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

Measurement of Nonlinear Mode Couplings in the Large Mirror
Device-Upgrade
Takuma YAMADA, Sanae –I. ITOH1) , Shigeru INAGAKI1) , Yoshihiko NAGASHIMA,
Shunjiro SHINOHARA2) , Takashi MARUTA2) , Kenichiro TERASAKA2) , Kunihiro KAMATAKI2),
Naohiro KASUYA3) , Masatoshi YAGI1) , Yoshinobu KAWAI1) , Akihide FUJISAWA3) andKimitaka ITOH3)
Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa 277-8561, Japan
1) Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan
2)
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga 816-8580, Japan
3) National Institute for Fusion Science, Toki 509-5292, Japan(Received: 1 September 2008 / Accepted: 10 December 2008)

Experimental study of plasma turbulence has been proving theoretical predictions by using multi-channel
diagnostics and analyzing techniques such as bi-spectral analysis. We performed a fluctuation measurement of
the ion saturation-current with a 64-channel poloidal Langmuir probe array in the Large Mirror Device-Upgrade.
A number of fluctuation peaks in the poloidal wave number–frequency space were observed, and they satisfied
the matching conditions of wave number and frequency with each other. The nonlinear mode couplings among
the fluctuation peaks were confirmed by two-dimensional bi-coherence analysis. By amplitude correlation technique, three original parent modes were found. All the other fluctuation peaks were quasi-modes, which weresuccessively generated by mode couplings from the three original parent modes.
Keywords: linear plasma, drift wave, turbulence, bi-coherence, amplitude correlation technique

1. Introduction

The poloidal wave numbers and frequencies of the fluctuation peaks satisfied the matching condition with each otherand suggested that these peaks were produced successively
by nonlinear mode couplings from three irreducible parent modes [14]. The mode couplings among the fluctuation peaks were confirmed by bi-spectral analysis withconsidering only the matching condition in the frequency
space [11]. The results from 2D bi-spectral analysis, which
was considering the matching conditions of the poloidal
wave number and frequency, was introduced [9].
In this paper, we present the results from the 2D bispectral analysis and investigation of the energy transfer
direction in detail. The successive generation of the quasimodes from the original parent modes is visualized.

Theoretical studies and simulations on drift wave turbulence predict interesting models that the drift waves arestrongly coupled with meso-scale waves such as zonal
flows and streamers, which have strong influences on transport rates [1]. Recent progress on measurement methods
such as multi-channel measurements [2–5], and analyzing
techniques such as bi-spectral analysis [6, 7] make it possible to observe these meso-scale structures in plasmas. For
example, zonal flow was measured in the Compact Helical System plasma [8], and streamer was measured in the
Large Mirror Device-Upgrade (LMD-U) linear plasma [9].
Experimental study has been proving theories and becoming more important for investigating plasma turbulence tocontribute to the ITER project.
In order to study the nonlinear mode couplings andmeso-scale structures in drift wave turbulence experimentally, multi-probe measurements were performed in
the LMD-U linear plasma. The LMD-U plasma allows multi-channel Langmuir probe measurement becauseof its low temperature. With the multi-channel measurement, many interesting features of plasma turbulence
have been observed in LMD-U. For example, drift
wave modes [10], route to drift wave turbulence [11],and broadband mode [12] were observed. With a 64channel poloidal Langmuir probe [13], two-dimensional(2D) power spectrum of the ion saturation-current fluctuation, which showed a number of fluctuation peaks inthe poloidal wave number–frequency space, was observed.

2. 2D Bi-coherence Analysis

Bi-spectral analysis [6] examines the relationshipamong three waves with wave numbers and frequencies of(kk 1 , ω1 ), (kk 2 , ω2 ) and (kk 3 , ω3 ), which satisfy the matchingconditions k 3 = k 1 + k 2 and ω3 = ω1 + ω2 . By using apoloidal probe array, the matching conditions of not only
the frequency ω, but also the poloidal wave number kθ canbe considered. A poloidal probe array can measure thepoloidal mode number m of the fluctuation wave (m is an
integer), which is related to the poloidal wave number kθ
by m = r p kθ , where r p is the radius of the probe array.
When the Fourier transformed expression of a 2D
waveform z(θ,t) is Z(m, f ), where θ is the poloidal angleof the probe and f = ω/2π, the bi-spectrum B of the threewaves is expressed as B = �Z(m1 , f1 )Z(m2 , f2 )Z ∗ (m3 , f3 )�.
When the three waves fluctuate independently, the absoluteauthor’s e-mail: takuma@k.u-tokyo.ac.jp

