25 Years of Self-Organized Criticality: Solar and Astrophysics

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

It is instructive to study the functional shape of waiting time distributions that result from non-stationary Poisson processes. In Fig. 6 we illustrate five cases, which each can be derived analytically: (1) a stationary Poisson process with a constant rate λ ₀; (2) a two-step process with two different occurrence rates λ ₁ and λ ₂; (3) a nonstationary Poisson process with a linearly increasing occurrence rate λ(t)=λ ₀ t/T, varying like a triangular function for each cycle, (4) a piecewise constant Poisson process with an exponentially varying rate distribution, and (5) a piecewise constant Poisson process with an exponentially varying rate distribution steepened by a reciprocal factor. For each case we show the time-dependent occurrence rate λ(t) and the resulting probability distribution P(Δt) of events. We see that a stationary Poisson process produces an exponential waiting-time distribution, while nonstationary Poisson processes with a discrete number of occurrence rates λ _i produce a superposition of exponential distributions, and continuous occurrence rate functions λ(t) generate powerlaw-like waiting-time distributions at the upper end. The analytical derivations of these five cases is given in Aschwanden (2011a).

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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

The fractal-diffusive self-organized criticality (FD-SOC) model predicts a powerlaw distribution \(N(T) \propto T^{-\alpha_{T}}\) of event durations T with a slope of α _T =[1+(d−1)β/2] (Eq. (11)) that derives directly from the scale-free probability conjecture N(L)∝L ^−d (Eq. (1)) and the random walk (diffusive) transport (L∝T ^β/2; Eq. (10)). For classical diffusion (β=1) and space dimension d=3 the predicted powerlaw is α _T =2. From this time scale distribution we can also predict the waiting time distribution with a simple probability argument. If we define a waiting time as the time interval between the start time of two subsequent events, so that no two events overlap with each other temporally, the waiting time cannot be shorter than the time duration of the intervening event, i.e., Δt _i ≥(t _i+1−t _i ). Let us consider the case of non-intermittent, contiguous flaring, but no time overlap between subsequent events. In this case the waiting times are identical with the event durations, and therefore their waiting time distributions are equal too, reflecting the same statistical probabilities,

$$ N(\Delta T) d\Delta t \propto N(T) dT \propto T^{-\alpha_T} dT \propto \Delta t^{-\alpha_{\Delta t}} d\Delta t , $$

(40)

with the powerlaw slope,

$$ \alpha_{\Delta t} = \alpha_T = 1 + (d-1)\beta/2 . $$

(41)

This statistical argument is true regardless what the order of subsequent event durations is, so it fulfills the Abelian property. Now we relax the contiguity condition and subdivide the time series into blocks with contiguous flaring, interrupted by arbitrarily long quiet periods when no event happens (Fig. 7). The contributions of waiting times from the subset of contiguous time blocks will still be identical to those of the event durations, while those time intervals from the intervening quiet periods add a few arbitrarily longer waiting times, which form an exponential drop-off in the case of random quiescent time intervals (Fig. 7). As long as the number of quiet time intervals is much smaller than the number of detected events, the modified waiting time distribution will still be similar to the one of contiguous flaring (Eq. (40)), which is α _Δt =2.0 for classical diffusion β=1 and space dimension d=3. Interestingly, this predicted slope is identical to that of nonstationary Poisson processes in the limit of intermittency (Fig. 6, bottom).

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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).

In Fig. 8 we display some forms of the Weibull distribution function for different shape parameters k=0.1,…,5. The distribution function turns into a powerlaw function for k↦0, into an exponential function for k=1, and into a Rayleigh distribution for k≫1, which is almost Gaussian-like. For k=1, the process is Poissonian or random and has no memory. For k<1 the flaring rate decreases over time, while for k>1 the flaring rate is increasing with time, indicating that the process has some memory and persistence, because a persistent driver with memory varies the flaring rate with a systematic trend, which causes also long correlation times among clusters of events. Thus, the Weibull distribution function allows to model random-like (Poissonian) processes as well as processes with memory and persistence.

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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.

