Proceedings of ITC/ISHW2007

Spatiotemporal Behavior of Drift Waves in LMD-U
Takuma YAMADA1) , Sanae -I. ITOH1) , Kenichiro TERASAKA2) , Naohiro KASUYA3) ,Yoshihiko NAGASHIMA1) , Shunjiro SHINOHARA2) , Takashi MARUTA2) , Masatoshi YAGI1) ,Shigeru INAGAKI1) , Yoshinobu KAWAI1) , Akihide FUJISAWA3) and Kimitaka ITOH3)
1)

2)

Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga 816-8580, Japan
3)
National Institute for Fusion Science, Toki 509-5292, Japan

In the LMD-U linear magnetized plasma, fluctuation measurements with multi-channel poloidal Langmuirprobe arrays have been performed. The mean poloidal mode number of the fluctuation and its spread suggest theexistence of broadband fluctuation, which is considered to be produced by nonlinear couplings. The broadband
fluctuation develops into high poloidal mode number and high frequency region, which do not satisfy the linear
dispersion relation of drift wave modes. The broadband fluctuation revealed small correlation time and poloidal
length than the fluctuation peaks. Poloidal mode decomposed axial coherence was measured with two poloidal
probe arrays. The axial coherence was strong both for the broadband fluctuation and fluctuation peaks.
Keywords: linear plasma, multi-probe, drift wave, turbulence, nonlinear coupling

1

Introduction

Recently, there has been an advance in the study of nonlinear interaction between drift wave turbulence and mesoscale structures such as zonal flows and streamers [1, 2]),and experiments in linear plasma devices have been inprogress [3–10]. In these studies, complex wave patternsin poloidal direction have been observed (see also reports
from toroidal plasma experiments, e.g., [11–14]). Theseadvancements highlight the need to measure fluctuations
at multiple spatial positions simultaneously. Fluctuation
measurements using poloidal multi-probe arrays have been
performed in linear plasmas and torus plasmas [3, 4, 6, 15].
In the LMD-U linear magnetized plasma [16], multipoint measurements of the ion saturation current and floating potential fluctuations using poloidal Langmuir probearrays have been in progress. Drift wave modes driven bysteep radial density gradient were identified, and a drift
wave turbulence regime was achieved [17]. The poloidal
mode numbers and frequencies of the drift wave modes
were compared with the calculated linear dispersion relation of drift wave [18], and they were in a good agreement.
Moreover, in a drift wave turbulence regime, quasi-modes
and broadband components that do not satisfy the dispersion relation were found and suggested that they were produced by nonlinear couplings of parent modes [19]. In thisarticle, we report the details about the broadband components found in the drift wave turbulence regime.

2

Poloidal Probe Arrays

We have performed a quasi-two-dimensional measurementof the ion saturation current fluctuation of the LMD-U linauthor’s e-mail: takuma@riam.kyushu-u.ac.jp

3.74 m0.4 m

1.885 m1.625 mPumping

PlasmaRf antenna
φ0.095 m

48ch 64ch

φ0.445 m
Poloidalprobe arrays
Magnetic coils
Pumping

Fig. 1 Schematic view of the LMD-U linear plasma device, andthe positions of the polodial probe arrays (64-channel and
48-channel).

ear plasma [16]. The schematic view of the LMD-U deviceis shown in Fig. 1. A linear magnetized plasma is createdby an rf (the frequency of 7 MHz and the power of 3 kW)
antenna in a quartz tube (the axial length of 0.4 m and theinner diameter of 0.095 m) with argon gas filled in. The
plasma is guided along straight magnetic field created bymagnetic coils surrounding the vacuum vessel to form acolumn shape. The axial length and inner diameter of thevacuum vessel are 3.74 m and 0.445 m, respectively.
There are two poloidal Langmuir probe arrays installed on LMD-U. A 64-channel poloidal probe array is
installed at the axial position of 1.885 m and a 48-channel
poloidal probe array is installed at 1.625 m. The tungsten probe tips of the 64-channel probe array are fixed at
the measuring radius of 40 mm (the probe tips are 3.9 mm
apart), and the position of the whole probe array is adjustable two-dimensionally in the plasma cross section.
Therefore, the precise poloidal mode number of the fluctuation is available by this probe array [20]. The 48-channel
probe array consists of 16 probe units, which are mov-Proceedings of ITC/ISHW2007
10-4

