J. Plasma Fusion Res. SERIES, Vol.2 (1999) 180-183

An Aspect of Discrete Dynamo Event in a Reversed-Field
Pinch Plasma as Self-Organized Criticality
YAGI Yasuyuki, BRUSKIN Leonid G.t, WATANABE Masayuki2, SOLZONELLA Tommaso3,SHIMADA Toshio, SAKAKITA Hajime and KOGUCHI Haruhisa
Electrotechnical ktboratory, Tsukuba-shi, 305-8568, JapanlPlasma Research Center, U. Tsukuba, Tsukuba-shi, 305-8577 Japan
2Fac. of Eng., Iwate Univ., Morioka-shi, 020-8551, Japan
3Consorzio RFX, Corso Stati Uniti 4, 35127 Padua, Italy(Received: 8 December 1998 / Accepted: 3 June 1999)

Abstract
A detailed look at a discrete dynamo event in a reversed-field pinch plasma (RFP), TPE-1RM20, isgiven. Dynamics of the magnetic signals are especially shown for a typical case with an analytic scenario
to understand the phenomena. An alternative, synthetic aspect is tried by checking the similarities to theSelf Organized Criticality.

Keywords:
reversed field pinch, TPE-1RM20, discrete dynamo event, self organized criticality

1. lntroduction
In reversed-field pinch (RFP) plasmas, we oftenobserve sawtooth-like repetitive oscillations in soft Xray signals. They are sometimes accompanied bypicture as is tried in the first half of this paper. It is also
tried in this paper ifthese events have any similarities tothe self-organized criticality (SOC) [1] which is a stateof complex systems. Sand pile crashes, earthquakes andmany other natural phenomena accompanied with some
sort of crash events might be understood by SOC.
In this paper, we give a close look at an example ofsawteeth in the global parameters as /0, F and @ whereI, F and @ are plasma current, reversal ratio and pinch
parameter, respectively. Amplitude of the sawtoothcrashes becomes prominent especially when @the discrete dynamo event from a middle @ discharges
in TPE-IRM20 device [2] featuring wave number (or nspectrum) and frequency spectra. Then we discuss aboutsimilarities of the discrete dynamo event to SOC.

increases. We call these large sawtooth crash events as

discrete 'dynamo' events since the toroidal flux
generation is often accompanied with the large sawtoothcrash events. These discrete dynamo events occur after agradual change of the current profile (peaking) in

2. Experimental Device and Operation

resistive time scale and appears as a rapid release of thestored energy as well as a rapid change of the current
profile (flattening). Soft X-ray signals are observed tocrash with n = 0 structure. i.e.. the crash is almost
instantaneous along the torus, where n is the toroidal
mode number. These natures of the discrete dynamo
events in RFP could be normally explained by a MHD

TPE-lRM20 is a medium sized RFP (Rla = 0.7510.192m) which was operated in 1992-1996, where Rand a are major and minor radii, respectively. Generalfeatures of the coherent oscillations and discrete dynamo

events are reported in [3]. The n-spectrum is obtained
from the toroidal pick-up coil array for B, component@1999 by The Japan Society of Plasrna
Science and Nuclear Fusion ResearchCorresponding author's e-mail: yagi@ etl.go.jp
180

yagi y . et at., An Aspect of Discrete Dynamo Event in a Reversed-Field Pinch Plasma as Self-Organized Criticality

It clearly shows a relatively long current diffusion

placed on the outboard and inboard side of thephase, where plasma is heated, and a crash event inmuch shorter time scale. Difference between in- andequatorial plane. There are 32 pairs of in- and outboard
pick-up coils so that we can resolve from n = 0 to n =15. To have much higher band width than the Br-anayoutboard side B, signals is raveled as m = I (odd)component and plotted in Fig. 2 (b) for the same time-outside the vacuum vessel, frequency spectrum (/spectrum) is obtained from an insertable B, probe placed
at 2 cm away from the plasma surface. Note that thetypical cut-off frequency of the vacuum vessel for the lz= lln = 0 component is 5kHz while the in-situ magnetic
probe has 25OkHz band width limited by the sampling

