The influence of magnetic islands on the impurity ion motion in the drift optimized stellarator
The motion of the charged test particles ensemble of tungsten ions in the drift optimized stellarator Wendelstein 7-X with the chain of 5 magnetic islands is studied. The numerical code solves the guiding center equations written in the Hamiltonian form. To simulate the Coulomb scattering the Mont...
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| Cite this: | The influence of magnetic islands on the impurity ion motion in the drift optimized stellarator / Zh.S. Kononenko, A.A. Shishkin // Вопросы атомной науки и техники. — 2008. — № 4. — С. 95-98. — Бібліогр.: 7 назв. — англ. |
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Kononenko, Zh.S. Shishkin, A.A. 2017-01-03T16:54:30Z 2017-01-03T16:54:30Z 2008 The influence of magnetic islands on the impurity ion motion in the drift optimized stellarator / Zh.S. Kononenko, A.A. Shishkin // Вопросы атомной науки и техники. — 2008. — № 4. — С. 95-98. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 52.55.Fa https://nasplib.isofts.kiev.ua/handle/123456789/110356 The motion of the charged test particles ensemble of tungsten ions in the drift optimized stellarator Wendelstein 7-X with the chain of 5 magnetic islands is studied. The numerical code solves the guiding center equations written in the Hamiltonian form. To simulate the Coulomb scattering the Monte Carlo collision operator is used. The scattering of the particles due to the collisions and magnetic islands is shown. The question of the possibility to use the magnetic islands for the prevention of impurity penetration to the plasma is discussed Вивчено рух ансамблю заряджених частинок іонів вольфраму у дрейфово-оптимізованому стелараторі Вендельштайн 7-Х з ланцюгом з п’яти магнітних островів. Рівняння ведучого центру, що записані у гамільтоновій формі, розв’язуються чисельно. Для моделювання кулонівського розсіяння використовується дискретний оператор Монте-Карло. Показано розсіяння частинок в полоїдальному перерізі під впливом зіткнень та магнітних островів. Обговорюється питання про можливість використання островів для запобігання проникання домішок у плазму. Изучено движение ансамбля заряженных тестовых частиц ионов вольфрама в дрейфово- оптимизированном стеллараторе Вендельштайн 7-Х с цепочкой из пяти магнитных островов. Уравнения ведущего центра, записанные в гамильтоновской форме, решаются численно. Для моделирования кулоновского рассеяния используется дискретный оператор Монте-Карло. Показано рассеяние частиц в полоидальном сечении под воздействием столкновений и магнитных островов. Обсуждается вопрос о возможности использования магнитных островов для предотвращения проникновения примесей в плазму. The authors would like to thank the Science and Technology Center in Ukraine (Project № 3685) for the partial support of this work. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Термоядерный синтез (коллективные процессы) The influence of magnetic islands on the impurity ion motion in the drift optimized stellarator Вплив магнітних островів на рух іонів домішки у дрейфово-оптимізованому стелараторі Влияние магнитных островов на движение примесных ионов в дрейфово-оптимизированном стеллараторе Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
The influence of magnetic islands on the impurity ion motion in the drift optimized stellarator |
| spellingShingle |
The influence of magnetic islands on the impurity ion motion in the drift optimized stellarator Kononenko, Zh.S. Shishkin, A.A. Термоядерный синтез (коллективные процессы) |
| title_short |
The influence of magnetic islands on the impurity ion motion in the drift optimized stellarator |
| title_full |
The influence of magnetic islands on the impurity ion motion in the drift optimized stellarator |
| title_fullStr |
The influence of magnetic islands on the impurity ion motion in the drift optimized stellarator |
| title_full_unstemmed |
The influence of magnetic islands on the impurity ion motion in the drift optimized stellarator |
| title_sort |
influence of magnetic islands on the impurity ion motion in the drift optimized stellarator |
| author |
Kononenko, Zh.S. Shishkin, A.A. |
| author_facet |
Kononenko, Zh.S. Shishkin, A.A. |
| topic |
Термоядерный синтез (коллективные процессы) |
| topic_facet |
Термоядерный синтез (коллективные процессы) |
| publishDate |
