Proceedings of ITC/ISHW2007

Confinement study on the reactor relevant high beta LHD plasmas
K.Y. Watanabe, S. Sakakibara, H. Funaba, S. Ohdachi, K.Tanaka, I. Yamada, L. Garciaa, Y. Narushima,
C. Michel, H. Yamada, Y. Suzuki, N. Nakajima, K. Narihara, T.Tokuzawa, K. Kawahata
and LHD experimental group
National Institute for Fusion Science,322-6 Oroshi-cho, Toki 509-5292, Japana

Univ. Carlos III, Avda. de la Universidad, 3028911 Leganes, Madrid, Spain

By improving the heating efficiency of the perpendicular NB injection due to the suppress the Shafranov shift by
the pellet injection and the NBI power modulation, in addition to the increment of the parallel NBI power, we had
the 5% beta plasma, and we had the 4.8% beta plasma in quasi-steady state with only gas-puffing and parallel NB
injection. According to an analysis of the resistive g-mode by using 3 dimensional resistive MHD analyzing code
for a typical high beta discharge with 4% beta and 106 magnetic Reynolds number, the radial mode widthnormalized by a plasma minor radius is ~5%. This fact supports that we have never observed disruptive phenomenain the LHD high beta discharges because the predicted MHD instabilities are located at peripheral surfaces with thehigh magnetic shear and their radial mode structure is quite narrow. According to the transport properties in thehigh beta discharges, the gradual degradation of the local transport with beta comparing with GB (Gyro-Bohm)
model is observed. An anomalous transport model based on a GMT (G-Mode Turbulence) model is fairly consistent
with the beta dependence of the experimental thermal transport. However, for a fusion reactor with LHD likeconfiguration, the anomalous transport based on the GMT is still important, but it would not be strong obstacle forthe production of the high performance plasmas because the magnetic Reynolds number in a reactor is much less by
300~400 times than that in the present LHD high beta discharges.
Keywords: Heliotron, LHD, high beta, MHD stability, Local

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 andinstall a well-defined divertor configuration. For an
economical fusion reactor, achievement and sustainment ofhigh beta plasma with <β>=5% is necessary. Then, the<β>=5% is a targeted value in the LHD projects from thedesign phase [1]. It has been considered to have adisadvantage with respect to pressure driven
654

LHD (JPN)
CHS (JPN)
W7AS (BRD)
ATF (USA)
HeliotronE (JPN)

32
101985 1990 1995 2000 2005

Year

Fig.1 The yearly progress of the achieved beta value inhelical devices
author’s e-mail: journal@jspf.or.jp

transport, g-mode turbulencemagneto-hydrodynamics (MHD) instabilities because themagnetic hill region exists. In order to predict the behaviorof reactor plasma, we have made big effort aimed atachieving <β>=5% by increasing the heating capabilities
and optimizing the operational conditions like theconfigurations, the heating efficiency of NBI and so on.
The achieved beta value is increasing yearly as shown in
Fig.1. Recently By improving the heating efficiency ofthe perpendicular NB injection due to the suppress the
Shafranov shift by the pellet injection and the NBI power
modulation, in addition to the increment of the parallel
NBI power, we had the 5% beta plasma, and we had the
4.8% beta plasma in quasi-steady state with only
gas-puffing and parallel NB injection. In both cases, wehave not observed the disruptive phenomena.
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 theoreticalmodels, such as the Gyro-Reduced-Bohm (GRB) and theresistive g-Mode Turbulence (GMT) in the LHD withdifferent magnetic hill configuration (plasma aspect ratio).

