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Plasma and Fusion Research: Regular Articles Volume 8, 2403096 (2013)
Effects of Neutral Particles on the Stability of the Detachment
Fronts in Divertor Plasmas∗)
Satoshi TOGO, Makoto NAKAMURA1), Yuichi OGAWA, Katsuhiro SHIMIZU2),
Tomonori TAKIZUKA3) and Kazuo HOSHINO1)
Graduate School of Frontier Sciences, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8568, Japan
1)Japan Atomic Energy Agency, 2-166 Oaza-Obuchi-Aza-Omotedate, Rokkasho, Kamikita, Aomori 039-3212, Japan
2)Japan Atomic Energy Agency, 801-1 Mukoyama, Naka, Ibaraki 311-0193, Japan
3)Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan
(Received 7 December 2012 /Accepted 22 May 2013)
Operation with detached divertor plasmas is considered to be a hopeful way in order to reduce the divertor
heat load in the next generation tokamaks. The physical mechanism of detached divertor plasmas, however, has
not fully been understood yet. We have studied them with a one-dimensional divertor model. Detached divertor
plasmas have been successfully reproduced by introducing a self-consistent neutral model. The flows of particles
and heat from the core plasma were investigated in the detached regime. Also investigated were the conditions
of the recycling rate and neutral loss time constant where the detached regime occurred.
c
2013 The Japan Society of Plasma Science and Nuclear Fusion Research
Keywords: divertor heat load, detached divertor plasma, detachment front, plasma fluid model, self-consistent
neutral model, recycling rate, neutral loss
DOI: 10.1585/pfr.8.2403096
1. Introduction
Reduction of the divertor heat load is one of the crucial
issues in designing the next generation tokamaks such as
ITER and DEMO. In order to resolve this issue, detached
divertor plasmas are considered to be a promising way [1].
The physical mechanism of them, however, has not fully
been understood yet.
In modeling SOL-divertor plasmas, two-dimensional
(2D) codes, such as SONIC [2] and SOLPS [3], and point
models have been used. It is considered, however, that 2D
codes are computationally massive to focus on studying
each physical phenomenon in plasmas. On the other hand,
the latter models are very easy, but have not reproduced
detached divertor plasmas so far. Thus we have been using
the one-dimensional (1D) codes [4, 5], which are compu-
tationally lighter than 2D ones, in order to gain physical
insights of detached divertor plasmas.
In our previous works [6–8], we reproduced par-
tially detached divertor (PDD) plasmas, which will be
adopted for ITER operation scenarios [9], with a ‘multi-
layer (ML)’ 1D model. It was shown that the cross-field
heat transport, which have been proven to significantly af-
fect behaviors of the PDD plasmas [10, 11], prevented the
detachment front in the inner flux tube from moving up-
stream and resulting in X-point MARFE. In order to focus
on the effects of the cross-field heat transport, we had in-
troduced a simple neutral model where the neutral flux,
author’s e-mail: togo@ppl.k.u-tokyo.ac.jp
∗)This article is based on the presentation at the 22nd International Toki
Conference (ITC22).
given by the recycling of the ion flux plus auxiliary gas
puffing, decayed exponentially with the local mean free
path of the ionization reaction or the geometric mean of
mean free paths of the ionization and charge exchange re-
actions. In this paper, in order to focus on the effects of
neutrals on detachment fronts, we adopted a time depen-
dent self-consistent neutral model. The neutral flux is as-
sumed to be composed of convection with a constant flow
velocity and diffusion involving the charge exchange re-
action [3]. Source term due to volume recombination re-
action and loss/source term due to the cross-field neutral
transport are involved in it. The total number of ion and
neutral particles becomes conserved.
The geometry of the 1D divertor model and the plasma
fluid equations are shown in Sec. 2.1. Comparison be-
tween old neutral model and new one is shown in Sec. 2.2.
As simulation results, we first show the difference in the
neutral density profiles resulted from each transport mech-
anism introduced in the neutral model in Sec. 3.1. In
Sec. 3.2, we show a simulation result of the detached
regime and the flows of particles and heat from the core
plasma. Also is shown there the conditions of the recy-
cling rate and neutral loss time constant where the detached
regime occurs. Finally, in Sec. 4, we present a conclusion.
2. Model
2.1 Geometry and plasma fluid equations
The 1D divertor model is used to analyze a SOL-
divertor plasma along the magnetic field shown in Fig.1.
