25 Years of Self-Organized Criticality: Solar and Astrophysics
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DOI: 10.1007/s11214-014-0054-6
- Cite this article as:
- Aschwanden, M.J., Crosby, N.B., Dimitropoulou, M. et al. Space Sci Rev (2016) 198: 47. doi:10.1007/s11214-014-0054-6
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Abstract
Shortly after the seminal paper “Self-Organized Criticality: An explanation of 1/fnoise” by Bak et al. (1987), the idea has been applied to solar physics, in “Avalanches and the Distribution of Solar Flares” by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme.
Keywords
Instabilities Methods: statistical Sun: flare Stars: flare Planets and satellites: rings Cosmic raysAcronyms
- 1D, 2D, 3D
1-, 2-, 3-dimensional
- ACE
Advanced Composition Explorer (spacecraft)
- AE
Auroral Electron jet index
- AGILE
Astro-Rivelatore Gamma a Immagini LEggero (spacecraft)
- AGN
Active Galactic Nuclei
- AIA
Atmospheric Imaging Assembly (on SDO)
- AlMg
Aluminium-Magnesium filter (on Yohkoh spacecraft)
- AU
Auroral Upper geomagnetic index
- AU
Astronomical Unit (Sun-Earth distance)
- BATSE
Burst And Transient Source Experiment (on CGRO)
- BC
Box Counting (fractal dimension)
- BCS
Bent Crystal Spectrograph (on SMM)
- BTW
Bak, Tang, and Wiesenfeld (SOC model)
- COBE
COsmic Background Explorer (spacecraft)
- CGRO
Compton Gamma Ray Observatory (spacecraft)
- CLUSTER
Magnetospheric mission with 4 spacecraft
- CME
Coronal Mass Ejection
- DCIM-P
Decimetric pulsation radio burst
- DCIM-S
Decimetric spike radio burst
- DNL
Distant Neutral Line (in geotail)
- EIT
Extreme ultra-violet Imager Telescope (on SOHO)
- EM
Emission Measure
- ESA
European Space Agency
- EUV
Extreme Ultra-Violet
- EUVE
Extreme Ultra-Violet Explorer (spacecraft)
- EXOSAT
European X-ray Observatory SATellite
- FD-SOC
Fractal-Diffusive Self-Organized Criticality model
- Fermi
Hard X-ray spacecraft
- FSOC
Forced Self-Organized Criticality
- GEOTAIL
Magnetospheric mission (spacecraft)
- GOES
Geostationary Orbiting Earth Satellite (spacecraft)
- GRANAT
Gamma Ray Astronomical observatory (Russian spacecraft)
- HSP
High Speed Photometer (instrument on HST)
- HST
Hubble Space Telescope
- HXR
Hard X-Rays
- HXRBS
Hard X-Ray Burst Spectrometer (on SMM)
- ICE
International Cometary Explorer (spacecraft)
- IMAGE
Magnetospheric spacecraft
- IMF
Interplanetary Magnetic Field
- IMP
Interplanetary Monitoring Platform (spacecraft)
- ISEE-3
International Cometary Explorer (spacecraft)
- ISSI
International Space Science Institute, Bern, Switzerland
- IT
Intermittent Turbulence
- keV
Kilo electron Volt
- LA
Linear size vs. Area method (fractal dimension)
- LASCO
Large-Angle Solar COronagraph (on SOHO)
- LIM
Local Intermittency Measure (method)
- MHD
Magneto-HydroDynamics
- MeV
Mega electron Volt
- MG
Magnetogram
- MK
Mega Kelvin
- MW
Microwave Burst
- MW-S
Microwave Burst Synchrotron emission
- NASA
National Aeronautics and Space Administration
- NOAA
National Oceanic and Atmospheric Administration
- OFC
Olami–Feder–Christensen (SOC model)
- OGO
Orbiting Geophysical Observatory (spacecraft)
- OSO
Orbiting Solar Observatory (spacecraft)
- PA
Perimeter versus Area (fractal dimension)
Probability Distribution Function
- PHEBUS
Gamma ray burst instrument (on GRANAT)
- POLAR
Magnetospheric mission (spacecraft)
- PSR
Pulsar
- RHESSI
Ramaty High Energy Solar Spectroscopic Imager (spacecraft)
- RTV
Rosner–Tucker–Vaiana model (coronal heating)
- RXTE
Rossi X-ray Timing Explorer (spacecraft)
- SDF
Surviving Distribution Function
- SGR
Soft Gamma-ray Repeaters
- SDO
Solar Dynamics Observatory (spacecraft)
- SEP
Solar Energetic Particle event
- SMM
Solar Maximum Mission (spacecraft)
- SOBP
Self-Organized Branching Process
- SOC
Self-Organized Criticality
- SOHO
SOlar and Heliospheric Observatory (spacecraft)
- SST
Swedish Solar Telescope (observatory)
- STEREO
Sun TErrestrial RElations Observatory (spacecraft)
- SuperDARN
Super Dual Auroral Radar Network
- SXR
Soft X-Rays
- SXT
Soft X-ray Telescope (on Yohkoh spacecraft)
- TGF
Terrestrial Gamma-ray Flashes
- TRACE
TRAnsition region and Coronal Explorer (spacecraft)
- TV
TeleVision (camera)
- UCB
University of California, Berkeley
- ULYSSES
Interplanetary mission (spacecraft)
- UV
Ultra-Violet
- UVI
Ultra-Violet Image (on POLAR spacecraft)
- WATCH
Wide Angle Telescope of Cosmic Hard X-rays (on GRANAT)
- XMM
X-ray Multi-Mirror mission (spacecraft)
- WIC
Wideband Imaging Camera (on IMAGE spacecraft)
- WL
White Light
- WMAP
Wilkinson Microwave Anisotropy Probe (spacecraft)
- WTD
Waiting Time Distribution function
- Yohkoh
Japanese Solar-A mission (spacecraft)
1 Introduction
About 25 years ago, the concept of self-organized criticality (SOC) emerged (Bak et al. 1987), initially envisioned to explain the ubiquitous 1/f-power spectra, which can be characterized by a powerlaw function P(ν)∝ν^{−1}. The term 1/fpower spectra or flicker noise should actually be understood in broader terms, including power spectra with pink noise (P(ν)∝ν^{−1}), red noise (P(ν)∝ν^{−2}), and black noise (P(ν)∝ν^{−3}), essentially everything except white noise (P(ν)∝ν^{0}). While white noise represents traditional random processes with uncorrelated fluctuations, 1/f power spectra are a synonym for time series with non-random structures that exhibit long-range correlations. These non-random time structures represent the avalanches in Bak’s paradigm of sandpiles. Consequently, Bak’s seminal paper in 1987 triggered a host of numerical simulations of sandpile avalanches, which all exhibit powerlaw-like size distributions of avalanche sizes and durations. These numerical simulations were, most commonly, cellular automata in the language of complexity theory, which are able to produce complex spatio-temporal patterns by iterative application of a simple mathematical redistribution rule. The numerical algorithms of cellular automata are extremely simple, basically a one-liner that defines the redistribution rule, with an iterative loop around it, but can produce the most complex dynamical patterns, similar to the beautiful geometric patterns created by Mandelbrot’s fractal algorithms (Mandelbrot 1977, 1983, 1985). An introduction and exhaustive description of cellular automaton models that simulate SOC systems is given in Pruessner (2012, 2013), and a review of cellular automaton models applied to solar physics is given in Charbonneau et al. (2001).
Four years after introduction, Bak’s SOC concept was applied to solar flares, which were known to exhibit similar powerlaw size distributions for hard X-ray peak fluxes, total fluxes, and durations as the cellular automaton simulations produced for avalanche sizes and durations (Lu and Hamilton 1991). This discovery enabled a host of new applications of the SOC concept to astrophysical phenomena, such as solar and stellar flare statistics, magnetospheric substorms, X-ray pulses from accretion disks, pulsar glitches, and so forth. A compilation of SOC applications to astrophysical phenomena is given in a recent textbook (Aschwanden 2011a), as well as in recent review articles (Aschwanden 2013; Crosby 2011). The successful spreading of the SOC concept in astrophysics mirrored the explosive trend in other scientific domains, such as the application of SOC in magnetospheric physics (auroras, substorms; see review by Sharma et al. 2014), in geophysics (earthquakes, mountain and rock slides, snow avalanches, forest fires; see Hergarten 2002 and review by Hergarten in this volume), in biophysics (evolution and extinctions, neuron firing, spread of diseases), in laboratory physics (Barkhausen effect, magnetic domain patterns, Ising model, tokamak plasmas; Jensen 1998), financial physics (stock market crashes; Sornette 2003), and social sciences (urban growth, traffic, global networks, internet) or sociophysics (Galam 2012). This wide range of applications elevated the SOC concept to a truly interdisciplinary research area, which inspired Bak’s vision to explain “how nature works” (Bak 1996). What is common to all these systems is the statistics of nonlinear processes, which often ends up in powerlaw-like size distributions. Other aspects that are in common among the diverse applications are complexity, contingency, and criticality (Bak and Paczuski 1995), which play a grand role in complexity theory and systems theory.
What became clear over the last 25 years of SOC applications is the duality of (1) a universal statistical aspect, and (2) a special physical system aspect. The universal aspect is a statistical argument that can be formulated in terms of the scale-free probability conjecture (Aschwanden 2012a), which explains the powerlaw function and the values of the powerlaw slopes of most occurrence frequency distributions of spatio-temporal parameters in avalanching systems. This statistical argument for the probability distributions of nonlinear systems is as common as the statistical argument for binomial or Gaussian distributions in linear or random systems. In this sense, solar flares, earthquakes, and stockmarket systems have a statistical commonality (e.g., De Arcangelis et al. 2006). On the other hand, each SOC system may be governed by different physical principles unique to each observed SOC phenomenon, such as plasma magnetic reconnection physics in solar flares, mechanical stressing of tectonic plates in earthquakes, or the networking of brokers in stock market crashes. So, one should always be aware of this duality of model components when creating a new SOC model. There is no need to re-invent the universal statistical aspects or powerlaw probability distributions each time, while the modeling of physical systems may be improved with more accurate measurements and model parameterizations in every new SOC application.
There is another duality in the application of SOC: the numerical world of lattice simulation toy models, and the real world of quantitative observations governed by physical laws. The world of lattice simulations has its own beauty in producing complexity with mathematical simplicity, but it cannot capture the physics of a SOC system. It can be easily designed, controlled, modified, and visualized. It allows us to perform Monte-Carlo simulations of SOC models and may give us insights about the universal statistical aspects of SOC. Real world phenomena, in contrast, need to be observed and measured with large statistics and reliable parameters that have been cleaned from systematic bias effects, incomplete sampling, and unresolved spatial and temporal scales, which is often hard to achieve. However, computer power has increased drastically over the last 25 years, exponentially according to Gordon Moore’s law, so that enormous databases with up to ≈10^{9} events have been gathered per data set from some SOC phenomena, such as from solar small-scale phenomena for instance (McIntosh and Gurman 2005).
We organize this review by describing first some basics of SOC systems (Sect. 2), concerning SOC definitions, elements of a SOC system, the probability concept, geometric scaling laws, transport process, derivation of occurrence frequency distributions, waiting time distributions, separation of time scales, and the application of cellular automata. Then we deliver an overview on astrophysical applications (Sect. 3), grouped by observational results and theoretical models in solar physics, magnetospheres, planets, stars, galaxies, and cosmology. In Sect. 4 we capture some discussions, open issues and challenges, critiques, limitations, and new trends on the SOC subject, including also discussions of SOC-related processes, such as turbulence and percolation. The latter section mostly results from discussions during two weeks of dedicated workshops on “Self-organized Criticality and Turbulence”, held at the International Space Science Institute (ISSI) Bern during 2012 and 2013, attended by participants who have contributed to this review.
2 Basics of Self-Organized Criticality Systems
2.1 SOC Definitions
Dynamical systems with extended spatial degrees of freedom naturally evolve into self-organized critical structures of states which are barely stable. Flicker noise, or 1/fnoise, can be identified with the dynamics of the critical state. This picture also yields insight into the origin of fractal objects (Bak et al. 1987).
In this first seminal paper, the authors had already fractal structures like cosmic strings, mountain landscapes, and coastal lines as potential applications in mind and concluded: We believe that the new concept of self-organized criticality can be taken much further and might betheunderlying concept of dissipative systems with extended degrees of freedom (Bak et al. 1987). In this spirit, the application of the SOC concept has been broadened substantially over the last 25 years.
Self-organized criticality is regarded as scale invariance without external tuning of a control parameter, but with all the features of the critical point of an ordinary phase transition, in particular long range (algebraic) spatiotemporal correlations (Pruessner 2012).
SOC is a critical state of a nonlinear energy dissipation system that is slowly and continuously driven towards a critical value of a system-wide instability threshold, producing scale-free, fractal-diffusive, and intermittent avalanches with powerlaw-like size distributions (Aschwanden 2014).
2.2 The Driver
The driver is the input part of a SOC system. Without a driver, avalanching would die out and the system becomes subcritical and static. On the other side, the driver must be slowly and continuous, so that the critical state is restored in the asymptotic limit, while a strong driver would lead the system into a catastrophic collapse and may destroy the system. In the classical BTW model, sand grains are dripped under the action of gravity at a slow rate, at random locations of the sandpile, which re-fill and restore dents from previous avalanches towards the critical angle of repose. In astrophysical systems, the driver or energy input of a SOC system may be gravity (in galaxy formation, star formation, black holes, planet formation, asteroid formation), gravitational disturbances (in Saturn ring), or creation and stressing of magnetic flux (in solar flares, stellar flares, neutron stars, pulsars). The driver must bring the system back to the critical point after each major avalanche, which means that the system is locally pushed towards the instability threshold again, so that further avalanching can occur. In the slowly-driven limit, the time duration of an avalanche is much longer than the (waiting) time intervals between two subsequent events, which warrants a separation of time scales. In some natural systems the driver may temporarily or permanently stop, such as the solar dynamo during the Maunder minimum that stopped solar flaring, or the final stage of the sweep-up of debris left over from the formation of the solar system 4.0 billion years ago that stopped lunar cratering.
2.3 Instability and Criticality
2.4 Avalanches
Avalanches are defined as nonlinear energy dissipation events, which occur in our generalized SOC definition whenever and wherever a local instability threshold is exceeded. Avalanches are the output part of a SOC system, which balance the energy input rate in the time average for conservative SOC systems. Avalanches are detectable events, which can be obtained in astrophysical observations with large statistics, such as length scales (L), time scales or durations (T), fluxes (F), fluences or energies (E). The occurrence frequency distributions of these observables tend to be powerlaw-like functions, a hallmark of SOC systems, but deviations from powerlaw functions can be explained by measurement bias effects (such as incomplete sampling, finite system-size effects, truncations of distributions), or could reflect multiple physical processes. Unnecessary to say that these observables and their size distributions and underlying scaling laws provide the most important evidence and tests of SOC models.
The time evolution of avalanches contain essential information on the underlying spatio-temporal transport process (i.e., diffusion, fractal diffusion, percolation, turbulence, etc.). A generic time evolution is an initially nonlinear (i.e., exponential) growth phase, followed by a quenching or saturation phase (as expressed in the popular saying “No trees grow to the sky!”). In solar flares, for instance, the initial growth phase is called “impulsive phase”, and the subsequent saturation phase is called “postflare phase”. In earthquakes, the terms “precursors” and “after shocks” are common.
2.5 Microscopic Structure and Complexity
2.6 The Scale-Free Probability Conjecture
Common characterizations of SOC systems are statistical distributions of SOC parameters (also called “size distributions”, “occurrence frequency distributions”, or “log(N)–log(S) plots”). How do we derive a statistical probability distribution function (PDF) for SOC systems? This question has been answered in the original SOC papers (Bak et al. 1987, 1988) in an empirical way, by performing numerical Monte-Carlo simulations of avalanches in Cartesian lattice grids, according to the well-known algorithm with next-neighbor interactions (BTW model). Several theoretical attempts have been made to derive statistical probabilities, by considering avalanches as a branching process (Harris 1963; Christensen and Olami 1993), by exact solutions of the Abelian sandpile (Dhar and Ramaswamy 1989; Dhar 1990, 1999; Dhar and Majumdar 1990), by considering the BTW cellular automaton as a discretized diffusion process using the Langevin equations (Wiesenfeld et al. 1989; Zhang 1989; Forster et al. 1977; Medina et al. 1989), or by renormalization group theory (Medina et al. 1989; Pietronero and Schneider 1991; Pietronero et al. 1994; Vespignani et al. 1995; Loreto et al. 1995, 1996). Most of these analytical theories represent special solutions to a particular set of mathematical redistribution rules, but predict different powerlaw exponents for the probability distribution functions obtained with each method, and thus lack the generality to interpret the ubiquitous and omnipresent SOC phenomena observed in nature.
2.7 Geometric Scaling Laws
2.8 Fractal Geometry
“Fractals in nature originate from self-organized critical dynamical processes” (Bak and Chen 1989). The fractal geometry has been postulated for SOC processes by the first proponents of SOC. However, the geometry of fractals has been explored at least a decade before the SOC concept existed (Mandelbrot 1977, 1983, 1985). An extensive discussion of measuring the fractal geometry in SOC systems associated with solar and planetary data is given in Aschwanden (2011a, Chap. 8) and McAteer (2013a).
Fractals are measurable from the spatial structure of an avalanche at a given instant of time. Therefore, they enter the statistics of time-evolving SOC parameters, such as the observed flux or intensity per time unit, which is proportional to the number of instantaneously active nodes in a lattice-based SOC avalanche simulation.
2.9 Spatio-Temporal Evolution and Transport Process
Let us consider some basic aspects in the time domain of SOC avalanches. The spatio-temporal evolution of SOC avalanches has been simulated with cellular automaton simulations (Bak et al. 1987, 1988; Lu and Hamilton 1991; Charbonneau et al. 2001), which produced statistics of the final avalanche sizes L and durations T, but there is virtually no statistics on the spatio-temporal evolution of the instantaneous avalanche size or radius r(t) as a function of time t, which would characterize the macroscopic transport process. Statistics on this spatio-temporal evolution is important to establish spatio-temporal correlations and scaling laws between L and T, which defines the macroscopic transport process.
2.10 Flux and Energy Scaling
The original BTW model specified avalanche sizes by the total number of active nodes, which corresponds to the cluster area of an avalanche in a 2D lattice. If we want to characterize the area a(t) of an avalanche as a function of time, which is a highly fluctuating quantity in time, we can define also a time-integrated final area a(<t) that includes all nodes that have been gone unstable at least once during the course of an avalanche, which is a monotonically increasing quantity and quantifies the size of an avalanche with a single number A=a (t=T), which we simply call the time-integrated avalanche area.
2.11 Coherent and Incoherent Radiation
Self-organized criticality models can be diagnosed and tested by means of statistical distributions, e.g., by the omnipresent powerlaw or powerlaw-like size distributions, and by the underlying scaling laws that relate the powerlaw slopes of different observables to each other (see also McAteer et al. 2014 for a description of methods). The original paradigm of a SOC model, the BTW cellular automaton simulations (Bak et al. 1987, 1988), produced powerlaw distributions of two variables, the size S, and the time duration T. The size S is simply defined by the time-integrated area A of active nodes (pixels) in 2D lattice simulations, or by the time-integrated fractal volume V_{f} of active nodes (voxels) in 3D lattice simulations.
2.12 Waiting Times and Memory
Waiting times, also called “elapsed times”, “inter-occurrence times”, “inter-burst times”, or “laminar times”, are defined by the time interval between two subsequent bursts. The distribution of waiting times requires to break a continuous time series down into discrete events, for instance by using a threshold criterion. Consequently, waiting time statistics requires a separation of time scales, which means that the burst durations have to be shorter than the waiting times, otherwise multiple bursts are counted as a single one and the waiting time between two closely following bursts is missing in the statistics.
2.12.1 Stationary Poisson Processes
A waiting time distribution measured in a global system loses all timing information from individual local regions, so we can never conclude from the waiting times of a global system whether the waiting times in a local region is a random process or not. However, the opposite is true and can be mathematically proven, i.e., that the combination of time series with random time intervals produces a combined time series that has also random time intervals. This property is also called the superposition theorem of Palm and Khinchin (e.g., Cox and Isham 1980; Craig and Wheatland 2002) and is analogous to the central limit theorem (Rice 1995). An example that waiting times in local regions can be completely different from those of the global system was confirmed in earthquake statistics, where aftershocks (occurring in the same local region) exhibit an excess of short waiting times (Omori’s law; Omori 1895), compared with the overall statistics of (spatially) independent earthquakes.
2.12.2 Non-stationary Poisson Processes
Thus we learn from the last four examples that most continuously changing occurrence rates produce powerlaw-like waiting-time distributions P(Δt)∝(Δt)^{−p} with slopes of p≲2,…,3 at large waiting times, despite the intrinsic exponential distribution that is characteristic to stationary Poisson processes. If the variability of the flare rate is gradual (third and fourth case in Fig. 6), the powerlaw slope of the waiting-time distribution is close to p≲3. However, if the variability of the flare rate shows spikes like δ-functions (Fig. 6, bottom), which is highly intermittent with short clusters of flares, the distribution of waiting times has a slope closer to p≈2. This phenomenon is also called clusterization and has analogs in earthquake statistics, where aftershocks appear in clusters after a main shock (Omori’s law; Omori 1895). Thus the powerlaw slope of waiting times contains essential information whether the flare rate is constant, varies gradually, or in form of intermittent clusters.
Powerlaw-like waiting time distributions can also be produced by standard BTW sandpile simulations, when correlations exist in the slowly-driven external driver, producing a “colored” power spectrum, especially when only avalanches above some threshold are included in the waiting-time distribution (Sanchez et al. 2002).
2.12.3 Waiting Time Probabilities in the Fractal-Diffusive SOC Model
2.12.4 Weibull Distribution and Processes with Memory
As we stated in a previous section, we can never conclude from the waiting times of a global system whether the waiting times in a local region is a random process or not. Non-stationary Poisson processes may fit an observed waiting time distribution perfectly well, with an appropriate flaring rate function f(λ), but the best-fit solution is not unique. Local regions may have non-random statistics with clustering, memory, and persistence. Such non-Poissonian processes can, for instance, be characterized with the more general Weibull distribution, which originally has been used to describe particle size distributions (Weibull 1951). Here we outline the formalism according to an application to (solar) coronal mass ejections (Telloni et al. 2014).
