Self-organized criticality and observable features of avalanching systems Michal Bregman October 7, 2005

Self-organized criticality and observable features of
avalanching systems
Michal Bregman
October 7, 2005
Contents
1
Introduction
6
1.1
Characterization of the SOC state . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2
Numerical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.2.1
The basic sandpile model . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.2.2
Running sandpiles . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.2.3
Forrest-fire model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.2.4
Earthquakes model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
Analytical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.3.1
Renormalization group . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.3.2
Mean-field approach . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.4.1
Dynamics of sandpile . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.4.2
Dynamics of rice pile . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.4.3
Dynamics of water droplets . . . . . . . . . . . . . . . . . . . . . . .
15
1.4.4
Superconducting vortex avalanches . . . . . . . . . . . . . . . . . . .
18
Space plasma and magnetosphere . . . . . . . . . . . . . . . . . . . . . . . .
18
1.5.1
Solar flares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.5.2
Current sheet and magnetic storms . . . . . . . . . . . . . . . . . . . .
19
1.6
Observational relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.7
Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
1.3
1.4
1.5
2
Sandpile model
23
2.1
One-dimensional bi-directional sandpile . . . . . . . . . . . . . . . . . . . . .
23
2.1.1
Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
Two dimensional sandpile model . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.2.1
The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.2.2
Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.2.3
Non-constant grain transfer . . . . . . . . . . . . . . . . . . . . . . . .
36
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.2
2.3
2
CONTENTS
3
Dynamics of the burning model
3.1 One dimensional model . . . . .
3.1.1 The model . . . . . . .
3.1.2 Field presentation . . . .
3.1.3 Analytical treatment . .
3.1.4 Numerical results . . . .
3.2 Two dimensional burning model
3.2.1 The model . . . . . . .
3.2.2 Field presentation . . . .
3.2.3 Numerical result . . . .
3.3 Summary . . . . . . . . . . . .
3
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41
42
42
43
44
46
53
53
53
54
57
4
A comparative study
4.1 One dimensional models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Two dimensional models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
58
61
62
5
Comparison with observations
63
6
Conclusions
66
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List of Figures
1.1
One-dimensional sandpile. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2
Forest-fire model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.3
Sandpile experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.4
Rice pile experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.5
Experiment with water droplets. . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.6
Chain reaction in water droplet experiment. . . . . . . . . . . . . . . . . . . .
17
1.7
Solar flares as avalanches in the loop network. . . . . . . . . . . . . . . . . . .
19
1.8
Illustration of avalanching magnetotail. . . . . . . . . . . . . . . . . . . . . .
20
2.1
Sandpile activity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.2
Number of active sites as a function of time.(One dimensional case) . . . . . .
27
2.3
Distribution of clusters size for the high state. . . . . . . . . . . . . . . . . . .
28
2.4
Distribution of active and passive phase durations. . . . . . . . . . . . . . . . .
29
2.5
Distribution of active and passive phase durations for different drivings. . . . .
30
2.6
Distribution of clusters size for the high state. . . . . . . . . . . . . . . . . . .
32
2.7
2D sandpile: distribution of active and passive phase durations. . . . . . . . . .
33
2.8
2D sandpile: distribution of active and passive phase durations for different
drivings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2D sandpile: distribution of cluster sizes. . . . . . . . . . . . . . . . . . . . . .
35
2.10 Distribution of active phase duration for the constant grain number transfer case
and for the proportional transfer case. . . . . . . . . . . . . . . . . . . . . . .
37
2.11 Distribution of passive phase durations for the constant grain number transfer
case and for the proportional transfer case. . . . . . . . . . . . . . . . . . . . .
38
2.12 Distribution of cluster sizes for the constant grain number transfer case and for
the proportional transfer case in log linear scale. . . . . . . . . . . . . . . . . .
38
3.1
Energy release for various drivings. . . . . . . . . . . . . . . . . . . . . . . .
49
3.2
Individual avalanche structure. . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.3
Distribution of the cluster sizes. . . . . . . . . . . . . . . . . . . . . . . . . .
51
3.4
Mean temperature for various drivings. . . . . . . . . . . . . . . . . . . . . . .
51
2.9
4
LIST OF FIGURES
3.5
3.6
3.7
5
52
54
3.8
Duration of the active and passive phases. . . . . . . . . . . . . . . . . . . . .
Passive and active phase duration for high input probability and weak dissipation.
Passive and active phase durations for various input probabilities and weak dissipation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cluster size distribution for various input probabilities and dissipation. . . . . .
4.1
4.2
4.3
4.4
Active phase duration for the case of low driving.
Passive phase duration for the case of low driving.
Active phase duration for the case of low driving.
Passive phase duration for the case of low driving.
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59
60
61
62
5.1
5.2
Burst lifetimes for electrojet index. . . . . . . . . . . . . . . . . . . . . . . . .
Quiet times for electrostatic bursts. . . . . . . . . . . . . . . . . . . . . . . . .
64
65
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56
57
Chapter 1
Introduction
Many physical (Lu and Hamilton, 1991; Wang and Shi, 1993; Field et al., 1995; Paczuski et al.,
1996; Durian, 1997; Politzer, 2000; Charbonneau et al., 2001; Klimas et al., 2004) and nonphysical (McNamara and Wiesenfeld, 1990; Lee et al., 2004) systems exhibit avalanching behavior, where a small external perturbation may result in a large response of the system. This
response is of a finite spatial and temporal extent. After the response (avalanche) is over, the
system switches to the passive state, where it waits for another favorable external perturbation.
These avalanches seem in most cases unpredictable and uncorrelated. Moreover, distribution of
their sizes often does not seem to possess any characteristic scale, showing power-law slopes.
Fourier spectra of properly constructed times series (activity peaks, avalanche start time, etc.)
also are power-law (famous 1/f noise) (Bell, 1980; Hooge et al., 1981; Matthaeus and Goldstein, 1986; Restle et al., 1986; Rosu and Canessa, 1993; Maslov et al., 1994). These features
together brought to life the self-organized-criticality (SOC) paradigm Bak et al. (1987), which
suggests that systems consisting of many interacting constituents may exhibit some general
characteristic behavior. These systems are dynamical, that is, the state of each constituent is
time-dependent, and open, that is, external driving is always present and energy (or particle
number) is lost. The basic suggestion of SOC was that, under very general condition, such
dynamical systems may organize themselves into a macro-state with a complex but rather general structure. The systems are complex in the sense that many events (avalanches) of various
sizes are present and no single characteristic event size exists: there is no just one time and
one length scale that controls the temporal evolution of this system. Although the dynamical
response of the system is complex, the simplifying aspect is that the statistical properties (which
define the system macro-state) are described by simple power laws. SOC paradigm has attracted
great attention and was extended to a larger number of systems (Sornette and Sornette, 1989;
Obukhov, 1990; Yan, 1991; Cote and Meisel, 1991; Zaitsev, 1992; Alstrøm and Leão, 1994;
Rodriguez-Iturbe et al., 1994; Noever et al., 1994; Hwa and Pan, 1995; Elmer, 1997; Chang,
1999a; Tamarit, 1999; Lise and Paczuski, 2001; Fonstad and Marcus, 2001; Feigenbaum, 2003;
6
1.1. CHARACTERIZATION OF THE SOC STATE
7
Baker and Jacobs, 2004).
The claim by Bak et al. (1987) was that this typical behavior develops without significant
tuning of the system from outside. Further, the state into which system organized themselves
have the properties similar to those exhibited by equilibrium systems at a critical point. Therefore, Bak, Tang and Wiesenfeld (BTW) described the behavior of this system as Self Organized
Criticality (SOC). The idea was proposed as a kind of universal behavior of wide class of open
dynamical systems far from equilibrium, and has been applied to a large number of physical
systems with quite different underlying micro-physics.
1.1
Characterization of the SOC state
The term self organized criticality emphasizes two aspects of the system behavior. Self organization is used to described the ability by certain nonequilibrium systems to develop structures
and patterns, that is, enter a certain regime, in the absence of control or manipulation by an
external agent. The word criticality is used in order to emphasize the similarity with phase
transitions: a system becomes critical when all its elements react in a coordinated way, similar
to a domino effect, that is, the correlation length becomes infinite.
The difference between systems that exhibit non critical behavior and systems that exhibit
critical behavior is that, in non critical systems the reaction of the system to external perturbation
is described by a characteristic response time and characteristic length scale over which the
perturbation is felt, that is, we can predict what will be the avalanche size in the next perturbation
and when it will occur. Although the response of a non critical system may differ in details as the
system is perturbed at different position and at different time, the distribution of responding can
be described by the average response. For critical system, a perturbation that applied at different
position or at same position at different time can lead to a response of any size. It is important
to mention that till now there does not exist a clear and generally accepted definition of what
SOC is. Nor does a sufficiently clear picture exist of the necessary conditions under which SOC
behavior arises. The prevailing opinion is (see,e.g. Vespignani and Zapperi, 1998) that SOC
may occur in a system with clear time separation: the average time between subsequent external
perturbations is much larger than the time of the avalanche (even a large one) development. This
regime is also often referred to as a weak driving limit. It has to be mentioned, however, that
this opinion is not shared by everybody and sometimes it is claimed that the very presence of
the scale-free (power-law) distribution is itself a signature of SOC (Boettcher and Paczuski,
1997; Corral and Paczuski, 1999; Lise and Paczuski, 2001; Hughes et al., 2003; Paczuski and
Hughes, 2004). It is also worth mentioning that real avalanching systems do not necessarily
have to be in the state of SOC. Thus, the study of avalanching systems can be divided into two
different (but closely related) subjects: a) whether avalanching systems do evolve into universal
8
CHAPTER 1. INTRODUCTION
SOC behavior, and b) what is the relation of the macroscopic properties of avalanching systems
to the microprocesses occurring inside and to the external driving.
1.2
Numerical modeling
Ever since the first proposal of SOC, one of the basic tools in studies of avalanching systems is
numerical modeling of cellular automata models, among which sandpile models are the simplest
and have drawn the greatest attention so far. In these cellular automata models a system is
represented as an array of cells possessing some property. This property is changed by external
driving and due to the internal redistribution mechanism, transferring it from one cell to another.
Such transfer depends on some criticality conditions and goes on until the system relaxes to a
subcritical state. The simplest sandpile model is a one-dimensional array of cells. Each cell
contains an integer number of grains. Grains can be added to any cell from outside (according to
the driving rules). Redistribution of grains inside the array depends on fulfilling some condition,
usually when a local gradient (slope) exceeds some critical value.
1.2.1
The basic sandpile model
The original model by Bak et al. (1987, 1988) introduced the first basic sandpile model, as we
mentioned previously. A one-dimensional pile of sand is essentially a one-dimensional array of
integers, hn , n = 1, . . . , N . These integers represent numbers of sand grains (height) in each
cell n. These heights change due to the external input (addition of grains) as well as certain rules
of the sand redistribution. At any step of time one grain is added to a randomly chosen cell so
that the pile gets a slope with time. If the slope of the sandpile becomes too steep, exceeding
some critical value, the pile collapses until its slope reaches a barely stable position with respect
to small perturbations. Bak at el basic model was a one dimensional sandpile model with one
open and one closed boundary. This means that there is a wall at the left edge, n = 1, of the
pile and sand can freely exit from the right edge, n = N . Let Zn = h(n) − h(n + 1) be the
height difference (slope) between successive positions along the sandpile. The external driving,
i.e. adding of one grain of sand at nth position is described by the following rules: hn = hn + 1
and no change for other cells, or
Zn → Zn + 1
(1.1)
Zn−1 → Zn−1 − 1
(1.2)
1.2. NUMERICAL MODELING
9
When the height difference reaches some given critical value Zc , the site is relaxed and one
grain of sand moves to the lower site:
Zn → Z n − 2
(1.3)
Zn±1 → Zn±1 + 1
(1.4)
Figure 1.1 illustrates the behavior of the sandpile model proposed by Bak et al. (1988).
Figure 1.1: One-dimensional sandpile automaton. The state of the system is specified by an
array of integers representing the height difference between neighboring plateaus. From (Bak
et al., 1988)
As mentioned earlier there is an open boundary at the right edge and a close boundary at the
left edge of the pile:
Z0 → 0
(1.5)
ZN → Z N − 1
(1.6)
ZN −1 → ZN −1 + 1
(1.7)
After the unstable site collapses other sites may become unstable (slope exceeds the critical
one), in principle, so that the redistribution occurs once again, without adding grains from the
outside. Thus, an avalanche develops. The process continues until all the height differences
Zn are below the critical value, Zn < Zc . Then another grain of sand is added with the same
fixed probability to the system (at a randomly chosen site). There is complete time separation
between the external driving and avalanche development.
10
CHAPTER 1. INTRODUCTION
Bak et al. (1987, 1988) suggested that statistical analysis of the cluster (snapshot of an
avalanche) size distribution, and well as of the avalanche size (lifetime) should provide the
basic information about the system behavior. Numerical measurements of these quantities have
been made and it was suggested that a Fourier-spectrum of the lifetime distribution has the
power-law shape 1/f α , with α ≈ 1. The latter statement was later shown to be wrong but the
whole idea stimulated extensive research in the field.
Bak et al. (1987, 1988) extended their analysis onto two-dimensional models of the same
type. They found that the two-dimensional cluster size in a sandpile has a fractal structure.
