Self-organized criticality and observable features of avalanching systems. Michal Bregman

Transcription

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

2 Contents 1 Introduction Characterization of the SOC state Numerical modeling The basic sandpile model Running sandpiles Forrest-fire model Earthquakes model Analytical approach Renormalization group Mean-field approach Experiments Dynamics of sandpile Dynamics of rice pile Dynamics of water droplets Superconducting vortex avalanches Space plasma and magnetosphere Solar flares Current sheet and magnetic storms Observational relations Objectives Sandpile model One-dimensional bi-directional sandpile Numerical analysis Two dimensional sandpile model The model Numerical analysis Non-constant grain transfer Summary

3 CONTENTS 3 3 Dynamics of the burning model One dimensional model The model Field presentation Analytical treatment Numerical results Two dimensional burning model The model Field presentation Numerical result Summary A comparative study One dimensional models Two dimensional models Summary Comparison with observations 63 6 Conclusions 66

4 List of Figures 1.1 One-dimensional sandpile Forest-fire model Sandpile experiment Rice pile experiment Experiment with water droplets Chain reaction in water droplet experiment Solar flares as avalanches in the loop network Illustration of avalanching magnetotail Sandpile activity Number of active sites as a function of time.(one dimensional case) Distribution of clusters size for the high state Distribution of active and passive phase durations Distribution of active and passive phase durations for different drivings Distribution of clusters size for the high state D sandpile: distribution of active and passive phase durations D sandpile: distribution of active and passive phase durations for different drivings D sandpile: distribution of cluster sizes Distribution of active phase duration for the constant grain number transfer case and for the proportional transfer case Distribution of passive phase durations for the constant grain number transfer case and for the proportional transfer case Distribution of cluster sizes for the constant grain number transfer case and for the proportional transfer case in log linear scale Energy release for various drivings Individual avalanche structure Distribution of the cluster sizes Mean temperature for various drivings

5 LIST OF FIGURES 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 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 Burst lifetimes for electrojet index Quiet times for electrostatic bursts

6 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

7 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 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 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, h n, 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 Z n = 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: h n = h n + 1 and no change for other cells, or Z n Z n + 1 (1.1) Z n 1 Z n 1 1 (1.2)

9 1.2. NUMERICAL MODELING 9 When the height difference reaches some given critical value Z c, the site is relaxed and one grain of sand moves to the lower site: Z n Z n 2 (1.3) Z n±1 Z n±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: Z 0 0 (1.5) Z N Z N 1 (1.6) Z N 1 Z N (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 Z n are below the critical value, Z n < Z c. 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 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 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) 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

11 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 = and L = 800. Trees are grey, burning trees are black, and empty sites are white. Figure 1.2: Snapshot of the forest fire model in the steady state. Trees are gray, burning trees are black and empty site is white. From Clar et al. (1994). g = g C (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 g C (p) and near g C (p). At g C (p), the fire just percolates through the system. Since sites are not permanently immune, this kind of percolation is different from usual site percolation and is in the same universality class as directed percolation in d + 1 dimensions (the preferred direction corresponds to the time) [22]. The critical immunity g C (p) increases with increasing p, since the fire then can return sooner to sites where it already has been. Above g C (p) the steady state of the system is a completely dense forest. A similar model, using the language of spreading diseases has been studied independently in [23]. Consolini and De Michelis (2001) proposed a modifies forest model named Revised Forest Fire Model (RFFM). Their simple model was introduced in order to simulate the occurrence of sporadic localized relaxation phenomena (current disruption events) in the geotail neutral plasma sheet. The model consists of a two dimensional lattice and involves periodic boundary conditions. Each site is in one of the following states that we characterized before: site with 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 set the immunity to zero, but we will show later that the SOC state persists for g > 0. The ratio p/f is a measure for the number of trees growing between two lightning strokes and therefore for the mean number of trees destroyed per lightning stroke. In the limit plasma conditions are locally stable, while the site with the trees are locally unstable regions, and the burning sites are connected to those regions where a relaxation phenomenon is taking 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 model is SOC when tree growth is so slow that fire burns down even large clusters before the following rules: 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 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 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

13 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 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 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 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 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 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