15

Frequency f (kHz)Poloidal angle θ (rad)

2π

π

10-5
10
10-6
5
10-7

0
0

0.5

1
Time t (ms)1.5

2

Fig. 2 Spatiotemporal behavior of the ion saturation currentI(θ, t) measured with the 64-channel poloidal probe array.

able in the radial direction. Each probe unit has 3 tungsten probe tips. The probe tips line up to equal distancewhen the measuring radius is set to 40 mm (the probe
tips are 5.2 mm apart). This probe array can measure thepoloidal wave number and radial profile of the plasma fluctuation [17]. By these two poloidal probe arrays, the detailsof the broadband fluctuation and fluctuation peaks such aspoloidal mode numbers, poloidal correlation lengths andpoloidal mode decomposed axial correlations are measurable.

3

Spectral Analysis

The spatiotemporal behavior of the ion saturation current
fluctuation varies by changing parameters from periodical coherent wave structure to turbulence regime, which
consists of many fluctuation components with different
poloidal mode numbers and frequencies [4]. In LMD-U,
increasing the magnetic field (over 0.04 T) and decreasing
the filled argon pressure (under 0.4 Pa) make the plasma
into turbulence regime by affecting the density gradientand collision with neutrals [16]. Figure 2 shows the spatiotemporal behavior of the ion saturation current I(θ, t),where θ is the poloidal angle and t is the time, in the turbulence regime measured with the 64-channel probe array.
The magnetic field was 0.09 T and the argon pressure was
0.27 Pa. The increasing direction of the poloidal angle θcorresponds to the electron diamagnetic direction. The
spatiotemporal waveform consists of a number of fluctuation components. The main fluctuation is the flow in theelectron diamagnetic direction with the frequency of about
3 kHz.
The spatiotemporal waveform in the turbulence
regime can be decomposed into the poloidal mode number and frequency spaces by spectral analysis. By twodimensional Fourier transformation (poloidal angle θ topoloidal mode number m, and time t to frequency f ),

10-8

0
-5

0
5
Poloidal mode number m

10

Fig. 3 Contour plot of the power spectrum S(m, f ) (arb. unit)
over plotted with the calculated linear dispersion relation of drift waves (white asterisks). The solid black line
shows the mean poloidal mode number, and the broken
black lines show its spread.

ˆI(θ, t) is transformed to I(m,f ). Figure 3 shows theˆpower spectrum S(m, f ) = |I(m,f )|2 /df , where df isthe frequency resolution and is 0.1 kHz in this case. The
power spectrum S(m, f ) was an ensemble of 300 time
windows with 10 ms (= df −1 ) span each. One of the advantages of poloidal multi-point detection to single point
measurement is that the poloidal flow direction can be determined. When the frequency is set to f ≥ 0, the fluctuations with positive poloidal mode numbers flow in the electron diamagnetic direction and that with negative poloidal
mode numbers flow in the ion diamagnetic direction.
The white asterisks in Fig. 3 show the calculated linear
dispersion relation of drift waves with a uniform rotation
by dc radial electric field assumed. The fluctuation peaksat (m, f ) = (1, 2.8 kHz) and (2, 3.2 kHz) satisfy the dispersion relation, and they are considered to be drift wave
modes. The peak at (m, f ) = (–1, 0.9 kHz) is a flute wavelike mode in the ion diamagnetic direction. Other fluctuation peaks such as (m, f ) = (2, 5.6 kHz) and (3, 4.7 kHz)do not satisfy the dispersion relation, and they are quasimodes. The solid and broken lines in Fig. 3 show the frequency dependences of the mean poloidal mode number
and its spread, respectively. They are defined bymS(m, f )S(m, f ), (1)m(f ) =m

2∆m(f )=mm
2[m − m(f )] S(m, f )S(m, f )

.