Both distributions, of the the cluster size and the lifetime as well, show a power law distribution
indicating that the system is in critical point as expected. Although two-dimensional systems
seem more close to reality, one-dimensional sandpiles still remain in the center of research,
partly because of the ease of numerical treatment.
1.2.2
Running sandpiles
The obvious drawback of the Bak et al. (1987, 1988) model is that it requires complete time
separation. However, in real systems, as we know, the external driving does not stop when
avalanche is in action. The SOC hypothesis applicability to a number of systems, like plasma,
was severely criticized (Krommes, 1999; Krommes and Ottaviani, 1999; Krommes, 2000, 2002)
on the grounds that the real characteristic time ratio is inverse, that is, driving is faster than the
avalanche development. In order to get rid of the time separation limitation running sandpile
models were proposed (Hwa and Kardar, 1992) where the external driving does not depend
on the absence of an avalanche. It might reinforce a fading avalanche, make two avalanches
run simultaneously and even overlap. Running sandpiles and, more generally, running cellular
automata, are the basic models for studies of avalanching systems. It is often claimed that SOC
is possible only for infinitely weak driving (see, e.g., Vespignani and Zapperi, 1998), although
this opinion is not accepted by everybody (see, e.g., Hughes and Paczuski, 2002).
1.2.3
Forrest-fire model
Forest fire models were proposed as another kind of model exhibiting avalanching behavior and
presumably possessing SOC behavior. The first forest fire model was introduced by Chen et al.
(1990) as another realization of a simple SOC system. The fire forest model is also a cellular
automaton which is defined on a d dimensional lattice. Each cell can be at three different
positions: occupied by a tree, burning tree and empty site. At each time step trees can growth in
the empty site with the probability p, and burn either due to lightnings with the probability f , or
if at last one of his nearest neighbors is burning. In the limit of slow tree growth p and without
lightnings, fire only spreads from burning trees to their neighbors. The fire front become more
1.2. NUMERICAL MODELING
11
and more regular and spiral shaped with decreasing p. A snapshot of this state is shown in
Figure 1.2
Forest fires and other examples of self-organized criticality
6805
Figure 1. Snapshot of the Bak et al forest-fire model in the steady state for p = 0.005 and
Figure 1.2: Snapshot
of the
model
theandsteady
state.
Trees are gray, burning trees
L = 800.
Treesforest
are grey, fire
burning
trees arein
black,
empty sites
are white.
are black and empty site is white. From Clar et al. (1994).
g = gC (p) the fire density becomes zero and the forest density becomes one. Figures 2 and
3 show two snapshots of the system for values of g far below gC (p) and near gC (p). At
Consolini
and De Michelis (2001) proposed a modifies forest model named Revised Forest
gC (p), the fire just percolates through the system. Since sites are not permanently immune,
this kind
of percolation
is different
usualwas
site percolation
andin
is in
the same
universalitythe occurrence
Fire Model
(RFFM).
Their
simplefrom
model
introduced
order
to simulate
class as directed percolation in d + 1 dimensions (the preferred direction corresponds to the
of sporadic
localized
relaxation
(current
disruption
events)
with increasing
p, since
the fire in
thenthe
cangeotail neutral
time) [22].
The critical
immunityphenomena
gC (p) increases
the steady
state
of the system
sooner
sites where
it already
been.
Above gC (p) lattice
plasmareturn
sheet.
Thetomodel
consists
of ahastwo
dimensional
and
involves
periodic boundary
is a completely dense forest. A similar model, using the language of spreading diseases has
conditions.
Each independently
site is in one
of the following states that we characterized before: site with
been studied
in [23].
a tree, burning site and empty site. The analog between this model and the magnetic field
1.3. Self-organized critical (SOC) behaviour
conditions
in the Earth’s magnetotail is that empty sites can be associated to those regions where
The SOC behaviour occurs when the lightning probability is non-zero. For simplicity, we
plasma conditions are locally stable, while the site with the trees are locally unstable regions,
set the immunity to zero, but we will show later that the SOC state persists for g > 0. The
p/f is sites
a measure
for the number
trees regions
growing between
lightning strokes
and
and theratio
burning
are connected
to ofthose
where two
a relaxation
phenomenon
is taking
therefore for the mean number of trees destroyed per lightning stroke. In the limit
place. Here we call by relaxation phenomenon any kind of event during which dissipation takes
f !p
(1.3)
place, like a current disruption event. In each step the state of the sites are change according to
there exist consequently large forest clusters and correlations over large distances. The
the following
rules:
model is SOC when tree growth is so slow that fire burns down even large clusters before
1. A tree is growing in an empty site with the probability p.
2. A burning tree becomes stable at the next step.
3. A tree starts to burn with the probability f or if at least one of his neighbors is burning.
The growing probability can be connected to the growing rate of macroscopic configuration
12
CHAPTER 1. INTRODUCTION
instability that is a result of external processes. The lightning strike probability f should be
related to the occurrence probability of a local reconnection event.
The purpose of this model was to present a new SOC model that seems to be more related to
the nature. After years it discover that RFFM also shows critical behavior that is very similar to
the ordinary critical phenomena, that is, this model is also missing the self organized ingredient
and there is a control parameter that depends on the relation between the growing probability
and the lightning probability.
1.2.4
Earthquakes model
Earthquakes occur as a result of relative motion between tectonic plates. Friction between the
plates prevents a smooth motion and the plates stick together until the stress between this two
plates exceeds a critical value and is then released within second or minutes. The stress between
the two plates is built over years so we can say that this external driving is very slow. There are
two types of earthquakes, small and big ones, and both of them exhibit power law distribution
for their size. Olami et al. (1992) suggested Earthquakes model that based on Burridge-Knopoff
spring block model (Burridge and Knopoff , 1967). Their model is a generalized continuous,
nonconservative cellular automaton model. Their two-dimensional model is based on dynamical system of blocks that are connected by springs. Each block is connected to four nearest
neighbors. Each block is connected to a single plate by another set of springs. When the force
on one of the blocks exceeds some critical value, the block starts to slip. The slip of one block
will redefine the forces on its nearest neighbors. This movement causes a chain reaction. The
boundary condition for this system is that the force on the boundary is equal to zero. The time
interval between earthquakes is much larger than the actual duration of an earthquake (as is supposed for SOC systems). The model focused on measuring the probability distribution of the
size of the earthquakes. It was found that this size is proportional to the energy that is released
during an earthquake. The model is different from the Bak model in the following points: (1)
after relaxation the strain on the critical site is set to zero. (2) the relaxation is not conservative.
(3) the redistribution of strain to the neighbors is proportional to the strain in the relaxing site.
The simulation results show a power law distribution for the energy release (that is the earthquake size), that is SOC system in the case of nonconservative model (conservative earthquake
models did not show any SOC behavior).
1.3
Analytical approach
The research of self organized criticality is mostly based on numerical approach, in the sense
of developing the sandpile model and introducing new cellular automata models. In addition, a
1.3. ANALYTICAL APPROACH
13
number of experiments have been performed to establish the connection between the numerical
results and the realistic world in order to check if SOC systems do exist in nature. The difficulty
of developing analytical treatment is based on the fact that such systems are complex systems
that consist of many interacting cells. In many cases the corresponding physical systems to be
explained are continuous. Dynamical description of such systems is not available at present.
The obvious way is to turn to the statistical mechanics approach. Even the ability to introduce
these models into the statistical mechanics language in a more or less rigorous way is rather
difficult and Dhar and Ramaswamy (1989); Dhar (1990) made it possible in some way for only
part of the BTW (Bak, Tang and Wiesenfeld) model. Until now there are two basic approaches,
the renormalization group analysis and the mean-field theory and the studies are concentrated
mostly on the SOC regime.
1.3.1
Renormalization group
The renormalization method (RG) has for the last 25 years or so been the flagship among techniques applied by theoretical physicists. This method was used to describe a large kind of systems in quantum theory of high energy physics, thermodynamics of phase transition, etc. This
success has stimulated also the use of this method in connection with self organized criticality.
In the RG approach it is assumed (sometimes implicitly) that the system is already in a
SOC state and that there is no characteristic size/scale. Respectively, there should be a kind of
scaling symmetry (if the size is multiplied by a number nothing physically essential changes),
and all distributions are power law. This allows to establish relations between various powerlaw indices. Nevertheless, determination of all indices requires knowledge of dynamics, that is,
a solution of the RG equations, which has been done only in several exactly solvable models
(Dhar and Ramaswamy, 1989; Maslov and Zhang, 1995; Helander et al., 1999). RG describes
behavior in the critical point (SOC) and does not describe approach to SOC. Corrections for
non-SOC regimes when driving is not weak are either unknown or difficult to find. These
drawbacks of the method resulted in it’s quick substitution with the mean field theories.
1.3.2
Mean-field approach
The mean field approach (in terms of a branching process) was first proposed by Zapperi et al.
(1995) in an attempt to provide a more general and comprehensive theoretical understanding of
SOC. Previous works such as the renormalization group method were too restrictive in doing
so since they didn’t take the external driving into consideration but only the critical avalanche
behavior. Furthermore, most of previous studies focused on conceiving a particular SOC model
such as sandpile or forest-fire instead of attempting to find a full conceptual framework for the
general SOC phenomena. The mean field theory succeed in dealing with high dimensional sys-
14
CHAPTER 1. INTRODUCTION
tems where most of the other theories fail. Mean field theory consists of estimating the average
behavior of many interacting degrees of freedom. The specific details of the surrounding is replaced by the typical average behavior. In doing so it is capable of incorporating symmetries and
conservation laws and establish general connection with SOC behavior. Zapperi et al. (1995)
proposed a connection between the mean field theory and the SOC theory based on the concept
of branching processes. In the mean field theory spreading of avalanche can be approximated
by an evolving front consisting of non interacting particles that can either trigger subsequent
activity or die out. This is the branching process. For a branching process to be critical one
must fine tune a control parameter to a critical value. This, by definition, cannot be the case for
SOC system, where the critical state is approached dynamically without the need to fine tune
any parameter. Zapperi et al. resolved this paradox by introducing a new mean field model
call self organized branching process (SOBP) that acts like a branching process but without the
control parameter. This can be done by explicitly incorporating the boundary condition. Their
numerical model is based on several models which are different in their size. The analytical
calculation of the avalanche distribution agreed with their simulation results. Also, they got
that the branching process can be exactly mapped into SOC models in the limit d → ∞ i.e it
provides a mean field theory of self organized criticality systems.
1.4
Experiments
Numerically studied sandpiles and cellular automata develop avalanches in an ideal world.
Whether these models are appropriate for the real world should be verified in experiments and
observations. So far a number of laboratory experiments have been performed, mostly with
sand and rice grains.
1.4.1
Dynamics of sandpile
A trivial sandpile experiment was suggested by Held et al. (1990). In Figure 1.3 we can see
the experimental system of the sandpile model. Their main objective was to find out whether
there is a power law behavior in the avalanche size distribution of sandpiles and if there is any
difference between small and large systems. Their model was a pile with the base ranged from
10-50 grains in diameter. An individual grains (driving) were intermittently dropped on the
apex of a conical sandpile. The number of grains that participated in the avalanche process
was measured as the total mass that leaves the pile. The diameter of the sandpile was made
variable to study sandpiles of various sizes. It was found that in the small sandpile the mass
fluctuations show a critical finite size scaling similar to that associated with the second order
phase transition. The larger sandpile showed relaxation oscillations and did not exhibit scale
1.4. EXPERIMENTS
15
Figure 1.3: Schematic illustration of the experimental apparatus. From Zapperi et al. (1995).
invariance. Several years after this experiment was suggested, Rosendahl et al. (1993) repeated
it and suggested that in the sandpile system most of the avalanches are of small size, while
the mass transfer in the large sandpile (the sand that measured to get out from the system) is
related to the larger avalanches. Thus, the results by Held et al. (1990) do not properly describe
the avalanching (or SOC) behavior of the studied system. It worth mentioning though that the
definition that self organized criticality is define by only small avalanches is still argued.
1.4.2
Dynamics of rice pile
The turn to a rice model was based on the basic differences between these two models, that is,
the shape of the grain. In sand each grain has a different shape, while in rice we can find strong
similarity between the different grains. One of the three dimension rice model was suggested by
Aegerter et al. (2003). The pile was grown up from a uniform line source. This uniform source,
as we can see in Figure 1.4, was a custom built mechanical distributor based on a nail board
producing a binary distribution. Therefore the external driving was impact in this line and drop
rice at an average rate of 5g/s. They measured the size of the displaced volume (grains) in each
time step for three experimental setups and found that there is no intrinsic size characteristic
for the avalanches. It was found also that the model is self organized into a critical state with
distribution function for the size of the avalanche that behaves like a power law, that becomes
more and more exact when the length of the model was increased.
1.4.3
Dynamics of water droplets
The deposition, growth and motion of fluid is a subject of enormous interest to many disciplines
in science. One of the first experiments was suggested by Plourde et al. (1993). They presented
16
CHAPTER 1. INTRODUCTION
AEGERTER, GÜNTHER, AND WIJNGAARDEN
FIG.
1. A schematic
setup. The
distribution
Figure 1.4: A
schematic
image of theimage
setup. of
Thethe
distribution
board
can be seenboard
on top, where
can befrom
seena on
top,point
where
is dropped
from
single
point and Within
rice is dropped
single
and rice
subsequently
divided
intoa even
compartment.
the wooden
box
bounding
the
rice
pile.
