Proceedings of ITC/ISHW2007

3-D ideal MHD stability of super dense core plasma in LHD
Y. Narushima, K.Y. Watanabe, R. Sakamoto, Y. Suzuki, S. Sakakibara, S. Ohdachi,
H. Yamada and LHD experimental group
W.A. Cooper a
National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, JapanaCentre de Recherches en Physique des Plasmas,
Association Euratom / Confederation Suisse, EPFL, 1015 Lausanne, Switzerland

The characteristics of confinement properties and magnetohydrodynamics (MHD) stabilities of the super dense core(SDC) plasmas are intensively studied in the Large Helical Device (LHD) experiment. The SDC plasmas are produced inthe configuration with outward shifted magnetic axis and rapid fueling into the core region by pellet injection. The SDC
plasmas have high electron pressure due to the large electron density (ne~1021[m-3]) in the core region which induces alarge Shafranov shift (∆/aeff>0.5). It is useful to apply the production method of the SDC plasmas for obtaining high-beta
plasmas with high pressure in the core region. In this study, the characteristics of the ideal MHD instabilities of the SDC
plasmas are analyzed by using the 3-D ideal MHD code TERPSICHORE. The calculated equilibrium of the typical SDC
plasma with the central beta of β0 = 5[%] solved as the result of the HINT/HINT2 codes shows that a deep magnetic welldepth appears around the core region due to the large Shafranov shift. The Mercier criterion (DI) shows that the plasma isstable all over the plasma region and consequently the low-n interchange modes do not appear. Although the typical SDC
plasma is stable to the interchange mode, it is worthwhile to investigate the other pressure profiles and the magnetic field
configurations to know the role of the MHD instabilities on the experimentally obtained SDC plasmas. In the experiment,
the SDC plasmas cannot be produced in the configuration with inward shifted magnetic axis. Such configuration tends to
be MHD unstable, in which the core resonant interchange modes become unstable in case of relatively low-beta plasma
with finite pressure gradient in the core region. We will investigate how the MHD instabilities affect the production of theSDC plasma with steep pressure gradient in the core region in the configuration with inward shifted magnetic axis.

Keywords: Large Helical Device, MHD, Internal Diffusion Barrier,

81. Introduction

author’s e-mail:narusima@LHD.nifs.ac.jp

6β0max[%]The characteristics of confinement properties andmagnetohydrodynamics (MHD) stabilities of the superdense core (SDC) plasmas are intensively studied in theLarge Helical Device (LHD) experiment [1]. Figure1
shows the attainable central beta β0 against to the positionof the magnetic axis Rax. The SDC plasmas are producedin the outward-shifted configuration Rax > 3.75[m]. On
the other hand, in the inward-shifted configuration Rax <3.70[m], SDC plasmas are not obtained. The steeppressure gradient maintains at the core region for Rax =3.75m case whereas the gradual gradient appears in theinward-shifted configuration of Rax = 3.6m as shown inFig.2. We focus on stabilities of magnetohydrodynamics(MHD) of SDC plasmas in this study. MHD stabilities
can be categorized into the ideal mode and resistive mode.
Each mode, furthermore, can be classified into the low-n4
20

3.60

3.65

3.70

3.75

Rax[m]

3.80

3.85

Fig.1 Magnetic axis (Rax) dependence of central beta(β0). Threshold is seen at between Rax = 3.7m
and 3.75m. The β0 seems to be suppressed ininward shifted configuration (Rax < 3.7).

Proceedings of ITC/ISHW2007

7β[%]

PeakedBroad(a)

(e)

(i)

(b)

(f)

(j)

(c)

(g)

(k)

6 (d)

(h)

(l)

Rax = 3.85[m]6ParabolicRax[m]
3.60

54

3.65

β0[%]

3Rax = 3.60[m]

23.70

10
2.5

3.0

3.5

4.05.03Fig.2 Normalized pressure profiles. Filled and open
circles indicate the case of Rax = 3.60[m] and3.85[m] respectively. The steep gradient is seenin Rax = 3.85[m].

