J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)366–370
000–000

Configuration Effect on LH Transition in Tohoku University Heliac
TAKAHASHI Hiromi, KITAJIMA Sumio, YOKOYAMA Masayuki1, TANAKA Yutaka, UTOH Hiroyasu,
HASHIZUME Hidetoshi, SASAO Mamiko and TAKAYAMA Masakazu2
Department of Quantum Science and Energy Engineering, Tohoku University, Miyagi, 980-8579, Japan
1
National Institute for Fusion Science, Gifu, 509-5292, Japan2
Department of Electronics and Information Systems, Akita Prefectural University,
Akita, 015-0055, Japan(Received: 9 December 2003 / Accepted: 29 July 2004)

Abstract
Electrode biasing experiments in three types of magnetic configuration were carried out to investigate the influenceof poloidal ion viscosity on L-H transition in Tohoku University Heliac (TU-Heliac). Poloidal ion viscosity values
estimated from experiments had local maxima at a poloidal Mach number Mp ~ –2.4, where plasmas showed negative
differential resistance, in all configurations studied. However, the ion viscosity at the local maximum is dependent on
the configuration. The ion poloidal viscosity was calculated based on neoclassical theory and it showed good agreementwith that estimated from experiment. The results of calculations showed local maxima at Mp ~ –2.4 in all
configurations, and indicated configuration dependence. These tendencies agreed with the experimental results.
Keywords:
plasma, heliac, stellarator, electrode, biasing, radial electric field, poloidal rotation, H mode, L-H transition,
viscosity

1. Introduction

measure the ion orbit loss and the radial electric field with
sufficient precision in both temporal and spatial resolution.
Motivated by the present state of the H mode study
described above, we performed an electrode-biasing
experiment under the active control of the electrode current,
where the bifurcation phenomena with multiple solutions aredegenerated. In the present study, the L-H transition with three
types of magnetic configuration, which have different Fouriercomponents, was studied experimentally to investigate theinfluence of poloidal ion viscosity in TU-Heliac. We report
here that the critical driving force for the H mode transition
is dependent on the configuration, and compare theexperimental results with neoclassical theory where theShaing model [12,13] is applied to ion viscosity.

Although the H mode is widely observed in various
tokamaks, such as ASDEX [1], DIII-D [2], TFTR [3], etc.,
there have been only a few reports in helical/stellarator
devices, such as Wendelstein 7-AS [4] and CHS [5]. One of
the characteristics of the H mode is poloidal rotation induced
by a radial electric field. The shear poloidal rotation
suppresses fluctuation levels in plasma and results in adecrease in anomalous transport coefficient. These phenomena
were confirmed in both tokamaks and helical/stellarator
devices. In TU-Heliac, the influence of a radial electric field
on plasma parameters was investigated by electrode-biasing
experiments. In both positive and negative biasing
experiments, the improvement of plasma confinement was
clearly observed when the radial electric field was formed [68].
The neoclassical theory points out the criterion of L-H
transition from the viewpoint of ion viscosity. In this theory,the ion viscosity has a local maximum against the rotation
velocity. When the driving force in poloidal direction exceeds
a critical value, the poloidal rotation velocity increases rapidly
and a plasma transits into the H mode. So far, various models
for ion viscosity are proposed to explain L-H transition
phenomena [9-11]. However, these models have not yet been
examined in detail by experiment because it is difficult to

2. Experimental setup
The TU-Heliac is a small heliac device of n = 4, where
n is the toroidal period [14,15]. The heliac configurations areproduced by three sets of coils: toroidal field coils, a center
conductor coil, and vertical field coils. Various magnetic
configurations can be formed easily by selecting ratios of coilcurrents, and Fourier components can be varied widely. The
configuration parameters for typical cases of TU-Heliac aresummarised in Table 1. The target plasma in biasing
experiments is produced by alternative ohmic heating of 18.8

Corresponding author’s e-mail: h.takahashi@ppl2.qse.tohoku.ac.jp
1366©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Takahashi H. et al., Configuration Effect on LH Transition in Tohoku University Heliac