From
Aegerter
et
al.
(2003).
subsequently divided into even compartments. Within the wooden
box bounding the rice pile, a reconstruction of its surface is shown,
a systemas
with
dynamics
of avalanching
type in continuously driven water. The experimental FIG. 2. A
it the
is used
in further
analysis.
system is shown in Figure 1.5. An avalanche of a tilted sprayed surface occurs when a droplet
From the dis
grows instructure
size and eventually
the critical
mass,
which
time it runs down
the surface,
of the reaches
pile and
the size
ofat the
avalanches,
is disThe dark spo
cussed
in Sec.droplets
II. Thein avalanche
sizecreating
distributions
and their
triggering
other stationary
its path and thus
a chain reaction.
We can see a
and blue%,
scaling
are in
presented
Sec.
III A, together
with
snapshotfinite
of suchsize
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Figure 1.6. in
In this
experiment
they changed
the viscosanalysis. Th
determination
criticalwater
exponents.
In Sec.
IIIslow
B, driving,
the that
ity of thethe
droplet
and also the rateofof the
the entering
that is the analog
to the
age surface
surface
roughness
analyzed
and theasnecessary
is, driving
the system
toward theisthreshold
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in avalanchestechniques
study. Their results
tion of proj
arethat
briefly
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inavalanche
this section,
theavalanche
scalinglifetime
rela- behave
are shown
the distribution
function
of the
size and
Fig. 2, whe
tionslaw.between
the itroughness
exponents
andthat
thethecritical
expolike a power
Furthermore,
was shown (for
the first time)
distribution
of the time
extracted fr
are avalanches
introduced.
These
resultsdistribution.
are also put
into
a wider
betweennents
successive
behaves
like Poisson
It was
found
that higher viscontexta larger
and range
compared
with
the results
obtained
from
KPZ
cosity provided
of power
law behavior,
and that
increasing
the flow
rate led in
to this way
face can be
roughening
systems that
!27".
exponential
decay of distribution,
is, there are characteristic length and time scale (no self
reconstruct
organized criticality).
II. EXPERIMENTAL DETAILS
The experiments were carried out on long grained rice
with dimensions of typically #2!2!7 mm3 , similar to rice
A of Ref. !12". The pile was grown from a uniform line
source that is 1 m wide. This uniform distribution was
achieved via a custom built mechanical distributor based on
a nail board producing a binary distribution !28". The actual
setup consists of a board with an arrangement of triangles, as
field of vie
mm, as we
both structu
curacy wer
is roughly m
well suited
In a sing
period of #
experiment
1.4. EXPERIMENTS
17
Figure 1.5: Schematic diagram of the experimental apparatus: distilled water is sprayed through
the spray mister into a transparent plastic then the streams run down the dome, and drop onto
the slanted annular impact ring. From Plourde et al. (1993).
Figure 1.6: Typical digitalized output from the image-processing system. From Jánosi and
Horváth (1989)
18
1.4.4
CHAPTER 1. INTRODUCTION
Superconducting vortex avalanches
In Field et al. (1995) experiment it was suggested that a superconductor system with a critical
external field can behave like a sandpile model. In the experimental setup the magnetic field was
increaed slowly until the external field reached a value such that flux first entered the interior
of the tube. If the flux enters the tube (as a vortex), this would induce a voltage pulse on the
coil. Each pulse represents the sudden influx of many spatially correlated vortices, that is, a flux
avalanche. In brief, one can state that an avalanche is represented by the number of vortices. At
high rate there is non-negligible overlapping between the avalanches. The experiment results
were that in the marginally stable state a hard superconductor exhibits flux avalanches with a
power low distribution of sizes (number of vortices in the avalanche).
1.5
Space plasma and magnetosphere
One of the fields where the avalanching system and SOC concepts are widely exploited last
years, is the space plasma physics. There only remote and/or partial observations are available.
1.5.1
Solar flares
A solar flare is believed to be a rapid change in a strong, complicated coronal magnetic field
(due to magnetic reconnection) with a subsequent prompt energy release. According to the
classical picture by Parker (Parker, 1989) the turbulent plasma flows below the photospheric
surface drive the anchored flux tubes into complex, stressed configuration. The magnetic lines
are stretched out from the sun and its overlying arcade rise as shown in Figure 1.7. There are
lots of close arcades that become closer and closer. When outward line of magnetic field meet
inward line they reconnect. When the lines are reconnecting (an outward going line with an
inward directed line) they release energy impulsively. The reconnection process causes particle
acceleration with the subsequent release of the observed X radiation. An observer can measure
the statistics of the energy released, peak x-ray flux distribution and the passive time interval
between solar flares. Except for the passive time that act like Poisson distribution, the other
parameters are found to be all characterized by power law distribution. These statistics probably
indicate that the solar corona may be in a state of self organized criticality (Lu and Hamilton,
1991; Nakagawa, 1993; Boffetta et al., 1999; Charbonneau et al., 2001). Such a view implies
that the x-ray flares are avalanche of reconnection events. One of the proposed models of the
solar flares suggests (Hughes et al., 2003; Paczuski and Hughes, 2004) a SOC-behaving net of
flux tubes.
have generated substantial progress in the theoretical
quiescent time interunderstanding and computational modeling of local reall characterized by
connection in laboratory, space, and solar plasmas.
y distribution is parHowever, it remains computationally prohibitive to use
e-free behavior over
the equations of plasma physics to describe extended
y [5]. These statistics
y be in a state
self- PLASMA AND MAGNETOSPHERE
1.5.ofSPACE
19
ch a view implies that
all transient brightenre not fundamentally
avalanches of recontics of solar coronal
ures with other interas earthquakes, forest
60
nce [10,11].
40
l model of multiple
20
driven at their foot0
0
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ootpoints, of opposite
to a two-dimensional
50
50
ere. Footpoints of the
100
100
egation on the surface.
150
150
footpoints when they
nnection, can lead to
200
200
trigger a cascade of
des of magnetic loop
FIG. 1 (color online). Snapshot of a configuration of loops in
Figure
1.7:
Snapshot
of a configuration
of 200
loops
in m
the!steady
ar flares.
the steady
state with L !
and
1 (seestate.
text).From Hughes et al. (2003)
13)=131101(4)$20.00
1.5.2
 2003 The American Physical Society
131101-1
Current sheet and magnetic storms
In the early 1960s, spacecraft observation established the existence of the geomagnetic tail.
This geomagnetic tail is the name of special regime in the Earth magnetosphere that is stretched
away from the sun behind the Earth. In the center of the geomagnetic tail there is a layer of
current sheet. This current sheet can be defined as a thin surface (thin is relative term but the
sheet thickness is relative small so we can describe this sheet as a plane) across which the
field strength or/and direction can change substantially, which means that this layer must carry
substantial electric current. As we mentioned above, these magnetic field lines on the night side
(magnetic tail) are stretched due to the solar wind that carries plasma from the sun to Earth. This
stretch can be further enhanced because of the additional plasma pressure on the magnetic lines
due to the large disturbances of the solar wind coming to the Earth. This additional pressure
forces the magnetic lines to get close to each other and because of their different direction their
breaking occurs. After the breaking the lines reconnect and in this process send plasma to the
Earth, eventually causing what is known as a magnetic storm. Magnetic storms and more local
events (substorms) have been a subject of intensive studies, and ideas of avalanching systems
and SOC have been applied for quite a while (see, e.g., Klimas et al., 2000; Surjalal Sharma
et al., 2001; Uritsky et al., 2001).
20
CHAPTER 1. INTRODUCTION
702
A. T. Y. Lui: Testing the hypothesis of the Earth’s magnetosphere
(a)
(b)
0
10
Figure 1.8: Illustration of avalanching
magnetotail. Left - processes in the current sheet are
compared to snow avalanches. Right - enhanced aurora during substorm is a result of the reconnection in the current sheet.10From Lui (2004).
-1
-2
10
1.6
Observational relations
-3
10
-4
10
-5
10
For models and laboratory experiments
the identification for SOC or SOC-like is more simple
10
10
10
10
10
than dealing with remote observations, like those
(c) available in space. In numerical systems the
1. Auroral power
exhibits a scale-free
density
distribution
similar to an avalanching
system:directly,
(a) a schematic since
of a snow all details are
results of theFig.
external
driving
is theprobability
output
that
an observer
can see
avalanche and magnetotail activity sites responsible for observed auroral dissipation, (b) an example of a global auroral observation from
Polar UVI, and (c) a representative distribution of auroral area showing a power law dependence suggestive of its scale-free nature.
immediately available. But in real systems, like the magnetosphere, we can’t say for sure that an
observer can see only the system response to the driving and not mixture of this response with
the driving himself or other processes occurring between the system and observer and affecting,
e.g., propagation of accelerated particles or radiation. Auroral indices (measure of the magnetic
field deviation from the daily averaged values along the base of the auroral oval) have received
considerable attention for testing the prediction of model showing complexity. The broken
power law of the burst lifetime and size distribution (Consolini and Marcucci, 1997; Consolini
and De Michelis, 1998; Consolini and Lui, 1999) have been taken to be strong indicator for
complexity and SOC in the magnetosphere’s dynamics evolution. Since that, Consolini and
Lui (1999) have shown presence of a small ”bump” with characteristic values of burst with
duration. Lui (2002) found a power law slope plus a ”bump” (The measure was done in the
quiet times where no substorms can interrupt and finding a power law behavior.) at large values
corresponding to substorm breakups.
6
7
8
9
10
1.7. OBJECTIVES
1.7
21
Objectives
Despite intensive and extensive studies of avalanching systems and self-organized criticality,
in particular, during last twenty years, some issues remain weakly analyzed or even were not
touched essentially. Until now almost always the research of avalanching systems was based
on the concept of weak driving, that is, by popular definition, SOC systems. As we know
nature can not wait until the avalanche will fade away to enter a new perturbation and also can
not control the perturbation to be weak (although in some cases, like probably earthquakes, it
may occur). Therefore, studies of strongly driven systems which could possess avalanching
behavior, is not less important. Our objective in the present work was to study numerically and
analytically avalanching systems without the restriction of weak driving, in order to know to
what extent (if any) the behavior of such systems approaches the self-organized regime within
the usual SOC definition, or whether this definition should be revised to make it less restrictive
(from a physicist point of view).
The problem with some of the avalanching systems is that we do not always know what are
the physical microprocesses of the system and it is not so easy (if possible at all) to examine
these microprocesses. In order to learn about this kind of systems we measure the results in
statistical way (like active time and passive time) and derive implication from the results for the
physical processes. It is important to add that sometimes we can not be sure that the results that
we are measuring are not affected by the medium and other interruptions in the way from the
system to the observer.
In order to achieve progress in the issues described above, we perform extensive analysis
of the traditional sandpile model and exam the influence of the driving change on the system
behavior (we leave the area of weak driving). Our system is made to be more related to existing systems by looking at bi-directional avalanches (no closed boundary is allowed). The
results provide the clues to the understanding whether the classical definition of SOC systems
(infinitely weak driving) is physically meaningful or it makes sense to understand SOC in a
broader way by weakening this condition.
Taking into account that one of the main applications should to be to the space plasma reconnecting systems, in the course of our analysis we propose a new avalanching model which is,
in our opinion, more related to the, e.g., localized reconnection processes in the current sheet in
the Earth magnetotail (Milovanov et al., 2001; Milovanov and Zelenyi, 2002; Lui et al., 2005).
In this new model we focus on the physical properties which were not taken care of in the previous works. We study this system numerically in the way similar to the sandpile and perform
a comparative analysis of the two systems. In addition, we propose a new method of analytical
study of avalanching systems and derive certain statistical properties for our suggested model.
We also perform some analysis of two-dimensional systems in order to understand the effect of
dimensionality of statistical distributions and SOC behavior.
22
CHAPTER 1. INTRODUCTION
The work is organized as follows. In Chapter 2 we deal with the one and two dimensional
sandpile model. In Chapter 3 we introduce the new burning model analytically and numerically,
and extend the model to the two dimensional case. In Chapter 4 we compare the results of these
two models. In Chapter 5 we provide a very brief preliminary comparison of the analyzed
models with some observations of real physical systems. Chapter 6 briefly summarizes the
results of the work and suggests directions of further studies.
Chapter 2
Sandpile model
Sandpile models have been proposed for explanation of various phenomena, including solar
flares (Lu and Hamilton, 1991; Boffetta et al., 1999; McIntosh et al., 2002; Hughes et al., 2003)
and reconnection in the Earth current sheet (Takalo et al., 1999; Milovanov et al., 2001; Klimas
et al., 2004). Sandpile models also serve as a basic class of prototype models for studies of
the onset and features of self-organized criticality (see, for example, Bak et al., 1987; Kadanoff
et al., 1989; Dhar, 1990; Hwa and Kardar, 1992; Benza et al., 1993; Christensen and Olami,
1993; Ghaffari et al., 1997; Dendy and Helander, 1998; Chapman et al., 2001) It is, therefore,
natural to start the analysis with a properly adjusted sandpile model. In this section we consider
a bi-directional sandpile model, which is a direct generalization of the directed model studied by
Carreras et al. (2002); Newman et al. (2002); Sánchez et al. (2002) in a slightly different context. The reason for using this simple model is that it is possible to compare immediately some
results with those published in literature. On the other hand, apparent simplicity of the model
did not prevent from applying it to rather complicated physical systems. It is also relatively easy
to adapt the model introducing various features of driving and redistribution mechanism.
