24C12
Observation of nonlinear coupling of plasma turbulencein linear cylindrical plasma
直線磁化プラズマにおける乱流の非線形結合の観察	 
Tatsuya Kobayashi1, Shigeru Inagaki2,3, Yoshihiko Nagashima2,3, Stella Oldenbürger3,
Akihide Fujisawa2,3, Sanae-I. Itoh2,3, Kimitaka Itoh3,6, et al.
小林達哉1,	 稲垣滋2,3,	 永島芳彦2,3,	 Stella	 Oldenbürger3,	 荒川	 弘之4,	 佐々木真2,3,	 	 
山田琢磨3,5,	 矢木雅敏2,3,6,	 糟谷直宏3,7,	 藤澤彰英2,3,	 伊藤早苗2,3,	 伊藤公孝3,71Interdisciplinary Graduate School of Engineering Sciences, Kyushu University,
6-1 Kasuga-koen, Kasuga, Fukuoka 816-8580, Japan1
九州大学総合理工学府 〒816-8580 福岡県春日市春日公園6-1
2
Research Institute for Applied Mechanics, Kyushu University, 6-1 Kasuga-koen, Kasuga, Fukuoka 816-8580, Japan2
九州大学応用力学研究所 〒816-8580 福岡県春日市春日公園6-1
3
Itoh Research Center for Plasma Turbulence, Kyushu University,
6-1 Kasuga-koen, Kasuga, Fukuoka 816-8580, Japan
3
九州大学伊藤極限プラズマ研究連携センター 〒816-8580 福岡県春日市春日公園6-1
4
Japan Atomic Energy Agency, 801-1 Mukoyama, Naka, Ibaraki 311-0193, Japan4日本原子力研究開発機構 〒311-0193 茨城県那珂市向山801-1
5Graduate School of Frontier Sciences, The University of Tokyo,5-1-5 Kashiwanoha, Kashiwa, Tokyo 277-8561, Japan5
東京大学新領域創成科学研究科 〒277- 8561 千葉県柏市柏の葉5-1-5
6Japan Atomic Energy Agency, 2-166, Omotedate, Obuchi, Rokkasho-mura, Aomori, 039-3212 Japan6日本原子力研究開発機構 〒039-3212	  青森県六ヶ所村尾駮表舘2番166,
7
National Institute for Fusion Science, 322-6 Oroshi-cho, Toki, Gifu 509-5292, Japan7
核融合科学研究所 〒509-5292 岐阜県土岐市下石町322-6
Density fluctuations propagating in the azimutal direction are observed in the Plasma Assembly forNonlinear Turbulence Analysis (PANTA), under various neutral gas pressure conditions. Nonlinear
waveform and their frequency-mode number Fourier power spectrum entirely change depending on theneutral gas pressure. Nonlinear causal relation is estimated by using bispectral analysis and the Amplitude
Correlation Technique (ACT). Finally we discuss about relation between magnitude of amplitude andstrength of nonlinear coupling.

1. Introduction
Understanding of plasma turbulence is one of
the important issues to realize thermonuclear
fusionreactors.
Many
theoreticalandexperimental works have contributed to revealthat turbulent transport can be a serious obstacle
to achieve good confinement state of magnetizedplasmas [1]. It is well known that turbulence canform structures (e.g., zonal flows) throughnonlinear wave-wave coupling process, whichstrongly influence plasma confinement ability.
Recently, some theoretical works have reported
impact of neutral gas pressure on the turbulent
structure formation [1,2]. Neutral-ion and/or
neutral-electron
collisions
werepossible
candidates for a damping force in turbulent
structure formation. Neutral gas pressure
dependence on the structure formation wasdiscussed. Here, we focus on the impact ofneutrals in low temperature linear magnetizedplasma. Characteristic features of nonlinear
coupling are precisely investigated in different
neutral gas pressure regimes, and mechanismsthat determine the turbulent-state are discussed.
Bispectral analysis has been widely used toexperimentally observe nonlinear interactions[3,4]. The squared bicoherence, i.e., the squared
normalized third-order spectral correlation, shows
strength of nonlinear correlation among threemodes, which satisfy the matching condition,
f3=f1 f2, with a value from zero to unity. Even if
one uses bispectrum analysis, causal relation,such as direction of energy transfer among modes,
cannot be determined. For this purpose, we apply
the ACT [5], which is defined as,(2)
CAB (τ ) = A˜ (t) B˜ (t − τ ) A˜ 2 (t) B˜ 2 (t) ,where A˜ (t) [ B˜ (t) ] is amplitude modulation,
calculated by running FFT and A (t) [ B (t) ]€€f=6.8kHz

10.4

-6PSD
f=1.4kHz(counter-prop.)

f=8.2kHz0.2b2-7

10 05Frequency [kHz]10Power [A.U.]10(a)f=5.4kHz-52

10-4bPSD

10Fig.1. Observed power spectrum density (blackcurve) and squared bicoherence with respect to amode at f3~1.4 [kHz] (red curve).

