23P060-P
Flute instabilities and associated radial transport in the divertor
and dipole regions of the GAMMA10 A-divertor
GAMMA10 A-divertor におけるダイポール部を含めた系でのフルートモー
ドと径方向輸送
Isao Katanuma, Shun Masaki, Shuhei Sato, Kazuto Sekiya, Youhei Okuyama, Toshiki Horita,
Makoto Ichimura, Tsuyoshi Imai
片沼伊佐夫、真崎俊、佐藤周平、関谷和人、奥山洋平、堀田俊樹、市村真、今井剛
Plasma Research Center, University of Tsukuba
1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8577, Japan
筑波大学・プラズマ研究センター、〒 305-8577 茨城県つくば市天王台 1-1-1
The numerical simulation on the ﬂute instabilities was performed in the GAMMA10A-divertor magnetic conﬁguration. Here the dipole region is included in the calculations as well as the divertor region. The dipole region has been found to have anstabilizing eﬀects on the ﬂute instabilities in the GAMMA10 A-divertor.

1. Introduction
The GAMMA10 tandem mirror is planning
to introduce an axisymmetric divertor mirror
cell in it, where one of the anchor mirror cellis replaced by the divertor mirror as shown inFig.1. The main purpose to containing the divertor is to perform such a simulation experiment as a divertor installed in big torus. The
divertor mirror is an axisymmetric mirror stable to the ﬂute mode, where the stability is dueto the plasma compressibility[1]. The pressureradial proﬁle of plasma has to be controlled inorder to achieve the stable plasma in the divertor mirror such that ν(ψ)U γ (ψ) = const radially, where γ the speciﬁc heat index, U thespeciﬁc volume of a magnetic ﬁeld line deﬁned
by[2],U (ψ) =p̂⊥ (χ) + p̂ (χ)
dχB2Here plasma pressure is assumed to be represented by a separation of variables p(ψ, χ) =p̂(χ)ν(ψ) and B = ∇ψ × ∇θ = ∇χ. The stability criterion of ﬂute modes in a mirror is givenby∂U (ψ)1=−∂ψ2

(p̂⊥ + p̂ )κψdχ ≤ 0B2

Here κψ is the normal component of magnetic
ﬁeld line curvature κ = κψ ∇ψ + κθ ∇θ. Thecriterion ∂U/∂ψ ≤ 0 can be applied to a nonparaxial device such as a divertor as well as
the GAMMA10 tandem mirror with a long thinmagnetic ﬁeld.
The divertor mirror has an x-point, where
the magnitude of magnetic ﬁeld is zero, so that
there is a large classical radial transport acrossthe separatrix. Here separatrix is a magneticﬂux tube, the ﬁeld lines on which pass throughx-point. The stability condition νU γ = const,
therefore, is broken near the separatrix, andthe ﬂute modes become unstable. The ﬂute instability and the associated radial transport insuch a situation has been investigated[3,4]. The
divertor simulations[3,4] and other divertor stability analysis such Ref.[1] assume the boundary condition φ̃ = 0 on the separatrix, whereφ̃ is the perturbed electrostatic potential. Thisboundary condition is used in all of the arti-p⊥
1296B

flat region

30z-3-6(m)-9-12Figure 1: GAMMA10 A-divertor
cles on the divertor mirror to the ﬁrst author’s
knowledge. The condition φ̃ = 0 on the separatrix results from the electron short circuit effects along the azimuthal magnetic null line at
x-point. However, the resistivity can generate
due to the chaotic orbits of particles[5], so thatthe boundary condition φ̃ = 0 at x-point may
not be applicable. Therefore, it is necessary tocarry out the simulation on the ﬂute instabilities without the boundary condition of φ̃ = 0at x-point.
2. Simulation Results
In order not to use the boundary conditionφ̃ = 0 at x-point, we include the dipole region
as well as the divertor region.
(a.u.)

U

pA=1

0.3
0.2

(a)

0.1
0.0

xnull

0.2

0.4

x

0.6

0.8

1.0

Figure 2: Radial proﬁle of speciﬁc volume.
Figure 2 plots the radial proﬁle of thespeciﬁc volume of the GAMMA10 A-divertor.
There is the x-point around x  0.63, wherethe dipole region is x >∼ 0.63 and the divertor
region is x <∼ 0.63. Therefore the boundary
condition φ̃ = 0 at x  0.63 is not used in thesimulation here.
The basic equations are the reduced MHD
equations which are described in Refs.[3,4]. The
boundary condition is that φ̃ = 0 at x = 1,
where is near the inner coils inside the dipole
region. In the simulation the plasma pressurein the anchor mirror cell was made the same asthat in the central cell and the divertor mirror

cell, so that the system is unstable to the ﬂute
modes.
τ = 80.0

φfmax = 5.0×10-2
min = -5.0×10-2
da = 0.5×10-3

Tf / U2/3

max = 5.0
min = -5.0
da = 0.5(Ti+Te)xnullxnull

(a)

(b)

Figure 3: Contour plots of φ̃ and Ti + Te .
The ﬂute instability grew in the linear phaseand then entered into the non-linear saturation
phase. Figure 3 plots φ̃ and Ti + Te at the endof linear phase of normalized time τ = 80. It
is found that the ﬂute modes are localized justinside x-point (xnull ). The remarkable featureis that φ̃  0 in the dipole region because themagnetic ﬁeld lines in the dipole region have agood curvature for the divertor region as seenin Fig.2.
3. Conclusion
We carried out the numerical simulation ofﬂute instability in the GAMMA10 A-divertor
including the dipole region. It has been foundthat φ̃  0 in the dipole region even if the ﬂute
modes are unstable. Therefore it is concluded
that φ̃ = 0 at x-point can be used as a boundary
condition in the divertor[6].
[1] Y.Sasagawa, I.Katanuma, et.al. Phys. Plasmas,13, 122506 (2006).
[2] I.Katanuma, V.Pastukhov, et.al. J. Plasma Fus.
Res. 84, 279 (2008).
[3] I.Katanuma, et.al. Phys. Plasmas, 17, 032303(2010).
[4] I.katanuma, et.al. Phys. Plasmas, 17, 112506(2010).
[5] R.Numata and Z.Yoshida, Phys. Rev. Lett. 88,045003 (2002).
[6] I.Katanuma, et.al. Phys. Plasmas, 18 (2011) inpress.