J. Plasma Fusion Res. SERIES, Vol. 8 (2009)

Reynolds Stress Measurements for Investigation ofNonlinear Processes of Turbulence in the Large MirrorDevice and in the Large Mirror Device-Upgrade
Yoshihiko NAGASHIMA, Sanae-I. ITOH1) , Kimitaka ITOH2) , Akihide FUJISAWA2) ,Shigeru INAGAKI1) , Yoshinobu KAWAI1) , Shunjiro SHINOHARA1) , Masayuki FUKAO3) ,Takuma YAMADA, Kenichiro TERASAKA1) , Takashi MARUTA1) , Kunihiro KAMATAK1) ,Hiroyuki ARAKAWA1) , Masatoshi YAGI1) , Naohiro KASUYA2) , George R. TYNAN4) ,Patrick H. DIAMOND4) , and Yuichi Takase
The University of Tokyo, Kashiwa, Chiba 277-8561, Japan
1)
Kyushu University, Kasuga, Fukuoka 816-8580, Japan
2)
National Institute for Fusion Science, Toki, Gifu 509-5292, Japan
3)
Myojocho, Uji, Kyoto 611-0014, Japan
4)
University of California San Diego, La Jolla, CA 92093, USA(Received: 1 September 2008 / Accepted: 25 December 2008)

Based on success of low-frequency zonal potential measurement (∼400 Hz) in the large mirror device [Y. Nagashima JPSJ 77 114501 (2008)], a multi-channel Reynolds stress probe system has been developed in the large mirror device-upgrade. The system can simultaneously detect ﬂuctuation Reynolds
stress at poloidally diﬀerent 8 locations. Preliminary results from the system demonstrate that poloidally
symmetric potential oscillations exist in a frequency range of 200–600 Hz.
Keywords: zonal ﬂow, Reynolds stress, linear plasma deviceof a new probe system in the large mirror deviceupgrade (LMD-U) is introduced. Finally, we present
preliminary results from the probe system in experiments of the spectral transition.

1. Introduction
Understanding of turbulence and transport in selfregulated plasma structure is one of the most important research themes in plasma physics and nuclear fusion research. The drift wave turbulence isconsidered to dominate in magnetically conﬁned nonuniform plasmas [1]. Recent progress of turbulence
study shows that secondary instabilities driven by
nonlinearity of the drift wave turbulence may play
important roles in self regulation of turbulence [2].
Therefore, observation of self-regulation in turbulent
ﬂuctuations (i.e., nonlinear energy transfer among various ﬂuctuations) is a crucial task in turbulence research. In order to investigate the nonlinear energy
transfer between drift-wave ﬂuctuations and the secondary instabilities, it is necessary to evaluate oﬀdiagonal terms in momentum balance equations (i.e.,the Reynolds stress in the poloidal plane). In particular, the zonal ﬂow and its drive force are precisely deﬁned as poloidally averaged parameters. In
the large mirror device (LMD) [3], we measured thelocal Reynolds stress as well as the zonal potential [4].
In the next stage, we have developed a multi-channel
Reynolds stress probe in the large mirror deviceupgrade (LMD-U) [5]. In this manuscript, we introduce results from Reynolds stress measurement in linear devices of Kyushu University.
First, we brieﬂy describe results from the large
mirror device (LMD). Next, experimental apparatus

2.

Summary of the Reynolds StressMeasurement in the LMD

In LMD, secondary instabilities of the drift-wave
spectrum have been investigated intensively [4, 6, 7].
A qualitative transition of ﬂuctuation spectrum occurs from combination of coherent spectral peaks toa signature of broadband spectrum due to reducing ﬁlling Ar gas pressure PAr under the ﬁxed magnetic ﬁeld B of 0.12 T [8]. Previous researches on thetransition of ﬂuctuations have been studied in basic plasma experiments in detail, and relationships
between the transition and appearance of the lowfrequency ﬂuctuations were presented [9, 10]. The coexistence of low-frequency zonal potential (∼400 Hz)
and the drift-wave spectrum (7–8 kHz) is observed under the discharge conditions that PAr ∼3.5 mTorr,
RF input power is 2 kW, and B=0.12 T. The caseof PAr ∼3.5 mTorr is the boundary condition of thetransition. Nonlinear couplings between the Reynolds
stress of the drift-wave spectrum and the zonal potential are successfully detected by a Reynolds stressprobe.

author’s e-mail: nagashima@k.u-tokyo.ac.jp

50©2009 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Y. Nagashima et al., Reynolds Stress Measurements for Investigation of Nonlinear Processes of Turbulence in the Large Mirror Device and in the Large...

