Effects of MHD instabilities on confinement properties of high beta heliotron plasmas

K.Y.Watanabe, H.Funaba, S.Sakakibara, Y.Narushima, T.Yamaguchi, H.Yamada,
K.Narihara, K.Tanaka, T.Tokuzawa, I.Yamada, K.Kawahata
and LHD Experimental Group
National Institute for Fusion Science, 322-6 Oroshi, Toki, 509-5292, Japan
Abstract. In order to study the MHD instabilities on the confinement properties in heliotron plasmas, we make
comparative analyses between the observed beta gradients and the prediction of ideal MHD instabilities, and between the
experimental thermal conductivity and the prediction based on some theoretical models, such as the Gyro-Reduced-Bohm
(GRB) and the resistive g-Mode Turbulence (GMT) in the Large Helical Device (LHD) with different magnetic hill
configurations. Around the maximum beta region achieved experimentally up to now, the obvious and/or rapid change of
the beta gradients and the thermal conductivity is hardly observed even the low-n ideal MHD mode is predicted unstable,
where the mode width is less than 5% of the plasma minor radius. The gradual degradation of the local transport with
beta comparing with GRB model is observed. An anomalous transport model based on the GMT model is fairly
consistent with the beta dependence of experimental thermal transport in the high beta region and/or the high magnetic
hill configuration in the LHD.
Keywords: heliotron, helical, stellarator, MHD instability, confinement, transport,
PACS: 52.55.Hc, 52.55.Tn, 52.25.Fi
1. INTRODUCTION
A heliotron device is a helical type toroidal magnetic plasma confinement system and a probable candidate as
thermonuclear fusion reactor under steady-state operation because it can confine plasma with only external coils and
install a well-defined divertor configuration. It has been considered to have a disadvantage with respect to pressure
driven magneto-hydrodynamics (MHD) instabilities because the magnetic hill region exists. However, in recent
experiments of a heliotron (LHD [Large Helical Device] in NIFS, Japan), the
high beta with more than 4% as the volume averaged beta value, which is
relevant to that of a heliotron reactor design (5%), is achieved without
disruptive phenomenon [1]. A gradual degradation of confinement
performance is observed as the beta value increases as shown in Fig.1 [2].
Figure 1 shows the normalized global energy confinement time by the ISS95
empirical scaling [3] as the function of <βdia>. Here the normalized one is
defined as an improvement factor, HISS95, and <βdia> is a volume averaged beta
value based on the diamagnetic measurements [2]. It should be noted that
HISS95_kin is evaluated based on the profile measurements of the electron
temperature and density under the assumption with Ti=Te and Zeff=1.
In order to study the MHD instabilities on the confinement properties in
heliotron plasmas, we analyze the confinement property in 2 configurations
with different magnetic hill height because the interchange mode is considered
the most important as the MHD instabilities in heliotron plasmas. LHD can
change the magnetic hill height and the plasma aspect ratio by changing the
current ratio in the 3 layers of helical coils [4]. In Fig.1, HISS95_kin in 2
configurations with different magnetic hill and plasma aspects (Ap=6.2 and
FIGURE 1. The normalized
global confinement time by
ISS95 empirical scaling in
Ap=6.2 and 8.3 configuration.
0
0.5
1
1.5
2
0 1 2 3 4
H
ISS95_kin