Solar flare event catalogs containing the peak rate (P), fluences (E), and flare durations (T), have therefore been compiled from a number of spacecraft or balloon-borne hard X-ray detectors over the last three decades, such as from OSO-7 (Datlowe et al. 1974), a University of Berkeley balloon flight (Lin et al. 1984), HXRBS/SMM (Dennis 1985; Schwartz et al. 1992; Crosby et al. 1993), BATSE/CGRO (Schwartz et al. 1992; Biesecker et al. 1993, 1994; Biesecker 1994), WATCH/GRANAT (Crosby 1996; Georgoulis et al. 2001); ISEE-3 (Lu et al. 1993; Lee et al. 1993; Bromund et al. 1995); PHEBUS/GRANAT (Perez Enriquez and Miroshnichenko 1999). RHESSI (Su et al. 2006; Christe et al. 2008; Lin et al. 2001), and ULYSSES (Tranquille et al. 2009). Three examples of hard X-ray peak flux distributions are shown in Fig. 9. Note that the size distributions of peak counts have a sharp cutoff at the lower end due to a fixed count rate threshold that is generally used in the compilation of hard X-ray flare catalogs, and thus the powerlaw slope can be determined with the highest accuracy. Other parameters have generally a gradual rollover at the low end due to incomplete sampling and finite-resolution effects, which causes truncation effects in the histogram and hampers the accuracy of the powerlaw fit. The size distribution of solar flare hard X-ray counts, which has already been pointed out before the SOC concept came along (Dennis 1985), is still one of the “cleanest” powerlaw size distributions measured in astrophysics (Fig. 9).

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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

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.

Flare statistics from the GOES satellite could be sampled over a period of 37 years (1975–2011), which covers about three solar cycles. Since the soft X-ray flux from the Sun varies by about two orders of magnitude during each solar cycle, due to the variation of emerging magnetic fields and the resulting coronal plasma heating rate, which is driven by the solar magnetic dynamo, the Sun is an ideal system to study SOC systems with variable drivers. While the powerlaw of the soft X-ray peak rate was found to be invariant during different solar cycles, having a roughly constant value of α _F =1.98±0.11, the time durations were found to have a variable slope from α _T ≈2.0 during solar minima to α _T ≈2–5 during solar maxima (Fig. 10), which was explained in terms of a flare pile-up effect (Aschwanden and Freeland 2012). In other words, the separation of time scales, i.e., the waiting times and flare durations, is violated during the busy periods of the solar cycle maximum. In contrast, an opposite trend has been reported for a 158-day modulation of the flare rate (Bai 1993).

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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

Large solar flares (with energies of E≈10³⁰–10³² erg) exhibit heated plasma with peak temperatures of T _e ≈10–35 MK, most conspicuously detected in soft X-rays, which cools down to temperatures of T _e ≈1–2 MK that is readily detected in the postflare phase in extreme ultra-violet (EUV) wavelengths. Also small flares, microflares, and nanoflares (with energies of E≈10²⁴–10²⁷ erg) radiate mostly in the EUV temperature range. Combining these wavelengths, one can obtain statistics of solar flare energies extending over up to 9 orders of magnitude (Fig. 11), hence the term “nanoflares”. Therefore, gathering flare statistics in EUV is expected to complement the lower end of the size distribution sampled in the upper end in soft X-rays and hard X-rays.

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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.