3. Experimental results

kHz. The vacuum vessel is filled with the fuelling gas He
before discharge. Plasma parameters are measured using aLangmuir probe (triple probe) at toroidal angle φ = 0° and a
6 mm microwave interferometer at φ = 90°. The hot cathodemade of LaB6 , used as an electrode to form a radial electricfield, is inserted horizontally from the low magnetic field sideat φ = 270°. It is biased against the vacuum vessel by aconstant current power supply. In this study, we attempted tocontrol the poloidal driving force to clarify the role of ion
viscosity for L-H transition. The poloidal driving force is theLorentz force of J × B. Thus, it can be changed continuously
by the electrode current control. Figure 1 shows the magnetic
surfaces and the position of the hot cathode in three types ofconfigurations with magnetic axis Rax located at (a) 7.7 cm,(b) 7.9 cm (standard configuration), and (c) 8.4 cm,
respectively here the magnetic axis is defined in a poloidal
cross section θ = 0°. They have similar profiles of rotationaltransform.

Typical time evolutions of electrode current IE , bias
voltage VE and electron line density n̂e are shown in Fig. 2.
Here, Te ≈ 20 eV, ne ≈ 2 × 1012 cm-3 at ρ = 0.5 and B0 ≈ 0.3 T
in each configuration. IE is controlled with the constantcurrent power supply such that IE is maintained at –4.2 A
from t = 0 ms to 5 ms, and –IE is ramped down to 0 A from
5 ms to 10 ms. In each configuration, there is a region where–VE increases even though –IE decreases, indicating that theplasma shows negative differential resistance. In this region,
various characteristics were observed, such as a decrease inplasma pressure, an increase in fluctuation level, etc. [16]. As
shown in Fig. 3, IE follows an s-shape curve against VE ,showing bifurcation.
The upper side of Fig. 4 shows the profiles of radial
electric field Er at the H mode (open circles), just before
transition to the L mode (open diamonds) and at the L mode
(open squares) in each configuration and the lower side ofthat shows time evolutions of Er at different radial positions.
Here, ρ is the averaged minor radius normalised by theaveraged plasma radius a. The profiles of Er show that thesteeper gradient region shifts to the inside with –IE decreasingin each configuration. This indicated that Er collapses fromthe outer to the inner region at the transition from H mode toTable 1 The configuration parameters in four types ofconfiguration.

εmn (r /R0 = 0.05; Boozer)

Rax
(cm)

a
(cm)

Well depth
(%)

(0,1)

(1,0)

(1,1)

7.3

6.3

–0.99

–0.09

–0.051–0.060

7.5

6.6

–0.11

–0.11

–0.049–0.056

7.9

6.8

2.3

–0.13

–0.040–0.051

8.4

6.2

4.1

–0.16

–0.039–0.050Fig. 1

The magnetic surfaces and the positions of the hot
cathode in three types of configuration. The magnetic
axes are located at (a) 7.7 cm, (b) 7.9 cm (standard
configuration), and (c) 8.4 cm, respectively here θstands for the poloidal angle around the Center
Conductor Coil.

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Fig. 2

Typical time evolutions of electrode current IE , bias
voltage V E and electron line density n̂e for theconfigurations of (a) Rax = 7.7 cm, (b) 7.9 cm, and (c) 8.4
cm. IE is controlled with a constant current power
supply so that IE is maintained at –4.2 A during biasingtime t = 0 ms and 5 ms, and –IE is ramped down to 0 A
from 5 ms to 10 ms. The plasma shows negative
differential resistance in the circled region in eachconfiguration.

Fig. 3

The relation between bias voltage VE and electrodecurrent IE . IE show s-shape curves for VE . IE bifurcatesfor VE with multiple solutions in the circled regions.

Takahashi H. et al., Configuration Effect on LH Transition in Tohoku University Heliac

Fig. 4The profiles and time evolutions of E r for theconfiguration of (a) Rax = 7.7 cm, (b) 7.9 cm, and (c) 8.4
cm. As seen in time evolutions, the steeper gradientregion shifts to the inner region with –IE decreasing ineach configuration. Thus, Er collapses from the outer tothe inner region. The strong Er is maintained in the
inner region even after transition to the L mode.

L mode. It is understood from both Er profiles and time
evolutions that the strong Er is maintained in the inner region
even after transition to the L mode.

