Plasma and Fusion Research: Regular Articles

Volume 3, 044 (2008)

Experimental Study of Drift Wave Turbulence in Linear Plasmas∗)
Takuma YAMADA, Sanae -I. ITOH, Takashi MARUTA1) , Naohiro KASUYA2) ,Shunjiro SHINOHARA1) , Yoshihiko NAGASHIMA, Masatoshi YAGI, Kenichiro TERASAKA1) ,Shigeru INAGAKI, Yoshinobu KAWAI, Masayuki FUKAO3) , Akihide FUJISAWA2)and Kimitaka ITOH2)
1)

Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga 816-8580, Japan
2)
National Institute for Fusion Science, Toki 509-5292, Japan
3)
Myojocho, Uji 611-0014, Japan(Received 1 December 2007 / Accepted 18 June 2008)

Linear plasmas allow multi-point, low-temperature measurements with Langmuir probes. We measured
ion saturation-current fluctuations of a Large Mirror Device-Upgrade linear plasma using a poloidal Langmuirprobe array. By varying the discharge conditions, the spatiotemporal behavior showed a change from a coherent
sine wave to a turbulent waveform through a periodic, modulated sine wave. The two-dimensional (poloidal
wave number and frequency) power spectrum for each regime showed a single fluctuation peak, a peak andits harmonics, and a number of peaks in the poloidal wave number–frequency space. Bi-spectral analysis was
performed for the turbulent regime, and showed the existence of nonlinear couplings among fluctuation peaks
and broadband components.
c 2008 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: linear plasma, drift wave, turbulence, broadband fluctuation, bi-spectral analysis
DOI: 10.1585/pfr.3.044

1. Introduction

power spectrum [17]. In this article, we present a change
in the spatiotemporal behavior and two-dimensional power
spectrum from a coherent to a turbulent regime. In addition, we identify nonlinear couplings among the fluctuation peaks and broadband components. Bi-spectral analysis, which indicates nonlinear coupling among the threewaves, is explained in this section. Section 2 describes theexperimental setup, and Secs. 3 and 4 provide results ofspectral and bi-spectral analyses, respectively.
Bi-spectral analysis examines the relationship among
three waves with frequencies of ω1 , ω2 and ω3 , which satisfy the matching condition ω3 = ω1 + ω2 . When the
Fourier transformed expression of a temporal waveformz(t) is Z(ω), the bi-spectrum B of the three waves is expressed as

The study of nonlinear couplings in drift wave turbulence can help our understanding of anomalous transports. Theoretical prediction provides models that indicatethat drift waves couple with meso-scale structures, such as
zonal flows and streamers [1, 2]. Experimental studies onthese nonlinear couplings are in progress, both for linear
and toroidal plasmas [3–13]; however, linear plasmas offer the advantage of allowing multi-point, low-temperature
measurements with Langmuir probes, and their simple
cylindrical shape makes instability models easy to set up.
Drift waves are excited by the radial pressure gradient andpropagate in the poloidal direction of the plasma cross section. Therefore, multi-point measurements with a poloidal
probe array in a linear plasma are useful for understanding
drift wave turbulence.
In a Large Mirror Device-Upgrade (LMD-U) linear
plasma [14], drift waves and the route to drift wave turbulence were observed [15]. A 64-channel poloidal Langmuir probe array, which can measure the precise poloidal
wave numbers of the fluctuations [16], obtained a highquality, two-dimensional (poloidal wave number kθ andfrequency ω) power spectrum of drift wave turbulence. In
drift wave turbulence, a number of fluctuation peaks andbroadband components appeared in the two-dimensional

B = Z(ω1 )Z(ω2 )Z ∗ (ω3 ).