0.r0 (a)It 0.08

frequency.
To have a typical discrete dynamo event in TPElRM20, we operated the machine with a deeper reversal
at 1p = l30kA. The typical waveforms of a discharge inI6j

E o.oeXU)

Iro.&ta middle @ region (-1.7-1.8) is shown in Fig. 1. It
0.02
4.32

exhibits a couple of discrete dynamo events during thedischarge. We show an enlarged dynamics of the n- andfspectra for the narrow box in Fig. l.

time {msi4.96

300
250

3. Dynamics of a Discrete Dynamo Event
Soft X-ray signal enlarged for one of the typical

ID 200o"tf

discrete dynamo events in Fig. I is shown in Fig. 2 (a).

e 150
100

TPE-IRM20 middle Theta discharge

0.20

50

0.r5

15

s0.t01't20t-g rm(c):--t0.05?r!

:-t0

0.00

-50

-0.05

-'0t0!L2.OI

1.0P6xEFN&4o.o2o24681012

4"4 4.5 4.S 4.7 4"8 4.9time (ms)time (ms)Fig. 1 Typical waveforms of a middle @ discharge in
TPE-1RM20 are shown tor lo, <Br> (average
toroidal field), V.op (one turn loop voltage) andB*i" (toroidal magnetic field at the limiter surface)in (a) and F and @ in (b).

Fig. 2 Enlarged dynamics of the discrete dynamo eventis shown in SXR signal (a), toroidal distribution ofm = 1 magnetic signals (b), n-spectrum (c) and f-spectrum (d).

181

Yagi Y. et al., An Aspect of Discrete Dynamo Event in a Reversed-Field Pinch plasma as Self-Organizedinterval as Fig. I (a). The rotation and wall-locking of
the m = I mode is clearly seen. Figures 2 (c) and (d)show the n- andfspectra for the same discrete event. To

Criticality

Averaged Power100

have the well time-resolved spectrum, we used a
Wavelet technique for the /-spectrum [4]. Timel0-lresolution of this technique is about ten sampling points
so that it is about 20ps for the data shown here. It is.:sufficiently small compared with the rypical time of-_ l0',-change in the /-spectrum of about 200ps. Note that anormal FFT would not be appropriate for this fastdynamics. In Fig. 2 (d), the rotation of the mode at 70
kHz and its slowing down toward the wall-locking isseen. If there is no poloidal mode rotation, toroidal
rotation of the single mode (la = lln = 7) is l0kHz at t =l0-3

4.5ms.

The discrete dynamo event shown here is a typicalexample of those in the middle @ discharges in TpE-lRM20. The discrete dynamo event in Fig. 2 isunderstood by the following scenario. In the current
diffusion phase, as current profile is being peaked,(b) Estimation of the Hurst Exponent

single unstable mode (plausibly, near-axis resonant
tearing mode) starts to grow at t = 4.60ms in a time-scale of 200ps. The mode is initially rotating toroidally

but is slowing down and finally locks to the wall at / =4.75ms. Then the n-spectrum becomes broader to thehigher z side in r = 4.804.82ms. This may be either due
to the non-linear coupling of the modes or due to thechange of the current profile so that the higher zl-modes
become unstable. The magnetic islands overlap to all
over the plasma cross-section and an expulsion of thestored energy occurs conesponding to a crash of the soft
X-ray at t = 4.82ms. A relaxation phase follows in / =4.82-5.0ms with an enhancement of m = 0 with low-nnumbers. Then the current diffusion process resumes.

l0'(r')

Y

lou

l0'

102

ldnFig. 3 The magnetic signals of the insertable probe areanalyzed. Power spectrum of the 8, signal in (aland B/S versus sampled data for the fl signal toestimate the Hurst exponent t5l in (b).