4. Comparison with neoclassical theory
Neoclassical theory predicts the existence of a local
maximum, which has an influence on sudden transition
phenomena, in the ion viscosity for poloidal rotation. This is
one of the characteristics of the L-H transition. To investigate
the influence of ion viscosity on the L-H transition, theestimated poloidal ion viscosity was compared with theviscosity calculated from neoclassical theory. The poloidal
driving force is balanced with a poloidal ion damping force,
which consists of the poloidal ion viscosity term and a friction
term caused by charge exchange. In the steady state, thenormalised poloidal momentum balance equation for the ion[10,17], where the Shaing model [10,12,13] is applied to theion viscosity, can be written as:

2 <r>< Jρ > B 0
π pi
4
ν in
= − Π p,n −(1 + 2 q 2 )Θ Mp ,( v t / < r >)π

< Bp ⋅ ∇ ⋅Π i >Π p,n =

4

Mp = −( Eρ / B0 )
,
Θ vt

Θ =

The dependence of the ion viscosity on poloidal Mach
number. (a) Experimental ion viscosity evaluated frompoloidal driving force, (b) calculated ion viscosity
applied to the Shaing model, and (c) the comparison
between the experiment and the calculation for the
standard configuration.

The electric current density is estimated as:< Jρ > =

(1)

,

(2)

ε
,
q

(3)

π ni mi vt 2 (B0 / qR 0 )Fig. 5IE,
2π < r > L(4)where L is the length of the magnetic axis. On the other hand,
the right side of Eq. (1) indicates the poloidal ion damping
force; the first term is the neoclassical ion viscous force andthe second is the friction between the ion and the neutral
particle. The collision frequency in the friction term can be
written as:

νin = < σcx ui > nn ,(5)where σcx is the cross-section of the charge exchange between
the ion and a neutral particle [18,19], ui is the velocity of theion, and nn is the density of the neutral particle.
Figure 5 shows the dependence of ion viscosity on Mp :(a) experimental ion viscous force F EXP (the difference
between the left side of Eq. (1) and the second term on theright side) for the configurations of Rax = 7.7 cm (opencircles), 7.9 cm (open squares), 8.4 cm (open triangles), andthe closed symbols correspond to those in the region where

where angled brackets denote the flux surface average, Jρ isthe radial current density, B0 is the magnetic field on the axis,
ni is the ion density, mi is the mass of the ion, vt is the ion
thermal velocity, r is the minor radius, q is the safety factor,
νin is collision frequency between the ion and the neutral
particle, Mp is the poloidal Mach number, Bp is the poloidal
magnetic field, ∏i is the ion viscous stress tensor, R0 is the
major radius, and ε = < r >/R0 is the toroidal ripple. The left
side of Eq. (1) indicates the poloidal driving force of J × B.
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Takahashi H. et al., Configuration Effect on LH Transition in Tohoku University Heliac

the plasma shows negative differential resistance; (b)calculated ion viscous force FCAL , which is applied to the
Shaing model (the first term on the right side of Eq. (1)) forthe configurations of Rax = 7.3 cm (closed circles), 7.5 cm(closed squares), 7.9 cm (closed diamonds), 8.4 cm (closed
triangles); and (c) comparison between experiment andcalculation for the standard configuration. In eachconfiguration, the experimental driving force was estimatedusing the parameters measured at ρ ~ 0.5, and ion temperature
Ti was assumed to be 0.2Te . On the other hand, the theoreticalion viscosity was calculated under the following conditions;the ratio of the collision frequency to bounce frequency fortrapped particle, ∇p drift velocity to vt , and the ∇T drift
velocity to vt were assumed to be 36, 0.36, and –0.093,
respectively, and Ti was assumed to be 0.6Te . As shown in
Fig. 5, the dependence of experimental ion viscous force on
Mp shows good agreement with the neoclassical theory. As
shown in Fig. 5(a), the experimental ion viscosity has a local
maximum at Mp ~ –2.4, where the plasma shows negative
differential resistance in each configuration. This indicatesthat L-H transition occurs when the poloidal driving force
exceeds a critical value as the local maximum of ion viscosity.
The result agrees with the scenario of transition to H modepredicted by neoclassical theory. In the calculation (Fig. 5(b)),
a local maximum also exists at Mp ~ –2.4 where there is alocal maximum of estimated ion viscosity. As can be seen in
Fig. 5(c), the dependence of experimental ion viscous force
on Mp shows good agreement in |Mp| < 4 with the neoclassicaltheory in the standard configuration. The difference in |Mp| >4 between the experiment and the neoclassical theory wasthought to be due to ambiguity with regard to evaluation of
the friction term. In this paper, friction was evaluated under
the assumption that T i = 4 eV and that it has a linear
dependence on Mp . However, it is nonlinearly dependent on
Mp in practice because the ion temperature and the crosssection of the charge exchange are changed with changes inelectrode current. In the high Mp region where the plasma
confinement state is H mode, plasma pressure and electron
density increase such that the interaction between ions andneutral particles increases. Consequently, the value of friction
was thought to be larger than the subtracted value.
Figure 6 shows the dependence of critical ion viscosity
for L-H transition on configuration. The critical ion viscosity
evaluated from the experiment corresponds to that in thenegative differential region and the calculated value
corresponds to that at Mp = –2.4. The value of critical ion
viscosity is greatest in the standard configuration and lowest
in outward configuration in both the experiment andcalculation. The reason for the decrease in ion viscous force
in the outward configuration can be explained qualitatively
from the viewpoint of a ripple in the magnetic field and/or
the shape of the magnetic surface. As shown in Table 1,
toroidicity ε10 and helicity ε11 decrease with the shift inposition of the magnetic axis to the outward configuration.
Thus, the magnetic structure in the outward configuration has
good poloidal symmetry. Therefore, the ion viscous force

