Plasma and Fusion Research: Rapid Communications

Volume 11, 1203122 (2016)

Non-Ideal Ballooning Mode Instability with Real Electron Inertia
Haruki SETO, Masatoshi YAGI, Nobuyuki AIBA, Akinobu MATSUYAMA,Benjamin D. DUDSON1) and Xueqiao XU2)
National Institutes for Quantum and Radiological Science and Technology, Rokkasho 039-3212, Japan
1)
York Plasma Institute, Department of Physics, University of York, Heslington, York, YO10 5DD, UK
2)
Lawrence Livermore National Laboratory, CA 94550, USA(Received 11 October 2016 / Accepted 26 October 2016)
Impacts of electron inertia with an electron skin depth (ESD) longer than the realistic value used in earlynumerical studies on non-ideal ballooning modes (NIBMs) are√numerically investigated by a linearized 3-field∗
reduced MHD model. In this paper, 4 diﬀerent ESDs
de = 0, de , 10de , 10de are used for a resistivity dependence
study of the growth rate of NIBMs, where de = c0 ε0 me /ne e2 is the real ESD and de∗ = 10de corresponds to anorder of ESD used in a numerical study on collisionless ballooning mode (CBM) reported in [Kleva and Guzdar
Phys. Plasmas 6, 116 (1999)]. In the case with the real ESD de∗ = de , a transition from resistive ballooning mode(RBM) to CBM occurs in the edge relevant resistivity regime, while the electron√inertia eﬀect is overestimatedand the growth rate is almost independent of resistivity in the cases with de∗ = 10de and 10de . These results
indicate that the real ESD is one of key factors for the edge stability and turbulence analysis.
c 2016 The Japan Society of Plasma Science and Nuclear Fusion Research

Keywords: non-ideal ballooning mode, electron inertia eﬀect, reduced MHD model, linear analysis
DOI: 10.1585/pfr.11.1203122
It is one of the most important issues for predicting
global confinement performance of fusion devices to understand plasma instability and transport phenomena in theedge region where collisionality drastically changes fromcollisionless to collisional. In the collisional regime, resistivity destabilizes a ballooning mode below the ideal stability limit (ISL), so called resistive ballooning mode (RBM)[1, 2]. On the other hand, in the collisionless regime,
electron inertia also destabilizes collisionless ballooningmode (CBM) [3–5]. A CBM simulation with the real electron skin depth (ESD), however, has not been carried out
so far due to limitation of computational resources, since
its characteristic
scale length is associated with the ESD,de = c0 ε0 me /ne e2 ∼ 1.68 × 10−3 [m] for the electron
number density ne = 1.0 × 1019 [m−3 ]. A simulation ofCBM with an ESD de∗ longer than the real value by one
order de∗ ∼ 10de showed that the longer ESD makes thegrowth rate of CBM larger [6]. According to a local analysis, CBM with the real ESD can be destabilized only close
to the ISL or for extremely high toroidal mode numbers
for which fluid models are no longer valid [7]. In addition, electron inertia can be a key player in energy turbulent
transport in the edge region as well as current diﬀusion [8]so that the simulation of CBM with the real ESD is alsosignificant as a preliminary step for an edge turbulent simulation for L/H transition.
In this paper, impacts of the overestimated electron
inertia on CBM/RBM instabilities in the resistivity regime
relevant to the edge plasma are numerically investigatedauthor’s e-mail: seto.haruki@qst.go.jp

Fig. 1 The shifted circular equilibrium for CBM/RBM instability analysis: the radial profiles of p/pax and q (left) andthe radial profiles of s and α (right), where the beta value
at the magnetic axis is pax /(B2ax /2μ0 ) = 5 × 10−4 .

by BOUT++ code [9] which was applied to a linear instability analysis of ideal ballooning mode with the realelectron mass [10]. The resistivity regime relevant to theedge plasma lies roughly from η ∼ 10−9 for T e ∼ 103 [eV]to η ∼ 10−7 for T e ∼ 102 [eV], where T e is the electron
temperature and the Spitzer resistivity η ∝ T e−3/2 has beenemployed. Note that quantities without unit are normalizedwith the major radius Rax , the magnetic field intensity Bax
and the poloidal Alfvén time tA with the ion number density ni = 1.0 × 1019 [m−3 ] and the deuterium mass, wherethe subscript ax indicates values at the magnetic axis.
Figure 1 shows a shifted circular equilibrium with thepressure shear factor α  0.6 and the magnetic shear factor s  2.5 at r/a  0.9 used for CBM/RBM instability analyses. This equilibrium is based on a linearized
Grad-Shafranov equilibrium [11] and is far from the ISL
αcritical ∼ 0.6s ∼ 1.5 for s  2.5 [12]. In CBM insta-1203122-1

c 2016 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Rapid Communications

Volume 11, 1203122 (2016)

