RCAAM of the Academy of Athens Self-Organized Criticality in the Solar Atmosphere: Universal Property of Solar Magnetism, Or Merely One of Eruptive Active Regions? Manolis K. Georgoulis* RCAAM of the Academy of Athens * Marie Curie Fellow Bern, CH, 15 - 19 Oct 2012 SOC & TURBULENCE 1

OUTLINE What is the extent of SOC validity in solar magnetic structures? Courtesy: TRACE Observational facts in the solar active- region atmosphere Where do observables and “moments” stem from? SOC models of solar active regions X-CA approaches: revisions and enhancements of SOC models Open questions: how can we rigorously determine SOC in solar active regions? If SOC is at work, what can we gain from its application? Conclusions Falling grains of sand OUTLINE SOC & TURBULENCE BERN, 15 -19 OCT, 2012

SOLAR MAGNETIC FIELDS: COMPLEXITY AT WORK NOAA AR 10930 12/12/06, 20:30 UT Source: Hinode SOT/SP Ever-increasing spatial resolution leads to ever-increasing intermittency in the observed spatial structures SOLAR MAGNETIC FIELDS: COMPLEXITY SOC & TURBULENCE BERN, 15 -19 OCT, 2012

SOLAR MAGNETIC FIELDS: COMPLEXITY AT WORK Ever-increasing temporal resolution leads to ever-increasing intermittency in the observed dynamical response SOLAR MAGNETIC FIELDS: COMPLEXITY SOC & TURBULENCE BERN, 15 -19 OCT, 2012

OBSERVATIONAL FACTS: FRACTALITY, MULTIFRACTALITY, TURBULENCE SOLAE MAGNETIC FIELDS: MULTISCALING SOC & TURBULENCE BERN, 15 -19 OCT, 2012

HIGHER-ORDER MULTISCALING OBSERVATIONAL FACTS: HIGHER-ORDER MULTISCALING Wavelet transform modulus maxima (WTMM) method Solar magnetic fields “tested positive” to any mono- or multi-scaling method one might devise h, D(h) --> multifractal scaling spectra Conlon et al. (2010) BERN, 15 -19 OCT, 2012 SOLAR MAGNETIC FIELDS: MULTISCALING SOC & TURBULENCE

OBSERVED BEHAVIOR OF POTENTIALLY FLARING VOLUMES Dimitropoulou et al. (2009) Fractality and power laws in the volume and free energy of gradient-identified potentially unstable structures (Vlahos & Georgoulis 2004) BERN, 15 -19 OCT, 2012 SCALING OF FLARING VOLUMES: OBSERVATIONS SOC & TURBULENCE

MODELED BEHAVIOR OF FLARING VOLUMES MacIntosh & Charbonneau (2001) Clearly fractal flaring volumes in 3D Aschwanden & Aschwanden (2008) BERN, 15 -19 OCT, 2012 SCALING OF FLARING VOLUMES: MODELS SOC & TURBULENCE

STATISTICS OF SOLAR SUB-FLARES & FLARES Hannah et al. (2011) Power-law statistics of events Well-defined, extended power-law statistics reported for total flare energy, peak luminosity, and duration since the 1970’s Compilation of various statistical studies (Hannah et al. 2011) BERN, 15 -19 OCT, 2012 SOLAR FLARE STATISTICS SOC & TURBULENCE

WHAT LIES BEHIND THESE OBSERVABLES? -- Interpretations: BERN, 15 -19 OCT, 2012 RECAP & INTERPRETATIONS SOC & TURBULENCE

SELF-SIMILAR FLARE MODELS From a Poissonian flare-occurrence probability: Rosner & Vaiana (1978) To a power-law probability of flares with energy between E and E+dE, if E >> E0 : Dependence of the PDF index on mean flaring rate and the system’s stress rate α Criticism raised by future works, e.g. Lu & Hamilton - indeed, model was rather abstract Further works of the 1990’s (e.g., RCS - Litvinenko 1994; 1996; master equation - Wheatland & Glukhov 1998; logistic equation - Aschwanden et al. 1998, etc.), all had pros and cons Clearly, more is needed than a simple, “magical” equation! BERN, 15 -19 OCT, 2012 HISTORICAL, SELF-SIMILAR FLARE MODELS SOC & TURBULENCE

Flock of birds (Source: Youtube) SELF ORGANIZATION Self-Organization: Reduction of the many degrees of freedom exhibited by a complex system to a small number of significant degrees of freedom dictating the system’s evolution (e.g. Nicolis & Prigogine 1989) Flock of birds (Source: Youtube) Competition between at least two parameters, or probabilities System unstable, should any of these probabilities dominate BERN, 15 -19 OCT, 2012 SELF ORGANIZATION SOC & TURBULENCE

