24A09
Study of Nonlinear Processes of Fluctuations in Laboratory Magnetized Plasmas	 Stella Oldenbürgera, S. Inagakia,b, A. Fujisawaa,b, T. Kobayashic, S.-I. Itoha,b, G. Bonhommed
Stella Oldenbürgera,	 稲垣滋a,b,藤澤彰英a,b,小林達哉c,伊藤早苗a,b,G. BonhommedaItoh Research Center for Plasma Turbulence, Kyushu University, 6-1 Kasuga-koen,Kasuga 816-8580, Japana
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6-1bResearch Institute for Applied Mechanics, Kyushu University, 6-1 Kasuga-koen, Kasuga 816-8580, Japanb
九州大学	  応用力学研究所	  〒816-8580 福岡県春日市春日公園6-1c
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, 6-1 Kasuga-koen,
Kasuga 816-8580, Japanc
九州大学	  総合理工学府	  〒816-8580 福岡県春日市春日公園6-1
dInstitut Jean Lamour, UMR CNRS 7198, Faculté des Sciences et Technologies, Campus Victor Grignard, BP 70239
54506 Vandoeuvre-lès-Nancy Cedex, France
Experimental techniques used to study dynamics of turbulent fluctuations in two magnetized linear plasma
devices are presented. The characteristics and data requirements of wavelet-based bicoherence analysis areexplained and results obtained from fast imaging data on the Mirabelle device are shown. The results
reveal a temporal delay between coupling detection and power gain of a fluctuation component and show
the importance of taking temporal dynamics of nonlinear processes into account. Dynamics of particle
transport have been studied on PANTA and the discovered modulation of transport by coexisting modes
calls for a merging of nonlinear coupling and transport studies.

1. Nonlinearity and Dynamics of Fluctuations
Progress in the experimental studies ofturbulent processes in cylindrical magnetizedplasma devices is generally centered around threeaxis: the search for advanced analysis tools, theuse of new types of diagnostics to fulfill datarequirements for analysis and eventuallyinterpretation of results in terms of physicalprocesses.
Concerning detection of nonlinear coupling fromdata of plasma fluctuations, different analysis
approaches have been pursued. A recurrent
mathematical quantity is the bicoherence [1]. It
tests whether three fluctuation components havematching frequencies (or wave numbers) andmatching phase for a long enough set of ensembles,
usually taken as successive time windows.
Bicoherence indicates whether the necessary
conditions for nonlinear coupling are satisfied. To
know if energy or enstrophy is effectivelytransferred between fluctuation components, thetransfer coefficients can be computed for the given
fluctuation regime [2]. Other methods are yetnecessary to study the fine dynamics and interplayof fluctuation components. One of those is theamplitude correlation technique [3] (see also
contribution 24C12 by T. Kobayashi in thisconference), which is based on the assumption thatthe power of the nonlinearly driven mode willchange later than the power of the parent mode.
Another method consists in investigating sequenceswhere power of driven fluctuations increases whencoupling conditions are fulfilled. This last method
requires computing possible couplings within very
short time windows. This can be achieved by using
wavelet bicoherence [4]. Wavelets can extract thefluctuation components of a signal for short data
windows and significantly improve the temporal
resolution of the results. An additional possibility isto average over a second dimension such as spaceinstead of time, thus obtaining a quantity close to awave number wavelet bicoherence [5]. Theseadvanced methods require fluctuation data highly
resolved in time and space.
3. Experimental Access to Fluctuation Data
Linear magnetized plasma devices arecharacterized by relatively low temperatures thatallow a wide choice of diagnostics. Interesting data
dimensions are time, radius, azimuthal angle andaxial position. Arrays of Langmuir probes canprovide good temporal resolution for several spatial
points but can perturb the plasma. Azimuthal arrays,
useful for investigation wave number space, usually
have a fixed radial position. A non-perturbative,
flexible diagnostic on linear machines is fast
imaging using high sensitivity high-speed cameras.
It was found on the Mirabelle device that asignificant contribution to light emission resultsfrom excitation of neutral atoms by plasma
electrons [6] and that light intensity is
proportional to plasma density. With a high enoughtemporal and spatial resolution, camera data can beused instead of probe data to study turbulence.
4. Bicoherence Analysis on Camera Data
The wavelet wave number bicoherence has been
applied to fast camera data from the linear device
Mirabelle [7]. Figure 1 shows a specific moment intime when a fluctuation with azimuthal mode
number m=3 is beginning to develop in the powerspectrum S(m), after this mode appeared in thesummed bicoherence b2(m). This kind of temporalsequences can suggest an energy transfer to thegrowing mode via nonlinear coupling. The
corresponding camera frame is also shown in fig. 1.