During the relaxation phase, the toroidal flux isenhanced showing the 'dynamo activity'.

4. Self Organized Criticality (SOC)
Alternatively, the slow change ending with a rapid
crash might be understood by SOC. Mathematicalmethod to show that the system is SOC, is not welldefined. There are, however, some indicators which theSOC state should exhibit. Checking the ft- orfspectrum

and 40kHz (after) corresponding to the rotation of themode. This is one of the characteristics of the SOC,
though other causes may be possible. To check longrange or long-time correlation in fluctuations is anotherway. This method is usually done by calculating theHurst exponent, F1 [5]. The system is said to have along-time correlation when 0.5 < Il < 1.0. Figure 3

if they show a geometrical function; k-' or f-t is one way.
FFT is used for the averagedfspectrum of the B, signalof the insertable probe in I ms (Fig. 3(a)). It isshows the R/,S values (see [5]) versus sampled number

interesting to note that the obtained power spectrum has

ofdata for well before and after the crash event in Fig. 2for B, of the insertable probe. Obtained Il exponent is
0.64 before the crash and increases to 0.82 after thecrash. Both values show a long-range time correlation

almost linear trend in log-log plot and approximately
shows af I decay in lkHz < f < 25OkHz both before andafter the crash event. A hump appears at TOkHz (before)182

yagty. et al., An Aspect of Discrete Dynamo Event in a Reversed-Field Pinch Plasma as Self-Organized Criticality

which is also one of the characteristics of the SOC.

phase reminds us of a similarity of the phenomena to thethe value before the crash (-F1 = 0.64) agrees

SOC. A preliminary trial to obtain characteristics related
to the SOC state is tried by using a magnetic signal in

Especially,quite well with those values in various types of plasma
confinement systems summarized in Ref. [5]. The
increase of 11 after the crash means that the system
becomes more distant from the random state (F1 = 0.5).
Then a reduction of transport might be expected. It is ofinterest to note that the SXR increases to a larger levelin the next heating cycle than the peak before the crash.
It indicates that the electron stored energy increases,
which may have a relation to the increase of F1 value.
Finally, a caution needs to be noted for interpretation ofIl obtained here in a sense that the situation is notnecessary be stationary, the sampled points (1000 andthe scrape offregion. It shows a lfspectrum regardlessof the crash event. The 11 exponent is in the range from
0.64 before the crash to 0.82 after the crash. They are infact part of the features of the SOC, though notnecessarily proving it. They may be one of the various
aspects of the complex natures of the plasma.

Acknowledgment
Authors are grateful to Dr. K. Sugisaki forcontinuous encouragement and to Prof. T. Tamano forgiving us a chance for this collaboration. One of the500 points before and after the crash, respectively) may
not be large enough and the mode rotation might beauthors (L.G.B.) acknowledges support by NWO andFOM Institute of Plasma Physics of the Netherlands.

affecting the result. In fact, R/S profile is greatly

The experimental work of TPE-1RM20 is financiallysupported by Science and Technology Agency.

modified if B, is used probably because of an effect by
the mode rotation.

Referenceslll

P. Bak, et al.,Phys. Rev. Lett.,59,381 (1987).
12) Y.Yagi et a/., J. Plasma and Fusion Research 69,

5. Discussion and Summary
A typical example of the discrete dynamo event in
RFP plasma is shown in the real space signal, n- (ft-)

700 (l993).
[3] P.L. Brunsell, et aI.,Phys. Fluids,5, 885 (1993).

f spectrum. A scenario for the discrete event isgiven from an experimental observation. Alternatively tothis analytic approach to the phenomenon, a synthetic
way of understanding is also tried. A discrete dynamo
andt4l L.G. Bruskin et aI., Rev. Sci. Instruments, 70(1999) to appear.

t5l B.A. Carreras, et al.,Phys. Plasma, 5,3632 (1998).

event with a fast crash followed by a longer driven183