Self-consistent model of the RF plasma production in stellarator
Self-consistent model of the RF plasma production in stellarator that includes system of the balance equations and the boundary problem for the Maxwell’s equations is developed. The first numerical calculations of RF plasma production in the Uragan-2M stellarator are presented. Разработана самосогла...
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| Zitieren: | Self-consistent model of the RF plasma production in stellarator / V.Е. Moiseenko, Yu.S. Stadnik, A.I. Lyssoivan, M.B. Dreval // Вопросы атомной науки и техники. — 2010. — № 6. — С. 21-23. — Бібліогр.: 2 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859775960530812928 |
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| author | Moiseenko, V.Е. Stadnik, Yu.S. Lyssoivan, A.I. Dreval, M.B. |
| author_facet | Moiseenko, V.Е. Stadnik, Yu.S. Lyssoivan, A.I. Dreval, M.B. |
| citation_txt | Self-consistent model of the RF plasma production in stellarator / V.Е. Moiseenko, Yu.S. Stadnik, A.I. Lyssoivan, M.B. Dreval // Вопросы атомной науки и техники. — 2010. — № 6. — С. 21-23. — Бібліогр.: 2 назв. — англ. |
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| description | Self-consistent model of the RF plasma production in stellarator that includes system of the balance equations and the boundary problem for the Maxwell’s equations is developed. The first numerical calculations of RF plasma production in the Uragan-2M stellarator are presented.
Разработана самосогласованная модель ВЧ-создания плазмы в стеллараторе, включающая систему уравнений баланса и краевую задачу для уравнений Максвелла. Представлены первые результаты численных экспериментов по ВЧ-созданию плазмы в стеллараторе Ураган-2М с помощью разработанной модели.
Розроблено самоузгоджену модель ВЧ-створення плазми в стеллараторі, що складається з системи рівнянь балансу та крайової задачі для рівнянь Максвелла. Представлено перші результати числових експериментів з ВЧ-створення плазми в стеллараторі Ураган-2М за допомогою розробленої моделі.
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SELF-CONSISTENT MODEL OF THE RF PLASMA PRODUCTION
IN STELLARATOR
V.Е. Moiseenko1, Yu.S. Stadnik1, A.I. Lyssoivan2, M.B. Dreval1
1Institute of Plasma Physics NSC “Kharkov Institute of Physics and Technology", Kharkov, Ukraine;
2Laboratory for Plasma Physics - ERM/KMS, Association EURATOM - BELGIAN STATE,
1000 Brussels, Belgium
Self-consistent model of the RF plasma production in stellarator that includes system of the balance equations and
the boundary problem for the Maxwell’s equations is developed. The first numerical calculations of RF plasma
production in the Uragan-2M stellarator are presented.
PACS: 52.50.Qt, 52.55.Hc.
INTRODUCTION
Plasma production in the ICRF band (Ion Cyclotron
Range of Frequencies) is a possible way to build up dense
target plasma in a stellarator (see [1]). The self-consistent
model developed here simulates plasma production with
arbitrary ICRF antennas and includes system of the
particle and energy balance equations for the electrons
and neutrals and the boundary problem for the Maxwell’s
equations. Solution of the Maxwell’s equations allows
determining a local value of the electron RF heating
power, which influences on the ionization rate and, in this
way, on the evolution of plasma density.
At small values of the plasma density, the slow wave
(SW) is responsible for plasma production. With increase
of the plasma density the SW is strongly damped
propagating to the centre of the plasma column and is
absorbed in the antenna vicinity. At high values of the
plasma density, the Alfvén resonances come to play in the
plasma production. The electrons are heated by the RF
field owing to collisional and Landau wave damping.
NUMERICAL MODEL
The model of the radio-frequency (RF) plasma
production includes the system of the balance equations
and the boundary problem for the Maxwell’s equations. It
is assumed that the gas is atomic hydrogen. The system of
the balance equations of particles and energy reads:
( )
( ,v1
v
4
3
2
3
eeaeiHB
E
eeB
a
aeeHBRFe
eeB
Tnχnnσεk
τ
Tnk
)(C
nnσεkP
t
Tnk
∇⋅∇+−+−
−−=
∂
∂
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2010. № 6. 21
Series: Plasma Physics (16), p. 21-23.
