J. Plasma Fusion Res. SERIES, Vol.7 (2006) 102-105

Simulation Modeling of Impurity Transport in Toroidal Fusion Plasmas
YAMAZAKI Kozo∗ , AMANO Tsuneo, IGITKHANOV Yuri1 , GARCIA Jeronimo2 ,DIES Javier2 , SAMITOV Marat3 and MIKHAILOV Mhail3
National Institute for Fusion Science, Toki 509-5292, Japan
1 Max-Planck-Institut fur Plasmaphysik, IPP-EURATOM Ass., Greifswald, Germany
2 Universitat Politecnica de Catalunya, Barcelona, Spain
3 Russian Research Centre ’Kurchatov Institute’, 123182 Moscow, Russia（Received: 5 October 2004 / Accepted: 21 December 2005）

AbstractThe impurity transport simulation code coupled with 1-D (dimensional) transport / 3-D equilibrium TOTAL (Toroidal Transport Analysis Linkage) code is developed with self-consistent ambipolar electric field
due to both fuel and impurity ions. Multi-species of impurity ions are treated in the analysis of ignited fusion
plasmas controlled by feedback scheme with gas/pellet fueling and heating power. The present analysis withrather low impurity contents shows that the neoclassical ripple transport flux of impurity ions is small comparing with the axi-symmetric neoclassical impurity flux. When the anomalous diﬀusion is introduced, thiscancels the axi-symmetric neoclassical flux and makes the impurity density profile flat. This simulation code
can be utilized for the impurity transport analysis of various toroidal magnetic configurations.
Keywords:
impurity transport, helical plasma, toroidal fusion reactor, magnetic configuration, radiation loss∂ni
1 ∂+  V  Γi = S i ,Zi ni ≈ ne ,i∂t
V ∂ρ
 
3 ∂ne T e
1 ∂5+ V  q e + Γe T e
2 ∂tV ∂ρ
21．Introduction
In order to access to the ignition regime of the fusion reactor, the impurity radiation loss and plasma density limit are critical, as well as the energy confinement degradation and the plasma beta limitation. Thedensity limit might be related to the impurity radiation
power loss in helical systems, and be relevant to the radiation and plasma current instability eﬀects in tokamak systems. Moreover, to clarify the radial impurity
profile, the radial energy balance is important for theaccess to the ignited reactor plasma. In order to clarify these issues, reactor plasma analysis was performed
using the TOTAL (toroidal transport analysis linkage)code in Ref. [1]. In this paper, we include multi-species
impurity dynamics with self-consistent ambipolar electric field and study the ignited plasma operation dynamics of helical reactor systems.

(1)= PHe − Pei − Prad − Γe Er , 
3 ∂ni T i
1 ∂5

+ 
V q i + Γi T i
2 ∂tV ∂ρ2

(2)

= PHi + Pei − Pcx + zi Γi Er ,(3)using the normalized radius ρ and the volume V defined by the equilibrium magnetic surface. The plasmaequilibria with arbitrary 3-D shapes are calculated byVMEC code [3] for tokamak and helical systems including advanced shaped stellarators [4]. The radiation loss Prad is the summation of bremsstrahlung, line
radiation and synchorton radiation. The present synchrotron radiation loss model was compared with nonlocal CYTRAN synchrotron radiation results [5], however, the energy loss is small in the present reactor
model with assumed good reflection coeﬃcients.
For the impurity dynamics, the rate equation anddiﬀusion equation are solved using IMPDYN code [6]coupled with ADPAK atomic physics package [7],2．Transport models including impurity dynamics
For studying the transport of plasma fuel and impurity ions in fusion reactors, we used 2.0-D (1-Dtransport/3-D Equilibrium) time-dependent simulation
model [2] with low-Z gas and high-Z metal impuritydynamics. The plasma density ne , ni and temperature
T e , T i are described by

C 2006 by The Japan Society of Plasma

Science and Nuclear Fusion Research
Corresponding author’s e-mail: yamazaki@ees.nagoya-u.ac.jp∗ Present address: Department of Energy Engineering and Science, Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan102

Simulation Modeling of Impurity Transport in Toroidal Fusion Plasmas∂nk1 ∂= −  (V  Γk ) + [γk−1 nk−1 + αk+1 nk+1
∂tV ∂ρ

is modeled to be one tenth of thermal diﬀusivity as usedin Ref. [2]. The target alpha power is 450 MW (roughly1 GW-electric). The gas fueling and the external heating
power are feedback controlled to trace the time evolution of target fusion power. The previous analysis hasbeen given in Ref. [1] and the plasma with peaked temperature (central temperature 〜 20 keV) and flat density profile (average 1.5×1020 /m−3 ) are expected in thisreactor.

− (γk + αk )nk ]ne + S k , (4)Γk = ΓkNC s + ΓkNCa − Dk (ρ)∂nk+ Vk (ρ)nk ,∂ρ(5)with ionization rate γk , recombination rate αk and particle source term S k . The fluxes ΓkNC s and ΓkNCa are neoclassical symmetric and asymmetric ones, respectively.
Here, a constant diﬀusion coeﬃceint Dk and a simple
inward velocity model Vk = V(a) · (r/a) are adopted foranomalous impurity transport. The main fuel neutrals
are calculated by the AURORA Monte Calro code [8].
The ripple transport flux Γ NCa is given by the density
gradient term, temperature gradient term and the radial
electric field term as shown in Refs. [9,10],