2.13 The Separation of Time Scales
2.14 Cellular Automaton Models
Since the original BTW model has been a paradigm of SOC models for 25 years, we should evaluate its predictive potential, since every theory can only be validated when it is able to make quantitative predictions for future (or past) measurements. The original BTW model simulated a complex system by numerical lattice simulations of iterating a simple next-neighbor interaction redistribution rule (generally called a cellular automaton model, which produced a distribution with a powerlaw slope of α_{E}≈0.98 for avalanche sizes in 2D space, or α_{E}≈1.35 for avalanche sizes in 3D space Bak et al. 1987). These values are somewhat different from the predictions of the basic SOC model based on the scale-free probability conjecture (Sects. 2.6 and 2.7), which predicts α_{E}=9/7≈1.29 for avalanche sizes in 2D space, and α_{E}≈1.50 for avalanche sizes in 3D space. Other extensive BTW simulations with a variety of grid sizes find α_{E}≈1.42±0.01 for avalanche sizes in 2D space, and α_{E}≈1.47±0.02 for avalanche sizes in 3D space (Charbonneau et al. 2001). The latter values are actually almost consistent with the value α_{E}=1.55 (in 2D space) obtained from a pre-Bak simulation as a model for propagating brittle failure in heterogeneous media (Katz 1986). From these few examples it is already clear that various cellular automaton models produce different powerlaw slopes, and thus the question arises whether the obtained powerlaw slopes depend on the numerical details of the setup of lattice simulations, or whether they have universal validity that is independent of numerical redistribution rules and may even apply to observations in nature.
References | Dimension d | Powerlaw slope α_{S} | Powerlaw slope α_{T} |
---|---|---|---|
Ruelle and Sen (1992) | 1 | 1.0 | 1.0 |
Bak and Sneppen (1993) | 1 | 1.0–1.1 | |
Christensen et al. (1996) | 1 | 1.55 | 1.7–1.9 |
Aschwanden (2012a) | 1 | 0.88±0.09 | 1.17±0.02 |
FD-SOC prediction (Aschwanden 2012a) | 1 | 1.00 | 1.00 |
Bak et al. (1987) | 2 | 0.98 | 0.97 |
Zhang (1989) | 2 | 1.2–1.7 | 1.5 |
Dhar (1990) | 2 | 1.2–1.3 | 1.30–1.50 |
Manna (1990) | 2 | 1.22 | 1.38 |
Manna (1991) | 2 | 1.25–1.30 | 1.50 |
Christensen et al. (1991) | 2 | 1.21 | 1.32 |
2 | 1.20 | 1.16 | |
Drossel and Schwabl (1992) | 2 | 1.0–1.2 | 1.20–1.30 |
Olami et al. (1992) | 2 | 1.2–1.3 | |
Pietronero et al. (1994) | 2 | 1.25 | |
Priezzhev et al. (1996) | 2 | 1.20 | |
Lübeck and Usadel (1997) | 2 | 1.00, 1.29 | 1.48 |
Chessa et al. (1999) | 2 | 1.27 | |
Lin and Hu (2002) | 2 | 1.12–1.37 | |
Bonachela (2008) | 2 | 1.30 | |
Charbonneau et al. (2001) | 2 | 1.42±0.01 | 1.71±0.01 |
McIntosh et al. (2002) | 2 | 1.41±0.01 | |
Aschwanden (2012a) | 2 | 1.48±0.03 | 1.77±0.18 |
FD-SOC prediction (Aschwanden 2012a) | 2 | 1.29 | 1.50 |
Bak et al. (1987) | 3 | 1.35 | 1.59 |
Grassberger and Manna (1990) | 3 | 1.33 | 1.63 |
Christensen et al. (1991) | 3 | 1.37–1.47 | 1.60 |
Charbonneau et al. (2001) | 3 | 1.47±0.02 | 1.74±0.06 |
McIntosh et al. (2002) | 3 | 1.46±0.01 | 1.71±0.01 |
Aschwanden (2012a) | 3 | 1.50±0.06 | 1.76±0.19 |
FD-SOC prediction (Aschwanden 2012a) | 3 | 1.50 | 2.00 |
Based on the scale-free probability conjecture and the geometric scaling laws of the fractal-diffusive SOC model described in Sects. 2.6–2.10, we predict for classical diffusion (β=1) and a mean fractal dimension D_{d}=(1+d)/2 the following powerlaw slopes for avalanche size distributions (Eq. (22)): α_{E}=1 for 1D space, α_{E}=9/7≈1.29 for 2D space, and α_{E}=3/2=1.5 for 3D space, which agree with most of the measured slopes of avalanche sizes in cellular automaton simulations (Table 1). For event durations we predict: α_{T}=1 for 1D space, α_{T}=3/2=1.5 for 2D space, and α_{T}=2.0 for 3D space, which also roughly agrees with the simulations in Table 1.
Frequency distributions measured from solar flares in hard X-rays and γ-rays. The prediction is based on the FD-SOC model (Aschwanden 2012a)
Powerlaw slope of peak flux α_{P} | Powerlaw slope of fluence α_{E} | Powerlaw slope of durations α_{T} | Number of events n | Instrument and threshold energy | References |
---|---|---|---|---|---|
1.8 | 123 | OSO-7(>20 keV) | Datlowe et al. (1974) | ||
2.0 | 25 | UCB(>20 keV) | Lin et al. (1984) | ||
1.8 | 6775 | HXRBS(>20 keV) | Dennis (1985) | ||
1.73±0.01 | 12,500 | HXRBS(>25 keV) | Schwartz et al. (1992) | ||
1.73±0.01 | 1.53±0.02 | 2.17±0.05 | 7045 | HXRBS(>25 keV) | Crosby et al. (1993) |
1.71±0.04 | 1.51±0.04 | 1.95±0.09 | 1008 | HXRBS(>25 keV) | Crosby et al. (1993) |
1.68±0.07 | 1.48±0.02 | 2.22±0.13 | 545 | HXRBS(>25 keV) | Crosby et al. (1993) |
1.67±0.03 | 1.53±0.02 | 1.99±0.06 | 3874 | HXRBS(>25 keV) | Crosby et al. (1993) |
1.61±0.03 | 1263 | BATSE(>25 keV) | Schwartz et al. (1992) | ||
1.75±0.02 | 2156 | BATSE(>25 keV) | Biesecker et al. (1993) | ||
1.79±0.04 | 1358 | BATSE(>25 keV) | Biesecker et al. (1994) | ||
1.59±0.02 | 2.28±0.08 | 1546 | WATCH(>10 keV) | Crosby (1996) | |
1.86 | 1.51 | 1.88 | 4356 | ISEE-3(>25 keV) | Lu et al. (1993) |
1.75 | 1.62 | 2.73 | 4356 | ISEE-3(>25 keV) | Lee et al. (1993) |
1.86±0.01 | 1.74±0.04 | 2.40±0.04 | 3468 | ISEE-3(>25 keV) | Bromund et al. (1995) |
1.80±0.01 | 1.39±0.01 | 110 | PHEBUS(>100 keV) | Perez Enriquez and Miroshnichenko (1999) | |
1.80±0.02 | 2.2±1.4 | 2759 | RHESSI(>12 keV) | Su et al. (2006) | |
1.58±0.02 | 1.7±0.1 | 2.2±0.2 | 4241 | RHESSI(>12 keV) | Christe et al. (2008) |
1.6 | 243 | BATSE(>8 keV) | Lin et al. (2001) | ||
1.61±0.04 | 59 | ULYSSES(>25 keV) | Tranquille et al. (2009) | ||
1.73±0.07 | 1.62±0.12 | 1.99±0.35 | Average | All HXR observations | |
1.67 | 1.50 | 2.00 | FD-SOC prediction | Aschwanden (2012a) |
The diffusion or spreading exponent β and the fractal dimension D_{d} are essentially macroscopic parameters to describe the average dynamics and inhomogeneous spatial structure of avalanches, which are microscopically defined in terms of an iterative mathematical redistribution rule. The diffusion exponent β characterizes the macroscopic transport process (subdiffusive, classical diffusion, hyper-diffusion), and the fractal dimension describes the spatial inhomogeneity of an avalanche, in the spirit of Bak and Chen (1989): Fractals in nature originate from self-organized critical dynamical processes. Cellular automata exhibit a range of fractal dimensions and diffusion exponents, as the values in Table 1 demonstrate, and thus may not have universal validity for SOC systems. If we find the same disparity among astrophysical observations, as we will survey in the following sections, nature operates in SOC systems with different spatial inhomogeneities and transport processes, which may be related to the underlying physical scaling laws in each SOC system. The cellular automaton world may have (slightly) different SOC parameters (β,D_{d}) than the astrophysical world, but we are able to describe the nonlinear dynamics of complex systems with the same theoretical framework.
3 Astrophysical Applications
We subdivide the astrophysical phenomena that have been associated with SOC according to solar physics (Sects. 3.1, 3.2), the Earth’s magnetosphere and planets (Sect. 3.3), and stars and galaxies (Sect. 3.4). We tabulate the statistics of SOC parameters mostly in form of measured power law indices. In addition, we discuss briefly the theoretical interpretations in each case and summarize studies that contain modeling attempts of these SOC phenomena, often tailored to a specific astrophysical object.
3.1 Solar Physics: Observations
The applications of SOC theory to solar data outnumbers all other astrophysical applications. Therefore, we brake the subject down into observational statistics from different wavelengths (hard X-rays, soft X-rays, EUV, radio, etc.) in Sect. 3.1, and into various aspects of theoretical modeling (e.g., cellular automaton simulations, magnetic fields, magnetic reconnection, plasma magneto-hydrodynamics (MHD), coronal heating, particle acceleration, solar wind, Sun-Earth connection, etc.) in Sect. 3.2.
3.1.1 Statistics of Solar Flare Hard X-Rays
Solar flares provide the energy source for acceleration of nonthermal particles, which emit bremsstrahlung in hard X-ray wavelengths, once the non-thermal particles interact with a high-density plasma via Coulomb collisions. Most solar flares display an impulsive component in hard X-rays, produced by accelerated coronal electrons that precipitate towards the chromosphere and produce intense hard X-ray emission at the footpoints of flare loops. Therefore, hard X-ray pulses are a reliable signature of solar flares, often detected at energies ≳20 keV, but for smaller flares down to ≳8 keV.
A compilation of occurrence frequency distribution powerlaw slopes of solar hard X-ray flare peak fluxes (α_{P}), fluences or energies (α_{E}), and flare durations (α_{T}) is listed in Table 2. In this Table we combined both the powerlaw slopes α_{E} from the fluences (which is the time-integrated or total number of hard X-ray counts per flare) and nonthermal energies (which are computed from the hard X-ray energy spectrum assuming a low-energy cutoff at 10 or 25 keV), both representing a physical quantity in terms of energy. In Table 2 we indicate also the number of events, which constrains the accuracy of the fitted powerlaw slopes. Synthesizing the datasets with the largest statistics (HXRBS/SMM, BATSE/CGRO, RHESSI), the following means and standard deviations of the powerlaw slopes were found α_{P}=1.73±0.07 for the peak fluxes (Fig. 9), α_{E}=1.62±0.12 for the fluences or energies, and α_{T}=1.99±0.35 for the flare durations (Aschwanden 2011b). The uncertainties of the powerlaw slope quoted in literature generally include the formal fitting error only, while the standard deviations given here reflect methodical and systematic uncertainties also, since every dataset has been analyzed from different instruments and with different analysis methods. One of the largest systematic uncertainties results from the preflare background subtraction, because the preflare flux is often not specified in solar flare catalogs. Nevertheless, given these systematic uncertainties, the observed values are consistent with the theoretical predictions of the basic fractal-diffusive SOC model, based on an Euclidean space dimension of d=3, a mean fractal dimension of D_{3}=2, and classical diffusion β=1, which yields α_{P}=1.67 for peak fluxes, α_{E}=1.50 for energies, and α_{T}=2.00 for durations (Eq. (24)). Thus, the basic fractal-diffusive SOC model predicts the correct powerlaw slopes within the uncertainties of hard X-ray measurements.
Frequency-size distributions of solar flares are generally sampled from the entire Sun, and thus from multiple active regions that are present on the visible hemisphere at a given time. This configuration corresponds to a multi-sandpile situation, and the resulting powerlaw distribution is composed of different individual active regions, which may have different physical conditions and sizes. In particular, different sizes may cause an exponential cut-off at the upper end of the size distribution due to finite system-size effects. A study of flare statistics on individual active regions, however, did not reveal significant differences in their size distributions, and thus the size distributions of individual active regions seem to follow the universal powerlaw slopes that are invariant, individually as well as in a superimposed ensemble (Wheatland 2000c), except for one particular active region (Wheatland 2010).
Instead of testing powerlaw slopes of size distributions, an equivalent test is a linear regression fit among SOC parameters. For instance, statistics of WATCH/GRANAT data exhibited correlations of P∝E^{0.60±0.01}, T∝E^{0.53±0.02}, and T∝P^{0.54±0.03} (Georgoulis et al. 2001), which are consistent with the predictions of the standard model (Sect. 2.10), i.e., P∝E^{0.75}, T∝E^{0.50}, and T∝P^{0.67}, given the uncertainties of about ±0.15 due to data truncation effects that are not accounted for in the linear regression fits.
Time series analysis of solar hard X-ray bursts has been performed for a few flares with a variety of methods, such as wavelet analysis (Aschwanden et al. 1998a), search for quasi-periodic variations (Jakimiec and Tomczak 2010), search for sub-second time scales (Cheng et al. 2012), statistics of UV subbursts (used as proxies for the hard X-ray subbursts) during a flare that exhibit powerlaw distributions (Nishizuka et al. 2009a, 2009b), multi-fractal spectral analysis of a hard X-ray time profile (McAteer et al. 2007, McAteer 2013b), or wavelet and local intermittency measure (LIM) analysis (Dinkelaker and MacKinnon 2013a, 2013b). The size distributions N(t) of hard X-ray sub-burst durations during a flare were found to be mostly exponential (Aschwanden et al. 1998a), probably due to finite system-size effects in each flaring region. The LIM method can reveal scale-invariant time evolutions, such as the fragmentation of the energy release cascading from large to smaller structures (the “top-down” scenario), or a small flare event that is avalanching into a larger structure (the “bottom-up” scenario), but it was found that neither of the two extremes captures the totality of a flare time profile (Dinkelaker and MacKinnon 2013a, 2013b).
3.1.2 Statistics of Solar Flare Soft X-Rays
Frequency distributions measured from solar flares in soft X-rays. Measurements with no preflare background subtraction are marked with parentheses
Powerlaw slope of peak flux α_{P} | Powerlaw slope of total fluence α_{E} | Powerlaw slope of durations α_{T} | log range | Instrument | References |
---|---|---|---|---|---|
1.8 | 1 | OSO-3 | Hudson et al. (1969) | ||
1.75 | 1.44 | 2 | Explorer | Drake (1971) | |
1.64–1.89 | 1.5–1.6 | 2 | Yohkoh | Shimizu (1995) | |
1.79 | 2 | SMM/BCS | Lee et al. (1995) | ||
1.86 | 2 | GOES | Lee et al. (1995) | ||
1.88±0.21 | 3 | GOES | Feldman et al. (1997) | ||
1.7±0.4 | 2 | Yohkoh | Shimojo and Shibata (1999) | ||
1.98 | 1.88 | 3 | GOES | ||
1.98±0.11 | 2.02±0.04 | 5 | GOES | Aschwanden and Freeland (2012) | |
(2.11±0.13) | (2.03±0.09) | (2.93±0.12) | 3 | GOES | Veronig et al. (2002a) |
(2.16±0.03) | (2.01±0.03) | (2.87±0.09) | 3 | GOES | Yashiro et al. (2006) |
1.67 | 1.50 | 2.00 | FD-SOC prediction | Aschwanden (2012a) |
Size distributions of soft X-ray peak fluxes, fluences, and durations were mostly obtained from flare detections with the OSO-3 spacecraft (Hudson et al. 1969), the Explorer (Drake 1971), Yohkoh/SXT (Shimizu 1995; Shimojo and Shibata 1999), the SMM/BCS (Lee et al. 1995), and the GOES spacecraft (Lee et al. 1995; Feldman et al. 1997; Veronig et al. 2002a, 2002b; Yashiro et al. 2006; Aschwanden and Freeland 2012). Interestingly, the size distribution of the peak count rates in the range of α_{P}=1.64–1.98 is similar to the hard X-rays, and thus implies a proportionality between the hard X-ray counts and the soft X-ray fluxes, which is different from what is expected from the Neupert effect. Since the Neupert effect predicts that the time profile of soft X-rays approximately follows the time integral of the impulsive hard X-rays, one would expect that the soft X-ray peak flux distribution should be equal to the hard X-ray fluences, which is however not the case (Lee et al. 1995). The different powerlaw slopes indicate a special scaling law between flare temperatures and densities, i.e., n_{e}∝T^{−4/5} (Lee et al. 1995), while the Neupert effect must be considered as an oversimplified rule that neglects any temperature dependence.
Some of the size distributions of soft X-ray peak fluxes have been found to have values steeper than α_{P}≥2.0 (Veronig et al. 2002a; Yashiro et al. 2006), which in hindsight we can understand to be a consequence of neglecting the subtraction of the preflare background flux, which makes up a substantial amount of the total flux for small flares.
A power spectrum of a time series of the GOES 0.5–4 Å flux during a flare-rich episode of two weeks during 2000, containing about 100 GOES >C1.0 flares, has been found to follow a spectral slope of P(ν)∝ν^{−1} (Bershadskii and Sreenivasan 2003), which indeed confirms Bak’s original idea that the SOC concept provides an explanation for the 1/f-noise (Bak et al. 1987).
3.1.3 Statistics of Solar Flare EUV Fluxes
Frequency distributions measured in small-scale events in EUV, UV, and Hα
Powerlaw slope of peak flux α_{P} | Powerlaw slope of total fluence or energy α_{E} | Powerlaw slope of durations α_{T} | Waveband λ (Å) | References |
---|---|---|---|---|
2.3–2.6 | 171, 195 | Krucker and Benz (1998) | ||
1.19±1.13 | 195 | Aletti et al. (2000) | ||
2.0–2.6 | 171, 195 | Parnell and Jupp (2000) | ||
1.68–2.35 | 1.79±0.08 | 171, 195 | ||
2.31–2.59 | 171, 195 | Benz and Krucker (2002) | ||
2.04–2.52 | 171, 195 | Benz and Krucker (2002) | ||
1.71±0.10 | 2.06±0.10 | 171 | Aschwanden and Parnell (2002) | |
1.75±0.07 | 1.70±0.17 | 195 | Aschwanden and Parnell (2002) | |
1.52±0.10 | 1.41±0.09 | AlMg | Aschwanden and Parnell (2002) | |
1.54±0.03 | 171+195+AlMg | Aschwanden and Parnell (2002) | ||
2.12±0.05 | 6563 | Georgoulis et al. (2002) | ||
1.5–3.0 | 1–500 | Greenhough et al. (2003) | ||
1.4–2.0 | 171, 195, 284 | McIntosh and Gurman (2005) | ||
1.66–1.70 | 1.96–2.02 | EUV | Uritsky et al. (2007) | |
1.86±0.05 | 1.50±0.04 | 2.12±0.11 | EUV | Uritsky et al. (2013) |
1.5 | 2.3 | 1550 | ||
2.42–2.52 | 2.02–2.66 | STEREO 171 | Aschwanden et al. (2013b) | |
2.66–2.69 | 2.50–2.52 | STEREO 195 | Aschwanden et al. (2013b) | |
2.14–2.18 | 2.15–2.24 | STEREO 284 | Aschwanden et al. (2013b) | |
2.58–2.70 | 2.61–2.74 | STEREO 304 | Aschwanden et al. (2013b) | |
1.67 | 1.50 | 2.00 | FD-SOC prediction | Aschwanden (2012a) |
Nonetheless, flare statistics from different wavelength regimes start to converge, as shown in Fig. 11. What is still needed is an unified identical detection method that uniformly samples events from the largest giant flare down to the smallest nanoflare.
3.1.4 Statistics of Solar Flare Radio Fluxes
Solar radio bursts are usually subdivided into incoherent (gyroemission, gyrosynchrotron emission, free-free emission) and coherent emission mechanisms (electron beam instability, loss-cone instability, maser emission). Since incoherent emission mechanisms scale with the volume of the emitting source, which could be a solar flare region, we expect some proportionality between the flare energy and the radio burst flux, such as for microwave bursts and type IV bursts (produced by gyrosynchrotron emission). Consequently we expect powerlaw slopes of their size distributions that are similar to other incoherent emission mechanisms of flares (e.g., bremsstrahlung in hard X-rays or soft X-rays). On the other hand, since coherent emission mechanisms produce a highly nonlinear response to some wave-particle instability, their emitted intensity flux is expected to scale nonlinearly with the flare volume, and thus may produce quite different size distributions.
Frequency distributions measured from solar radio bursts, classified as type I storms (I), type III bursts (III), decimetric pulsation types (DCIM-P), decimetric millisecond spikes (DCIM-S), microwave bursts (MW), and microwave spikes (MW-S)
Powerlaw slope of peak flux α_{P} | Powerlaw slope of durations α_{T} | log range | Waveband frequency f | Radio burst type | References |
---|---|---|---|---|---|
1.8 | 2 | 3 GHz | MW | Akabane (1956) | |
1.5 | 2 | 3, 10 GHz | MW | Kundu (1965) | |
1.8 | 2 | 1, 2, 3.75, 9.4 GHz | MW | Kakinuma et al. (1969) | |
1.9–2.5 | 2 | 3.75, 9.4 GHz | MW | Kakinuma et al. (1969) | |
1.74–1.87 | 2 | 1–35 GHz | MW | Song et al. (2011) | |
1.65 | 2 | 2.8 GHz | MW | Das et al. (1997) | |
1.71–1.91 | 4 | 0.100–2 GHz | MW | Nita et al. (2002) | |
1.26–1.69 | 3 | 110 kHz–4.9 MHz | III | Fitzenreiter et al. (1976) | |
1.28 | 2 | 100 MHz–3 GHz | III | Aschwanden et al. (1995) | |
1.45±0.31 | 3 | 100 MHz–3 GHz | III | Aschwanden et al. (1998b) | |
1.22–1.25 | 2.5 | 650–950 MHz | III | Das et al. (1997) | |
1.33±0.11 | 3 | 100 MHz–3 GHz | DCIM-P | Aschwanden et al. (1998b) | |
2.9–3.6 | 1.5 | 164, 237 MHz | I | Mercier and Trottet (1997) | |
4.8±0.1 | 0.5 | 185–198 MHz | I | Iwai et al. (2013) | |
2.99±0.63 | 3 | 100 MHz–3 GHz | DCIM-S | Aschwanden et al. (1998b) | |
7.4±0.4 | 5.4±0.9 | 0.5 | 4.5–7.5 GHz | MW-S | Ning et al. (2007) |
1.67 | 2.00 | FD-SOC prediction | Aschwanden (2012a) |
Type III bursts, which are believed to be produced by plasma emission excited by an electron beam-driven instability, display flatter size distributions in the order of α_{P}≈1.2–1.5 (Fitzenreiter et al. 1976; Das et al. 1997; Aschwanden et al. 1995, 1998b), which can be explained by a nonlinear scaling F∝E^{γ} between radio peak flux P and flare energy E. For the radio peak flux distribution \(N(P) \propto P^{-\alpha_{P}}\), and assuming the standard volume scaling N(V)∝V^{−5/3} (Eq. (5)), we expect then, say for a nonlinear exponent γ=2, a powerlaw slope of α_{P}=(1+1/2γ)≈1.25. The fact that relatively flat powerlaw slopes have also been observed for other coherent radio bursts, such as α_{P}≈1.3 for decimetric pulsations (DCIM-P; Aschwanden et al. 1998b), may also indicate a nonlinear scaling to the flare volume.
