RF way of impurity fluxes control in tokamaks
We have studied the influence of the local heating on the impurity flows in tokamaks in the Pfirsch-Schluter regime. If the effect of the thermoforce on the impurity ions is included into consideration, the impurity flux can be reversed by heating the impurities. This concept can be realized in toka...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Cite this: | RF way of impurity fluxes control in tokamaks / D.L. Grekov, S.V. Kasilov // Вопросы атомной науки и техники. — 2000. — № 6. — С. 76-78. — Бібліогр.: 4 назв. — англ. |
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| author | Grekov, D.L. Kasilov, S.V. |
| author_facet | Grekov, D.L. Kasilov, S.V. |
| citation_txt | RF way of impurity fluxes control in tokamaks / D.L. Grekov, S.V. Kasilov // Вопросы атомной науки и техники. — 2000. — № 6. — С. 76-78. — Бібліогр.: 4 назв. — англ. |
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| description | We have studied the influence of the local heating on the impurity flows in tokamaks in the Pfirsch-Schluter regime. If the effect of the thermoforce on the impurity ions is included into consideration, the impurity flux can be reversed by heating the impurities. This concept can be realized in tokamak experiments using RF heating. We describe the scheme of the RF heating of impurities and present estimates of the power required.
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| first_indexed | 2025-11-24T08:17:21Z |
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76
UDC 533.951
RF WAY OF IMPURITY FLUXES CONTROL IN TOKAMAKS
D.L.Grekov, S.V.Kasilov
Institute of Plasma Physics,
National Science Center "Kharkov Institute of Physics and Technology",
61108, Kharkov, Ukraine
We have studied the influence of the local heating on the impurity flows in tokamaks in the Pfirsch-Schluter
regime. If the effect of the thermoforce on the impurity ions is included into consideration, the impurity flux can be
reversed by heating the impurities. This concept can be realized in tokamak experiments using RF heating. We
describe the scheme of the RF heating of impurities and present estimates of the power required.
Experimentally discovered degrading of plasma
parameters due to impurity ions stimulates the search for
diverse techniques of plasma cleaning. Passive
(divertors) and active methods can possibly be used to
act on impurity ions. Active methods, investigated
beginning from [1,2], may involve the bulk component
puffing (particle source), bulk plasma heating (heat
source) and momentum transfer e.g. due to high energy
particle injection. But the thermoforce, which affects the
impurity ions, was implied small and was not taken into
account as in [1,2], so in later papers. This automatically
moved beyond the scope the heating of impurities.
To reverse the impurity influx it is necessary to
provide the asymmetric (relative to the tokamak
equatorial plane) particle, heat or momentum source.
The asymmetric particle or momentum source may be
established very easy. But there were no any
propositions as to asymmetric heat source in tokamak
till now. This paper concerns the impurity flux reversal
in tokamaks in the Pfirsch-Schluter regime using the RF
heating of impurity ions.
Following [1,2], where the heavy impurity
behaviour in tokamaks is considered, write the MHD
equations
for bulk plasma ions (i) and impurity ions (I) in the form
∇ = + × + ||p n e E v B Ri i i( )
! ! ! !
,
∇ = + × − ||p n Z e E v B RI I I I( )
! ! ! !
, (1)
where
! !
R C
m n
u C n Ti i
iI
i i|| || ||= − − ∇1 2τ
. (2)
Here pα is the pressure,
!
vα is the velocity, nα is the
particle density, Z eα is the charge, m α is the mass,
Tα is the temperature of α -species ions (α = i,I ), τ iI is
the time of scattering ions off impurities,
! ! !
u v vi I|| = − ,
!
R|| is the friction force. When the sources of heat and
particles are present the radial flux of impurity ions,
averaged over the tokamak magnetic surface, has the
form
Γ I
i i
I iI i i
i
I I
I i i i
I iI i
t
i
i
Qi
i
n q
Z T
C
C
C n
p
r Z n
p
r
C
C
T
r
n q
Z T
eB R
cn
C
C
C
a
C
C
a
T
== +
−
−
− +
−
2 1 1 5
2
2 2
1
2
2
3
2
3
2 2
1
2
2
3
2
3
ρ
τ
∂
∂
∂
∂
∂
∂
ρ
τ τ . (3)
Here q rB R Bt p= / 0 , a, R0 are the small and large radii
of the torus Bt and Bp are the toroidal and poloidal
components of the confining magnetic field, ρi is the
ion gyroradius, aQ i, aτ i are the amplitudes of the sinϑ
components in the Fourier series of heat Q ri( , )ϑ and
bulk ion τ ϑi r( , ) sources, r, ϑ - radius and poloidal
angle at the minor cross-section of the torus (at the low
field side ϑ =0). To neutralize the first term in (3) and
nullify Γ I it is necessary that sgn( ) sgn( )a Bi tτ = − and
sgn( ) sgn( ).a BQi t= However, one obtains too large aQ
i values when heating bulk ions asymmetrically.