(a)

(b)

(c)
(d)

Fig. 2 Transition of spectrum during the discharge.
(a)Time evolution of the ﬂoating potential at
r=4 cm. Fluctuation spectra before (b) or after (c)the transition.

Fig. 1 (a, c) Schematic views and (b, d) photographs ofthe multi-channel Reynolds stress probe (MCRSP)
system. (a) Enlarged view of the probe head drawing, and (b) photograph of the manufactured probehead. The scale unit is mm. (c) Arrangement of
8 probes at the radial location r=40 mm, and (d)calibration scene of the arrangement of the MCRSP
system.

3.

tric ﬁeld ﬂuctuation E˜r and radial wave number kr
are measured by the electrodes #3 and #5. Then weobtain the ﬂuctuation Reynolds stress per mass density −�ṽr ṽθ � = �Ẽθ Ẽr �. The electrodes #1 and #2 areavailable for various purposes. We can check eﬀects ofspatial aliasing on ﬂuctuation measurements by use of
the electrodes #1-2.
Each probe can move radially, therefore, it is possible to measure poloidal structures of ﬂuctuations atarbitrary radial locations. Figures 1(c) and 1(d) show
an arrangement of the 8 probes at the radial location
r=4 cm. A schematic view of the arrangement is seenin Fig. 1(c), however, the real position of the 8 probes
is not the same as the view. Therefore, we calibrated
the arrangement by use of polar tick marks printed
on a circular transparent board. The center of thetransparent board should be set to be in the plasma
center. Plasmas in LMD-U are produced in a quartz
tube with a length of 40 cm, and we let the laser lightthrough two 0.1 cm holes located at the both ends ofthe tube. Then, the center of the transparent board isarranged to hit the laser. The laser light also hits thetip of the 2D probe electrode that is located at theplasma center. The 2D probe system detected thatour plasmas have circular proﬁles with poloidal symmetry [13], and this method for identifying the plasma
center is valid. The tick marks are adjusted horizontally and vertically by a plumb bob, as shown a yellow
vertical line in Fig. 1(d).

Experimental Apparatus in theLMD-U

Based on the success of the Reynolds stress measurement in LMD shown in section 2, a new probe system, multi-channel Reynolds stress probe (MCRSP)
system, has been developed in LMD-U, where thenonlinear dynamics of ﬂuctuations have been investigated in detail [11, 12]. A purpose of the system is tomeasure poloidal variations of the Reynolds stress forinvestigating the energy transfer among ﬂuctuations
with ﬁnite poloidal wave numbers. In addition, thesystem could observe temporal behaviors of poloidal
averaged zonal ﬂow and Reynolds stress.
The MCRSP system is located 125 cm away fromthe end of the vessel (source side). The MCRSP has 8probes, and each probe head houses six tungsten electrodes which covers simultaneous measurement at different six locations on the poloidal cross-section. The
cylindrical electrodes with a diameter of 0.8 mm arebent at tips of the electrodes, as shown in Figs. 1(a)and 1(b). In this experiment, we present observationof the ﬂoating potential ﬂuctuation (Φ̃f ). Neglecting the temperature ﬂuctuation, we assume that Φ̃fis corresponding to the electrostatic potential ﬂuctuation. The electrodes #3-6 are used mainly for measuring the ﬂuctuation Reynolds stress per mass density(RS). The poloidal and radial distances among theelectrodes are 10 mm. The poloidal electric ﬁeld ﬂuctuation E˜θ and poloidal wave number kθ are derived
from the electrodes #4 and #6, and the radial elec-

4.

Preliminary results from theMCRSP
4.1 Transition of spectrum during discharge
A qualitative transition of spectrum is also foundin LMD-U [14, 15]. A boundary condition of the tran-51

Y. Nagashima et al., Reynolds Stress Measurements for Investigation of Nonlinear Processes of Turbulence in the Large Mirror Device and in the Large...