Fig. 2 Linear growth rates of n = 40 CBM/RBM√ as functions ofresistivity for 4 diﬀerent de∗ = 0, de , 10de , 10de whichcorrespond to the computational electron masses m∗e =√0, me , 10me , 100me respectively for de∗ ∝ m∗e .

bility analysis, grid widths in the directions perpendicular
to the field line must be shorter than the ESD. BOUT++
code employs a field-aligned coordinates (x, y, z) [13] sothat the grid width in the flux surface label x is restricted
by Δx/RBp  de → Δx  de RBp ∼ 4 × 10−5 andthe grid width in the field line label z is restricted byΔzRBp /B  de → Δz  de B/RBp ∼ 2 × 10−2 respectively, where y is the parallel label. We employ an equally
spaced grid for a 1/40 sector of torus with Δx ∼ 3 × 10−6
for the radial domain corresponding to 0.8 ≤ r/a ≤ 1.0,Δy ∼ 5 × 10−2 for 0 ≤ y < 2π and Δz ∼ 6 × 10−4 for0 ≤ z < 2π/40, which is fine enough for the real ESD.
For linear instability analyses of CBM/RBM, a 3-field
reduced MHD (RMHD) model with resistivity and electron inertia is employed,∇2 Ã∂ ∇2⊥ φ̃= −B2 ∇ ⊥ + b × κ · ∇ p̃,∂t BB∂ p̃b · ∇⊥ φ̃ × ∇⊥ p=−,∂tB∂ Ã − de∗ 2 ∇2⊥ Ã = −∇ φ̃ + η∇2⊥ Ã ,
∂t

(1)
(2)
(3)

where φ is the electrostatic potential, B is the magnetic
field intensity, A is the magnetic potential, b is the unit
vector along the equilibrium magnetic field, κ is the magnetic curvature and ˜ indicates perturbed quantities respectively. In this model, the other kinetic eﬀects are neglected
to make the impact of the computationally large de∗ clear.
Resistivity dependences of growth rates √of n = 40
CBM/RBM for 4 diﬀerent ESDs de∗ = 0, de , 10de , 10deare summarized in Fig. 2, where the growth rate and theresistivity are normalized as γtA → γ and η/μ0 R2ax tA−1 → ηrespectively. We choose n = 40 as a typical toroidal mode
number which is relevant to fluid models including RMHDmodels [2, 6, 10]. In the case with de∗ = 0 where is only theRBM branch, the growth rate of RBM shows good agreement with a local theory γ ∝ η1/3 [1] in 10−8 ≤ η ≤ 10−7 .
The growth rate, however, becomes less sensible to the re-Fig. 3 Radial structure of poloidal harmonics (left) and eigen
function in parallel direction (right) of n = 40 mode ofφ̃ in the collisionless limit η = 1.0 × 10−10 with de∗ = de(upper row) and de∗ = 10de (lower row).

sistivity in η > 10−7 since the eigen function of RBM isstrongly localized in the parallel direction and its parallel
derivatives become no longer negligible [2]. A transition
from RBM to CBM occurs in the resistivity regime relevant to the edge plasma in the case with the real ESDde∗ = de . This result indicates that CBM with realistic
toroidal mode numbers for fluid models can be destabilized
in the edge region even when equilibrium√ is not close to theISL. Finally, in the cases with de∗ = 10de and de∗ = 10de ,CBM is overestimated and its growth rates are almost independent of resistivity in η < 10−7 . In addition, Fig. 3 showsthat the eigen function of CBM with de∗ = 10de goes tothe strong ballooning limit rather than the weak ballooning
limit in contrast to the case with the real ESD. These results indicate that the real ESD is one of key factors for thestability and turbulent analysis in the edge plasma whereresistivity lies in the CBM/RBM transient regime. Further linear spectrum analyses on NIBMs with other kinetic
eﬀects, especially ion diamagnetism which can stabilize
middle-n modes, are future works.
This work is partly supported by MEXT KAKENHI
Grant No. 16K18342. Computations were carried out on
the Helios supercomputer at the Computational Simulation
Centre of the International Fusion Energy Research Centre(IFERC-CSC).
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Plasma and Fusion Research: Rapid Communications

Volume 11, 1203122 (2016)[10] P.W. Xi et al., Phys. Plasmas 21, 056110 (2014).
[11] J.P. Graves, MHD equilibrium and stability in tokamaks(http://spc.epfl.ch/theory-course/).

[12] K. Miyamoto, NIFS-PROC-80 (2010).
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