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

Solar energetic particle (SEP) events represent a subset of large solar flares that produce protons, electrons, and helium ions, with energies of ≳25 keV to 1 GeV. It was noted early on that the size distribution of peak counts of SEP events is flatter (α _E ≈1.2–1.4) than those of flare electromagnetic emission (Hudson 1978). Size distributions of the fluences of SEP events were gathered in a typical range of α _E ≈1.2–1.4 (Van Hollebeke et al. 1975; Belovsky and Ochelkov 1979; Cliver et al. 1991; Gabriel and Feynman 1996; Smart and Shea 1997; Perez Enriquez and Miroshnichenko 1999; Miroshnichenko et al. 2001; Gerontidou et al. 2002; Belov et al. 2007). The powerlaw slopes of the size distributions of peak fluxes and fluences are listed in Table 6, which clearly exhibit a much flatter range (α _P ≈α _E ≈1.2–1.4) than those measured in hard (Table 2) and soft X-rays (Table 3), as noted earlier (Hudson 1978). Cliver et al. (2012) interpreted this discrepancy as a selection effect of SEP events being preferentially associated with larger flares, and thus the SEP events are drawn from a subset of flares that do not form a statistically representative sample. This interpretation has been demonstrated by sampling subsets of flares that are associated with >10 MeV proton events, or with ≥1000 km s⁻¹ CMEs, which exhibited a similar flat powerlaw slope as the SEP events themselves (Fig. 12; Cliver et al. 2012). Note that a powerlaw function fits the size distribution of SEP fluences only in the low-fluence part, while the high-fluence part is better fitted by an exponential cutoff function, i.e., \(N(E) \propto E^{-\alpha_{E}} \times\exp{(-E/E_{0})}\), based on SEP data from 41 solar cycles from 1561 to today (Miroshnichenko and Nymmik 2014).

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Table 6

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).

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).

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).

This mean-value theorem predicts a scaling law between the fractal avalanche area \(A_{f} \propto L^{D_{2}}\) and the fractal avalanche volume \(V_{f} \propto L^{D_{3}}\),

$$ V_f \propto A_f^{\delta} , \qquad\delta= {D_3 \over D_2} = {1+3 \over1+2} = {4 \over3} \approx1.33 , $$

(54)

which is lower than the Euclidean scaling law, V∝A ^3/2=A ^1.5. Cellular automaton simulations yield an intermediate value for the exponent, δ=1.41±0.04 (Fig. 13; McIntosh and Charbonneau 2001).

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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)).

A generalization to fractal areas, A _f ∝L ^D , is a multi-fractal system, which has a spectrum f(D) of fractal dimensions D,

$$ A_f \propto L^{f(D)} , $$

(56)

where D is called the strength or significance, and f(D) the singularity spectrum. This singularity spectrum has a peak at f(D) _max and a minimum of f(D) _min (Fig. 14), which is characterized by the terms contribution diversity C _div =D _max −D _min and dimensional diversity D _div =f(D) _max −f(D) _min , both being measures of the geometric complexity and richness of a fractal structure. Multi-fractal analysis of solar data has been carried out on magnetograms (Abramenko 2005; Conlon et al. 2008; Ioshpa et al. 2008; Hewett et al. 2008; Dimitropoulou et al. 2009). Flare-quiet active regions were found to have a lower degree of multi-fractality than flaring active regions (Abramenko 2005). Multi-fractality was also found to increase during magnetic flux emergence in active regions, while a decrease occurred when the active regions evolved to large-scale, coherent structures (Conlon et al. 2008; McAteer et al. 2010). While the exact energy scaling is not known, spatial complexity and flare productivity seem to be related, even when they have different fractal dimensions (Hewett et al. 2008). The 2D photospheric magnetic field contains the footprints of 3D magnetic structures in the solar corona, which explains numerous correlations between photospheric and coronal phenomena (e.g., Dimitropoulou et al. 2009; Uritsky et al. 2013). On the other side, solar active regions with major flares were not found to exhibit a higher level of fractality, multi-fractality, or non-Kolmogorov turbulence than non-flaring regions (Georgoulis 2012).

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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.

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.

The dynamics of the solar wind is often characterized by the MHD turbulent cascade model. The solar wind power spectrum exhibits fully developed turbulence of the Kolmogorov type, P(ν)∝ν ^−5/3, in interplanetary space and near Earth (Fig. 15), while the input spectrum in the lower corona is of the 1/f-noise type, P(ν)∝ν ⁻¹ (Matthaeus and Goldstein 1986; Nicol et al. 2009). The MHD turbulent cascade starts at the largest scales fed by MHD waves with a 1/f-noise spectrum in the lower corona, while turbulent interactions produce a cascade of energy through vortices and eddies to progressively smaller sizes with a spectrum of ν ^−5/3, and final energy dissipation at the smallest scales by heating of electrons, with a spectrum of ν ^−11/3 (Meyrand and Galtier 2010). The analysis of MHD turbulence in solar wind data includes determining power spectra and structure functions, waiting time distributions of solar-wind bursts, identifying the phenomenology or MHD turbulence (Kolmogorov 1941; Kraichnan 1974), characterizing self-similarity and intermittency, and identifying the most intermittent structures, such as shock waves, small random events, current cores, and 1D current sheets (e.g., Horbury and Balogh 1997; Veltri 1999).

