Proceedings of ITC/ISHW2007

Analysis of MHD stability in high-β plasmas in LHD
Noriyoshi NAKAJIMA1) , Satoru SAKAKIBARA1) , Kiyomasa WATANABE1) , Stuart HADOSON2) ,and Cris HEGNA3)
1)

National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, Japan
Princeton Plasma Physics Laboratory, PO Box 451, Princeton, NJ 08543, USA
3)
Department of Engineering University of Wisconsin-Madison, WI 53706, USA

2)()
Up to now, theoretically idealized MHD equilibria reflecting experimental conditions have been used inorder to examine the ideal MHD stability. This approach has been useful from the aspect of investing generalproperties of the ideal MHD stability. Since the properties of a three dimensional MHD equilibirium with large
Shafranov shift significantly change by the pressure profile, the current profile, and the boundary condition,
however, ideal MHD stability analysis based on theoretically idealized MHD equilibria is considered not to beenough to investigate the proper MHD atbility of experimentally obtained MHD equilibria. Indeed, it is shownthat ideal MHD stability based on the realistic reconstructed MHD equilibrium with fine structures is diﬀerent
from that based on the theoretically idealized MHD equilibrium. Especially, it is firstly reported that high-nballooning modes are destabilized in the magnetic well region with tokamak-like magnetic shear.
Keywords: high-n ballooning modes, LHD
DOI: 10.1585/pfr.1.001

1Introduction

In three-dimensional configurations, the confinement region is surrounded by the stochastic magnetic field lines
related to magnetic islands or separatrix, leading to thefact that the plasma-vacuum boundary is not so definite
compared with tokamaks that the various modulations of
the plasma-vacuum boundary will be induced around thestochastic region by synergetic eﬀects between a transport
around the stochastic region and a large Shafranov shift of
the whole plasma or a large Pfirsch-Schluter current, in especially high-β operations.
To examine such modulation eﬀects of the plasmaboundary on MHD instabilities, high-β plasmas allowing
a large Shafranov shift or a large Pfirsch-Schluter current
are considered in the inward-shifted LHD configurations
with the vacuum magnetic axis Rax of 3.6 m, so that it has
been found that the free boundary motion of MHD equilibrium or the whole plasma outward-shift due to a large
Pfirsch-Schluter current has significant stabilizing eﬀects
on ideal MHD instabilities, leading to partially resolving
the discrepancy on MHD stability between experimental
results and theoretical analyses [1].
Although experimental aspects on the boundary, thepressure profile, and the current condition are included in
the equilibria used in Ref.[1], such equilibria are still theoretically idealized judging from the experimental point ofview [2]. Thus, it is needed to use equilibria which are
more relevant to the experimental conditions, in order tomore clarify MHD stability in planar axis Heliotron configuration with a large Shafranov shift like LHD. The purauthor’s e-mail: nakajima.noriyoshi@nifs.ac.jp

pose of the present research is to clarify MHD stability especially in IDB-SDC plasma or high-β plasma of LHD bycomparing between theoretically idealized MHD equilibria and experimentally reconstructed MHD equilibria. For
such a purpose, especially, high-n ballooning local modestability analysis is performed, because such local mode
analysis does not need whole information of MHD equilibrium. The precise information of MHD equilibrium near
the plasma periphery is not needed, once the core MHD
equilibrium is consistently reconstructed to experimentalconditions. This research might lead to more deeper understanding of MHD equilibrium and stability in the planaraxis Heliotron configuration with a large Shafranov shift
like LHD.

2 in theoretically idealized MHD equilibria
In order to clarify dependence of the stability properties of
the ideal high-n ballooning on MHD equilibrium, firstly,
high-n ballooning stability analyses are performed for theoretically idealized MHD equilibria in the inward-shiftedvacuum configuration with Rax = 3.75 m.

