J. Plasma Fusion Res. SERIES, Vol.7 (2006) 89-93

Impurity Dynamics in Stellarator W7-AS Plasmas
IGITKHANOV Yuri, POLUNOVSKY Eduard1 , BEIDLER Craig D,
BURHENN Reiner and YAMAZAKI Kozo2
Max-Planck Institut fur Plasmaphysik, Greifswald, Germany
1 ITER, JWS Garching, Germany
2 Nagoya University, Nagoya 464-8603, Japan（Received: 5 October 2004 / Accepted: 9 February 2006）

Abstract
Numerical eﬀorts to understand the neoclassical transport of impurities in stellarator plasmas have been
undertaken. The new code solves the radial continuity equations for each ionization stage of the impurity ionsfor given background plasma profiles and magnetic configuration. An analytic description of the neoclassicaltransport coeﬃcients based on numerical results from the DKES (Drift Kinetic Equation Solver) code andmonoenergetic Monte-Carlo calculation (C.D. Beidler et al., EPS 1994), is here applied for impurity transport coeﬃcients. The transition between the diﬀerent charge states due to the ionization and recombination inbalance equation is described by using the ADAS (Atomic Data and Analysis Structure) database. The impurity behavior in some typical discharges from W7-AS with moderate (NC) and improved energy confinement(HDH) has been considered.
It is shown that the spatial distribution results from the competition between the radial electric field andthe thermal force (which together produce a convective flux), and the diﬀusive term, which flattens the radial
impurity distribution. The impurity ions are localized at the radial position where the convective flux goes
through zero. It is also shown that for typical stellarator discharges there is no pronounced temperaturescreening eﬀect as in tokamak plasmas.
Keywords:
Impurity dynamics, stellarator physics
1．Introduction
Development of the Stellarator Impurity Transport
code (SIT) has been started, aiming to have a numerical tool for better understanding of the existing experimental observations of impurities in stellarator plasmas,
and ultimately to prove the feasibility of the stellarator
as a reactor. The code solves the system of continuity
equations (averaged over the magnetic flux surfaces) for
impurity ions in each charge stage, coupled due to theionization and recombination. In the current stage it calculates the evolution in time and space of impurity ionscoming from the wall or due to the pellet ablation. In
this sense SIT code is similar to tokamak impurity codes
MIST [1] or STRAHL [2]. The new code evaluates impurity behavior in the frame of the stellarator-specificneoclassical transport, which unlike the tokamak configuration is strongly dependent on magnetic topology
and the radial electric field. In the case of a test impurity approximation, considered here, the radial electric field is evaluated from the ambipolarity conditionin the background plasma and depends on neoclassicaltransport coeﬃcients for electrons and ions. An analytical description of the neoclassical transport coeﬃcientfor the background plasmas (based on numerical resultsfrom the DKES code [3] and monoenergetic Monte
Carlo simulations) was generalized to impurity ions ofarbitrary mass and charge state and used in the code asa neoclassical transport model for impurities. The reduction of Pfirsch-Schluter convection due to the radial
electric field and its impact on impurity dynamics has
been included. Calculations were performed for given
plasma density and temperature profiles. This can be√justified because the impurity flux density is ∼ mi /metimes larger than the plasma diﬀusion rate and, in thecase of very small impurity concentration, when the radiation is negligible in the plasma energy balance.
2．Impurity density equations, transport coefficients and boundary conditions
The SIT code solves the set of coupled timedependent one-dimensional continuity equations for
C 2006 by The Japan Society of Plasma

Science and Nuclear Fusion Research

Corresponding author’s e-mail: yli@ipp.mpg.de

89

Impurity Dynamics in Stellarator W7-AS Plasmas

impurity density, nIj for each ionization stages, j of agiven impurity species, I, averaged over the magnetic
surfaces:
∂nIj

1 ∂
+
(rΓ Ij ) = S Ij ,∂tr ∂r

j = 1, 2 . . .

DI,PSj ≈(3)

In this coeﬃcient the dependence on the radial electricfield, Er is taken into account (the second term in (3)).
Here ΩE = Er /aB0 , Ω j = Z j eB0 /mI , x ≈ mI V 2 /2T, aand R0 are the average minor and major radii, ν I is thecollisional frequency for impurity ions, and bt is the “effective” toroidal curvature, V is the velocity of the impurity ion of mass mI , ı is the rational transform valueand t ≈ a/R. The physical meaning of the new term in(2) can be understood as follows. As the collisionality
increases, even modest values of the radial electric field
result in a poloidal rotation which is suﬃcient to suppress charge separation, and Pfirsch-Schlüter diﬀusion
is reduced to the classical level (the last term in (3)).
Calculation show, that the dependence on electric field
is important particularly for high Z impurity ions. The
radial electric field can be found from ambipolarity condition, assuming the tracer approximation for impurity
ions, Γ i (Er ) ≈ Γ e (Er ). Here we are also ignoring thevariation of electric field in poloidal direction (which isdiﬃcult to estimate), but which can have a noticeable
impact. Excluding the term with electric field from ambipolarity conditions (2), one obtains:
⎛ I 
⎞

