合同会合＠筑波大学 2014/08/01	

非等方イオン温度と仮想ダイバータモデルを導入したSOL-ダイバータプラズマシミュレーション
Satoshi Togo, Tomonori Takizukaa,
Makoto Nakamurab, Kazuo Hoshinob, Yuichi Ogawa	Graduate School of Frontier Sciences, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa 277-8568, Japan
aGraduate school of Engineering, Osaka University, 2-1 Yamadaoka, Suita 565-0871, JapanbJapan Atomic Energy Agency, 2-166 Omotedate, Obuchi, Rokkasho, Aomori 039-3212, Japan	Introduction	
Edge transport code packages, such as SOLPS and SONIC, arewidely used to predict performance of the scrape-off layer (SOL) anddivertor of ITER and DEMO. Simulation results, however, have notsatisfactorily agreed with experimental ones.
Comparison of results (EXP vs SIM)

M. Wischmeier et al., J. Nucl. Mater. 390-391, 250 (2009).	

Introduction	
The parallel ion viscosity term in the momentum transport equation isan approximated form of the ion pressure anisotropy which is derived
under the assumption that the temperature anisotropy is small enough.
rigorous equation with anisotropic∂m nV ∂+ (m nV + nT + nT ) = M
T and no viscosity term	∂t∂x2iii , //einTi = n(Ti , // + 2Ti ,⊥ ) 3

nTi, // = nTi + π i

π i = 2n(Ti , // − Ti ,⊥ ) 3nTi,⊥ = nTi − π i 2variabletransformation	

∂mi nV ∂∂π+
mi nV 2 + nTi + nTe + i = M
∂t
∂x
∂x()

∂mi nV ∂∂ ⎛∂V ⎞
conventional equation with+
mi nV 2 + nTi + nTe + ⎜ − ηi , //⎟ = M
isotropic Ti and viscosity term	
∂t∂x∂x ⎝∂x ⎠()

Kinetic simulations showed remarkableanisotropy in the ion temperature even forthe medium collisional regime.

A. Froese et al., Plasma Fusion Res. 5 (2010)026.

Boundary condition at the divertor plate is necessary tosolve this Eq. and the Bohm condition M = 1 is usuallyadopted, while rigorous Bohm condition only imposes thelower limit of M ≥ 1.

M = 1	

Method (plasma fluid equations)	
By introducing the anisotropic ion temperature
directly into the fluid modeling, the secondderivative parallel ion viscosity term can beexcluded because the ion pressure term can bewritten by its parallel component nTi,//.

Eq. of continuity∂mi n ∂mi nV+=S∂t∂x

Eq. of momentum transport∂mi nV ∂+mi nV 2 + nTi , // + nTe = M∂t∂x()

Eq. of parallel (//) ion energy transport∂qieff∂ ⎛ 113⎞ ∂ ⎛ 1⎞, //22⎜ mi nV + nTi , // ⎟ + ⎜ mi nV + nTi , // ⎟V + c∂t ⎝ 222∂x⎠ ∂x ⎝ 2
⎠n(T − T ) m n(Te − Ti , // )∂nTe= Qi , // + i ,⊥ i , // + e−Vτ rlxmiτe∂x

Momentum transport equation becomes
1st order differential equation.
Therefore the boundary condition at thesheath entrance becomes unnecessary.

Eq. of perpendicular (⊥) ion energy transport∂nTi ,⊥∂t+∂nTi ,⊥V∂x+ (1 − c )∂qieff,⊥∂x= Qi ,⊥ −n(Ti ,⊥ − Ti , // ) 2me n(Te − Ti ,⊥ )
+
τ rlx
mi
τe

Eq. of electron energy transport∂qeeff3m n(Ti − Te )∂nTe∂ ⎛ 3⎞ ∂ ⎛ 5⎞= Qe + e+V⎜ nTe ⎟ + ⎜ nTe ⎟V +∂t ⎝ 2∂xmiτe∂x⎠ ∂x ⎝ 2⎠

c = 0.5
Flux limiting coefficient:
αi,// = αi,⊥ = 0.5, αe = 0.2
ion temperature anisotropy
relaxation time: τrlx = 2.5τi(E. Zawaideh et al., Phys. Fluids 29 (1986) 463)

are used in the present study.

