Stochasticity of the magnetic field lines and high Z ion motion
Stochastic layer of the magnetic field lines is numerically analyzed for the realistic magnetic field of HELIAS reactor with use of the flux coordinate system. Analytical expressions for the stochastic diffusion coefficients are obtained for the simplified magnetic field model.
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irk-123456789-823612015-05-29T03:02:07Z Stochasticity of the magnetic field lines and high Z ion motion Shishkin, O.A. Shishkin, A.A. Wobig, H. Magnetic Confinement Stochastic layer of the magnetic field lines is numerically analyzed for the realistic magnetic field of HELIAS reactor with use of the flux coordinate system. Analytical expressions for the stochastic diffusion coefficients are obtained for the simplified magnetic field model. 2000 Article Stochasticity of the magnetic field lines and high Z ion motion / O.A. Shishkin, A.A. Shishkin, H. Wobig // Вопросы атомной науки и техники. — 2000. — № 3. — С. 19-21. — Бібліогр.: 6 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/82361 533.9 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 Shishkin, O.A. Shishkin, A.A. Wobig, H. Stochasticity of the magnetic field lines and high Z ion motion Вопросы атомной науки и техники |
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Stochastic layer of the magnetic field lines is numerically analyzed for the realistic magnetic field of HELIAS reactor with use of the flux coordinate system. Analytical expressions for the stochastic diffusion coefficients are obtained for the simplified magnetic field model. |
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Article |
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Shishkin, O.A. Shishkin, A.A. Wobig, H. |
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Shishkin, O.A. Shishkin, A.A. Wobig, H. |
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Shishkin, O.A. |
| title |
Stochasticity of the magnetic field lines and high Z ion motion |
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Stochasticity of the magnetic field lines and high Z ion motion |
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Stochasticity of the magnetic field lines and high Z ion motion |
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Stochasticity of the magnetic field lines and high Z ion motion |
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Stochasticity of the magnetic field lines and high Z ion motion |
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stochasticity of the magnetic field lines and high z ion motion |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2000 |
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Magnetic Confinement |
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Stochasticity of the magnetic field lines and high Z ion motion / O.A. Shishkin, A.A. Shishkin, H. Wobig // Вопросы атомной науки и техники. — 2000. — № 3. — С. 19-21. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT shishkinoa stochasticityofthemagneticfieldlinesandhighzionmotion AT shishkinaa stochasticityofthemagneticfieldlinesandhighzionmotion AT wobigh stochasticityofthemagneticfieldlinesandhighzionmotion |
| first_indexed |
2025-07-06T08:52:00Z |
| last_indexed |
2025-07-06T08:52:00Z |
| _version_ |
1836886965348204544 |
| fulltext |
UDC 533.9
19
Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 19-21
Stochasticity of the Magnetic Field Lines and High Z Ion Motion
Oleg A. Shishkin, Alexander A. Shishkin* and Horst Wobig**,
Plasma Physics Chair, Department of Physics and Technology,
Kharkov “V.N. Karazin” National University, Kharkov-77, 61077, UKRAINE,
*Institute of Plasma Physics, National Science Center, “Kharkov Institute of Physics and
Technology”, Academicheskaya str. 1, Kharkov-108, 61108, UKRAINE
and
Plasma Physic Chair, Department of Physics and Technology,
Kharkov “V.N. Karazin” National University, Kharkov-77, 61077, UKRAINE,
**Max-Planck-Institute-for-Plasma-Physics, D-85748, Garching, Germany
Stochastic layer of the magnetic field lines is numerically analyzed for the realistic magnetic field of HELIAS
reactor with use of the flux coordinate system. Analytical expressions for the stochastic diffusion coefficients are
obtained for the simplified magnetic field model.
1. INTRODUCTION
Magnetic islands are the part of the divertor
configuration in the modern fusion devices and future
reactor systems. The examples are the operating devices
Wendelstein-7AS and heliotron Large Helical Device,
the stellarator under construction Wendelstein-7X and
stellarator reactor system HELIAS. Overlapping of the
adjacent magnetic islands leads to the creation of the
stochastic layer. The properties of the stochastic layer
are usually analyzed by numerical calculations of the
magnetic field lines with the use of Biot-Savart law in
cylindrical coordinate system [1]. In this paper the
properties of the stochastic layer are studied by
numerical calculations of the drift motion equations
with the use of the expansion of magnetic field in the
series depending on coordinates. This approach also
allows considering the particle motion and MHD model
for the impurity ion dynamics in finite β plasma.
Numerical calculations are provided with use of the flux
coordinates for the reactor system HELIAS.