0R[m]4.5(global) mode and high-n (local) mode. Here, n means thetoroidal Fourier mode number. In this study, we haveinvestigated about low-n ideal mode, high-n ideal mode,and resistive high-n mode. The latter mode relates to theplasma transport. The analyses of low-n ideal mode arecarried out by using 3-D ideal MHD stability codeTERPSICHORE [2]. The Mercier index shows thestability of ideal high-n mode. The resistive high-nmode is examined from local parameters of specificvolume, Reynolds number and pressure gradient. The
equilibria analyzed here are solved as the result of theHINT/HINT2 codes [3,4].
This paper is organized as follows. Characteristics ofMHD stability of experimentally obtained plasmas aredescribed in the following section. In section 3, equilibria
with modeled pressure profiles are analyzed to orient theγ/ωA (m/n = 2/1) 1(a)

0.1
(a)

010.01(b)1E-3(b)000.5 ρ1.03.60 3.65 3.70 3.75 3.80 3.85

Rax[m]

Fig.3 Dependence on the growth rate of m/n = 2/1 modeof Rax. Typical mode structures in case of (a)Rax= 3.6[m] and (b)Rax = 3.65[m] are also shown.

00.5

1.03.75

ρFig.4 Contour lines of Mercier index and the position ofι/2π = 1/2 surface in ρ - β0 space. Dashed andsolid thin lines indicate DI = 0 and 0.2
respectively. Bold solid lines correspond to theposition of ι/2π = 1/2.

experimental results. Finally, we will discuss andsummarize in section 4.

2. Characteristics of MHD stability ofexperimentally obtained plasmas
Experimentally obtained SDC plasmas have a steeppressure gradient in the core region which corresponds toaround the position of the rotational transform ι/2π = 1/2.
We focus on, therefore, the m/n = 2/1 mode as a low-n
ideal MHD mode. Here, m means the poloidal Fouriermode number. The analyses of low-n ideal mode arecarried out by using 3-D ideal MHD stability code
TERPSICHORE. The relationship between thegrowth rate and Rax is shown in Fig.3. In case of Rax =3.6[m], the m/n = 2/1 mode is unstable with the growth-2rate γ/ωA = 1.62x10 and the wide mode structureappeasers as shown in Fig.3(a). In case of Rax = 3.65[m],-3
the growth rate is 1.74x10 and the width of the structure
is narrower than that in case of Rax = 3.6m. It is reported
that the growth rate of γ/ωA = 10-2 is defined as thestability boundary [5]. It can be, therefore, consideredthat the ideal low-n (m/n = 2/1) interchange mode isstable except for the configuration with Rax = 3.6[m].

3. MHD stability of analytical equilibrium
Equilibria with modeled pressure profiles areanalyzed to orient the experimental results. Here, threekinds of pressure profiles of parabolic (β = β0(1 - ρ2) (1 ρ8)), peaked (β = β0(1 - ρ2)2), and broad (β = β0(1 - ρ4) (1- ρ8)) are adopted. The contour lines of Mercier index DI
are shown in Fig. 4 where dashed and solid thin lines

(a) Peaked-2γ/ωA = 1.84x10β0 = 1.9%

6 (a) γ/ωA = 1x100(b) DI = 0β0[%]2(b)Parabolic
1.0 γ/ω = 1.10x10-2A
β=
0.5 0 2.6%0-2

PeakedParabolic
BroadExperiment

4Sξ [a.u.]

