J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)218–221
000–000

The Effect of Non-Axisymmetry of Magnetic Configurations
on Radial Electric Field Transition Properties in the LHD
YOKOYAMA Masayuki, IDA Katsumi, YOSHINUMA Mikiro, SHIMOZUMA Takashi,
WATANABE Kiyomasa, MURAKAMI Sadayoshi1, WAKASA Arimitsu2, KUBO Shin,
TAKEIRI Yasuhiko, NARIHARA Kazumichi, MORITA Shigeru, TANAKA Kenji,
YOSHIMURA Yasuo, NOTAKE Takashi, OSAKABE Masaki, ITOH Kimitaka,
KOMORI Akio, MOTOJIMA Osamu and LHD experimental G.
National Institute for Fusion Science, Toki, 509-5292, Japan
Department of Nuclear Engineering, Kyoto Univ., Kyoto, 606-8501, Japan
2
Graduate School of Engineering, Hokkaido Univ., Sapporo, 060-8628, Japan
1(Received: 9 December 2003 / Accepted: 12 March 2004)

Abstract
Transition property of the radial electric field (Er) in LHD have been theoretically investigated and also applied
to explain experimental results. Especially, effects of the helicity of the magnetic configuration on the condition torealize the electron root are examined. Larger helicity makes the threshold collisionality higher. This is attributed to
the nonlinear dependence of Γe(Er) in a low collisional regime. This interesting feature predicts that the threshold
temperature becomes higher for a case of smaller helicity. The variation of the threshold density anticipated from theanalysis for cases with different magnetic axis position is qualitatively verified in the density scan experiment.
Keywords:
LHD, radial electric field (Er), ion and electron root, effective helicity, magnetic configuration

1. Introduction

ambipolar Er . The density scan experiment was conductedmotivated by this prediction and predicted feature has beensuccessfully verified as described in Sec. 3. Finally, asummary is given in Sec. 4. Basic information on neoclassicaltransport calculation is available in ref. [6].

It has been theoretically predicted that plasma
confinement can be improved in non-axisymmetric helicalsystems by utilizing the electron root of the ambipolar radial
electric field, Er , for example [1-3]. Recently, the validity ofthis concept was also demonstrated experimentally in LHD[4], in which the neoclassical ion heat diffusivity reduces by
the large positive Er at the edge region of neutral-beam-heatedplasmas [5].
A comprehensive understanding of how the parameterregion of the electron root depends on plasma parameters isof vital importance for realizing improved confinement inLHD. In ref. [6], calculations in a wide parameter range were
performed for this purpose. It was clarified that the thresholdelectron temperature Te(b) for realizing the electron root
depends on n e0.4 and also on B0.4, here ne is the electron
density and B the magnetic field strength at the magnetic axis.
It is also shown that Te(b) depends strongly on ∇Te and only
slightly on ∇Ti and ∇ne . Here, Ti is the ion temperature. In
this paper, the effects of the non-axisymmetry (helicity) of amagnetic configuration on Te(b) is focused in Sec. 2 for LHD
by utilizing the configuration flexibility to change the helicityin LHD. The density threshold is predicted to vary dependingon the helicity based on the analysis of the neoclassical2. Effects of the helicity of a magnetic
configuration on the threshold electron
temperature for electron root regime inLHD
Calculations over a wide parameter range wereperformed in ref. [6], as briefly mentioned in Sec. 1. In thissection, the effect of the helicity of the magnetic configuration
is explained. Possibility of multiple Er solutions and thetransition from ion root to electron root based on theambipolar condition is attributed to the helicity of themagnetic configuration (for example, [1-3]). The effect of thehelicity on realizing the electron root has been alreadyinvestigated [7], and findings clarified that the required
temperature becomes lower as the helicity is increased as aparameter. This interesting feature is now re-considered inLHD configurations in order to provide information regarding
to experiments. Here, control of the vacuum magnetic axis
©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: yokoyama@nifs.ac.jp1218

Yokoyama M. et al., The Effect of Non-Axisymmetry of Magnetic Configurations on Radial Electric Field Transition Properties in the LHD