2.1

in currentless MHD equilibria with peakedpressure profile

The properties of currentless MHD equilibria with apeaked pressure profile; P(s) = P0 (1 s )2 , under the fixedboundary condition are shown in Fig.1, where s is thenormalized toroidal flux. As β increases by using P0 ,

Proceedings of ITC/ISHW2007
qualitatively similar to those in fixed boundary equilibria,change of iota in free boundary equilibrium is more significant than that in fixed boundary equilibrium, and formation of magnetic islands is suggested in shearless region
judging from the spikes of Di . The spike comes from thedivergence of the Pfirsch-Sch”ulter current indicating existence of the magnetic island. The corresponding normalized growth rates of high-n ballooning modes are shown inFig.4. Most significant diﬀerences between fixed boundary
equilibria and free boundary equilibria are that helical-likeballooning modes destabilized in the magnetic hill region
with helical-like magnetic shear extend to the magnetic
well region with tokamak-like magnetic shear. Since ballooning formalism breaks near shearless region, the global
mode analysis migh be needed for precise stability.

tokamak-like magnetic shear is created near the magnetic
axis. Although the magnetic hill still remains near theplasma periphery, the Mercier stability in the magnetic
hill region is improved as β increases. Boundary between magnetic well and hill exists in helical-like maggnetic shear region. The corresponding normalized growth
rates of high-n ballooning modes are shown in Fig.2. Highn ballooning modes are destabilized in the peripheral magnetic hill region with helical-like magnetic shear. As β increases, properties of the high-n ballooning modes change
from helical-like ones with strong magnetic field line dependence to tokamak-like ones with weak magnetic field
line dependence. Helical-like high-n ballooning modes
become unstable only near the magnetic field line with
α = ζ − qθ = 0 where the local magnetic curvature isbaddest. On the other hand, tokamak-like high-n ballooning modes become unstable independent of the magnetic
field line, even in the magnetic field line with α = π/M (Mis the toroidal field period of the MHD equilibrium) where
the local magnetic curvature is locally good.

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Fig. 3 β-dependences of the rotational transform , magnetic welland hill −V”, and Mercier criterion DI with pressure profile P in currentless MHD equilibria with the peaked pressure profile under the free boundary condition.

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Fig. 1 β-dependences of the rotational transform , magnetic welland hill −V”, and Mercier criterion DI with pressure profile P in currentless MHD equilibria with the peaked pressure profile under the fixed boundary condition.

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Fig. 4 β-dependence of the normalized growth rates γτA0 for
three diﬀerent plasma volumes. The most right column
corresponds to 3

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Fig. 2 β-dependence of the normalized growth rates γτA0 for
three diﬀerent plasma volumes. The most right column
corresponds to 1

2.2 in currentless MHD equilibria with broadpressure profile

In order to investigate eﬀects of the boundary condition, the MHD equilibria are created under the free boundary condition. In Fig.3, the properties of currentless MHD
equilibria under the free boundary condition is shown. Although the β-dependences of the magnetic shear, magnetic
well, and Mercier stability in free boundary equilibria are

In order to investigate eﬀects of the pressure profile, thecurrentless MHD equilibria are made with a broad pressure profile; P(s) = P0 (1 − s2 )5 . In Fig.5, the properties of
the currentless MHD equilibrium with the broad pressure
profile under the fixed boundary is shown. The steep pressure gradient near the plasma periphery coming from thebroad pressure prifile makes magnetic hill region narrow.

1.0

Proceedings of ITC/ISHW2007
The corresponding normalized growth rates are indicatedin Fig.6. As well as the MHD equilibria with peaked pressure profile, high-n ballooning modes are destabilized in
the peripheral magnetic hill region with helical-like magnetic shear. As β increases, properties of the high-n ballooning modes change from helical-like ones with strong
magnetic field line dependence to tokamak-like ones withweak magnetic field line dependence.

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Fig. 7 β-dependences of the rotational transform , magnetic welland hill −V”, and Mercier criterion DI with pressure profile P in currentless MHD equilibria with the broad pressure profile under the free boundary condition.

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Fig. 5 β-dependences of the rotational transform , magnetic welland hill −V”, and Mercier criterion DI with pressure profile P in currentless MHD equilibria with the broad pressure profile under the fixed boundary condition.

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Fig. 8 β-dependence of the normalized growth rates γτA0 for
three diﬀerent plasma volumes. The most right column
corresponds to 7

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Fig. 6 β-dependence of the normalized growth rates γτA0 for
three diﬀerent plasma volumes. The most right column
corresponds to 5

The eﬀacts of the free boundary are shown in Figs.7
and 8 for currentless MHD equilibria with a broad pressure
profile. The diﬀerences between fixed boundary and freeboundary are same as the case of currentless MHD equilibria with peaked pressure profile.

3in reconstructed MHD equilibria

In this section. the stability of high-n ballooning modes isinvestigated in the reconstructed MHD equilibria.
The Fig.9 denotes the properties of both the reconstructed MHD equilibrium and the variations corresponding to IDB-SDC plasma in the standard configuraion withRax = 3.75m. The corresponding normalized growth rates
are shown in Fig.10. The behaviors of all quantities ofequilibrium and stability are similar to those in the theoretically idealized MHD equilibria.

