J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)291–294
000–000

Avalanches Properties in a Self-Organized Critical Transport ModelGARCIA Luis and CARRERAS Benjamin A.1
1Universidad Carlos III, 28911 Leganés, Madrid, Spain
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8070, U.S.A.
(Received: 9 December 2003 / Accepted: 31 March 2004)

Abstract
We have proposed a one-dimensional transport model based on critical-gradient fluctuation dynamics to describe
some of the properties of plasma turbulence induced transport. This model has the characteristic properties of a selforganized critical (SOC) system. In this model, the flux is self-regulated by the stability properties of the fluctuations.
A high-gradient edge region emerges where transport dynamics is close to marginal stability. The core remains at thesubcritical gradient that is typical of a SOC system. Avalanches are quasi-periodic events triggered mostly near theedge region.
Keywords:SOC, avalanche, transport
In the fluctuation equation, γ is the linear growth rate of theinstability, µ is the coefficient of the nonlinear term that isresponsible for the saturation of the turbulence, and the thirdterm, S1, is a small source term to guarantee a minimal levelof seed fluctuations. It is implemented as a low-level random
noise that represents the trigger of a classical diffusion
mechanism. The transport equation, Eq. (2), includes a radial
diffusion term and a random source term, S0. This source isimplemented by the addition of an amount δ with probability
p0∆t/δ each time step. In the diffusion term, we assume thatthe diffusivity is proportional to the level of fluctuations andis given by µ0Φ.
The underlying instability is assumed to be a criticalgradient instability. Then the linear growth rate is

1. Introduction
Self-organized criticality (SOC) dynamics [1] can
explain some plasma transport phenomena that are difficult
to explain with a purely diffusive transport model. Sandpile
dynamics is an interesting paradigm for plasma transport [2,3]and several models have been based on this analogy. We haveproposed a one-dimensional transport model [4] that is anatural extension of the sandpile models applied to plasmatransport. The model is based on critical-gradient fluctuation
dynamics. It includes an evolution equation for the envelopof the plasma fluctuations and a transport equation with thediffusivity proportional to the fluctuation level.
When the fluctuation evolution is introduced, fluctuationlevels and fluxes of all magnitude are possible. We observed
large fluctuation amplitudes propagating mostly inward andcreating a wake of low-level stationary fluctuations. These
low-level fluctuations cause a diffusion-like process while thelarge-level propagating fluctuations cause an avalanche-liketransport. Here, we investigate the space-time structure of theavalanches. We show the existence of double quasi-periodic
processes with well-defined space-time structures.

γ = γ 0 – ∂h – Z c Θ – ∂h – Z c ,∂x∂xwhere Θ is the Heaviside function, and Zc is the absolutevalue of the critical gradient.
This model represents a generalization of the classicalsandpile model used to interpret plasma transport [3] by theaddition of fluctuation dynamics that regulates the amount oftransport, which couples back to the fluctuations through thegradient drive.
The numerical scheme used in solving these equations
is discussed in Ref. [4]. The boundary condition at the edge
is hedge = 0 (open end). At the origin, h is time advanced by
setting h0t+∆t = h0t – ∆tµ0Φ 0t+∆t Z0. The fluctuation level Φ isdefined at radial points midway between grid points [4] sothe values at the boundary do not enter in the scheme. For

2. ModelThe model proposed in Ref. [4] consists of two equations
describing the evolution of the root-mean-square fluctuations,Φ (x), and of the averaged particle density, h(x). They are∂Φ = Φ (γ – µΦ ) + S ,1∂t(1)∂h = ∂ µ Φ ∂h + S .
0∂t ∂x 0 ∂x

(2)

(3)

©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author’s e-mail: lgarcia@fis.uc3m.es
1291

Garcia L. et al., Avalanches Properties in a Self-Organized Critical Transport Modelthe results shown here, Zc = 5, γ 0 = 1, µ = 200, µ0 = 100, and∆t = δ = 0.05.