2. ANALYTICAL TREATMENT
In the reactor system HELIAS the magnetic field
configuration with five islands is realized. To create the
model of such configuration the main magnetic field
mB
with the field perturbation
1B are taken. To model
stochasticity of the magnetic island surfaces it is
necessary to use one more magnetic field perturbation
2B . Taking into account all this it is possible to write
the total magnetic field in the following form
2BBB += ∑
, (2.1)
where
1BBB +=∑ m
. (2.2)
Let us consider the toroidal magnetic flux ψ , which
concerns to the total magnetic field and is a function of
the coordinates. Using quasi-cylindrical coordinate
system one can write
ϕ
ϑ
ϑ
ψ
ϕ
ψ
ϕ
ψ
ϕ
ψ
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂= r
rd
d , (2.3)
where
2ψψψ += ∑
. (2.4)
Here
∑ψ is related to
∑B , and
2ψ is related to the
additional magnetic field perturbation which is
presented by
2B .
It is necessary to note that for the magnetic force line
equation (2.3) can be rewritten as follows
ϕ
ϑ
ϕ ϑ
ψψ
ϕ
ψ
ϕ
ψ
B
B
r
R
B
B
r
R
d
d r
∂
∂+
∂
∂+
∂
∂= . (2.5)
In the small perturbation approximation the toroidal flux
can be represented by expansion in Taylor series
ϕ
ψ
ϕ
ϑ
ψ
ϑ
ψ
ψψ
∂
∂
∆+
∂
∂
∆+
∂
∂
∆+= ∑∑∑
∑
R
R
r
r
r
r . (2.6)
Substituting expansion (2.6) in to (2.5) one can obtain
+
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂ ∑∑∑∑
2
2
3121
r
DA
r
A rr
ψ
ϑ
ψψ
ϕ
ψ (2.7)
0
2
2
2
=
∂∂
∂
+
∂
∂
+ ∑∑
ϑ
ψ
ϑ
ψ
ϑϑϑ r
DD r
.
This is the equation, which describes “diffusion” of the
magnetic field line in const=ϕ cross section.
Coefficients
ϑϑϑ rrr DDD ,, are interpreted as diffusion
coefficients [2,3] and have following form
( )
1
2
A
BBr
D rr
rr
+∆
= ∑ , (2.8)
( )
1
22
1
A
BB
r
r
D
ϑϑ
ϑϑ
ϑ +∆
=
∑ , (2.9)
( ) ( )
1
22
11
A
BBr
r
BB
r
r
D
rr
r
ϑϑ
ϑ
ϑ +∆++∆
=
∑∑ , (2.10)
where
20
Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 19-21
( ) +
∂
∂∆+∆
∂
∂= ∑
R
RR
RR
B
A r 11
1 ϕ
ϕϕ
ϕ
( ) ( ) +
∂
∂∆+∆
∂
∂++ ∑ Rr
RR
rR
BB rr
11
2 ϕϕ (2.11)
( ) ( )
∂
∂∆+∆
∂
∂++ ∑ R
RR
R
BB
r
111
2 ϑ
ϕϕ
ϑϑϑ
.
These analytical expressions for the stochastic diffusion
coefficients can be used for the optimization of the
magnetic field perturbation parameters that leads to
creation of the effective divertor configurations.
3. NUMERICAL STUDIES
As was mentioned above for the numerical studies of
the magnetic field properties HELIAS reactor system
was chosen. Parameters of HELIAS reactor system are
the following: large torus radius cmR 22000 = , plasma
radius cma p 180= , and magnetic field at the circular
axis of the torus TB 50 = . Finite β case is considering
( %3=β ). The main magnetic field is written as an
expansion with the use of the flux coordinate system [4]
( ) ( ) ++= ∑
∞
=0
,0
0
cos||1
k
pk Mkb
B
B
ζr
( ) ( )ϑζ lMkb p
l k
kl −+ ∑ ∑
∞
=
∞
−∞=
cos||
1
, r . (3.1)
Here B is magnitude of the magnetic field at the point
with coordinates (
pr , ζ , ϑ ), || pr is the radial
coordinate of the force line, ζ is the coordinate along
the torus, ϑ is the angle between the equatorial plane of
the torus and vector
pr . Now introduce magnetic field
perturbations of the form [5,6]
BB ii αδ ×∇= . (3.2)
Perturbation parameter is taken as a harmonic function
of coordinates, perturbation frequency and phase δ
( )δωϑζαα +−−= tmnm
pi sin|| r . (3.3)
Here α is an amplitude of perturbation, mn, are the
‘wave’ numbers. There is the sense to use a perturbation
frequency ω in the case, when the perturbation depends
on time. In this work static perturbations are considered
( 0=ω ). The equations of drift motion are written in
Hamiltonian form [5, 6]
ζζ ∂
∂−= H
P& ,
ϑϑ ∂
∂
−=
H
P& , (3.4)
ζ
ζ
P
H
∂
∂=& ,
ϑ
ϑ
P
H
∂
∂=& , (3.5)
ζ
ζ
ϑ
ϑ
ψψψ P
P
P
P
&&&
∂
∂+
∂
∂= , (3.6)
−
∂
∂
+
∂
∂
= ζ
ζ
ϑ
ϑ
ρρ
ρ P
P
P
P
cc &&&
||
ζ
ζ
αϑ
ϑ
αψψ
ψ
α
ζ
ζ
ϑ
ϑ
&&&&
∂
∂−
∂
∂−
∂
∂+
∂
∂
∂
∂− P
P
P
P
. (3.7)
Relationship ( )ζϑψαρρ ,,|| −= c
is also very important
to be introduced. The Hamiltonian for the drift motion is
Φ++= BBH µρ 22
||2
1 , (3.8)
where
||ρ is a normalized parallel gyroradius, µ is a
normalized magnetic moment and Φ is an electric
potential. On Fig.1 the model of the magnetic field
configuration that is realized as a main magnetic field
configuration in HELIAS reactor is presented.