(1)

When the three waves fluctuate independently, the absolutevalue |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 indicates the relationship among
the phases of the three waves, are expressed as
|B|2
,
(2)
|Z(ω1 )Z(ω2 )|2 |Z(ω3 )|2 
Im(B)
.
(3)
φB = tan−1Re(B)Bi-spectral analysis is important for investigating the coub2 =author’s e-mail: takuma@riam.kyushu-u.ac.jp∗) This article is based on the invited talk at the 24th JSPF Annual Meeting(2007, Himeji).

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c 2008 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Regular Articles

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pling among three waves. When the bi-coherence b is finite
for the three waves, it suggests that one wave is produced
by nonlinear coupling of the other two waves.

the poloidal mode number space by coupling with the realfluctuation modes [16].
Increasing the magnetic field or decreasing the argonpressure changes the regime from coherent to turbulent.
Figure 2 shows the spatiotemporal fluctuations of the ion
saturation-current measured with the 64-channel probe array. The discharge conditions of the magnetic field andargon pressure are (a) 0.02 T and 2 mTorr, (b) 0.09 T and
6 mTorr, and (c) 0.09 T and 2 mTorr, respectively. The increasing direction of the poloidal angle θ corresponds tothe electron diamagnetic direction. The waveform of (a) is
a coherent sine wave with poloidal mode number m = 3.

2. Experimental Setup
The LMD-U linear plasma has a cylindrical shape,with diameter of about 100 mm and axial (z) length of
3740 mm. Under the experimental conditions of 3 kWof 7 MHz rf power, 0.01-0.15 T of axial magnetic field(in the +z direction), and 1-6 mTorr of argon pressure in
the source region, the electron density and temperature of
the LMD-U plasma are about 1019 m−3 and 3 eV, respectively. The electron density gradient is steep in the radiusof r = 30-40 mm, and produces resistive drift wave instabilities, propagating in the poloidal direction (θ).
The ion saturation-current fluctuations in LMD-U
were measured with a 64-channel poloidal Langmuir probearray, shown in Fig. 1. The axial and radial positions of
the poloidal probe array are zp = 1885 and rp = 40 mm,
respectively. Each tungsten probe tip has a diameter of
0.8 mm and is 3 mm long. The 64 probe tips are aligned in
the poloidal direction (θ) with 3.9 mm spacing. Mounting
the array on a single metal plate aligns the tips precisely,with misalignments of less than 0.1 mm. The probe array measures the poloidal mode number m of fluctuations
up to ±32, where m is related to the poloidal wave number kθ by m = rp kθ . The minimum measurable poloidal
mode number, m = ±1, corresponds to the poloidal wave
number kθ = ±0.025 mm−1 , and the maximum measurable
mode number, m = ±32, corresponds to the wave number kθ = ±0.8 mm−1 . The metal plate with the probes isconnected to the vacuum flange through a welded bellows.
Changing the lengths of the screws around the bellows
moves the metal plate in two dimensions in the plasma
cross section, and the probe array center can be adjusted tothe plasma column center. This is important since m = 1
dominant shift error induces spurious fluctuation modes inFig. 2 Spatiotemporal behavior of ion saturation-current fluctuations (arb. unit) measured with the 64-channel poloidal
probe array. The discharge conditions (magnetic field
and argon pressure) are (a) (0.02 T, 2 mTorr), (b) (0.09 T,
6 mTorr), and (c) (0.09 T, 2 mTorr).

Fig. 1 Schematic view of a 64-channel poloidal Langmuir probearray. The measuring radius is 40 mm. The probe array isconnected to a vacuum flange through a welded bellows
so that it moves in two dimensions.

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Only a single instability mode of m = 3 in the electron diamagnetic direction is excited in the plasma. The straightlines in the figure show that the probe array measurements
are performed with a good symmetry in the plasma crosssection, due to the fine positioning of the probe array. The
waveform of (b) is a periodic waveform with m = 1. Periodic positive spikes and some modulations with m ≥ 2
are seen. The waveform of (c) shows that the plasma is ina turbulent regime and is not a periodic waveform. Thereis a main m = 1 fluctuation in the electron diamagnetic
direction, together with complex structures separating andcombining.
In conclusion, varying the discharge conditions
changes the plasma from (a) a coherent regime to (b) aperiodic regime, to (c) a turbulent regime. Further analysis
is performed by two-dimensional spectral analysis.