2008 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Вплив магнітних островів на рух іонів домішки у дрейфово-оптимізованому стелараторі Влияние магнитных островов на движение примесных ионов в дрейфово-оптимизированном стеллараторе |
| description |
The motion of the charged test particles ensemble of tungsten ions in the drift optimized stellarator Wendelstein
7-X with the chain of 5 magnetic islands is studied. The numerical code solves the guiding center equations written
in the Hamiltonian form. To simulate the Coulomb scattering the Monte Carlo collision operator is used. The scattering
of the particles due to the collisions and magnetic islands is shown. The question of the possibility to use the
magnetic islands for the prevention of impurity penetration to the plasma is discussed
Вивчено рух ансамблю заряджених частинок іонів вольфраму у дрейфово-оптимізованому стелараторі
Вендельштайн 7-Х з ланцюгом з п’яти магнітних островів. Рівняння ведучого центру, що записані у
гамільтоновій формі, розв’язуються чисельно. Для моделювання кулонівського розсіяння використовується
дискретний оператор Монте-Карло. Показано розсіяння частинок в полоїдальному перерізі під впливом
зіткнень та магнітних островів. Обговорюється питання про можливість використання островів для
запобігання проникання домішок у плазму.
Изучено движение ансамбля заряженных тестовых частиц ионов вольфрама в дрейфово-
оптимизированном стеллараторе Вендельштайн 7-Х с цепочкой из пяти магнитных островов. Уравнения
ведущего центра, записанные в гамильтоновской форме, решаются численно. Для моделирования
кулоновского рассеяния используется дискретный оператор Монте-Карло. Показано рассеяние частиц в
полоидальном сечении под воздействием столкновений и магнитных островов. Обсуждается вопрос о
возможности использования магнитных островов для предотвращения проникновения примесей в плазму.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/110356 |
| citation_txt |
The influence of magnetic islands on the impurity ion motion in the drift optimized stellarator / Zh.S. Kononenko, A.A. Shishkin // Вопросы атомной науки и техники. — 2008. — № 4. — С. 95-98. — Бібліогр.: 7 назв. — англ. |
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2025-11-26T23:37:03Z |
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| fulltext |
THE INFLUENCE OF MAGNETIC ISLANDS ON THE IMPURITY ION
MOTION IN THE DRIFT OPTIMIZED STELLARATOR
Zh.S. Kononenko, A.A. Shishkin1
Kharkiv V.N. Karazin National University, Kharkоv, Ukraine
1National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: kononenko_zh@mail.ru
The motion of the charged test particles ensemble of tungsten ions in the drift optimized stellarator Wendelstein
7-X with the chain of 5 magnetic islands is studied. The numerical code solves the guiding center equations written
in the Hamiltonian form. To simulate the Coulomb scattering the Monte Carlo collision operator is used. The scat-
tering of the particles due to the collisions and magnetic islands is shown. The question of the possibility to use the
magnetic islands for the prevention of impurity penetration to the plasma is discussed.
PACS: 52.55.Fa
1. INTRODUCTION
The control of impurity ions plays the important role
for the plasma confinement in the present-day toroidal
fusion magnetic devices. Impurities come to the plasma
from the walls and divertor plates and lead to the sig-
nificant increase of the bremsstrahlung radiation losses.
The most dangerous for the confinement are the impuri-
ties of heavy metals such as molybdenum and tungsten.
In this paper, the motion of the ensemble of tungsten
impurity ions in the magnetic field of Wendelstein 7-X
is studied. W7-X is the modular stellarator being con-
structed at Max-Planck-Institut für Plasmaphysik in
Greifswald, Germany. The W7-X coil system consists
of 50 modular and 20 planar non-circular coils com-
bined in M=5 magnetic periods [1]. There are also 10
control coils used to control the magnetic island posi-
tion. The coil system of W7-X is developed in such a
way that the drift losses are minimized in this configura-
tion. The main W7-X parameters are the following: the
major plasma radius R0=5.5 m, minor plasma radius
a=0.52 m, the average magnetic field on the magnetic
axis B
___________________________________________________________
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2008. № 4.