2. Achievement of reactor relevant high beta
plasmaProceedings of ITC/ISHW2007

Fig.2 The waveform of the <βdia>=5.0% plasma. (a) the Fig.3 The waveform of the <βdia>=4.8% plasma. (a) the
averaged beta value and the line averaged electron
averaged beta value and the beam component. (b)-(d)density. (b) the normalized magnetic axis shift. (c)the mode amplitude of the magnetic fluctuations.
the port-through NBI power.
tangential NBI. It suddenly decreases as the magnetic axis
shift torus- outwardly because most of the ionized NB is
At first, we briefly mention the progress of thetrapped helically. Up to last experimental campaign, theextension of the operational high beta regime up to thecontribution of the perpendicular NBI to the beta value wasexperimental campaign before last [2-4]. The heating
very small in the high beta discharges where the large
efficiency of the NB and the MHD stability are the verymagnetic axis shift was observed. In the high beta andimportant key parameters to extend the operational highlow-density discharges, the beam component of thebeta regime. In early high beta operation in LHD, thepressure is fairly large [2]. Then, it is expected that, justplasma was heated only with the tangentially injected
after the pellet injection, the beam pressure suddenly
neutral beam (NB). Because the tangential radius of thedecreases, and that the magnetic axis shift decreases.
NB is located at ~3.65m, the maximum of the heating
Moreover, the reduction of the NBI power leads to theefficiency is located around Rax=3.6m. As well known, thereduction of the magnetic axis shift due to beta. In order tomagnetic axis shifts torus-outwardly as the beta increases.
control the magnetic axis position before perpendicular
In order to keep the magnetic axis ~3.6m in the high beta
NB injection, we apply the pellet injection and a short
regime, the vacuum magnetic (pre-set) configurations withbreak down of the tangential NBI. Due to the suppress thethe much torus-inward shifted magnetic axis and/or the
Shafranov shift by the pellet injection and the NBI powerhigh aspect plasma ratio are favorable. On the contrary, themodulation and the addition to the perpendicular NBI
both configurations with the more torus-inward shifted
power, we transiently had the 5.0% beta plasma, where
pre-set magnetic axis and the higher aspect plasma ratio
plasma with more than 90% of maximum <βdia>, <βdia>max,
are more unfavorable for the property of MHD stability.
ismaintained for a short time, ~10τE. Here <βdia> is theAfter optimizing the magnetic axis position and the plasma
volumeaveraged beta value [2] and τE is the energy
aspect ratio, we found the Rax=3.6m and Ap=6.6
confinement time based on the diamagnetic measurement.
configuration the most optimum for the high beta operation,
Figure 2 shows the waveform of the <βdia>=5.0% plasma.
and we obtained a <βdia>=4.5% plasma in F.Y.2005 [4].
In
Fig.2(a), the averaged beta value and the line averaged
In last experimental campaign in F.Y.2006, we finely
electrondensity are shown, the normalized magnetic axis
study the heating efficiency of perpendicular NBI in theshift in Fig.2(b), and the port-through NBI power inhigh beta discharges to use it more effectively for theFig.2(c), where #1+2+3 corresponds to the sum of theextension of the operational high beta regime than evertangentially NB injected power and #4 to thebefore. According to a calculation, the heating efficiency of
perpendicularly NB injected power. The pellets are injected
perpendicular NBI is much more sensitive to the magnetic
during t=0.6-0.8s. Before the pellets inject, the normalized
axis position of the configurations than that of theProceedings of ITC/ISHW2007
shift of the magnetic axis exceeded 40%. During pellet
injections and a NBI line break down for 0.1s, the shift ofthe magnetic axis approaches to zero. The perpendicular
NB injection starts just after all pellet injections. After thelast pellet injection, the electron density suddenly
decreases and the beta value increases and reaches the
maximum value of the beta, when the shift of the magnetic
axis is more than 40% of the minor radius. After the beta
maximum it decreases gradually. It should be noted that
according to the comparison the discharges with andwithout perpendicular NBI, the contribution of theperpendicular NBI to the beta value is ~10% of the total
beta value.
Figure 3 shows the waveform of the <βdia>=4.8%
plasma, which corresponds to the maximum beta value
without pellet injection and the perpendicular NBI. The
extension of the beta range is mainly due to the incrementof the tangential NBI power comparing with theexperimental campaign before last. In Fig.3(a), theaveraged beta value and the beam component of the stored
energy are shown, the magnetic fluctuations in Fig.3(b)-(d).
The high beta plasma with <βdia> > 0.9x<βdia>max ismaintained for more than 80 times of τE (in quasi-steady).
And the MHD activities corresponding to the peripheral
rational surfaces are observed and its levels are not so large.
The above properties are almost same with those in the
m/n=1/1 unstableρ=0.9 (ι~1)
-2
12
γ/ω
=10
dβ /dρ
A
kin-2

0.3x10A<βdia>=4.5% plasma [2-4].