3. Spectral Analysis
The power spectrum of the ion saturation-current fluctuation in each regime, explained in the previous section,
was calculated by two-dimensional Fourier transformation(from poloidal angle θ to poloidal mode number m, andfrom time t to frequency f = ω/2π). Figure 3 shows thetwo-dimensional power spectra S (m, f ) of Figs. 2 (a)-(c).
An average of the results from 300 time windows (eachtime window is 10 ms long) during discharges was used toderive each power spectrum S (m, f ). A two-dimensional
Fourier transformation can determine the poloidal propagating directions of the fluctuations by the signs of thepoloidal mode number m (when the frequency is set tof ≥ 0). Positive and negative mode numbers correspond to
the electron and ion diamagnetic directions, respectively.
This feature is a great advantage over single-probe measurements.
The power spectrum in the coherent regime (Fig. 3 (a))
shows only a single fluctuation peak at (m, f ) =(3, 4.2 kHz); therefore, a coherent sine wave is observedin the spatiotemporal behavior. The density fluctuation(the ion saturation-current fluctuation) leads the potential
fluctuation (the floating potential fluctuation) by 10-30 degrees, and the normalized density fluctuation level is comparable to the normalized potential fluctuation level. This
indicates that the observed instability mode was a drift
wave mode [15]. Drift wave instabilities with m  3 were
damped and only the m = 3 mode remained.
The power spectrum in the periodic regime (Fig. 3 (b))
shows some fluctuation peaks with the relationship off [kHz] = 1.15 m, which is shown by a dashed line. The
peak at (m, f ) = (1, 1.15 kHz) is considered the standard
drift wave mode excited in the plasma, and the others areharmonics of the standard wave. Up to the eighth harmonic are seen in the figure. The harmonics are excitedby self-nonlinear couplings of the standard mode. Therefore, the spatiotemporal behavior shows a quasi-periodic
structure with a modulated sine wave. These harmonics

Fig. 3 Contour plots of two-dimensional power spectra S (m, f )(arb. unit) for each discharge shown in Figs. 2 (a)-(c).
The dashed line in (b) shows the relationship f [kHz] =1.15 m. The solid lines in (b) and (c) show linear dispersion relations of drift waves, calculated by numericalcode analyses.

are non-mode peaks, since drift waves in plasmas do notshow straight linear dispersion relationships in m- f space,
and the harmonics do not satisfy the linear dispersion relationship. The solid line in the figure indicates the linear
dispersion relationship of drift waves calculated by numerical code analysis [18]. The line shows the case with no dc
radial electric field. In reality, there is a dc radial electricfield inside the plasma, which rotates the plasma by E × B
drift with a frequency of fE×B ; it is not currently accurately
measurable at LMD-U. The linear dispersion relation ismodified by fE×B by subtracting m · fE×B from the eigen044-3

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drift wave mode [15]. The peaks at (m, f ) = (2, 5.6 kHz)and (3, 8.4 kHz) must be harmonics of the peak (m, f ) =(1, 2.8 kHz) produced by self-nonlinear couplings, sincethese satisfy the relationship of f [kHz] = 2.8 m. Other
peaks are excited instability modes or peaks excited by
nonlinear couplings. For instance, the fluctuation peak at(m, f ) = (−1, 0.9 kHz) in the ion diamagnetic direction is
not a drift wave mode, since its intensity is strong in theplasma edge region, rather than the steep gradient region(see Fig. 4 (a)). The solid line in the figure again shows
the calculated linear dispersion relation of drift waves withno dc radial electric field. Several unstable modes, including (m, f ) = (1, 2.8 kHz) in the m- f plane (except (m, f )= (−1, 0.9 kHz)), must satisfy the dispersion relation byconsidering the dc electric field.
In addition, there are broadband fluctuation components other than the fluctuation peaks. These are also considered to be produced by nonlinear couplings. Identification of the nonlinear couplings by bi-spectral analysis isdiscussed in the next section.