On the other hand, some very steep size distributions have been observed, such as α_{P}≈3–5 for type I bursts (Mercier and Trottet 1997; Iwai et al. 2013), or α_{P}≈3–7 for decimetric and microwave millisecond spike bursts (Aschwanden et al. 1998b; Ning et al. 2007), which implies either a strong quenching effect that inhibits high levels of radio fluxes, or a pulse-pileup problem that violates the separation of time scales (i.e., the inter-burst time intervals or waiting times are shorter than the burst durations). The latter effect is most likely to occur in the statistics of fine structure in complex patterns of radio dynamic spectra, where a multitude of small pulses occur in clusters. Such peculiar types of clustered radio emission are, for instance, type I bursts (Mercier and Trottet 1997; Iwai et al. 2013), or decimetric millisecond spikes (Aschwanden et al. 1998b). Low-resolution radio observations tend to cause an exponential cutoff at large radio flux values, even when the actual distribution has a powerlaw shape (Isliker and Benz 2001), and thus explains the trend of steeper powerlaw slopes. Also stochastic models of clustered solar type III bursts produce powerlaw-like size distributions with an exponential cutoff (Isliker et al. 1998b).
From the statistics of solar radio bursts we learn that we can discriminate between three diagnostic regimes (as grouped in Table 5): (1) the incoherent regime where the radio burst flux is essentially proportional to the flare volume (α_{P}≈1.7–1.9); (2) the coherent regime that implies a nonlinear scaling between the radio peak flux and the flare volume P∝V^{γ} with γ≈2 and α_{P}≈1.2–1.5; and (3) the exponential regime with clustered bursts that violate the separation of time scales with steep slopes α_{P}≈2–7 and have an exponential cutoff. Thus, the powerlaw slopes offer a useful diagnostic to quantify scaling laws between the radio flux (emissivity) and the flare volume.
3.1.5 Statistics of Solar Energetic Particle (SEP) Events
Frequency distributions of solar energetic particle (SEP) events
Powerlaw slope of peak flux α_{P} | Powerlaw slope of total flux or total energy α_{E} | Spacecraft | Energy range E_{min} | References |
---|---|---|---|---|
1.10±0.05 | IMP4-5 | 20–80 MeV protons | Van Hollebeke et al. (1975) | |
1.40±0.15 | >10 MeV protons | Belovsky and Ochelkov (1979) | ||
1.13±0.04 | IMP8 | 24–43 MeV protons | Cliver et al. (1991) | |
1.30±0.07 | IMP8 | 3.6–18 MeV electrons | Cliver et al. (1991) | |
1.32±0.05 | IMP, OGO | >10 MeV protons | Gabriel and Feynman (1996) | |
1.27±0.06 | IMP, OGO | >30 MeV protons | Gabriel and Feynman (1996) | |
1.32±0.07 | IMP, OGO | >60 MeV protons | Gabriel and Feynman (1996) | |
1.47–2.42 | >10 MeV protons | Smart and Shea (1997) | ||
1.27–1.38 | >10 MeV protons | Mendoza et al. (1997) | ||
1.00–2.12 | IMP | >10 MeV protons | Miroshnichenko et al. (2001) | |
1.35 | >10 MeV protons | Gerontidou et al. (2002) | ||
1.34±0.02 | >10 MeV protons | Belov et al. (2007) | ||
1.46±0.03 | >100 MeV protons | Belov et al. (2007) | ||
1.22±0.05 | >10 MeV protons | Belov et al. (2007) | ||
1.26±0.03 | >100 MeV protons | Belov et al. (2007) | ||
1.56±0.02 | >10 MeV protons | Crosby (2009) | ||
1.67 | 1.50 | FD-SOC prediction | Aschwanden (2012a) |
Because of the high energies of SEP events, which can harm astronauts or electronic equipment in space, statistical information that improves their predictability is highly desirable (Gabriel and Patrick 2003), but statistical studies demonstrate that it is not possible to predict the time of occurrence of SEP events within narrow limits (Xapsos et al. 2006).
3.1.6 Statistics of Solar Flare Waiting Times
In the simplest scenario we could envision that solar flares occur randomly in space and time. However, there are subsets of flares that occur simultaneously at different locations (called ”sympathetic flares”; Fritzova-Svestkova et al. 1976; Pearce and Harrison 1990; Wheatland 2006; Moon et al. 2002, 2003), as well as flare events that subsequently occur at the same location (called “homologous flares”; Fokker 1967), which indicates a spatial or temporal clustering that is not random. We outlined the concept of random processes in Sect. 2.12, which can produce an exponential waiting time distribution (for stationary Poisson processes), as well as powerlaw-like distributions of the waiting time (for non-stationary Poisson processes; Fig. 6). Moreover, both exponential and powerlaw distributions can be generated with the Weibull distribution (Sect. 2.12.3). The functional shape of the waiting time distribution depends moreover on the definition of events in a time series, where powerlaws are found more likely to occur when a threshold is used (Buchlin et al. 2005). Allowing an overlap of time scales between burst durations and quiet times, agreement was found between the waiting time distributions sampled with different thresholds (Paczuski et al. 2005; Baiesi et al. 2006). In summary, the finding of powerlaw-like waiting time distributions has no unique interpretation, because it can be consistent with both a random process without memory (in the case of a non-stationary Poisson process) or with a non-random process with memory (in the case of a Weibull distribution with k≠1). This dichotomy of stochasticity versus persistence or clustering has been noted in SOC processes before, for earthquakes that have aftershocks with an excess of short waiting times (Omori’s law; Omori 1895).
Waiting time distributions measured from solar flares hard X-ray events, soft X-ray events, coronal mass ejections, and radio bursts. The waiting time distribution (WTD) functions are abbreviated as: PL=powerlaw, E=exponential, PE=powerlaw with exponential rollover, DE=double exponential
Observations: Spacecraft or instrument | Number of events | Time range Δt | WTD | Powerlaw α_{Δt} | References |
---|---|---|---|---|---|
HXRBS/SMM | 8319 | 1–100 min | PL | 0.75±0.1 | Pearce et al. (1993) |
BATSE/CGRO | 6596 | 2–400 min | E | Biesecker (1994) | |
WATCH/GRANAT | 182 | 10–300 min | PE | 0.78±0.13 | Crosby (1996) |
ICE/ISEE-3 | 6916 | 0.01–20 hrs | DE | Wheatland et al. (1998) | |
SMM/HXRBS | 12,772 | 0.01–500hrs | PL | 2.0 | Aschwanden and McTiernan (2010) |
BATSE/CGRO | 4113 | 0.01–200 hrs | PL | 2.0 | Aschwanden and McTiernan (2010) |
BATSE/CGRO | 7212 | 1–5000 hrs | PL | 2.14±0.01 | Grigolini et al. (2002) |
RHESSI | 11,594 | 2–1000 hrs | PL | 2.0 | Aschwanden and McTiernan (2010) |
GOES 1–8 Å | 32,563 | 1–1000 hrs | PL | 2.16±0.05 | Wheatland (2000a) |
GOES 1–8 Å | 32,563 | 1–1000 hrs | PL | 2.4±0.1 | |
GOES 1–8 Å | 4645 | 1–1000 hrs | PL | 2.26±0.11 | Wheatland (2003) |
GOES 1–8 Å | (sol min) | 1–1000 hrs | PL | 1.75±0.08 | Wheatland (2003) |
GOES 1–8 Å | (sol max) | 1–1000 hrs | PL | 3.04±0.19 | Wheatland (2003) |
SOHO/LASCO | 4645 | 1–1000 hrs | PL | 2.36±0.11 | Wheatland (2003) |
SOHO/LASCO | (sol min) | 1–1000 hrs | PL | 1.86±0.14 | Wheatland (2003) |
SOHO/LASCO | (sol max) | 1–1000 hrs | PL | 2.98±0.20 | Wheatland (2003) |
FD-SOC prediction | (sol min) | PL | 2.00 | Aschwanden (2012a) |
In addition, the powerlaw slope of α_{Δt}=2 is also predicted by the fractal-diffusive model (Sect. 2.12.3) in the slowly-driven limit, while steeper observed slopes are consistent with the predicted modification for strongly-driven systems (Eq. (43)). The results compiled in Table 7 indeed yield higher values of α_{Δt}≈3 during periods of high flare activity, as it occurs during the solar cycle maximum.
In soft X-rays, a similar powerlaw slope of α_{Δt}≈2.1–2.4 was found, which could be fitted with a non-stationary Poisson process (Wheatland 2000a; Moon et al. 2001), with a shell model of turbulence (Boffetta et al. 1999), or with a Levy function (Lepreti et al. 2001). These different interpretations underscore the ambiguity of powerlaw distributions, which do not allow to discriminate between SOC and turbulence processes. Moreover, the powerlaw slope of waiting time distributions varies during the solar cycle, which implies a time-variable SOC driver (Wheatland and Litvinenko 2002). The flaring rate was found to vary among different active regions, as well as during the disk transit time of a single active region (Wheatland 2001). The variability in the flare rate was found to correlate with the sunspot number, however with a time lag of about 9 months, which reflects the hysteresis of the coronal response to the solar dynamo (Wheatland and Litvinenko 2001), a result that can be used for statistical flare forecasting (Wheatland 2004; Wheatland and Craig 2006). Additional tests whether the waiting time of solar flares is random (multi-Poissonian) or clumped in persistent clusters (with some memory) have been carried out with a Hurst analysis, finding a Hurst exponent of H=0.74±0.02 (compared with H=0.5 for a pure stochastic process) (Lepreti et al. 2000), or by fitting a Weibull distribution (Sect. 2.12.3), finding two statistical components for coronal mass ejections, a continuous random process during solar minima, and another component with temporary persistence and memory during solar maxima (Telloni et al. 2014), similar to the FD-SOC scenario (Fig. 7 and Sect. 2.12.3), or the aftershocks in earthquake statistics (Omori’s law).
Does the waiting time give us some information about the energy build-up in solar flares? Early studies suspected that the waiting time is the longer the more energy is built up, which predicts a correlation between the waiting time and the energy of the flare (Rosner and Vaiana 1978). However, several observational studies have shown that no such correlation exists (e.g., Lu 1995b; Crosby 1996; Wheatland 2000b; Georgoulis et al. 2001; Moon et al. 2001), not even between subsequent flares of the same active region (Crosby 1996; Wheatland 2000b). The original SOC model of BTW assumes that avalanches occur randomly in time and space without any correlation, and thus a waiting-time interval between two subsequent avalanches refers to two different independent locations (except for sympathetic flares), and thus bears no information on the amount of energy that is released in each spatially separated avalanche. In solar applications, flare events seem to deplete only a small amount of the available free energy, and thus no correlation between waiting times and flare magnitudes are expected to first order. In contrast, however, recent studies that analyze the probability differences of subsequent events from the GOES flare catalog, and compare them with randomly re-shuffled data, find non-trivial correlations between waiting times and dissipated energies. Flares that are close in time tend to have a second event with large energy. Moreover, the flaring rate as well as the probability of other large flares tends to increase after large flares (Lippiello et al. 2010), similar to the clustering of coronal mass ejections (CMEs) (Telloni et al. 2014), and aftershocks of earthquakes (Omori 1895).
3.1.7 Solar Fractal Measurements
“Fractals in nature originate from self-organized critical dynamical processes (Bak and Chen 1989). In principle, SOC avalanches could be non-fractal and encompass space-filling solid volumes, as the sandpile analogy suggests. However, using the BTW model as a paradigm for SOC avalanches, it is quite clear from inspecting numerical simulations that the next-neighbor interactions propagate in “tree-like” patterns that can indeed be quantified with a fractal dimension (e.g., Aschwanden 2012a). Also the EUV images of solar flares show highly fragmented postflare loops that can be characterized with a fractal dimension (Aschwanden and Aschwanden 2008a).
Area fractal dimension D_{2} of scaling between length scale L and fractal area \(A(L) \propto L^{D_{2}}\) measured from various solar phenomena observed in different wavelength regimes: WL=white light, H-α=visible spectral line in the Balmer series produced in hydrogen at 6562.8 Å, MG=magnetogram measured with Zeeman effect, e.g., from Fe XIV 5303 Å line; EUV=extreme ultra-violet, SXR=soft X-rays. The methods are: PA=perimeter vs. area, LA=linear size vs. area, and BC=box-counting
Phenomenon | Wavelength regime | Method | Area fractal dimension D_{2} | References |
---|---|---|---|---|
Granules | WL | PA | 1.25 | Roudier and Muller (1987) |
Granules | WL | PA | 1.30 | Hirzberger et al. (1997) |
Granular cells | WL | PA | 1.16 | Hirzberger et al. (1997) |
Granules | WL | PA | 1.09 | Bovelet and Wiehr (2001) |
Super-granulation | MG | PA | 1.25 | Paniveni et al. (2005) |
Super-granulation | MG | PA | 1.2, 1.25 | Paniveni et al. (2010) |
Small scales | MG | PA | 1.41±0.05 | Janssen et al. (2003) |
Active regions | MG | LA | 1.56±0.08 | |
Plages | MG | LA | 1.54±0.05 | Balke et al. (1993) |
Active regions | MG | LA | 1.78–1.94 | Meunier (1999) |
MG | PA | 1.48–1.68 | Meunier (1999) | |
Active regions | MG | 1.71–1.89 | Meunier (2004) | |
–Cycle minimum | MG | 1.09–1.53 | Meunier (2004) | |
–Cycle rise | MG | 1.64–1.97 | Meunier (2004) | |
–Cycle maximum | MG | 1.73–1.80 | Meunier (2004) | |
Quiet Sun | MG | multifractal | Lawrence et al. (1993) | |
Active regions | MG | multifractal | Lawrence et al. (1993) | |
Active regions | MG | BC | multifractal | Cadavid et al. (1994) |
Active regions | MG | BC | multifractal | Lawrence et al. (1996) |
Active regions | MG | BC | 1.25–1.45 | McAteer et al. (2005) |
Active regions | MG | multifractal | Conlon et al. (2008) | |
Active regions | MG | multifractal | Hewett et al. (2008) | |
Active regions | MG | multifractal | Conlon et al. (2010) | |
Quiet Sun network | EUV | BC | 1.30–1.70 | Gallagher et al. (1998) |
Ellerman bombs | Hα | BC | 1.4 | Georgoulis et al. (2002) |
Nanoflares | EUV 171 Å | BC | 1.49±0.06 | Aschwanden and Parnell (2002) |
Nanoflares | EUV 195 Å | BC | 1.54±0.05 | Aschwanden and Parnell (2002) |
Nanoflares | SXR | BC | 1.65 | Aschwanden and Parnell (2002) |
Flare 2000-Jul-14 | EUV 171 Å | BC | 1.57–1.93 | Aschwanden and Aschwanden (2008a) |
Flares | EUV | BC | 1.55±0.11 | Aschwanden et al. (2013a) |
FD-SOC prediction | d=2 | 1.50 | Aschwanden (2012a) |
For the application of SOC models to solar flares, which have a 3D geometry, we cannot measure the volume fractal dimension D_{3} directly. If we rely on the simple mean-value theorem, D_{d}=(1+d)/2 (Eq. (8)), we expect a volume fractal dimension of D_{3}=2.0. Attempts have been made to determine the 3D volume fractal dimension D_{3} from observations of 20 large-scale solar flares, using a fractal loop arcade model, which yielded a mean value of D_{3}=2.06±0.48 (calculated from Table 1 in Aschwanden and Aschwanden 2008b). Thus, we can conclude that the solar flare observations are consistent with the volume fractal dimension predicted by the standard SOC model, i.e., D_{3}=(1+d)/2=2.0 (for d=3).
The volume fractal dimension is important to derive the correct scaling law between the length scale r(t) of a SOC avalanche at a given time t and the instantaneous fractal avalanche volume V_{f}(t) (Eq. (12)), being proportional to the observed flux f(t), as well as for the total time-integrated energy e(t) (Eq. (14)). It affects the powerlaw slopes of the size distributions of avalanche areas (α_{A}), avalanche volumes (α_{V}), flux (α_{F}), and total energy (α_{E}) (see Eqs. (22) and (36)).
While the previous discussion applies to fractal geometries in 2D space with two spatial dimensions, the concept of fractals has also been applied to a time series f(t), where a fractal dimension is measured in the 2D space of f versus t. We can easily imagine that a constant function f(t)=const represents a straight line in a 2D box [t,f], and thus has the fractal dimension of D_{2}=1, while an erratically fluctuating noise time series renders a plotted box [t,f] almost black, and thus has an almost space-filling Euclidean dimension D_{2}=2. So, a fractal (or multi-fractal) dimension of a time series is essentially a measure of the time variability, and has been applied to solar radio burst data (Higuchi 1988; Watari 1996), or daily flare indices (Watari 1995; Sen 2007). A multi-fractal spectrum of the hard X-ray time profile of a solar flare was used to discriminate thermal and non-thermal emission based on their different temporal signatures (McAteer 2013b). In principle, such a dimensional time variability analysis could also be applied to SOC simulations, and this way could characterize the predicted waiting time distribution, but we are not aware of such studies.
3.1.8 Flare Geometry Measurements
The most fundamental assumption in the SOC standard model is the scale-free probability conjecture, i.e., N(L)∝L^{−d} (Eq. (1)), which should be easy to test with imaging solar observations, but there is surprisingly little statistics available. For solar flare observations we expect that SOC systems have an Euclidean dimension of d=3, and thus the prediction for the size distribution of flare length scales is N(L)∝L^{−3}. The directly measured quantity in solar flares is usually the Euclidean flare area A, which relates to the length scale by L∝A^{1/2} (Eq. (2)), and thus a size distribution of N(A)∝A^{−2} is expected.
Measurements of powerlaw slopes of solar flare size distributions of geometric parameters: length scales (α_{L}), flare areas (α_{A}), and flare volumes (α_{V}). The references are: B1998=Berghmans et al. (1998); A2000=Aletti et al. (2000); A2000b=Aschwanden et al. (2000b); AP02=Aschwanden and Parnell (2002); A2012b=Aschwanden (2012b); A2013=Aschwanden et al. (2013a); and A2012a=Aschwanden (2012a); L2010=Li et al. (2012)
Instrument | Wavelength or energy λ,ϵ | Number of events N | Length exponent α_{L} | Area exponent α_{A} | Volume exponent α_{V} | References |
---|---|---|---|---|---|---|
SOHO/EIT | 304 Å | 13,067 | 2.7 | B1998 | ||
SOHO/EIT | 195 Å | 13,607 | 2.0 | B1998 | ||
SOHO/EIT | 195 Å | 1.26±0.04 | A2000 | |||
SOHO/EIT | 195 Å | 1.36±0.05 | A2000 | |||
TRACE | 171–195 Å | 281 | 2.10±0.11 | 2.56±0.23 | 1.94±0.09 | A2000b |
TRACE/C | 171–195 Å | 3.24±0.16 | 2.43±0.10 | 2.08±0.07 | AP2002 | |
TRACE/A | 171 Å | 436 | 2.87±0.24 | 2.45±0.09 | 1.65±0.09 | AP2002 |
TRACE/B | 171 Å | 436 | 2.77±0.17 | 2.34±0.10 | 1.75±0.13 | AP2002 |
TRACE/A | 195 Å | 380 | 2.59±0.19 | 2.16±0.18 | 1.69±0.05 | AP2002 |
TRACE/B | 195 Å | 380 | 2.56±0.17 | 2.24±0.04 | 1.63±0.04 | AP2002 |
Yohkoh/SXT | AlMg | 103 | 2.34±0.27 | 1.86±0.13 | 1.44±0.07 | AP2002 |
TRACE + SXT | 171, 195, AlMg | 919 | 2.41±0.09 | 1.94±0.03 | 1.55±0.03 | AP2002 |
AIA/SDO | 335 Å | 155 | 1.96 | A2012b | ||
AIA/SDO | 94 Å | 155 | 3.1±0.6 | 2.0±0.1 | 1.5±0.1 | A2013 |
AIA/SDO | 131 Å | 155 | 3.5±0.5 | 2.2±0.2 | 1.7±0.2 | A2013 |
AIA/SDO | 171 Å | 155 | 3.5±1.2 | 2.1±0.5 | 1.7±0.2 | A2013 |
AIA/SDO | 193 Å | 155 | 3.5±0.9 | 2.0±0.3 | 1.7±0.2 | A2013 |
AIA/SDO | 211 Å | 155 | 2.7±0.6 | 2.1±0.3 | 1.6±0.2 | A2013 |
AIA/SDO | 304 Å | 155 | 2.9±0.6 | 2.1±0.2 | 1.7±0.1 | A2013 |
AIA/SDO | 335 Å | 155 | 3.1±0.4 | 1.9±0.2 | 1.6±0.1 | A2013 |
AIA/SDO | 94–335 Å | 155 | 3.2±0.7 | 2.1±0.3 | 1.6±0.2 | A2013 |
RHESSI | 6–12 keV | 1843 | 2.65±0.08 | L2012 | ||
FD-SOC prediction | 3.00 | 2.00 | 1.67 | A2012a |
Area measurements have also been carried out for supra-arcade downflows during flares, which were found not to be compatible with a powerlaw distribution (McKenzie and Savage 2011), a result that is not surprising given the small range of measured areas (covering about a half decade).
The geometric measurements are also of fundamental importance for deriving and testing physical scaling laws, which are generally expressed by a length scale (i.e., a coronal loop length), or by a volumetric emission measure (which is proportional to the total flare volume), or thermal energy (which is also proportional to the total flare volume). We will discuss such theoretical scaling laws in Sect. 3.2.7.