Fig.1. Wave number bicoherence and power spectrum
computed from camera data on Mirabelle (left side),corresponding camera frame (right side).

The presented method’s weak point, just as for thetemporal wavelet bicoherence, is the doubt about
independence of the considered ensembles. A
repetitive record of the same sequence of events
could guarantee real ensemble averaging.
5. Nonlinear interactions and transport
Energy transfers between different scales caninfluence other phenomena in plasmas such as
fluctuation-induced transport. A prominent
example how nonlinear interactions can influencetransport is the drive of zonal flows by drift wave
turbulence. A study on the PANTA showed thattransport is also a non-stationary phenomenon insmall linear devices even without explicit
observation of detached coherent structures [8].
Figure 2 shows the time and frequency
decomposed radial particle transport in thePANTA in a regime featuring a solitary-wave-likefluctuation. Transport on the frequency of themode m=2 is modulated in time and fluctuates
following an m=0 structure. The alteration oftransport on one mode by another mode requires
an interaction between those modes that could bedetected by methods searching for nonlinear
couplings. Transport and nonlinear couplingstudies are planned in a wide range of discharge
conditions, by using amongst others biasing of a

newly installed isolated endplate.

Fig.2. Frequency and time resolved transport for thePANTA solitary-wave-like regime.

Acknowledgments
This work was partly supported by the European
Community under the EURATOM and CEA
contract LRC-DSM #99.18, and the French
Research Federation for fusion studies. It was partly
carried out within the framework of the European
Fusion Development Agreement. The views andopinions expressed herein do not necessarily reflect
those of the European Commission. Further
financial support was received from the French
National Research Agency through contracts
ANR-08-JCJC-0068-01 and ANR-07-BLAN-018701 and by a Bonus Qualité Recherche (BQR) from
Nancy-Université and Région Lorraine. This was
also partly supported by a Grant-in-Aid for
Scientific Research S from JSPS (21224014), theGrant-in-Aid for Scientific Research C (20560767),
the collaboration programs of RIAM Kyushu
University and NIFS (NIFS10KOAP023).
References[1] Y. C. Kim and E. J. Powers: IEEE Trans. Plasma Sci.
7 (1979) p.120.
[2] P. Manz, M. Xu, S. C. Thakur and G. R. Tynan:
Plasma Phys. Control. Fusion 53 (2011) 095001.
[3] T. Yamada, S.I. Itoh, S. Inagaki, Y. Nagashima, s.
Shinohara, N. Kasuya, K. Terasaka, K. Kamataki, H.
Arakawa, M. Yagi, A. Fujisawa and K. Itoh: Phys
Plasmas 17 (2010) 052313.
[4] B. P. van Milligen, C. Hidalgo, and E. Sanchez:
Phys. Rev. Lett. 74 (1995) p. 395.
[5] F. Brochard, T. Windisch, O. Grulke and T. Klinger:
Phys. Plasmas 13 (2006) 122305.
[6] S. Oldenbürger, C. Brandt, F. Brochard, N. Lemoineand G. Bonhomme: Rev. Sci. Instrum. 81 (2010)
063505.
[7] S. Oldenbürger, F. Brochard and G. Bonhomme:
Phys. Plasmas 18 (2011) 032307.
[8] S. Oldenbürger, S. Inagaki, H. Arakawa, T.
Kobayashi, K. Uriu, K. Kawashima, A. Fujisawa, K.
Itoh and S.-I. Itoh: Proc. of the 38th Conf. on Plasma
Physics (2011) P2.025.