J. Plasma Fusion Res. SERIES, Vol. 2 (1999) 8-l I

Self-Organized Criticality and Plasma Fluctuation Dynamics
CARRERAS Benjamin A., LYNCH Vickie E. and NEWMAN David E.
Oak Ridge National l-oboratory
Oak Ridpe. Tennessee 37831-8070, U.S.A.
(Received: 7 December 1998 / Accepted: 19 May 1999)

Abstract
Characteristic self-organized criticality (SOC) tll dynamics can explain some of the properties oftransport in magnetically confined plasmas, and offers a new perspective on how plasma simulations
should be carried out. To understand the implications of SOC dynamics in magnetically confinedplasmas, new analysis techniques have been developed and applied to the results of numericalsimulations. These studies have provided the basis for the application of these techniques to fluctuation
data from several confinement devices, including tokamaks, stellarators, and reversed field pinches. The
results of the analysis reveal the self-similar character of the edge plasma fluctuations and the existenceof long-range time correlations that are consistent with SOC dynamics.

Keywords:
transport, fluctuations, self-organized criticality

Theoretical models of plasma turbulence andinduced transport are based on local plasma instabilities.
These instabilities have characteristic scale lengths ofthe order of a few Larmour radii. While radial
correlation measurements of plasma fluctuations agreewith such scale lengths, local transport coefficients in
the L-mode regime do not scale with gyroradius as
expected from such local instability models [2]. The socalled Bohm scaling of the plasma diffusivities implies
that they carry information on the overall plasma size.
Furthermore, perturbative plasma experiments also seemto be inconsistent with local transport properties, andthey cannot be modeled by a simple diffusion operator
in the transport equation [3].
If we tum our attention to numerical calculations ofplasma turbulence, we find that they have often been

carried out with constant averaged profiles. In such asituation, transport calculations only show local
properties of fluctuations. Such calculations have been
useful in investigating the dynamic saturation
Corresponding author's e -mail: ob3 @ ornl. gov

mechanisms of turbulence, but they fail to present a
proper transport picture consistent with the experimental
results described above. On the other hand, for criticalgradient (or gradient scale length) instability models,when those numerical calculations are carried out with afixed flux, rather than a fixed gradient, and for longenough times (of the order of transport time scales),changes in the transport properties are indeed observed[4,5]. Local fluctuations at adjacent flux surfaces can betriggered in succession as a result of the quasilinear
modification of the averaged plasma profiles, and anavalanche-like transport phenomenon occurs. Theseavalanches have scale lengths that vary from thefluctuation scale length to the plasma size. They canoften be visualized and identified (Fig. l). They also
lead to fluctuation spectra with a l/f-like region infrequency, and display the plasma behavior that is very
close to that observed in a self-organized critical system.

Such a spectrum is shown in Fig. 2 at two radial@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCarreras B.A. et aL, Self-Organized Criticality and Plasma Fluctuation Dynamics

positions for a pressure-gradient-driven turbulencecalculation [4]. The decay index is about -1.3 for
frequencies between 20 and 200, close to -1. Note thatthe frequency range with a decay index of the order ofI varies with radial position.

-Self-organized criticality (SOC) dynamics Il] may

explain some of the properties of transport inmagnetically confined plasmas [6,71. To understand thisrole of SOC dynamics in plasma transport, it isimportant to find specific properties that may be used
for its experimental identification. One of the main suchcharacteristics is transport by avalanches. This leads tocorrelation functions with algebraic tails, which signifythat long-range time and space correlations exist.
Accurately detecting those tails is a problem. For
instance, in Fig. 3 the fluctuation autocorrelation
function for the same numerical data as shown in Fig. 2
shows a long tail that may have an algebraic decay. The
oscillating character of the tail makes it difficult todetermine the decay index. This difficulty is further
enhanced in the experiment by the presence of noise.
Therefore, to experimentally determine such a tailrequires new techniques for quantifying the asymptotic
behavior of the correlation functions and probability

distribution functions. One such technique is therescaled range (R/S) analysis [8], and the scaled window

variance (SWV) technique [9]. These methods
determine the Hurst exponent [10], H, which is directly
related to the fractal dimension of the time series and to
the exponent of the algebraic tail of the autocorrelation
function. Applying these techniques to the previously
described numerical results and for a range of time lags
larger than the.decorrelation time of the turbulence, we
see that the R/S is a straight line in the log-log plot overmore than an order of magnitude in time lag (selfsimilarity range) with a Hurst exponent of about 0.8(Fig. a). This gives us a signature to look for in thetlme
Fig. 1 Avalanche propagation. Sequence of density
fluctuations triggered by a single small-size
density perturbation from a pressure-gradientdriven turbulence calculation [4].