3.1.9 Solar Wind Measurements
The solar wind is a turbulent magneto-fluid, consisting of charged particles (electrons, protons, alpha particles, heavy ions) with typical energies of 1–10 keV, which escape the Sun’s gravity field because of their high kinetic (supra-thermal) energy and the high temperature of the solar corona. The solar wind has two different regimes, depending on its origin, namely a fast solar wind with a speed of v≲800 km s^{−1} originating from open-field regions in coronal holes, and a slow solar wind with a speed of v≲400 km s^{−1} originating from low latitudes in the surroundings of coronal streamers. The dynamics of the solar wind was originally explained by Parker (1958) as a supersonic outflow that can be derived from a steady-state solution of the hydrodynamic momentum equation. Later refinements take the super-radial expansion of the coronal magnetic field, the average macro-scale and fluctuating meso-scale electromagnetic field in interplanetary space, and the manifold micro-scale kinetic processes (such as Coulomb collisions and collective wave-particle interactions) into account. The properties of the solar wind that can be measured from the solar corona throughout the heliosphere are plasma flow speeds, densities, temperatures, magnetic fields, wave spectra, and particle composition, which all exhibit complex spatio-temporal fluctuations. Most of the observations of the solar wind were made in-situ (with the Mariner, Pioneer, Helios, ISEE-3, IMP, Voyager, ACE, WIND, Cluster, Ulysses, or STEREO spacecraft), complemented by remote-sensing imaging (with STEREO) and radio scintillation measurements.
Powerlaw slopes measured in the size distributions of magnetic field fluctuations in the solar wind. The burst energy E is defined as the area-integrated and time-integrated Poynting flux, derived from the Akasofu parameter
SOC systems produce fractal spatio-temporal structures. The fractal nature of magnetic energy density fluctuations in the solar wind has been verified observationally (Hnat et al. 2007; Rypdal and Rypdal 2010a, 2010b). Moreover, solar wind turbulence is found to be multi-fractal, requiring a generalized model with multiple scaling parameters to analyze intermittent turbulence (Macek and Szczepaniak 2008; Macek and Wawrzaszek 2009; Macek 2010), although a single generalized scaling function is sometimes sufficient too (Chapman and Nicol 2009; Rypdal and Rypdal 2011). However, the fractal geometry of solar wind bursts seems not to be self-similar, since the ratio of kinetic (E_{k}) to magnetic energy (E_{B}∝B^{2}) is frequency-dependent, with a magnetic energy spectrum of \(\propto E_{B}^{-5/3}\) and a kinetic energy spectrum of \(\propto E_{k}^{3/2}\) (Podesta et al. 2006a, 2006b, 2007). It was suggested that the interplanetary magnetic field (IMF) is clustered (self-organized) by low-frequency magnetosonic waves, leading to a fractal structure with a Hausdorff dimension of D=4/3 and a turbulent power spectrum with ν^{−5/3} (Milovanov and Zelenyi 1999).
In the end, can we claim that the dynamics of the solar wind is consistent with a SOC system? Observationally we find that magnetic field and kinetic energy fluctuations measured in the solar wind exhibit powerlaw distributions, which is consistent with a SOC system. One argument against the SOC interpretation is the observed powerlaw distribution of waiting times (Boffetta et al. 1999), but this argument applies only with respect to the original BTW model, while it presents no obstacle for nonstationary Poisson processes. Another (Occam’s razor) argument was that a SOC interpretation is not needed when turbulence can already explain solar wind spectra (Watkins et al. 2001). Considering the spatial structure of the solar wind, a fractal (or multi-fractal) property was identified, another hallmark of SOC models. What about the driver, instability, and avalanches expected in a SOC system? The driver mechanism is the acceleration of the solar wind in the solar corona itself, a process that basically follows the hydrodynamic model of Parker (1958), and may be additionally complicated by the presence of nonlinear wave-particle interactions, such as ion-cyclotron resonance (e.g., for a recent review see Ofman 2010). Then, the instability threshold, triggering extreme bursts of magnetic field fluctuations, the avalanches of solar wind SOC events, can be caused by dissipation of Alvén waves, onset of turbulence, or by the ion-cyclotron instability. Thus, in principle the generalized SOC concept can be applied to the solar wind, if there is a system-wide threshold for an instability that causes extreme magnetic field fluctuations. On the other side, the MHD turbulent cascade model explains naturally two particular spatial scales with enhanced energy dissipation (i.e., the proton the electron gyroradii), which is in contrast with the scale-freeness of energy dissipation in classical SOC models. Nevertheless, the solar wind dynamics can be described by multiple models that do not exclude each other: (1) the MHD turbulent cascade model describes the power spectrum of the solar wind, (2) kinetic theory captures the microscopic physics of wave-particle interactions and the evolution of particle velocity distributions in the solar wind, and (3) SOC models quantify the statistics and macroscopic size distributions of extreme events in the solar wind.
3.1.10 Solar-Terrestrial Effects
A solar-terrestrial effect that has been modeled in terms of SOC models is the connection between solar flare occurrence and temperature anomalies on Earth. The scaling of the Earth’s short-term temperature fluctuations and solar flare intermittency was analyzed in terms of the spreading exponent and the entropy of diffusion, finding that both have a Lévy flight statistics with the same exponent α_{Δt}=2.1 in the waiting-time distribution (Scafetta and West 2003). The same data were re-analyzed by Rypdal and Rypdal (2010a), who found that only the integrated solar flare index is consistent with Levý flight, while the global temperature anomaly follows a persistent fractional Brownian motion. The persistence (long-range memory) of solar activity was investigated further and it was found that the sunspot number and the total solar irradiance are long-range persistent, while the solar flare index is very weakly persistent, with a Hurst exponent of H<0.6 (Rypdal and Rypdal 2012). A stochastic theory to model the temporal fluctuations in avalanching SOC systems has been developed to understand these solar-terrestrial observations (Rypdal and Rypdal 2008a, 2008b). Three other Earth climate factors (average daily temperature, vapor pressure, and relative humidity) were analyzed and found to exhibit power-law distributions and thus believed to constitute a SOC system (Liu et al. 2013).
The prediction of solar-terrestrial effects, such as geoeffective solar eruptions and SEP events, resulting in space-weather storms and magnetospheric disturbances, are of course of highest interest for our society. Statistics of the most extreme events need to be derived from the rarest events at the upper end of the size distributions, where a powerlaw extrapolation is often questionable, and thus has been modeled with different cutoff functions, often associated with finite-system size effects. The best relevant data we have at hand is the solar flare statistics from the last 40 years, while geological tracers (nitrate concentrations in polar ice cores or select radionuclides) extend over millenia, but are not reliable proxy records of solar flares or SEP events (Schrijver et al. 2012), because nitrate spikes in ice cores can also be caused by biomass burning plumes (Wolff et al. 2012). Theoretical studies focus on extreme value and record statistics in heavy-tailed processes with long-range memory (Schumann et al. 2012). The inclusion of memory and persistence is obviously very important, because the predicted number of extreme events during a clustered time interval can be much larger than predicted in a purely stochastic SOC model, such as in the original BTW model (Strugarek and Charbonneau 2014).
3.2 Solar Physics: Theoretical Models
3.2.1 Solar Cellular Automaton Models
The first applications of BTW cellular automaton simulations to solar flares were made by Lu and Hamilton (1991), who interpreted the avalanches in terms of small magnetic reconnection events, where unstable magnetic energy is dissipated, and demonstrated that the powerlaw slopes of numerically simulated avalanche sizes, durations, and instantaneous peak sizes match the observed frequency distributions of hard X-ray fluences E, flare durations T, and peak fluxes P. The powerlaw slopes were found to be essentially invariant when the size of the system (i.e., the Cartesian lattice grid) was changed (Lu et al. 1993).
While the BTW model arranges an isotropic redistribution (in all next-neighbor directions), with the magnetic field strength B being the redistributed quantity, in the application to solar flares (Lu and Hamilton 1991), an anisotropic cellular automaton model with a one-directional redistribution along the direction with the largest magnetic field gradient was proposed by Vlahos et al. (1995), in order to mimic the inhomogeneity of active regions in general, and the directivity of the dominant magnetic field in the solar corona in particular. The anisotropic BTW model produced steeper powerlaw distributions than the isotropic standard model, a property that was utilized to construct a hybrid model with a steep powerlaw slope for nanoflares and a flatter slope for large flares (Vlahos et al. 1995; Georgoulis et al. 1995, 1998; Georgoulis and Vlahos 1996, 1998), which was believed to match the observations (Fig. 11). However, the anomalously steeper powerlaw slopes reported for nanoflares early on (Benz and Krucker 1998; Parnell and Jupp 2000), have been downward-corrected later on due to inadequate modeling effects (McIntosh and Charbonneau 2001; Benz and Krucker 2002; Aschwanden and Parnell 2002), and are now more consistent with the size distribution of larger flares (Fig. 11). Moreover, an anomalously steep powerlaw slope α_{P}>2 for the energy cannot be reconciled with the standard SOC model based on the scale-free probability conjecture and diffusive transport (Sects. 2.6–2.11).
The Sun often displays multiple sunspot groups or active regions at the same time, at least during the solar maximum. This implies, in Bak’s sandpile analogy, that solar flare statistics originates from multiple simultaneous sandpiles. Consequently the size distributions of active regions has to be folded into the event distributions, an effect that still produced size distributions close to a single powerlaw (Wheatland and Sturrock 1996; Wheatland 2000c).
3.2.2 Analytical Microscopic Solar SOC Models
While cellular automaton models are most powerful in simulating SOC processes, the iterative numerical scheme is generally non-deterministic and unpredictable (with some exceptions, e.g., see Strugarek and Charbonneau (2014) for a discussion on prediction from numerical SOC models), while analytical models are deterministic and give us direct physical insights into the dynamics of a SOC system. Let us review a few of the analytical approaches that have been employed to model solar SOC processes.
The spatio-temporal transport process of a SOC avalanche can macroscopically be approximated by a fractal-diffusive relationship, r(t)=κ(t−t_{0})^{β/2} (Eq. (9)), where β=1 corresponds to classical diffusion or random walk. The process of a random walk of particles through a fractal environment in 3D space was analytically described in Isliker and Vlahos (2003). Particles propagate freely in space not occupied by the fractal, but are scattered off into random directions when they hit a boundary of a fractal structure. This spatio-temporal transport process turns into a classical random walk in the limit of very sparse fractals, but produces enhanced diffusion (hyper-diffusion) with β>1 for fractal dimensions D_{d}>2. Since the diffusive spreading exponent β is a free parameter in the standard SOC model (Sect. 2.9), the analytical derivations of particle transport in fractal structures can give us physical insight into the nature of the diffusion process and the values of the spreading exponent β.
3.2.3 Analytical Macroscopic Solar SOC Models
Variations of this original model in terms of powerlaw-like growth (rather than exponential), or logistic growth (Aschwanden et al. 1998b), predict strong deviations from powerlaw distributions of flare energies and durations (Aschwanden 2011a, Chap. 3).
3.2.4 Solar Magnetic Field Models and SOC
While the original SOC models have random drivers that incur little disturbances at random places, solar SOC models became more realistic by prescribing drivers that mimic the photospheric magneto-convection (at the lower boundary of the computation box) and drive MHD turbulence in 2D (Georgoulis et al. 1998), drivers that lead to collision of large-amplitude torsional Alfvén wave packets (Wheatland and Uchida 1999), drivers that conserve helicity (Chou 1999; 2001), by calculating linear force-free fields (Vlahos and Georgoulis 2004), by calculating an initial nonlinear force-free field from an observed magnetogram (Dimitropoulou et al. 2011), by using a sequence of observed vector magnetograms as an initial condition (Dimitropoulou et al. 2013), or by designing divergence-free (∇⋅B=0) redistribution rules (Fig. 16; Morales and Charbonneau 2008a, 2008b, 2009). Several of these SOC simulations were designed to mimic coronal heating according to the field line braiding scenario postulated by Parker (1988), where the SOC driver is represented by the photospheric convection-driven random motion of coronal loop footpoints, while SOC avalanches are triggered by magnetic reconnection above some critical threshold angle of magnetic field misalignments (Krasnoselskikh et al. 2002; Morales and Charbonneau 2008a, 2008b, 2009; Uritsky et al. 2013). In one recent study, the photospheric statistics of avalanches (measured from magnetograms) and coronal statistics (measured from extreme-ultraviolet images) was performed simultaneously and scaling relationships were found between these two types of events, i.e., \(L_{cor} \propto L_{phot}^{1.39}\) and \(T_{cor} \propto T_{phot}^{0.87}\), a correlation that implies a stochastic coupling between photospheric magnetic energy injection (into the corona) and coronal heating events (Uritsky et al. 2013). This stochasticity corroborates the findings of Dimitropoulou et al. (2009) on the lack of correlations between fractal properties of the photosphere and corona.
All these recent studies clearly demonstrate an advancement from the simple original cellular automaton algorithms to more sophisticated data-driven physical models. These physical models often are able to reproduce the standard size distributions and waiting time distributions that are predicted from the standard SOC model (Sects. 2.6–2.12). For instance, the dynamic data-driven integrated flare SOC model of Dimitropoulou et al. (2013) obtains the following powerlaw slopes: α_{P}=1.65±0.11 for peak energies, α_{E}=1.47±0.13 for energies, and α_{T}=2.15±0.15 for the duration of large flares, which agrees well with the standard model (α_{P}=1.67, α_{E}=1.5, α_{T}=2.0; Eq. (24)). It proves the robustness of the generic standard SOC model, regardless of the specific physics that is involved in a particular phenomenon. Vice versa, deviations from the predicted powerlaw size distributions of the standard model can reveal crucial hints which assumptions of the standard SOC model are violated, implying possible refinements to the model.
3.2.5 Magnetic Reconnection in Solar Flares and SOC
An alternative SOC reconnection model applied to solar flares is the separator reconnection scenario (Longcope and Noonan 2000), where currents flowing along the network of magnetic field separators are sporadically dissipated. Scaling this system to solar length scales and inductances yields typical energies of E≈4×10^{28} ergs, waiting times of Δt≈300 s, and a heat flux of F≈2×10^{6} ergs s^{−1} cm^{−2}. The observed flare energy distribution N(E)∝E^{−3/2} requires a probability of P(L)∝L^{−1} for separator length scales L (Wheatland 2002; Wheatland and Craig 2003), which corresponds to a size distribution of N(L)dL∝L^{−2}dL. Generalizing the flare geometry to d=1,…,3 dimension depending on the reconnection topology (E∝L^{d}), size distributions of 4/3≤α_{E}≤2 were predicted (Craig 2001).
Solar flares produced by cascades of reconnecting magnetic loops were simulated in form of a SOC model by Hughes et al. (2003). This model produces a powerlaw distribution of flare energies with a slope of α_{E}=3.0±0.2. This prediction disagrees with most flare observations, which find α_{E}≈1.5, but it corroborates anisotropic SOC models. Despite discrepancies, the model still gives us some insight into the topology of energy dissipation regions. The standard model predicts a probability distribution of N(L)∝L^{−3} for length scales (Eq. (1)), and thus the model of Hughes et al. (2003) can be reconciled with the standard SOC model if the dissipated energy volume is proportional to the length scale, i.e., E∝L, which requires a 1D geometry of the dissipation region, such as separators of magnetic domains.
3.2.6 Particle Acceleration in Solar Flares and SOC
We can consider a hierarchy of SOC systems in our universe: our universe may be just one particular event in a multi-verse; galaxies are singular events in our universe; stars are singular events in a galactic system; planets are singular events in a solar system; solar flares are individual events in the solar corona; and accelerated particles are singular events of a solar flare hard X-ray burst. In the latter example we would consider the energy spectrum of accelerated particles as the energy distribution in a SOC system, while the acceleration process of each particle is an avalanche, driven by some electro-magnetic field in a magnetic reconnection region or shock structure. The threshold for particle acceleration is the run-away regime in a thermal plasma, which requires a velocity of a few times the thermal speed. Once the particle gets accelerated out of the thermal bulk distribution, either by a DC electric field, by wave-particle interactions, or by a quasi-parallel shock structure, it ends up with a final energy E≫E_{th} when it leaves the acceleration region, and the ensemble of all accelerated particles in a solar flare produce an energy spectrum that is often close to a powerlaw, \(N(E) \propto E^{-\epsilon_{E}}\). What powerlaw slope does the standard SOC model predict? The scale-free probability conjecture, N(L)∝L^{−d} (Eq. (1)), would still be applicable, since the probability to accelerate a particle in a subvolume with length scale L is reciprocal to the volume size. Also the fractal-diffusive transport process, L∝T^{β/2} (Eq. (9)), could still yield an appropriate model for any stochastic and diffusive (wave-particle or shock) acceleration process. However, the fractal dimension could vary from a straight trajectory with D_{d}≳1 and β≈2 to a random path with D_{d}≈(1+d)/2 and β≈1. Consequently, we predict powerlaw slopes for the energy spectrum in the range of ϵ_{E}=1+1/(γD_{3}/2+1/β)≈1.5,…,1.67 (Eq. (36)), either for D_{3}=1,…,2 or β=1,…,2. This is a relatively narrow range that should be testable. However, finite system-size effects are expected in relatively small magnetic reconnection regions, which will lead to a gradual cutoff at the upper end of the energy spectrum, with a steeper powerlaw slope if the energy spectrum is fitted with a double powerlaw function. Nevertheless, the standard SOC system predicts a lower limit of α_{E}≥1.5 for all particle spectra.
3.2.7 Hydrodynamic Flare Models and SOC
Measuring powerlaw slopes of different physical parameters in SOC systems provides a direct diagnostics or test of physical scaling laws. Hydrodynamic simulations or scaling laws were employed in a few studies in the context of SOC systems.
A shell model of MHD turbulence was used to demonstrate that chaotic dynamics with destabilization of the laminar phases and subsequent restabilization due to nonlinear dynamics can reproduce the observed waiting time distribution of N(Δt)∝(Δt)^{−2.4}, implying long correlation times, in contrast to classical SOC models that predict Poisson statistics of uncorrelated random events (Boffetta et al. 1999). A numerical simulation of a 1D MHD model of coronal loops was able to produce a similar waiting time distribution, N(Δt)∝(Δt)^{−2.3}, a result that was also used to underscore the existence of sympathetic flaring (Galtier 2001).
3.2.8 The Role of Nanoflares
3.3 Planets
Now we start our journey to review SOC interpretations in planetary atmospheres and solar system bodies, starting with the Earth’s magnetosphere (Sect. 3.3.1) and atmosphere (Sect. 3.3.2), and then continuing to lunar craters (Sect. 3.3.3), the asteroid belt (Sect. 3.3.4), Mars (Sect. 3.3.5), Saturn’s ring system (Sect. 3.3.6), Jovian and Neptunian Trojans (Sect. 3.3.7), Kuijper belt objects (Sect. 3.3.8), and extrasolar planets (Sect. 3.3.9).
3.3.1 The Earth’s Magnetosphere
In the Earth’s magnetosphere, a number of phenomena have been interpreted as features of a SOC system, such as geomagnetic substorms, current disruptions, magnetotail current disruptions and associated magnetic field fluctuations, bursty bulk flow events, and auroras seen in UV and optical wavelengths. Some of these are discussed briefly in the following, while a more detailed treatment is given in the review by Sharma et al. (2014). Magnetospheric SOC phenomena have also been reviewed previously (Aschwanden 2011a: Chaps. 1.6, 5.5, 7.2, 9.4, 10.5).
Frequency distributions measured from magnetospheric phenomena. Values determined with non-standard methods are marked with parentheses. The data sets of Uritsky et al. (2002) refer to different observing periods, the data sets of Kozelov et al. (2004) to different luminosity threshold levels, and the data sets of Uritsky et al. (2008) to different latitude zones (HL=high latitude events, LLs=low-latitude small-scale events, and LLl=low-latitude large-scale events). The predictions (marked in boldface) are based on the FD-SOC model (Aschwanden 2012a)
Phenomenon | Powerlaw slope of area α_{A} | Powerlaw slope of peak flux α_{P} | Powerlaw slope of fluence α_{E} | Powerlaw slope of durations α_{T} | References |
---|---|---|---|---|---|
Geotail flow bursts | 1.59 ± 0.07 | Angelopoulos et al. (1999) | |||
AE index | 1.24 | ||||
AU index | 1.3 | Freeman et al. (2000b) | |||
Aurora UV (substorms) | (1.21±0.08) | (1.05 ± 0.08) | Lui et al. (2000) | ||
Aurora UV (quiet) | (1.16±0.03) | (1.00 ± 0.02) | Lui et al. (2000) | ||
Aurora UV Jan 1997 | 1.73 ± 0.03 | 1.66 ± 0.03 | 1.46 ± 0.04 | 2.08 ± 0.12 | Uritsky et al. (2002) |
Aurora UV Feb 1997 | 1.74 ± 0.03 | 1.68 ± 0.03 | 1.39 ± 0.02 | 2.21 ± 0.11 | Uritsky et al. (2002) |
Aurora UV Jan 1998 | 1.81 ± 0.04 | 1.73 ± 0.02 | 1.62 ± 0.03 | 2.24 ± 0.11 | Uritsky et al. (2002) |
Aurora UV Feb 1998 | 1.92 ± 0.04 | 1.82 ± 0.03 | 1.61 ± 0.04 | 2.39 ± 0.11 | Uritsky et al. (2002) |
Aurora UV | 1.85 ± 0.03 | 1.71 ± 0.02 | 1.50 ± 0.02 | 2.25 ± 0.06 | Kozelov et al. (2004) |
Aurora TV 2.0 kR | 1.98 ± 0.04 | 2.02 ± 0.02 | 1.74 ± 0.03 | 2.53 ± 0.07 | Kozelov et al. (2004) |
Aurora TV 2.5 kR | 1.85 ± 0.04 | 1.92 ± 0.02 | 1.66 ± 0.04 | 2.38 ± 0.05 | Kozelov et al. (2004) |
Aurora TV 2.R kR | 1.86 ± 0.05 | 1.84 ± 0.03 | 1.60 ± 0.02 | 2.33 ± 0.06 | Kozelov et al. (2004) |
Aurora UV HL | 1.87 ± 0.05 | 1.81 ± 0.02 | 1.57 ± 0.02 | 2.30 ± 0.11 | Uritsky et al. (2008) |
Aurora UV LLs | 2.11 ± 0.16 | 2.16 ± 0.09 | 1.83 ± 0.04 | 3.21 ± 0.33 | Uritsky et al. (2008) |
Aurora UV LLl | 1.09 ± 0.14 | 1.32 ± 0.14 | 1.04 ± 0.12 | 1.26 ± 0.44 | Uritsky et al. (2008) |
Outer radiation belt | 1.5–2.1 | 1.5–2.7 | Crosby et al. (2005) | ||
Ionospheric disturbances | 1.8–2.5 | Bristow (2008) | |||
FD-SOC prediction: | 2.00 | 1.67 | 1.50 | 2.00 | Aschwanden (2012a) |
The SOC interpretation of magnetospheric phenomena has also stimulated cellular automaton simulations and alternative aspects of SOC modeling, such as finite system-size effects (Chapman et al. 1998, 1999; Chapman et al. 2001), powerlaw robustness under varying loading (Watkins et al. 1999), the discretization in terms of MHD equations (Takalo et al. 1999a, 1999b), renormalization group analysis (Tam et al. 2000; Chang et al. 2004), the scaling of the critical spreading exponents (Uritsky et al. 2001), phase transition-like behavior (Sitnov et al. 2000, 2001; Sharma et al. 2001), aspects of percolation and branching theory (Milovanov et al. 2001; Zelenyi and Milovanov 2004), chaotic turbulence models (Kovacs et al. 2001), forced SOC models (Consolini 2001; Chang et al. 2003), modeling of energetic particle spectra in magnetotail (Milovanov and Zelenyi 2002), MHD modeling of the plasma sheet dynamics near a SOC state (Klimas et al. 2004), aspects of complexity systems (Dendy et al. 2007), the framework of thermodynamics of rare events (Consolini and Kretzschmar 2007), kinetic theory of linear fractional stable motion (Watkins et al. 2009b), avalanching with an intermediate driving rate (Chapman and Watkins 2009; Chapman et al. 2009), and multi-fractal and fractional Lévy flight models (Zaslavsky et al. 2007, 2008; Rypdal and Rypdal 2010b).