dia> (%)
Ap=6.2
Ap=8.3
8.3) is shown. Higher plasma aspect corresponds to higher magnetic hill
(worse curvature) as shown in Fig.2. It should be noted that the achieved
maximum beta values of configurations with Ap=6.2 and 8.3 are 4.1% and
2.6%, respectively.
In this paper, we make comparative analyses between the observed beta
gradient and the prediction of ideal MHD instabilities, and between the
experimental thermal conductivity and the prediction based on some
theoretical models, such as the Gyro-Reduced-Bohm (GRB) and the resistive
g-Mode Turbulence (GMT) in the LHD with different magnetic hill
configuration (plasma aspect ratio).
2. CONFINEMENT PROPERTY IN HIGH BETA REGIME
Here we review the confinement properties in high beta range up to
<βdia>~4% in Ap=6.2 configuration according to previous works [2,5].
Generally speaking, in heliotron devices, the global MHD mode in the core
region is unstable in the intermediate beta range and it is stable in the high
beta range. On the other hand, because the instability in the peripheral region
is more unstable as the beta becomes higher, it is considered that the
behavior of the instability in the peripheral region limits the operational beta
range, and the impact of the confinement properties in the peripheral region
is quite large because the their volume is larger than one of the core. Here
we mention the comparative analysis between the achieved pressure
gradients and the prediction of ideal MHD instability in the peripheral region,
ρ=0.9, and the local transport properties. Here it should noted that a loworder
rational surface, ι~1, is located around ρ=0.9. Figure 3(b) shows the
experimental thermal beta gradients superposed on the contours of the low-n
ideal MHD unstable modes by terpsichore code [6] that resonate with a
peripheral rational surface. Here the thermal beta gradients are estimated by
the profile measurement of the electron temperature and density with
Thomson scattering and FIR under the assumption of Zeff=1 and Ti=Te. The
data is obtained in 0.45-1.75T operations. The envelope of the observed
thermal pressure gradients in the beta range of <βdia>=3-4% looks to
coincide with a contour of the growth rate of the m/n=1/1 ideal MHD mode,
γlow-n/ωA=10-2, where the radial mode width corresponds to 5% of the plasma
minor radius.
Figure 3(a) shows the thermal conductivities normalized by the GRB
model, χGRB, as a function of <βdia>. Here χGRB=ρ*χB, ρ*=ρs/ap, ρs and ap are
the ion gyro-radius measured with the electron temperature and a plasma
minor radius, χB is the thermal conductivity of Bohm model. The GRB
model has the similar property of the global energy confinement with the
ISS95 empirical scaling. The normalized χeff gradually increases with <βdia>.
However, the disruptive degradation of χeff is not observed around <βdia>=3-
4%, where the global ideal MHD modes is predicted unstable. Figures 3(a)
and (b) suggest that the low number ideal MHD mode does not affect
seriously the confinement up to <βdia>~4% when the mode width is small, ex.
it is less than 5% of the plasma minor radius. The localized ideal MHD
instabilities might affect the local transport. But up to now, the effect of
ideal modes on the transport process has hardly been studied. Next we
introduce an anomalous transport model based on the resistive g-mode
(interchange) turbulence (GMT) proposed by B.Careras et al. [7], and
compare with the experimental data. The thermal conductivity of the GMT
model are written as the following,
( ) ( ) ( ) ( ) B
0.33
*
0.67
*
1
0 GMTe
4 3 4 3
0
4 3 2
0
7 3
GMTe χ q sˆ κ R a βR L S v R G β ν ρ χ n eff p Te ∝ − ∝ (1)
FIGURE 2. The curvature
radial profile in Ap=6.2 and 8.3
configuration.
0
2
4
6
0.6 0.8 1
κ
n (m-1)
ρ
6.2
Ap=8.3
FIGURE 3. The observed beta
gradients (b) and the normalized
thermal conductivities by GRB
model (a) and GMT model (c) in
Ap=6.2 configuration.
0.1
1
10
100
0 1 2 3 4

dia
> (%)
χ
eff/χ
GMTe @ρ=0.9
0.1
1
10
100
χ
eff/χ
GRB @ρ=0.9
0
4
8
12
Mercier
m/n=1/1 unstable
γ/ω
A =10-2
0.3×10-2
ρ=0.9 (ι~1)

kin/dρ
(a)
(b)
(c)
Here GGMTe=(q/s^)7/3( κ
nR0)4/3aeff
2 is defined as a geometric factor. It should be noted that the GMT model scales with
β1 not like the GRB model, which scales β0. Figure 3(c) shows the thermal conductivities normalized by the GMT
model, χGMTe, as a function of <βdia>. χeff/χGMTe in the beta range of <βdia > < 1% is quit large, which occurs because
there the effect of the GMT is quite small. In the beta range of <βdia > >1%, the beta dependence of the χeff looks
consistent with the GMT model. That main reason is considered the β dependence of the thermal conductivity.
3. CONFINEMENT PROPERTY IN HIGH MAGNETIC HILL CONFIGURATION
In the previous section, the possibility is shown that the resistive g-mode (GMT) plays an important role in the
confinement properties in high beta plasmas. In order to confirm it, we analyze the confinement property in higher
magnetic hill configuration (Ap=8.3). Figure 4(b) shows the experimental thermal beta gradients superposed on the
contours of the Mercier parameter. The data is obtained in 0.5-1T operations. Here it should be noticed that the
information of the stability boundary of the low-n ideal MHD mode is not shown in Fig. 4(b) because there is no
rational surface with low order at around ρ=0.9 (ι~1.3, the nearest low-order rational surfaces are m/n=3/4 and 4/5,
and both modes hardly observed in LHD experiments). In the beta range with more than <βdia > ~0.5%, Mercier
mode is predicted unstable. In discharges with a high magnetic hill configuration, minor collapses (the plasma stored
energy and the temperature inside of the m/n=1/1 rational surface) are
observed. The discharges do not terminate after the collapses, the perturbed
radial magnetic field of the m/n=1/1 structure rapidly grows when
collapses appear. Here it should noted that a ι=1 surface is located at
ρ~<0.7. In Fig.4, the data with minor collapses is excluded. The obvious
change of the beta gradients does not appear up to <βdia >~2.5%. Figure
4(a) shows the thermal conductivities normalized by the GRB model as a
function of <βdia>. The normalized thermal conductivity looks to increase
with beta from a low beta range. Figure 4(c) shows the thermal
conductivities normalized by the GMT model as a function of <βdia>. The
beta dependence of the χeff looks consistent with the GMT model though
the dispersion of χeff/χGMTe is fairly large.
4. SUMMARY AND DISCUSSION
In order to study the MHD instabilities on the confinement properties in
heliotron plasmas, we make comparative analyses between the observed
beta gradient and the prediction of ideal MHD instabilities, and between
the experimental thermal conductivity and the prediction based on some
theoretical models, such as the Gyro-Reduced-Bohm (GRB) and the
resistive g-Mode Turbulence (GMT) in the LHD with different magnetic
hill configurations. Here we focus the behavior in the peripheral region
because the instability in the peripheral region becomes unstable with the
beta value and the volume element of the peripheral region is quite larger
than one of the core. Around the maximum beta region achieved
experimentally up to now, obvious and/or rapid change of the beta
gradients and the thermal conductivity is hardly observed even the low-n
ideal MHD mode is predicted unstable, where it is predicted that its growth
rate is finite, but its mode width is less than 5% of the plasma minor radius.
The gradual degradation of the local transport with beta comparing with
GRB model, which is almost consistent with the ISS95 empirical global
energy confinement time scaling [3], is observed. It should be noted that
the ISS95 scaling well expresses the confinement properties of many
helical devices. The beta dependence of the anomalous transport model
based on the GMT model [7] is fairly consistent with the experimental
thermal transport in the high beta region and/or the high magnetic hill
configuration in LHD, which suggests a possibility that the GMT model
play a main role in the transport process. Here it should be noted that in
FIGURE 4. The observed beta
gradients (b) and the normalized
thermal conductivities by GRB
model (a) and GMT model (c) in
Ap=8.3 configuration.