PERCOLATION MODELS OF SOLAR ACTIVE-REGION EMERGENCE & EVOLUTION Simulated magnetic flux emergence and active-region formation in solar atmosphere - Implementation on cellular automata (CA) models -- Two primary competing probabilities: Pst : stimulation probability D : diffusion probability -- Two secondary probabilities: Psp : probability of spontaneous emergence Pm : moving probability Seiden & Wentzel (1996) Wentzel & Seiden (1992) (follow-up percolation models by MacKinnon, MacPherson, Vlahos, etc.) Third-decimal digit changes in stimulation and/or diffusion probability enough for the system to collapse (either die out or fill the entire grid with magnetized cells) BERN, 15 -19 OCT, 2012 PERCOLATION MODELS SOC & TURBULENCE

FLARE STATISTICS BY PERCOLATION MODELS? Power-law PDFs in event energy with index ~ -1.55 Plausible timeseries of magnetic energy release Lower-boundary formed by percolation and LFF extrapolation model defining the overlaying field (Fragos et al. 2004) Energy “release” episodes due to lower-boundary dissipation and subsequent extrapolation changes (Fragos et al. 2004) BERN, 15 -19 OCT, 2012 PERCOLATION MODELS SOC & TURBULENCE

SELF-ORGANIZED CRITICALITY (SOC) MODELS Introducing criticality via a critical threshold, one alleviates the need for fine tuning The system is not followed from initial formation, but it is allowed to evolve to the SOC state. SOC is a robust state of statistical stationarity exhibited by systems that are “far from equilibrium” By definition, SOC applies to randomly triggered instabilities for which, however, there are deterministic relaxation rules BERN, 15 -19 OCT, 2012 SOC MODELS SOC & TURBULENCE

ORIGINAL SOC SOLAR-FLARE MODELS -- Characteristics: -- The first model (Lu & Hamilton 1991; Lu et al. 1993; Lu 1995), in accordance with BTW Constant driver δΒ << critical threshold |Bc| Randomly chosen point i perturbed Local gradient calculated & compared to |Bc| Isotropic redistribution rules if |dBi| > |Bc| Elementary energy release per instability: A state in which the system enters, but has no way of exiting. If external forcing ceases, the system remains static. Flux conservation, avalanches, and a spectral power of the form S(f) ~ f-2 BERN, 15 -19 OCT, 2012 SOC FLARE MODELS SOC & TURBULENCE

REVISED SOC SOLAR-FLARE MODELS -- The second “Statistical Flare” model (Vlahos, Georgoulis, et al.) Constant driver δΒ << critical threshold Bc SOC survived in this “unorthodox” model, too! Randomly chosen point i perturbed Local gradient calculated & compared to Bc, both isotropically, and anisotropically Isotropic redistribution rules if dBi (isotropic) > Bc and anisotropic rules in case dBi (anisotropic) > Bc Elementary energy release per instability: BERN, 15 -19 OCT, 2012 SOC FLARE MODELS SOC & TURBULENCE

DIFFERENT FACES OF SOC --> Different avalanche attributes Vlahos et al. (1995) --> Different avalanche attributes --> Single power-law PDF (Lu et al. 1993) --> Double power-law PDF (Georgoulis & Vlahos 1996) BERN, 15 -19 OCT, 2012 SOC FLARE MODELS SOC & TURBULENCE

For α ∈ [1.0, 2.5], SOC manages to survive FURTHER SOC CA REVISIONS: VARIABLE DRIVER Event PDFs for α=1.6 Constant Variable driver δΒ with PDF with α --> free parameter For α ∈ [1.0, 2.5], SOC manages to survive Output power-law indices variable, and controlled by α! Georgoulis & Vlahos (1998) BERN, 15 -19 OCT, 2012 SOC FLARE MODELS SOC & TURBULENCE

EXTENDED CELLULAR AUTOMATA (X-CA) FLARE MODELS --> First effort to discretize the resistive term of the MHD induction equation (Vassiliadis et al. 1998) Inferred resistivity of isotropic SOC models (Lu & Hamilton 1991 [LH91]; Lu et al. 1993 [L93], Georgoulis & Vlahos (iGV) Inferred resistivity of the anisotropic anisotropic SOC model of Georgoulis & Vlahos (aVG) Model setup Isotropic SOC CA models resemble hyper-resistivity conditions (η ∇2J), while nonlinear resistivity (η J) is exhibited by anisotropic SOC CA models BERN, 15 -19 OCT, 2012 X-CA SOC FLARE MODELS SOC & TURBULENCE

