24C06
Analysis of Partially Detached Divertor Plasmas with Multi-Layer 1D Model
多層型一次元モデルを用いた部分非接触ダイバータプラズマ解析
aSatoshi Togo, bMakoto Nakamura, cKatsuhiro Shimizu, cTomonori Takizuka, bKazuo Hoshino andaYuichi Ogawaabc
東郷 訓, 中村 誠， 清水勝宏，c滝塚知典，b星野一生，a小川雄一

aDepartment of Advanced Energy, Graduate School of Frontier Science, the University of Tokyo 5-1-5, Kashiwanoha,
Kashiwa 277-8568, Japanb
Japan Atomic Energy Agency, 2-166 Oaza-Obuchi-Aza-Omotedate, Rokkasho, Kamikita, Aomori 039-3212
cJapan Atomic Energy Agency, 801-1 Mukoyama, Naka, Ibaraki 311-0193
a277-8568
5-1-5b039-3212
2-166
c311-0193
801-1

東京大学大学院新領域創成科学研究科 〒千葉県柏市柏の葉
日本原子力研究開発機構 〒青森県上北郡六ヶ所村大字尾駮字表舘
日本原子力研究開発機構 〒茨城県那珂市向山

We have developed a "multi-layer (ML)" 1D model for investigating behaviors of partially detacheddivertor (PDD) plasmas. The basic idea is to put the detached and attached flux tubes adjacent each other,so that PDD plasmas that are multi-dimensional phenomena can be represented in one-dimensional results.
Cross-field transport terms are evaluated as source terms in each partial differential equation. We show thecontents of the ML 1D model and explain briefly the role of each section. We also show some results ofpreliminary simulations that insure the validity of the code.

1. Introduction
Reduction of the divertor heat load is one of thecrucial issues in designing the next generation
tokamaks such as ITER and DEMO. In order toresolve this issue, operating under partiallydetached divertor (PDD) plasmas is considered tobe essential [1]. It has been shown by experimentsthat PDD plasmas are preferred to fully detacheddivertor plasmas with respect to thermal stability[2], so that PDD plasmas have been adopted for
ITER operation scenarios [3].
In order to model PDD plasmas, many
two-dimensional codes have been developed. It isconsidered, however, that somewhat simpler codessuch as one-dimensional codes [4, 5] are adequate
to gaining physical insight of PDD plasmas because
2D codes are computationally massive andsometimes inconsistency with experimental results
was found [6]. While 1D codes are useful andcomprehensible because of its simplicity, they are
essentially unable to model PDD plasmas due totheir multi-dimensionality.
The ‘multi-Layer (ML)’ 1D model we propose
here has been developed for the purpose of solving
this dimensional problem by treating cross-fieldtransport terms as source terms in each partial
differential equation. Detached and attached flux
tubes are put adjacent to each other as shown in
Fig.1, so that interactions between these tubes canbe evaluated. In addition, the particle and energy
transport to the first wall can be taken into account.
We make brief explanation of this model and show

some results of preliminary simulations.
2. Model2.1 Plasma fluid model
The 1D transport equations are given as follows[7];∂ (mn ) ∂ (mnv )
+= mS
∂t
∂x
∂ (mnv ) ∂ mnv 2 + P+=M
∂t
∂x
∂ 1

2
 mnv + 3nT ∂t  2


()

(1)
(2)

(3)

∂T ∂  1
+  mnv 2 + 5nT v − κ e=Q∂x ∂x  2Here, the density n , the flow velocity v , thetemperature T of ions and electrons are assumedto be equal, respectively. P (= 2nT ) is the plasmapressure and κ e is the parallel electron heat
conductivity.
At the stagnation point (x=0), we use thefollowing symmetric boundary conditions:∂n∂T= 0, v = 0,=0
∂x
∂x(4)

At the divertor target (x=L), we use the following
boundary conditions:

Ms =v=1cs(5)

Fig.1. Schematic picture of the multi-layer 1D model.

1∂Tq heat =  5nT + mnv 2 v − κ e= γnTc s(6)2∂xHere, M s is the Mach number, c s is the soundspeed, q heat is the heat flux and γ ( ≈ 6.5) is thesheath energy transmission factor. These boundary
conditions are the same as B2 code.
2.2 Neutral diffusion model
We use the simple 1D neutral diffusion model:(nn vn ) j −1 2 = (nn vn ) j +1 2 exp − ds  λion , j (7)Here, s is the coordinate along the poloidal
direction and λion is the mean free path ofionization.
At the divertor target, we use the following
condition:(8)(nn vn ) = η trap (nv )div sin θdivHere, η trap is the recycling rate and θ is theFig.2. Correlation between plasma parameters and input
flux amplitude factor Rin. The subscripts ‘0’, ‘x’, ’d’represent the stagnation point, the X-point, the divertor
plate, respectively.

pitch of the magnetic field.
consistent with that of two-point model [8].
3. Results
Prior to simulation studies using the ML 1D
model, we checked validity of the ML1D plasma
fluid code. In this simulation, ASDEX-like plasma
parameters are assumed. The particle and heat flux
from core plasma are set to be 6.0×1021 [s-1] and4.0MW, respectively. The pitch of magnetic field isfixed at 2.0 degrees. The calculated density,temperature and output flux amplitude factor asfunctions of the input flux amplitude factor
Rin = 1 1 − η trap are shown in Fig.2.

( ())

The output flux amplitude factor is consistent
with the input flux amplitude factor. The correlation
between plasma parameters and Rin is also

References[1] ITER Physics Expert Group on Divertor, Nucl.
Fusion 39 (1999) 2391.
[2] G.F. Matthews, J. Nucl. Mater. 220–222 (1995) 104.
[3] L. Loarte et al., Nucl. Fusion 47 (2007) S203.
[4] R. Goswami et al., Phys. Plasmas 8 (2001) 857.
[5] W. Fundamenski et al., J. Nucl. Mater. 290–293(2001) 593.
[6] M. Wischmeier et al., J. Nucl. Mater. 390–391(2009) 250.
[7] S. Nakazawa et al., Plasma Phys. Control Fusion 42(2000) 401.
[8] K. Shimizu et al., J. Plasma Fusion Res. 80 (2004)
189.