The Earth’s magnetosphere is a large-scale natural system driven by the turbulent solar wind and exhibits non-equilibrium phenomena (Sharma and Kaw 2005), including SOC discussed here. In general the properties of such systems are characterized as a combination of global and multiscale features, and have been studied extensively using the techniques of nonlinear dynamics and complexity science (Sharma 1995; Klimas et al. 1996; Vassiliadis 2006). The first evidence of global coherence of the magnetosphere was obtained from time series data of AE index in the form of low-dimensional dynamics (Vassiliadis et al. 1990; Sharma et al. 1993). This result is consistent with the morphology of the magnetosphere derived from observations and theoretical understanding (Siscoe 1991), and simulations using global MHD models (Lyon 2000; Shao et al. 2003). The recognition of the low dimensional dynamics of the magnetosphere has stimulated a new direction in the studies of the solar wind-magnetosphere coupling and such systems in nature. Among these is the forecasting of the global conditions of space weather, viz. the AL and AE indices for substorms (Vassiliadis et al. 1995; Ukhorskiy et al. 2002, 2003, 2004; Chen and Sharma 2006) and the disturbance time index Dst for magnetic storms (Valdivia et al. 1996; Boynton et al. 2011). The forecasting of regional space weather requires data from the spatially distributed stations around the globe (Valdivia et al. 1999a, 1999b; Chen et al. 2008) and the predictability is largely determined by the availability of long time series data from the network of observing stations. The spatio-temporal dynamics of many systems are studied using such data, including the images obtained from satellite-borne imagers, by defining new variables computed from the data. For example, the fragmentation parameter (Rosa et al. 1998, 1999) represent the complexity of the spatial structure and has been used to model the dynamics of the solar atmosphere using the hard X-ray images from SOHO spacecraft. Further, the low-dimensionality of the magnetosphere has stimulated the development of models with a small number of equations (Vassiliadis et al. 1993, Horton and Doxas 1996).
The multiscale nature of the magnetosphere, expressed in many ways including the power law dependence of the scales, is a reflection of turbulence and plays an essential role in the accuracy of the forecasts. An early recognition of this was in the analogy of the dynamics of the magnetosphere to turbulence generated by a fluid flow past an obstacle (Rostoker 1984). The power law dependence of the AE index and of the solar wind provided quantitative measures of the power law indices and also the differences (Tsurutani et al. 1990). The scaling laws, which have been studies in detail using techniques such as the structure functions (Takalo et al. 1993), have many implications. The first is the characterization in terms of SOC, as discussed earlier in this section. The second is that the predictability of a multiscale system could not quantified readily in terms of the characteristic quantities such as the Lyapunov exponents (Vassiliadis et al. 1991) in a low-dimensional dynamical system. The presence of many scales as well as the non-equilibrium nature imply that predictions should be based on the statistical properties of the dynamical trajectories, e.g., using a mean-field approach (Ukhorskiy et al. 2004). Further, this approach is suitable for analyzing the predictability of extreme events (Sharma et al. 2012).
In summary, let us ask: What is the merit of the SOC concept in the context of magnetospheric phenomena? The standard fractal-diffusive SOC model (Sects. 2.6–2.11) predicts the probability distribution functions for each parameter as a function of the dimensionality (d), diffusive spreading exponent (β), fractal dimension (D_{d}), and type of (coherent/incoherent) radiation process (γ). The waiting time distributions are predicted by the FD-SOC model to follow a powerlaw with a slope of α_{Δt}≈2 during active and contiguously flaring episodes, while an exponential cutoff is predicted for the time intervals of quiescent periods. This dual regimes of the waiting time distribution predict both persistence and memory during the active periods, and stochasticity during the quiescent periods. All these predictions of the FD-SOC model provide useful constraints of the physical parameters and underlying scaling laws. Significant deviations from the size distributions predicted by the FD-SOC model could imply problems with the measurements or data analysis, such as indicated by the contradicting results of Lui et al. (2000) and Uritsky et al. (2002) in the case of auroral size distributions.
Let us emphasize again that the generic FD-SOC model is considered to have universal validity and explains the statistics and scaling between SOC parameters, but does not depend on the detailed physical mechanism that governs the instabilities and energy dissipation in a particular SOC process. The physical process may be well described by a number of established models, such as turbulence theory, kinetic theory, wave-particle interactions, and other branches of plasma physics. There was also a debate whether magnetospheric substorms are SOC or forced-SOC (FSOC) (e.g., Chang et al. 2003), an issue that largely disappears in our generalized FD-SOC concept, where a slow driver is required to bring the SOC system continuously near to the instability limit, but is does not matter whether the driver is internally, externally, or is globally organized.
3.3.2 Terrestrial Gamma-Ray Flashes
Terrestrial gamma-ray flashes (TGF) are gamma-ray bursts of terrestrial origin that have been discovered with the Burst and Transient Experiment (BATSE) onboard the Compton Gamma Ray Observatory (CGRO) and have been studied with RHESSI, Fermi, and AGILE since. These TGF bursts are produced by high-energy photons of energy >100 keV and last up to a few milliseconds. They have been associated with strong thunderstorms mostly concentrated in the Earth’s equatorial and tropical regions, at a typical height of 15–20 km (Fishman et al. 1994; Dwyer and Smith 2005; Smith et al. 2005). The physical interpretation is that the TGF bursts are produced by bremsstrahlung of high-energetic electrons that were accelerated in large electric potential drops within thunderstorms. However the gamma-rays produced in thunderstorms (at 5 km) can not readily propagate to higher altitudes due to atmospheric absorption. A mechanism for the generation of gamma rays that can reach the satellite-borne instruments is through the excitation of whistler waves by the relativistic electrons generated in the thunderstorms (Kaw et al. 2001; Milikh et al. 2005). The whistler waves form a channel by nonlinear self-focusing and the relativistic electrons propagate in this channel to higher altitudes (30 km). The gamma-ray generated at this altitude can escape the atmosphere and thus account for the BATSE/CGRO results.
A size distribution of the gamma-ray emission from TGF events needs to be corrected for the distance from the TGF-producing thunderstorm to the detecting spacecraft (in Earth orbit). In a combined analysis of TGF data from the RHESSI and Fermi satellites, corrected for their different orbits, different detection rates, and relative sensitivies, a true fluence distribution was derived, which was found to have a powerlaw shape of α_{E}=2.3±0.2 if a sharp cutoff was assumed, or a slope of α_{E}≤1.3–1.7 when a more realistic roll-over of the RHESSI lower detection threshold is assumed (Ostgaard et al. 2012).
We can consider a part of the Earth’s atmosphere that contains a thunderstorm as a SOC system of finite size, where the electrostatic charging process represents the driver, the critical condition for electric discharging is given by an electric conductivity threshold, and the spontaneously triggered gamma-ray flashes or lightenings represent the avalanches. The PDF is then given by the scale-free probability conjecture (Eq. (1)), which together with the fractal-diffusive transport predicts an energy or fluence distribution with a powerlaw slope of α_{E}=1.5 in 3D space, which matches the observed and corrected fluence distribution with a slope of α_{E}≈1.3–1.7. The agreement with the standard FD-SOC model is consistent with an incoherent process for gamma-ray production, where the gamma-ray flux is proportional to the emitting volume of a TGF.
3.3.3 Lunar Craters and Meteorites
An amazingly straight powerlaw size distribution has been found for the sizes of lunar craters (Fig. 25), with a cumulative powerlaw slope of \(\alpha_{L}^{cum}=2.0\) over a size range of L=0.65–69,000 m, which covers 5 orders of magnitude (Cross 1966), derived from crater statistics measured in pictures of the lunar probes Ranger 7, 8, 9 combined with a lunar map of Wilkins (1946). Since a cumulative size distribution is flatter than a differential size distribution (by a value of one), this corresponds to a powerlaw slope of \(\alpha_{L} = \alpha_{L}^{cum} + 1 = 3.0\). A similar powerlaw index of α_{L}=2.75 was found for the size distribution of meteorites and space debris from man-made rockets and satellites in the range of L=10 μm–10 cm (Fig. 3.11 in Sornette 2004).
The size distribution of meteorites and planetesimals may also be generated by a SOC process in the first place. The slow driver that provides the trickling of sand grains is the gravity-driven formation process of the solar system itself, which clumps the local molecular cloud into meteorites and planets. The aspect of self-organized criticality, which is a balance between the gravity and the frictional force that controls the critical angle of repose in Bak’s sandpile, can be understood as a critical point between the condensation rate of planetesimals or meteorites (by self-gravity) and the diffusion rate (driven by thermal pressure and external gravitational disturbances). This critical threshold given by the balance of the condensation rate and the diffusion rate has to be exceeded in order to initiate the gravitational collapse that forms a solar system body. The gravitational collapse is the underlying instability in a physical SOC concept (Fig. 1, right frame).
Hence, from such a generalized point of view, we might consider the meteorite formation as a SOC pr ocess and the resulting lunar cratering as the imprint of this process. The main benefit of the FD-SOC framework is the direct prediction of the scale-free size distribution of crater sizes, i.e., N(L)∝L^{−3} (Eq. (1)), which can also be used as a prediction for any other targets in the solar system, such as cratering on Earth, Mars, or Mercury. This allows us, for instance, to predict the collisional probability of an asteroid hitting our Earth, although we have to take into account the variability of the impact rate, which varied drastically during the lifetime of our solar system. Both the Moon and the Earth were subject of intense bombardment between 4.0 and 3.7 billion years ago, which was the final stage of the sweep-up of debris left over from the formation of the solar system (Bottke et al. 2012). The impact rate at that time was thousands of times higher than it is today.
3.3.4 The Asteroid Belt
The asteroid belt is a large accumulation of irregular small solar system bodies orbiting the Sun between the orbits of Mars and Jupiter. The largest of these small bodies is Ceres, with a diameter of 1020 km, followed by Pallas (538 km), Vesta (549 km), Juno (248 km), and extends down to the size of dust particles. While most planetesimals from the primordial solar nebula formed larger planets under the influence of self-gravity, the gravitational perturbations from the giant planets Jupiter and Saturn prevented a stable conglomeration of planetesimals in the zone between Mars and Jupiter. This fragmented soup of primordial planetesimals makes up the asteroid belt. The larger asteroids (≥120 km) are believed to be primordial, while the smaller ones are likely to be a byproduct of fragmentation events (Bottke et al. 2005).
Frequency distributions measured from planetary phenomena
Phenomenon | Instrument | Powerlaw slope of length α_{L} | Powerlaw slope of fluence α_{E} | References |
---|---|---|---|---|
Terrestrial γ-ray flashes | 1.3–1.7 | Ostgaard et al. (2012) | ||
Lunar craters | Ranger 7, 8, 9 | 3.0 | Cross (1966) | |
Meteorites, space debris | 2.75 | Sornette (2004) | ||
Asteroid belt | Spacewatch Surveys | 2.8 | Jedicke and Metcalfe (1998) | |
Asteroid belt (<5 km) | Sloan Survey | 2.3 | Ivezic et al. (2001) | |
Asteroid belt (>5 km) | Sloan Survey | 4.0 | Ivezic et al. (2001) | |
Asteroid belt | Subaru Survey | 2.3 | ||
Jovian Troyans (<40 km) | Hawaii 2.2 m | 3.0±0.3 | Jewitt and Trujillo (2000) | |
Jovian Troyans (>40 km) | Hawaii 2.2 m | 5.5±0.9 | Jewitt and Trujillo (2000) | |
Neptune Trojans | Subaru Survey | 5 ± 1 | Sheppard and Trujillo (2010) | |
Kuiper belt objects | 4.3 | Fraser et al. (2008) | ||
Saturn ring | Voyager 1 | 2.74–3.11 | Zebker et al. (1985) | |
Saturn ring | Voyager 1 | 2.74–3.11 | French and Nicholson (2000) | |
Extrasolar planets | Kepler | 2.48 | Catanzarite and Shao (2011) | |
FD-SOC prediction: | 3.00 | 1.50 | Aschwanden (2012a) |
If the small bodies in the asteroid belt are formed by a SOC process, the scale-free probability conjecture predicts a size distribution of N(L)∝L^{−3}, which is indeed close to what is observed (Fig. 25). However, there are slight deviations from a single powerlaw distribution for small and large bodies, which indicate some additional effects. Nevertheless, an almost scale-free behavior is observed for a range of L≈0.4–50 km, which makes it appropriate to consider the formation process in terms of a SOC system. As we discussed for the formation of meteorites above (Sect. 3.3.3), the aspect of self-organized criticality can be understood as a critical point between the condensation rate of planetesimals or meteorites by self-gravity, and the diffusion rate driven by external gravitational disturbances, mostly from the giant planets Jupiter and Saturn. If this critical threshold of the ratio of the condensation rate to the diffusion rate exceeds the value of unity, the self-gravity force takes over and forms a small solar system body, which represents an avalanche process with a well-defined instability threshold.
3.3.5 Mars
It has also been suggested to apply SOC dynamics to Martian fluvial systems (Rosenshein 2003). The motivation was that complexity theory provides powerful methods to analyze, interpret, and model terrestrial fluvial systems, including the fractal structure of meandering, sediment dynamics, bedrock incision, and braiding.
Another application of SOC systems to Mars is the statistics of dust storms, especially the interannual variability of Mars global dust storms (Pankine and Ingersoll 2004a, 2004b). Previously it was thought that the threshold for wind speed for starting saltation and lifting dust from the Martian surface was a finely tuned process. In the study of Pankine and Ingersoll (2004a, 2004b), however, it was shown that the fine-tuning of this parameter could be the result of a negative feeback mechanism that lowers the threshold of the wind speed. In this way, the Martian atmosphere/dust system could organize itself as a SOC system, and no fine-tuning of a critical threshold is required.
3.3.6 Saturn’s Ring System
Saturn and Jupiter are the most massive planets in our solar system with a gravity that is sufficiently strong to keep numerous moons, rings, and ringlets in their strong gravitational field. The Saturn ring extends from 7,000 km to 80,000 km above Saturn’s equator, consisting of particles ranging from 1 cm to 10 m, with a total mass of 3×10^{19} kg, which is comparable with the mass of its moon Mimas. Theories about the origin of Saturn’s ring range from nebular material left over from the formation of Saturn itself, collisional fragmentation (Greenberg et al. 1977), to the tidal disruption of a former moon.
3.3.7 Jovian and Neptunian Trojans
The Jovian Trojans are two swarms of asteroids, which lead or trail Jupiter by ±60^{∘} on its orbit, known as the Lagrangian L4 and L5 point. The Jovian Trojans contain some 250 members. Their origin has been interpreted in terms of trapping of asteroidal fragments. A statistical analysis yielded a differential size distribution of N(L)∝L^{−3.0±0.3} in the size range of L=2–30 km, and N(L)∝L^{−5.5±0.9} in the size range of L=50–84 km (Jewitt and Trujillo 2000).
Similarly, Trojans have been detected in the L4 and L5 regions of the planet Neptune, with a size distribution that approaches a powerlaw slope of α_{L}=5±1 at the upper end (Sheppard and Trujillo 2010), while a flatter slope is found at the lower end. The scarcity of intermediate- and smaller-sized Neptune Trojans (≤45 km), which is also found for other objects in the Kuiper Belt, Jovian Trojans, and main belt asteroids, was interpreted in terms of a primordial origin, rather than a collisional or fragmentational origin, for which a size distribution of N(L)∝L^{−3} is expected in the SOC model. However, the smaller bodies of the Neptunian Trojans in the range of L=2–30 km could still be consistent with a SOC origin, if they have the same distribution as Jovian Trojans (with N(L)∝L^{−3.0±0.3}; Jewitt and Trujillo 2000). Their size range and distribution is close to that of asteroids (Fig. 25).
3.3.8 Kuijper Belt Objects
The Kuijper belt is a region of our solar system beyond the orbit of Neptune (at 30 AU) out to ≈50 AU, consisting of many small bodies. A size distribution of N(L)∝L^{−4.3} was found for objects with L≳100 km (Fraser et al. 2008; Fraser and Kavelaars 2008; Fuentes and Holman 2008). A comparison of the cumulative size distributions of Kuiper Belt objects, Neptunian Trojans, Jovian Trojans, and asteroids is shown in Fig. 25. Obviously, there is a paucity of objects in the zone of L≈30–45 km that shows up in the Neptunian Trojans and in the Kuiper belt objects (Fig. 25). The data seem to be consistent with the predicted powerlaw slope of α_{L}≈3 only for small length scales of L≈1–30 km.
3.3.9 Extrasolar Planets
The oligarchic growth of protoplanets has been brought into the context of a self-organized protoplanet-planetesimal system (Kokubo and Ida 1998). The growth and orbital evolution of protoplanets embedded in a swarm of planetesimals has been simulated with a 3D N-body code, which shows the relative distribution of large planets that grow oligarchically, while most of the planetesimals remain small (Kokubo and Ida 1998).
Using the Kepler space telescope for search of Sun-like stars and (extrasolar) planets, a sample of over 150,000 stars was measured during the first 4 months of the mission. The Kepler science team determined sizes, surface temperatures, orbit sizes, and periods for over a thousand new planet candidates. From a size distribution of 1176 Earth-sized planet candidates within a range of L=2,…,20 Earth radii, a powerlaw distribution was found in the range of L≈2–10 Earth radii (Fig. 25), with a powerlaw slope of \(\alpha_{L} = \alpha_{L}^{cum} + 1 = 1.48 + 1 = 2.48\), while the relatively narrow distribution falls of steeply between L≈10–20 Earth radii (Catanzarite and Shao 2011).
This sample from 1176 different stars can be considered as a galactic SOC system, in which case a size distribution of N(L)≈L^{−3} is predicted by the FD-SOC model, which is close to the observed value of N(L)≈L^{−2.5} for a subset of Earth-like planets. The accretion of an Earth-like planet represents then an avalanche event, triggered by a gravitational instability in each stellar system.
3.4 Stars and Galaxies
We can obtain information on spatial scales and spatio-temporal scaling laws from SOC phenomena in our solar system (i.e., from the Sun, the planets, the magnetosphere), while such information from the rest of the universe is concealed by distance and cosmological time scales. Nevertheless, a number of stellar phenomena have been attributed to SOC phenomena. The observables are mostly time durations T, peak fluxes P, and fluences E of electromagnetic emission in some wavelength range, measured with some automated event detection algorithm from time series of a stellar object. We will compile such observations from stellar flares, pulsars, soft gamma-ray repeaters, blazars, and black-hole objects in the following, and compare them with the predictions of the FD-SOC model.
3.4.1 Stellar Flares
Time series with rapidly fluctuating emission in soft X-rays, EUV, and visible light from individual stars have been gathered with EXOSAT (Collura et al. 1988; Pallavicini et al. 1990), the Hubble Space Telescope (HST) (Robinson et al. 1999), the Extreme Ultraviolet Explorer (EUVE) (Osten and Brown 1999; Audard et al. 1999, 2000; Kashyap et al. 2002; Güdel et al. 2003; Arzner et al. 2007), the X-ray Multi-Mirror Mission (XMM) or Newton (Stelzer et al. 2007), and most recently with the surveys of the Kepler mission (Walkowicz et al. 2011; Maehara et al. 2012; Shibayama et al. 2013). Impulsive bursts detected in the time series in excess of the noise level have been interpreted as stellar flares, because they show similar temporal and wavelength characteristics as solar flares, except that they exceed solar flares in their luminosity by several orders of magnitude (Aschwanden et al. 2008c). Therefore, they should be considered as “giant flares” by solar standards. These stellar flares have been observed mostly in solar-like G-type stars (Notsu et al. 2013; Maehara et al. 2012; Shibayama et al. 2013), and in cool dwarf (dMe) stars (Robinson et al. 1999; Audard et al. 2000; Kashyap et al. 2002; Güdel et al. 2003; Arzner et al. 2007; Stelzer et al. 2007; Walkowicz et al. 2011; Maehara et al. 2012). From soft X-ray and EUV spectroscopy, flare temperatures of T_{e}≈10–100 MK have been determined in some of the stellar flares, exceeding solar flare temperatures (T_{e}≈5–35 MK). Consequently, the same physical interpretation in terms of magnetic reconnection with subsequent heating of chromospheric plasma has been proposed for stellar flares, in analogy to their solar analogs, although their total emission measure is a few orders of magnitude larger than for solar flares (Aschwanden et al. 2008c).