2.1
One-dimensional bi-directional sandpile
We consider a one-dimensional sandpile defined on a L long array of sites. The sandpile array
is the array of integers N (i), which are heights (number of grains at the site i). The system is
open at both boundaries, that is, grains are freely dropping from the sites i = 1 and i = L,
provided the slope is sufficiently steep (see below). This allows, in principle, development of
avalanches in both directions and is the main difference with the model of Sánchez et al. (2002).
The system is externally driven by adding Nd grains at each step to each site with the probability
p.
The average total amount of grains added to the system at each time step will be Jin =
pLNd . Activity of each site is determined by the local slope Z(i) = max(N (i) − N (i −
23
24
CHAPTER 2. SANDPILE MODEL
1), N (i) − N (i + 1), that is, the maximum gradient of the height. If this slope exceeds some
critical slope, Z(i) > Zc , the site becomes active and collapses to one or both its neighbors. This
collapse is characterized by the transfer of 2Nf grains to the neighbors. If there was a collapse
to only one neighbor it will get all the grains but if there was collapse to two neighbors the
number of grains will be divided equally between the two. Thus, the collapse process depends
on the site itself and also on the state of its nearest neighbors. To summarize all this we get that
there will be a collapse of one site i if:
θ(N (i) − N (i + 1) − Zc ) + θ(N (i) − N (i − 1) − Zc ) > 0
(2.1)
where the step function θ(x) = 1 when x ≥ 0 and zero otherwise. If this condition is satisfied
then the heights are changed as follows
N (i) → N (i) − 2Nf
(2.2)
N (i + 1) → [N (i + 1) + 2Nf ]θ(N (i) − N (i + 1) − Zc )θ(−N (i) + N (i − 1) + Zc )
+ [N (i + 1) + Nf ]θ(N (i) − N (i + 1) − Zc )θ(N (i) − N (i − 1) − Zc )
(2.3)
and similarly for N (i − 1). Updating of all sites is done simultaneously after checking the
state of all sites. At the next step the whole procedure repeats, including random adding of
grains and collapse of active sites. Open boundary conditions are given by N (1) = N (L) = 0,
which means that sand leaves the system at the boundary whenever the nearest neighbor, N (2)
or N (L − 1), collapses. Woodard et al. (2005) suggested that the strength of the external
driving can be described in terms of two parameters: the measure of the temporal overlap of
avalanches x1 = Nd L2 p/Nf (overlapping is not negligible when x1 & 1) and the measure of
spatial merging of clusters x2 = Nd Lp (merging is not negligible is x2 & 1). It has to be noted
that x1 assumes that the average avalanche length is ∼ L, otherwise x1 should be reduced by
the factor L/average avalanche length.
2.1.1
Numerical analysis
Since we are interested in the stationary regime (see below) we prepare our initial distribution
so that it is near the critical point, in order to save simulation time. Of course, a sandpile can
also be built by dropping grains in a random way onto an empty array, but in this case we have
to wait a long time until stationary regime is achieved. Even for the prepared initial sandpile
stationary regime is achieved in about 50,000 steps. Stationarity is assessed by visual inspection
of the times series of the number of active sites. The stationary regime is characterized by the
balance between the external driving (average number of grains added to the system per unit
time) and losses at the edges (average number of grains that leave the system from the boundary
2.1. ONE-DIMENSIONAL BI-DIRECTIONAL SANDPILE
25
per unit time).
The numerical analysis proceeds in the following way. We drop grains to the system in
random way (at each step each site gets Nd grains with the probability p). After the grains are
dropped we scan the system searching for the sites where the slope become more then the critical value. The sites with the slope above the critical value are relax and transfer 2Nf grains to
their neighbors. The grains transfer occurs simultaneously at all relevant sites (synchronous updating). In the present numerical analysis we use the following parameters: L = 201 ,Nd = 10
,Zc = 200 and Nf = 15. These parameters were chosen by Sánchez et al. (2002) for their
one-directional model. We choose the same set of parameters in order to be able to easily compare our results to those of Sánchez et al. The probability p was used to control the avalanching
activity, if p increases there will be more active sites, and therefore more avalanches.
We have chosen to measure the distribution functions for the active phase duration, passive
phase duration and the size of avalanche per iteration. One of our main purpose is to understand whether very limited measurements done by a remote observer may provide sufficient
information to distinguish between two physically different avalanching systems (the SOC is
only bonus). These three main measurements that we introduce above are very simple to observe from distance and, because of that, can give us useful and simple information. On the
other hand, in many cases more sophisticated measurements are not available because of insufficient resolution or not useful because of the unknown properties of the medium between the
avalanching system and the remote observer.
As we mentioned above, this model is different from Sánchez et al. (2002) model in few
elements, one of them being the bi-directionality, which removes the artificially closed boundary inappropriate for comparison with such physical systems like solar flares or current sheet.
In this bi direction case we expect that the avalanches main direction will be down the slope,
so there is relatively weak activity in the top of the pile. Most of the activity should occur
between the middle and the bottom of the pile, so that there should be substantial similarity
with the ordinary one-directional sandpile. Thus, we can expect for a power law distribution
for the duration of the active time, that is, we expect for system to act like a self organized
one and no correlation between the avalanches duration and length. We also expect to Poisson
distribution for the passive time (like Sánchez et al.) because we expect that the passive time
depend only on the nature of the external driving. This driving is random and we believe that
the probability of avalanche to start is random too. If we will increase this probability we will
get more and more avalanches, that is the system will be more active, so that temporal overlapping of avalanches become more and more important. With further increase of the driving
spatial merging of avalanches should become non-negligible. It should be noted also that large
avalanches, spanning more than a half of the system, should occur on the both sides of the top,
that is, some activity at the upper part should be observed.
26
CHAPTER 2. SANDPILE MODEL
To start with, we show the activity of the system in Figure 2.1 (upper panel). As expected,
200
180
160
140
site
120
100
80
60
40
20
1
2
3
4
5
time
6
7
8
9
10
x 104
80
70
60
site
50
40
30
20
10
9.94
9.945
9.95
time
9.955
9.96
9.965
x 104
Figure 2.1: View of the sand system activity: 106 time steps on the top panel, enlarged part on
the bottom panel. (One dimensional case)
in the middle of the pile, that is near i = 100, there is almost no activity. In the bottom panel
of this Figure we provide a closeup of the activity graph to show a developing avalanche in
more detail. Figure 2.2 shows the time series of the number of active sites (mentioned above).
From both figures it is easily seen that the system is in a regime of non-weak driving, that is,
far from the self-organized-critical state (see, for example, Vespignani and Zapperi, 1998), and
(temporal) avalanche overlapping is substantial. Yet cluster merging is negligible. There are
two kinds of avalanches, as we can see in the bottom panel of Figure 2.1: one that expands in
2.1. ONE-DIMENSIONAL BI-DIRECTIONAL SANDPILE
27
120
number of active sites
100
80
60
40
20
0
0.5
1
1.5
time
2
2.5
3
x 105
Figure 2.2: Number of active sites as a function of time.(One dimensional case)
both directions (two-dimensional shape) and the other that expands only in one direction (linear
shape).
To quantify the difference (if any) in the system behavior for various drivings, we consider
the low (p = 0.00005) and high (p = 0.00015) states. Physically the two states differ according
to the percentage of the time which the system spend in the active state (see below). The
merging parameters for the two cases are: x2 = 0.1 (low) and x2 = 0.3 (high), so that spatial
merging should be negligible in both cases. On the other hand, x1 = 1.3 (low) and x2 = 3.9
(high) mean that temporal overlapping may be significant.
Figure 2.3 shows the cluster size distribution in the high state. The complete distribution
consists of two sub-distributions. The lower sub-distribution corresponds to the odd-length
clusters which are possible only when the cluster grows until it touches the edge of the sandpile
(we can see it in two clusters at Figure 2.1). A typical cluster size is even because of the
punctuated nature of the clusters (a site that collapses in some iteration becomes a hole - inactive
site - in the next iteration) and our method of counting the holes together with the surrounding
active sites. The distribution of cluster sizes is Poisson like with the mean cluster size w̄ = 12.
The corresponding distributions of the active and passive phase duration are shown in Figure 2.4. The active phase duration is calculated as the difference between the moment of the first
appearance of an active site and the moment when the last active site becomes passive. Respectively, the passive phase duration is the time between the end of activity (previous avalanche)
and the beginning of activity (next avalanche). Obviously, overlapping avalanches contribute to
the same active phase duration. While the passive phase durations are distributed according to
the Poisson statistics (as should be expected in the absence of correlation between avalanches),
28
CHAPTER 2. SANDPILE MODEL
100
10−1
pdf
10−2
10−3
10−4
10−5
10−6
0
20
40
60
80
100
cluster size
120
140
160
180
Figure 2.3: Distribution of clusters size for the high state in log linear scale. Upper trend is for
even size clusters, lower for odd size ones.(One dimensional case)
the active phase duration show shape of power low behavior at large duration. The mean passive and active duration, respectively, are Tp = 58 and Ta = 23. The transition to power law
for active phase duration occurs near the mean value. Figure 2.5 shows the distribution for active and passive phase duration for different driving - low and high. In the passive and active
phase durations we change the driving from low to high state in order to check if there are any
influence on the system behavior. As the results show, the driving not only did not change the
characteristic of the behavior (power law, Poisson distribution), but also did not even change
the slope of the curves.
It is difficult to conclude unambiguously whether the system is in the self-organized critical
state, especially in view of the controversy in the definition of what SOC is. It is often claimed
that self-organized criticality is achieved only with very slow driving when complete time separation is ensured (Vespignani and Zapperi, 1998). From this point of view our system cannot
be in a SOC state because we do not limit ourselves with slow driving. Another point of view is
that SOC means more presence of power law distributions in the system which indicates lack of
any characteristic length/time scale (Hughes and Paczuski, 2002; Hughes et al., 2003; Paczuski
and Hughes, 2004), and there is no correlation between the avalanches. Power-law distributed
active phase durations may be considered a kind of indication of SOC or quasi-SOC behavior.
2.1. ONE-DIMENSIONAL BI-DIRECTIONAL SANDPILE
29
100
10−1
pdf
10−2
10−3
10−4
10−5 0
10
101
10 2
active phase duration
103
100
10−1
pdf
10−2
10−3
10−4
10−5
0
100
200
300
400
500
passive phase duration
600
700
Figure 2.4: Distribution of active (top-log log scale) and passive (bottom-log linear scale) phase
durations.(One dimensional case)
30
CHAPTER 2. SANDPILE MODEL
100
10−1
pdf
10−2
10−3
10−4
10−5 −1
10
10 0
10 1
active phase duration
10 2
100
10−1
pdf
10−2
10−3
10−4
10−5
0
2
4
6
8
passive phase duration
10
12
Figure 2.5: Distribution for active (top-log log scale) and passive (bottom-log linear scale) phase
duration for high state (circles) and low state (stars). The distribution are normalized with the
mean values.(One dimensional case)
2.2. TWO DIMENSIONAL SANDPILE MODEL
2.2
2.2.1
31
Two dimensional sandpile model
The model
One dimensional systems seem somewhat exceptional (although they proved to be useful in
numerical as well as analytical studies). It is therefore natural to generalize the previous model
for two dimensions. This model is very similar in the basic rules to the one dimensional model.
Let there be a L × L array of sites. The system is open and there is external random driven
similar to that of the one-dimensional case: Nd grains are dropped at each site at each time step
with probability p. The total amount of energy added to the system at each time step will be
Jin = pL2 Nd . If the height of the site exceeds by more than Zc the height of at least one of the
four closest neighbors, the site becomes active and collapses to those its neighbors for which the
height difference exceeds the critical one. This collapse is characterized by the transfer of 4Nf
grains to the neighbors - if there was a collapse to only one neighbors it will get all the grains
but if there was collapse to more then one neighbors the number of grains will divided equally
between the accepting neighbors. To summarize all this we get that there will be a collapse if:
θ(N (i, j) − N (i + 1, j) − Zc ) + θ(N (i, j) − N (i − 1, j) − Zc )
+ θ(N (i, j) − N (i, j + 1) − Zc ) + θ(N (i, j) − N (i, j − 1) − Zc ) > 0
(2.4)
where the step function θ(x) = 1 when x ≥ 0 and zero otherwise. All the boundaries are open,
that is, N (1, 1 : L) = N (L, 1 : L) = N (1 : L, 1) = N (1 : L, L) = 0.
At this stage we introduce a modification of the model considering also the case when the
transferred grain number Nf is not constant but is some percentage of the height difference,
Nf → ηZ, where Z = max(N (i) − N (neighbors). In order to compare the two models (constant transferred grain amount and variable transferred grain amount) we choose the percentage
so that for the height difference equal to the critical one, the number of transferred grains be
the same in both cases. Namely, in the numerical analysis below Nf = 3 for Zc = 20, so that
η = 0.15.
2.2.2
Numerical analysis
We again start our simulation with the proper preparation of the initial distribution near the
critical point, building the slope which is marginally stable, and begin our analysis when the
system enters stationary regime. We drop grains to the system in the random way, as described
above, and then we scan where the slope become steeper then the critical value. The grain
transfer occurs simultaneously at all relevant sites (synchronous updating). In the present numerical analysis we use the following parameters: L = 101 ,Nd = 1 ,Zc = 20 and Nf = 3.