becomes apparent through the increase of Γe around Er ≈ 0 inthe 1/ν regime. Based on this fact, multiple solutions ofambipolar Er becomes possible for cases with larger εh,eff (Rax= 3.75 m and 3.90 m) while the only ion root is allowed for aposition Rax is considered for varying the helicity of magnetic
configurations. Figure 1 shows the radial ( ρ) profile of theeffective helicity, εh,eff , for three cases with different Rax : 3.60m, 3.75 m, and 3.90 m. The ε h,eff is calculated by the⎯ 2π/16)(ν/vd2)D]3/2 with the GIOTA code
definition, εh,eff = [(9√[8], which is based on the bounce-averaging method. Here, νis the collision frequency, vd is the drift velocity and D is theparticle diffusion coefficient in the 1/ν regime, respectively.
It is seen that εh,eff increases as Rax is increased, that is,as the configuration is outwardly shifted in this range of Rax .
This εh,eff is utilized for the calculation of neoclassical particle
fluxes based on the analytical formulae using the single
helicity model of a magnetic field (such as in ref. [2]). Thus,
the effect of the helicity of a magnetic field is taken into
account through the different value of εh,eff depending on Rax .
The Er is determined with the ambipolar condition, Γe = Γ i ,where Γj denotes the particle flux of plasma species with j =e for electron and j = i for ion.
The Er diagrams for these three cases are shown in Fig.
2 on (Ti , Te) plane. Calculations are performed at ρ = 0.8 with
B = 1.5 T with assumed ne and Te,i profiles of ne( ρ) = ne(0)(1– ρ8) with ne(0) = 1 × 1019 m-3 and Te,i( ρ) = Te,i(0)(1 – ρ2).
This diagram is obtained with varying Te,i(0) while fixing
ne(0). The hydrogen plasma is assumed. The Er is positive
above the upper boundary and negative below the lowerboundary. The surrounded region corresponds to the regionof multiple Er solutions. The εh,eff is estimated to be about
0.05, 0.14, and 0.27 for Rax = 3.60 m, 3.75 m, and 3.90 m,respectively. It can be clearly recognized from Fig. 2 that Te(b)(Te for lower boundary) increases as εh,eff is decreased, whichis consistent with the dependence reported in ref. [7]. This
result indicates that the density threshold for realizing theelectron root becomes higher for a case with larger εh,eff ,based on the result shown in Fig. 2 and the fact that Te(b) ∝ne0.4 if the temperatures are almost the same among different
Rax cases.
The condition for Te = Ti = 1 keV is now considered as
an example (shown by the dot in Fig. 2). This condition
locates in the region of multiple solutions for the case of Rax= 3.90 m, marginally enters to the region of multiple solutions
for the case of Rax = 3.75 m, and still locates in the region ofion root for the case of Rax = 3.60 m. The normalized
collisionality, νh,e* is plotted in Fig. 3. The νh,e* is defined asνh,e* = νe /[εh,eff3/2vT ι/2πR] with νe being the electron-ion
collision frequency, ι/2π the rotational transform, and vT theelectron thermal velocity. This definition is introduced justby replacing the value of the single helicity to the value ofthe effective helicity on the collision frequency separating theplateau and 1/ν regimes [9]. The νh,e* = 1 corresponds theboundary between the plateau and 1/ν regimes. As forinformation, εh,eff is also plotted. It is recognized that νh,e*becomes effectively lower as εh,eff is increased even with thesame real collision frequency, that is, effectively locates
deeper in the low collisional regime for larger εh,eff case. Thisaffects the behavior of neoclassical particle flux as shown in
Fig. 4. As νh,e* is decreased, the nonlinear dependence of Γe2219

Fig. 1

Radial profile of εh,eff for Rax = 3.60 m, 3.75 m and3.90 m.

Fig. 2

Dependence of Er diagram for Rax = 3.60 m, 3.75 m and3.90 m on (Ti , Te) plane (at ρ = 0.8 with B = 1.5 T). As for
information, ε h,eff , is denoted in the bracket,respectively.

Fig. 3The νh,e* and εh,eff at ρ = 0.8 with Te = Ti = 1 keV for Rax =3.60 m, 3.75 m and 3.90 m.

Yokoyama M. et al., The Effect of Non-Axisymmetry of Magnetic Configurations on Radial Electric Field Transition Properties in the LHD

Fig. 4

The neoclassical particle fluxes, Γe and Γi , as a function of Er for Rax of (a) 3.60 m, (b) 3.75 m, and (c) 3.90 m. The calculations
are based on the same conditions as those for Fig. 3.

Fig. 5

Dependence of Er diagram for Rax of (a) 3.60 m, (b) 3.75 m and (c) 3.90 m on (ne , Te) plane. The calculations are based on thesame conditions as those for Fig. 3 except fixing Te = Ti .

case of smaller εh,eff (Rax = 3.60 m). The enhancement of thenon-linearity of Γe(Er) by enlarging the helicity of a magnetic
configuration plays the key role for this mechanism. This factsuggests that the realization of the electron root (or enteringthe region of multiple solutions) becomes easily possible forlower Te for a case with larger εh,eff .
Now, the Er diagram is considered based on the different
aspect, that is, the density dependence on (ne , Te) plane.
Figure 5 shows Er diagrams for cases with Rax = 3.60 m, 3.75m, and 3.90 m. The density scan calculations are performedby assuming Te = T i . Other conditions are same as thoseutilized for Fig. 3. The dependence of T e(b) on ne can berecognized as Te(b) ∝ ne0.4, which reproduces the dependence
already been derived in ref. [6]. These Er diagram alsosuggests that the density threshold for entering region ofmultiple solutions and/or electron root becomes higher forcases with larger εh,eff . For example, in the case of Te(0.8) = 1
keV, it is possible to enter region of multiple solutions ispossible above ne(0.8) > 1 × 1019 m-3 for a case of Rax = 3.90m, while the density has to be as low as around 0.3 × 1019
m-3 for a case of Rax = 3.60 m.
These density and temperature dependence of thresholdto enter the regime of multiple solutions and electron root
can be summarized as follows. The higher helicity makes thethreshold collisionality larger (higher n e and lower
temperature, especially Te). This feature is attributed to thenonlinear dependence of Γe(Er) in a low collisional regime
induced by the helicity of a magnetic configuration.