Frequency distributions measured from stellar flares. The predictions (marked in boldface) are based on the FD-SOC model (Aschwanden 2012a)
Star | Instrument | Number of events | Powerlaw slope of peak flux α_{P} | Powerlaw slope of fluences α_{E} | References |
---|---|---|---|---|---|
13 M dwarfs | EXOSAT | 17 | 1.52 ± 0.08 | Collura et al. (1988) | |
22 M dwarfs | EXOSAT | 20 | 1.7 ± 0.1 | Pallavicini et al. (1990) | |
RS CVn | EUVE | 25 | 1.5–1.7 | Osten and Brown (1999) | |
47 Cas, EK Dra | EUVE | 28 | 1.8–2.3 | Audard et al. (1999) | |
YZ Cmi | HSP/HST | 54 | 2.25 ± 0.10 | Robinson et al. (1999) | |
HD 2726 | EUVE | 15 | 1.9–2.6 | Audard et al. (2000) | |
47 Cas | EUVE | 12 | 2.0–2.6 | Audard et al. (2000) | |
EK Dra | EUVE | 16 | 1.8–2.3 | Audard et al. (2000) | |
κ Cet 1994 | EUVE | 5 | 1.9–2.6 | Audard et al. (2000) | |
κ Cet 1995 | EUVE | 10 | 2.2–2.5 | Audard et al. (2000) | |
AB Dor | EUVE | 16 | 1.8–2.0 | Audard et al. (2000) | |
ϵ Eri | EUVE | 15 | 2.4–2.5 | Audard et al. (2000) | |
GJ 411 | EUVE | 15 | 1.6–2.0 | Audard et al. (2000) | |
AD Leo | EUVE | 12 | 1.7–2.0 | Audard et al. (2000) | |
EV Lac | EUVE | 12 | 1.8–1.9 | Audard et al. (2000) | |
CN Leo 1994 | EUVE | 14 | 1.9–2.2 | Audard et al. (2000) | |
CN Leo 1995 | EUVE | 14 | 1.5–2.1 | Audard et al. (2000) | |
FK Aqr | EUVE | 50 | 2.60 ± 0.34 | Kashyap et al. (2002) | |
V1054 Oph | EUVE | 70 | 2.74 ± 0.35 | Kashyap et al. (2002) | |
AD Leo | EUVE | 145 | 2.1–2.3 | Kashyap et al. (2002) | |
AD Leo | EUVE | 261 | 2.0–2.5 | Güdel et al. (2003) | |
AD Leo | EUVE | 2.3 ± 0.1 | Arzner and Güdel (2004) | ||
HD 31305 | XMM | 22 | 1.9–2.5 | Arzner et al. (2007) | |
TMC | XMM | 126 | 2.4 ± 0.5 | Stelzer et al. (2007) | |
G5-stars | Kepler | 1538 | 1.88 ± 0.09 | 2.04 ± 0.13 | Shibayama et al. (2013) |
FD-SOC prediction | 1.67 | 1.50 | Aschwanden (2012a) |
Since solar flares show the trend of a steeper powerlaw slope α_{E} in the fluences measured in soft X-rays and EUV, compared to hard X-rays, we suspect also that the prolonged thermal emission in soft X-rays and EUV, due to plasma cooling, boosts the time-integrated fluence so that the total dissipated energy is overestimated, unlike the fluences in hard X-rays, where thermal emission is completely negligible at electron energies E≥25 keV. Unfortunately, current hard X-ray detectors are not sensitive enough to detect hard X-ray emission from stellar flares.
Thus, we conclude that hard X-rays provide the most accurate measurements of dissipated energies during flares, which are also consistent with the predictions of the FD-SOC model, while soft X-rays, EUV emission, and white-light (bolometric) emission exhibits a nonlinear scaling with the emitted energy. The fluence measured in soft X-rays and EUV emission are boosted due to plasma heating and cooling processes. The reconciliation of measurement methods of the total dissipated energy in hard X-rays, soft X-rays, and EUV is still an open problem, which could be resolved with multi-wavelength statistics of solar data, and by modeling the scaling laws between dissipated energies and the fluxes in different wavelengths. Apparently the bias in the soft X-ray and EUV wavelengths affects the energy distributions measured from (giant) stellar flares to a larger degree than those of solar flares.
3.4.2 Star Formation
The formation of stars is initiated by gravitational collapses of molecular clouds. Such a gravitational collapse can be triggered by collisions of two molecular clouds, by the explosion of a nearby supernova, which ejects shocked matter, or even by galactic collisions, which cause compression of matter and tidal forces. If there is a critical mass reached, which is quantified by the Jeans mass criterion, which mostly depends on the initial size of the unstable galactic fragment, the collapsing molecular cloud will build up a dense core by self-gravity, which forms a star with nuclear burning. Smaller sizes develop into non-radiating brown dwarfs.
Considering star formation as a SOC process, the situation is similar to the formation of planetesimals and planets, where a critical condition is given by the balance between the forces of self-gravity and diffusion. A collapsing molecular cloud gains kinetic energy from the gravitational potential according to the conservation of angular momentum. However, tidal forces, external gravitational disturbances, and thermal pressure represent forces that contribute to the local diffusion of the molecular cloud. Therefore there is a threshold for the instability of a gravitational collapse, which is self-organizing by the given balance between the opposing forces of contraction and diffusion. This process could possibly also be modeled in terms of a percolation model.
SOC avalanches have a fractal structure, and hence fractals are expected for star-forming regions also. Indeed, fractal and self-fimilar patterns have been observed in the Milky Way from dense cores to giant molecular clouds in a range of 0.1<L<100 pc (Elmegreen and Scale 2004; Bergin and Tafalla 2007), as well as in star-forming regions in the Andromeda nebula M33 (Sanchez et al. 2010). The fractal dimension in the interstellar medium has a value of D_{3}≈2.3±0.3 (Elmegreen and Falgarone 1996), in bright young stars and molecular gas is D_{2}≈1.9, and in fainter stars and HII regions is D_{3}≈2.2–2.5. The predictions of the FD-SOC model is D_{3}≈2.0. The fractal structure has generally been attributed to interstellar turbulence, which however does not exclude a generalized description in terms of a SOC process. It has been argued that the interstellar mass function (IMF) of starbursts is independent of local processes governing star formation and thus can be considered as a universal self-organized criticality process (Melnick and Selman 2000).
3.4.3 Pulsars
A pulsar is a highly magnetized, rapidly-rotating neutron star that emits a beam of electromagnetic radiation. Since the beamed emission is aligned with the magnetic axis, we observe rotationally modulated pulses whenever the beam axis points to the Earth (line-of-sight direction) during each period of its rapid rotation. Besides these regular periodic pulses on time scales of milliseconds, which are measured with high accuracy, there occur sporadic glitches in pulse amplitudes and frequency shifts, probably caused by sporadic unpinning of vortices that transfer momentum to the crust (Warszawski and Melatos 2008). Conservation of angular momentum produces then a tiny increase of the angular rotation rate, called “positive spin-ups” of the neutron star.
Frequency distributions measured from giant pulses of pulsars (Crab, Vela, PSR), soft gamma-ray repeaters (SGR), black-hole objects (Cygnus X-1, Sgr A^{∗}), and a blazar (GC 0109+224). The size distributions were reported in units of (cumulative) pulse energies (Argyle and Gower 1972), in radio flux densities (Lundgren et al. 1995), (cumulative) pulse amplitudes (Cognard et al. 1996), electric fields (Cairns 2004), fractional increase of the spin frequency (Δν/ν) (Melatos et al. 2008), or peak fluxes (Ciprini et al. 2003). Powerlaw slopes of peak fluxes are marked with parentheses. Uncertainties (standard deviations) are quoted in brackets
Object | Waveband | Number of events | Powerlaw slope of energies α_{S},(α_{P}) | References |
---|---|---|---|---|
Crab pulsar | 146 MHz | 440 | 3.5 | Argyle and Gower (1972) |
Crab pulsar | 813–1330 MHz | 3×10^{4} | 3.06–3.36 | Lundgren et al. (1995) |
PSR B1937+21 | 430 MHz | 60 | 2.8 ± 0.1 | Cognard et al. (1996) |
PSR B1706-44 | 1.5 GHz | 6.4 ± 0.6 | Cairns (2004) | |
Vela pulsar | 2.3 GHz | 6.7 ± 0.6 | Cairns (2004) | |
PSR B0950+08 | 0.4 GHz | 6.2 ± 0.5 | Cairns (2004) | |
Crab pulsar | 0.8 GHz | 5.6 ± 0.6 | Cairns (2004) | |
PSR B1937+214 | 0.4 GHz | 4.6 ± 0.2 | Cairns (2004) | |
PSR B1821-24 | 1.5 GHz | 9.0 ± 2.0 | Cairns (2004) | |
PSR 0358+5413 | 6 | 2.4 [1.5, 5.2] | Melatos et al. (2008) | |
PSR 0534+2200 | 26 | 1.2 [1.1, 1.4] | Melatos et al. (2008) | |
PSR 0537+6910 | 23 | 0.42 [0.39, 0.43] | Melatos et al. (2008) | |
PSR 0631+1036 | 9 | 1.8 [1.2, 2.7] | Melatos et al. (2008) | |
PSR 0835+4510 | 17 | −0.13 [−0.20, +0.18] | Melatos et al. (2008) | |
PSR 1341+6220 | 12 | 1.4 [1.2, 2.1] | Melatos et al. (2008) | |
PSR 1740+3015 | 30 | 1.1 [0.98, 1.3] | Melatos et al. (2008) | |
PSR 1801+2304 | 9 | 0.57 [0.092, 1.1] | Melatos et al. (2008) | |
PSR 1825+0935 | 8 | 0.36 [−0.30, 1.0] | Melatos et al. (2008) | |
SGR 1806-20 | 1.6 | Chang et al. (1996) | ||
SGR 1900+14 | >25 keV | 1.66 | Gogus et al. (1999) | |
SGR 1806-20 | >21 keV | 1.43, 1.76, 1.67 | Gogus et al. (2000) | |
Gamma-ray bursts | 83 | 1.06 ± 0.15 | Wang and Dai (2013) | |
GC 0109+224 | optical | (1.55) | Ciprini et al. (2003) | |
Cygnus X-1 | 1.2–58.4 keV | (7.1) | Mineshige and Negoro (1999) | |
Sgr A^{∗} | 2–8 keV | 1.5, (1.0) | Nielsen et al. (2013) | |
FD-SOC prediction | 1.50, (1.67) | Aschwanden (2012a) |
A cellular automaton model has been developed for pulsar glitches, based on the superfluid vortex unpinning paradigm (Warszawski and Melatos 2008, 2012; Melatos and Warszawski 2008). The lattice grid in this model simulates the collective behavior of up to 10^{16} vortices in the interior of the pulsar. The cellular automaton generates scale-free avalanche distributions with powerlaw slopes of α_{S}=2.0–4.3 for avalanche sizes, and α_{T}=2.2–5.5 for avalanche durations. This numerical model produces size distributions that are not too far off the predictions of the FD-SOC model (α_{E}≈1.5, α_{T}=2.0), but covers an intermediate range between the flatter slopes reported by Melatos et al. (2008) and the steeper slopes observed in radio wavelengths earlier. Larger observational statistics and a consistent definition of avalanche energies is needed to settle the pulsar SOC problem.
3.4.4 Soft Gamma Ray Repeaters
A class of gamma-ray bursts that were detected with the Compton Gamma Ray Observatory (CGRO), the Rossi X-ray Timing Explorer (RXTE), and International Cometary Explorer (ICE) in hard X-rays ≈20–40 keV (a wavelength regime that is also called soft gamma-rays), with repeated detections from the same source location, has been dubbed Soft Gamma Ray Repeaters (SGR). These gamma-ray bursts are believed to originate from slowly rotating, extremely magnetized neutron stars (magnetars) that are located in supernova remnants (Kouveliotou et al. 1998, 1999), where neutron star crust fractures occur, driven by the stress of an evolving, ultrastrong magnetic field in the order of B≳10^{14} G (Thompson and Duncan 1996). We should be aware that repeated bursts from the same source are the exception rather than the rule for gamma-ray bursts.
The size distributions of the fluences of sources SGR 1900+14 and SGR 1806-20 were found to exhibit powerlaw distributions with slopes of α_{E}=1.66 (Gogus et al. 1999) and α_{E}=1.43, 1.76, and 1.67 (Gogus et al. 2000), extending over a range of about 4 orders of magnitude in fluence. The waiting time distributions were found to be consistent with a log-normal distribution (which is approximately a powerlaw function in the upper tail). Based on these observational statistics, SGR bursts have been interpreted in terms of a SOC process (Gogus et al. 1999; 2000). Since the source location is identical for an object that produces SGR bursts, we can identify it with a single SOC system, an assumption that cannot be made for other gamma-ray bursts, which are non-repetitive and often do not have an unambiguous source identification with known distance. Moreover we find that the fluence or energy distribution of the bursts matches the prediction of the fractal-diffusive SOC model, with α_{E}=1.5. In the magnetar model, the triggering mechanism for SGR bursts is a hybrid of stress-induced starquakes and magnetically powered flares (Thompson and Duncan 1996), and thus has some similarity with the physical process of earthquakes.
A recent study was carried out with data from the Swift satellite, which has a rapid response, suitable for detecting afterglows of gamma-ray bursts. In a sample with 83 localized sources for which the redshift was known (and thus the distance), a size distribution of (distance-corrected) energies could be constructed, and a powerlaw distribution with slope of α_{E}=1.06±0.15 was found (Wang and Dai 2013). The size distribution of time duration was found to have a slope of α_{T}=1.10±0.15. These results were interpreted in terms of a 1D SOC system (Wang and Dai 2013), for which the FD-SOC model predicts α_{E}=1 and α_{T}=1. This 1D interpretation for gamma-ray bursts with afterglows appears to be different from soft gamma-ray repeaters, which are consistent with a 3D SOC system.
3.4.5 Blazars
Blazars are very compact objects associated with super-massive black holes in the center of active, giant elliptical galaxies. They represent a subgroup of active galactic nuclei (AGN), which emit a relativistic beam or jet that is aligned or nearly-aligned with the line-of-sight direction to Earth. Due to this particular geometry, blazars exhibit highly variable and highly polarized emission in radio and X-ray emission. Optically violent variable (OVV) quasars are a subclass of blazars.
The optical variability of blazar GC 0109+224 was monitored from 1994 onwards and the light curve exhibited a power spectrum P(ν)≈ν^{−p}, with 1.57<p<2.05 (Ciprini et al. 2003), which is consistent with the 1/f or flicker noise characteristics of SOC avalanches in the BTW model (Bak et al. 1987; Hufnagel and Bregman 1992). The frequency distribution of radio peak fluxes of flaring events from blazar GC 0109+224 was found to be a powerlaw distribution (over about one order of magnitude), N(P)∝P^{−1.55} (Ciprini et al. 2003), which is consistent with the prediction of the FD-SOC model, i.e., N(P)∝P^{−1.67}, within the uncertainties of the measurements. Interpreting blazars as a SOC phenomenon, the critical threshold for a pulse is given by the geometric coalignment condition between the emitted beam direction (of accelerated particles producing gyrosynchrotron emission) and the observer’s line-of-sight direction from Earth. The intermittency of blazar bursts observed on Earth is believed to be caused by sporadic bursts of energy releases, created by internal shocks that occur within AGN jets.
3.4.6 Black Holes and Accretion Disks
The first Galactic X-ray source that has been identified as a black-hole candidate, Cygnus X-1, emits hard X-ray pulses with a time variability down to 1 ms. These hard X-ray pulses are attributed to inverse Compton scattering of soft photons by hot electrons heading toward the event horizon within the black hole’s accretion disk.
Statistics of the fluctuations in the light curve from Cygnus X-1, observed in hard X-rays with Ginga and Chandra, exhibit complex 1/f noise spectra and size distributions of peak fluxes with very steep powerlaw slopes of α_{P}≈7.1 (Negoro et al. 1995; Mineshige and Negoro 1999), which have been interpreted in terms of SOC models applied to accretion disks (Mineshige et al. 1994a, 1994b; Takeuchi et al. 1995; Mineshige and Negoro 1999). A SOC interpretation was also suggested for the VY Scl-type cataclysmic variable KR Aurigae (Kato et al. 2002; Dobrotka et al. 2012), UU Aqr (Dobrotka et al. 2012), and for the broad-line radio galaxy 3C-390.3 (Leighly and O’Brien 1997), for the Seyfert I MCG-6-30-15 (Sivron and Goralski 1998; Sivron 1998), or for the extreme narrow-line Seyfert 1 galaxy IRAS 13224-3909 (Gaskell 2004), which all exhibit a highly intermittent variability on top of a shot noise background like Cygnus X-1.
In contrast, a total of 39 X-ray flares observed with Chandra from Sgr A^{∗}, the 4×10^{6} M_{⊙} black hole at the center of our Galaxy, revealed powerlaw distributions with slopes of α_{P}=1.9±0.4 for the peak luminosity (of the 2–8 keV flux) and α_{E}=1.5±0.2 for the fluence (Nielsen et al. 2013), which is perfectly consistent with the predictions of the FD-SOC model (α_{P}=1.67 and α_{E}=1.5).
Cellular automaton models were constructed to mimic mass accretion by avalanches that are triggered when the mass density of the disk exceeds some critical value, which could reproduce the 1/f power spectra N(ν)∝ν^{−1.6} and produced size distributions with powerlaw slopes of α_{E}=2.8 for energies and α_{T}=1.4 for durations (Mineshige et al. 1994a, 1994b; Yonehara et al. 1997). A BTW-related model produced an energy distribution of α_{E}=1.35 (Mineshige et al. 1994a, 1994b) that is closer to the FD-SOC prediction (α_{E}=1.5). Adding gradual diffusion to the SOC avalanches in the cellular automaton simulations produced a steeper (exponential) energy size distribution that was closer to the observations (Takeuchi et al. 1995). Further modified cellular automaton models were developed that include reservoirs of different capacities (Negoro et al. 1995), hydrodynamic models of advection-dominated accretion disks (Takeuchi and Mineshige 1997), relativistic effects (Xiong et al. 2000), non-local transport of angular momentum in terms of the kinematic viscosity of magnetic loops in the accretion disk corona (Pavlidou et al. 2001), and boson clouds around black holes (Mocanu and Grumiller 2012).
Most of the various cellular automaton models designed to mimic a physical mechanism operating in black-hole objects have difficulty to reproduce the observed steep size distributions, while most of them seem to produce 1/f power spectra without special assumptions. The observed steep size distributions may represent deviations of the accretion disk system from a pure SOC system. The notion of SOC may still be useful to understand the observations, but it cannot explain all properties of the fluctuations.
3.4.7 Galactic Structures
What physical mechanism produces galactic structures? A nonlinear theory was proposed in which the structure of spiral galaxies arises from percolation phase transition (Schulman and Seiden 1986a, 1986b; Seiden and Schulman 1990). The differential rotation of the galaxy triggers propagating patterns of star formation. This scenario is very similar to a SOC model, since it has a critical point at the second-order phase transition associated with the percolation threshold, which causes avalanches of star formations. Percolation processes, however, require fine-tuning, in contrast to SOC systems. The process of stochastic self-propagating star formation was simulated with a cellular automaton model that provides a representation of the percolation process operating in spiral galaxies (Seiden and Schulman 1990).
The formation of galaxies has been modeled with two opposite scenarios, the top-down scenario that starts with a monolithic collapse of a large cloud (Eggen et al. 1962; Zeldovich 1970), versus the now more widely accepted bottom-up scenario, where smaller objects merge and form larger structures that ultimately turn into galaxies (Searle and Zinn 1978; Peebles 1980). The second scenario is more widely accepted now and corresponds also closer to a SOC-driven avalanching scenario. In most models of galaxy formation, thin, rotating galactic disks result as a consequence of clustering of dark matter halos, gravitational forces and disturbances, and conservation of angular momentum. The fractal-like patterns of the universe from galactic down to solar system scales is thought to be a consequence of the gravitational self-organization of matter (Da Rocha and Nottale 2003). Fractal structures are observed throughout the universe (Baryshev and Teerikorpi 2002). It is conceivable that gravitational forces in an expanding universe lead to sporadic density fluctuations or waves that initiate a local instability of self-gravitating matter like an avalanche in a sandpile SOC model, in case a critical threshold exists without need of fine-tuning.
3.4.8 Cosmology
The spatial flatness, homogeneity, and isotropy of the universe at cosmological scales can be considered as a critical point that would require an extreme fine-tuning, unless there is a self-organizing principle that creates such a special state in a natural way. Moffat (1997) proposes that the universe evolves as a SOC system (in the sense of a BTW model), where the Hubble expansion undergoes “punctuated equilibria” like the SOC scenario of intermittent evolution (Bak and Sneppen 1993). The inflationary scenario, which predicts a rapid expansion of the early universe to explain the flatness and the horizon problem, could be the manifestation of a major SOC avalanche, while a SOC scenario would predict many intermittent inflationary phases (Moffat 1997). The critical point of a cosmological system would be the critical density Ω=1 that discriminates between an open (Ω<1) and a closed (Ω>1) universe, independent of the initial conditions and without fine tuning of the parameters. A related SOC concept has also been applied to quantum gravity (Ansari and Smolin 2008). With the recent advent of string theory and multi-verses, we might even consider our universe being only one single avalanche episode in a multi-verse SOC scenario.
3.4.9 Cosmic Rays
High-energy particles can be accelerated by a number of physical mechanisms, e.g., by electric fields, by shock waves, or by stochastic wave-particle interactions, such as by cyclotron resonance, which requires magnetic fields. In Sect. 3.2.6 we discussed how a first-order Fermi process as well as a fractal reconnection model can produce the observed powerlaw spectra of high-energy particles (Nishizuka and Shibata 2013). Cosmic rays, which travel through a large part of the universe, probably undergo many local acceleration processes, and thus their trajectories may look like a diffusive random walk. The acceleration process of cosmic rays has been interpreted in terms of a SOC process (Aschwanden 2014). The critical threshold is the “runaway regime” (e.g., Holman 1985) of a charged particle in a thermal distribution, which is a critical velocity, i.e., v_{crit}≳4v_{th}, that is necessary to enable efficient acceleration out of the thermal distribution. Considering the subsequent acceleration process as a SOC avalanche, which can be achieved by an arbitrary number of localized acceleration steps, the particles are likely to undergo a diffusive random walk, as it is characterized by the fractal-diffusive SOC model. The FD-SOC model predicts than a powerlaw distribution for the energy spectrum of accelerated particles, which is approximately fulfilled for cosmic rays (as well as for nonthermal particles in solar flares). The FD-SOC model predicts an energy spectrum of N(E)≈E^{−1.5}, which is however different from the observed cosmic ray spectrum with N(E)≈E^{−3.0}. This discrepancy has been interpreted in terms of an incomplete sampling effect of cosmic-ray avalanches (Aschwanden 2014). Since cosmic rays are in-situ measurements in a very localized target region (i.e., the Earth surface), only a small 1-D cone of an isotropic cosmic-ray avalanche is sampled, leading to an energy gain that is proportional to the traveled length scale, i.e., L∝E, and thus to an energy spectrum N(E)∝N(L)∝L^{−3}∝E^{−3}. Solar flare observations, in contrast, provide remote-sensing of a complete SOC avalanche of accelerated particles, and thus are expected to have an energy spectrum of N(E)∝E^{−1.5}, which is indeed an asymptotic limit for the hardest solar flare spectra (e.g., Dennis 1985; Miller et al. 1997).
4 Discussion: SOC Concepts, Critiques, New Trends, and Open Problems
4.1 A Dual Approach of Self-Organized Criticality Systems
A theory or a physical model is only useful (or acceptable) if it can make quantitative predictions, and if these predictions can be tested by observations, and hence the theory is falsifyable. What is the current status of a SOC theory or a SOC model? In this review we stress the dual nature of SOC models, in the sense that they include (i) universal statistical aspects that apply to all SOC systems, and (ii) special physical mechanisms that are idiosyncratic to a particular SOC phenomenon. There is a consensus that the powerlaw function of the size distribution of a SOC observable is a universal statistical aspect that is common to all SOC systems, regardless whether we sample statistics of solar flares or earthquakes, while the underlying physical mechanisms are completely different, such as magnetic reconnection in solar flares, or mechanical stressing in earthquakes. If we accept this dichotomy, we should be able to build a generalized SOC theory that predicts the universal statistical properties, which should be purely of “mathematical nature” and “physics-free”, while the nonlinear energy dissipation process of a SOC event still can be described with (single or multiple) specific physical SOC models that are different for every SOC manifestation. In this spirit we reviewed the basic elements of a generalized SOC theory in Sect. 2, while we touched on possible interpretations in terms of particular physical mechanisms that produce a SOC phenomenon in Sect. 3.