4. Bi-Spectral Analysis
Nonlinear couplings in the turbulent regime are confirmed by bi-spectral analysis, as shown below. Whether
three waves with frequencies of f1 , f2 , and f1 + f2 fluctuate independently or dependently is judged by bi-spectral
analysis. Bi-spectral analysis was applied to prove the nonlinear couplings among the fluctuation peaks that appear in
Fig. 3 (c). Figure 5 shows the squared bi-coherence of theion saturation-current fluctuation measured with a single
probe. The frequency range is 0-10 kHz. The result was an
ensemble of 300 time windows (each time window is 10 mslong). The white region indicates that the three waves haveno relationships, and the dark region indicates that a phaserelationship exists among the three waves; therefore, one
wave is excited by nonlinear coupling of the rest of the
waves.
The dark lines, f1,2 = 0.9 and 2.8 kHz, indicate that thefluctuation peaks (m, f ) = (1, 2.8 kHz) and (−1, 0.9 kHz)in Fig. 3 (c) couple with many other fluctuation peaks andalso with broadband components. Self-nonlinear couplingof (m, f ) = (1, 2.8 kHz) is weak, but above the confidence level of 0.003 (= 1/300). Coupling between (m, f )= (1, 2.8 kHz) and (2, 5.6. kHz) to produce the third harmonic is significantly strong. Other strong and weak couplings are seen everywhere in f1 - f2 space. For instance,
the non-drift wave mode (m, f ) = (−1, 0.9 kHz) couples
strongly with the frequency of 4.7 kHz, whose poloidal
mode number is estimated to be three by Fig. 3 (c). This
coupling indicates the three wave couplings in Fig. 3 (c)of (m, f ) = (−1, 0.9 kHz) + (3, 4.7 kHz) = (2, 5.6 kHz),
which are expected from previous work [15]. (An exhaustive analysis of peak decomposition, shown in Fig. 5, will
be discussed in a future report.) The weak and broad couplings in the f1,2 > 4 kHz region indicate couplings among

Fig. 4 (a) The radial profile of power spectrum (arb. unit) ofrelative ion saturation-current fluctuation in a turbulence
regime. A radially movable Langmuir probe was used.
The discharge conditions (magnetic field and argon pressure) are (0.02 T, 2 mTorr). (b) The power spectrum (arb.
unit) at r = 40 mm.

frequencies. The radial electric field is of the order of O
(10 V/m), so that fE×B is of the order of O [kHz]. Therefore, the standard mode can be understood as a drift wavemode.
The power spectrum in the turbulent regime(Fig. 3 (c)) shows a number of fluctuation peaks in
m- f space. Figure 4 shows the results of the samedischarge conditions measured with a single probe.
Figure 4 (a) shows the radial profile of the power spectrum
S (r, f ) measured with a radially movable probe [15], andFig. 4 (b) shows the power spectrum S ( f ) at r = 40 mm
measured with one probe of the 64-channel probe array.
The two-dimensional power spectrum S (m, f ) contains
more information than the power spectrum S ( f ) measured
with a single probe. The broadband spectrum seen in
Fig. 4 (b) reveals some fluctuation peaks in m- f space bydecomposing into poloidal mode numbers.
The fluctuation peaks in Fig. 3 (c) do not satisfy a single proportional relationship between m and f . Therefore,
more than one unstable mode is excited in the plasma.
Some of the peaks are excited by nonlinear couplings between other peaks. The strongest peak in m- f space is at(m, f ) = (1, 2.8 kHz). This fluctuation peak has a strong
intensity at a steep gradient region in the plasma electron density profile (see Fig. 4 (a)), and is considered a044-4

