24P030-P
Thermal Stability Analysis of Detached Divertor Plasma
非接触ダイバータプラズマの熱的不安定性に関する解析
aa

Masanori Ito, aSatoshi Togo, bMakoto Nakamura and aYuichi Ogawa
伊藤政範，東郷 訓，中村 誠，小川雄一

The Graduate School of Frontier Science, Tokyo University5-1-5, Kashiwanoha, Kashiwa 277-8568, Japan
Japan Atomic Energy Agency, 2-166 Oaza-Obuchi-Aza-Omotedate, Rokkasho, Kamikita, Aomori 039-3212a
東京大学大学院新領域創成科学研究科 〒277-8568 柏市柏の葉5-1-5b
日本原子力研究開発機構 〒039-3212 青森県上北郡六ヶ所村大字尾駮字表舘2-166b

One-dimensional transport equations for plasma fluid are used to investigate the thermal
stability in the region of divertor. Equilibrium equations and perturbation equations are combined
into the linear differential equations for each perturbation. We consider effects of radiation from
impurities, charged particle-neutral interactions, recombination and ionization. Neutral
density and the rate parameters of charged particle-neutral interactions and recombination arerelated to the stability of divertor plasma.
1. Introduction
Divertor heat loads should be reduced forfusion reactors with a large amount of heatingpower. One of the solutions is the plasma
detachment. The feature of the detachment is a
sharp reduction of the temperature in the divertor
region. It has been observed in some experiments.
[1] The Plasma detachment is triggered by effectsofradiation
fromimpurities,
chargedparticle-neutral interactions, recombination andionization. Recent study has simulated thedivertor plasma with the 1-D fluid equation, andshown the attached and detached states in thedivertor region.
[2] The interactions with
Neutral particles can cause instability of thedetached divertor plasma. Figure 1 shows theposition of the detachment front as a function of
the neutral particle density at the divertor plate.
When neutral density is high( nn  3.98 1019 m-3),
plasma detachment occurs.

instability for the plasma detachment in thedivertor region.
2. Formulation
In past studies [3], the analysis was based on theheat balance equation. Perturbation of temperature
wasintroduced to investigate the MARFE
instability.
We use the fluid equations for plasma and theneutral particle diffusion equation as follows.
nn  n    Dn n, T  n   n 2 v rec  nnn v ion
t x x (1)n nv  n 2 v rec  nnn v ion
tx

(2)

mnv  mnv2  2nT
 mnv nn v CX  n v rectx




1
 
1
T 
2
2
 3nT  mnv    5nT  mnv v   || et 22x  x 

(4)

3 
1
1
  mv2  T nnn v CX   mv2  3T n 2 v rec  I ionnnn v ion  nnn L(T )2 22From these equations, a set of equations of the
1st order perturbations of neutral density, plasma
density, momentum, and temperature are derived.
Equilibrium equations (5) and perturbation
equations (6) are combined into the linear
differential equations (8) for each perturbation.

f nn , n, v, T   0~
f n  n~ , n  n~, v  v~, T  T  0

X
 AX
t
Fig.1.movement of detachment front

In this study, we will propose a stability model
based on the 1-D fluid equations. It describes(3)nn

 n~n  a11~n a21X
Where v~  , A   a ~ 31
T a  41a12a22a32
a42(5)(6)

a13 a14 a23 a24 (7)a33 a34 a43 a44 

Spatial perturbations are assumed as follows.

~n~n  eiknnx , n~  eikn x , v~  eikv x , T  eikT x

Here, wavenumber k is assumed as follows.

(8)k nn 

1
1 ,
1 ,
1 ,
kT kv kn LTLv
Ln
Lnn

(9)

64 x 103

固有値最大値

Lb is the decay length of the variable b on thedetachment front. is a free parameter and here
assumed to be 1/10 when maximum eigenvalue
become zero in the case of steady detachment front
as shown in fig.2.

the case of varying charge exchange, smaller rates
reduce instability. Larger rates of recombination
reduce the eigenvalues, but there is a limit to thesmall instability. The effect of varying charge
exchange rates is larger than recombination rates.

210-10246<v>cx / <v>cx*810Fig.4. Eigenvalue for varying charge-exchange rate
53.5 x 10

Fig.2. Factor of wavenumver

We investigate the instability of plasma
detachment by evaluating the eigenvalue ofequations (7). In this study, ITER-like plasma
parameters are assumed.

固有値最大値

32.5
2
1.5
10.5
00

1

2

3

4

5

<v>rb / <v>rb*

3. Result
3-1 Neutral particle density
Figure 3 shows the maximum eigenvalues near
the divertor plate. In the case of small neutral
particle density ( nn  3.98 1019 m-3), these arenegative values. These indicate the attached orsteady detachment plasma in the condition. Whenthe neutral density is higher than 3.98 1019 m-3,
eigenvalues become positive. In this condition,
plasma detachment becomes unstable.
2.5x 105

2
1.5
1
0.5
0-0.5
3.8

3.9

4
4.1
4.2
neutral density（m-3）

4.3
4.4
x 1019Fig.3. Eigenvalues for varying neutral density

3-2 Charge exchange and recombination
In order to investigate the effects of rate
parameters, each rate parameter was changed as
free parameter. Figures 4 and 5 shows theeigenvalues as functions of the rate parameters ofcharge exchange and recombination. Ionization andimpurity radiation loss were also taken into account,but these were found not affect the eigenvalues. InFig.5. Eigenvalue for varying recombination rate

4. Discussion & Conclusion
We proposed the analytical model of thethermal stability of the detachment fronts in which
perturbations of neutral density, plasma density,momentum, and temperature were taken intoaccount. This model enables to evaluate Instability
in the divertor region. In the state of the attach
plasma or steady detached plasma, the systemequation represent the negative eigenvalues; Thestates are stable. In the state of the unsteadydetached plasma, eigenvalue of the systemequation is positive; the state is instable. The
Neutral density and the rate parameters of charged
particle-neutral interactions and recombination
are related to the stability of divertor plasma.
The stability analysis in this paper takes into
account the time perturbation and assumed
wavenumber. On the other hand, Wesson et al.
proposed another analysis. The stability problem istime-independent and omits the space variable. We
apply Wesson’s theory to the plasma fluid
equations and neutral transport equation.
References[1] G. F. Matthews: J. Nucl. Mater. 220-222 (1995) 104[2] M. Nakamura et al.: J. Nucl. Mater. (2011).
[3] J. A. Wesson et al.: Nuclear Fusion. 33 (1993) 1019
[4] N. Ohyabu: Nuclear Fusion. 9 (1979) 1491