15 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 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 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 16 AEGERTER, GÜNTHER, AND WIJNGAARDEN CHAPTER 1. INTRODUCTION Figure 1.4: AFIG. schematic 1. A schematic image of theimage setup. of Thethe distribution setup. The board distribution can be seenboard on top, where rice is dropped can befrom seena on single top, point where and rice subsequently is dropped divided fromintoa even single compartment. point and Within the wooden subsequently box bounding divided the riceinto pile. even From Aegerter compartments. et al. (2003). Within the wooden box bounding the rice pile, a reconstruction of its surface is shown, a systemas with the is used dynamics in further of avalanching analysis. type in continuously driven water. The experimental FIG. 2. A 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 of the reaches pile and the critical the size mass, ofatthe which avalanches, time it runs down is discussed other stationary in Sec. droplets II. Thein avalanche its path and thus size creating distributions a chain reaction. and their We can see a the surface, The dark spo triggering snapshotfinite of suchsize developing scaling reaction are in presented Figure 1.6. in In this Sec. experiment III A, together they changed with the viscosity of the the droplet determination and also the rateof of the entering critical water exponents. that is the analog In Sec. to the IIIslow B, driving, the that analysis. Th and blue, is, driving surface the system roughness toward theisthreshold analyzed of instability and theasnecessary in avalanches techniques study. Their results age surface are shown arethat briefly the distribution introduced. function Also, of the inavalanche this section, size and theavalanche scalinglifetime relations law. between Furthermore, it roughness was shown (for exponents the first time) andthat the the critical distribution expo- of the time behave tion of proj like a power Fig. 2, whe betweennents successive areavalanches introduced. behaves These like Poisson resultsdistribution. are also put It was into found a wider that higher viscosity provided context a larger and range compared of power with law behavior, the results and that obtained increasing from the flow KPZ rate led extracted fr in to this way roughening systems 27. face can be exponential decay of distribution, that is, there are characteristic length and time scale (no self reconstruct organized criticality). field of vie II. EXPERIMENTAL DETAILS mm, as we The experiments were carried out on long grained rice both structu with dimensions of typically mm 3, similar to rice curacy wer A of Ref. 12. The pile was grown from a uniform line is roughly m source that is 1 m wide. This uniform distribution was well suited achieved via a custom built mechanical distributor based on In a sing a nail board producing a binary distribution 28. The actual period of setup consists of a board with an arrangement of triangles, as experiment

17 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 18 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 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.

19 quiescent time interall characterized by understanding and computational modeling of local re- have generated substantial progress in the theoretical distribution is par- connection in laboratory, space, and solar plasmas. -free behavior over However, it remains computationally prohibitive to use [5]. These statistics the equations of plasma physics to describe extended be in a state 1.5. ofspace self- PLASMA AND MAGNETOSPHERE 19 h a view implies that ll transient brightene not fundamentally avalanches of reconics of solar coronal res with other inters earthquakes, forest ce [10,11]. model of multiple driven at their foototpoints, of opposite to a two-dimensional re. Footpoints of the gation on the surface. ootpoints when they nection, can lead to trigger a cascade of es of magnetic loop r flares FIG. 1 (color online). Snapshot of a configuration of loops in Figure 1.7: Snapshot the steady of a configuration state with L of200 loops and in mthe steady 1 (seestate. text). From Hughes et al. (2003) )=131101(4)$ The American Physical Society 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 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) Observational relations For models and laboratory experiments the identification for SOC or SOC-like is more simple 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

21 1.7. OBJECTIVES 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 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.

23 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 N d 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 J in = pln d. Activity of each site is determined by the local slope Z(i) = max(n(i) N(i 23

24 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) > Z c, the site becomes active and collapses to one or both its neighbors. This collapse is characterized by the transfer of 2N f 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) Z c ) + θ(n(i) N(i 1) Z c ) > 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) 2N f (2.2) N(i + 1) [N(i + 1) + 2N f ]θ(n(i) N(i + 1) Z c )θ( N(i) + N(i 1) + Z c ) + [N(i + 1) + N f ]θ(n(i) N(i + 1) Z c )θ(n(i) N(i 1) Z c ) (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 x 1 = N d L 2 p/n f (overlapping is not negligible when x 1 1) and the measure of spatial merging of clusters x 2 = N d Lp (merging is not negligible is x 2 1). It has to be noted that x 1 assumes that the average avalanche length is L, otherwise x 1 should be reduced by the factor L/average avalanche length 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