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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).

3.2.1 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).

Variants or alternatives to the BTW model that have been applied to simulate the size distributions of solar flares, to name a few, include 3D vector fields with periodic, constant, and symmetric boundaries (Galsgaard 1996), a BTW model with additional nonlocal (remote) triggering that accommodates sympathetic flaring (MacKinnon and MacPherson 1997; Macpherson and MacKinnon 1999), lattice models with non-stationary driving that reproduce the observed waiting time distributions (Norman et al. 2001), emergence of magnetic flux in evolving active regions (Vlahos et al. 2002), lattice models with deterministic drivers based on the BTW model (Strugarek et al. 2014) or on a finite driving rate version of the Olami–Feder–Christensen (OFC, Olami et al. 1992) model (Hamon et al. 2002), which usually is not in a SOC state but rather “on the edge of SOC”, prediction of solar flares by data assimilation to the BTW model (Belanger et al. 2007), a divergence-free field braiding cellular automaton model (Fig. 16; Morales and Charbonneau 2008a, 2008b, 2009), or drivers with diffusive characteristics that mimic a turbulent substrate (Baiesi et al. 2008). The 3D vector field simulations revealed two necessary criteria for the generation of powerlaws: a contiuous driver that produces large scale regions with coherent tension, and a partial (rather than a complete) release of the triggering quantity (Galsgaard 1996).

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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.

A 1D cellular automaton model was constructed in terms of a branching probability p, which yields a probability of N(s)=sp ^s−1(1−p)² after s time steps, and this way leads to a size distribution of N(s)∝s ⁻² (MacKinnon et al. 1996). This branching-type model was extended to a 2D version by introducing some ad hoc functions that could produce powerlaw-like size distributions (Macpherson and MacKinnon 1999). This exercise demonstrated that the statistical redistribution rule of a cellular automaton model with higher dimensions (d≥2) cannot easily be formulated in terms of analytical branching probabilities. In an attempt to generalize this 1-D branching process to higher dimensions, Litvinenko (1998) points out a result from a tree branching process (Fig. 17), for which an asymptotic limit was found with the following analytical expression (Otter 1949),

$$ \bigl\langle N(s)\bigr\rangle \propto s^{-3/2} \exp{ \biggl(-{s \over s_0} \biggr)} , $$

(58)

that is close to observed frequency distributions of flare energies, if we identify the size s with the energy E of flares. A similar description of activation in a forest-fire model was adopted by Christensen et al. (1993).

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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).

A better matching macroscopic description of SOC systems was developed in terms of a fractal-diffusive transport process (Aschwanden 2012a, 2012b), which can accommodate a fluctuating time profile of the energy dissipation rate or observed flux, a fractal spatial structure, diffusive transport (Fig. 18), and spatio-temporal scaling laws that predict powerlaw functions of all physical SOC variables. These scaling laws are in agreement with virtually all measurements made in solar flares. The time evolution of the avalanche radius r(t), the fractal dimension D _d (t) (in Euclidean space with dimension d), the average energy dissipation rate or flux f(t), the peak energy dissipation rate or peak flux p(t), and time-integrated energy or fluence e(t) are,

$$\begin{aligned} r(t) =& \kappa(t - t_0)^{\beta/2} , \end{aligned}$$

(65)

$$\begin{aligned} D_d(t) =& 1 + (d-1) \rho(t), \end{aligned}$$

(66)

$$\begin{aligned} f(t) =& f_0 t^{D_d(t) \beta/2}, \end{aligned}$$

(67)

$$\begin{aligned} p(t) =& p_0 t^{d \beta/2} , \end{aligned}$$

(68)

$$\begin{aligned} e(t) =& \int_{t_0}^t f(t) dt \approx f_0 t^{\langle D_d\rangle \beta/2} , \end{aligned}$$