sition is that PAr is 3.6-3.9 mTorr and B is ﬁxed at0.09 T. The transition in LMD-U occurs much fasterthan that in LMD. The reﬂected RF power changes
just after the transition, and returns gradually tothe power level of discharges without the transition(less than 10% of the forward RF power). Figure 2(a)demonstrates the transition during a discharge. The
DC ﬂoating potential decreases at about 0.15 s, andthe ﬁrst back transition can be seen at about 0.24 s.
Two distinctive features of spectrum are found. One
is a qualitative change of the spectrum. The spectrum
in the beginning of the discharge looks like a combination of coherent spectral peaks in the frequency rangeof 1–10 kHz as shown in Fig. 2(b). After the transition,
the spectrum tends to be closed to broadband one inFig. 2(c). The other is a behavior of low frequency
ﬂuctuations (<1 kHz). The low frequency components
increase after the transition. If nonlinear couplings between the low (<1 kHz) and high (1–10 kHz) frequency
components occur, it is natural that the bandwidth of
the spectrum broadens. Here, we abbreviate the interval before or after the transition as phase B or phase
A, respectively. We use the word “the transition” inthe case that the transition occurs from the phase B
to the phase A, and the inverse transition is called“the back transition” in this manuscript.

4.2

Fig. 3 (a, b) Radial proﬁles of the ﬂoating potential. Proﬁles of (a) stationary component and (b) ﬂuctuation (f >1 kHz) component. In (a, b), plots inblack or red mean proﬁles before or after the transition. (c, d) Polar plots of the ﬂoating potential at r=4 cm. Plots of (c) stationary componentand (d) ﬂuctuation component. Diamond plots orsolid lines indicate measurements by each probe or
poloidal averaged values at the same radial locations, respectively. Plots in black indicate data obtained at the electrode#1,2,4,6. Points in blue or
red are measured at the electrode #3 or #5, respectively.

Arrangement of the probes in thepoloidal planeWe calibrated the arrangement of the 8 probes atthe same radial location. Based on the observation,
here we check the validity of the calibration by meanof polar plots (plot radii or angles are corresponding
to potential intensities or probe location angle, respectively) of the stationary Φf which reveal diﬀerence ofthe center location of between plasmas and the arrangement [13].
First, Figure 3(a) shows radial proﬁles of the stationary ﬂoating potential Φf . The proﬁles were measured by shot-to-shot radial scans of one probe in theMCRSP system. The Φf after the transition becomes
more negative than that before the transition inside
the plasma (r <5 cm). Figure 3(b) shows proﬁles of
the amplitude of the ﬂoating potential ﬂuctuation in
the frequency range higher than 1 kHz |Φ̃f,>1 kHz |. The|Φ̃f,>1 kHz | increases after the transition at the radial
location r >3 cm. Next, Figure 3(c) shows polar plotsof the Φf at r=4 cm after the transition. Black data
indicates plots at the electrodes #1, 2, 4, 6, blue plots
are measured at #5, and red plots are the data at#3, respectively. The proﬁle of the Φf in Fig. 3(a)shows a positive gradient at r=4 cm, and the relationship among black, blue and red plots corresponds tothe positive gradient of the Φf proﬁle. Considering
the variance of the Φf ’s at the same radial locations,discrepancy of the center position between the plasmas and arrangement is roughly ∼0.5 cm. In contrast
to this, the proﬁle of the |Φ̃f,>1 kHz | in Fig. 3(b) does
not show a clear gradient at r=4 cm, therefore, polarplots of the |Φ̃f,>1 kHz | in Fig. 3(d) could not discriminate blue and red plots from black plots in terms ofthe gradient.

4.3

Low-frequency potential oscillations and the Reynolds stress

The zonal potential is observed in LMD discharges
where the transition occurs. Existence of the lowfrequency zonal potential as well as the Reynolds
stress are also examined in LMD-U. The zonal potential is related to the Reynolds stress through the vorticity equation ∂t Δφ/B = ∂rr �Eθ Er �/B 2 − μΔφ/B.
Therefore, the relationship between the zonal potential and the Reynolds stress can test the causality
in the drift wave-zonal ﬂow system. Figure 4 shows
spatio-temporal structures of ﬂuctuations in the frequency range of 200–600 Hz. The aliasing in thepoloidal direction does not occur. Poloidally symmetric potential oscillations are dominant in Fig. 4(a).

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Y. Nagashima et al., Reynolds Stress Measurements for Investigation of Nonlinear Processes of Turbulence in the Large Mirror Device and in the Large...

Fig. 4 Temporal behaviors of ﬂuctuations in the frequency
range of 200–600 Hz at r=4 cm. (a) Φf (θ, t), and(b) Eθ Er /B 2 (θ, t). In calculating the Reynolds
stress, the electric ﬁeld ﬂuctuations are selected in
the frequency range of 1–40 kHz. Poloidal proﬁles
are measured by the 8 probes, and void points areinterpolated in the contour plots.