Серия: Плазменная электроника и новые методы ускорения (6), с.95-98. 95
B0=2.5 T.
In our previous paper [2] it was shown how the
magnetic islands influence on the impurity transport
when the particles do not form the drift islands. In this
paper, we concentrate our attention on a such collision-
ality regime when the particles form the drift island and
study the influence of the magnetic islands on the parti-
cle dynamics.
2. MAGNETIC FIELD MODEL
For the analytical description and numerical calcula-
tions we use Boozer magnetic coordinates ),,( φθψ ,
where ψ is the label of the magnetic surfaces, θ and φ
are the poloidal and toroidal angles, respectively. The
strength of the equilibrium magnetic field, which has
the nested magnetic surfaces is used in the following
form:
∑ −⋅+=
nm
mn nMmrbBB
,
0 ))(cos)(1( φθ , (1)
where BB0 is the average magnetic field on the magnetic
axis. The Fourier coefficients as the functions of
the normalized radius r are presented in Fig.1 for the so-
called vacuum standard configuration of W7-X. For the
simplicity we consider the model magnetic field where
only the diamagnetic , mirror , toroidal
and helical harmonics are non-zero. For
the different magnetic configurations of W7-X these
harmonics are much larger than the rest of the harmon-
ics [1]. In axisymmetric configurations the toroidal
harmonic is the leading and it provides the main
contribution to the Shafranov shift of the magnetic sur-
faces. In W7-X it is significantly reduced compared to
the geometrical inverse aspect ratio. The mirror har-
monic characterizes the ripple of the magnetic
field strength on the magnetic axis. It can be used for
the suppression of the bootstrap current in finite beta
plasmas to improve the confinement [3]. It should be
noted that the sign of this harmonic is opposite to the
sign of the helical and toroidal harmonics. The helical
harmonic leads to the existence of the helically
trapped particles and 1/ν diffusion regime in a long
mean free path limit.
)(rbmn
)(rb00 )(rb01
)(rb10 )(rb11
)(rb10
)(rb01
)(rb11
Fig.1. The Fourier harmonics of B for the standard
configuration of W7-X
The magnetic islands are modeled by adding to the
equilibrium magnetic field the perturbation in the fol-
lowing form [4]:
)( 0BB
rr
αδ ×∇= , (2)
where the function α , which has the unit of the length,
presents a structure of the destroyed magnetic field. The
perturbation function α is chosen in the form
)sin( θφαα mnr m
0 −= . (3)
The total magnetic field can be presented as the sum of
the equilibrium magnetic field and the perturbation B
r
δ
BBB 0
rrr
δ+= . (4)
For the W7-X configuration (with M=5 magnetic
periods along the torus) the perturbation with the wave
numbers n=5, m=5 was studied. This perturbation splits
the rational magnetic surface , where the
rotational transform is
).( 950rres =
1=ι , into the chain of
5 magnetic islands (Fig.2). For the case of consideration
the radial islands width was .. cm94≈Δ
We should note that the form of magnetic surfaces in
W7-X stellarator in real geometry changes from the
bean to the triangle depending on the toroidal angle φ .
In Fig. 2 the magnetic surfaces are circular because we
operate in Boozer magnetic coordinates where the mag-
netic lines are straight in ),( φθ plane.
3. GUIDING CENTER EQUATIONS
To study the particle dynamics in the complicated
stellarator magnetic field the guiding center equations
written in the Hamiltonian form [4,5]:
are solved numerically using the Runge-Kutta method.
Here, )( gIIgIg c ′−′++= ριγ and αρρ += ||c . The
system of equations (5) is written in the normalized
form. All lengths are normalized to the major plasma
radius , the time is given in the units of 0R 0/1 ω , where
mc
ZeB0
0 =ω is the particle gyrofrequency on the axis,
the energy is normalized to , and the magnetic
field strength is normalized to its average value on the
magnetic axis . The functions I and g are propor-
tional to the toroidal and poloidal currents inside and
outside the flux surface, respectively. The rotational
transform profile
2
0
2
0 Rmω
0B
)(rι is taken in the form presented in
Fig.3.
