23P133-B
Numerical analysis of axisymmetric toroidal equilibria with flow
流れをもつ軸対称トロイダル平衡の数値解析
Atsushi Ito and Noriyoshi Nakajima
伊藤 淳, 中島徳嘉
National Institute for Fusion Science322-6 Oroshi-cho, Toki 509-5292 Japan
核融合科学研究所 〒509-5292 土岐市下石町322－6
Numerical analysis of axisymmetric toroidal equilibria with flow is performed based on single-fluid andtwo-fluid magnetohydrodynamic (MHD) models. Effects of toroidal and poloidal flow comparable to thepoloidal sound velocity, two-fluid, ion finite Larmor radius (FLR), pressure anisotropy and parallel heatfluxes on high-beta toroidal equilibrium is studied by solving reduced MHD equilibrium equations. Higher
order terms of quantities like the pressures and the stream functions show the shift of their isosurfaces fromthe magnetic surfaces due to effects of flow, two-fluid and pressure anisotropy.

1. Introduction
We investigate small scale effects on equilibriumwith flow based on reduced magnetohydrodynamic(MHD) models. At the sharp boundary of awell-confined region in magnetically confinedplasmas where high-beta is achieved by shear-flow
suppression of instability and turbulent transport,
small-scale effects cannot be neglected. A reduced
set of Grad-Shafranov (GS) type equilibrium
equations for high-beta tokamaks with flow
comparable to the poloidal sound velocity, ion
finite Larmor radius (FLR), pressure anisotropy andparallel heat fluxes on high-beta tokamaks
equilibrium has been derived from the fluid
moment equations for collisionless, magnetizedplasmas [1]. We show the results of numericalsolutions of the reduced GS equations by means of
the finite element method. The two-fluid effects
induce the diamagnetic flows, which result inasymmetry of the equilibria with respect to the signof the E×B flow. Higher order terms of quantities
like the pressures and the stream functions show theshift of their isosurfaces from the magnetic surfaces
due to effects of flow, two-fluid and pressure
anisotropy.

Grad-Shafranov (GS) equation of which the firstorder is same as that for static equilibria [1],
 ∂2
∂2 
 2 + 2 ψ 1
∂Z  ∂R
(1)
2 ′
Ix=− µ0 R02   ∑ ( ps′||1 + ps′⊥1 ) + g*′  −  1  , R0  s =i , e  2 and the second order is given by ∂2∂2 
 2 + 2 ψ 2∂Z  ∂R x 
 I 2 ′′+  µ0 R02   ∑ ( ps′′⊥1 + ps′′||1 ) + µ0 R02 g*′′ +  1  ψ 2
 2 
 R0  s =i , e∇ψ 1 ∂21 ∂ψ 1
∂2 =+ F  2 + 2 ψ 1 + F ′R ∂R2
∂Z 
 ∂R2


 x − µ0 R02  E*′ +   ∑ ( Ps′⊥ 2∗ + Ps′||2∗ ) R0  s =i , e(2)21 x +   ∑ ( ps′⊥1 + ps′||1 + Cs′⊥1 + Cs′||1 )  ,2  R0  s =i , ewhere2. Equilibrium Equations−1
 B02  
The reduced set of equilibrium equations are
λ − λiF (ψ 1 )    mi n0 R02  Φ1′ + Hpi′⊥1 derived from the fluid moment equations for =en0
 µ0  collisionless, magnetized plasmas with the(3)slow-dynamics (drift) ordering [2] by theλHpi′⊥1  + ∑ ( ps1 − ps ⊥1 ) ×  Φ1′ +asymptotic expansions for large aspect ratio,en0 s =i , ehigh-beta tokamaks with poloidal sonic flow. The
includes contributions from the E×B and the ion
FLR effects appear as the gyroviscosity and thediamagnetic poloidal flows with the gyroviscousperpendicular (diamagnetic) heat fluxes in the fluid
cancellation and the pressure anisotropy. The
moment equations. The equilibrium equations
coefficients C…(ψ1) are obtained by solving theconsist of the first two orders of theequations for the higher-order quantities asfunctionals of the lowest order quantities while
those in the second terms, denoted by ‘*’, arearbitrary functions of ψ1. The pressure and otherquantities are determined once Eqs. (1) and (2) aresolved. This set of equilibrium equations is anextension of previous models [3,4] to include
pressure anisotropy with parallel heat flux. The GS
equation for ψ1, (1), is same as for the single-fluid,static case. In spite of its complexity, the GS
equation for ψ2, (2), is a linear, elliptic partial
differential equation once the solution for ψ1 of (1)is substituted and, thus, is easy to solve. However,singularity in the GS equation for ψ2 occurs when
the poloidal flow velocity equals the poloidal soundvelocities. It arises because higher order terms not
negligible in its vicinity are ordered out in theasymptotic expansions. The present model is tostudy the extension of regular, elliptic solution forsingle-fluid MHD equilibria with flow. One has tochoose the profiles of free functions that do notinclude the vicinity of singularity to get regularsolutions.
1.2
10.8
0.6
0.4
0.2
00.7
0.6
0.5
0.40.3
0.2
0.1
00.1

(a) ψ

3. Numerical Solutions
The reduced GS equations are solved numericallyby means of the finite element method. The circularcross-section, the fixed boundary condition at thenormalized minor radius r=1 and the up-down
symmetry are assumed. Figure 1 shows the radial
profiles at the midplane of (a) the magnetic flux ψ,(b) the total pressure p, (c) the ion stream function
Ψ and (d) the parallel and perpendicular pressures
for ions and electrons. The two-fluid effects induce
the diamagnetic flows, which result in asymmetryof the equilibria with respect to the sign of the E×B
flow. Higher order terms of quantities like thepressures and the stream functions show the shift oftheir isosurfaces from the magnetic surfaces due toeffects of flow, two-fluid and pressure anisotropy.
The parallel and perpendicular components of thepressures for ions and electrons [Fig. 1 (d)] areself-consistently determined and show their peaks
in different positions from those of the magnetic
flux and each other. The shift of the isosurfaces of
the ion stream function from the magnetic flux iscaused by the breaking of the frozen-in condition
due to the two-fluid effect. However, we havefound that it also depends on the FLR effect. Figure
2 shows that these shifts in the FLR two-fluid andthe Hall MHD models are in opposite radial
directions.
1(b) p

0.06

(b)

(a)

0.04

0.5

0.02

Z 00-0.02

-0.5

-0.04

-1

(c) Ψ-0.06
-1

-0.5

0x

00.5

1-1-0.5

0x

0.5

1-0.1

Fig. 2. Profiles of (∇ψ ×∇Ψ)⋅(R∇ϕ) in the poloidal
cross-section for (a) the FLR two-fluid and (b) the Hall
MHD models. This quantity is always zero in thesingle-fluid MHD model since Ψ= Ψ(ψ).

-0.2
-0.3
0.6
0.5
0.4
0.3
0.2
0.10(d)pe||pe⊥pi||pi⊥
-1

-0.5

00.5

1xFig. 1. Radial profiles in the midplane obtained fromnumerical solutions for the FLR two-fluid model. The
solid (dashed) lines show the case where the poloidal
directions of the E×B and the ion diamagnetic flows are
the same (opposite).

References[1] A. Ito and N. Nakajima, Nucl. Fusion, in press.
[2] J.J. Ramos, Phys. Plasmas 12 (2005) 052102.
[3] A. Ito and N. Nakajima, AIP Conf. Proc. 1069 (2008)
121.
[4] D. Raburn and A. Fukuyama, Phys. Plasmas 17(2010) 122504.