3. Effect of global MHD mode on high beta
confinement
According to the analysis of the MHD equilibrium
properties in the high beta configuration, the stability in thecore region is improved as the beta increases. On thecontrary, the stability in the peripheral region becomesworse as the beta increases. This characteristics is
consistent with the fact shown in the previous section that
only the MHD activities corresponding to the peripheral
rational surfaces are observed in the LHD high beta
plasmas. In this section, the effect of the global MHD
mode resonating with the peripheral surface like m/n=1/1(m and n are the poloidal and toroidal mode numbers,
respectively) on the confinement is studied.
Figure 4 shows the thermal beta gradients at the
m/n=1/1 rational surface (ρ~0.9) as faction of <βdia> in the
Ap=6.2 configuration, where the achieved <βdia>max is

(%)
1.5

#55397

<β >
dia

1.0
0.5

P

0.0

NB

0

0.5

1
time (s)1.5

2

Fig.6 The evolution of the beta value and the NBI
port-through power in Ap=8.3 configuration.

8

Mercier

0.84s
0.87s
0.90s

2.0
4

0.93s
0.97s

(a)

4

4.4

1.0
0

0

1

2

3

4

<β > (%)
dia

Fig.4 The peripheral beta gradients as the function of<βdia> in Ap=6.2 configuration. The contours of thegrowth rate of the low-n ideal MHD unstablemodes are over plotted.

0.0
2.8

3.2

1.0

Ideal

0.0
0.7

(b)

Ideal

1.0

0.5

3.6
R(m)

S=10

6

S=10

5

0.8

ρ

S=10

0.0

ξ(m/n=1/1)

0.0

ξ(m/n=1/1)
0.9

6

0.5

0.2

0.4

ρ

0.6

0.8

1.0

1.0

Fig.5 The radial mode structure of the global mode(m/n=1/1) for the plasma of ‘A’ in Fig.4 taking
the resistivity into account by FAR3D. .

Fig.7 (a) The evolution of the electron temperature incollapse. (b) The radial mode structure of the
global mode (m/n=1/1) at the just before collapse
calculated by FAR3D. .

Proceedings of ITC/ISHW2007
4.1% [2]. In Fig.4, the contours of the growth rate of the
low-n ideal MHD unstable modes by terpsichore code [5]are superposed. It should be noted that the gradients are
estimated as the averaged value for ∆ρ=0.1. Around
Mercier unstable region, the gradients slightly changes, but
the strong reduction of the gradients are not observed in theregion where the ideal global mode is predicted unstable.
According to the analysis of the dependence the amplitudeof magnetic fluctuation on the magnetic Reynolds number,
S around <βdia >~3%, it scales S-0.69 [3]. This fact
supports that the observed magnetic fluctuation in thehigh beta plasmas is due to the resistive interchange
instability (g-mode) because the mode width estimated bya simple model, w ~ b~θ Bt , is consistent with thetheoretically predicted linear mode width of the g-mode.
Figure 5 shows the mode structure of the g-mode
calculated by FAR3D code [6] for the plasma presented bythe ‘A’ in Fig.4. For the plasma consistent with theexperimental S, ~106, the mode width is expected narrow,~5% of the plasma minor radius. This fact supports that
we have never observed disruptive phenomena in the
LHD high beta discharges because the predicted MHD
instabilities are located at peripheral surfaces with thehigh magnetic shear and their radial mode structure isquite narrow. It is noticed that, in the LHD high beta
plasmas, the mode width of the resistive mode is a little
bit larger than that of the ideal mode.
In order to confirm the above statement, we study
the comparative analysis between the theoreticalpredicted mode structure and the electron temperature
profile in the high aspect configuration, Ap=8.3, which is
expected the much more unstable than the Ap=6.2 due thelow magnetic shear and the high magnetic hill, there theachieved <βdia>max is 2.6% and the collapse phenomena
are observed shown in Fig.6, where the beta drops
suddenly at t~0.85s. Figure 7(a) shows the evolution of
the electron temperature profiles just before collapse andafter it. The electron temperature at the core region
significantly decreases due to the collapse. Figure 7(b)shows the linear mode structure predicted by FAR3D
code. In this case the mode width is more than 10% of theminor radius and the amplitude of the mode at center isfairly large, which is much larger than that in Fig.5. It
supports that the radial mode width is the very important
key parameter against the apparent effect of the global
MHD instabilities on the plasma confinement like thecollapse. The identification of the threshold value should
be investigated more.