Fig.2. Poicaré plot of magnetic surfaces for 0φ =
Fig.3. The profile of the rotational transform ( )rι
4. COLLISIONAL OPERATOR
To model a collisional kick for a test particle in the
pitch angle space, the discrete collision operator is ap-
plied after each integration time step. The Lorentz colli-
sion operator has the form [6]:
,)(1()1()()( 1
2
1 ttttt dndnn ΔνλΔνλλ −− −±−= (6)
where tΔ should satisfy the condition 1<<tdΔν . The
symbol “ ± ” indicates that the sign should be chosen
randomly, but with the equal probabilities for plus and
minus.
The deflection can be calculated through the plasma
parameters as [6,7]:
))()(( /// βαβα
αα
αβββαβα ΨΦ
υ
Λπ
ν xx
m
nee4
32
22
d −= (7)
Here , , and αe αm αυ are the charge, mass, and the
velocity of a test particle, and are the charge and
the density of background particles (ions and electrons),
βe βn
αβΛ is the Coulomb logarithm, and
β
βαβα υ
m
T2
x =/
is the ratio of the test particle velocity to thermal veloc-
ity of the background particles.
96
b a
Fig.4. The scattering of the ensemble of 500 impurity ions for the case without magnetic islands (in the poloidal
cross-section). (a) The initial positions of test ions; (b) Particle positions after 5 collision times
a b
Fig.5. The scattering of the ensemble of 500 impurity ions in case of existence of the chain of 5 magnetic islands
(in the poloidal cross-section). (a) The initial positions of test ions; (b) Particle positions after 5 collision times
b a
Fig.6. The temporal dependence of the mean-square displacement C2(t).
(а) The configuration without magnetic islands; (b) The configuration with the chain of 5 magnetic islands
The functions Φ(x) and Ψ(x) are given by
∫ −=
x
0
2 dtt2x ,)exp()(
π
Φ (8)
./))()(()( 2x2xxxx ΦΦΨ ′−= (9)
In this paper, we study the collisionality regime when
the normalized deflection frequency is . 5100.1~ −×=dν
5. PARTICLE SCATTERING
The particles of the monoenergetic ensemble of
1000N = impurity ions with the energy keV3E =
with randomly distributed initial pitch angles starts their
motion from the same initial radial coordinate 950r0 .= ,
where 1=ι . The initial poloidal and toroidal angles of
particles are distributed randomly but uniformly. We
97
study the motion of tungsten impurity ions with the
atomic mass number A=184 and the charge state Z=30.
Each particle evolves independently of others according
to equations (5) and its initial conditions. The pitch an-
gle of a particle is changed at each time step using the
Monte Carlo collision operator (6).
As the measure of statistical properties of the en-
semble, the mean square displacement is calculated:
>><−=< 2
2 trtrC ))()(( δδ , where 0rtrtr −= )()(δ is
the particle radial displacement. Brackets mean the en-
semble average: ∑
=
=><
N
1i
iX
N
1X , where N is the
number of particles of the ensemble. The number of
particles of the ensemble was chosen to satisfy 2 con-
tradictory conditions: the number of particles should be
enough large to obtain good statistics but not very large
in order to preserve the CPU time.
Fig.4 presents the scattering of the ensemble of 500
impurity ions for the case without magnetic islands. The
particles do not strongly deviate from the initial mag-
netic surface after 5 collision times in the
radial direction. Fig.6,a shows the linear increase in
time of C
dE /t ν= 5
2(t) for this case. It means that we have the
normal diffusive process with the diffusion coefficient
defined as
dt
tdC
2
1tD 2 )()( = [6].
Fig.5 presents the scattering of the similar ensemble
but for the case when the chain of 5 magnetic islands
exists in the plasma. In the considered collisionality
regime the particles form the drift islands. Thus, parti-
cles are allowed to have large radial deviations. As it is
shown in Fig.5,b, the particles nearly uniformly fill the
islands region after 5 collision times. In this case the
mean-square displacement C2(t) saturates to the constant
value, as it is shown in Fig.6,b.