1800 2200 2600
R (cm)
-400
0
400
Z
(
c
m
)
Fig.1 Vertical cross section of the main magnetic field
configuration in HELIAS reactor system represented in
flux coordinates.
In a real device five magnetic islands have to appear
without any additional perturbation. But for the
analytical and numerical calculations these islands,
which have to be crossed by divertor plates, are
modeled by magnetic field perturbation term
1B with
‘wave’ numbers 5,5 == nm . Divertor plates are the
strong source of high charged tungsten ions and other
types of heavy ions. Such ions can be considered as
impurity ions, which can appear at outside magnetic
surfaces of the confinement volume (Fig.2).
1800 2200 2600
R (cm)
-400
0
400
Z
(
cm
)
Fig.2 Trajectory of highly charged tungsten ion in the
main magnetic field configuration (vertical cross
section).
21
Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 19-21
As the test particle the tungsten ion with charge number
Z=30, the energy W=1keV and 5.0|| =V
V is taken.
If stochasticity of outside magnetic surfaces and island
magnetic surfaces is caused by some effect, specific
conditions can be created that leads to escape of
impurity ions from outside magnetic surfaces and
confinement volume. As was mentioned in section 2,
such stochasticity of the magnetic field is modeled by
using additional perturbation term
2B with ‘wave’
numbers 10,9 == nm (Fig.3).
1800 2200 2600
R (cm)
-400
0
400
Z
(
cm
)
Fig.3 Vertical cross section of the magnetic field
configuration with stochastic layer caused by additional
perturbation with ‘wave’ numbers (m=9,n=10).
On Fig.4 the trajectory of the tungsten ion in the
stochastic magnetic field configuration is shown.
1800 2200 2600
R (cm)
-400
0
400
Z
(
cm
)
Fig.4 Trajectory of highly charged tungsten ion in
magnetic field configuration with stochastic layer
(vertical cross section).
.
As one can see after several rounds along the small
radius of the torus the tungsten ion follows the
resonance structure and stochastic magnetic field lines
in the divertor region and moves outside from the last
close magnetic surface. Such effect can be used to
remove impurity ions from the edge of plasma back to
the divertor plates and to the wall of the vacuum vessel.
CONCLUSIONS
1. Analytical expressions for the stochastic
diffusion coefficients that can be used for the
optimization of the magnetic perturbations
parameters (amplitudes, “wave” numbers,
phases) to create effective divertor
configurations are obtained.
2. It is shown that the high Z impurity ion follows
the resonance structure (magnetic islands) and
stochastic magnetic field lines in divertor region
of HELIAS configuration.
REFERENCES
[1]. P. Bachmann, J. Kißlinger, D. Sünder, H. Wobig,
Bifurcation of Temperature in the Boundary Region of
Advanced Stellarator // IPP-Report, Max-Planck-Institut
fur Plasmaphysik, IPP III/262 Mai 2000.
[2]. M.N. Rosenbluth, R.Z. Sagdeev, J.B. Taylor,
G.M. Zaslavski, Destruction of Magnetic Surfaces by
Magnetic Field Irregularities // Nuclear Fusion 6 (1966)
297.
[3]. A.A. Shishkin, Estafette of Resonance
Stochasticity and Control of Particle Motion // (this
Conference).
[4]. C.D. Beidler, Neoclassical Transport Properties
of HSR // Proceedings of the 6th Workshop on
WENDELSTEIN 7-X and Helias Reactors, IPP 2/331,
January (1996) 194.
[5]. R.B. White and M.S. Chance, Hamiltonian
Guiding Center Drift Orbit Calculation for Plasmas of
Arbitrary Cross Section // Phys. Fluids 27 (10), October
(1984) 2455.
[6]. A.A.Shishkin, I.N. Sidorenko and H. Wobig,
Magnetic Islands and Drift Surface Resonances in
Helias Configurations // Journal of Plasma and Fusion
Research SERIES, v.1 (1998) 480.
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