(2)

m

The mean poloidal mode number traces the strong fluctuation peaks, and the spread decreases in the existence of thestrong peaks. In the high frequency region with no remarkable fluctuation peaks (f > 10 kHz), the mean poloidal

Proceedings of ITC/ISHW2007
10 -3

0

10 -5

m -7.0

10 -6
10 -7
10

(a)

-4

f -7.0-8

10 -9

Auto-correlation

Power spectrum S

10

1

(b)

(a)

-1
0

10

15

20

1

(b)

0
1
10
Frequency f (kHz)110Mode number m-1
0

Fig. 4 One-dimensional power spectra (a) S(f ), (b) S(m)(squares) and S(−m) (triangles) (arb. unit). Dashed lines
show the relationship S ∝ f −7.0 and S ∝ m−7.0 .

Figure 4 shows the logarithmic plots of power spectra S(f ) and S(m) calculated by integrating S(m, f )with m and f , respectively. Both spectra have relations
S(f ) ∝ f −7.0 and S(m) ∝ m−7.0 in broadband regions(f > 10 kHz or m > 5). It is interesting that the decay laws are the same in the frequency and poloidal mode
number spaces. It is also interesting that the power law ofS(m) is nearly equal to that led from the previous work
(i.e., S(m)0.5 ∝ m−3.6 [3]). Although different experimental conditions induce individual eigen functions, thepower laws in broadband regions become almost the same.

Broadband Fluctuation

Figure 5 shows the auto-correlation functions of the ion
saturation current fluctuation in the frequency ranges offull-range and f > 10 kHz. Owing to the main fluctuation peaks, the auto-correlation time of the frequency
of full-range is long (about the order of 10 ms). On theother hand, the auto-correlation time of the frequency off > 10 kHz, which is in the broadband fluctuation region,is short (about the order of 1 ms). It means that the broadband fluctuation has a short time life than the main fluctuation peaks. Many short time fluctuations with various
frequencies accumulate to form a broadband fluctuation.
Correlation length (poloidal angle) was calculated
from the 64-channel data. Coherence coh of two time data

0.5

1

1.5

2

Time t (ms)Fig. 5 Auto-correlation functions of the ion saturation current
fluctuation in the frequency ranges of (a) full-range and(b) broadband region (f > 10 kHz).
1

15

Frequency f (kHz)mode number increases with the relationship m ∝ f andthe spread is an increasing function of f . These facts imply the existence of broadband components. The quasimode peaks and broadband components appear nearly on
the line m ∝ f , and they are away from the linear dispersion relation curve of the drift wave. This is because
nonlinear couplings between the parent modes force to excite quasi-mode peaks and broadband components in theposition away from the dispersion relation. Thus, energy
cascade to high m and high f region is expected [19].

4

5

10
0.5
5

0
-π

0
0
Poloidal angle θ (rad)πFig. 6 Frequency dependence of the coherence in the poloidal
angle space. Correlation length is long for fluctuation
peaks and short for broadband fluctuation.

Ix and Iy is defined bycoh 2 (f ) =|Sxy (f )|2,
Sxx (f )Syy (f )(3)where Sxy = Iˆx Iˆy∗ /df . Correlation length is the distance
where the coherences become e−1 . Figure 6 shows the frequency dependence of the coherence in the poloidal angle
space. The coherence between different poloidal channelsof the 64-channel probe array was calculated to produce
Fig. 6. The correlation length in the poloidal direction islong for fluctuation peaks such as 0.9 kHz and 2.8 kHz, andis small (under π/4) for broadband fluctuation. The correlation length gradually decreases as the frequency becomes
high. From this result, strong fluctuation peaks are produced globally in the poloidal direction, while broadband
fluctuation is produced locally in the poloidal direction.
One of the features of our experiment is that two
poloidal probe arrays are used for fluctuation measure-Proceedings of ITC/ISHW2007

Frequency f (kHz)