87©2009 by The Japan Society of Plasma
Science and Nuclear Fusion Research

T. Yamada et al., Measurement of Nonlinear Mode Couplings in the Large Mirror Device-Upgrade

b

2

φb

10

J

8

G

Frequency f (kHz)value |B| becomes 0. When the phases of the three waves
are connected by a certain relationship, |B| becomes finite.
The bi-coherence b, which is a normalized value of B, andthe bi-phase φb , which shows the relationship among thephases of the three waves, are expressed as|B|2,=
�|Z(m1 , f1 )Z(m2 , f2 )|2 ��|Z(m3 , f3 )|2 �
Im(B)
,
(1)
= tan−1Re(B)respectively. Bi-coherence analysis is important for investigating the coupling among three waves. When the bicoherence b is finite for the three waves, it suggests that
one wave is produced by nonlinear mode coupling of theother two waves.
2D bi-spectral analysis is effective when the plasma
turbulence has many fluctuation peaks in the m– f plane.
The mode couplings among the waves become clearer bydecomposing the bi-coherence analysis into the m plane.

�3

10

I
6

�4

10

D

H

F

4

B

A

�5

10

2

E

0
C

�2
0

2
4
6
Poloidal mode number m

�6

10
8

Fig. 1 Contour plot (arb. unit) of 2D power spectrum of ion
saturation-current fluctuation in LMD-U. The discharge
conditions (magnetic field, argon pressure) are (0.09 T,
0.27 Pa). Positive f indicates the propagation in the electron diamagnetic direction. The frequency resolution is
0.1 kHz. A number of fluctuation peaks, which satisfy
the matching conditions with each other, are observed.

3. Spectral Analysis in LMD-U

The LMD-U [10] vacuum vessel has the axial lengthof 3740 mm, with a straight axial magnetic field (0.01–
0.15 T). The LMD-U plasma has the plasma radius of
about 50 mm, since the plasma is generated by radiofrequency wave (7 MHz / 3 kW) inside a quartz tube withan inner diameter of 95 mm. The quartz tube is filled withargon gas with a pressure of 0.2–0.8 Pa. The electron density and temperature of the plasma are about 1019 m−3 and
3 eV, respectively. Resistive drift wave mode is excited in
the LMD-U plasma by its density gradient. By increasing the magnetic field (over 0.04 T) or decreasing the argon pressure (under 0.4 Pa), the excited mode changes into
drift wave turbulence [12]. A 64-channel poloidal Langmuir probe array [13] at r p = 40 mm is installed on LMDU.
The 2D waveform z(θ,t) of the ion saturation-current
fluctuation generated in LMD-U was measured with the
64-channel poloidal probe array. The sampling frequency
was 1 MHz. In the discharge conditions of the magnetic
field of 0.02 T and the argon pressure of 0.27 Pa, the 2D
power spectrum S(m, f ) = |Z(m, f )|2 showed a single drift
wave mode with (m, f ) = (3, 4.2 kHz). In this paper, m ≥ 0,
and positive f indicates the propagation in the electron diamagnetic direction. By increasing the magnetic field to
0.09 T, z(θ,t) changed into drift wave turbulence. In thiscase, S(m, f ) showed a number of fluctuation peaks in them– f plane (see Fig. 1). The small extension of the mode
numbers of the fluctuation peaks is caused by the misalignment of the probe tips, which was discussed in Ref. [13].
(In addition, broadband fluctuation with the same decay
laws in the m and f space was also observed [12].)
According to Fig. 1, the (m, f ) of the fluctuation peaksare (A: 1, 2.8 kHz), (B: 2, 3.2 kHz), (C: 1, −0.9 kHz),(D: 2, 5.6 kHz), (E: 2, 1.9 kHz), (F: 1, 4.1 kHz),(G: 3, 8.4 kHz), (H: 3, 4.7 kHz), (I: 2, 6.9 kHz) and(J: 3, 9.7 kHz). Examining the mode numbers and frequencies of the fluctuation peaks, we can find out thatthese fluctuation peaks satisfy the matching condition witheach other. For example, the peaks (D) and (G) are thehigher harmonics of the peak (A), i.e., (α: D = A + A)
and (β: G = A + D). Other matching conditions are(γ: E = A + C), (δ: B = C + F), (ε: H = C + D),(ζ: H = A + E), (η: I = A + F) and (κ: J = A + I). The
Greek letter is the label of each mode coupling.
When the matching condition exists, there is a possibility that the three waves are nonlinearly coupled, and one
wave is forced to be excited by the other two waves. The
wave excited in this way is called quasi-mode [15]. By
this argument, we can suspect that many of the fluctuation
peaks are quasi-modes, which are excited by the origin of
three irreducible real modes. The combination of the three
original parent modes is considered to be (A, B, C) or (A,C, F), for example. The mode coupling of the waves isverified by bi-coherence analysis, and the energy transfer
direction from parent mode to child mode is estimated byamplitude correlation technique.