1.0

The properties of the reconstructed MHD equilibrium
corresponding to IDB-SDC plasma in the outward-shiftedvacuum configuraion with Rax = 3.85m are denoted in
the upper row of Fig.11. The corresponding normalized
growth rate is shown in the left column of Fig.??. The
most significant feature of stability of the high-n ballooning modes is that helical-like ballooning modes appear in
the both magnetic hill region with helical-like magnetic
shear and magnetic well region with tokamak-like magnetic shear. Moreover, high-n ballooning modes in themagnetic well region with tokamak-like magnetic shear
are more tokamak-like ballooning modes than those in themagnetic hill region with helical-like magnetic shear, because the magnetic field lines where the mode is unstable are wider in the magnetic well region with tokamaklike magnetic shear than in the magnetic hill region withhelical-like magnetic shear. As is understood from thepressure profile shown in upper row of Fig.11, the reconstructed pressure profile has fine structures, namely, slightstair-case like structures. Although those fine structures
are not so significant, it is considered that such fine structure changes the stability criterion of the high-n ballooning
modes through the balance between stabilization eﬀect due
to the local magnetic shear and destabilization eﬀect due tothe local magnetic curvature. High-n ballooning modes in
the magnetic well region with tokamak-like shear might

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Proceedings of ITC/ISHW2007
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Fig. 9 β-dependences of the rotational transform , magnetic welland hill −V”, and Mercier criterion DI with pressure profile P in reconstracted MHD equilibria and the variationsin the stabdard vacuum configuration with Rax = 3.75.
The upper (lower) row includes the β-variations of whole(core) region.
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Fig. 10 β-dependence of the normalized growth rates γτA0 for
three diﬀerent plasma volumes. The left (right) column
includes the β-variations of whole (core) region.

lead to core density collapse experimentally reported.
The properties of the reconstructed MHD equilibrium corresponding to high-β plasma in the inward-shiftedvacuum configuraion with Rax = 3.60m are denoted in
the lower row of Fig.11. The corresponding normalized
growth rate is shown in the right column of Fig.??. As wellas the above case of IDB-SDC, fine structures of the pressure profile makes the significant change in the Mercier
criterion, leading to the non-monotonic change in the normalized growth rate as os shown in the right column ofFig.??.

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Fig. 11 β-dependences of the rotational transform , magnetic welland hill −V”, and Mercier criterion DI with pressure
profile P in reconstracted MHD equilibria. The upper(lower) row corresponds to the reconstructed equilibriumin the outward-shifted (inward-shifted) vacuum configuration with Rax = 3.85 (3.60) m.
3.02.0< 0.50.0

Summary

Up to now, theoretically idealized MHD equilibria reflecting experimental conditions have been used in order to examine the ideal MHD stability. This approach has been
useful from the aspect of investing general properties of
the ideal MHD stability. As is well know, however, theproperties of a three dimensional MHD equilibirium withlarge Shafranov shift significantly change by the pressure
profile, the current profile, and the boundary condition. Indeed, high-n ballooning stability is completely diﬀerent

Fig. 12 β-dependence of the normalized growth rates γτA0 . The
left (right) column corresponds to the reconstructed equilibrium in the outward-shifted (inward-shifted) vacuum
configuration with Rax = 3.85 (3.60) m.

between equilibria under fixed boundary and those underfree boundary. Although free boundary equilibria are morestable than fixed boundary ones in the inward-shifted vacuum configuration with Rax = 3.60m, free boundary equilibria are more unstable than fixed boundary ones in thestandard vacuum configuration with Rax = 3.75m. Moreover, it is shown that ideal MHD stability based on therealistic reconstructed MHD equilibrium with fine structures is diﬀerent from that based on the theoretically idealized MHD equilibrium. Especially, it is firstly shownthat high-n ballooning modes are destabilized in the magnetic well region with tokamak-like magnetic shear, whichmeans that high-n ballooning stability is quite sensitive toMHD equilibrium. Stability analyses based on idealized
MHD equilibria might not be enough to interpret experimental results on MHD stability. More extensive stability
analyses based on reconstructed MHD equilibria will beneeded.
[1] N. Nakajima, et al., Nucl. Fusion 46 (2006) pp.177-199.
[2] K.Y. Watanabe, et al., Plasma Phys. Control. Fusion, 49(2007) pp.605-618.