⎟
⎜⎜ (n j )Tn⎟I, j I ⎜IΓ j = −D11 n j ⎜⎜⎝ I + β + γT ⎟⎟⎟⎠
(4)
n
T
nj

(1)

Here the impurity flux Γ Ij can be written as Γ Ij =−DIj ∇nIj + nI V Ij . In general the radial diﬀusion coefI, j
I
ficient DIj = D1,1
+ Dan
and the convective velocity,I
I
I
V j = VWare +Van consist of a neoclassical and an anomalous term. Since in stellarators the Ware pinch can beneglected, the neoclassical impurity flux Γ Ij can be written as (with a slightly diﬀerent definition of the neoclasI, j
compared to Ref. [1]):sical transport coeﬃcients Dnk

 
DI, j
I, j I Γ Ij = −D11
n j + nIjI 11T⎧⎞ ⎫⎛ I, j⎪⎜⎪⎪⎜⎜⎜ D12 3 ⎟⎟⎟⎟  ⎪⎨⎬− ⎟ T ⎪ ≡ Γ di f f + Γ conv ,Z E −⎜⎪⎪⎩ I r ⎝ DI, j 2 ⎠ ⎪⎭ 2
4 VR0 bt
νI ,3 Ω j t ı

(2)

11

The first term in equation (2) describes the impurity diffusion flux, whereas the rest terms constitute the convective flux. Here and further we assume that all impurity ions have the same temperature, T I ≈ T i ≡ T
and the prime denotes the radial derivative of density
and temperature. The sign and the value of the neoclassical convective flux (two second terms in (2)) depend on the radial electric field and the thermal force.
We assume that functional dependence of the neoclasI, j
for impurity species onsical transport coeﬃcients Dn,kplasma parameters is the same as for the backgroundplasma coeﬃcients. For background plasma species theI(withmono-energetic radial transport coeﬃcients, Dn,kn, k = 1, 2 and I = e, i) are obtained by energy convolution of the mono-energetic transport coeﬃcients with a
Maxwellian distribution function. For various magnetic
field configurations, the databases of these coeﬃcients
calculated by DKES code are fitted based on traditional
analytic theory, with axisymmetric contributions in theplateau collisionality regime taken into account. Finally, the functional dependence of these coeﬃcientson plasma parameters is expressed analytically, whereas
the radial dependence for a given magnetic field configuration is calculated by using the least-squares fittingof numerical results [4,5]. This semi-analytic description makes possible a rapid numerical simulation of the
impurity transport.
For the collisional case we are using a new expression for the Pfirsch-Schluter monoenergetic coefficients, DI,PSj (Er , V, r) [6] :⎛⎛⎞−1  2 ⎞⎛2 ⎞2 ⎟
⎜⎜⎜⎜⎜
Ω
R
3ν
⎟
⎟⎟
⎜
t ı ⎟⎟⎟⎟⎟
I
E
⎟
⎜
⎜
DI,PSj (Er , V, r) ≈ DI,PSj ⎜⎜⎜⎜⎝⎜⎜⎜⎝1 + ⎜⎝⎜ 2 2 0 ⎟⎠⎟ ⎟⎟⎟⎠ +
⎟,
bt ⎟⎠
ıV

where n = ni ≈ ne  Z j nIj , and
 e

 i

D
D12
3
3
i
⎛ I
⎞
De11 D12
D
−
−
Z
e −
i
i
11 D11
2
2⎜⎜⎜ D12 3 ⎟⎟⎟11γT = Z j+ ⎜⎝ I − ⎟⎠D11 2
De11 + Zi2 Di11β = Zj

De11 − Zi Di11
De11 + Zi2 Di11

(5)

In equation (4) the first term is the diﬀusive flux
driven by the impurity density gradient. The two lastterms here are proportional to the plasma density andtemperature gradients, respectively. These terms represent impurity convection. Whether impurity will localize at the plasma edge or at the center depends on thesign of γT and β coeﬃcients in (4). The second termin (4) describes impurity convection due to the electron
density gradient (driven by the radial electric field) andis proportional to β ≈ −Z j /Zi , since De11  Di11 . HereZi is the charge number for background ion, and Z j for
impurity ions. This term is always negative that corresponds to inward flux. The third term in (4) is proportional to the thermal diﬀusion coeﬃcient, γT . WhenγT > 0, an outward drift (in positive radial direction)
happens and the “temperature screening eﬀect” takesI
coeﬃcients are positive).
place (notice, that D1k
90Impurity Dynamics in Stellarator W7-AS Plasmas