4. Mode Couplings and Energy Transfer

To evaluate the mode couplings expected above, bicoherence analysis was applied to the ion saturationcurrent fluctuation. The 2D Fourier transformed expression Z(m, f ) was calculated first, and m1 , f1 , m2 and f2
were chosen to investigate the bi-coherence b among the88

T. Yamada et al., Measurement of Nonlinear Mode Couplings in the Large Mirror Device-Upgrade

�: 0.13

�: 0.11

�

�

0

0.2

Cross-correlation5(m1, m2, m3) = (1, 1, 2)

�
0.04

�

0.1

�

Cross-correlation

Frequency f2 (kHz)

10

0.54

�5

10
�10

�5
0
5
Frequency f1 (kHz)

10

0.0

0.4

(a)

B �> F

0.0
0.2 (b)A �> H

0.3
0.2
0.1

0.1

0.0
�15

�10

�5

0
5
Time � (ms)1015

Fig. 2 Contour plot of squared bi-coherence among three waves(m1 , f 1 ), (m2 , f 2 ) and (m3 , f 3 ), which satisfy the matchingcondition (m1 , m2 , m3 ) = (1, 1, 2). Frequency ranges
are (–10 kHz ≤ f1 , f 2 ≤ 10 kHz) and (–20 kHz ≤ f3 ≤20 kHz). Circles indicate the mode couplings α, γ, δ andη. Values are the squared bi-coherence b2 .

Fig. 4 Cross correlation functions between time evolutions ofwave powers; (a) CBF (τ) and (b) CAH (τ). τ of the peaks
indicate time delays of (a) (F) from (B) and (b) (H) from
(A). They are ensembles of ten discharges.

(m1, m2, m3) = (1, 2, 3)

is an ensemble of 300 time windows (each time window is
10 ms long), so that the confidence level is 0.003 (= 1/300).
Self-nonlinear coupling of (α: D = A + A), which is shownat f1 = f2 = 2.8 kHz, is weak (b2 = 0.04) but above the confidence level. Other combinations of the mode couplings
can be confirmed in Fig. 2. The values of the squared bicoherence b2 are (γ: E = A + C: 0.54), (δ: B = C + F:0.13) and (η: I = A + F: 0.11). Figure 3 shows the squaredbi-coherence in the case of (m1 , m2 , m3 ) = (1, 2, 3). The
mode couplings among waves with m = 1, 2 and 3 are clarified. The values of the squared bi-coherence b2 are (β:G = A + D: 0.27), (ε: H = C + D: 0.37), (ζ: H = A + E:0.16) and (κ: J = A + I: 0.11). Thus, the expected mode
couplings were confirmed and it was proved that the number of the irreducible parent modes was three.
Bi-coherence analysis indicates only the existence ofthe mode coupling among three waves. The quasi-mode
which is generated by the mode coupling of the other twomodes is not determined by bi-coherence analysis. Therefore, the three original parent modes are not clarified yet.
The powers of the fluctuation peaks (A) and (C) are relatively strong and stable in time, while other peaks appears
intermittently with short lifetimes. The correlation lengthsof (A) and (C) in the poloidal direction are long and they
are produced globally in the poloidal direction [12]. Fromthese facts, it is natural to conclude that (A) and (C) are
the original parent modes. The left problem is whether thethird parent mode is. (B) and (F) are the candidates.
To explore the parent mode between two modes, theenergy transfer direction is required. Amplitude correlation technique [16, 17] can support to find the direction. This technique is effective when the amplitude of the

10

0.2

Frequency f2 (kHz)�: 0.115

�: 0.37

�: 0.27
�: 0.16

0

0.1

�5

10
�10

�5
0
5
Frequency f1 (kHz)

10

0.0

Fig. 3 Contour plot of squared bi-coherence among three waves(m1 , f 1 ), (m2 , f 2 ) and (m3 , f 3 ), which satisfy the matchingcondition (m1 , m2 , m3 ) = (1, 2, 3). Frequency ranges are
the same as Fig. 2. Circles indicate the mode couplings
β, ε, ζ and κ. Values are the squared bi-coherence b2 .

