On influence of external low frequency helical perturbation on tokamak edge plasma
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2002
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| Zitieren: | On influence of external low frequency helical perturbation on tokamak edge plasma / I.M. Pankratov, A.Ya. Omelchenko, V.V. Olshansky // Вопросы атомной науки и техники. — 2002. — № 5. — С. 3-5. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-778152015-03-08T21:40:24Z On influence of external low frequency helical perturbation on tokamak edge plasma Pankratov, I.M. Omelchenko, A.Ya. Olshansky, V.V. Magnetic confinement 2002 Article On influence of external low frequency helical perturbation on tokamak edge plasma / I.M. Pankratov, A.Ya. Omelchenko, V.V. Olshansky // Вопросы атомной науки и техники. — 2002. — № 5. — С. 3-5. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 52.55.Fa http://dspace.nbuv.gov.ua/handle/123456789/77815 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 Pankratov, I.M. Omelchenko, A.Ya. Olshansky, V.V. On influence of external low frequency helical perturbation on tokamak edge plasma Вопросы атомной науки и техники |
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Article |
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Pankratov, I.M. Omelchenko, A.Ya. Olshansky, V.V. |
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Pankratov, I.M. Omelchenko, A.Ya. Olshansky, V.V. |
| author_sort |
Pankratov, I.M. |
| title |
On influence of external low frequency helical perturbation on tokamak edge plasma |
| title_short |
On influence of external low frequency helical perturbation on tokamak edge plasma |
| title_full |
On influence of external low frequency helical perturbation on tokamak edge plasma |
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On influence of external low frequency helical perturbation on tokamak edge plasma |
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On influence of external low frequency helical perturbation on tokamak edge plasma |
| title_sort |
on influence of external low frequency helical perturbation on tokamak edge plasma |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2002 |
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Magnetic confinement |
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http://dspace.nbuv.gov.ua/handle/123456789/77815 |
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On influence of external low frequency helical perturbation on tokamak edge plasma / I.M. Pankratov, A.Ya. Omelchenko, V.V. Olshansky // Вопросы атомной науки и техники. — 2002. — № 5. — С. 3-5. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT pankratovim oninfluenceofexternallowfrequencyhelicalperturbationontokamakedgeplasma AT omelchenkoaya oninfluenceofexternallowfrequencyhelicalperturbationontokamakedgeplasma AT olshanskyvv oninfluenceofexternallowfrequencyhelicalperturbationontokamakedgeplasma |
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2025-07-06T02:01:22Z |
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2025-07-06T02:01:22Z |
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1836861130684760064 |
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MAGNETIC CONFINEMENT
ON INFLUENCE OF EXTERNAL LOW FREQUENCY HELICAL
PERTURBATION ON TOKAMAK EDGE PLASMA
I.M. Pankratov, A. Ya. Omelchenko, V.V. Olshansky
Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics and
Technology” Akademicheskaya str., 1, 61108 Kharkov, Ukraine
PACS: 52.55.Fa
1. INTRODUCTION
The Dynamic Ergodic Divertor (DED) of TEXTOR is
installed to control plasma edge behaviour [1]. The DED
helical coils create a specific topology of magnetic field at
the plasma edge, where external DED helical
perturbations with poloidal number m and toroidal
number n are resonant on the magnetic surfaces ( )resrq =
nm (q(r) - safety factor) (see, e.g., [2, 3]). However, this
topology was investigated using vacuum DED field
perturbations without the plasma response. Remind, that
the m = 12, n = 4 perturbation field structure is chosen as
a standard DED operation regime.
The interaction of an external helical field with a plas-
ma was investigated also in the CSTN- IV tokamak [4].
In the present paper the influence of plasma response
to DED helical perturbation penetration is considered in
cylindrical geometry. Analytical solutions of
perturbations are found and their numerical investigation
is carried out.
2.
BASIC EQUATIONS
We start from magnetohydrodynamic equations
( ) [ ]BJVVV ×+− ∇=
∇+
∂
∂
c
p
t
1ρ , (1a)
tc
rot
∂
∂−= BE 1
, JB
c
rot π4= (1b)
and Ohm’s law ( σ - conductivity)
[ ]
×+= BVEJ
c
1σ . (2)
We consider a current carrying cylindrical plasma whose
axis is taken to be as the z direction. The external axial
magnetic field 0zB is large with respect to the poloidal
magnetic field 0θB produced by the axial current. The
perturbation values depend on the azimuthal angle θ , the
coordinate z and the time t as ( )[ ]tkzmi ωθ −−exp ,
Rnk = , R plays the role of the tokamak major radius, ω is
the frequency of the external perturbation.