) (1)
e
E
e
aei
e nD
n
nn
dt
dn
∇⋅∇+−=
τ
σ v ,
constVnVndVn VVae ==+∫ 0 ,
where is the plasma density, is the neutral gas
density, is the electron temperature, is the RF
power density, which is coupled to the electrons, is
the Boltzman’s constant,
en an
eT RFeP
Bk
eVH 6.13=ε is the ionization
potential threshold for the hydrogen atom, χ is the heat
transport coefficient, is the diffusion coefficient, D Eτ
is the particle confinement time, is the vacuum
chamber volume,
VV
veσ , viσ are the excitation and
ionization rates and 5.3/ ≈Φ= eaa TeC is the ratio of
the ambipolar potential energy to the electron
temperature. Dependence of the excitation and ionization
rates of atoms by the electron impact on electron
temperature is approximated by formulas:
)
4
3exp(2v
e
H
se T
k
ε
σ −= ,
)exp(8.3v
e
H
H
e
si T
T
k
ε
ε
σ −= .
(2)
Balance of the electron energy includes the RF
heating, energy losses for the excitation and ionization of
atoms and losses caused by the heat transport. The
balance of the charged particles includes accounts for the
ionization and diffusion losses of particles. The last
equation in the system (1) reflects the global balance of
the particles. It is assumed, that the neutral gas is
uniformly distributed in the vacuum chamber volume,
including the plasma column.
The RF field can produce plasma inside the
confinement volume and outside it as well. The losses of
the charged particles in the outside region have a
convection character: the particles escape to the wall
along the lines of force of the magnetic field. Such losses
of particles outside the confinement volume are described
in τ -approximation. The τ -approximation is also used
inside the plasma column to describe energy exchange
with ions. Out of the confinement region the particle
confinement time is given by the following formula:
s
n
RL
v2
=τ , (3)
where is the local mirror ratio, is the connection
length of the magnetic field line, is the ion sound
velocity in plasma.
R L
sv
The problem is solved in cylindrical geometry. The
plasma is assumed to be azimuthally symmetrical and
uniformly distributed along plasma column. The length of
plasma cylinder is RL π2= and the ends are assumed to
be identical.
The account of the diffusion and the thermal
conductivity effects require application of the conditions
of regularity at the axis of the cylinder:
0
0
=
∂
∂
=r
e
r
n , 0
)(
0
=
∂
∂
=r
ee
r
Tn , (4)
and boundary conditions
0=
=aren , 0=
=areeTn (5)
at the chamber wall.
To make the system of the equations (1) closed, it is
necessary to determine the single external quantity in it,
(RF power density). This quantity can be found
from the solution of the boundary problem for the
Maxwell’s equations:
RFeP
( ) extir
c
jEE 02
2
ˆ ωμεω
=⋅−×∇×∇ , (6)
where is the electric field, is the external RF
currents. The dielectric tensor reads:
, where
E extj
( )
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−= ⊥
⊥
//00
0
0
,ˆ
ε
ε
ε
ε ig
ig
tr
2
−−++
⊥
+
=
εε
ε ,
22
( ) ( )[ ]ααα
α
α π
νω
ωε zWziz
ikz
P +
+
+= 121
2
// ,
2
−−++ −
=
εεg ,
( )ααα
α
νωωω
ω
ε
icc
iP
−−
+=++
2
1 ,
( )ααα
α
νωωω
ωε
icc
P
++
+=−−
2
1 .
The Maxwell’s equations are solved at each time
moment for current plasma density and temperature
distributions. The Maxwell’s equations solution allows
determining the value of local RF heating power of the
electron plasma component which influences on the
ionization rate and, in this way, on the increase of plasma
density. The RF power density in cylindrical system of
coordinates reads:
).Im2
Im(
2
*
//
2
220
gEEE
EEP
rz
rRF
ϕ
ϕ
ε
εεωε
++
++= ⊥⊥
(7)
The Crank-Nicholson method is used for the solving
of system of the balance equations (1). The Maxwell’s
equations (6) are solved in the 1D geometry using the
Fourier series in the azimuthal and the axial coordinates.