Therefore in the following consideration we'll take into
account the thermoforce which is due to the impurity
temperature gradient along the magnetic field lines
R n TT
I
I I
II
II Ii
∝ ∇
+||
τ
τ τ
. This friction force is
considerably less then the thermoforce due to the
temperature gradient of bulk ions
R n TT
i
i i
ii
ii iI
∝ ∇
+||
τ
τ τ
( τ αβ is the collision time):
R RT
i
T
I/ changes from ZI
2 1>> to Z m mI I i
2 1 2( / ) / on
increasing nI from n m m n ZI i I i I≤ ( / ) //1 2 to
n n ZI i I≥ / . Nevertheless the account of RT
I allows
one to include into consideration the heat source acting
on impurities. To find ∇ ||TiI, use the expressions for
heat fluxes
! !
q C nT u C
nT
m
Ti i i
i i
i
ii iI
ii iI
i|| || ||= −
+
∇2 3
τ τ
τ τ
!
q C
n T
m
TI
I I
I
II Ii
II Ii
I|| ||= − ′
+
∇3
τ τ
τ τ
. (4)
Inserting ∇ ||TiI, from (4) to (2) with account of RT
I
yields
77
! !
! !
R
m n
Ñ
Ñ
Ñ
u
C
C
q
nT
C
C
q
n T
I
i i
iI
i
i i
I
I I
|| ||
|| ||= +
+ −
′
′
τ π
1
2
2
3
2
3
2
3
4
3
.
Now let us consider the action of the heat source of the
type Q Q rα α ϑ= ′ ( ) sin on the impurity flow. Note that
other components of the QI Fourier series expansions
over sin( )pϑ , cos( )pϑ do not influence Γ I in this
approximation. From the divq divq Q r
! !
α α α ϑ|| ∧= − + ( , )
(here
! !
q
n T
e B
B Tα
α α
α
α∧ = × ∇
5
2 2 is the inclined heat flux)
we have
! !
q B
n T
e B B
T
r
rQ r
B
t|| = − +
′
α
α α
α ϑ
α α
ϑ
∂
∂
ε ϑ
5
0
( )
cos .
After averaging over the magnetic surface one
obtains
Γ ΓI i I
i i
I iI i i
i
I I
I i I
i i
I iI i
t i
i
Qi
i i
Z
n q
Z T
C
C
C n
p
r Z n
p
r
C
C
T
r
C
C
T
r
n q
Z T
eB R
c
C
C
C
a
n
C
C
a
n T
C
= − = +
−
− +
′
′
−
− +
− +
′
/
2 1 1 5
2
5
2
2 2
1
2
2
3
2
3
2
3
2 2
1
2
2
3
2
3
ρ
τ
∂
∂
∂
∂
∂
∂
∂
∂
ρ
τ
τ 2
3
1
′
C
a
n T
Q
I I
(6)
Thus, having B Q rI0 0′ <( ) , one may reverse the
impurity flux using the asymmetric heat source acting on
impurities.
Then we analyze the possibility of asymmetric heat
source managing by means of Alfven resonance
heating. We suppose that antenna is placed at the high
field side of the torus. The antenna launches wave
magnetic field B|| , which is parallel to the confining
magnetic field. The bulk ions are deuterium. The
frequency ω is chosen so, that ω ω= cD R( ) at the outer
side of the torus (ωcD R( ) is cyclotron frequency of
deuterium, R R r= +0 cos ϑ ). As for impurities
Z MI I/ /< 1 2 , so properly changing system
parameters we may position the cyclotron resonance of
impurity ions at R R≈ 0 (Fig.1). Then the Alfven
resonance (AR) zone, where ε1
2( ) ( )r N r= || , is located
at the inner side of minor cross-section relative to
ω ω= cD zone. When the plasma density is within the
common values 10 1013 14 3÷ −cm , the AR occupies
rather narrow region ∆a at the plasma periphery.