The probability p was used to control the avalanching activity. In what follows we focus on the
32
CHAPTER 2. SANDPILE MODEL
passive and active phase durations and the avalanche size. We expect to see whether there is
any effect of the dimensionality of the system on the distribution shapes for these variables. We
again consider the low p = 0.000005 and high p = 0.00005 states. The corresponding merging parameters for the two-dimensional case are x1 = pL3 Nd /Nf and x2 = pL2 Nd . For our
parameters we have in the low state x1 = 1.7 and x2 = 0.05, while in the high state x1 = 17.
and x2 = 0.5. Although x1 seems to mean strong temporal overlapping, it should be taken into
account that for a more proper estimate one should substitute L2 by the average avalanche size,
which may be substantially smaller.
Figure 2.6 shows the distribution of individual avalanche sizes.
PDF − size of avalanche (per iteration)
0
10
−1
10
−2
pdf
10
−3
10
−4
10
−5
10
−6
10
0
10
20
30
size of avalanch
40
50
60
Figure 2.6: Distribution of clusters size for the high state in log linear scale. Upper trend is for
even size clusters, lower for odd size ones.(Two dimensional case)
In deriving this distribution overlapping avalanches were counted separately. We can see
that the avalanche size in the high driving case exhibit a Poisson-like distribution. The results
are similar to what was obtained in the one dimensional system, in the sense that here we also
get odd- and even-sized avalanches. The main difference is that in the one dimensional case
the two different kinds of avalanches join together in the large avalanche size limit, while in the
two dimensional case these two distributions remain well separated even for largest clusters.
The different between these two kind is the size of the system. In the one dimensional system
there is large probability that the avalanche develops until it gets to the boundary. This can be
seen in the sense that in large avalanches this two curves join into one (large in sense that have
size comparable to the system size). In the two dimensional model the system is very big and
the chances that even a large avalanche reaches the boundaries are low. We can also see that
2.2. TWO DIMENSIONAL SANDPILE MODEL
33
in the two dimensional model the odd-sized avalanches (that is, the lower distribution) are less
common compared to the one dimensional system. In Figure 2.7 we introduce the passive phase
duration (log-linear scale, upper panel) and active phase duration (log-log scale, lower panel)
of this model for the case of high driving.
PDF − passive time
−1
10
−2
pdf
10
−3
10
−4
10
−5
10
0
20
40
60
80
100
passive phase duration
120
140
PDF − system active time
0
10
−1
10
−2
pdf
10
−3
10
−4
10
0
10
1
10
2
10
active phase duration
3
10
Figure 2.7: Distribution of passive (top-log linear scale) and active (bottom-log log scale) phase
duration.(Two dimensional case)
The passive distribution behaves like Poisson except for long avalanches where statistics
become poor. From this distribution one can conclude that the evolution of this two dimensional
34
CHAPTER 2. SANDPILE MODEL
sandpile model does not show any time correlation among the avalanche event. The active
distribution behaves like Poisson at small duration but looks more like a power law for larger
durations. This change of the behavior with the avalanche duration increase is not clear yet and
further analysis is required.
It is worth noting that, in contrast with most previous works, we do not restrict ourselves
with very slow driving. Figure 2.8 shows the comparison of the passive and active phase duration distributions for two different input probabilities.
PDF − passive time
−1
10
−2
pdf
10
−3
10
−4
10
−5
10
0
100
200
300
400
500
600
passive phase duration
700
800
900
PDF − system active time
0
10
−1
10
−2
pdf
10
−3
10
−4
10
−5
10
0
10
1
10
2
10
active phase duration
3
10
Figure 2.8: Distribution of passive (top-log linear scale) and active (bottom-log log scale) phase
duration. Blue line for low state and green line for high state.(Two dimensional case)
The green line describes the case where the input probability p = 0.00005 and the blue
2.2. TWO DIMENSIONAL SANDPILE MODEL
35
line describes the input probability of p = 0.000005. In the case for lower input probability
the system was in the passive state most of the time. These two kinds of distributions look the
same in the active phase duration graph, that is, the active time does not depend strongly on the
driving probability. In the passive phase duration we can clearly see that the system with the
low probability is less active (the blue line). The largest duration of the passive time in the case
of high probability is around 100 while the largest duration in the case of the low probability is
around 600.
Figure 2.9 shows the comparison of the avalanche sizes for the regimes of two different
drivings. We can see that in the low state the statistics becomes very poor. While the even
PDF − size of avalanche (per iteration)
0
10
−1
10
−2
pdf
10
−3
10
−4
10
−5
10
−6
10
0
10
20
30
40
50
size of avalanch
60
70
80
Figure 2.9: Distribution of cluster sizes in log linear scale. Blue line for low state and green line
for high state.(Two dimensional case)
size avalanches are around the Poisson distribution, the odd size avalanches do not show any
particular distribution.
36
2.2.3
CHAPTER 2. SANDPILE MODEL
Non-constant grain transfer
As is mentioned above, in this model we allow transfer of non-constant number of grains.
Namely, (integer part of) the 15% of the maximum height difference is transferred, which corresponds to 3 grains for the height difference exactly equal to the critical value. We expect that
the proportional transfer be more important for larger events (longer avalanches or active phase
durations) since it means more efficient smoothing for higher gradients. In all subsequent plots
the red line corresponds to the constant number of grain transfer while the cyan line corresponds
to the proportional transfer.
In Figure 2.10 we compare the active phase durations for the two cases. The upper bottom
(red curve) shows the active phase duration for the constant transfer case and driving probability
p = 0.00005 (log-linear scale). The lower panel (cyan curve) shows the active phase duration
for the proportional transfer case and the same driving probability p = 0.00005 (log-linear
scale). We can see that in the constant transfer case the distribution deviates from the Poissonlike and somewhat resembles a power law dependence at larger durations. In the proportional
transfer case we can see Poisson behavior for larger durations too, which means suppression
of larger events. The mean active value is Ta = 11 which is identical to the mean active
value that we get in the constant model. Figure 2.11 shows the passive phase durations for the
two models. Both distributions exhibit Poisson behavior, but the proportional transfer curve is
steeper than the constant transfer curve indicating that in the case of the proportional transfer
system is more stable than the constant transfer system (the passive time is the time that the
system accumulates energy until it get to the critical state). The mean passive time Tp = 16.
This value is much bigger than the value in the constant model and it supports the conclusion
that in the non-constant system avalanches end much faster.
Figure 2.12 show the distribution of avalanche size for this two above cases The effect on the
avalanche size seems negligible. Deviations for odd-sized avalanches are related to relatively
poor statistics.
2.2. TWO DIMENSIONAL SANDPILE MODEL
37
PDF ! system active time
0
10
!1
10
!2
pdf
10
!3
10
!4
10
!5
10
0
20
40
60
80
100
120
active phase duration
140
160
180
PDF − system active time
0
10
−1
10
−2
pdf
10
−3
10
−4
10
−5
10
0
20
40
60
80
100
active phase duration
120
140
160
Figure 2.10: Distribution of active phase duration for the constant grain number transfer case
(top panel, log-linear scale) and for the proportional transfer case (bottom panel, log-linear
scale).(Two dimensional case)
38
CHAPTER 2. SANDPILE MODEL
PDF − passive time
−1
10
−2
pdf
10
−3
10
−4
10
−5
10
0
20
40
60
80
100
passive phase duration
120
140
160
Figure 2.11: Distribution of passive phase duration in log linear scale. Red curve for the constant grain number transfer case and cyan curve for the proportional transfer case.(Two dimensional case)
PDF − size of avalanche (per iteration)
0
10
−1
10
−2
pdf
10
−3
10
−4
10
−5
10
−6
10
0
10
20
30
size of avalanch
40
50
60
Figure 2.12: Distribution of cluster size. Red curve for constant number of grains state and cyan
curve for percent number of grains.(Two dimensional case)
2.2. TWO DIMENSIONAL SANDPILE MODEL
39
In Table 2.1 we summarize the mean values for the bi-directional sandpile model:
Mean cluster size
w̄
Mean
active
phase duration Ta
Mean
passive
phase duration Tp
Table 2.1: Mean values for bi-directional sandpile.
High state
Low state
High state
Low state
p
= p
= p
= p
=
0.00015 1D 0.00005 1D 0.00005 2D 0.000005
Constant
2D
Constant
12
12
7
11
High state
p
=
0.00005 2D
Percent
8
23
21
11
14
11
58
176
12
81
16
40
2.3
CHAPTER 2. SANDPILE MODEL
Summary
In this chapter we examined the influence of the dimensionality on the behavior of typical distributions (avalanche size, active and passive phase durations) in the sandpile model. The main
shapes of these distributions do not change noticeably when the dimensionality is increased
(from one-dimensional to two-dimensional). It is important to mention that we focus here only
on one- and two-dimensional cases because of our intention to eventually apply the results to
certain physical systems (like the magnetospheric current sheet or solar flux tube carpet). At
this stage we do not attempt any generalization of our conclusions to higher dimensions. The results for the one-dimensional model have been published in Gedalin et al. (2005d). The results
for the two-dimensional model with constant transfer are being prepared for publication. The
results for the proportional transfer are of preliminary character, and we are planning further
studies.
Chapter 3
Dynamics of the burning model
Running sandpiles are most ubiquitous, being based on the slope controlled rules and immediate passive-active transitions. SOC, and sandpile models, in particular, have been widely
applied to plasma system, especially to those which are thought to be governed by localized
reconnection(Chang, 1999b; Chapman et al., 1998; Charbonneau et al., 2001; Boffetta et al.,
1999; Consolini and De Michelis, 2001; Klimas et al., 2003; Krasnoselskikh et al., 2002; Lu
and Hamilton, 1991; Takalo et al., 1999; Valdivia et al., 2003; Uritsky et al., 2001). The more
sophisticated field reversal model (Takalo et al., 2001; Klimas et al., 2004) is based on the hysteresis behavior of the resistivity (diffusion). It is unclear whether it can be directly applied to
collisionless localized reconnection. It should be mentioned, though, that there is no definitive
observational evidence relating SOC models to the localized reconnection processes in plasma
sheet, and there is no general agreement regarding the dynamical nature of current sheet reconnection. Our new model has a number of features characteristic of the localized reconnection
process in the current sheet (Milovanov et al., 2001; Milovanov and Zelenyi, 2002; Zelenyi et al.,
2002) that are not, in our opinion, properly treated in other models. These features are:
• the transition passive-to-active depends on the local threshold, resembling what happens
in a current layer, when its width become less than some critical value, or, alternatively,
when the current exceed the critical current;
• there is a finite life time of an active site, that is, an active site which excites its neighbor,
does not become passive immediately (like in the sandpile model) but remains active for
some time;
• a part of energy dissipates into ”radiation” which can be observed by remote observer.
The first declaration of the locality of the threshold can be also introduced even in the most
simple sandpile model. This can be done by changing of the variables, that define a site, to a
slope. This works, however, only for one dimensional directed models (otherwise a local vector
41
42
CHAPTER 3. DYNAMICS OF THE BURNING MODEL
field of ”slopes” has to be defined). Dissipative models are rather ubiquitous. We, however,
propose that the dissipated energy is the energy which comes to a distant observer directly from
the active site, and can be used for identification of the system state. The finite lifetime feature
is the most important since it removes the unphysical condition of immediate energy release by
an active site with the transition to the passive regime. The proposed feature not only allows to
study the behavior of the avalanching system in the fast driving regime, but also make possible
resolution of the temporal behavior of the active regions.
3.1
3.1.1
One dimensional model
The model
Our discrete model is one dimensional array of L sites. Each site i, 1, ..., L is characterized by
his temperature Ti this temperature is the analog to the number of sand grains in the familiar
sandpile model. We choose to work with temperature because temperature is a continuously
varying parameter. The system is open in the sense that energy can get out from the system,
like in normal systems. There is external random input that brings energy from outside to the
system. At each time step the system gets the amount of heat q with the probability p. Thus,
the average heat input into a single site is qp and the total time average driving into the system
is qpL. This heat that a site gets results to increasing of the site temperature until its start
burning. This happens when the site temperature exceeds the upper critical value, that is the
condition Ti ≥ Tc indicate the burning beginning. During this burning the site looses energy
(heat). Part of this heat goes to the closest neighbors and part of it dissipates (the last one is
the energy that we can see from large distance). The rate of the energy loss (energy per time
step) is Ji = kT . In other words we can say that if this site were left along, its temperature
would change according to Ṫ = −kT or T = T (0)exp(−kt). Because of the heat losses the
site temperature decreases until the temperature drops below the lower critical value, that is, the
condition Ti < T0 = sTc , s < 1 defines the point where the burning ceases. Burning does not
start again until Ti ≥ Tc . Physically, it corresponds to the idea that energy release from a site
will persist until it is exhausted, and can start only after it exceeds the upper critical value again.