For perturbations of radial components of plasma velocity
~
rV and magnetic field ( ) π ρ4~
rrBrB = the linearized
version of Eqs. (1), (2) take the form:
( ) ( ) ( ) ( ) ( ) ( )rBrF
dr
drF
dr
drrFriVrFrimrV
dr
dr
dr
d
rr
++=
+−
22222
~
2
2
2
2
2~
4
3
444
2
π ρ ωπ ρ ωπ ρ ωδπ ρ ωδ
, (3)
( ) ( ) ~
222
2
2
2
4
)(1
rrVrFirBik
r
mrB
dr
dr
dr
d
r π ρ ωδδ
−=
−+− , ( π σ ωδ 4c= , ( ) 000 zkBB
r
mrF −== θkB ). (4)
The perturbations ~
zV and ~
zB are small and for
simplicity we put 0~~ == zz BV . We use approximation of
an incompressible plasma motion 0~ =Vdiv , neglect the
p∇ term, variations of the plasma density ρ and
conductivity σ (compare with [5]).
The value ( )rF is equal to zero inside the plasma, ( ) 0=resrF
,
when ( ) nmrq res = ( ( ) 00 θRBrBrq z= ). The region near
resrr ≈ is the resonant (interaction) zone.
Inside and near the interaction zone Eq. (3) have the
next general solution normalized to the value
( ) ( ) ( )π ρπ ρ 44 ⋅== rkrICrBV m
vac
rrA ( ( )krI m -
modified Bessel and ( ) ( )zH 2,1
41 - Hankel functions):
( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
( )
,rrfor
4
3exp
4
3exp
4
3exp
4
3exp15.0
4
3exp15.0
4
3exp
24
res
)1(
41
22
41
)2(
41
0
0
0
1
41
0
)1(
41
21
41
2
2
2
22
≥
+
−
−
−−
+
=
∫∫
∫∫
++
−++
z
az
z
zazm
r
uRiuuHduizHuRiuHudu
uRiuuHduiuRiuuHduiizH
krI
zr
rV
πππ
ππππ
(5)
Problems of Atomic Science and Technology. 2002. № 5. Series: Plasma Physics (8). P. 3-5 3
( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
( )
,rrfor
4
3exp
4
3exp
4
3exp
4
3exp15.0
4
3exp15.0
4
3exp
24
res
0
)1(
41
22
41
)2(
41
0
0
1
41
0
0
)1(
41
21
41
2
2
2
22
≤
+
−
−
−−
+
=
∫∫
∫∫
−−
+−−
z
z
z
azzm
r
uRiuuHduizHuRiuHudu
uRiuuHduiuRiuuHduiizH
krI
zr
rV
πππ
ππππ
(6)
where
( ) ( )
+
⋅
+
⋅
±=±
45222
2
2
243
21
1
4
3
42
1
u
F
dr
drF
dr
dr
r
r
Q
i
uQ
r
r
krI
uR resresm
π ρ ωπ ρ ω
δ
δ
, (7)
( ) ( ) ( ) resresres rrrQrrz −= 212δ , RnSVQ zA ω= , π ρ40zzA BV = , ( )
resrrqrqS == |' . (8)
In the ( )uR ± term the radius r is a function of u :
( ) ( ) uQrrur resres δ21 ±= , a - the minor plasma
radius. Outside the resonant zone
( ) ( )rFrVr
24π ρ ω−≈± . The same result we obtain
from Eqs. (5), (6) in the case 12 > >z . We assume that the
radial vacuum perturbation of magnetic field ~
rB
dominates in the plasma and in the right side of Eq. (3)
we take ( ) ( )krCIrB m= (the vacuum perturbation of
the magnetic field) .
From Eq. (3), (5), (6) it follows that the half width of
the interaction (resonant) zone r∆ is of the order of
( ) 212~ Qrr res⋅∆ δ . (9)
From Eq. (4) we obtain the contribution to the radial
magnetic field perturbation of the plasma motion response
( ( ) ( ) ( )rBrBrB r
vac
rr 1
~ += ) with
( ) ( ) ( )rVrFrW r
±+=
24
1
π ρ ω
(10)
( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( )
′′′′′−′′′′= ∫∫
r
a resm
m
mm
r
resm
m
m
res
res
vac
r
r rW
krI
rkIrkKrrdkrIrW
krI
rkIrrdkrK
r
ri
rB
rB
0
2
2
1 1
δ
. (11)
The same for the poloidal component ( ( ) ( ) ( )rBrBrB vac
1
~
θθθ += , ( )zKm′ = dzdK m , ( ) dzdIzI mm =′ )
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
′′′′′−′′′′′= ∫∫
r
resm
m
m
r
a resm
m
mm
res
res
vac
r
rW
krI
rkIrrdkrKrW
krI
rkIkrKrrdkrI
m
kr
rB
rB
0
2
2
1 '1
δ
θ . (12)
3. COMPUTATIONAL RESULTS
3.1 Tokamak CSTN-IV
First, we present calculations for the CSTN-IV
experiment [4] ( R =0.4 m, a =0.1 m, resr =7,5 cm, m=6,
n=1, 0zB =0.086 T, pln =1.5⋅ 1018 m-3).