Fig. 6The configuration effect on poloidal ion viscosity
required for L-H transition. The circles are FEXP and thesquares are FCAL . The critical ion viscosity is the largest
in the configuration of R ax = 7.9 cm in both theexperiment and the calculation.

decreases because it is correlated with the amplitude of themagnetic field ripple. The bend in the magnetic surface inthe outward configuration is less sharp than the other two atθ = 90° and 270° as shown in Fig. 1. The plasma is thought
to rotate easily for the poloidal direction and this is consistent
with the lower poloidal ion viscosity.
The mechanism of L-H transition in TU-Heliac is thebifurcation phenomenon arising from the local maximum in
the ion viscosity for poloidal flow velocity, and it can be
explained by the neoclassical theory. The experiment showed
good agreement with neoclassical theory. However, there wereslight differences between the experimental and calculated
values for the absolute ion viscosity. There are several
possible explanations for this as follows: (i) The ion
temperature was not measured. In this paper, the ion
temperature was assumed to be Ti = 0.2Te . However, the ion
temperature should be measured for more quantitativeevaluation of ion viscous force because ion viscosity andpoloidal Mach number are proportional to Ti-1 and T i-1/2,
respectively. (ii) The evaluated friction includes ambiguity ofTi and cross-section. For evaluation of friction, measurementof the ion temperature is also required. The friction must show
nonlinear dependence on Mp as discussed above. (iii) The
experimental ion viscosity might include the effect ofanomalous viscosity. However, anomalous viscosity is notgoverned by ion viscosity because ion temperature is very low
as Ti ~ 4 eV in TU-Heliac. (iv) Finally, the poloidal driving
force does not correspond perfectly to the changes in local
plasma parameters. However, the poloidal driving force andthe poloidal Mach number were evaluated by plasma
parameters measured locally using a Langmuir probe, thedriving force also includes the electrode current as a global
plasma parameter. Consequently, the driving force is averaged
as a whole.

5. Summary
The hot-cathode-biasing experiment with three types ofmagnetic configuration in TU-Heliac yielded the following
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findings: (i) Plasmas show negative resistance after transition
from the H mode to the L mode in all configurations. (ii) The
radial electric field collapses from the outer to the inner region
with transition from the H mode to the L mode. (iii) The
poloidal ion viscosity estimated from the experiment shows
good agreement with neoclassical theory in |Mp| < 4. (iv) The
poloidal ion viscosities estimated from the experiments have
local maxima at Mp ~ –2.4 in all configurations studied here.
(v) The poloidal ion viscosities at local maximum correspondto those in the negative differential resistance region. (vi) The
regions where the local maximum of ion viscosity exists agreewith the predictions from neoclassical theory. (vii) In bothexperiment and calculation, the value of the local maximumof the ion viscosity is largest in the standard configuration.

Acknowledgment
Authors wish to acknowledge Prof. K. Itoh for useful
discussion. This work was supported in part by the LHD Joint
Planning Research program at National Institute for Fusion
Science.

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