Let us review how the definition of a generalized SOC theory evolved over the last 25 years. The BTW model essentially defined a SOC process by simulating a cellular automaton, which demonstrated that a powerlaw size distribution resulted for avalanche sizes and durations. Since 1/f noise has a power spectrum in the form of a powerlaw function, the claim was made that both phenomena may be related. Many of the subsequent studies came up with different cellular automaton models, which produced a range of powerlaw slopes (see Table 1 and Pruessner 2012), some of them produced exact powerlaw size distributions over many orders of magnitudes (which demonstrated “universality” with regard to the scale-free size range, such as the Manna and Oslo model), while others exhibited significant deviations from exact powerlaw distributions (and thus cannot claim universality). The next important insight concerned the relationships between the powerlaw slopes of different SOC parameters, which depend on the nonlinear scaling laws between the SOC parameters. Further progress was made by predicting the statistical probability distributions of SOC parameters, using branching theory, percolation theory, discretized diffusion models, or renormalization group theory. A more detailed review on these theoretical and mathematical efforts is given in the article by Watkins et al. in this volume. A very simple theoretical framework that unifies many features of previous SOC models is the fractal-diffusive SOC model, based on the scale-free probability conjecture (Eq. (1)), which is able to predict probability distributions of observable SOC parameters and the underlying scaling laws between the SOC parameters. This basic SOC model has no free parameters for the most common case of 3D Euclidean space and classical diffusion transport, and offers a prediction for most of the astrophysical observations of SOC systems reviewed here. The model can also be adjusted to a different space dimension, fractal dimension, and type of diffusive transport. The FD-SOC model should be considered as a macroscopic approximation of the complex micro-dynamic processes in a SOC system.
Is this SOC theory complete? By no means, there is still a lot of statistics and data analysis required to pin down the scaling laws, statistical truncation bias, event selection bias, and other unknown effects for those SOC phenomena where the generic FD-SOC model yields a different prediction than what is observed. In addition, there are a number open questions in SOC models that try to reproduce real-world data, such as the time variability of the driver, effects that cause deviations from ideal powerlaw distributions, predictive capabilities, alternative SOC-related processes, which are discussed in the following sections.
4.2 Universal Aspects of SOC Systems
The universal aspects that are common to all SOC phenomena define a SOC theory. In Fig. 1 we sketched the basic characteristics of a SOC system: (i) a critical threshold for instabilities, (ii) a statistically slow driver that continuously nudges the system toward a critical point, and (iii) a nonlinear energy dissipation process when an avalanche is triggered. The most crucial and testable predictions of a SOC theory are the statistical probability distributions. The central key feature of the FD-SOC theory is the statistical probability argument for geometric length scales, the so-called scale-free probability conjecture, N(L)dL∝L^{−d}dL. This statistical argument is derived by the same principle as a binomial distribution is derived for a stochastic process by enumerating all possible outcomes of dice combinations. This conjecture can easily be tested by extensive statistics of length scales, such as we demonstrated for lunar craters, asteroid sizes, Saturn ring particle sizes, magnetospheric aurora sizes, solar flare sizes, and can be done in the same way for other SOC processes, such as earthquake rupture areas, for instance. An additional assumption of the FD-SOC model is the fractal-diffusive transport, which involves random walk statistics for the avalanche transport, i.e., L∝T^{1/2} for classical diffusion, and a fractal geometry of the instantaneous avalanche size \(V_{f} \propto L^{D_{d}}\), where the mean fractal dimension can be estimated from the mean-value dimension D_{d}≈(1+d)/2. Integrating such a fractal-diffusive avalanche in time yields then the total size S of an avalanche. If energy dissipation of an avalanche is proportional to the time-integrated size of an avalanche, we obtain the total dissipated energy E∝S, the energy dissipation rate F∝V_{f}, and the peak energy dissipation rate P∝V. In astrophysical applications, the energy dissipation rate F is generally measured by the flux or intensity of electromagnetic radiation in some wavelength, but the universal meaning of the energy dissipation rate is simply the instantaneous avalanche size during a snapshot, while the total energy is the time-integrated avalanche volume. Thus this generalized SOC concept is still universally applicable to every SOC system, regardless if it is observed by an astronomical instrument, by a geophysical monitor, by financial statistics, or by computer lattice simulations.
4.3 Physical Aspects of SOC Systems
The physics comes in once we identify the avalanche, the threshold, and the dissipated energy with a particular instability in the real world (Table 16). For solar flares, for instance, the threshold may be given by a critical stressing angle between a potential and non-potential magnetic field line in an active region, the avalanche may be manifested by a solar flare emitting in all wavelengths, triggered by a magnetic reconnection process of the over-stressed magnetic field lines, and the dissipated energy can be measured by the change of magnetic energy before and after the flare, or by the thermal energy of the heated plasma, or by the total kinetic energy of accelerated particles. If we consider an earthquake, the threshold may be given by the limit of elastic stressing of tectonic plates, the instability is the slip-stick motion of the tectonic plates, the avalanche is the spatio-temporal pattern of the rupture area on the Earth’s surface, and the measured energy is the magnitude indicated by the vibrations detected by a seismometer. The advantage of separating the universal aspects from the physical aspects of a SOC system is that we can understand the statistics of SOC parameters independently of the physical model of a SOC phenomenon. For instance, we have very vague ideas about the exact physical process that occurs in pulsar glitches, in giant pulses from black-hole candidates, or in the bursts from soft gamma ray repeaters, but the FD-SOC model can predict the distributions and basic scaling laws between spatial and temporal parameters. In the case of imaging observations, where we can measure both spatial and temporal scales, the FD-SOC model can place absolute values on the diffusion coefficients, which may help to identify the physical transport process that occurs during an avalanche. We should also be aware that the universal FD-SOC model assumes a proportionality between the avalanche size S and total dissipated energy E, which may not always be the case, such as for coherent emission mechanisms (e.g., laser or maser emission), which requires a specific physical model.
4.4 Powerlaws and Deviations
The functional shape of size distributions of SOC parameters is generally expected to be a powerlaw function, i.e., N(x)dx∝x^{−α}dx, which is a consequence of the scale-free nature of SOC processes. Numerical simulations of cellular automaton models were indeed capable to reproduce an exact powerlaw probability distribution function for avalanche sizes over many orders of magnitude, such as the Manna model (Manna 1991) or the Oslo model (Christensen et al. 1996), while substantial deviations from ideal powerlaw functions have been found in real-world observations, which raises the question how well the ideal powerlaw distributions predicted by standard SOC models characterizes real-world data. Taken to the extreme, sceptics doubt whether powerlaws have any relevance at all (Stumpf and Porter 2012).
Starting from first principles, a powerlaw function of length scales is predicted from the scale-free probability conjecture (Eq. (1)) in our generic standard model, which is fundamentally based on the principle of statistical maximum likelihood, and does not depend on any other assumption. However, this ideal distribution function is always limited within a finite range of spatial sizes [x_{1},x_{2}], given by the spatial resolution limit or lower limit of complete sampling x_{1}, and the finite system size or maximum avalanche size x_{2} that happened during the observed time interval x_{2}. So, the powerlaw function is expected only over this limited range [x_{1},x_{2}], while there is generally a rollover at the lower end and an exponential-like drop-off at the upper end. However, this range can be enlarged by lowering the lower limit x_{1} by more sensitive instruments, and by increasing the upper limit x_{2} by extending the total observing time (in case the largest avalanche does not exceed the finite system size).
Starting from the powerlaw function of the length scale distribution N(L)∝L^{−d}, the FD-SOC model predicts powerlaw distribution functions for all other parameters, such as the area A, the volume V, the fractal area A_{f}, the fractal volume V_{f}, the flux F, the peak flux P, the fluence S, and energy E, because these SOC parameters are all related to each other by powerlaw relationships, such as by the definition of the Hausdorff dimension D_{d}, or the diffusive transport with spreading exponent β. Even the intermittent waiting times are predicted to be a powerlaw for contiguous flaring periods, with the only exception of quiescent time intervals, which may follow an exponential distribution (if they are produced randomly).
There are a number of additional effects that cause deviations from ideal powerlaw distribution functions. The most obvious deviations occur from the truncation of distribution functions. If we have statistics over many orders of magnitude, the truncation effects are less severe, but are crucial for small samples. Let us explain this with an example that is illustrated in Fig. 20. Solar flare statistics is usually limited by a peak count threshold, i.e., complete sampling is only achieved for P≥P_{thresh}. In this case we expect a perfect powerlaw for the differential size distribution of peak fluxes in the range of P_{thresh}≤P≤P_{max}, where P_{max} is the count rate of the largest observed flare (for instance see Fig. 20e). However, if we sample the statistics of a related parameter, such as the thermal energy (Fig. 20f), the peak count threshold causes a truncation effect that extends over the lower half (logarithmic) range of energies, where sampling is not complete in energy and thus produces a broken powerlaw with a flatter slope in the lower half (logarithmic) range. The same truncation effect affects also linear regression fits. Nevertheless, these truncation effects can be numerically simulated or analytically calculated (e.g., see example in Aschwanden and Shimizu 2013, Appendix A), and this way can be taken into account in the prediction of the probability distribution functions of SOC parameters.
4.5 The Meaning of Self-Organized Criticality
After we have reviewed a large number of astrophysical observations (Sect. 3) with powerlaw behavior, the question arises whether all of these observed phenomena are SOC systems, and which are not consistent with a SOC interpretation. To answer this question we remind again our pragmatic generalized definition of a SOC system: SOC is a critical state of a nonlinear energy dissipation system that is slowly and continuously driven towards a critical value of a system-wide instability threshold, producing scale-free, fractal-diffusive, and intermittent avalanches with powerlaw-like size distributions (Aschwanden 2014). This definition is independent of any particular physical mechanism, but describes only some universal system behavior that is common to virtually all threshold-operated nonlinear energy dissipation processes, in the limit of slow driving. Given this much larger perspective of a SOC definition, we can ask whether the term “self-organized crtiticality” is still justified in this context, which includes also “critical points” now that define a threshold for an instability. The terms “self-organizing”, “self-tuning”, or “self-adjusting” mean in this context only that the system is continuously driven towards a critical threshold, without necessity of external control. If we have a self-sustaining slow driver, the continuous pushing of the system towards an instability threshold is automatically organized. In case the driver stops, the triggering of instabilities stops too, and the system becomes static. For instance, the cratering of the Moon has almost stopped, and thus the observed craters are only remnants of a dynamical state. However, during the times of heavy lunar meteorite bombardment, a SOC system with a critical (relative-velocity) threshold that triggered impacts on the Moon like earthquakes, or a scale-free distribution of meteorites could be an alternative source. The same is true for solar flares: there are quiescent static periods during the solar cycle minimum when nearly no magnetic flux is generated by the solar dynamo, while flaring during the maximum of the solar cycle constitutes a highly dynamic period of a continuously driven SOC system.
4.6 SOC and Turbulence
What is the relationship between a SOC system and a turbulent system? Because both systems exhibit powerlaw functions in the power spectrum, scale-free size distributions, and many degrees of freedom, there are commonalities that make their distinction difficult. Vortices are in turbulence what avalanches are in SOC. A first difference was noted in the predicted waiting time distribution. The original BTW model considered SOC avalanches as statistically independent and thus predicted an exponential waiting time distribution, while turbulent media exhibit long correlation times and predict powerlaw-like waiting time distributions (Boffetta et al. 1999; Giuliani et al. 1999; Freeman et al. 2000a). This argument, however, is alleviated by alternative SOC models (Sect. 2.12), such as non-stationary flaring rates (Wheatland et al. 1998), the fractal-diffusive SOC model (Aschwanden 2014), or models with persistence and memory as modeled with the Weibull distribution (Telloni et al. 2014).
How is SOC different from turbulence? Both processes may produce similar statistics for slow driving, but start to differ when we move from slow to intermediate driving, when the smallest avalanches are “swamped”, but the large avalanches persist, so that intermittent turbulence shows only finite-range powerlaw scaling (Chapman et al. 2009; Chapman and Watkins 2009; Chapman and Nicol 2009).
Instead of considering single vortices in fully developed turbulence as the equivalent of a SOC avalanche, a more satisfactory concept may be the notion of SOC in the state of near-critical turbulence, which is in the transition between the laminar state and the fully developed turbulence state. In this regime, the system profiles that store the free energy exciting the turbulence (i.e., pressure or temperature gradients, in a fusion plasma for instance) are very close to their local threshold values for the onset of instability. As a result, the local perturbations excited when these thresholds are overcome (giving rise to local eddies) may propagate to nearby locations (other eddies) as the former are relaxed, and the local profiles are brought back below critical. This is similar to the BTW sandpile. This regime is considered to be important in tokamak plasmas, because the local turbulent fluxes that bring the profiles back below a marginal state are strongest at higher temperatures. Thus, the equivalent to an avalanche is not a single eddy, but a chain of eddies at different locations connected in time, very much as a sand avalanche would happen in a sandpile. And the system is still in a turbulence-dominated regime, although turbulence is fully-developed only locally, not globally. These ideas were proposed in the mid-90s in the lab fusion community (Carreras et al. 1996; Newman et al. 1996; Mier et al. 2008; Sanchez et al. 2009), and have been given rise to a large body of work in the area of SOC, dealing with self-similarity, long-temporal correlations, and non-diffusive transport.
Intermittent turbulence (IT) and self-organized criticality (SOC) seem to co-exist in the magnetic field fluctuations of the solar wind at time scales of T=10–10^{3} s (Podesta et al. 2006a, 2006b, 2007), and in the solar corona (Uritsky et al. 2007). It was proposed that the coexistence of SOC and IT may be a generic feature of astrophysical plasmas, although the explicit complementarity between SOC and IT in astrophysical observations has not been demonstrated (Uritsky et al. 2007), IT phenomena can be explained without invoking SOC (Watkins et al. 2009a), and may need multi-fractal scaling (Macek and Wawrzaszek 2009), or three turbulence regimes (Meyrand and Galtier 2010).
The extent to which SOC and turbulence phenomena are really separable in complex systems is subject to a few conditions and topological constraints, also involving the ambient dimensionality. In two embedding dimensions, there is a theoretical possibility that SOC couples to turbulence via the inverse cascade of the energy, giving rise to large-amplitude events beyond the range of applicability of the conventional SOC (Milovanov and Rasmussen 2014). It has been discussed that the phenomenon occurs universally in two-dimensional fluid (as well as fluid-like, such as the drift-wave and drift-Alfven) turbulence and requires time scale separation in that the Rhines time of the vortical system must be small compared with the instability growth time. Then the typical avalanching behavior associable with SOC will be amplified by the inverse cascade, which acts as to fuel the SOC avalanches “on-the-fly” with the energy. The energy reservoir for this behavior is only limited to the finite size of the system. It has been suggested that this new complexity phenomenon, the SOC-turbulence coupling, has serious implications for operational stability of big fusion confinement devices such as for instance the future power plants, where it may trigger transport events of potentially a catastrophic character (Milovanov and Rasmussen 2014). In this regard, it was argued that SOC was not really an alternative to the notion of turbulence and that there is kind of SOC-turbulence duality instead, coming along with the condition for time scale separation. A hybrid SOC-turbulence model has also been developed based on statistical arguments, using nonlocal transport and the formalism of a space-fractional Fokker–Planck equation (Milovanov and Rasmussen 2014). According to the hybrid model, the processes of amplification taking place will manifest themselves in the form of algebraic tails on top of the typical log-normal behavior of the probability distribution function of the flux-surface averaged transport. This suggestion finds further justification in the general properties of log-normal behavior in hierarchical systems with subordination (Montroll and Shlesinger 1982). In the realm of solar physics, SOC predicts scale-free distributions for large avalanche events (e.g., in solar flaring active regions) down to the smallest avalanche events (e.g., in nano-flaring or non-flaring active regions), which implies also the same turbulence characteristics for flaring and non-flaring active regions, as it has been observationally verified (Georgoulis 2012). Related unifications of SOC processes, intermittent turbulence, and chaos theory include analysis of dynamical complexity via nonextensive Tsallis entropy (Milovanov and Zelenyi 2000; Balasis et al. 2011; Pavlos et al. 2012), fractional transport models (Zelenyi and Milovanov 2004; del Castillo-Negrete 2006), and the formalism of fractional Ginzburg–Landau equation (Milovanov and Rasmussen 2005; Milovanov 2013).
4.7 SOC and Percolation
A recent discussion of the SOC concept versus the percolation problem is given in Milovanov (2013). Both SOC and percolation systems share the implications of threshold behavior, the spatial self-similarity, and fractality. One essential difference is that percolation is a purely geometrical model, while SOC involves also the temporal fractality, i.e., the 1/f noise. Another difference is the role of fine-tuning, which needs an externally manipulated control parameter in a percolation system, while it is automatically self-organizing in a SOC system. However, some nonlinear phenomena have been modeled with both SOC and percolation models, such as the spread of diseases or forest fires, which indicates a strong commonality between the two models, as well as some ambiguity in the choice of the most suitable model for a given observed phenomenon (e.g., Grassberger and Zhang 1996). Regarding numerical simulations, both models can be represented with iterative lattice-grid simulations, using similar mathematical re-distribution rules in each iterative step. It has been discussed that SOC and percolation systems can be both represented with cellular automation models, but having different re-distribution rules. In the basic theoretical perspective, though, this lattice-grid approach seems to overly simplify the integral picture of the self-organization, as it tends to disregard the peculiar role of nonlinearity behind the phenomena of SOC. Generally, standard percolation processes can be made self-organized by including a feedback loop generating self-organization in a marginally stable state. Then marginal dynamical stability of systems with spatio-temporal coupling will also require marginal topological connectedness (Milovanov 2013), so that in the presence of many dynamical degrees of freedom the operation of nonlinear feedback will automatically lead the system into a state of critical percolation. This general theoretical framework has been demonstrated on a lattice model using random walks to represent the microscopic re-distribution rules and the idea of “holes” or missing occupied sites which by themselves could participate to the random walk and dynamically generate a feedback (Milovanov 2010; 2011).
From a practical (or observational) perspective, the question arises whether the percolation and SOC models predict the same, or different, size distributions, after adjustment of the optimum control parameters. However, since any automation model is only an idealized representation of microscopic physics in complex systems, none of the two models is expected to mimic microscopic transport to an accurate level, but may rather approximate the microscopic size distributions. In this regard, the advantage of the random walk approach once again lies in a theoretically consistent picture of the dynamics, making it possible to obtain non-Markovian kinetic equations at criticality in terms of fractional calculus (Milovanov 2009; 2011). The main idea here is that fractional generalizations of the diffusion and Fokker–Planck equations (e.g., Metzler and Klafter 2000 for review) incorporate via a Laplace convolution the key signatures of non-Gaussianity and long-time dependence characteristic of the dynamical systems at or near SOC. One by-product of the fractional model is the prediction that the relaxation of a super-critical system to SOC is of Mittag–Leffler type (similar to the Cole-Cole behavior in glassy systems and polymers: see Milovanov 2011). The Mittag–Leffler relaxation implies that the behavior is multi-scale with a broad distribution of durations of relaxation events consistently with a description in terms of fractional relaxation equation (e.g., Metzler and Klafter 2000; Sokolov et al. 2002) and at odds with a single-exponential relaxation dynamics of the Debye type (Coffey 2004 for an overview; references therein). We should stress that the notion of feedback plays a very important role in the phenomena of SOC, as it ensures a steady state, where the system is marginally stable against a disturbance (Kadanoff 1991). For instance, in sandpiles, the unstable sand slides off to decrease the slope and reinstall stability, thus providing a feedback of the particle loss process on the dynamical state of the pile. Following Sornette (1992), we also note that, using the idea of feedback, it is possible to convert the standard critical phenomena into self-organized criticality dynamics, thereby extending considerably the span of models exhibiting SOC. One example of this conversion is localization-delocalization transition on a separatrix system of nonlinear Schrödinger equation with disorder and self-adjusting nonlinearity, giving rise to a percolation structure in wave-number space, which is critical and self-organized (Milovanov and Iomin 2012; 2014).
In solar and astrophysics, percolation models have been applied to the formation of galaxies (Schulman and Seiden 1986a, 1986b; Seiden and Schulman 1990), to magnetotail current systems and the phenomena of tail current disruption (Milovanov et al. 1996; Milovanov et al. 2001; Milovanov 2013; Arzner et al. 2002), to the solar dynamo (Schatten 2007), to photospheric magnetic flux concentrations (Balke et al. 1993), and to the emergence of solar active regions (Wentzel and Seiden 1992; Seiden and Wentzel 1996). Each of these phenomena can also be modeled with a threshold-operated instability in a SOC system. Hence, the jury is still out which model describes the real-world observations better.
4.8 SOC and Branching Theory
A branching process is a Markov process (i.e., a memory-less process) that models a population with a random distribution at time step n to predict the number of individuals in the next generation or time step n+1 according to some probability distributions. To some degree, the branching process during a single time step has the same purpose as the re-distribution rule in a cellular automaton simulation. The question is whether the two processes have the same probability distributions for the spatio-temporal evolution of an avalanche event. The branching theory was mostly applied to the evolution of a population, which ended either in infinite growth or in global extinction. SOC avalanches end always after a finite time interval, and thus can only evolve as a branching process with final extinction. What is common to both processes is a critical threshold or critical probability for next-neighbor or next-generation propagation. Therefore, a self-organized branching process with critical probabilities (Zapperi et al. 1995; Corral and Font-Clos 2013) has much in common with a SOC system of the BTW-type. Again, for practical purposes to model observations, we may ask whether the two models predict equal or different size distributions, using some suitable critical probabilities.
In astrophysics, self-organizing branching theory has been applied to magnetotail current systems (Milovanov et al. 2001), to solar soft X-rays (Martin et al. 2010), and to solar flares (MacKinnon and MacPherson 1997; Macpherson and MacKinnon 1999; Litvinenko 1998). The branching theory applied to solar flares (MacKinnon and MacPherson 1997; Macpherson and MacKinnon 1999; Litvinenko 1998) as well as the self-organized branching process (SOBP) model (Zapperi et al. 1995; Hergarten 2012) predict both a size distribution of N(S)∝S^{−3/2}, which is also predicted by the fractal-diffusive self-organized criticality (FD-SOC) model, and thus indicates an equivalent description of the multiplicative avalanche growth characteristics, and makes these two models indistinguishable with regard to their size distributions.