To summarizing all this, the energy flux is:
Ji = kTi [θ(Ti − Tc ) + θ(Tc − Ti )θ(Ti − T0 )θ(−Ṫi )],
(3.1)
where the step-function θ(x) = 1 when x ≥ 0 and zero otherwise. The last term θ(J) in (3.1) is
related to the idea that a site should be considered burning when the site temperature is below
the upper critical value, only if its temperature is above the lower critical value and this site
was burning at the previous iteration, that is, its temperature was decreasing. Note that the
3.1. ONE DIMENSIONAL MODEL
43
description is imprecise since the random input may occasionally cause some reheating even
during the burning stage. This problem is easily avoided during the discretization, as we shall
see a little later. It is worth mentioning that the life time of a lonely burning site, tl , can be
estimated as tl ≈ ln(Tc /T0 )/k. As we mentioned earlier, the energy that released from the site
is partly dissipated and most of it transferred isotropically to the neighbors. Let a < 1 be the
part of energy that is going to the neighbors, so if at the moment t the temperature of the site
was Ti at the next time step the site temperature would be
Ti (t + 1) = Ti (t) − Ji (t) + (a/2)(Ji−1 (t) + Ji+1 (t)) + η(i, t).
(3.2)
The last term reflects the possibility that the site will get input energy from the driving with the
average hηi = pq, the distribution of the random heat η will be usually taken as uniform if not
specified otherwise. Now it is easy to see that the flux can be properly rewritten as
Ji (t) = kTi (t)[θ(Ti (t) − Tc ) + θ(Tc − Ti (t))θ(Ti (t) − T0 )θ(Ji (t − 1))],
(3.3)
The two equations (3.2) and (3.3) completely determine the model. It is worth to reiterate
that the proposed model has the following features which are usually absent (or incomplete) in
other models: a) the active-passive transitions depend only on the local conditions, that is, the
temperature of the site and the energy release of the same site determine whether it is active or
passive, b) a site which becomes active does not fade away immediately once it transfers energy
to neighbors, but lives for some time, and c) there is some dissipation along with the energy
flow inside the system. These features make the system somewhat resembling the current sheet
with localized reconnection.
3.1.2
Field presentation
While discretization is the natural and only way for performing numerical simulations, in reality
there is nothing which requires to break a continuous system into a number of discrete sites,
although some minimal scale is always present in physical systems. In this section we transfer
our model to a field model. We introduce the respective temporal and spatial reference lengths,
τ and l. Thus, the transfer to the field presentation would be t + 1 → t + τ , and i ± 1 → x ± l.
Now we can rewrite the temperature equation:
τ
al2 ∂ 2 J(x, t)
∂T (x, t)
= −(1 − a)J(x, t) +
+ η(x, t).
∂t
2
∂x2
(3.4)
Respectively, the equation for the flux is
J(t) = kT [θ(T (t) − Tc ) + θ(Tc − T (t))θ(T (t) − T0 )θ(J(t − τ ))].
(3.5)
44
CHAPTER 3. DYNAMICS OF THE BURNING MODEL
We require that in the interval T0 < T < Tc all the variables like the temperature and the flux
be changing continuously, so that we can Taylor expand to get
τ J˙ = −J + k(T + τ Ṫ )[θ(T + τ Ṫ − Tc ) + θ(Tc − T − τ Ṫ )θ(T + τ Ṫ − T0 )θ(J)
(3.6)
or
˙
J = kT [θ(T − Tc ) + θ(Tc − T )θ(T − T0 )θ(J − τ J)].
(3.7)
Since the discontinuous θ-functions can be substituted with tanh:
x 2n+1 1
θ(x) = 2 1 + tanh
L
where L is sufficiently small and the integer n ≥ 0 is sufficiently large, the derived field equations can be written in an explicitly smooth way.
3.1.3
Analytical treatment
In this section we would like to present a calculation of the cluster size distribution. Let Nav
be the number of active sites, in the case for weak to moderate driving. The temperature of a
burning site decreases from Tc to T0 , so that we can estimate that the energy, which is released
in the burning process from a single site, equals to Tc − T0 in the time interval tl . The average
power is Pav ≈ (Tc −T0 )/tl . The total energy losses rate is dE/dt ≈ −(1−a)Nav (Tc −T0 )/tl +
(dE/dt)b , the last term describe the energy that is lost from the boundaries. In a sufficiently
large system the effect of the boundary losses can be neglected in all cases. We examine our
system (our numerical results) in the stationary state, that is when the energy that enters the
system is equal to the total energy losses. In this case we get
Nav ≈
Lpqtl
.
(1 − a)(Tc − T0 )
(3.8)
when Lpq represent the input energy. This approximation should be valid for Na v L, so that
pqtl (1 − a)Tc .
(3.9)
To get a more precise prediction we will look into kinetic description of the cluster distribution. Let N (w) be the number of clusters with length w. The evolution of the cluster number in
time is given by
dN (w)
= −[P (w → w + ∆) + P (w → w − ∆)]N (w)
dt
+ P (w − ∆ → w)N (w − ∆) + P (w + ∆ → w)N (w + ∆),
(3.10)
3.1. ONE DIMENSIONAL MODEL
45
where P (w → w + ∆) and P (w → w − ∆) are the probabilities (per unit time) of growth and
shrinking, and ∆ is the typical change of length in one step. In our case ∆ = 1 or ∆ = 2, that
means, the cluster can grow at the end of the cluster or at both ends (this is one dimensional
model). Since we are interested in estimates only and will not solve the (discretized) kinetic
equation (3.10) exactly, we simply put ∆ ≈ 1. We proceed by Taylor expanding to obtain
dN
∆2 d2
d
=
((P+ − P− )N ) ,
((P+ + P− )N ) − ∆
2
dt
2 dw
dw
(3.11)
where P+ and P− are the growth and shrinking probabilities, respectively. In the stationary
state, (dN/dt) = 0, one has
∆ d
((P+ + P− )N ) = ((P+ − P− )N ) + C,
2 dw
(3.12)
where C = const is the probability of the spontaneous appearance of a cluster. Since only
clusters of size one are born from the passive background, we have to put C = 0. Then
A
N (w) = exp(−
α
Z
βdw),
(3.13)
where A = const, α = P+ + P− , and β = (P− − P+ )/α.
Now we would like to describe the growth and shrinking probability. First we introduce
the average temperature Tp , we can say that this is also the average temperature of a passive
site if we assume that Na v L. The heat flux that transfers from a site that exists on the
cluster boundary to its passive neighbor is J = kT during about its life time. The passive site
becomes active if the heat flux exceeds the difference between the critical temperature and the
site temperature, which can happen during the time tg when J > (Tc − Tp ). Estimating the
initial temperature of the active site as Tc we have tg = (1/k) ln[kTc /(Tc − Tp )]. The growth
probability then will be P+ ≈ tg /tl and does not depend on the cluster size but depend on the
dynamics of the system like strong driving. If the driving is strong the mean temperature Tp
gets close to the critical temperature and, as we can see, the time tg during which that passive
site may become active depends on Tc − Tp . The shrinking probability is simply the probability
to find the active site at the boundary in the end of its life, so that P− ∼ 1/tl . It also does not
depend on the cluster size. We got α = (1 + tg )/tl and β = (1 − tg )/(1 + tg ), so that
N (w) ∝ exp[−(1 − tg )w/(1 + tg )] ⇒ ln N ∝ −w.
(3.14)
This result appears dependent on the average temperature Tp which itself should depend on the
system dynamics.
In order to establish the relation of Tp to the system parameters we should consider the
46
CHAPTER 3. DYNAMICS OF THE BURNING MODEL
smallest size, w = 1, clusters. For such clusters (3.11) is not applicable. Instead, we have
dN1
= −P− N1 + γN0 ,
dt
(3.15)
where the last term describes the spontaneous (driving determined) conversion of passive sites
into active ones. Since the average driving heat flux into a passive site is qp, and the gap
Tc − Tp should be exceeded to make the site active, we can estimate the birth probability γ ∼
qp/(Tc − Tp ). In the stationary state we would then have
N1
qpN0
=
.
Tc − Tp
tl
(3.16)
Since N (w) ∝ exp(−βw), we have
Z
L
N (w)dw ≈ N1 /β,
Nav =
(3.17)
1
Substituting (3.17) into (3.16), with the use of (3.8) and taking into account that N0 ≈ L, we
get eventually
β
Tc − Tp
=
.
(3.18)
Tc − T0
1−a
Since β itself depends on Tp , the relation (3.18) is in fact a (nonlinear) bootstrap equation for
the average temperature.
As a by-note, the above analysis allows to predict the maximum size of the cluster. Indeed,
N (w) = N (1) exp[−β(w − 1)] ≈
≈
Nav
exp[−β(w − 1)]
β
Lpqtl
exp[−β(w − 1)],
β(1 − a)(Tc − T0 )
(3.19)
and one has N (wmax ) = 1, so that
wmax ≈
1
Lpqtl
ln
.
β β(1 − a)(Tc − T0 )
(3.20)
Of course, this relation (as all previous) is of approximate character only. It is clear, however,
that the average number of active sites depends on the driving more strongly than the maximum
cluster size.
3.1.4
Numerical results
One of the objectives of our research is to propose an avalanching model that would describe
the reconnection phenomena more properly, in our opinion, then the avalanching models that
3.1. ONE DIMENSIONAL MODEL
47
are already suggested. Another objective is to find observable characteristics of avalanching
systems that will enable us to map those system to one of our avalanche models (sandpile or
burning) so we will able to learn about the micro process of those system by remote observer
who is limited in his methods of observational data processing. In order to compare these two
models we are going to perform analysis similar to what we have done in the case of the one
dimensional sandpile model.
Until now we focused on the description of the model and analytical treatment. For such
kind of models, however, existing analytical methods are not sufficiently developed yet. On the
other hand, numerical methods proved to be efficient. Our numerical model uses the following
parameters: the length of the system L = 100, the critical temperature Tc = 50, the low critical
temperature T0 = 0.3Tc , the fraction of energy release going to neighbors a = 0.9, the inverse
relaxation time k = 0.3, and the heat amount input at each step q = 2. The driving strength
is then determined by the probability of the input p. In our analysis we examine the system
for various drivings and fractions of energy released that is going to the neighbors. Figure 3.1
shows how the energy release pattern changes with the driving increase by a factor of ten. For
the driving probability p = 0.005 (bottom panel) the system seems to be active most of the time.
For lower driving (upper panel) the system is essentially passive with rare bursts of activity.
Although the difference in the activity level is quite clearly seen visually from Figure 3.1, it
is instructive to introduce quantitative tools of comparisons. Let J(t, i) be the two-dimensional
P
array of intensities. Then Jav =
i,t J(t, i)/(LNt ), where Nt is the number of time steps,
would have the meaning of the mean site intensity. The mean radiated energy would be (1 −
P
a)Jav . Respectively, nav = i,t θ(J)/(LNt ) would give the mean fractional number of active
sites. We remind the reader that the mean energy input per site is qp. In the stationary regime
(1 − a)Jav ≈ qp (the equality cannot be precise because of the losses at the boundaries - see
complete analysis in section 3.1.3). The results of this comparison are given in Table 3.1, where
we list the following parameters: p is the driving probability, qp is the average driving input
per site, Jav is the average power released by active sites, nav is the mean fractional number
of active sites, wmax is the largest cluster size (from all cluster sizes measured at all times),
and tmax is the longest avalanche duration (measured for all sites). The last two are defined
as follows. First we determine the largest cluster size max(w(t)) for each time t and then
wmax = maxt (max(w(t))). The longest duration max(t(i)), on the other hand, is determined
for each site separately, and then tmax = maxi (max(t(i))). Thus, wmax corresponds to the timeaverage spatial pattern which an observer would see instantaneously, while tmax corresponds to
the spatially-average time evolution which an observer, sitting at some site, would see.
In Figure 3.2 individual avalanches are shown. The stronger driving case exhibits first indications of avalanche spatial merging (cluster merging). At stronger drivings such merging
would become more important, thus spoiling our analytical treatment. There is a very clear
48
CHAPTER 3. DYNAMICS OF THE BURNING MODEL
Table 3.1: Quantitative comparison of activity level.
p = 0.0005 p = 0.001 p = 0.005 p = 0.01
qp
0.001
0.002
0.01
0.02
Jav
0.011
0.026
0.12
0.24
nav
0.0013
0.0032
0.0145
0.029
wmax 15
18
21
22
tmax 32
37
37
36
picture of the change of the flux that get out from a site in the cluster as function of time.
For the analysis of the distribution N (w) the size of the system was increased to L = 400, in
order to reduce the effects of the edges. Figure 3.3 shows the distribution for various values of
the probability p. As expected, the maximum cluster size depends only weakly on the driving,
and the functional dependence ln N ∝ −w remains the same.
Figure 3.4 shows the behavior of the mean temperature. The plots are artificially shifted
since the dependence of the temperature on the driving is negligible. It is seen that the system
is in the stationary regime, since the temperature fluctuates around some constant value. With
increase of driving the fluctuations become more frequent.
The statistics for the passive and active phase duration are given in Figure 3.5. The distribution of the passive phase durations is Poisson in a wide range, as could be expected. The
Poisson nature of the PDF of passive phases suggests that the evolution of the burning model
as described here does not show any time correlation among the avalanche events. However,
further work is needed in order to investigate the emergence of time correlation as a function of
the driving strength and/or different updating rules including diffusion effects. The distribution
of the active phase durations deviates from Poisson toward smaller and larger durations.