The source term in equation (2) describes the atomic
processes of recombination and ionization between thediﬀerent charge states:(6)
S Ij = ne S j−1 nIj−1 − (S j + α j )nIj + α j+1 nIj+1
Here the ionization S j and recombination α j rates for
impurity ions are taken from ADAS. In the case of trace
impurity concentrations, considered below, the radial
electric field E r is determined mainly by backgroundplasma and can be evaluated from the diﬀerential equation, which permits a smooth transition in case of different roots of the ambipolarity condition [7,8].
⊥  2 ∂η∂Er
2 ∂ 2 ηẼr
Ẽr
− 2
ρ 2 +
∂t
B
∂Er
ρ ∂ρ
B
⎛
⎞
⎜⎜
⎟⎟
= e ⎜⎜⎜⎝ zi Γ(Er )i − Γe (Er )⎟⎟⎟⎠ ,iẼr ≡ Er − Er /ρ

(7)

Here Γe,i are the radial fluxes of background components, and ⊥ is the dielectric constant. The E r dependence of the viscosity coeﬃcient η(E r ) has beentaken to be in accordance with as η ≈ const. for
Er ≤ Eres ≡ ıνT e Bρ/R and η ∝ E r−β for Er ≥ Eres , β ≈ 4[7]. The boundary conditions are the symmetry condition at the center E r = 0 and at the edge either E r = 0 inlimiter plasmas, otherwise Γe ≈ zi Γi at the last closed
magnetic surfaces.
The impurity density equations (1) are integrated
from the plasma centre (ρ = 0, ρ ≡ r/a) to the edge(ρ = 1). The boundary conditions at the edge dependon the thermal velocity of impurity ions, VTI , and therecycling coeﬃcient, R I
Γ Ij = 0.25RI nIj VTI

Fig. 1 Background density (a) and temperature (b) proﬁles in HDH and NC (dashed) modes of W7-AS[9,10].

Fig. 1) [9]. Observation show that the NC discharge ischaracterized by impurity accumulation at the centre,
while in the HDH discharge impurity is mainly radiatedfrom the edge region [9]. A source of neutral impurity atoms was specified at the wall and can be varied intime. The calculated radial electric field (see Fig. 2) isin both cases negative, but for HDH is rather small, particularly at the edge. The thermal diﬀusion coeﬃcientsfor Fe impurity ions for the HDH mode are shown in
Fig. 3. γT for each charge state is negative, which predicts the accumulation at the centre. A steady state impurity distribution is achieved, when the source term isswitched oﬀ and the convective flux turns to zero. A
comparative analysis of the diﬀusive and the convective
fluxes shows that, in steady state impurities are localized radially, where these fluxes are compensate eachother. In the HDH mode of operation the flat density
profile in the bulk plasma slows down the inward impu-(8)

The boundary conditions at the centre are Γ Ij (0, t) =0. The density of neutral atoms entering the plasmaedge, r = a decays inside the plasma as,  a
ne s 0adr ,(9)n0 (r) = n0 (a) exp −r
V0r
where V0 is the thermal velocity and s0 is the ionization
rate of the injected impurity atoms. The impurity influx
can last during the all operation time or be terminatedafter some short time, simulating the case of pellet injection at the edge region.
3．Results and conclusions
Calculations have been performed for several impurity species (Fe, C) for background plasmas profiles typical for the HDH and NC configuration in W7-AS (see91

Impurity Dynamics in Stellarator W7-AS Plasmas

beyond a critical collisionality of background plasma,νe f f > ıV/3R0 ΩE , the poloidal rotation due to the radial
electric field is suﬃcient to suppress the charge separation responsible for Pfirsch-Schlüter diﬀusion, and thelosses are then reduced to the classical level. So, as thecollisionality increases, D IPS drops to its classical value
and then become independent of electric field. Thisdrop in DIPS value results in a lowering of the electricfield value.
The thermal diﬀusion coeﬃcient is negative, forall considered profiles, thus excluding the temperaturescreening eﬀect, except for the case with very low andflat density profiles and low charge states. Calculations
show that the impurity convection is very sensitive tothe value of the electric field and, hence to the plasma
profiles. In the HDH mode, convection in the bulk region is smaller than in the NC mode. At 0.05 sec impurities are still at the edge, whereas in the NC mode
impurities are already distributed in the centre. In bothregimes impurity ions finally accumulate eventually at
the centre and the “temperature screening” coeﬃcient isalways negative (no “screening”). However, the inward
convective drift in the case of the HDH mode is smaller
than in the NC mode, because of the flat density gradient and, ultimately, due to the smaller value of theelectric field. Figure 4 shows, that in both cases Fe impurity ions are accumulating in the centre. Calculationshows that, in the frame of correct stellarator-relevant
neoclassical theory, it is impossible to explain retentionof impurities at the plasma edge, seen in the HDH modeof W7-AS. However, the typical timescales needed toreach steady state ∼ 0.1 sec in the HDH plasma, and3 – 5 times less in the case of NC, are of the same orderFig. 2 The radial electric ﬁeld for NC (dashed) and HDH
modes of the W7-AS.