4.9 Challenges and Open Questions in Solar SOC Models
Attempting to connect the idealized analytical or numerical SOC models with real-world (astro)physical systems, one faces a host of questions that remain unanswered. We briefly touch on a few issues that arised from solar SOC models.
Evolving SOC Drivers:
The driver of a SOC system may naturally evolve in and out of a SOC state, vary cyclically or intermittently, or oscillate between low and high states. Well-known examples are the solar dynamo that cyclically modulates the solar flare rate, or the variability of low and high states in the black-hole object Cygnus X-1. Another example is a time-variable driver of solar active regions, as described in McAteer et al. (2014), which emulates how a dissipative, nonlinear dynamical system enters a SOC state. Standard SOC models, such as the BTW model, assume a steady driver and do not take into account the particular system behavior of variable SOC drivers. Real-world SOC systems are operated by time-variable drivers that are never exactly constant, which may alter the statistical distributions that are predicted from a constant driver. During the decay phase of a SOC driver, the dissipative properties of the system may possibly diffuse the available energy in a gradual (non-intermittent) fashion, and this way reduce the system’s control parameter to a value below the critical threshold, and this way inhibit intermittent instabilities (avalanches).
In a tectonic system, for example, earthquakes in an area would stop when inter-plate stresses are somehow mollified by repelling mantle motions below. While this is a hypothetical and hardly observable fact, at least within reasonable geological timescales, a solar active region, even a fiercely flaring/eruptive one, emerges, evolves, and disappears within weeks. If this system evolves into a SOC state, as amply argued in this review, then the discontinuation of magnetic-flux emergence from the solar interior signals the start of this active region’s demise. It is both, the time-dependent proper motions within the region (e.g., shear, sunspot rotation, outflows), as well as the overall solar differential rotation, which apparently quench the SOC-decaying driver in a gradual and non-intermittent manner by exhausting the region’s free magnetic energy. Flux emergence and related motions, on the other hand, become the realization of the classical SOC-building driver. The implementation of distinct SOC-building and SOC-decaying drivers, with the first being dominant during the SOC phase of the system, but weak or absent during the system’s decay, can characterize the finite lifetime of SOC states.
In the limit of a statistically slow SOC-building driver, another conceivable way for a system to exit SOC is by a “catastrophic” quenching of the SOC state by a single, system-wide instability that dissipates a substantial part of the system’s available energy. While this is in principle not prohibited in a SOC system and could occur when the entire system becomes a network of marginally stable configurations, observations suggests otherwise: earthquake clusters (e.g., Corral 2004) reveal that only a relatively small portion of the stress-accumulated free energy is released, regardless how powerful the earthquake is. In other words, a seismic fault does not disappear after any single earthquake. The same is qualitatively the case for solar active regions, where the total available free energy can be calculated or estimated (e.g., Tziotzou et al. 2013). This rule of thumb seems to indicate that SOC states fade away gradually only, rather than by a catastrophic event at once.
Hidden, Anisotropic, and Composite SOC States:
A potentially interesting finding on the evolution of a time-variable SOC driver was obtained from experiments with a static data-driven (S-IFM) and a dynamically driven (D-IFM) flare model of Dimitropoulou et al. (2011; 2013), described in McAteer et al. (2014). A given nonlinear force-free extrapolated magnetic field of an observed solar active region is first evolved into a SOC state, yielding a random but valid divergence-free magnetic configuration due to the S-IFM’s random driving. Next, the system is evolved back to the initial extrapolated state via D-IFM, while monitoring tests confirmed that the SOC state has not been destroyed. Therefore, one cannot rule out that the initial extrapolated-field state, despite being a force-free-equilibrium state, may in fact be a SOC state. The subject active region for the test happened to be an eruptive one; however, the eruptive property was not used in the test. Therefore, unless eruptive solar active regions have a topologically or otherwise distinct magnetic structure compared to non-eruptive ones, the same test might possibly work equally well with a non-eruptive active region. Should this be confirmed, it would be evidence that solar active regions, regardless of an eruptive or non-eruptive nature, may be in a SOC state. The question then arises, besides active regions, whether the quiet-Sun (or global stellar) magnetic field is in a SOC state also? This remains to be assessed. The lack of major flares and eruptions from non-eruptive active regions and the quiet Sun may be due to the lack of available free-energy density accumulation, a much weaker SOC-building driver, or a critical threshold of a different nature, heuristically proposed as an “anisotropic” SOC threshold by Vlahos et al. (1995) and subsequent works. It is now observed from exceptionally high-resolution solar observations that small-scale energy-release events resembling the hypothesized nanoflares occur in the active and the quiet solar corona (Cirtain et al. 2013; Winebarger et al. 2013). If the entire solar corona is in a “composite” SOC state, albeit with different critical thresholds and drivers in different regions, then there is a possibility to extend SOC validity over the global magnetic configurations of magnetically active, main-sequence stars.
Robustness of Power Laws:
Probability distributions of sizes and durations exhibit generally a powerlaw function with a specific slope for a given observable (such as the peak count rate, fluence, rise time, or decay time). The value of the powerlaw slope becomes the more robust, the larger the statistics is, gathered over sampling times as long as possible. Even for small statistics and short sampling times, the value of the powerlaw slope may be robust, as long as the driver is constant and the sample is statistically representative. However, this robustness is lost when subsets of data are histogrammed that contain some selection bias. This loss of robustness has been demonstrated in a study by Crosby et al. (1998), using a sample of some 1500 X-ray flares from the WATCH/GRANAT satellite, when subsets were selected by groups with different event durations: the power laws were found to be steeper for subsets with short duration, while they progressively flattened for longer events. A similar result was found for total-count distribution functions of these flares by Georgoulis et al. (2001), which was also used for a “statistical flare SOC cellular automaton model” (Georgoulis and Vlahos 1998).
The effect of a selection bias in time durations T on the size distribution function of an observable, such as the peak flux P, can easiest be understood from a scatterplot between the parameters P and T. If there would be an exact correlation with a correlation P∝T^{a}, the distribution N(P;T=T_{i}) of a subset with duration T_{i} would be a δ-function N(P=P_{i}) at the value \(P_{i} \propto T_{i}^{a}\). In reality, the correlations have a substantial scatter, which broadens the size distributions of each subset, but the trend that they are clustered around the value \(P_{i} \propto T_{i}^{a}\) persists. A consequence of this scatter between correlated parameters is also that the threshold in an observable (say in the peak count rate, P≥P_{0}), causes a truncation bias in the correlated parameter (say T≳T_{0}). Therefore, even when the peak rate distribution N(P) exhibits an exact powerlaw down to the threshold value P_{0} of the sample, the correlated time duration distribution N(T) will have a smooth rollover, which is a significant deviation from an ideal powerlaw. At the upper end of size distributions, finite-size effects cause an additional fall-off, which is another deviation from an ideal powerlaw distribution. These well-understood effects should be taken into account in arguments countering power laws and their validity and interpretation, as expressed by Stumpf and Porter (2012).
Hybrid SOC Models and Multi-Fractal Effects:
There is also a controversy about the hypothesized “soft” nanoflare population (Parker 1988) that must be sufficiently steep (α_{E}>2) to allow the bulk of the dissipated energy to originate from the lower end of the distribution, via a mostly thermal energy release, thus balancing the coronal energy losses and maintaining a hot corona (Hudson 1991; see also Sect. 3.2.8). This review presents evidence that nanoflares share the same powerlaw distribution of energies as microflares and large flares do. Therefore, the bulk of the released energy stems from large flares in the upper end of the distribution, which is debated by some studies to be insufficient to maintain the corona at its observed temperature. Indeed, statistical properties of small-scale events have been revisited to correct for multiple selection biases and have been shown to obey flatter power laws than originally found. Given the ever-improving but always finite observational sensitivity, however, it is conceivable that such a soft population, if existing, may still be eluding observation or may be partially suppressed by the better sampled intermediate and large events, as it appears to be the case with the results of Crosby et al. (1998) and Georgoulis et al. (2001). In addition, the prediction of the statistical flare model (Georgoulis and Vlahos 1998) for a dual population of instabilities and a “knee” between them, moving from a steeper (softer) to a flatter (harder) power law (Georgoulis and Vlahos 1996), has yet to be confirmed or ruled out. The statistical flare model remains the only SOC model that produces double scaling owning to a double instability criterion featuring “isotropic” and “anisotropic”, directional relaxation (see, however, Fig. 4 of Hughes et al. (2003) and relevant discussion).
Hybrid models can explain broken-powerlaw distributions, which imply also multi-fractality, a property that has been measured in a number of solar active region studies on the magnetic flux distributions (Lawrence et al. 1993; Cadavid et al. 1994; Gallagher et al. 1998; McAteer et al. 2005; Abramenko 2005; Conlon et al. 2008, 2010; Hewett et al. 2008; Abramenko and Yurchyshyn 2010).
Predictability in a SOC System:
Are large events resulting from a SOC system predictable? This remains a widely open question with profound geophysical (i.e., earthquake prediction) and space-weather (i.e., solar-flare/eruption prediction) implications. The question can naturally be linked to the question of inter-event, or waiting times. Extensive discussion on waiting times and their distribution in this review (Sect. 2.12 and references therein) has established that the form of the SOC waiting-time distribution is not an invariant SOC property such as the power-law distribution functions of event size. The degree of memory, intrinsic and different in each SOC system, determines the form of the waiting-time distribution. The opposite is not true generally, because the form of the waiting-time distribution cannot uniquely specify the degree of memory of the SOC system that created it. In addition, an instability—regardless how intense—tends to release only a small fraction of the system’s available energy, hence always imposing a finite degree of stochasticity that is complementary to the finite memory of the system. In case of no memory, that gives rise to a classical BTW exponential waiting-time distribution, events are purely random and cannot be predicted. In particular cases—such as, e.g., deterministically driven models—Strugarek and Charbonneau 2014 showed that the memory of the SOC system could be raised up to a level where large events can be forecasted systematically. Though, it must be noted that predictions from a SOC system necessarily rely on different realizations of the stochastic process and by such are intrinsically probabilistic. Achieving the most significant prediction probabilities then depends on the memory level of the model and is a matter of the specific physics of the SOC system in question (discussion below).
Helicity Conservation in Solar SOC Models:
What physical quantity is conserved in a SOC system? Two of the telltale SOC features are metastability and marginal stability. Metastability typically involves a conservative property of the system in the course of driving as it occurs in the original BTW concept, while marginal stability reflects the mere result of an upper accumulation limit for the conserved parameter, hence defining the critical threshold. Perturbing a low-beta, magnetized environment of a solar active region, for instance, one builds electric currents while conserving magnetic flux. Using a flux critical threshold, however, would be misleading, as large, severely flux-imbalanced active regions (e.g., a single compact sunspot surrounded by scattered opposite-polarity flux) do not flare or erupt in general. Electric current density could constitute a critical threshold for magnetic reconnection and hence for an instability, but it is not a conserved quantity: when stopping the SOC-building driver, the free magnetic energy due to electric currents will be gradually dissipated via a SOC-decaying driver, returning the system to eruption-free stability reflected in a current-free, potential state (e.g., Contopoulos et al. 2011). Although a few non-conservative SOC models have been proposed (Vespignani and Zapperi 1998; Pruessner and Jensen 2002 and references therein), the greatly larger number of conservative SOC models implies that one should perhaps look into a conservative control parameter first to identify a critical threshold: an attractive concept is that of magnetic helicity, a physical quantity that is roughly conserved in high magnetic Reynolds-number plasmas even during reconnection (e.g., Berger 1999). Magnetic helicity could indeed provide a critical threshold, complemented by a minimum free magnetic energy necessary to keep in pace with the accumulated helicity (Tziotziou et al. 2012). This may lead to an unbiased interpretation of eruptions as instabilities occur not because of magnetic reconnection primarily, but because a part or the entire magnetic structure reached its limit in terms of accumulated helicity. Uncovering the crucial physical details of this and similar mechanisms, including how the control quantity of the system (magnetic helicity in this example) consistently accumulates until the system becomes unstable, may potentially achieve closure between physical models and statistical interpretations of complexity systems governed by SOC.
5 Summary and Conclusions
The literature on self-organized criticality (SOC) models counts over 3000 refereed publications at the time of writing, with about 500 papers dedicated to solar and astrophysics. Given the relatively short time interval of 25 years since the SOC concept was born (Bak et al. 1987), the productivity in this interdisciplinary and innovative field speaks for the generality, versatility, and inspirational power of this new scientific theory. Although there exist some previous similar concepts in complexity theory, such as phase transitions, turbulence, percolation, or branching theory, the SOC concept seems to have the broadest scope and the most general applicability to phenomena with nonlinear energy dissipation in complex systems with many degrees of freedom. Of course there is no such thing as a single “SOC theory”, but we rather deal with various SOC concepts (that are more qualitative rather than quantitative), which in some cases have been developed into more rigorous quantitative SOC models that can be tested with real-world data. Computer simulations of the BTW type provide toy models that can mimic complexity phenomena, but they generally lack the physics of real-world SOC phenomenona, because their discretized lattice grids do not reflect in any way the microscopic atomic or subatomic structure of real-world physical systems.
- 1.
A general working definition of a SOC system that can be applied to the majority of the observed astrophysical phenomena interpreted as SOC phenomena can be formulated as: SOC is a critical state of a nonlinear energy dissipation system that is slowly and continuously driven towards a critical value of a system-wide instability threshold, producing scale-free, fractal-diffusive, and intermittent avalanches with powerlaw-like size distributions (Aschwanden 2014). This generalized definition expands the original meaning of self-organized “criticality” to a wider class of critical points and instability thresholds that have a similar (nonlinear) dynamical behavior and produce similar (powerlaw-like) statistical size distributions.
- 2.
A generalized (macroscopic description of a) SOC model can be formulated as a function of the Euclidean space dimension d, the spatio-temporal spreading exponent β, a fractal dimension D_{d}, and a volume-flux scaling (or radiation coherency) exponent γ. For standard conditions [d=3, D_{d}≈(1+d)/2, β=1, and γ=1], this SOC model predicts (with no free parameters) powerlaw distributions for all SOC parameters, namely α_{L}=3 for length scales, α_{A}=2 for areas, α_{V}=5/3 for volumes, α_{F}=2 for fluxes or energy dissipation rates, α_{F}=5/3 for peak fluxes or peak energy dissipation rates, and α_{E}=3/2 for time-integrated fluences or energies of SOC avalanches.
- 3.
The underlying correlations or scaling laws are: A∝L^{2} for the maximum avalanche area, \(A_{f} \propto L^{D_{d}}\) for the fractal avalanche area, V∝L^{3} for the maximum avalanche volume, \(V_{f} \propto L^{D_{d}}\) for the fractal avalanche volume, T∝L^{(2/β)} for the avalanche duration, \(F \propto L^{(\gamma D_{d})}\) for the flux or energy dissipation rate, P∝L^{(γd)} for the peak flux or peak energy dissipation rate, \(E \propto L^{(\gamma D_{d}+2/\beta)}\) for the fluence or total energy.
- 4.
Moreover, the FD-SOC model predicts a waiting time distribution with a slope of α_{Δt}=2 for short waiting times, and an exponential drop-off for long waiting times, where the two waiting time regimes are attributed to intermittently active periods, and to randomly distributed quiescent periods. The contiguous activity periods are predicted to have persistence and memory.
- 5.
Among the astrophysical applications we find agreement between the predicted and observed size distribution for 10 out of 14 reported phenomena, including lunar craters, meteorites, asteroid belts, Saturn ring particles, auroral events during magnetospheric substorms, outer radiation belt electron events, solar flares, soft gamma-ray repeaters, blazars, and black-hole objects.
- 6.
Discrepancies between the predicted and observed size distributions are found for solar energetic particle (SEP) events, stellar flares, pulsar glitches, the Cygnus X-1 black hole, and cosmic rays, which require a modification of the standard FD-SOC model or improved data analysis. The disagreement for SEP events is believed to be due to a selection bias for large events, or could alternatively be modeled with a different dimensionality of the SOC system. For stellar flares we conclude that the bolometric fluence is not proportional to the dissipated energy and flaring volume. Pulsar glitches are subject to small-number statistics. Black hole pulses from Cygnus X-1 have an extremely steep size distribution that could be explained by a suppression of large pulses for a certain period after a large pulse. For cosmic rays, the energy distribution appears to be subject to incomplete uni-directional sampling by in-situ observations, rather than omni-directional sampling by remote-sensing methods.
- 7.
Some of the SOC-associated phenomena have also been modeled with alternative models regarding their size or waiting time distributions and were found to be commensurable, such as in terms of turbulence, percolation, branching theory, or phase transitions. All these theories have some commonalities in their concept and can often not be discriminated based on their observed size distributions alone. Some of the physical processes may coexist and not exclude each other, such as SOC and turbulence in the solar wind.
Summary of theoretically predicted and observed powerlaw indices of size distributions in astrophysical systems
Length α_{L} | Area α_{A}, α_{th,A} | Duration α_{T} | Peak flux α_{P} | Energy α_{E} | Waiting time α_{Δt} | |
---|---|---|---|---|---|---|
FD-SOC prediction | 3.0 | 2.33 | 2.0 | 1.67 | 1.50 | 2.0 |
Lunar craters: | ||||||
Mare Tranquillitatis^{1} | 3.0 | |||||
Meteorites and debris^{2} | 2.75 | |||||
Asteroid belt: | ||||||
Spacewatch Surveys^{3} | 2.8 | |||||
Sloan Survey^{4} | 2.3–4.0 | |||||
Subaru Survey^{5} | 2.3 | |||||
Saturn ring: | ||||||
Voyager 1^{6} | 2.74–3.11 | |||||
Magnetosphere: | ||||||
EUV auroral events^{7} | 1.73–1.92 | 2.08–2.39 | 1.66–1.82 | 1.39–1.61 | ||
EUV auroral events^{8} | 1.85–1.98 | 2.25–2.53 | 1.71–2.02 | 1.50–1.74 | ||
Outer radiation belt^{9} | 1.5–2.1 | |||||
Solar Flares: | ||||||
HXR, ISEE-3^{10} | 1.88–2.73 | 1.75–1.86 | 1.51–1.62 | |||
HXR, HXRBS/SMM^{11} | 2.17±0.05 | 1.73±0.01 | 1.53±0.02 | 2.0^{a} | ||
HXR, BATSE/CGRO^{12} | 2.20–2.42 | 1.67–1.69 | 1.56–1.58 | 2.14 ± 0.01^{b} | ||
HXR, RHESSI^{13} | 1.8–2.2 | 1.58–1.77 | 1.65–1.77 | 2.0^{a} | ||
SXR, Yohkoh^{14} | 1.96–2.41 | 1.77–1.94 | 1.64–1.89 | 1.4–1.6 | ||
SXR, GOES^{15} | 2.0–5.0 | 1.86–1.98 | 1.88 | 1.8–2.4^{c} | ||
EUV, SOHO/EIT^{16} | 2.3–2.6 | 1.4–2.0 | ||||
EUV, TRACE^{17} | 2.50–2.75 | 2.4–2.6 | 1.52–2.35 | 1.41–2.06 | ||
EUV, AIA/SDO^{18} | 3.2±0.7 | 2.1±0.3 | 2.10±0.18 | 2.0±0.1 | 1.6±0.2 | |
EUV, EIT/SOHO^{19} | 3.15±0.18 | 2.52±0.05 | 1.79±0.03 | 1.47±0.03 | ||
Radio microwave bursts^{20} | 1.2–2.5 | |||||
Radio type III bursts^{21} | 1.26–1.91 | |||||
Solar energetic particles^{22} | 1.10–2.42 | 1.27–1.32 | ||||
Stellar Flares: | ||||||
EUVE flare stars^{23} | 2.17±0.25 | |||||
KEPLER flare stars^{24} | 1.88±0.09 | 2.04±0.13 | ||||
Astrophysical Objects: | ||||||
Crab pulsar^{25} | 3.06–3.50 | |||||
PSR B1937+21^{26} | 2.8±0.1 | |||||
Soft Gamma-Ray repeaters^{27} | 1.43–1.76 | |||||
Cygnus X-1 black hole^{28} | 7.1 | |||||
Sgr A^{∗} black hole^{29} | 1.9±0.4 | 1.5±0.2 | ||||
Blazar GC 0109+224^{30} | 1.55 | |||||
Cosmic rays^{31} | 2.7–3.3 |
Physical mechanisms operating in self-organized criticality systems
Phenomenon | Energy Input (steady driver) | Instability threshold (criticality) | Energy output (intermittent avalanches) |
---|---|---|---|
SOC-related Systems: | |||
Sandpile | gravity (dripping sand) | angle of repose | sand avalanches |
Superconductor | magnetic field change | phase transition | vortex avalanches |
Ising model | temperature increase | phase transition | atomic spin-flip |
Tea kettle | temperature increase | boiling point | vapour bubbles |
Earthquakes | tectonic stressing | dynamical friction | rupture area |
Forest fire | tree growth | fire ignition point | burned area |
BTW cellular automaton | input at random nodes | critical threshold | next-neighbor redistribution |
ASTROPHYSICS: | |||
Lunar craters | meteorite production | lunar collision | lunar impact craters |
Asteroid belt | planetesimals | critical mass density | asteroids |
Saturn ring | gravitational disturbances | collision rate | Saturn ring particles |
Magnetospheric substorm | solar wind | magnetic reconnection | auroral bursts |
Radiation belt | solar wind | magnetic trapping/untrapping | electron bursts |
Solar flares | magnetic stressing | magnetic reconnection | nonthermal particles |
Stellar flares | magnetic stressing | magnetic reconnection | nonthermal particles |
Pulsar glitches | neutron star spin-up | vortex unpinning | neutron starquakes |
Soft gamma-ray repeaters | magnetic stressing | star crust fracture | neutron starquakes |
Black-hole objects | gravity | accretion and inflow | X-ray bremsstrahlung pulses |
Blazars | quasar jets | jet direction jitter | optical radiation pulses |
Cosmic rays | galactic magnetic fields | (run-away) acceleration threshold | high-energy particles |
Acknowledgements
The author team acknowledges the hospitality and partial support for two workshops on “Self-Organized Criticality and Turbulence” at the International Space Science Institute (ISSI) at Bern, Switzerland, during October 15–19, 2012, and September 16–20, 2013, as well as constructive and stimulating discussions (in alphabetical order) with Sandra Chapman, Paul Charbonneau, Henrik Jeldtoft Jensen, Maya Paczuski, Jens Juul Rasmussen, John Rundle, Loukas Vlahos, and Nick Watkins. This work was partially supported by NASA contract NNX11A099G “Self-organized criticality in solar physics” and NASA contract NNG04EA00C of the SDO/AIA instrument to LMSAL. MKG acknowledges partial support by the EU Seventh Framework Marie-Curie Programme under grant agreement No. PIRG07-GA-2010-268245.