Proceedings of ITC/ISHW2007

0.2

0.4

ρS0
3.6

0.6

0.8

1.0

Fig.5 Typical structure of ξ of m/n = 2/1 mode. (a)Peaked pressure profile (b) Parabolic pressure
profile.

indicate DI = 0 and 0.2 respectively. Bold solid lines
correspond to the position of ι/2π = 1/2. In equilibria withRax = 3.60[m] (Fig.4(a)(e)(i)), Mercier unstable regions
dominate the ρ - β0 space. Mercier unstable region
generally narrows with increasing Rax. In particular, theequilibrium with peaked profile in Rax = 3.75[m] is
Mercier stable all over the region (Fig.4(d)).
In order to investigate the global ideal mode, wehave analyzed the stability of m/n = 2/1 mode whichmainly resonates at core region. The low-n ideal modetends to be unstable when the resonant surface obviously
enters the Mercier unstable region. The ideal low-ninterchange mode is destabilized (as above mentioned,growth rate becomes γ/ωA > 10-2) in Rax = 3.60[m] withpeaked and parabolic profiles, in which the modestructures are shown in Fig.5. In each cases, (peaked(Fig.5 (a)) and parabolic Fig.5 (b)), the growth rate isγ/ωA = 1.84x10-2 and 1.10x 10-2 respectively. The shapeof mode structure is similar to each other, in which thefull width at half maximum, ∆ρ, is ∆ρ ~ 0.1 and theposition of the peak corresponds to around ρ = 0.54. In
the other equilibria (Rax >3.6[m]), the low-n mode (m/n =2/1) does not appear.
The orientation of experimental results from theviewpoint of ideal MHD instability is shown in Fig.6.
The left figure (Fig.6 (a)) shows the global ideal MHD
unstable region by solid lines and the right one (Fig.6 (b))
indicates the unstable region of the local MHD mode.
The experimental results of central beta β0 aresuperimposed. The experimentally obtained plasmas
generally locate far from the MHD unstable (m/n = 2/1mode) region as shown in Fig.6 (a). Some ofexperimental results (Rax = 3.6[m] and 3.65[m]), on the

3.7
Rax[m]

Fig.6 Ideal MHD unstable region of (a) global mode and(b) local mode. Black, red and blue solid line
indicates pressure profile of peaked, parabolicand broad respectively. Closed circles shows
experimental result of central beta.

other hand, enter the Mercier unstable region as shown in
Fig.6(b). These results suggest that the ideal global mode
does not affect the beta (gradient) in outward-shifted
configuration. High-n ideal mode might influence theMHD characteristics because the unstable region ofhigh-n ideal mode covers the region where theexperimental data exists.

4. Discussion and summary
The ideal MHD stability of SDC plasma isinvestigated. The results show that the SDC plasmas arestable against to the ideal MHD mode. In order toinvestigate the other MHD stability, we will discuss thepossibility of the resistive mode here. The resistive MHD
instability is well correlated with the transport via theresistive pressure gradient driven turbulence. The
coefficient of the particle diffusion D is defined by themagnetic curvature (κn), Reynolds number (S) andpressure gradient (dβ/dρ) as follows [6-7].

2.5
2.0(V''/S)(dβ/dρ)

1.5

ρ

0.5
0.6
0.7

1.0
0.5
0.0

3.6

3.7

3.8 R [m]ax

Fig.7 Qualitative behavior of diffusion by resistive
pressure gradient driven turbulence.

Proceedings of ITC/ISHW2007

D∝κ n dβS dρ(1)The qualitative behavior of the diffusion against tothe Rax is shown in Fig.7. The deferential of specificvolume V'' is used here instead of κn to estimate thecoefficient of the particle diffusion. The plot at ρ = 0.7 of
Rax = 3.6[m] indicates the small value, which comes fromthe local flatting of the pressure gradient. Figure 7 showsthat the diffusion by the resistive pressure gradient driven
turbulence generally continuously decreases with the Rax.
The diffusion at ρ = 0.5 and 0.6, in particular, goes tozero around Rax = 3.7[m] where the threshold is seen as
shown in Fig.1. In other word, diffusion tends to becomelarge in the inward-shifted configuration. Furtheranalyses about resistive mode are required.

Acknowledgement
This study was supported by a Grant-in-Aid for
Scientific Research (A)(No.17206094) from the Ministryof Education, Culture, Sports, Science and Technology ofJapan. This work was supported by NIFS under Contract
No.NIFS07ULPP535.

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