The chain of the magnetic islands acts as the trap for
the particles. The particles move just inside and are ac-
cumulated in the region of islands. The control of parti-
cle motion using the magnetic islands can be used to
prevent the penetration of the impurities to the central
regions of the plasma from the walls.
SUMMARY
In this paper, the impurity ion motion is studied for
the standard configuration of W7-X stellarator when the
chain of 5 magnetic islands exists at the plasma periph-
ery. The mean square displacement of the ensemble of
impurity ions is calculated for the case without magnetic
islands and with them. It is shown that the magnetic
islands can prevent the penetration of impurity ions into
the centre of the plasma and lead to the change of the
regime of the particle transport.
Acknowledgements. The authors would like to thank
the Science and Technology Center in Ukraine (Project
№ 3685) for the partial support of this work.
REFERENCES
1. T. Andreeva. Vacuum magnetic configurations of
Wendelstein 7-X. IPP-Report 2002, III/270.
2. Zh.S. Kononenko and A.A. Shishkin. Impurity ion
dynamics near magnetic islands in the drift opti-
mized stellarator configuration of the Wendelstein 7-
X // Ukr. J. Phys. 2008, v.53, №5, p.438-442.
3. M. Yokoyama, Y. Nakamura and M. Wakatani. An
optimized helical axis stellarator with modulated l=1
helical coil // J. Plasma Fusion Res. 1997, v.73, №7,
p.723-731.
4. R.B. White and M.S. Chance. Hamiltonian guiding
center drift orbit calculation for plasmas of arbitrary
cross section // Phys. Fluids. 1984, v.27, №10,
p.2455-2467.
5. A.A. Shishkin, I.N. Sidorenko, H. Wobig. Magnetic
islands and drift surface resonances in Helias con-
figurations // J. Plasma Fusion Res. Ser. 1998, v.1,
p.480-483.
6. A.H. Boozer and G. Kuo-Petravic. Monte Carlo
evaluation of transport coeffiсients // Phys. Fluids.
1981, v.24, №5, p.851-859.
7. E. Strumberger. Deposition patterns of fast ions on
plasma facing components in W7-X // Nucl. Fusion.
2000, v.40, №10, p.1697-1713.
Статья поступила в редакцию 15.05.2008 г.
ВЛИЯНИЕ МАГНИТНЫХ ОСТРОВОВ НА ДВИЖЕНИЕ ПРИМЕСНЫХ ИОНОВ
В ДРЕЙФОВО-ОПТИМИЗИРОВАННОМ СТЕЛЛАРАТОРЕ
Ж.С. Кононенко, А.А. Шишкин
Изучено движение ансамбля заряженных тестовых частиц ионов вольфрама в дрейфово-
оптимизированном стеллараторе Вендельштайн 7-Х с цепочкой из пяти магнитных островов. Уравнения
ведущего центра, записанные в гамильтоновской форме, решаются численно. Для моделирования
кулоновского рассеяния используется дискретный оператор Монте-Карло. Показано рассеяние частиц в
полоидальном сечении под воздействием столкновений и магнитных островов. Обсуждается вопрос о
возможности использования магнитных островов для предотвращения проникновения примесей в плазму.
ВПЛИВ МАГНІТНИХ ОСТРОВІВ НА РУХ ІОНІВ ДОМІШКИ
У ДРЕЙФОВО-ОПТИМІЗОВАНОМУ СТЕЛАРАТОРІ
Ж.С. Кононенко, O.O. Шишкін
Вивчено рух ансамблю заряджених частинок іонів вольфраму у дрейфово-оптимізованому стелараторі
Вендельштайн 7-Х з ланцюгом з п’яти магнітних островів. Рівняння ведучого центру, що записані у
гамільтоновій формі, розв’язуються чисельно. Для моделювання кулонівського розсіяння використовується
дискретний оператор Монте-Карло. Показано розсіяння частинок в полоїдальному перерізі під впливом
зіткнень та магнітних островів. Обговорюється питання про можливість використання островів для
запобігання проникання домішок у плазму.
98
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