2. Results
We briefly overview observed fluctuation power
spectrum density, as an example. We show powerspectrum density and squared bicoherence withrespect to a mode at f3=1.4 [kHz] for the case Pn =1 mTorr in Fig. 1. Rather large amplitude peaks areobserved in low frequency range, f < 17 [kHz]. In
relatively low neutral gas pressure cases (Pn < 2mTorr), somewhat broad peaks exist at f~4-10[kHz]. A counter-propagating (ion diamagneticdirection) mode is also observed at f=1.4 [kHz].
Bicoherence between the broad spectral peaks andcounter-propagating mode is significant.
To clarify direction of energy flow, we apply theACT. Figure 2 (a) shows time evolutions ofnormalized mode amplitude of distinguishable
spectral peaks shown in Fig. 1. They are calculatedwith a time-window of 5 ms and the time-window
is shifted by steps of 0.5 ms. Figure 2 (b) shows
cross correlation function of mode amplitude atthree different frequencies with that of the referencemode at f=1.4 [kHz]. The mode at f=6.8 [kHz] has
opposite correlation to the counter-propagating
mode with zero time lag. This might indicate thecounter-propagating mode and the mode at f=6.8[kHz] are coupled (or in predator-prey relation).
Fluctuations at f=5.4, 8.2 [kHz] have positive
correlation with the reference mode. Taking into
account the time lag, primary modes which are

f=1.4kHz (counter prop.)
f=5.4kHz
f=6.8kHz
f=8.2kHz
0.4
0.42
Time [s]-1

01

C.C.F.

indicates long-time average ofA(t) [ B(t) ].
Directions of energy flows are estimated fromcorrelation and time lag in the time evolution ofamplitude modulation.
€ in €
Experiments are performed
the PANTA, alinear device with a rf helicon discharge of 3 kW.
Excited plasma has a typical electron density of ne ~1 1019 m-3 and electron temperature of Te ~ 3 eV.
Neutral gas pressure is measured with an ion gauge
and controlled by a mass flow controller just before
each discharge. A multichannel (64ch) azimuthal
Langmuir probe array measures spatiotemporal
evolution of density fluctuation. We define theelectron diamagnetic direction as the positive
direction.

0(b)0
f=1.4kHz v.s.
f=5.4kHz
f=6.8kHz
f=8.2kHz
-1
-0.02
-0.01

0! [s]

0.01

0.02

Fig.2. (a) Time evolution of fluctuation mode
amplitude (normalized by its long-time average), (b)amplitude correlation with respect to counter
propagating mode.

linearly unstable can be identified as the mode at
f=6.8 [kHz] and the mode at f=8.2 [kHz] and/or thecounter-propagating mode. Same analyses are also
applied in high neutral cases. In addition, recentlyendplate-biasing experiment is running in thePANTA. We externally give perturbation to mode
amplitudes and observe the time lag of responses
in fluctuating modes to obtain another evidence
for detection of the direction of energy flow.
3. Conclusion
We calculate integrated fluctuation power,
which is almost complementally determined bylinear mode power, and total squared bicoherence
in a low frequency region for various Pnconditions. Similarity of these shapes is found,clearly indicating that the nonlinear coupling
process is quite essential to determine turbulentstates.
Acknowledgments
This work is partly supported by a Grant-in-Aidfor Scientific Research S from JSPS (21224014),
the Grant-in-Aid for Scientific Research B fromJSPS (23360414), the collaboration programs ofRIAM
KyushuUniversityandNIFS(NIFS10KOAP023).
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47 (2005) R35.
[2] N. Kasuya, et. al: Phys. Plasmas 15 (2008) 052302.
[3] Y. C. Kim, et. al: IEEE Trans. Plasma Sci. 7 (1979)
120-131[4] Nagashima, et. al: Phys. Rev. Lett. 95 (2005) 95002[5] Yamada, et. al: J. Phys. Soc. Jpn. 79 (2010) 085001