Method (1D system with Virtual Divertor Model)	
In order to express the effect of thedivertor plate, a “virtual divertor(VD)” model is introduced which has
an artificial sinks of particle,
momentum and energy.
This VD model makes the periodic
boundary condition available.
The model of sink in the VD region:
S VD = −QiVD, // = −QiVD,⊥ = −mi nτ VDM VD = −mi nVτ VD+∂ ⎛∂V ⎞⎜η VD⎟∂x ⎝∂x ⎠

1 ⎛ 11⎞2⎜ mi nV + g i , // nTi , // ⎟τ VD ⎝ 22⎠
g i ,⊥ nTi ,⊥τ VD

QeVD = −1 ⎛ 3⎞⎜ g e nTe ⎟τ VD ⎝ 2⎠

Artificial viscosity is necessary for the velocity profile toanalytically connect in the divertor region.

decay time τVD = 5×10-6 s
artificial viscosity coefficientηVD = 1×10-5 Pa·s
VD cooling coefficient:gi,// = 1, gi,⊥ = 1.2, ge = 2.5(in center of VD, gi,// >> 3)
are used in the present study.

Results (Bohm condition and Ti anisotropy)	
Parameters of a symmetric SOL-divertor system	 Profiles of SOL-div. plasmas
Length of plasma region, VD region, divertor region
L = 44 m, LVD = 6 m, Ldiv = 4.4 m(X-point ~ plate)	
SOL width, Surface area of the separatrix	
dSOL = 2 cm , Ssep = 40 m2	

n (1019 /m3)	

Mesh size, Time step, Calculation time (quasi-stationary)Δx = 2 cm, Δt = 2×10-8 s, t = 2×10-3 s 	M	Γsep (particle flux) and Psep (heat flux)
are given in a wide range	Ti,// (eV)	

Bohm condition is satisfied automatically.
Ion temperature
quickly shifts toanisotropic evenfor λmfp/L > 0.01.
λmfp = vth ,i , //τ i

Ti , //(at x = 20.01 m)	
collisional caseTi,⊥ (eV)	

Ti,⊥
weakly collisional caseTe (eV)	

Results (validity of the approximated ion viscosity)	
Two kinds of approximated parallel ion viscosity,source-less(x = 20.01 m)	

source region(x = 17.59 m)	

πBRi(η

i , //∂V= −ηi , //∂x= 0.96nTiτ i )⎛ 11 ⎞⎟⎟
π ilim = ⎜⎜ BR +βnTi ⎠⎝ π i−1

β = 0.5 :viscosity limiting factor

are compared to the ion pressure anisotropy,
πidef = 2n(Ti,//-Ti,⊥)/3, in the source-less and(x = 20.01 m)	
source region.
(x = 17.59 m)	

In the source region, the correlation between
πilim and πidef becomes worse.

with source	 w/o source	In the collisional regime, the limited
viscosity πilim is dominated by the Braginskii
expression πiBR and much affected by thegradient of V which is brought by theparticle source.

πilim/πidef	Conclusions	・A one-dimensional SOL-divertor plasma simulation code has been developed byintroducing the anisotropic ion temperature and a virtual divertor (VD) model andwe have been successfully obtained the SOL-divertor plasmas satisfying therigorous Bohm condition, M ≥ 1.
・Simulations for various normalized ion MFPs, λmfp/L, have been conducted andshow that the ion temperature shifts quickly from isotropic to anisotropic evenfor λmfp/L ~ 0.01.
・The validity of the approximated parallel ion viscosity has also been investigated.
The limited parallel ion viscosity shows a good proportional relation to the definitionof the parallel ion viscosity in the source-less region, but the correlation becomesworse in the source region.
There remain some issues which are not treated with in the present study.
★Introduction of the plasma current, which does not flow in the symmetric system,
to study the general asymmetric system★Introduction of the radiation model and neutral model to investigate the highrecycling cold-and-dense plasmas and detached plasmas.
These issues will be discussed in the future papers.