J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)523–526
000–000

Relationships between the Prediction ofLinear MHD Stability Criteria and the Experiment in LHD
WATANABE Kiyomasa Y., NARUSHIMA Yoshiro, SAKAKIBARA Satoru,
OHDACHI Satoshi, TOI Kazuo, COOPER Antony W.1, NAKAJIMA Noriyoshi,
NARIHARA Kazumichi, TANAKA Kenji and LHD experimental group1

National Institute for Fusion Science, Toki 509-5292, Japan
Centre de Recherches en Physique des Plasmas, Association Euratom / Confederartion
Suisse, EPFL, 1015 Lausanne, Switzerland(Received: 12 December 2003 / Accepted: 3 August 2004)

Abstract
We analyze the relationship between the experimentally observed pressure gradients at resonant rational surfaces
and the theoretically predicted ideal magnetohydrodynamics (MHD) unstable region of global modes in the large
helical device (LHD). According to the stability analysis of the ideal MHD modes with a low toroidal mode number,
we find that the ideal MHD mode gives a constraint on the operational regime of the pressure gradients in the core. In
the edge, a clear saturation of the pressure gradients due to the ideal MHD instability has not been observed up to thehigh beta regime around 3% as the volume-averaged toridal beta value, where global ideal MHD modes are predicted
to be unstable.
Keywords:
heliotron, MHD stability, beta limit

1. Introduction

respectively. In the above previous works, they paid attention
to an averaged beta, not to pressure profile so much instudying the operational regime. The stability of pressure
driven MHD depends on the pressure gradients at theirresonant rational surfaces. Moreover the pressure gradient anda net toroidal current affect the stability of pressure driven
MHD modes through the change of MHD equilibrium infinite beta, for example, Shafranov shift, magnetic wellformation and so on. This type of change in MHD equilibrium
becomes significant in the devices with low aspect ratio and/or low rotational transform like CHS and the Large Helical
Device (LHD) [5]. In order to clarify the role of pressure
driven MHD instabilities on operational regime, it isnecessary to analyze the relationship between the unstablecondition of the pressure driven MHD modes and theexperimentally observed pressure gradients at every resonant
rational surface based on the consistent MHD equilibriumwith the measured profile data like density, temperature andcurrent of plasmas.
LHD is a heliotron device, where the experiments started
in 1998. The device major radius is 3.9 m, and the plasma
minor radius is 0.64 m in a typical operation [5]. The aspectratio, Ap and the central rotational transform, ι0 are fairly close
to CHS (Ap ~ 5 and ι0 ~ 0.3). The Reynolds numbers in LHD

Heliotron device is a probable candidate of toroidal
magnetic confinement systems as thermonuclear fusion
reactor under steady-state operation because it can confine
plasma with only external coils and install the well-defineddivertor configuration. However, it is theoretically predictedthat it has a disadvantage for pressure driven magnetohydrodynamics (MHD) instabilities, which may limit theoperational regime of the plasma parameters such as beta,pressure gradient and/or so on [1]. In tokamaks, it is wellknown that the operational beta limits are quite consistent
with theoretical predictions of ideal linear MHD theory [2].
On the contrary, in helical plasmas, the MHD instability
effects on the operational beta range have not been clear. A
limited number of experimental researches about the effectof pressure driven MHD instabilities on the operational beta
range in heliotron devices have been reported, for examples,on Heliotron DR [3] and the compact helical system (CHS)[4]. There is difference of results in the previous works. The
experimentally achieved beta is consistent with the beta limit
theoretically predicted by a low-n (m/n = 5/4) ideal MHD
instability in Heliotron DR. On the contrary, the discharges
are maintained in unstable region predicted by low-n ideal
MHD stability calculations in CHS. Here m and n are thepoloidal and toroidal mode number of fluctuations,©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author’s e-mail: kiyowata@LHD.nifs.ac.jp1523