outward transport of particles. If we look at the time trace ofthe fluxes at a fixed radial position x, the flux is bursty forx < xT. It is practically zero most of the time, and suddenly aflux burst occurs. Above the transition radius, there is acontinuous flux with what looks like a superimposed noise.
In Fig. 2, we have done a two-dimensional (2-D) plot of
the contours of the flux. The plot shows eight major eventsin which the first avalanche reaches all the way to the centerand every successive avalanche runs to a shorter distanceinward. The inward penetration point of the avalanches
appears to have a sawtooth-like envelope. We can see thatmost of the avalanches follow the pattern of having thetermination point moving outwards after each avalanche.
When the termination point of the avalanches has moved all
the way to the edge pedestal, a new avalanche goes all theway to the center and the process starts again.
During an avalanche, the functional form of the flux at agiven time is a front-like structure and it propagates inwards
at a practically constant speed. Near the termination point,
the flux propagation slows down. These properties areillustrated in Fig. 3, where we have plotted a sequence ofradial profiles of the flux during an avalanche. The plots aredone at equal time intervals. This front-like structure is driven
by a supercritical value of Z at a single cell, just at theposition of the front.
We have used the functional form of the avalanche as away to identify them and determine their basic parameters.
In a radial position where no avalanche is present, we follow
the time evolution of the flux. The appearance of a maximum
in the flux is an indication that an avalanche starts. We lookat the evolution of its neighboring cells to determine thedirection of propagation. Then we follow the evolution of themaximum of the flux as a way of tracking the avalanche.
Parameter such as initiation point, penetration point, length,duration, and total flux can be determined in this way for eachavalanche. These measurements allow us to study thestatistical properties of the avalanches.

3. Description of the avalanches
Numerical results using the previously described modelshow first a transient phase followed by a steady state regime.
In steady state, the function h can be characterized by its slope
Z = –dh/dx. An example is shown in Fig. 1 for a system withsize L = 3,200. There are four well-defined regions. Nearx = 0, transport is dominated by slow diffusion caused by thebackground fluctuations, the S1 term in Eq. (1). In this region
the profile is close to parabolic and Z is a linear function ofx. There is also an edge pedestal region, for x > xT, where Z isvery close to the critical value and the transport isquasicontinuous. This transition point, xT, depends on both p0
and the system size, L. The emergence of such pedestal andits properties was discussed in Ref. [4]. In the core, there are
two regions. One has a nearly constant value for Z andremains subcritical. In this region, transport is dominated byintermittent avalanche-like events. Between this region andthe edge pedestal there is a region with a linear increase in
the slope. This suggests that diffusion may play a significant
role in this region. We work within a range of parameter space
where the profiles have a broad avalanche dominated region,and the jump in the slope stays just at the edge, as shown inFig. 1.
In investigating the dynamics of this system, it isinteresting to focus on the time evolution of the particle flux
Γ = ΦZ. There is a different dynamical behavior of the fluxes
above and below the transition point, xT. Between thetransition point and the edge, there is continuous activity. In
the inner region, there are quasi-periodic flux bursts which
we call avalanches. They are the dominant transportmechanism. These avalanches are triggered in the outer region(x > xT), and they propagate inward (x < x T). They canpenetrate all the way to the center of the pile. Few avalanches
start in the inner region, but they are rare, and it takes a longtime for them to build up. Although most avalanches
propagate inward, all particle fluxes are positive, which causes

0
5.5

200000
III

III

5Time

400000<Z>

4.5

6000004IV

800000

3.5p0 = 5×10–630Fig. 1

800

1600
x

2400

03200

Time-averaged slope of h, showing the four characteristic transport regions.

Fig. 22292

255075 100 125 150 175
xA 2-D plot of the contours of the flux in the radius-timeplane.

Garcia L. et al., Avalanches Properties in a Self-Organized Critical Transport Model

10-2-1(a)Probability-2ln(Flux)-3-4-5

10-3

10-4

10-5

0400

-6

Probability

10-4

10-5

5.5
Initiation
Penetration

-1

10

Slope

10-6

5

0

5000

10-2

4

10-3

3.5

10000 15000
Duration

20000

Probability distribution function of the core avalanche
length and duration for L = 1,600 system.

310-4

2.5-5

0400

800
x

1200

5.5

21600

54.5

Distribution of initiation points of the avalanches and of
the points of maximum inner penetration for L = 1,600system.

ZFig. 4Fig. 5<Z>

Probability

4.5101600(b)Logarithm of the flux versus radial position at eightequally separated times during the evolution of anavalanche.

100

1200

10-3-720 40 60 80 100 120 140 160xFig. 3800
Length43.5

The distribution of initiation points of the avalanches andof the points of maximum inner penetration show clearly thequalitative features described in the previous section. An
example of these distributions for L = 1,600 is shown inFig. 4. We can see that most avalanches start at the boundarywith the edge barrier region. They can penetrate all the way
in and the probability of avalanches starting in the inner
region decays exponentially inward relatively fast. The tworegions in the PDF correspond to the two regions in the slope
profile as can be seen in the figure.
We are interested in the avalanches that penetrate thecore region. As shown in Fig. 5, the probability distributionfunction of the length and duration of these avalanches ispractically flat. This is consistent with the detail evolution of
the avalanches shown in Fig. 2.