high beta region in Figs. 3 and 4, plasmas are in Mercier unstable region.
The expression applied in this paper of the GMT [7] was derived under
the assumption that the resistive g-modes are only unstable. The thermal
conductivity model under the assumption, that the ideal MHD modes are
unstable, should be studied in future.
Figure 5 shows a contour of the thermal conductivity based on the
GMT model in S-R0dβ/dr space. Here S is the magnetic Reynolds
number. The thermal conductivity becomes large with the decrease of S
and increase of R0dβ/dr. Especially in high beta range, decrease of S
leads to significant increase of the thermal conductivity. In Fig.5, the
operation rage of S-R0dβ/dr for the data of Fig.2 is also shown in Fig.5.
In LHD, the decreasing the operational magnetic field strength extends
the operational beta range. Then in LHD high beta operation, S is small,
which leads to the prediction of large thermal conductivity. Here we shall
consider a Fusion reactor. Its geometrical factor on the GMT model, such
as magnetic shear and magnetic curvature, and the normalized beta
gradient are almost same with those in present LHD high beta operations.
On the other hand, the magnetic Reynolds number would be much larger by 300~400 times than that in the present
LHD high beta operations because the magnetic field strength would be larger by around 10 times and the device
size would be larger by ~3 times than the present LHD. When S is 300~400 times larger comparing with present
LHD high beta operation, the predicted thermal conductivity would be ~1m2/s. For a fusion reactor with LHD like
configuration, the anomalous transport based on the GMT is still important, but it would not be strong obstacle for
the production of the high performance plasmas.
As mention in section 3, in the high magnetic hill configuration, minor collapses are observed. The perturbed
radial magnetic field of the m/n=1/1 structure rapidly grows when collapses appear. According to the analysis of
Mercier parameter, DI, at ι=1 magnetic surface (ρ~0.7) in high magnetic hill configuration, large Mercier parameters
with DI>1 are often obtained just before collapses happen. On the other hand, as shown in section 2, the minor
collapse is never observed even the low-n ideal MHD mode is predicted unstable, where the mode width is less than
5% of the plasma minor radius [2]. The main difference between high magnetic hill configuration and low magnetic
hill configuration is the location of the rational surface with m=1 mode, where the magnetic shear strength in high
magnetic hill configuration is much smaller than in low magnetic hill configuration. On the contrary, the amplitude
of the magnetic curvature at ι=1 is almost same in both configurations as shown in Fig,1. The small magnetic shear
might lead to the large radial mode width. The relationship between the on-set condition of the minor collapse and
the mode width, magnetic shear strength is the important issue on the effects of MHD instabilities on confinement
property in future.
ACKNOWLEDGMENTS
We special thank the LHD technical staff for their great effort of LHD operation and maintenance. We are
grateful to Dr. A.Cooper for permitting use of the terpsichore code, to Drs. S.P.Hirshman and P.Merkel for the
VMEC code, to Prof. T.Hayashi for the HINT code and to Dr. Y.Nakamura for KMAG and KSPDIAG codes. We
acknowledge continuous encouragement by Prof. O.Motojima. This work is supported by NIFS under Contract No.
NIFS05ULHH519.
REFERENCES
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