We introduce x-axis along the magnetic field and set the
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2013 The Japan Society of Plasma
Science and Nuclear Fusion Research
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Plasma and Fusion Research: Regular Articles Volume 8, 2403096 (2013)
Fig. 1 Schematic picture of (multi-layer) 1D divertor model.
stagnation point and the divertor plate to be x=0 and
x=L, respectively. The particle and heat flux from the
core plasma are considered in the SOL region. We intend
to simulate PDD plasmas with ML 1D model, however, we
did only single tube analyses in this paper.
The 1D transport equations are given as follows [12];
∂(mn)
∂t+∂(mnV)
∂x=mS,(1)
∂(mnV)
∂t+∂(mnV2+P)
∂x=M,(2)
∂
∂t1
2mnV2+3nT
+∂
∂x1
2mnV2+5nT V−κe
∂T
∂x=Q.(3)
Here, the density n, the flow velocity V, the temperature
Tof ions and electrons are assumed to be equal, respec-
tively. P(=2nT ) is the plasma pressure and κeis the paral-
lel electron heat conductivity. The source terms are given
as follows;
S=Score +σizvnnne−σrc vnine,(4)
M=−mV σcxvninn−mV σrc vnine,(5)
Q=Qcore −Eiz σizvnnne−Lznimp ne
−mV2
2+3
2Tσcxvninn
−(Erc −13.6[eV])σrc vnine.(6)
Here, Score and Qcore are the input particle and heat flux
from the core plasma, respectively. Eiz represents the ion-
ization energy (13.6 eV) plus the radiation loss from ex-
cited atom so that it is set to be 30 eV. The impurity is
assumed to be carbon and its cooling rate Lzis treated in
non-coronal equilibrium [13] and the impurity density pro-
file is given by nimp =rimpniwith rimp set to be 3%. Erc
represents the energy loss by the volume recombination
involving radiative recombination and three-body recom-
bination. For the electron temperature lower than 5.25 eV,
the three-body recombination dominates so that the vol-
ume recombination reaction acts as net heat source.
We omit the explanation of the boundary conditions
here since they are the same as our previous works [6–8].
2.2 The neutral model
In our previous works [6–8], in order to focus only
on the effects of cross-field transport, we had introduced a
simple neutral model as follows;
nn,j=nn,j+1exp(−Δs/λ),(7)
λ=λiz or λizλcx .(8)
The subscript jrepresents the mesh number and Δsis
the mesh width in the poloidal direction. The ionization
and charge exchange mean free paths are denoted by λiz
and λcx, respectively. Equation (7) is based on the steady
state continuity equation without volume recombination
source. The neutral decay length can be chosen from λiz
and √λizλcx as Eq. (8).
In order to focus on the effects of neutrals in turn, we
have introduced a time dependent self-consistent neutral
model instead of our previous simple neutral model as fol-
lows;
∂nn
∂t+∂Γn
∂s=−σizvnnne+σrc vnine−nn
τn
,(9)
Γn=α(nnvFC)+β−λcx vth
∂nn
∂s.(10)
Here, Γnis the flux of neutral particle, vFC is the veloc-
ity of neutral particles which have Franck-Condon energy
(3.2 eV) which has negative value and vth is the local ther-
mal velocity of the plasma. The neutral flux is assumed
to be composed of convection with a constant flow veloc-
ity vFC and diffusion involving the charge exchange reac-
tion [3] and each effect is controlled by changing the val-
ues of input parameters αand β. The third term in the right
hand side of Eq. (9) represents cross-field neutral particle
loss term whose time constant τnis an input parameter. By
adding the volume recombination source term newly, the
number of ions and neutrals become self-consistently bal-
anced.
At the stagnation point and the divertor plate we use
the following boundary conditions, respectively;
Γn,stag =0,(11)
Γn,div =−ηtrap(nV)div.(12)
Here, ηtrap is the recycling rate which is also an input pa-
rameter.
3. Results
In the following simulation results, the parameters
are chosen to be ASDEX Upgrade like [14]. The con-
nection length Lis 22 m and the position of X-point is
17.6 m. The particle and heat flux from the core plasma
are 6.0×1021 s−1and 4 MW, respectively. The area of the
separatrix magnetic surface is 40 m2and the thickness of
the SOL is uniformly 2 cm.