Watanabe K. et al., Relationships between the Prediction of Linear MHD Stability Criteria and the Experiment in LHD

plasmas are larger by two orders than those in the medium
sized heliotron devices such as CHS [6]. It is considered thatthe ideal pressure driven MHD mode plays an important rolein MHD properties, in especially core region. In the earlyphase of LHD experiments, high beta discharge over 2% wasachieved in an unfavorable magnetic configuration for MHD
stability with inwardly shifted vacuum magnetic axis, Rax =3.6 m [6]. Recently, we obtain higher beta plasma around 3%
at the low field operation of 0.5 T in unfavorable configurations on MHD stability, the Rax = 3.6 m configuration, without
obvious degradation of the global energy confinement [7]. In
this paper, in order to study the role of ideal pressure driven
MHD modes on the operational regime in LHD, we compare
the experimentally observed pressure gradients at resonant
rational surfaces with the theoretically predicted unstableregion for ideal pressure driven MHD instabilities in the Rax= 3.6 m configuration, carefully taking account of the toroidal
current effects.

2. Experimental setup
LHD has powerful measurement systems of profiles; forexample, electron temperature profiles are measured at over
100 radial positions by Thomson scattering measurement [8],which leads to the detailed comparative analysis betweenexperiment and calculation. We consider the m/n = 2/1 and1/1 modes as typical low-n MHD unstable modes at the coreand edge region, because the rational surfaces of 1/2 and 1/1
modes exist at ρ = 0.5 and ρ = 0.9, respectively, in the Rax =3.6 m configuration. Here ρ is a radial variable. ρ2 is
proportional to the toroidal flux and it is equal to unity at a
plasma edge. Experimental beta profile is estimated under anassumption that it is proportional to electron pressure profile.
Electron pressure profile is product of electron temperatureby Thomson scattering measurement and electron density by
FIR measurement. Volume averaged beta 〈β 〉 is defined as〈β 〉 ≡ (2/3)(Wp /Vp)(Bav2/2µ0). Here Wp is the plasma stored
energy by a diamagnetic measurement and µ0 is thepermeability in vacuum. Vp and Bav are the plasma volume
and the volume-averaged toroidal magnetic field strength invacuum, respectively. The high beta discharges in LHD aremaintained by NBI (Neutral Beam Injection), where a finite
net toroidal current is observed due to bootstrap current andOhkawa current [9]. The net toroidal current is measured by
the Rogowskii coil loop. Up to now, however, we dot not havea technique to measure the toroidal current profile, so that
we use a model current profile in the following calculations.

Fig. 1Pressure gradients observed experimentally at ρ = 0.5(core) in the Rax = 3.6 m configuration in 〈β 〉-dβ /dρdiagram under (a) co- and balanced-NBI, and (b) cntr.NBI. The theoretical prediction based on low-n (m /n =2/1) ideal pressure driven MHD mode under currentlessplasma is shown by solid and dashed lines.

correspond to the observed pressure gradients under co- andbalanced NBI. Observed net toroidal current is 0~25 kA/T,
where the 1/2 rational surfaces exist. Because the positive
toroidal current makes worse the stability condition thancurrentless cases, the currentless condition for MHD
equilibria gives an upper limit of the stability condition in
the discharges under co- and balanced NBI. Solid and dashedlines in Figs. 1 and 2 denote the contours of the growth rateof low-n (m/n = 2/1 and m/n = 1/1) ideal MHD modes (withglobal mode structure), γlow-n /ωA = 10–2 and 1.5 × 10–2, forcurrentless equilibria. The growth rate is calculated by a MHD
stability analyzing code (TERPSICHORE [10]) for various
assumed pressure profiles. Here ωA = vA0 /R0, vA0 and R0 are