Instantaneous
Time averaged

30Fig. 6400

800x1200

1600

Time-averaged profile with an instantaneous profile ofthe slope Z for L = 1,600. The vertical line indicates theposition of xm.

averaged profile with an instantaneous profile at a given time,
excluding the avalanches. For the same time, we have plottedin Fig. 7 the instantaneous profile of Φ .
We see that the instantaneous Z profile has a region withstrong radial fluctuations in the inside and a relatively smoothprofile outside. Both regions are separated by a deep
minimum in Z at x = xm. The reason for that is that theminimum is the point where most of the avalanches comingfrom the edge finish. The position xm of the minimum moves
outwards with time, it is what determines the edge of thesawtooth in Fig. 2. The functional dependence of xm with t ispractically linear.

4. Quasi-periodic events
Let us consider with more detail the dynamical processof the avalanches. In Fig. 6, we have plotted the profile ofthe slope Z for L = 1,600. In the figure, we compare the time3293

Garcia L. et al., Avalanches Properties in a Self-Organized Critical Transport Model

10-3

peak is associated with the frequency of the combined events
like the ones shown in Fig. 2. The second peak corresponds
to the frequency of the individual avalanches. For frequencies
below this second peak, the spectrum shows the 1/f decay
characteristic of SOC systems.
An estimate of the frequency of the avalanches can bemade by assuming that the transport at the core is totally
dominated by these quasi-periodic events. The flux during afull event can be estimated by calculating the time behaviorof the fluctuations triggered by one of the front-like events
such as the ones in Fig. 3. After the event is been triggered,γ = 0 in Eq. (1) and we can neglect the effect of the source
over a short time equal to the period T between avalanches.
Then, integrating Eq. (1), we obtain

10-4

Φ

10-5
10-6
10-7
10-8Fig. 7

0400

800
x1200

1600

Instantaneous profile of Φ for the same time as theinstantaneous profile of Z in Fig. 6.

Φ (x , t ) =100
Power spectrum

10-1(4)

where Φ 1 is an integration constant. Then the flux balance atthe transition point is approximately given byf –1-2

10µp0 x T T 2 = µ0 Z ln (1 + µΦ1 T 2 ) .

10-3
10-4

(5)

Since xT is close to L, the frequency of the avalanches, thesecond peak in the spectrum of Fig. 8, is a function of Lp0.
On the other hand, the frequency of the combined events,
because there is a linear dependence on the penetration points
must be proportional to the frequency of the avalanches
divided by L. Therefore, the frequency of the combined events(first peak in the spectrum) is only a linear function of p0. We
have done a sequence of calculations varying L but keeping
p 0 L = 0.016 constant, where these properties of thefrequencies of the two peaks of the flux spectrum have been
confirmed.

10-5
10-6
10-7
10-8
10-7 10-6 10-5 10-4 10-3 10-2 10-1 100
Frequency

Fig. 8

Φ1,
1 + µΦ1t

Power spectrum of the flux across x = 200 for L = 400,
p0 = 5 × 10–5.

In the inner region, there is no activity and Φ remains
very low. Therefore, diffusion is low and the spikes generated
by the avalanches stopping are not eroded. In the outer region,
the avalanches travel through, and they leave behind a tail offluctuations that decays slowly as 1/t. This fluctuation tailgives an average fluctuation level that enhances diffusion andcauses the profile to be smooth.
When xm reaches at its maximum value is set back tozero. As the minimum in Z moves outward it becomesshallower and some avalanches can go through all the way to
x = 0. When they do so, they also smooth out the inner region.
At the point that the whole core is smooth the avalanches
penetrated all the way through and the process start again.
The maximum value of xm is about the position where thetime averaged Z profile changes from being constant toincreasing with radius.
The dominant avalanche processes in the region where
the profile is smooth have a quasi-periodic character. Thisquasi-periodicity of the avalanches is similar to the one
observed in the diffusive sandpile for small values of theroughness of the profile [5]. The quasi-periodicity is evident
in the power spectrum of the flux across a fixed radialposition. An example of the spectrum is shown in Fig. 8 for
L = 400 at x = 200, in the avalanche dominated region. There
are two clear peaks in the spectrum. The lowest frequency

5. Conclusions
For the range of parameter space considered in thispaper, the transport model based on critical gradientfluctuation dynamics leads to results very similar to thesandpile with diffusion. The dynamics of this model is close
to the sandpile dynamics when the roughness of the profile isslow. In this situation the profile is smoothed enough by thediffusion and avalanches became quasi-periodic events thatare triggered near the edge of the system and penetrate
inwards. Because the diffusive term is created by the wake ofthe avalanches, there is strong coupling between the size ofthe diffusion in a radial region and the frequency of theavalanches crossing that region. This leads to a double
periodicity of the avalanches.

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