23P053-P
Simulation of two-dimensional transport in tokamak plasmas for
integrated analysis of core and peripheral plasmas
トカマクプラズマの統合解析に向けた二次元輸送シミュレーション
Haruki Seto, Atsushi Fukuyama
瀬戸春樹, 福山淳
Department of Nuclear Engineering, Kyoto University, Sakyo-ku, Kyoto 606-8501, JAPAN京都大学工学研究科原子核工学専攻 〒 606-8501

京都市左京区吉田本町

In order to describe the behavior of tokamak plasmas in both core and peripheral regions selfconsistently, two-dimensional transport modeling is desirable and becoming feasible. In the presentstudy, we have formulated more rigorous transport equations with poloidal-angle dependence from
Braginskii’s equations for two-dimensional transport analysis. The set of equations is composedof continuity equation, equation for velocity including the neoclassical viscosity, and equation forenergy transport for each species. Preliminary numerical results of two-dimensional transport
analysis will be presented.

1 Introduction
In most of conventional core transport simulations in tokamaks, the particle density andthe temperature are assumed to be almost constant on a magnetic surface, and flux-averaging
method is employed to describe transport phenomena as a one-dimensional problem, since thetransport along the field line is very fast. On
the other hand, transport in a peripheral SOLdivertor plasma is usually described as a twodimensional problem with simplified transportmodels and plasma flow, for example SOLDOR[1] and B2.5 [2], since variation of quantities
along the field line is large and important tounderstand transport process in peripheral region. Recent remarkable progress in computational resources, however, has made more rigorous two-dimensional simulation of tokamak plasmas feasible. In the present study, we formulate a
set of two-dimensional transport equations with
the neoclassical viscosity [3] in magnetic flux coordinate system (MFCS) (ξ1 , ξ2 , ξ3 ) = (ρ, χ, ζ)from Braginskii’s equations [4] to develop an
integrated two-dimensional transport simulation
code for the entire tokamak plasmas including
both core and peripheral plasmas.

2 Assumptions
In the present study, the following four as-sumptions have been made to derive the transport equations. The first is that the plasma has
toroidal axisymmetry, which means that all physical quantities are independent of the toroidal
angle variable. The second is that the quantities related to MHD equilibrium depend only
on the flux label ρ; the poloidal flux function
ψ, the toroidal flux function ψT , the toroidal
current function I, the electrostatic potential φ,
the total plasma pressure p, and the rotationaltransform ι! . This assumption makes it possible
to reduce the magnetic equilibrium problem totwo-dimensional. The third is that phenomena
with the Alfvén time scale are much faster than
the relaxation process such as diffusion of magnetic field and transport phenomena, which implies that the MHD equilibrium is attained much
faster than the relaxation processes. Finally, thefourth is that the time derivatives of basis vectors are small enough to be ignored for simplicity
when we take the time derivative of vector quantities.

3 Modeling of transport equations
The transport equations are derived from Braginskii’s equations [4] and consist of the equation for particle density, momentum, and energy
transport for both electron and ion in MFCS.
The equation for particle density is the equation

of continuity of Braginskii’s equations.
The equation for momentum is derived fromequation of motion of Braginskii’s equation. Forthe compatibility with the neoclassical transport
theory [3], we take three components of vector quantities (ξ1N , ξ2N , ξ3N ) = (ρ, !, ζ), which indicates the radial direction, the field line direction and the toroidal direction respectively,NNN(eξ1 , eξ2 , eξ3 ) = (∇ρ, B/B, ∇ζ). And we assume that the force balance is attained in the radial and toroidal direction in the transport timescale. Therefore, taking the scalar product of theNequation of motion and eξi , we obtain equations
for momentum in each direction as follows.
1. radial direction
33!
∂pa ! 1 1i ∂Naneo
0 = −Fakin,1 −g 1i−g∂ξi3
∂ξii=1+ Naneo κρ − g 11 ea na

∂φ+∂ρ

i=13!

ξN

4 Preliminary simulation results

i=1

(1)

2. parallel direction3!
#
ψ " ∂pa∂ "ma na ua! = −Cakin,i Fakin,i − √∂tgB ∂χ
i=1∂ ln B 2 ψ " ∂Naneoψ"− √ Naneo+ √gB∂χ3 gB ∂χ
#me n e "ψ"∂Te∓
ue! − ui! ∓ 0.71 √ neτegB ∂χ(2)3. toroidal direction∂ ln BIψ " ∂NaneoIψ "0 = −Fakin,3 − √ 2 2 Naneo+√ 2 2gB R∂χgB R ∂χ"ea ψ+ Naneo κζ + 2 na uρa$ R'&#me n e % ζI "∓ue − uζi − 0.49
u−ui!
τeBR2 e!
()3Iψ "∂Te ! 3 ψ " g 1i ∂Te∓ 0.71 √ 2 2 ne+negB R∂χ2Ωe τe BR2 ∂ξii=1+ ma Sa uζa(3)√kin,i,iLor
where Fa , g ij , g, Naneo , Ca and Ca are thecontravariant component of kinetic stress force,the contravariant metric tensor, Jacobian, quantity from the neoclassical viscosity force, the coefficient of Lorentz force term and the coefficientof kinetic stress force in MFCS respectively.
kin ,iThe set of transport equations is coupled witha set of equations for electromagnetic field inorder to develop an integrated two-dimensionaltransport simulation code. The set of electromagnetic equations consists of Grad-Shaftranov
equation, magnetic diffusion equation [5], andPoisson equation for static electric field. The twosets of equations are reduced to two-dimensionalwith axisymmetry and the finite element methodis used to discretize the differential equations.

CaLor,i na uai

3I∂Te∓+ ma Sa uρa√ ne2Ωe τe gB ∂χ+ ma Sa ua!

The equation for energy transport is obtainedby transforming Braginskii’s equation for energy
transport into the advection-diffusion form%&↔
3 ∂pa= −∇ · pa upa − na χ a · ∇Ta + Qpa (4)2 ∂twhere upa ≡ 52 ua + p−1
a q ua is the energy flow ve3+ a−↔
π a · ∇ · ua + Qa
locity, Qpa ≡ 2 Ta Sa + ua · ∇p↔
the energy source, and χ a the diffusion tensor.

We will present preliminary numerical resultsof two-dimensional transport analysis, which weemploy toroidal coordinate system (-, θ, ϕ) instead of magnetic flux coordinate system.

Acknowledgment

This work is supported by Grant-in-Aid for
Science Research (S) (202260171) from JSPS and
G-COE program (J-051) from MEXT

Reference[1] K. Shimizu et. al, J. of Nucl. Mater. 313316(2003) 1277-1281[2] V. A. Rozhansky et. al, Nucl. Fusion Vol. 41387-401 (2001)[3] S.P. Hirshman and D.J. Sigmar, Nucl. Fusion
Vol.21 No.9 (1981) 1079
[4] S. I. Braginskii, in Reviews of Plasma Physics,
Vol. 1 (Leontovich, M. A., Ed.), Consultants
Bureau, New York (1965) 205.
[5] F. L. Hinton and R. D. Hazeltine, Rev. of Mod.
Phys. Vol. 48 (1976) 239