However, the poloidal correlation of the oscillations
are transiently small, and ﬂuctuations with m ≤1
may be sometimes seen. Fluctuations in the frequency
range of 200–600 Hz looks like mixture modes in which
the m=0 mode is dominant. Figure 4(b) demonstrates
that the amplitude of the Reynolds stress oscillation
signiﬁcantly has variation in the poloidal direction.
In reality, to investigate the relationship between
the zonal potential and the Reynolds stress, the ensemble average of the vorticity equation performed in
the poloidal direction should be considered. We investigated relationship between the transition and thewaveforms including poloidally averaged ones. Figure 5 shows waveforms of ﬂuctuations in the frequency
range of 200–600 Hz. The contour maps in Figs. 5(a)and 5(c) are the transient behaviors of the poloidal
proﬁles. The waveforms in Figs. 5(b) and 5(d) areaveraged poloidally. The transition occurs back andforth, and temporal behaviors of the waveforms arewell correlated with the transition phase. All of theﬂuctuation amplitudes in the phase A increase clearlyrelative to those in the phase B.
Our main interest is the correlation of the lowfrequency �Φf �(t) with the �Eθ Er /B 2 ��Φf �(t). Theamplitude of the cross-product of them in Fig.5(e)(black) also increases after the transition. The redline in Fig.5(e) is the 10 ms temporal average of thecross-product, and supports the increase. It should
be noted that signs of the �Eθ Er /B 2 ��Φf �(t) are notunique transiently. The positive sign indicates thatthe �Φf �(t) and the �Eθ Er /B 2 ��Φf �(t) are in phase,and the negative sign means the opposite. The positive sign of the �Eθ Er /B 2 ��Φf �(t) tends to be seen
just after the transition, and the negative one can beseen just after the back transition. Further studies arenecessary to conclude the tendency of the appearance

Fig. 5 Relationships between the transition and the waveforms in the frequency range of 200–600 Hz at
r=4 cm. (a) Φf (θ, t), (b) �Φf �(t), (c) Reynolds
stress Eθ Er /B 2 (θ, t), (d) �Eθ Er /B 2 �(t), and (e)�Eθ Er /B 2 ��Φf �(t). The black line in (e) is a raw
plot of the �Eθ Er /B 2 ��Φf �(t), and the red line isgiven by multiplying the 10 ms temporal average ofthe black line by 16.

of the positive and negative signs.

5. Summary and Future Direction
In summary, to investigate nonlinear dynamics
of ﬂuctuations poloidally and radially in detail, themulti-channel Reynolds stress probe system has been
installed in the large mirror device-upgrade. The system is successfully arranged in the poloidal direction
at the same radial location, and validity of the arrangement is conﬁrmed by a test using the experimental proﬁle data. We applied the system to experimentsof the discharge in which the spectral transition occurs. The preliminary results from the system demonstrates that the low frequency (200–600 Hz) ﬂoating
potential ﬂuctuations with m=0 are observed. In addition, ﬂuctuations with m ≤1 at the same frequency
as the m=0 modes may be observed transiently. Oscillation of the poloidally averaged Reynolds stress iscorrelated with the m=0 modes. and The correlation
after the transition becomes larger than that beforethe transition.
To identify the nonlinear energy transfer between
the zonal ﬂow and the Reynolds stress gradient in53

Y. Nagashima et al., Reynolds Stress Measurements for Investigation of Nonlinear Processes of Turbulence in the Large Mirror Device and in the Large...

terms of the poloidal average, it is necessary to measure poloidal proﬁles of the ﬂuctuations at two different radial locations. Therefore, installation of theadditional 8 probes as well as diagnostics for the direct
ion ﬂow such as the Mach probe is under considerationin the future.

Acknowledgment

We are grateful to Prof. T. Ohkawa andProf. F. Wagner for useful discussions.
Thiswork was partially supported by Grant-in-Aids
for Specially Promoted Research (16002005) [Itohproject] and for Young Scientists (B) (18760637) ofMEXT, Japan, and by the collaborations of NIFS
(NIFS07KOAP017), RIAM, Kyushu University,
and The University of Tokyo. We also appreciate the experimental supports of Mr. T. Nishijima,Mr. M. Kawaguchi.
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