3.1 Effects in the neutral model
In order to investigate how the neutral profile is af-
fected by the effects introduced newly. First, we investi-
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Plasma and Fusion Research: Regular Articles Volume 8, 2403096 (2013)
Fig. 2 Neutral profiles for different cases; (a) diffusion-
dominant case, (α, β)=(0,1) (red), and convection-
dominant case, (α, β)=(1,0) (green), (b) with cross-field
neutral loss term (red) and without it (green).
gated the effect of time dependence. If we clear Eq. (9)
of the diffusion term, neutral loss term and the volume re-
combination term, it almost coincides with our previous
neutral model Eq. (7) except for the time derivative term.
In a steady state, the neutral profile in the new model gave
close agreement with that in our previous model.
Second, we compared the neutral decay length for
convection-dominant case, (α, β)=(1,0) with that for
diffusion-dominant case, (α, β)=(0,1) without the cross-
field neutral loss term in attached regime. Figure 2 (a)
shows that the neutral decay length becomes longer in the
diffusion-dominant model. This was caused by the accel-
eration of the neutral particles by the charge exchange re-
action.
Third, we compared the neutral profile with the cross-
field neutral loss term with that without it. The time con-
stant of the neutral loss term τnwas set to be 10−4sso
that it is comparable to the time constants of other terms.
Figure 2 (b) shows that the neutral density decreases in the
high density region near the divertor plate with the cross-
field neutral loss term.
3.2 Simulation of the detached regime
In the following results, we considered only diffusion
process in the neutral flux by setting (α, β)=(0, 1). In order
to simulate the detached plasma regime, we made the re-
cycling rate ηtrap higher. The time variation of spatial pro-
files of the plasma parameters are shown in Fig. 3. Here,
the neutral loss term is not included and ηtrap is set to be
95.4%. The detachment front reaches the X-point in a few
Fig. 3 Time variation of the spatial profiles of plasma parame-
ters in the detached regime.
milliseconds. The divertor density becomes about 2 orders
lower than the peak density. The divertor temperature be-
comes much lower than 1 eV. The neutral does not decay
because of the very low temperature in the detached region.
The flows of particles and heat in the detached regime
are also investigated. Figure 4 shows the spatial (a) particle
and (b) heat flux profile for t=0.69 ms. The particle flux
increases by the flux from the core plasma and ionization
reaction. Near the detachment front, because of the ade-
quately low temperature, the volume recombination domi-
nates the ionization so that the particle flux decreases. The
heat flux increases by the flux from the core in upstream
region, however, after the sum of ionization, impurity ra-
diation and charge exchange energy loss dominate it at x
=16.8, the heat flux starts to decrease gradually. At x=
20.1, the temperature becomes lower than 5.25 eV and the
volume recombination reaction changes into heat source.
Near the detachment front, the ionization and impurity ra-
diation energy loss rapidly decrease, but charge exchange
energy loss remains. In the detached region, the charge ex-
change energy loss dominates the volume recombination
energy source so that the heat flux decreases.
In these simulations of detached regime, steady state
has not been yet achieved, so that the detachment fronts
move upstream beyond the X-point. If we introduce the
cross-field heat transport, we might be able to make the
detachment front more stable [6–8].
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Plasma and Fusion Research: Regular Articles Volume 8, 2403096 (2013)
Fig. 4 Spatial (a) particle and (b) heat flux profile. Red lines
represent local particle and heat flux. Blue input from the
core plasma, pink ionization, aqua (absolute) recombina-
tion, yellow charge exchange, black impurity radiation.
Detachment front is denoted by the vertical broken green
line.
Fig. 5 Conditions of recycling rate and cross-field neutral loss
time constant for detached regime to occur.
We also investigated the conditions of the recycling
rate ηtrap and cross-field neutral loss time constant τnwhere
the detached regime occurred. The conditions are shown in
Fig. 5. If we set τnto be negative value, the third term in
the right hand side of Eq. (9) changes from loss term into
source term, so that less recycling rate is needed for the de-
tached regime to occur. It implies that, in multi-layer anal-
yses, PDD plasmas might be reproduced by setting condi-
tions of the cross-field neutral transport. This is also our
future work.
4. Conclusion
In the one-dimensional SOL-divertor model, a self-
consistent neutral model has been introduced. Detached
divertor plasmas have been successfully reproduced and
the flows of particles and heat from the core were investi-
gated in single tube analyses. The conditions of recycling
rate and time constant of cross-field neutral loss term for
the detached regime to occur were also presented.
Current neutral model doesn’t consider that the veloc-
ity of neutrals produced by volume recombination is af-
fected by the velocity of original plasma particles and that
neutrals are reflected at the divertor plate. Also neutrals
should be divided into two or more generations according
to whether they have experienced the charge exchange re-
action or not. Such improvements are our future works.
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