-->experimental analysis.
We have applied the R/S statistics and the SWV

technique to the analysis of experimental data. The
analysis of electrostatic plasma edge fluctuations for a100

1.2

E ro"Ic).E o.s

to*.FL* o.u
E 0.4
I o.zio

I to'u* lo-tl0-l-0.2

loo lorrc2 lo3
Frequency (t*,)

lo4-0.4

0.05 0.1 o.ls 0.2
Time lag (t*)

0.25

Fig. 2 Density fluctuation spectra at two radial positionsof electrostatic potential fluctuations from apressure-gradient-driven turbulence calculation
t4l.

Fig. 3 Autocorrelation function of electrostatic potential
fluctuations for the same fluctuation data as Fig.
1.

Caneras B.A. et aI., Self-Organized Criticality and Plasma Fluctuation Dynamics

broad range of magnetic configurations has led to a self-t04

similarity range from the turbulence decorrelation time
to a time of the order of the transport time scale. The
value of Fl is between 0.62 and 0.75 [11]. These results
are evidence of the existence of long-range correlationsin the plasma edge turbulence in those confinement
devices. The narrow range of variation of the Hurst
exponent suggests a universal character of the plasmaaDedge turbulence dynamics.

Additionally, this narrow range of values for F1

l0r

found in different plasma confinement devices is also anindication of the similarity of the low-frequency range

l0-3Time lag (t*)

of the fluctuation spectra in those experiments.
However, the similarity of the spectra goes beyond thelow-frequency range. Using a rescaling transformation

l0'r

l0-2

100

Fig. 4 R/S for the same fluctuation data as Fig. 2 and forvalues of the time lag above the turbulenceof the spectra, we find clear evidence of the similarity ofdecorrelation time.

the plasma edge electrostatic fluctuation spectra over thewhole frequency range [2].
Some of the techniques used in the detection of thelong-range time dependencies can be extended to thedetection of cross-correlations for long time lags [13].

t.2IThey are important because the avalanches should show
up as long radial correlation in the long time-lag region.

()
L

Additionally, an avalanche should show radial()

propagation. In the resistive pressure-gradient-driven
turbulence calculations [4], there was clear evidence ofradial propagation. Here, we apply the new diagnostic tothe same data to study radial correlations. We detect arapid decrease of the correlation with increasing radial(J0.60,4-^e-\\V4tv/-\_^

0.20

separation at low time lags, but a much slower decrease-200< fime lag <l0.8(n(tlimc lag, ,()0.05 0.1 0.l s 0.2 0.25
Va

is observed at larger time lags. In particular, in theregion where the long-range time correlations areFig. 5 Radial cross-correlation of the density fluctuations
fluctuations as a function of the radial separation.
The data are the same as Fig. 2.

significant, the radial correlation is many times longer
than the radial correlation length of the fluctuations (Fig.
5). This extension of the analysis techniques allows thedetermination of radial correlations for noisy signals
and, in the case of experimental results, comparison
with the radial correlations expected from avalanchetransport. Analysis of multichord experimental data isunder way [14].
Finally, if long radial correlations in the fluxes
exist, they suggest the possibility of a superdiffusive
component in the plasma transport [l5]. Superdiffusivity
may result from both magnetically and electrostaticallyinduced SOC plasma transport. Such a superdiffusive
component is consistent with the Bohm-type scaling
observed in tokamak plasma transport and can offer an
explanation for some of the observed features thought tobe inconsistent with theory.
In conclusion, plasma turbulence models whenevolved under a fixed flux condition develop some of
the characteristic properties of SOC systems. The
analysis techniques presented here are effective in
detecting these SOC properties when applied to datafrom these turbulence calculations. This result puts theapplication of these techniques to experimental data on
relatively solid ground.

Acknowledgments
We want to acknowledge many fruitful discussions
with R. Balbin, P.H. Diamond, C. Hidalgo, B.P. vanMilligen, M.A. Pedrosa, and G.M. Zaslavsky.
Research was sponsored by the Oak Ridge National

Laboratory, managed by Lockheed Martin Energy
10carreras B.A. et al., self-organized criticality and plasma Fluctuation Dynamics

Research Corp. for the U.S. Department of Energy

t8l B.B. Mandelbrot and J.R. Wallis, Water Resources

under contract number DE-AC05-96OR22464.

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