15

1

10
0.8
5

0

0.6
-5

0
5
Poloidal mode number m

10

Fig. 7 Poloidal mode number decomposed axial coherence between the two poloidal probe arrays. The coherence isstrong not only for fluctuation peaks but also for broadband fluctuation. The solid white line shows the meanpoloidal mode number, and the broken white lines show
its spread.

ments and so that quasi-two-dimensional measurement isavailable. By setting all of the measuring radii of the 48channel probe array tips to 40 mm, poloidal analyses ofthe plasma column in separate of 0.26 m in the axial direction are performed. The 64-channel probe array canmeasure the poloidal mode number of |m| ≤ 32, and the
48-channel probe array can measure |m| ≤ 24. Although
the two probe arrays have different numbers of probes, andmany of the channels measure at different poloidal angles,
the fluctuations of the poloidal mode number in the region|m| ≤ 24 can be compared, and the poloidal mode number
decomposed axial correlations can be calculated. Figure 7
shows the coherence between the two poloidal probe arraysin the poloidal mode number–frequency space. Comparedwith Fig. 3, it can be said that the coherence is strong not
only for the fluctuation peaks such as drift wave modes,
but also for broadband fluctuation even the spectral poweris small. This fact suggests quasi-two-dimensional characteristics of the magnetized plasma turbulence5

Summary

In summary, we have performed a measurement of drift
wave turbulence in the LMD-U linear magnetized plasma
by use of poloidal probe arrays. The obtained twodimensional power spectrum S(m, f ) showed the excitation of drift wave modes and cascade to quasi-mode
peaks and broadband components (m ∝ f and S ∝f −7.0 , m−7.0 ), which do not satisfy the linear dispersion
relation of drift wave modes. The correlation time of thebroadband fluctuation was shorter than that of the fluctuation peaks. The correlation length in the poloidal direction was short (under π/4) for broadband fluctuation,

which suggests that this fluctuation is produced locally inthe poloidal direction. Poloidal mode number decomposed
axial coherence between the two poloidal probe arrays wascalculated and showed that the broadband fluctuation andfluctuation peaks had strong axial coherences.

Acknowledgments
This work was supported by Grant-in-Aid for SpeciallyPromoted Research of MEXT (16002005) (Itoh Project),by Grant-in-Aid for Young Scientists (B) of JSPS
(19760597), and by the collaboration programs of NIFS
(NIFS07KOAP017) and of 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] A. Latten, T. Klinger and A. Piel, Rev. Sci. Instrum. 66,
3254 (1995).
[4] T. Klinger et al., Phys. Rev. Lett. 79, 3913 (1997).
[5] M. Kono and M.Y. Tanaka, Phys. Rev. Lett. 84, 4369
(2000).
[6] C. Schröder et al., Phys. Rev. Lett. 86, 5711 (2001).
[7] T. Kaneko, H. Tsunoyama and R. Hatakeyama, Phys. Rev.
Lett. 90, 125001 (2003).
[8] V. Sokolov and A.K. Sen, Phys. Rev. Lett. 92, 165002(2004).
[9] M.J. Burin et al., Phys. Plasmas 12, 052320 (2005).
[10] G.R. Tynan et al., Plasma Phys. Control. Fusion 48, S51(2006).
[11] C. Lechte, S. Niedner and U. Stroth, New J. Phys. 4, 34(2002).
[12] Y. Nagashima et al., Phys. Rev. Lett. 95, 095002 (2005).
[13] Y. Hamada et al., Phys. Rev. Lett. 96, 115003 (2006).
[14] F.M. Poli et al., Phys. Plasmas 13, 102104 (2006).
[15] U. Stroth et al., Phys. Plasmas 11, 2558 (2004).
[16] S. Shinohara et al., Proc. 28th Int. Conf. on Phenomena in
Ionized Gases (Institute of Plasma Physics AS CR, Prague,2007), 1P04-08.
[17] K. Terasaka et al., Plasma Fusion Res. 2, 031 (2007).
[18] N. Kasuya et al., J. Phys. Soc. Jpn. 76, 044501 (2007).
[19] T. Yamada et al., submitted to Plasma Fusion Res.
[20] T. Yamada et al., submitted to Rev. Sci. Instrum.