3.1. ONE DIMENSIONAL MODEL
49
100
90
80
site number
70
60
50
40
30
20
10
0.2
0.4
0.6
0.2
0.4
0.6
0.8
1
time
1.2
1.4
1.6
1.8
2
x 104
100
90
80
site number
70
60
50
40
30
20
10
0.8
1
time
1.2
1.4
1.6
1.8
2
x 104
Figure 3.1: Energy release for various drivings: top p = 0.0005 and bottom p = 0.005.(One
dimensional model)
50
CHAPTER 3. DYNAMICS OF THE BURNING MODEL
45
16
14
40
site number
12
35
10
8
30
6
4
25
2
20
1.3
1.3005 1.301 1.3015 1.302 1.3025 1.303 1.3035 1.304 1.3045
time
x 104
50
16
48
14
46
12
site number
44
42
10
40
8
38
6
36
34
4
32
2
30
1.648
1.649
1.65
1.651
time
1.652
1.653
1.654
x 104
Figure 3.2: Individual avalanche structure: top p = 0.0005 and bottom p = 0.005.(One dimensional case)
3.1. ONE DIMENSIONAL MODEL
51
10 6
10 5
f
10 4
10 3
10 2
10 1
10 0
0
5
10
15
20
w
25
30
35
40
45
mean temperature
Figure 3.3: Distribution of the cluster sizes N (w) (log-linear scale) for p
0.00025, 0.0005, 0.001, 0.0025, 0.005.(One dimensional case)
time
Figure 3.4: Mean temperature for various drivings (shifted).(One dimensional case)
=
52
CHAPTER 3. DYNAMICS OF THE BURNING MODEL
10 0
10 −1
pdf
10 −2
10 −3
10 −4
10 −5
0
20
40
60
80
100
active time duration
120
140
160
10 −1
pdf
10 −2
10 −3
10 −4
10 −5
0
50
100
150
200
250
passive time duration
300
350
Figure 3.5: Duration (pdf) of the active (top-log linear scale) and passive (bottom-log linear
scale) phases.(One dimensional case)
3.2. TWO DIMENSIONAL BURNING MODEL
3.2
53
Two dimensional burning model
It is natural to extend this model to more realistic, that is, two dimensional systems. The main
idea is the same and the only change is that now an active site transfers energy to its four nearest
neighbors and not only two as in the previous model.
3.2.1
The model
In this system we also have open boundaries, and the process a site passing is still the same, that
is, a site starts to burn when Ti,j ≥ Tc and loses energy in this burning process at the rate Ji,j =
kT , so that its temperature is changing in time according to T = T (0)exp(−kt). The burning
ceases when the site temperature drops below the lower critical value Ti,j < T0 = sTc , s < 1.
So, if at the moment t the site temperature was T (t) at the next step its temperature would be:
Ti,j (t + 1) = Ti,j (t) − Ji,j (t) + (a/4)(Ji−1,j (t) + Ji+1,j (t) + Ji,j−1 (t) + Ji,j+1 (t)) + η(t). (3.21)
The rate of energy loss would be
Ji,j (t) = kTi,j (t)[θ(Ti,j (t) − Tc ) + θ(Tc − Ti,j (t))θ(Ti,j (t) − T0 )θ(Ji,j (t − 1))].
(3.22)
As we can see this equation is similar to the one dimensional one. This similarity is quite
obvious, since the rate of the energy loss depends only on the site itself and not on its neighbors.
3.2.2
Field presentation
Similarly to what has been done earlier for the one-dimensional model, here we represent the
two-dimensional model in the continuous form, following the same reasoning as earlier. The
temporal and spatial reference scales, τ and l, substitute t + 1 → t + τ and i ± 1 → x ± l,
j ± 1 → y ± l. Now we can rewrite the temperature equation as follows:
τ
al2 ∂ 2 J(x, y, t) ∂ 2 J(x, y, t)
∂T (x, y, t)
= −(1 − a)J(x, y, t) +
(
+
) + η(x, y, t).
∂t
4
∂x2
∂y 2
(3.23)
As we mentioned and explained earlier, the equation for the flux does not change when we
increase the dimensionality of the model, so, like the one dimensional model, we would get for
the interval T0 < T < Tc the same flux description
˙
J = kT [θ(T − Tc ) + θ(Tc − T )θ(T − T0 )θ(J − τ J)].
when θ(x) = 1 if x ≥ 0 and zero otherwise.
(3.24)
54
3.2.3
CHAPTER 3. DYNAMICS OF THE BURNING MODEL
Numerical result
We start with illustrating the avalanching process by presenting avalanche patterns for various
driving. We use the following parameters: length of the system L = 100 (the size is L × L),
upper critical temperature Tc = 30, low critical threshold T0 = 0.3Tc , fraction of energy release
going to neighbors a = 0.9 or a = 0.97 (that is in order the check the influence of this parameter
on the results), inverse relaxation time k = 0.3, and the heat amount input at each step q = 0.5.
The driving strength is then determined by the probability of the input p. Figure 3.6 show the
passive and active phase duration for input probability equal to 0.0001 that is high driving.
Fraction of energy that going to the neighbors equal to 0.9. In the top panel we can see that
PDF − passive time
0
10
−1
10
−2
pdf
10
−3
10
−4
10
0
50
100
150
passive phase duration
200
250
PDF − system active time
0
10
−1
10
−2
pdf
10
−3
10
−4
10
1
10
2
10
active phase duration
3
10
Figure 3.6: Passive (log-linear scale) and active (log-log scale) phase duration for high input
probability and weak dissipation.(Two dimensional case)
3.2. TWO DIMENSIONAL BURNING MODEL
55
the passive time behaves like Poisson distribution while the active times behave like a power
law. In the active time panel (bottom) we can see the change of the slope in the transition from
short avalanches to longer ones (this transition can be seen more clearly in the next Figure 3.7).
The mean value for the active phase duration is Ta = 19 and the mean passive time is Tp = 23
(we can say that the system is active for about 82% of the time, that is, the driving is not weak
and we should not be close to SOC, according to the widely-accepted criterion). Figure 3.7
shows the results of the runs with various driving probabilities and dissipations. In the passive
phase duration plot, there is merging of the curves that describe the same input probability
(cyan - p = 0.0003a = 0.9, green - p = 0.0003a = 0.97, red - p = 0.00003a = 0.9 and blue
p = 0.00003a = 0.97), without any relation to the dissipation parameter. It seems very natural
that the passive time has no dependence on the dissipation, because this parameter only changes
the amount of energy in the system, that is changes the active time (makes it longer). The higher
the input probability the smaller is the system passive time and the system is more active.
We would like to check this conclusion in one more way, looking at the mean passive time
parameters. The mean passive time for p = 0.0003a = 0.9 is Tp = 11, for p = 0.0003a = 0.97
is Tp = 13, for p = 0.00003a = 0.9 is Tp = 79 and for p = 0.00003a = 0.97 the mean passive
is Tp = 65. We can see that the passive times for the same input probability are very similar in
the case of different dissipation parameters.
In the active phase duration plot we can see the influence of the both input probability and
dissipation on the activity of the system. The higher the input probability the more active the
system is (for p = 0.0003a = 0.9 the mean value is Ta = 29, p = 0.0003a = 0.97Ta = 77,
p = 0.00003a = 0.9Ta = 17 and p = 0.00003a = 0.97Ta = 26), and the weaker the dissipation
the more active the system is (more energy stays in the system). We also can say that the system
distribution behaves like a power law and the transition between the two slopes is more clear.
56
CHAPTER 3. DYNAMICS OF THE BURNING MODEL
PDF − passive time
0
10
−1
10
−2
pdf
10
−3
10
−4
10
50
100
150
passive phase duration
200
250
PDF − active time
0
10
−1
10
pdf
−2
10
−3
10
−4
10
1
10
2
10
active phase duration
Figure 3.7: Passive (log-linear scale) and active (log-log scale) phase durations for various input
probabilities and weak dissipation.(Two dimensional case)
While the fit is quite good for the first two simulations, it seems to me that in the case
of the model #3 we are observing the emergence of a certain characteristic dimension
3.3. SUMMARY
57
of cluster size. To compare the 3 different distributions I have collected all the PDFs
The cluster size distribution is shown in Figure 3.8: Mod 1 and mod 2 have the same dis-
in one single figure.
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ResultsFigure
seem3.8:toThe
becluster
independent
on for
driving
but strongly
dependent on
size distribution
various changes
input probabilities
and dissipation.(Two
dimensional case). Courtesy Consolini.
dissipation factor.
Now I
sipation parameter a = 0.9 while in mod 3 (blue line) the dissipation is a = 0.93. We can see
that the cluster size distribution is independent of the driving changes but strongly depends on
am
simulatingWeoncana large
see whatof happens.
the dissipation.
see alsomatrix
that theto
distribution
the cluster size changes from power
law distribution for small sizes to the Poisson for larger sizes. If we compare these results to
the one dimensional burning model we can say that the cluster size distribution may provide
information about the dimensionality of the system.
3.3
Summary
In this chapter we examined the influence of the dimensionality on the behavior of typical
distributions (avalanche size, active and passive phase durations) in the burning model. The
main shapes of the passive phase duration do not change noticeably when the dimensionality
is increased (from one-dimensional to two-dimensional). It can be seen that increasing the
system dimension changes the active time distribution from Poisson in the one dimensional
case to the power law in the two dimensional case. In the cluster size we can see that changes
in the dimensionality create a substantial change in the distribution. The results for the onedimensional model have been published in Gedalin et al. (2005c,d). The results for the twodimensional model are preliminary and require further analysis to understand the details of the
transition and the relative role of the dimensionality and internal dynamics.
Chapter 4
A comparative study
In this section we analyze the two different simple avalanching models that were introduced in
the previous chapters from a point of view of a remote observer who is limited in his methods
of observational data processing. The underlying physical processes are very different in those
two systems. We are interested to know whether the statistical properties are similar or there is
clear manifestation of the difference in the microphysics.
4.1
One dimensional models
The above results already show that there are differences in the remote observation of both
systems. In order to make it more clear we visualize the distribution of passive and active phase
duration for the both models together. For this comparison we choose the low state, p = 0.00005
bi direction sandpile model and the low state p = 0.0003, strongly dissipative, a = 0.9, burning
model. For both models the system is active for about 10 percent of the time. The mean active
and passive phase duration for the sandpile are Ta = 21 and Tp = 176. The mean active phase
duration for the burning model, Ta = 11, is strongly affected by the lower limit ≈ 3 on the
avalanche lifetime, so,in order to insure proper visual comparison, we truncate the distribution
from below, excluding from the analysis the shortest avalanches (which are most probable). In
real system observations such short avalanches would be most probably filtered out because of
the fluctuating background (noise). With this truncation, the renormalized mean values for the
burning model are Ta = 23 and Tp = 145, which is pretty similar to those for the sandpile.
Thus, for the chosen parameters, both models exhibit similar level of activity.
In Figure 4.1 we compare the sandpile model (stars) and burning model (circles). While
the two distributions seem to well coincide for shorter avalanches, at longer values the sandpile
model clearly leaves the Poisson curve toward a power law slop.
Figure 4.2 show the passive phase duration In this figure both models behave similarly
except for the longest avalanches where the statistics becomes poor.
58
4.1. ONE DIMENSIONAL MODELS
59
100
pdf
10−1
10−2
10−3
10−4 −1
10
10 0
10 1
active phase duration
10
15
20
active phase duration
10 2
100
pdf
10−1
10−2
10−3
10−4
0
5
25
30
Figure 4.1: Active phase duration for the case of low driving, shown in two different presentations: log-log scale (top panel) and log-linear scale (bottom panel). Distributions for the
sandpile model are marked with stars, those for the burning model are marked with circles.(One
dimensional case)
60
CHAPTER 4. A COMPARATIVE STUDY
100
pdf
10−1
10−2
10−3
10−4
0
2
4
6
8
passive phase duration
10
12
Figure 4.2: Passive phase duration for the case of low driving, sandpile (stars) and burning
(circles) models in log linear scale.(One dimensional case)
4.2. TWO DIMENSIONAL MODELS
4.2
61
Two dimensional models
The two-dimensional results of the active and passive phase duration for both burning model
and sandpile model are gathered to one system of axes in order to visualize the differences
between these two models (similarly to what we have done for the one-dimensional case). We
present the results for these two systems in the case of low driving, that is, the system is active
for about 20% of the time. The sandpile results (red curve) are connect with system probability
p = 0.000005 and the burning results (cyan curve) are connect with system values p = 0.00003,
a = 0.9. The mean active and passive phase duration for the sandpile are Ta = 14 and Tp = 80
and the mean values for the burning model are Ta = 17 and Tp = 80. Both plots are normalized
with the mean values.
In Figure 4.3 we compare the active time for the sandpile model (stars) and burning model
(circles) in log-log scale. As we know from the results in Chapters 2 and 3, both distributions
PDF − active time
0
10
−1
pdf
10
−2
10
−1
10
0
1
10
10
active phase duration
2
10
Figure 4.3: Active phase duration for the case of low driving for the sandpile (stars) and burning
(circles) models in log log scale.(Two dimensional case)
behave like a power law distribution for at least intermediate and large avalanches, although
there is some difference between the two distributions. The active phase duration distribution
for the burning model exhibits some kind of a transition between two slopes which occurs at
intermediate values of the durations. At present the nature of this transition is not clear to us. It
does not seem to be directly related to the finite lifetime of a burning site nor to the size of the
system. This issue requires further investigation.
Figure 4.4 shows the passive phase duration. Like in the one dimensional model the passive
time durations do not help us to distinguish between the two models (as expected).