(69)

where ρ(t) is a random function varying in the range of [0,1]. The fractal dimension D _d (t) and the flux time profile f(t) can fluctuate randomly, but the Euclidean radius r(t) and the total (time-integrated) energy e(t) are monotonically increasing quantities during an avalanche. The scaling laws between these parameters and the resulting size distributions are defined in terms of the fractal-diffusive transport and the scale-free probability conjecture (Sects. 2.6–2.11), and thus all powerlaw indices are predicted from first principles in this model. The same SOC model has been applied to a host of astrophysical phenomena (Aschwanden 2014).

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3.2.4 Solar Magnetic Field Models and SOC

The transformation of a cellular automaton redistribution rule into a discretized MHD differencing scheme started with Takalo et al. (1999a) for an application to a magnetotail field model, and with Vassiliadis et al. (1998) for an application to solar flares. The curl of the current j at the cell boundaries is defined in terms of the magnetic field vectors in each neighbor cell, as shown in Fig. 19 and defined by Ampère’s and Ohm’s law (Eqs. (75) and (76)). This way, a resistivity can be defined as a function of the current at the flux tube boundary, as expected from a current-driven instability. Anisotropic cellular automata correspond to a nonlinear resistivity, while isotropic ones can be associated with hyper-resistivity (Vassiliadis et al. 1998). In the continuum limit, however, singularities can arise (Isliker et al. 1998a), which largely disappear in 3D models (Isliker et al. 2000). For solar flare applications, a threshold of a critical current j _c was found to be physically more appropriate (Isliker et al. 2001), than a threshold of a critical magnetic field B _c as used in the original SOC models (Lu and Hamilton 1991).

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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≈4×10²⁸ ergs, waiting times of Δt≈300 s, and a heat flux of F≈2×10⁶ 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).

While the original RTV law was applied to a single coronal loop, supposedly to be in approximate energy balance between the heating rate (H) and the conductive loss rate (−E _cond ) and the radiative loss rate (−E _rad ),

$$ H - E_{cond} - E_{rad} = 0 , $$

(87)

the same scaling law can also be applied to the peak time t _peak of a flare, just at the turnover time between dominant heating and dominant cooling when energy balance occurs for a brief moment (Aschwanden and Shimizu 2013),

$$ \begin{aligned} H(t) - E_{cond}(t) - E_{rad}(t) &\ge0 \quad \mbox{for}\ t < t_{peak} \\ H(t) - E_{cond}(t) - E_{rad}(t) &= 0 \quad \mbox{for}\ t = t_{peak} \\ H(t) - E_{cond}(t) - E_{rad}(t) &\le0 \quad \mbox{for}\ t > t_{peak} \end{aligned} $$

(88)

Since the emission measure EM _p , peak temperature T _p , and length scale L _p can be directly measured with multi-wavelength imaging observations in solar flares, such as with AIA/SDO, the RTV law can then be tested and the resulting size distributions of the other physical parameters (n _p ,E _th ,H) can be predicted. An example is shown in Fig. 20, where the size distributions of the parameters L,T _p ,n _p ,H,EM _p ,E _th are shown, as log–log histograms of the observed and derived variables, and in form of scatterplots as a function of the length scale L to visualize the truncation effects caused by the flux threshold detection limit (EM≥EM _thresh ), and then compared with Monte-Carlo simulations of the size distributions (red curves in Fig. 20). A powerlaw index of α _H ≈1.8 is inferred for the unknown size distribution of heating rates H.

