУ РЕЗОНАНСНЫМИ ПОВЕРХНОСТЯМИ И
ПРОСТРАНСТВЕННЫМИ ГАРМОНИКАМИ ВНЕШНИХ ВОЗМУЩЕНИЙ В ТОКАМАКЕ
И.Б. Семенов, С.В. Мирнов, Е.Д. Фредриксон, В.А. Вознесенский
Резонансная магнитная поверхность в токамаке может усиливать пространственные гармоники возмущений
других резонансных поверхностей, возмущения пространственных гармоник обмоток полоидального и
тороидального полей (Error field) или обмоток обратных связей (Feedback field). Поведение этой активной
резонансной среды грубо можно аппроксимировать системой связанных генераторов Ван дер Поля. Эффект захвата
частоты (или захвата пространственных гармоник возмущений в системе координат, связанной с плазмой), является
типичным для подобных нелинейных систем. Он происходит в том случае, когда амплитуда одной из мод
увеличивается и эта мода становится доминантной модой. Переход в состояние захвата (синхронизации) частоты
происходит за времена ~ 50 -100 µsec. В этот момент устойчивое состояние крупномасштабных МГД-возмущений
может скачком стать неустойчивым вследствие появления положительной обратной связи между резонансными
поверхностями (или между резонансными поверхностями и системой обратных связей). Этот эффект возможно
определяет взрывной характер развития неустойчивости срыва.
ЕФЕКТИ ЗВОРОТНОГО ЗВ'ЯЗКУ МІЖ РЕЗОНАНСНИМИ ПОВЕРХНЯМИ І ПРОСТОРОВИМИ
ГАРМОНІКАМИ ЗОВНІШНІХ ЗБУРЮВАНЬ У ТОКАМАЦІ
І.Б. Семенов, С.В. Мирнов, Е.Д. Фредріксон, В.А. Вознесенський
Резонансна магнітна поверхня в токамаці може підсилювати просторові гармоніки збурювань інших резонансних
поверхонь, збурювання просторових гармонік обмоток полоідального і тороідального полів (Error field) або обмоток
зворотних зв'язків (Feedback field). Поводження цього активного резонансного середовища грубо можна
апроксимувати системою зв'язаних генераторів Ван дер Поля. Ефект захоплення частоти (або захоплення
просторових гармонік збурювань у системі координат, зв'язаної з плазмою), є типовим для подібних нелінійних
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http://links.jstor.org/sici?sici=1364-503X(19960115)354%3A1704<143%3ATROPBI>2.0.CO%3B2-J
http://links.jstor.org/sici?sici=1364-503X(19960115)354%3A1704<143%3ATROPBI>2.0.CO%3B2-J
http://links.jstor.org/sici?sici=1364-503X(19960115)354%3A1704<143%3ATROPBI>2.0.CO%3B2-J
систем. Він відбувається в тому випадку, коли амплітуда однієї з мод збільшується і ця мода стає домінантною
модою. Перехід у стан захоплення (синхронізації) частоти відбувається за часи ~ 50 -100 µsec. У цей момент стійкий
стан великомасштабних МГД-збурювань може стрибком стати нестійким унаслідок появи позитивного зворотного
зв'язку між резонансними поверхнями (або між резонансними поверхнями і системою зворотних зв'язків). Цей ефект
можливо визначає вибуховий характер розвитку нестійкості зриву.
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