62
CHAPTER 4. A COMPARATIVE STUDY
PDF − passive time
0
10
−1
pdf
10
−2
10
0
5
10
15
passive phase duration
20
25
Figure 4.4: Passive phase duration for the case of low driving. Sandpile-stars, burning-circles
in log linear scale.(Two dimensional case)
4.3
Summary
In this chapter we presented the results of the comparison of simple statistical properties of the
two physically different systems. These statistical properties should be available to a distant
observer. In the one dimensional case it can be seen very clearly that there is a significant
difference between these two models. While in the sandpile model the active phase duration
distribution behaves like a power law distribution, in the burning model the active phase duration distribution behaves like Poisson distribution. In the two-dimensional case the situation
changes. The difference between the systems is lost and in both systems the active phase duration distribution behaves like a power law, although there is still some difference in that the
burning model shows transition between two power laws when moving from short durations to
longer ones. In both cases, one and two dimensional, the passive phase duration behaves like
Poisson distribution and it is not possible to distinguish between the two models from the passive time observations only. The results obtained for comparison of one-dimensional models
are published in Gedalin et al. (2005d). Two-dimensional comparison is in progress.
Chapter 5
Comparison with observations
In the previous chapters we presented analytical and numerical treatment of two different avalanching models. Models, however, remain just a toy unless there is application to real physical systems. As we mentioned earlier, one of our objectives is to find a relation between observations
that can be made remotely (from large distance or without possibility to analyzed the interior of
the system) and the physical microprocesses in the system. In order to examine the applicability
of our results, a comparison is desirable of the found numerically distribution behavior of the
passive and active phase duration with observations of some real physical systems. It appears
that most of the existing observations are not the measurements the proposed kind. Because
of that it appeared difficult to find in literature a description of physical system observations
that would provide us with both passive and active phase durations. Yet we able to find two
examples of observations where (separately) active and passive phase durations were measured.
The first example is the avalanching system of the current sheet of the Earth magnetotail.
The solar wind is the trigger (driving) for this system and the results (output) is the auroral
activity. This output is measured by the chain of ground-based stations that are located at
the base of the auroral oval. The auroral activity is represented as deviations of the magnetic
field strength from the daily averaged value. The distribution of the durations of the bursts of
the electrojet index AU (maximum positive disturbance) have been studied by Freeman et al.
(2000). This distribution is presented in Figure 5.1. It can be considered as the analog of
our active time durations. A set of thresholds was applied that corresponded to the 10, 25,
50, 75 and 90 percentiles of the cumulative probability distribution of the measurement, AU.
This thresholding was used in order to separate the background noise, which is present in all
physical systems, from true avalanching activity. Several thresholds were used in order to verify
the priori assumption that the power law slopes are independent of the choice of threshold.
63
64
1088
CHAPTER 5. COMPARISON WITH OBSERVATIONS
FREEMAN ET AL.: LIFETIMES OF AE IN
Figure 1. Burst lifetime PDFs for AU , |AL|, vB
Figure 5.1: Burst lifetime PDF for AU. From Freeman et al. (2000).
slopes are independent of the choice of threshold. The cor-
Figure 5.1 is to be compared with Figure 3.7 (bottom panel), which shows similar power
responding burst lifetime PDFs are shown in Figure 1. The
law for the active phase durations
in the two-dimensional
burning
model.
major component
of all the
PDFs
is a power law function
over 1-3 decades from just above the smallest measurable
lifetime. The power law range varies systematically with
the threshold but the power law exponent is approximately
independent of threshold.
One can fit a power law to the whole AE burst lifetime
PDF [Takalo, 1993; Takalo, 1999] but Consolini [1999] has
recently argued that the PDF is fitted better by the sum of
a power law with an exponential cut-off plus a lognormal.
Defining the burst lifetime PDF, D(T ), as the probability
of burst lifetimes in the range T to T +dT divided by the
interval dT , the PDF of a power law with an exponential
cut-off plus a lognormal is:
D(T )
=
+
A
−T
exp
α
T
Tcut
#
$
−(log T − µ)2
B
√
exp
2σ 2
2πσT
!
"
(1)
Figu
PDF
74 n
Bot
func
and
erro
cont
of a
T
bur
fit o
lifet
m−1
exp
an e
pow
of A
I
mag
lifet
m & 16. In Fig. 1, the raw and smoothed signals are
within the plasma. The
compared. Clearly, in the latter, most LFEs have merged
scarce, so that survival
into a rather continuous band on top of which clearly
stead of pdfs. The surseparated longer events prevail. This band can now be
y of interest q gives the
eliminated by amplitude thresholding. Before proceedur case, q will be either
65
ing, it must be clarified that the AVEs discussed here are
Therefore, q & 0 and
found within the last closed magnetic surface (LCS), in
al data, Sq "s# is easily
contrast to evidence
the so-called
found
in the
scrape-off
d values of q in decreasThe other experimental
is for‘‘blobs’’
the passive
phase
durations
that were measured in
layer
[19].plasmas
While the
connection,
if it exists,
be-experiments turbu; then a rankthenumber
is
experiments
on hot(SOL)
Tokamak
(Sánchez
et al., 2003).
In these
tween them is not clear, the physics governing their time
n T, q(k , by using rk !
lent electric
fluctuations,
of instabilities
the hot plasma, were
may bewhich
very developed
different, because
since SOL
magnetic in
field
er of appearances
of q(k fieldscales
as a function
of open.
time. The electric field was found to be of a bursty character, and the
lines are
ion is thenmeasured
given by:
q
Results
W7-AS
will be
presentedflux”.
first (discharge
o the pdf by
p
"s#
!
fluctuations were describedfrom
in terms
of ”bursts
of turbulent
The time between two subseNo. 35427 [5]). The signal has been sampled at %s !
ries the same informaquent turbulent fluxes is the quiet time, according to Sánchez et al. terminology, and correspond
2 MHz and has 200 000 usable points. The probe tips
ential or power-law beto
our
passive
phase
Thecm
details
of the
micro-process
are not known although
are duration.
located 0–2
within
thephysical
LCS. Examples
of quieter to detect power laws,
time
survival Thus,
functions
with
the m !are
32available. The obthis is a laboratory
experiment.
again obtained
very limited
observations
smoothed
signal
areisshown
2. Clearly,
thelower
survival
=s1
served distribution
of the quiet
times
shownininFig.
Figure
5.2 by the
curve (the upper curve
function
obtained
without
selecting
the
bursts
according
;
(1)
=s2 #k
q;fit
"q(k #,2 ;
(2)
0
10
q
Survival function [S ]
s, Sq "s# ) s$k for scales
des of power-law behavo claim power-law bequired. Furthermore, to
n artifact of the type of
relaxing the condition
o a pure exponential (s2
st q(k and s1 gives an
) and it is similarly
rded. As goodness-ofe merit function [17]:
-1
10
-2
10
Power law
region
All bursts
included
-1.38
~q
Only bursts
lasting more
than 20 µs
-3
10
Exp. fit: e
-0.017q
t q values.
t be solved to compute
-4
10 -4
ntification of the events
-3
-2
-1
0
10
10
10
10
10
sing the probe location
Quiet time [ms]
will be convoluted and
ated with faster local
FIG. 2 (color online). Examples of quiet-time survival funcFigure 5.2:
timefor
survival
function
W7-AS shot No.35427. From Sánchez et al. (2003).
s). One approach
is toQuiettions
W7-AS
shot No.for
35427.
corresponds to the proposed mode of the data processing using substantial
185005-2thresholding and is
not related directly to our subject). The distribution of the quiet timess (passive phase durations)
behaves like Poisson distribution. This suggests that avalanches crossing by the probe location
do so randomly.
Chapter 6
Conclusions
In the present work we have studied two internally different but externally similar systems
possessing avalanching behavior. The two systems were the bi-directional sandpile (which is a
generalization of the classical sandpile model) and the newly proposed burning model. The two
systems differ by the underlying microphysics, but should be similarly measured by a distant
observer.
The systems were analyzed in the case of moderate (non-weak) driving in order to examine
the influence of the driving on the system behavior. One of the objectives was to understand
whether the widely-accepted definition of SOC as the behavior in limit of negligible driving and
total time separation is too restrictive for the description of avalanching systems. The driving
applied in our studied models was intermediate to strong, but in all cases we cared to not make it
to strong so that the systems are only a part of the time in active state. Strongly driven systems,
which are most of the time in active state, cannot be expected to exhibit behavior similar to
SOC.
We have found for both systems that driving intensity does not change the statistical behavior of the system, that is, statistical distributions of the observed parameters (see below).
The only change is in the mean parameters, like the time that takes to the system to move from
active to passive state. We conclude, therefore, that since changing the driving does not change
the statistical behavior, physically (or observationally) there is no much difference between the
system which is strictly self-organized critical (according to the popular definition) and system which is moderately driven. Thus, it makes sense to adopt the SOC terminology even for
systems with non-weak driving, provided they exhibit scale-free behavior.
Change the dimensionality of the system, on the other hand, results in the change of the
statistical behavior for the same measurement in both models, sandpile and burning as well.
Apparently, the conclusion whether a system behavior is scale-free, that is, whether, SOC sets
on, is dimensionality sensitive. It is possible that some of the earlier found power-law distributions are due to the systems dimensionality, in which case it is not clear what is the relation
66
67
to SOC and whether the last is determined by the avalanching mechanism, dimensionality, or
both.
In order to allow more or less reliable remote observations, we propose to measure the passive and active phase durations and the cluster sizes in both models. We believe that this three
basic measurements are the only ones (at present) that are not influenced substantially by the
medium between the observed system and the remote observer, and for now give us the information that helps us to diagnose the system. It is possible, however, that future studies will provide
us with more parameters to measure. As we saw, the passive phase duration distributions did
not differ for both models, not do they depend on the system dimensions. This is reasonably
based on the fact that this measure provides us with the information about the driving nature.
Avalanches and SOC systems have been studied for quite a while. Most of the studies are
based on numerical and analytical treatment on a simple one dimension sandpile model. We
generalized this by extending it to bi-directional sandpile model (thus eliminating the closed
boundary) and further proceeded to the two dimensional case. Our numerical results show that
the active phase duration distribution is power-law like and the passive time duration distribution
behaves like the Poisson. This results are consistent with the found earlier by Carreras et al.
(2002); Newman et al. (2002); Sánchez et al. (2002). In the two dimensional model we get
Poisson for passive phase duration and Poisson for short avalanches and power law for large
avalanches for the active phase durations. In order to improve the model and make it more
applicable for other physical systems, we introduced a non-constant redistribution mechanism
where the number of grains which is transferred in the relaxation step constitutes certain part
(15 percent in our simulations) of the height difference. In this simulation we got the same
results for the passive phase duration and for the active phase duration we get a shape that is
indicative of Poisson distribution. The power law distribution points to the fact that avalanche
can develop in any temporal and spatial length. In the last case of the sandpile model the active
phase duration behaves like Poisson, which points to the fact that larger avalanches are rare:
their appearance is suppressed by the enhanced grain transfer for steeper slopes.
Our second (burning) model is newly proposed. It is built in order to provide better relation
of the avalanching models to the reconnection phenomenon. The main difference between these
two models is that in the burning model the underlying processes are more physically related to
the continuous systems of the type of the current sheet, in the sense that is takes some time to the
energy to be transferred from site to site while this process is immediate in the sandpile model.
We have also included the dissipation which is unavoidable in real reconnecting systems due to
particle acceleration. In the burning model the site relaxation depends on the state of the site
itself (critical current is a local parameter for reconnecting systems) and not on the difference
between the site and its neighbors. This model has been also studied in the one-dimensional and
two-dimensional cases. In the one dimensional burning model we found Poisson distribution
68
CHAPTER 6. CONCLUSIONS
for the passive time durations, for the driving which also was random as in the sandpile model.
The active phase duration distribution is also Poisson-like, and not power-law. Thus, according
to the SOC criteria, the one-dimensional burning model is not self-organized critical. However,
in the two-dimensional case the active phase durations become power-law distributed, which is
usually considered as clear indication of scale-free dynamics and SOC. Thus, dimensionality
may play a crucial role in our interpretation of whether the system is self-organized critical or
not, even if the underlying microphysics remains the same.
On the basis of the numerical analysis we propose the active phase durations may be used (at
least in a number of cases) by a remote observer to distinguish between systems with different
microphysics. Observations of passive phase durations do not provide information about the
redistribution mechanism but could be, probably, useful in understanding of the driving features.
The last issue requires further research.
Besides the numerical analysis of the burning model we have treated the model analytically
using and developing further the novel method proposed in Gedalin et al. (2005a). The method
is based on the developing a ”kinetic” equation for cluster growth and shrinking. In comparison
with the previous analytical works, we have been able to predict not only the average cluster
size (like in the mean field theory) but also the statistical distribution of the clusters sizes. The
predictions are in good agreement with what has been found numerically.
In the spirit of this work, in the future research we plan to develop the burning model in
several directions. We are going to extend the analytical treatment onto the two dimensional
burning model and establish relation between the distribution of clusters and the active passive
phase duration. We propose to study the effects of external driving by allowing various nonuniform distributions and including also large disturbances. We also plan to add more physical
ingredients to the internal redistribution mechanism, other than the burning, such like weak
diffusion. We also plan to further investigate the issue of reliably remote measurements, in
particular, related to the cluster distribution.
The results of this work have been published in Gedalin et al. (2005c,d) and presented at
the Annual Meeting of the European Geophysical Union in Vienna, 24-30 April, 2005, by M.
Bregman (Gedalin et al., 2005b).
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