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The most interesting size distribution or scaling relationship concerns the heating rate H, which holds the secret of the coronal heating process and/or energy dissipation process of flares. From full-Sun simulations of the corona composed as an ensemble of myriads of individual 1D loops, a scaling law of the heating flux F _H ∝BL ⁻¹ was found (Schrijver et al. 2004), which corresponds to a volumetric heating rate of,

$$ H \propto{F_H \over L} \approx B L^{-2} . $$

(89)

A different scaling is obtained from a magnetic reconnection scenario in Petschek’s theory, by assuming that the loop apex temperature is balanced by conductive cooling, T _e ∝(2HL ²)^2/7 (Shibata and Yokoyama 1999, 2002),

$$ H \approx \biggl( {B^2 \over4 \pi} \biggr) {v_A \over L} . $$

(90)

A similar scaling law was derived for magnetic reconnection processes with the Sweet-Parker reconnection scenario (Sweet 1958; Parker 1957; Cassak et al. 2008). Obviously, statistics on the size distributions N(B) of the magnetic field are required in order to infer the heating rate distribution N(H). The flow chart in Fig. 21 summarizes how the observed distributions are related to the model assumptions and physical scaling laws for solar flare events or coronal heating events.

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3.2.8 The Role of Nanoflares

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).

Most magnetospheric phenomena result from the interaction of the Earth’s (or some other planet’s) magnetic field with the ambient solar wind in the heliosphere, at the magnetopause, in the magneto-tail, and in the polar regions of the planet. The solar wind brings to Earth disturbances associated with solar flares, coronal mass ejections, shock waves and solar energetic particles, causing magnetospheric storms, substorms, and auroral activities. The solar wind is thus the driver of the processes in space weather. Substorms involve many processes, including magnetic reconnection, ballooning-mirror modes, current disruption, etc., which cause a fast unloading of the highly stressed geotail system (Papadopoulos et al. 1993; Baker et al. 1996, Horton and Doxas 1996). In addition, multi-scale intermittent turbulence of overlapping plasma resonances play an important role in substorms (Chang 1999a). Brief magnetospheric disturbances occur when the interplanetary magnetic field (IMF) flips southward, which triggers magnetic reconnection at the dayside magnetopause and transfers momentum and energy from the solar wind to the magnetosphere. Part of the transferred energy is stored in the magnetotail, where also magnetic reconnection and field relaxation events can occur during magnetospheric substorms. A magnetospheric substorm has three phases (Fig. 22): (1) the growth phase (when energy from the solar wind is transferred to the dayside magnetosphere), (2) the substorm expansion phase (when the energy stored in the magnetotail is released, the magnetosphere relaxes from the stretched tail, and the tail snaps into a more dipolar configuration and energizes particles in the plasma sheet), and (3) the recovery phase (during which the magnetosphere returns to its quiet state). The whole process causes changes in the auroral morphology (Fig. 22) and induces currents in the polar ionosphere, with the resultant heating leading to the auroral displays. The frequency of substorms is about 6 per day on average, but larger during geomagnetic storms.

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The agreement between the statistics of large auroral events observed with POLAR/UVI (Uritsky et al. 2002) and small auroral events observed with a TV camera (Kozelov et al. 2004) is excellent (Fig. 23) and covers a combined (but not overlapping) range of 10 orders of magnitude in energy. However, we can see in Table 11 a glaring discrepancy between the measurements made by Lui et al. (2000) and by Uritsky et al. (2002), while the latter agree surprisingly well with the predictions of the standard fractal-diffusive SOC model (Aschwanden 2012a). The measurements by Lui et al. (2000) yield much flatter powerlaw slopes, close to unity. This discrepancy has been convincingly explained in terms of a different methodology and event definition: The statistics carried out by Lui et al. (2000) refers to equidistant time snapshots that count large avalanche events multiple times, while the analysis of Uritsky et al. (2002) determines the time-integrated avalanche areas and energies, consistent with standard definitions of SOC parameters (also used in the FD-SOC model: see Sect. 2.10 and Eqs. (14) and (20)). Another anomaly that was found is the latitude dependence of the size distribution of auroral events (see Uritsky et al. 2008 in Table 11), which indicates substantially different scaling regimes of bursty energy dissipation in the inner and outer portion of the geotail plasma sheet (Uritsky et al. 2008, 2009).

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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).

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