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Volume 3, 044 (2008)
fluctuation peaks with more than one primary mode.
The nonlinear couplings among the fluctuation peaks andbroadband components in the turbulent regime were confirmed by bi-spectral analysis.
Bi-spectral analysis can be extended to two dimensions by considering the matching condition of the poloidal
wave number. The 64-channel poloidal probe array isable to measure the precise poloidal wave number. Therefore, two-dimensional bi-spectral analysis in kθ -ω space isplanned for future work.

Acknowledgments
This work was supported by a Grant-in-Aid forSpecially-Promoted Research of MEXT (16002005) (ItohProject), by Grant-in-Aid for Young Scientists (B) of JSPS
(19760597), and by the collaboration programs of NIFS
(NIFS07KOAP017) and RIAM, Kyushu University.
[1] P.H. Diamond et al., Plasma Phys. Control. Fusion 47, R35(2005).
[2] A. Yoshizawa, S.-I. Itoh and K. Itoh, Plasma and Fluid Turbulence, Theory and Modelling (Institute of Physics Publishing, Bristol and Philadelphia, 2003).
[3] T. Klinger et al., Phys. Rev. Lett. 79, 3913 (1997).
[4] M. Kono and M.Y. Tanaka, Phys. Rev. Lett. 84, 4369
(2000).
[5] C. Schröder et al., Phys. Rev. Lett. 86, 5711 (2001).
[6] T. Kaneko, H. Tsunoyama and R. Hatakeyama, Phys. Rev.
Lett. 90, 125001 (2003).
[7] V. Sokolov and A.K. Sen, Phys. Rev. Lett. 92, 165002(2004).
[8] M.J. Burin et al., Phys. Plasmas 12, 052320 (2005).
[9] G.R. Tynan et al., Plasma Phys. Control. Fusion 48, S51(2006).
[10] C. Lechte, S. Niedner and U. Stroth, New J. Phys. 4, 34(2002).
[11] Y. Nagashima et al., Phys. Rev. Lett. 95, 095002 (2005).
[12] Y. Hamada et al., Phys. Rev. Lett. 96, 115003 (2006).
[13] F.M. Poli et al., Phys. Plasmas 13, 102104 (2006).
[14] S. Shinohara et al., Proc. 28th Int. Conf. on Phenomena in
Ionized Gases (Institute of Plasma Physics AS CR, Prague,2007), 1P04-08.
[15] K. Terasaka et al., Plasma Fusion Res. 2, 031 (2007).
[16] T. Yamada et al., Rev. Sci. Instrum. 78, 123501 (2007).
[17] T. Yamada et al., Plasma Fusion Res. 2, 051 (2007).
[18] N. Kasuya et al., J. Phys. Soc. Jpn. 76, 044501 (2007).

Fig. 5 Squared bi-coherence of ion saturation-current fluctuation in f - f space. The discharge conditions are same as
those for Fig. 4. The confidence level is 0.003. Dark regions indicate nonlinear couplings among three fluctuations with frequencies f1 , f2 , and f1 + f2 .

broadband components. Thus, non-mode excitations froma few excited modes were confirmed by bi-spectral analysis. Drift wave turbulence is produced by nonlinear couplings among a few parent modes.

5. Summary
We measured the spatiotemporal behavior and twodimensional power spectra of the ion saturation-current
fluctuations in an LMD-U linear plasma using a 64channel poloidal probe array. By varying the discharge
conditions, i.e., the magnetic field and argon pressure, thebehavior changed from a coherent to a turbulent regime,
passing through a periodic regime. The coherent regime
showed a single fluctuation peak in m- f space. The
periodic regime showed a standard fluctuation peak andits harmonics. The turbulent regime showed a number of044-5