3. Relationships between the prediction oflinear MHD stability and theexperiment
Figure 1 shows the experimentally observed pressure
gradients at ρ = 0.5 in the Rax = 3.6 m configuration in〈β 〉-dβ /dρ diagram under (a) co- and balanced NBI, and (b)cntr.-NBI. Here β is a local beta value estimated by thevolume-averaged toroidal magnetic field strength. The data
were obtained in 0.5 T to 1.5 T operation. Circles in Fig. 1(a)2524Watanabe K. et al., Relationships between the Prediction of Linear MHD Stability Criteria and the Experiment in LHD

Fig. 2in spite of low net toroidal current. As a result, thetheoretically predicted unstable region disappears in 〈β 〉-dβ /dρ diagram. Under cntr.-NBI, even below 〈β 〉 = 1.8%, theexperimentally observed pressure gradients exceed the upper
limit of those under co- and balanced NBI. The deposition
profiles under co- and cntr.-NBI are almost same in the Rax =3.6 m configuration. According to observation of magnetic
fluctuation, m/n = 2/1 mode is dominant comparing withhigher harmonic modes like m/n = 4/2. These experimental
results suggest that the global ideal MHD modes limit theoperational regime through the pressure gradient limit in thecore region nevertheless the achieved pressure gradients reachthe global ideal MHD unstable region with γlow-n = 1.5 × 10–2
ωA.
In Fig. 2, the observed pressure gradients at ρ = 0.9
exhibit slight saturation with the beta increase. However, theclear saturation of the pressure gradients in the edge has notbeen observed in the global MHD unstable region.
Nevertheless, the achieved pressure gradients are more deeply
in the unstable region compared with results in the core.
These results from Figs. 1 and 2 suggest that a criterion basedon linear MHD analyses, where the ideal MHD instabilities
affect the operational regime, depends on the plasma
parameters such as the magnetic shear or the magnetic
Reynolds number, which are quite different in between thecore and the edge. As another possibility, the beam pressure
effect is considered to explain the reason why the effect of
the ideal MHD instability on the local pressure gradients isdifferent in between the core and the edge. In low field
operation, where the high beta discharges are operated, thebeam pressure is probably large. The qualitative estimationof beam pressure effects is one of our future research topics.
As another approach to make clear whether global ideal MHD
modes limit the pressure gradients in the edge like in the core,
it is considered that the observed pressure gradients could beextended to other magnetic configurations. This is also an
important subject for study.

Pressure gradients observed experimentally at ρ = 0.9(edge) in the Rax = 3.6 m configuration in 〈β 〉-dβ /dρdiagram.

the Alfven velocity and the major radius at the magnetic axis.
The dotted lines are the stability boundary of Mercier modes(with a highly localized mode structure / high-m limit) [11].
The maxima of the achieved pressure gradients seem tosaturate against the contour of γlow-n /ωA = 1.5 × 10–2 in theintermediate beta range of 1~1.8%. When 〈β dia〉 exceeds~1.8%, the maximum of the pressure gradient more thandoubles. These experimental observations coincide withviolation of low-n modes and stabilization due to spontaneous
generation of a magnetic well due to the Shafranov shift.
Triangles in Fig. 1(b) correspond to the observedpressure gradients under cntr.-NBI, where the observed nettoroidal current is –6~8 kA/T. According to a numericalanalysis [9], the bootstrap current is estimated to be about
13 kA at 〈β 〉 = 1.8% and 0.75 T. Therefore, we shouldcarefully treat the effect of the toroidal current on MHD
equilibria and stabilities. We use a model toroidal current
profile asj = j1*(1 – ρ2)2 + j2*(1 – ρ2)*ρ2,