3.2 Impurity transport in reactor plasmas

First, we checked the impurity profile in the LHR-S
background plasma with fixed density and temperature
profile, n ∝ (1 − ρ8 ) and T (keV) = 20 × (1 − ρ2 )2 .
The tokamak-like plasma with only symmetric neoclassical loss is analyzed as well as helical plasma transportwith symmetric and asymmetric neoclassical impurity
losses. In the case of carbon impurities, they are fully
stripped and the behavior near the edge region is diﬀer-Γ NCa ∝ Drip∇n ∇n/n + Drip∇T ∇T/T − ZEr /T
The first and second terms also depend on the radialelectric field.
Many elaborate works on neoclassical impurity dynamics have been done for tokamak plasmas. However,for helical systems only a few papers [11] related to impurity transport theory have been published. Here, weadopted the neoclassical formula of fuel ions to include
the impurity ions in addition to the tokamak-like axisymmetric neoclassical flux [12].
The radial electric field E r is determined by the neoclassical asymmetric transport flux Γ NCa [9,10] including impurity ions in helical system as follows,zk ΓkNCa (Er ) − ΓeNCa (Er ) = 0.
(6)
kor adding the time-derivative term of the radial electricfield in the right hand side instead of zero. Here, thesubscript k denotes fuel ions (deuteron & triton), helium
and impurity ions.
As an input of simulation code, impurity density ofeach spieces is given and the time-dependent contentsof these impurities are determined by their recycling
rate, and diﬀusive and convective transport terms.
3．Simulation results
3.1 Ignited plasma of LHD-type Helical Reactor (LHR)

In the present analysis, the LHR-S (Large HelicalReactor-Standard) system [1] having R = 16.5 m, B =5 T is adopted as a reference with H-factor based on
New LHD confinement scaling models [13]. The radialprofile of electron anomalous thermal transport coeﬃcient is assumed here as χ ∝ C(1 + kρm ) with k = 5,m = 4. The coeﬃcient C is determined to fit the required H-factor. Here, the particle transport coeﬃcientFig. 1 Steady-state ion impurity charge-state density
proﬁles in the LHR-S plasma with 20 keV peakedtemperature and 1020 /m3 ﬂat density. (a) without anomalous transport, and (b) with constantanomalous transport (DAN = 1 m2 /s).

103

Simulation Modeling of Impurity Transport in Toroidal Fusion Plasmas

ent from tokamak-like transport due to strong edge electric field. The anomalous transport with D AN = 1 m2 /sand v(a) = 0 flattens the impurity density profile. Iniron impurity case, the maximum density was obtained
for the charged state of Z = 24 (k = 25) at the radius ofρ 〜 0.8 (Fig. 1).
3.3 Operation dynamics of ignited LHR-S plasma
with impurity ions

The radiation loss and the relevant density limitation
are serious especially during the start-up phase of reactor plasmas. Here, we analyze the operational scenarios of LHR-S plasma from the low density startup tothe final high-density steady-state operation of the reactor (Fig. 2). Using less than 200 MW heating power
control, about 20 keV peaked temperature profile withrather flat density is expected (Fig. 3). In this operation
scenario, the whole plasma is in the negative-electricfield (ion-root) regime (Fig. 3 (a)). The initial carbon
and iron impurity contents in this case are 1 % and0.1 %, respectively. Finally, the eﬀective charge number
is 1.2 〜 1.5, and the radiation loss related to iron density is mainly at ρ 〜 0.8 and near the edge (Fig. 3 (b)Fig. 3 (a)Temperature, density and radial electric ﬁeld
proﬁles and (b) power density proﬁles of ignited
LHR-S plasma at t = 100 s in Fig. 2.

& 4 (b-1)). The carbon impurities are fully stripped andrather flat (Fig. 4 (a-1)). In this simulation, the anomalous term (AN in Fig. 4) compensates symmetric neoclassical term (NC sy) (Fig. 4 (a-2) for carbon, and (b2) for ion impurity), and the asymmetric term (NC as)
is not dominant in comparison with toroidal symmetricloss term. Only asymmetric neoclassical transport depends on the radial electric field, therefore, the eﬀect ofelectric field on impurity transport is small.
When the impurity percentage becomes larger, we
expect the change in the radial electric field profile related to the impurity transport dynamics. Also, the effect of magnetic configurations on the impurity transport will be analyzed using the present TOTAL code in
the future.
4．Summary(1) We developed the impurity simulation code coupledwith 1-D transport / 3-D equilibrium TOTAL (ToroidalTransport Analysis Linkage) code, which is applicableto helical and tokamak systems.
(2) Ambipolar electric filed due to both fuel and impurity ions are treated self-consistently. Multi-species ofimpurity ions are included in the analysis of feedback

Fig. 2 Time evolution of feedback controlled LHR-S reactor plasma with impurity ions.

104Simulation Modeling of Impurity Transport in Toroidal Fusion Plasmas

Fig. 4 Carbon (a-1) and Iron (b-1) impurity charge-state density proﬁles and their particle ﬂuxes (a-2, b-2) as a functionof normalized radius. The total ﬂux Γ is the summation of neoclassical symmetric part (NC sy), asymmetric part(NC as) and anomalous part (AN).

controlled ignited fusion plasmas.
(3) The present analysis with low impurity contentsshows that the neoclassical ripple transport dynamics
of impurity ions are small in comparison with axisymmetric contribution of neoclassical theory. Whenthe anomalous diﬀusion is introduced, it cancels theneoclassical flux and makes the impurity density profile flat.
(4) Eﬀects of the higher impurity density and the various helical magnetic configurations will be clarified using this TOTAL code in the near future.

[1]

[2]
[3]
[4]

[5]
[6]

[7]
[8]
[9]

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