three waves (m1 , f1 ), (m2 , f2 ) and (m3 = m1 + m2 , f3 =f1 + f2 ). Figure 2 shows an example of the bi-coherence
analysis applied to Z(m, f ). In this figure, m1 is fixed to
1 and m2 is fixed to 1 (therefore, m3 = 2). The horizontal and vertical axes are f1 and f2 , respectively. Thus, thesquared bi-coherence b2 in the case of (m1 , m2 , m3 ) = (1,1, 2) in the range of (–10 kHz ≤ f1 , f2 ≤ 10 kHz) and (–20 kHz ≤ f3 ≤ 20 kHz) is plotted in this figure. The result89

T. Yamada et al., Measurement of Nonlinear Mode Couplings in the Large Mirror Device-Upgrade

waves fluctuate with time. Time evolutions of the powersof two waves S1 (t) = S(m1 , f1 ,t) and S2 (t) = S(m2 , f2 ,t)are calculated by short-time Fourier transformation, andthe cross-correlation function between S1 (t) and S2 (t) areevaluated as�[S1 (t) − S¯1][S2 (t + τ) − S¯2]�
.
(2)
C12 (τ) = ��[S1 (t) − S¯1]2 ��[S2 (t) − S¯2]2 �

10
0.11

Frequency f (kHz)8

When the two waves fluctuate similarly in time, a peak inthe cross-correlation function appears near τ = 0 and thetime of the peak indicates the response of S2 (t) to S1 (t).
Figure 4(a) shows the cross-correlation function CBF (τ)
between the time evolutions of the powers of (B) and (F)(time window is 10 ms and move step is 0.2 ms). The peakof CBF (τ) is seen at τ = 2 ms, which indicates that (F) responds in 2 ms after the variation of (B). This delay timemeans the typical duration which is needed to complete theenergy transfer from (B) to (F). Therefore, (B) is considered to be the parent mode of (F). The matching condition
can be rewritten as (F = B − C). Figure 4(b) shows anotherexample; the cross-correlation function CAH (τ). It has apeak at τ = 1.6 ms. Therefore, (H) responds in 1.6 ms after(A), and (A) must be the parent mode of (H).
By these analyses, the three original parent modes are
considered to be (A, B, C). All the other waves (F–J) are
quasi-modes, which are successively generated by mode
couplings from the original parent modes (A, B, C). (A)and (B) have the features of drift wave [10], while (C) is
not a drift wave since its intensity is strong at the plasmaedge region and weak at the steep density gradient region [11]. Figure 5 is a summarized chart of the provided
results. The figure visually shows the successive generation of the quasi-modes from the original parent modes.
(A, B, C) are the original parent modes, (D, E, F) are thequasi-modes of the first generation, (G, H, I) are those of
the second generation, and (J) is that of the third generation. Focused into (H), there are two paths to create (H);(α → ε) and (γ → ζ). It can be said that (H) is created from(A) and (C) by multi-path; one is by way of (D) and theother is by way of (E).

6

0.11

4
2

�

D

� 0.04F
AI

�

0.13 B

�

0.54

E

J

�

G

� 0.27
0.37

�

H
�

0.16

�

0
C

�2

0

1

2
3
4
Poloidal mode number m

5Fig. 5 A chart that shows successive generation of quasi-modes
from the three original parent modes (A, B, C). Arrows
go from the parent mode to the child mode. The arrows
from (A) and (C) are omitted except the cases of α and γ.
The value near the arrow is squared bi-coherence of eachmode coupling.

original parent modes. A situation was observed that aquasi-mode of the second generation was created by multipath.

Acknowledgments

This work was supported by a Grant-in-Aid forSpecially-Promoted Research of MEXT (16002005) (ItohProject), Grant-in-Aid for Young Scientists (B) of
JSPS (19760597), the collaboration programs of NIFS
(NIFS07KOAP017) and RIAM, Kyushu University, andAsada Eiichi Research Foundation.
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5. Summary

In summary, by using a 64-channel poloidal Langmuir probe array, the 2D (poloidal mode number m andfrequency f ) power spectrum of the ion saturation-current
fluctuation was measured. 2D bi-coherence analysis,
which considers not only the matching condition of f but
also the matching condition of m, was applied for the observed fluctuation peaks, and the existence of the nonlinear mode couplings among the fluctuation peaks was confirmed. The energy transfer directions among the fluctuation peaks were estimated by amplitude correlation technique. Thus, three original parent modes were found out
to be (m, f ) = (1, 2.8 kHz), (2, 3.2 kHz) and (1, −0.9 kHz).
Other fluctuation peaks were quasi-modes, which weregenerated successively by mode couplings from the three90