4. Summary and discussion
We study the role of ideal MHD modes on theoperational regime with high beta discharges in LHD bycomparing between the experimentally observed pressure
gradients and the theoretically predicted unstable region ofideal pressure driven MHD modes. Low-n (n = 1) idealpressure driven MHD instabilities give a constraint on theoperational regime of the pressure gradients at the core region
and in the intermediate beta range of 1~1.8%. Even for thesame vacuum configuration, the net toroidal current
conditions together with the pressure profile significantly
change the ideal MHD stability through the variation of MHD
equilibria. The freedom of the pressure and the toroidal
current profiles extends the operational regime, which maylead to achievement of high beta discharges under theunfavorable magnetic configuration for MHD stabilities likethe Rax = 3.6 m configuration in LHD experiments. However,there are a lot of low order rational surfaces in addition to ι =(1)where 1st and 2nd terms are considered as the Ohkawa
current ( j1 > 0 under co-NB and j1 < 0 under cntr-NB) andbootstrap current ( j2 > 0), respectively [9]. In eq. (1), we
assume the bootstrap current 13.3 kA/T (10 kA at 0.75 T),
which is independent of 〈β 〉 for simplification, and determine
such j1 as the net toroidal current is –6 kA/T, which is theminimum of the observed toroidal current. Under cntr.-NBI,
the Shafranov shift at edge is suppressed by positive toroidal
current and that at core is enhanced due to negative toroidal
current, which leads to deeper magnetic well formation andlarger magnetic shear in core region comparing with theequilibrium with monotonic negative toroidal current profile3525

Watanabe K. et al., Relationships between the Prediction of Linear MHD Stability Criteria and the Experiment in LHD

1/2 and 1. Since Mercier modes do not affect the operational
regimes (see Figs. 1 and 2), it is considered that there is anupper limit of the wave number of the ideal MHD modes,
which affects the operational beta range and the pressure
gradients. Here we analyze the operational beta range basedon the ideal MHD theory. We know that the observed pressure
gradients are in the nonlinear saturation phase. However, since
it has not been clear how the pressure driven MHD instability
affects the experimental operation regimes of the helicalsystems, our approach (evaluating the experimentallyachievable pressure gradients by the linear growth rate and/orMercier parameter) would be useful, because it could be areference for more complicated nonlinear analyses, and acriterion for a reactor design.
We analyze only gas-puffing discharges in order to study
the condition on the operational regime achieved in stationary.
In the transient state like as just after pellet injection, theminor internal disruption with the saw-tooth-like fluctuation
is observed [12], and the observed pressure at core gradientsexceed the boundary of low-n (m/n = 2/1) ideal MHD stability[13], shown by lines in Fig. 1. Up to now, such state is not
maintained in stationary though it does not go to a majordisruption.
The authors deeply acknowledge the LHD deviceengineering group for their great effort of LHD operation andmaintenance. One of them (K.Y.W) would like to thank forfruitful discussion to Dr. Yuji Nakamura.

References[1] M. Wakatani, Y. Nakamura and K. Ichiguchi, FusionEng. Des. 15, 395 (1992).
[2] ITER Physics Basis, Nucl. Fusion 39, 2137 (1999).
[3] N. Yanagi et al., Nucl. Fusion 32, 1264 (1989).
[4] S. Okamura et al., Nucl. Fusion 30, 1337 (1999).
[5] A. Iiyoshi et al., Nucl. Fusion 39, 1245 (1999).
[6] S. Sakakibara et al., Nucl. Fusion 41, 1177 (2001).
[7] O. Motojima et al., Nucl. Fusion 43, 1674 (2003).
[8] K. Narihara et al., Rev. Sci. Instrum. 72, 1122 (2001).
[9] K.Y. Watanabe et al., J. Plasma and Fusion Res. Ser, 5,124 (2002).
[10] W.A. Cooper, Plasma Phys. Control. Fusion 34, 1011
(1992).
[11] A.H. Grasser, J.M. Green and J.L. Johnson, Phys. Fluids
18, 143 (1975).
[12] S. Ohdachi et al., in Proceedings of 13th stellarator
workshop, 2001, Canberra (http://wwwrsphysse.
anu.edu.au/admin/stellarator/proceedings.html).
[13] K.Y. Watanabe et al., Fusion Sci. Technol. (to bepublished).

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