For the discretization in the radial coordinate, the uniform
finite elements method is employed that uses a special set
of weight (test) and basis (shape) functions [2].
EXAMPLE OF CALCULATIONS
The following parameters of calculations for the
Uragan-2M stellarator are chosen: the major radius of the
torus is , the radius of the plasma column is
, the radius of the metallic wall is
21.7 10 cmR = ×
cm22=r cm34=a ,
the radial coordinate of the antenna is , the
toroidal magnetic field is
cm23=antr
kG5=B . The frame-type
antenna (Fig. 1) with the azimuthal angle 1a =ϕ and the
toroidal angle 08.0=aϑ was used in the calculations. The
current in the antenna is assumed not varying along the
conductors.
U
qa
ϕa ∼
Fig. 1. Frame antenna layout
The first results of calculations of RF plasma production
in the Uragan-2M stellarator are presented. Figs. 2, 3
display profiles of plasma density, electron temperature
and power deposition at the time moment .
Figs. 4-6 display the time evolution of electron
temperature, plasma density and density of neutral gas.
s100.45 -2⋅=t
0.0E+000
1.0E+012
2.0E+012
3.0E+012
4.0E+012
5.0E+012
ne
, c
m
-3
0 10 20 30 40
r, cm
Fig. 2. Profile of plasma density in s100.45 -2⋅=t
0
1
2
3
4
Te
, e
V
0 10 20 30 40
r, cm
0
0.4
0.8
1.2
1.6
po
w
0 10 20 30 40
r, cm
Fig. 3. Profile of electron temperature (upper chart) and
power deposition profile (lower chart) in time moment
s100.45 -2⋅=t
A characteristic feature of the calculations is higher
plasma temperature outside the confinement volume
(Fig. 3). Since the heating power-per-particle is higher at
lower plasma densities (Fig. 2) the electron temperature
increases at the edge of the plasma column (Fig. 3) where
the particle losses are faster.
23
0 0.001 0.002 0.003 0.004 0.005
t, s
0
4
8
12
16
<T
e>
, e
V
Fig.4. Time evolution of average electron temperature
0 0.001 0.002 0.003 0.004 0.005
t, s
0.0E+000
1.0E+012
2.0E+012
3.0E+012
<n
e>
, c
m
-3
Fig. 5. Time evolution of average plasma density
0 0.001 0.002 0.003 0.004 0.005
t, s
0
2E+012
4E+012
6E+012
8E+012
1E+013
<n
a>
, c
m
-3
Fig. 6. Time evolution of average neutral atoms density
At the initial stage of the plasma production, sharp peaks
in temperature are observed (Fig. 4). These peaks are
associated with the sharp increase of the antenna loading
resistance. It occurs when the wave global resonance
conditions in a plasma column is met. At the initial stage
of plasma production a slow wave damping is small, and
the peaks of the global resonances are more narrow and
high.
In the chosen regime, the plasma density rise saturates
and the plasma production process has a tendency to
stagnate because the power is insufficient to complete a
burnout of the neutral atoms (Figs. 5, 6).
CONCLUSIONS
For the self-consistent description of ICRF plasma
production in stellarators, the numerical model including
a system of the balance equations for particles and energy
and the boundary problem for the Maxwell's equations is
developed.
The Crank-Nicholson method is used for solving the
system of the balance equations. The Maxwell’s equations
are solved using the Fourier series in the azimuthal and
the longitudinal coordinates. Fortran90 computer code is
developed.
Using the self-consistent model for the ICRF plasma
production in stellarators the first numerical calculations
for the Uragan-2M stellarator are carried out.
ACKNOWLEDGEMENT
This work is supported in part by the STCU project №
4216.
REFERENCES
1. A.I. Lysojvan, V.E. Moiseenko, O.M. Schvets,
K.N. Stepanov. Analysis of ICRF ( ciωω < ) plasma
production in large-scale tokamaks // Nuclear Fusion.