Fig. 3 The thermal diffusion coefﬁcients, γT , for Fe impurity ions in HDH mode of W7-AS

rity penetration in this region. In the NC mode, the convective term dominates the transport and brings about
a rapid accumulation of impurities in the bulk plasma.
In both operational modes impurities are predominantlyin the Pfirsch-Schlüter regime, νeI f f  1, so that thecorrect Pfirsch-Schlüter description was important. As
it was mentioned above, the Pfirsch-Schlüter diﬀusion
should include the dependence on the radial electricfield (see Eq. (3)). In this expression the reduction of
the Pfirsch-Schluter convection due to the poloidal rotation caused by the radial electric field has been taken
into account and, its impact on the impurity dynamics
I
coeﬃcients and electric field
due to the change in Dn,kwas investigated. As one can see from equation (3),Fig. 4 Evolution of Fe ions proﬁles in the HDH & NC discharges of W7-AS.

92

Impurity Dynamics in Stellarator W7-AS Plasmas

of magnitude as in experiments [9].
Calculations show that the radial electric field plays
a dominant role in impurity dynamics and localization.
In case of low density and high temperatures (positive
radial electric field) one can probably expect impurity
retention at the plasma edge.
The diﬀerent impurity behavior in NC and HDH
modes could be probably explained as a result of change
in sign of the radial E-field at the edge from negative (inNC or H∗ mode) to positive, with pronounce screening
eﬀect. The main signature of HDH is the increase of average beta value, which causes a strong ergodization of
the edge area. At some higher beta value, a stochastic
area is setting up in the edge region beyond the pedestal
region. Calculations done in [10] for Helias finite-beta
equilibrium case confirm, that with increasing beta thewidth of the macroscopic islands at the edge increases,
showing the trend to overlapping of magnetic islands
and ergodization. The ergodization could change a signof the electric field in this area, making it positive theelectron losses become dominant. In NC mode beta issmall and the E-field is negative in entire area. With increase of average density we are approaching H∗ mode,
which has higher beta value, than in NC mode, but
not suﬃcient for eﬀective ergodization. When average pressure (beta) is increasing further the edge region
is ergodized stronger, and the positive E-field prevails
over the diﬀusivity. This model is however not yet included into code.
In conclusion: the main goal of our consideration
was to simulate impurity behavior in the case ofstellarator W7-AS plasmas in the frame of neoclassicaltheory. For this purpose an analytical description of theneoclassical transport coeﬃcient for the backgroundplasmas (based on numerical results from DKES codeand monoenergetic Monte Carlo simulations) has been

applied for the impurity ions of arbitrary mass andcharge state. The code describes the evolution intime and space of impurity concentration of diﬀerent
charge states, the corresponding convective and diﬀusive fluxes, and radiation intensities for given plasma
profiles. It was confirmed that, in contrary to tokamak,a temperature screening eﬀect does not appear in stellarator configurations. In the HDH mode the flat density profile in the bulk plasma slows down the inward
impurity penetration. In the NC mode, the convective
term dominates and brings about a rapid accumulationof impurities in the bulk plasma. However, the impurity
screening in the HDH mode can not be understood on
the base of the pure neoclassical transport, used here.
Acknowledgements
The authors thank R. Dux, K. McCormick, P. Grigull
and A. Weller for helpful discussions.
References[1] R.A. Hulse, Nucl. Technology / Fusion 3, 269(1983).
[2] K. Berhinger, JET Report No. JET-R (87) 08,
1987.
[3] W.I.van Rij and S.P. Hirshman, Phys. Fluids B 1,
56 (1989).
[4] C.D. Beidler, W.D. D’haeseleer, Plasma Phys.
Control. Fusion 37, 463 (1995).
[5] H. Maassberg et al., Phys. Plasmas 7, 1 (2000).
[6] Yu. Igitkhanov, C.D. Beidler et al., in Contribution of Plasma Physics, 2006 (submitted).
[7] K. Shaing et al., Phys. Fluids 26, 3315 (1983).
[8] H. Maassberg and K. Dyabilin, EPS (1993).
[9] K. McCormick et al., Phys. Rev. Lett. 89,
015001 (2002).
[10] E. Strunberger, Nucl. Fusion 37, 19 (1997).

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