is observed as the beta value increases as shown in ref.[2].
Here we focus the study of the peripheral transport
property because it affects a large effect on the global
confinement. Figure 8 shows the dependence of thenormalized thermal conductivity, χeff/χGB, by the GB(Gyro-Bohm) model in the peripheral region on the beta
value for the Ap=6.2 configuration. It should be noted that
the GB model has the similar property of ISS95
confinement scaling [7], and it is proportional to β0. In the
low beta regime, the χeff/χGB is insensitive to the beta value.
In the high beta regime, χeff/χGB looks proportional to β1[8].
As the transport model proportional to β1, the MHD
driven turbulence model is known. Here as a MHD driven
turbulence model, we introduce an anomalous transportmodel based on the resistive interchange turbulence(g-mode turbulence, GMT) proposed by Careras et al. [9].
The thermal conductivity of the GMT model are written
as the following,χ GMTe ∝ GGMTe β 1 ν * 0.67 ρ * 0.33 χ B .
Here GGMTe is defined as a geometric factor, which
increases with the bad curvature and decreases themagnetic shear, and χB is the thermal conductivity ofBohm model. In heliotron devices as LHD, the g-mode isalways unstable due to the magnetic hill in the peripheral
region.
In order to study the GMT on the confinement
properties in high beta LHD plasmas, we analyze theconfinement property in 2 configurations with different
magnetic hill height, and compare with the experimental
data the local transport analysis for the 2 configurations
with the different magnetic hill height. Figure 9 shows
the thermal conductivities normalized by the GMT model,
χGMTe, as a function of <βdia>. Here we focus on the
peripheral region, ρ=0.9. (a) and (b) in Fig.9 correspondto the Ap=6.2 and 8.3 configurations, respectively. The
amplitude of the bad curvature in Ap=8.3 is larger thanthat in Ap=6.2, that is, GGMTe is larger by twice. In
Fig.9(a), χeff/χGMTe in the beta range of <βdia > < 1% isquit large, which occurs because there the effect of the
GMT is quite small. In the beta range of <βdia > >1%, the100χeff/χGRB@ρ=0.9
GB
10

1

0.1
0.2

4. Transport properties of high beta plasmas
As shown in Fig.4, the strong reduction of the beta
gradients is not observed in the high beta plasma. However,a gradual degradation of global confinement performance

0.4 0.60.8 1<β > (%)

3dia

Fig.8 The normalized thermal conductivities at ρ =0.9
on the beta value in the Ap=6.2 configurations.
χGB denotes the gyro-Bohm model.

Proceedings of ITC/ISHW2007100

10χgmte=1m2/s

7

3

10

Ap=6.2
6

10

1

0.1
100

0.3

S

χeff/χGMTe @ρ=0.9

(a)
5

10

0

χeff/χGMTe @ρ=0.9

Ap=8.3

10

10

<β >
dia
<1%
~2%
~3%
~4%
20

30

40
R dβ/dr (%)

60

eff

Fig.10 The predicted thermal conductivity by a t modelin S-dβ/dρ diagram with the experimental data.
1

(b)
0.1

0.2

0.5
1
2
<βdia> (%)