1992, v. 32, p. 1361.
2. V.E. Moiseenko. Numerically stable Modeling of
Radio-Frequency Fields in Plasma // Problems of
Atomic Science and Technology. Series “Plasma
Physics” (7). 2002, N 4, p.100.
Article received 12.10.10
САМОСОГЛАСОВАННАЯ МОДЕЛЬ ВЧ-СОЗДАНИЯ ПЛАЗМЫ В СТЕЛЛАРАТОРЕ
В.Е. Моисеенко, Ю.С. Стадник, А.И. Лысойван, М.Б. Древаль
Разработана самосогласованная модель ВЧ-создания плазмы в стеллараторе, включающая систему уравнений
баланса и краевую задачу для уравнений Максвелла. Представлены первые результаты численных
экспериментов по ВЧ-созданию плазмы в стеллараторе Ураган-2М с помощью разработанной модели.
САМОУЗГОДЖЕНА МОДЕЛЬ ВЧ-СТВОРЕННЯ ПЛАЗМИ У СТЕЛЛАРАТОРІ
В.Є. Моісеєнко, Ю.С. Стаднік, А.І. Лисойван, М.Б. Древаль
Розроблено самоузгоджену модель ВЧ-створення плазми в стеллараторі, що складається з системи рівнянь
балансу та крайової задачі для рівнянь Максвелла. Представлено перші результати числових експериментів з
ВЧ-створення плазми в стеллараторі Ураган-2М за допомогою розробленої моделі.
|
| id | nasplib_isofts_kiev_ua-123456789-17447 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-02T08:48:03Z |
| publishDate | 2010 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Moiseenko, V.Е. Stadnik, Yu.S. Lyssoivan, A.I. Dreval, M.B. 2011-02-26T18:56:14Z 2011-02-26T18:56:14Z 2010 Self-consistent model of the RF plasma production in stellarator / V.Е. Moiseenko, Yu.S. Stadnik, A.I. Lyssoivan, M.B. Dreval // Вопросы атомной науки и техники. — 2010. — № 6. — С. 21-23. — Бібліогр.: 2 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/17447 Self-consistent model of the RF plasma production in stellarator that includes system of the balance equations and the boundary problem for the Maxwell’s equations is developed. The first numerical calculations of RF plasma production in the Uragan-2M stellarator are presented. Разработана самосогласованная модель ВЧ-создания плазмы в стеллараторе, включающая систему уравнений баланса и краевую задачу для уравнений Максвелла. Представлены первые результаты численных экспериментов по ВЧ-созданию плазмы в стеллараторе Ураган-2М с помощью разработанной модели. Розроблено самоузгоджену модель ВЧ-створення плазми в стеллараторі, що складається з системи рівнянь балансу та крайової задачі для рівнянь Максвелла. Представлено перші результати числових експериментів з ВЧ-створення плазми в стеллараторі Ураган-2М за допомогою розробленої моделі. This work is supported in part by the STCU project № 4216. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Магнитное удержание Self-consistent model of the RF plasma production in stellarator Самосогласованная модель ВЧ-создания плазмы в стеллараторе Самоузгоджена модель ВЧ-створення плазми у стеллараторі Article published earlier |
| spellingShingle | Self-consistent model of the RF plasma production in stellarator Moiseenko, V.Е. Stadnik, Yu.S. Lyssoivan, A.I. Dreval, M.B. Магнитное удержание |
| title | Self-consistent model of the RF plasma production in stellarator |
| title_alt | Самосогласованная модель ВЧ-создания плазмы в стеллараторе Самоузгоджена модель ВЧ-створення плазми у стеллараторі |
| title_full | Self-consistent model of the RF plasma production in stellarator |
| title_fullStr | Self-consistent model of the RF plasma production in stellarator |
| title_full_unstemmed | Self-consistent model of the RF plasma production in stellarator |
| title_short | Self-consistent model of the RF plasma production in stellarator |
| title_sort | self-consistent model of the rf plasma production in stellarator |
| topic | Магнитное удержание |
| topic_facet | Магнитное удержание |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/17447 |
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