Further we neglect the effect of rotational transform for
fast Alfven wave propagation. Assuming ε ε1 3 0/ = ,
we derive from Maxwell equations
( ) ( )ω ε
ε ε
ε
ε ε
2
2
2
2
2
2
1
2
2
2
2 2
1 1 0
c
B rot
i
R
RB Rdiv
N
R
RB|| || ||
||
||−
−
∇
+
−
−
∇
= , (7)
where N
ñl
R|| =
ω
, l is toroidal wave number,
ε ε= − ||1
2N , ε
ω
ω ϑ ω
1
2
2 2
1= +
−
∑ pi
cii
r
r
( )
( , )
,
[ ]ε
ωω
ω ϑ ω ϑ ω
2
2
2 2
=
−
∑ pi
ci cii
r
r r
( )
( , ) ( , )
.
Using Stox and Gauss theorems we convert (7) into
( ) ( )[ ] ( )
( ) ( ) ( )[ ] ( ) ( )
( )( )
( )
( ) ( ) ( )[ ]∂
∂
ϑ ϑ
ε ϑ
ε ϑ
∂
∂ϑ
ϑ ϑ
ε ϑ ε ϑ
ε ϑ
ϑ
ϑ
∂
∂
ϑ ϑ
r
R r B r
i r
r r
R r B r
r r
r N
aR r
rR a a
R a B a, ,
,
,
, ,
, ,
,
,
,
, ,|| ||
||
||− = −
−
−
−2
2
2
2
2 1
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )[ ]
( ) ( )
−
−
′ ′ + ′
′
′ ′ ′
′ − ′
+
||
′ ||
∫∫
R r
r
r r
r c
rB r dr
dr
R r
i r R r B r
r r
r
a
r
a
r, , ,
,
,
,
, , ,
, ,
ϑ ε ϑ ε ϑ
ε ϑ
ω ϑ ∂
∂ϑ ϑ
ε ϑ ϑ ϑ
ε ϑ ε ϑ
∂
∂
2
2
2 2
2
2
2
2
2
( )
( ) ( ) ( )[ ]
( ) ( )
+ ′
′ ′
′ ′ ′
′ − ′
||
∫
∂
∂ϑ ϑ
ε ϑ ϑ ϑ
ε ϑ ε ϑ
∂
∂ϑdr
rR r
r R r B r
r ra
r
,
, , ,
, ,2
2
2
. (8)
To get solution of (8) we apply the sharp layer
approach. This approach was initially developed in [3]
for the case of one dimensional plasma inhomogeneity.
Then it was generalized for the case of ( )r− ϑ
inhomogeneity in [4]. So we imply that B a||( , )ϑ and
∂
∂
ϑ
a
B a||( , ) are known at the plasma boundary. Note,
that for ∆a the inequalities ∆a a a m k<< ||, / , /1 are well
fulfilled. Taking into consideration that
∂ε
∂ϑ
∂ε
∂
∂ε
∂ϑ
∂ε
∂r r
a
a
∼ ∼ <<2 2 1
∆ , we expand B|| into
the series of small parameter ∆a a :
B B B|| || ||= + +0 1 " . Then B||0 is defined by
( ) ( )
( )
i
r
RB RB
r N
R a B a
a
ε
ε
∂
∂ϑ
∂
∂
ε ε
ε
∂ ϑ ϑ
∂
2 0 0
2
2
2
2 1
|| ||
||
||− =
−
−
( , ) ( , )
78
The solution of this equation is found by the method of characteristics
R r B r R B RBa a( , ) ( , ) ( ),ϑ ϑ δ|| ||= +0
( )δ ∂
∂ϑ
ε ϑ
ε ϑ
( )
( , )
( , )
RB i R B
r
r r
dra a
a
r
=
′
′ ′
′ +|| ∫ 2 ( ) ( )
∂
∂
ε ϑ ε ϑ
ε ϑa
R B
r r
N r
dra a
a
r
||
||
′ − ′
− ′
′∫
2
2
2
21
( , ) ( , )
( , )
. (9)
Here R a B a R Ba a( , ) ( , )ϑ ϑ|| ||= . Thus all features of
wave field in the thin layer between the plasma
boundary and neighbourhood of the AR are described
by these formulas. Using (9) the poloidal flux of RF
power is calculated:
( ) ( )S
c
R
B
N a
R B
i
a
R B
a
a a a aϑ πω ε
ε ∂
∂
∂
∂ϑ
=
−
+
||
||
|| ||
2
2
28 1
Re
*
. (10)
Then we analyze (10) supposing that at the plasma
boundary ( )B a im|| , ~ exp( )ϑ ϑ . The second term is
dominant at the vicinity of the boundary and
S
c m
a
B
N
a
ϑ πω
≈
−
||
||
2
2
28 1
. The RF power flux is directed
clockwise for m<0 modes and counter-clockwise for
m>0 modes. As far the wave penetrates into the plasma,
the first term increases. Taking into account
( )Re *B
a
R Ba a a|| ||
<∂
∂
0 , the counter-clockwise power
flux increases and clockwise power flux decreases. This
is the reason of the up-down asymmetric RF power
distribution in the plasma minor cross-section. The
radial RF power flux is Sr = Sr0 + δSr, where
( )S
c
R
B
N a
R Br
a
a a0
2
28 1
=
−
||
||
||πω
∂
∂
Im
*
and
( )δ ω ω
ε ϑ ω
ω
∂
∂
S
c
R
d
dr a
R B
m
a
R Br
pi
r a
t
ci
a a a a=
− −
=
−
|| ||
2
2
2
1 2
4
1( cos ) . (11)
Here δSr is the “absorbed” at the AR power. As it
follows from (11) too, absorbed in the down part of the
torus RF power exceeds the RF power, absorbed in the
upper part of the torus. In the vicinity of the AR the fast
Alfven wave is converted into the small scale kinetic