3

5

Fig.9 The normalized thermal conductivities at ρ =0.9
on the beta value. (a) and (b) correspond to theAp=6.2 and 8.3 configurations, respectively. χGMTe
denotes a g-mode turbulence model.
beta dependence of the χeff looks consistent with the
GMT model. As shown in Fig.9(b), the beta dependenceof the χeff looks also consistent with the GMT model inthe higher magnetic hill configuration though thedispersion of χeff/χGMTe is fairly large. It should be notedthat at the same <βdia> (~2%), the χeff in Ap =8.3 is larger
by 6~7 times than that in Ap=6.2, and that the magnetic
Reynolds number, S (~β0.5ν*-1ρ*-2), in Ap=8.3 is smaller
by ~10 times than that in Ap=6.2. The above facts support
the probability that the peripheral thermal transport in thereactor relevant high beta plasma in LHD is governed by
the g-mode turbulence.
According to the analysis of the beta dependence ofthe density fluctuation with relatively long wavelength,
le>30mm for the Ap =6.2 configuration, the amplitude of
the fluctuation is quite small in the low beta regime with<βdia > < 1% and it suddenly increases with beta value inthe beta range of <βdia > >1%. This behavior looks
synchronized with Fig.8. And according to ref.[10], theprobable poloidal mode number of the turbulence is ~10,
which is consistent with the observation of the density
fluctuation. This result would be another collateralevidence for the effect of the g-mode turbulence on the
LHD high beta plasma.
Next we consider the effect of the g-mode turbulenceon the confinement in reactor relevant plasmas. Figure 10
shows a contour of the thermal conductivity based on the
GMT model in S-R0dβ/dr space. The thermal
conductivity becomes large with the decrease of S andincrease of R0dβ/dr. Especially in high beta range,
decrease of S leads to significant increase of the thermal
conductivity. In Fig.10, the operation rage of S-R0dβ/dr

for the data of Fig.9(a) is also shown in Fig.10. In LHD,
the decreasing the operational magnetic field strength
extends the operational beta range. Then in LHD highbeta operation, S is small, which leads to the prediction oflarge thermal conductivity. Here we shall consider a
Fusion reactor. Its geometrical factor on the GMT model,such as magnetic shear and magnetic curvature, and thenormalized beta gradient are almost same with those inpresent LHD high beta operations. On the other hand, themagnetic 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 belarger by around 10 times and the device size would belarger by ~3 times than the present LHD [11]. When S is300~400 times larger comparing with present LHD highbeta operation, the predicted thermal conductivity wouldbe ~1m2/s. For a fusion reactor with LHD likeconfiguration, the anomalous transport based on the GMT
is still important, but it would not be strong obstacle forthe production of the high performance plasmas.

5. Summary
By improving the heating efficiency of theperpendicular NB injection due to the suppress the
Shafranov shift by the pellet injection and the NBI power
modulation, in addition to the increment of the parallel
NBI power, we had the 5% beta plasma, and we had the
4.8% beta plasma in quasi-steady state with only
gas-puffing and parallel NB injection, where the duration
time with <βdia>/<βmax> > 90% is longer than 100τE.
The characteristics of the high beta plasmas with
β~5% is as the followings;
1. As MHD activities of global modes, only the modes
resonated with peripheral rational surfaces are observed.
2. The apparent pressure-flattening region is not observed
according to the electron pressure profile measurements.
3. According to linear global MHD stability analyses

Proceedings of ITC/ISHW2007
taking resistivity effect into account, the global mode
resonated with peripheral rational surfaces is predictedunstable, but its radial mode width is ~5% of the plasma
minor radius.
From results of the comparative analyses between
achieved pressure gradients and linear MHD numericalanalyses for the high beta and the high aspectconfigurations, it is supported that the radial mode width
is the very important key parameter against the apparent
effect of the global MHD instabilities on the plasma
confinement like the collapse.
From the comparative analyses between
experimentally obtained thermal conductivities and sometheoretically predictions in high beta plasmas, there ispossibility that a theoretical model based on g-mode
turbulence (GMT) explains the beta dependence of theperipheral thermal conductivity in high beta plasmas.
This fact is supported by the similar analyses against
other configuration and the beta dependence of theobserved density fluctuation amplitude with relatively
long wavelength.
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. Because theanomalous transport based on the GMT strongly depends
on the S value, and the S value of a reactor is much larger
by 300~400 times than that in the present LHD high beta
discharges.

Acknowledgments
We special thank the LHD technical staff for theirgreat effort of LHD operation and maintenance. We are
grateful to Dr. A.Cooper for permitting use of theterpsichore code, to Drs. S.P.Hirshman and P.Merkel for
the VMEC code, to late Prof. T.Hayashi for the HINT codeand to Dr. Y.Nakamura for KMAG and KSPDIAG codes.
We acknowledge continuous encouragement by Prof.
O.Motojima. This work is supported by NIFS underContract No. NIFS06ULHH519.

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