3. Experimental verification in LHD
Motivated by the neoclassical analysis described in Sec.
2, verification experiment was conducted in LHD in a mannerof the density scan for three magnetic configurations with Rax= 3.50 m, 3.75 m, and 3.90 m. The Er is evaluated throughthe measurement of the poloidal velocity of Ne by the ‘fast’charge exchange recombination spectroscopy [10]. The
deduced values of Er for ρ ≈ 0.9 are shown in Fig. 6(a) for
three configurations. Te is in the range of 0.5 keV at ρ ≈ 0.9
for these discharges. It is observed that, for the same density
range of ne ≈ 1 × 1019 m-3, Er is already positive for a case ofRax = 3.90 m, while still negative for a case of Rax = 3.50 m.
It is interesting to see that transition from ion to electron root
is observed for a case of Rax = 3.75 m, between Rax = 3.50 m
and 3.90 m.
Corresponding density scan analysis of neoclassicalambipolar Er is also carried out. Figure 6(b) shows solutionsof Er as a function of ne . The calculation is performed byassuming the temperature ratio, Te /Ti , as same as that at the3220

Yokoyama M. et al., The Effect of Non-Axisymmetry of Magnetic Configurations on Radial Electric Field Transition Properties in the LHD

plasma center, since the radial profile of Ti is not available,unfortunately. The ratio is in the range of 1~1.25 for thesedischarges. Below a particular value of ne , there appear threeroots (middle root is unstable), which correspond to the regionof multiple solutions. Once n e becomes as low as thisthreshold value, the electron root seems to be realized among
three roots [5,6]. Thus, this threshold density could beconsidered as the threshold to realize the electron root. As
generally explained in Sec. 2, this threshold density becomes
smaller as εh,eff is decreased (corresponding to more inward
shift of a configuration).
Comparing to experimental results shown in Fig. 6(a),for the density range scanned in the experiment, the multiple
and/or electron root is anticipated for a case of Rax = 3.90 m,
while the ion root is predicted for a case of Rax = 3.50 m.
The experimental data for a case of Rax = 3.50 m shows thetendency of the gradual increase of ⏐Er⏐ as the density isdecreased on the contrary to the behaviour of the calculation
result. This might be due to the change of the temperature(especially, Te [6]) and/or the density profiles, which areassumed to be unchanged for the calculation. For a case ofRax = 3.75 m, scanned range of density rather well match thethreshold density anticipated from Fig. 6(b). Thus, thevariation of the threshold density anticipated from the analysisof the neoclassical ambipolar Er for cases with different Rax
is qualitatively (or even quantitatively) verified in LHD
experiment.

4. Summary
Radial electric field (Er) properties in LHD have beentheoretically investigated and also applied to LHDexperimental results. The ambipolar Er is calculated based on
the ambipolar condition in the framework of a neoclassicaltransport theory. The effects of the helicity of the magnetic
configuration on the condition to realize the electron root areexamined. Larger helicity makes the threshold collisionality
higher. This is attributed to the nonlinear dependence of Γe(Er)in a low collisional regime induced by the helicity of amagnetic configuration. This interesting feature predicts thatthe threshold temperature becomes higher for a case ofsmaller helicity, for example, by shifting the vacuum magnetic
axis position inward. The variation of the threshold density
anticipated from the analysis for cases with different Rax isqualitatively (or even quantitatively) verified in density scan
experiment conducted in LHD. This is a unique feature in
non-axisymmetric configurations; it is not the case for an
axisymmetric configuration such as tokamaks. The
comparison between different devices with different magnetic
field structure will be of vital importance for morecomprehensive understanding [11].

Acknowledgements
This work is performed with the support and under theauspices of the NIFS Collaborative Research Program. This
work has been supported by a grant-in-aid for young scientists(B) from the MEXT of Japan to one of the authors (MY).

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Fig. 6

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(a) The deduced Er for ρ ~ 0.9 for density scan experiment conducted in Rax = 3.50 m, 3.75 m, and 3.90 m
and (b) the neoclassical ambipolar Er for ρ ~ 0.9 fordensity scan calculation for Rax = 3.50 m, 3.75 m, and3.90 m.

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