wave (KW). Since the AR is placed at the plasma
periphery the input of ions (~ )k Li⊥
2 2ρ in the KW
dispersion is negligible for nowadays tokamak
parameters at ze<<1 and at ze>>1 ( z k ve Te= ||ω / 2 )
too. The KW propagates along the confining magnetic
field lines to the inner part of the torus, undergoing
small deviation to the low density side. Those KW,
which started in the − < <π ϑ π/ /2 2 region, intersect
the ion cyclotron resonance of impurity ions. Thus the
asymmetric heating of impurity ions can be provided
with a proper sign of B Q rI0 ′ ( ) . The portion of “useful”
KW is κ π πu = − +( ) / ( )2 2 .To stop the impurity
influx, the total RF power P Ra r Qtot u I= ′2π κ∆ is
necessary. Here ∆r is the radial size of region, where
RF field is absorbed by impurity ions.. For estimations
∆ ∆r a~ was adopted. Then for the tokamak of middle
size (a=50 cm, R0=150 cm) we get P kWtot ≈ 100 .
Thus it is shown in this paper that there exists an
effective RF method for reversing the flux of heavy
impurities in tokamaks.
References
1. P.H.Rutherford. Impurity transport in the Pfirsch-Schluter
regime // The Physics of Fluids, 1974, 17, №9, с.1782-1784.
Fig. 1
2. K.T.Burrell. Effect of particle and heat sources on
impurity transport in tokamak plasmas // The Physics of
Fluids, 1976, 19, №3, с.401-405.
3. К.N.Stepanov. To the influence of plasma resonance
on the propagation of the surface waves in
inhomogeneous plasma // Journal of technical physics,
1965, 35, в.6, с.1002-1014 (in Rus.).
4. S.V.Kasilov. Theory of ion cyclotron resonance in
plasma in an inhomogeneous magnetic field in
tokamaks and open traps. Ph. D. thesis. Kharkov, 1989,
21 p. (in Rus.).
References
Fig. 1
|
| id | nasplib_isofts_kiev_ua-123456789-78508 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-24T08:17:21Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Grekov, D.L. Kasilov, S.V. 2015-03-18T16:42:40Z 2015-03-18T16:42:40Z 2000 RF way of impurity fluxes control in tokamaks / D.L. Grekov, S.V. Kasilov // Вопросы атомной науки и техники. — 2000. — № 6. — С. 76-78. — Бібліогр.: 4 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/78508 533.951 We have studied the influence of the local heating on the impurity flows in tokamaks in the Pfirsch-Schluter regime. If the effect of the thermoforce on the impurity ions is included into consideration, the impurity flux can be reversed by heating the impurities. This concept can be realized in tokamak experiments using RF heating. We describe the scheme of the RF heating of impurities and present estimates of the power required. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Magnetic confinement RF way of impurity fluxes control in tokamaks Article published earlier |
| spellingShingle | RF way of impurity fluxes control in tokamaks Grekov, D.L. Kasilov, S.V. Magnetic confinement |
| title | RF way of impurity fluxes control in tokamaks |
| title_full | RF way of impurity fluxes control in tokamaks |
| title_fullStr | RF way of impurity fluxes control in tokamaks |
| title_full_unstemmed | RF way of impurity fluxes control in tokamaks |
| title_short | RF way of impurity fluxes control in tokamaks |
| title_sort | rf way of impurity fluxes control in tokamaks |
| topic | Magnetic confinement |
| topic_facet | Magnetic confinement |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/78508 |
| work_keys_str_mv | AT grekovdl rfwayofimpurityfluxescontrolintokamaks AT kasilovsv rfwayofimpurityfluxescontrolintokamaks |