Fig. 1. Profiles ( ) ( )res
vac
rr rBrB ~ , ( ) ( )res
vac
r rBrB ~
θ
The tendency in the ~
rB and ~
θB behavior is the same
as it is in the CSTN-IV experiment ( f=20 kHz, δ =2 cm).
Fig. 2. The radial profile of the velocity rV
4
0
1
2 ~
rB
~
rB
vac
rB
5 6 7 8 9
0
1
2
r (cm)
~
θB
~
θB
vacBθ
0 1 2 3 4 5 6 7 8 9 10
-0.4
0
0.4
r (cm)
rVRe
rVIm
rV
A very wide interaction region width r∆ is observed.
In Ref. [4] the theoretical estimate r∆ ~ 4 mm was
declared. In the figures the vertical dashed line shows the
resonant radius position.
3.2 TEXTOR-DED
Here the calculations for the TEXTOR-DED tokamak
are presented ( R =1.75 m, a =0.47 m, resr =43 cm, m=12,
n=4, 0zB =2.25 T, pln =1019 m-3).
CONCLUSIONS
It is shown that for the high frequency ( ~> 10 kHz) the
radial component of the perturbation field ~
rB is
amplified inward of plasma from the interaction zone.
This theoretical result confirms the CSTN-IV tokamak
measurements.
For a lower frequency ( ~< 1kHz) ~
rB is only
attenuated in the plasma between the resonant zone and
antenna.
Note, that for TEXTOR-DED the poloidal magnetic
field component of the vacuum perturbation is practically
5
40 41 42 43 44 45
-0.5
0.0
0.5
r (cm)
-0.5
0.0
0.5
-0.5
0.0
0.5
35 40 45 50 55
0
5
10
0
5
10
0
5
10
r (cm)
rV
rVIm
rVRe
rVRe
rVIm
rVRe
rVIm
~
rB
~
rB
vac
rB
~
rB
vac
rB
vac
rB
~
rB
a) a)
b) b)
c) c)
Fig. 3. The radial profile of the velocity rV :
a) f =10 kHz, δ =0.7 cm; b) f =1 kHz, δ =2.2 cm;
c) f =100 Hz, δ =6.96 cm.
Fig. 4. Profiles ( ) ( )res
vac
rr rBrB ~ :
a) f =10 kHz, δ =0.7 cm; b) f =1 kHz, δ
=2.2cm;c) f =100 Hz, δ =6.96 cm.
completely compensated by the plasma perturbation
response at resrr = .
The width of the resonant zone r∆ for TEXTOR-
DED is of the order of 0.5 cm (or larger). It is much larger
than the ion gyroradius. For the CSTN-IV experiment the
width of the interaction region is very wide.
This work was carried out in the frame of the WTZ
project UKR-01/003 between Germany and Ukraine.
REFERENCES
1. Fusion Engineering and Design.//Special issue:
Dynamic Ergodic Divertor (37). 1997.
2. K.H. Finken, S.S. Abdullaev, A. Kaleck, G.H. Wolf//
Nucl. Fusion.(39),1999,p. 637.
3. M.V. Jakubowski, S.S. Abdullaev, K.H. Finken,
M. Kobayashi// Problems of Atomic Science and
Technolog.Series:Plasma Physics.(7),2002, N4,p. 42
4. M. Kobayashi,T.Tuda,K.Tashiro et al.// Nucl.
Fusion (40), 2000, p.181.
5. B. Basu, B. Coppi// Nucl. Fusion. (17), 1977,p. 1245.
6
I.M. Pankratov, A. Ya. Omelchenko, V.V. Olshansky
Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics and Technology” Akademicheskaya str., 1, 61108 Kharkov, Ukraine
1.INTRODUCTION
BASIC EQUATIONS
We start from magnetohydrodynamic equations
3.COMPUTATIONAL RESULTS
CONCLUSIONS
REFERENCES
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