EXTENDED CELLULAR AUTOMATA (X-CA) FLARE MODELS --> Solar-like magnetic field B introduced, complete with knowledge of vector potential A, where ∇ x A = B (Isliker et al. 2000; 2001) --> Resistivity calculated, giving rise to current sheets and Ohmic dissipation Isliker et al. (2001) Envisioned (top) and achieved (bottom) magnetic field configuration Distribution of (sub-critical) electric current density in the simulation box Both isotropic and anisotropic SOC rules were implemented, and SOC managed to survive BERN, 15 -19 OCT, 2012 X-CA SOC FLARE MODELS SOC & TURBULENCE

EXTENDED CELLULAR AUTOMATA (X-CA) FLARE MODELS --> SOC in a 2D loop geometry (Morales & Charbonneau 2008; 2009) Morales & Charbonneau (2008) Basic model setup and driving SOC manages to survive and, moreover, assigning typical coronal-loop values, physical energy units appear for the first time. Inferred avalanche energies range between 1023 and 1029 erg. Other successful SOC X-CA approaches: -- Using helicity and its conservation (Chou 1999) -- Using separator reconnection (Longcope & Noonan 2000) -- Using a statistical fractal-diffusive model (Aschwanden 2012) The model’s dynamical response BERN, 15 -19 OCT, 2012 X-CA SOC FLARE MODELS SOC & TURBULENCE

Any initial valid field solution can be brought into the SOC state DATA-DRIVEN CA FLARE MODELS --> SOC achieved using a single vector magnetogram of an observed solar active region - “Static” integrated flare model (S-IFM) Dimitropoulou et al. (2011) NOAA AR 10247 2003-01-13 18:26 UT Haleakala IVM --> NLFF field solution as an initial condition --> LH driving and isotropic instability criterion --> Critical threshold on the local field Laplacian --> Physical units of energy released achieved Any initial valid field solution can be brought into the SOC state Avalanche relaxation in NOAA AR 10247 BERN, 15 -19 OCT, 2012 DATA-DRIVEN SOC FLARE MODELS SOC & TURBULENCE

DATA-DRIVEN CA FLARE MODELS --> SOC survived evolving via a timeseries of SOC-state magnetic field solutions - “Dynamic” integrated flare model (D-IFM) Dimitropoulou et al. (2012), A&A, submitted --> S-IFM applied to each magnetogram --> Discrete, Alfvén-timescale of spline-interpolated evolution from one magnetogram to the other --> Real time units for events’ onset Single-point driving abandoned in D-IFM. The entire grid receives perturbations, yet SOC survives BERN, 15 -19 OCT, 2012 DATA-DRIVEN SOC FLARE MODELS SOC & TURBULENCE

Therefore, is SOC at work in solar active regions? SUMMARY AND OVERARCHING QUESTIONS Therefore, is SOC at work in solar active regions? To find out, we should consider the following fundamental questions: 1. Critical threshold: what is/are the critical threshold(s) for solar flare occurrence? Does the system (i.e., an active region) reach minimal stability with respect to that/those thresholds? 2. Turbulence and SOC: can a turbulent system exhibit avalanche behavior? Is active-region evolution reminiscent of this behavior? 3. Waiting-time distributions: what distribution form(s) should be expected from a SOC system? Are observed distributions reminiscent of this? BERN, 15 -19 OCT, 2012 TOP-LEVEL SUMMARY SOC & TURBULENCE

CRITICAL THRESHOLD: HOW ARE FLARES TRIGGERED? Georgoulis, PhD Thesis (2000) Well-defined mean event (all sizes) frequency in SOC state In real solar active regions we do not yet know (i) whether this feature exists, and (ii) at what timescale (<< 11-year solar cycle) we should look for a well-defined mean flare number Mean gradient or slope in SOC models manages to stabilize in SOC state In real solar active regions we do not yet know (i) which parameter exhibits this property, if any, and (ii) whether this parameter can be calculated BERN, 15 -19 OCT, 2012 CRITICAL QUESTIONS FOR SOC APPLICABILITY SOC & TURBULENCE

Can we achieve avalanches in a fully compressible turbulent system? TURBULENCE & SOC: COMPATIBLE OR CONFLICTING CONCEPTS? Timeseries of mean Joule dissipation rate 2.5D reduced MHD (RMHD - incompressible) system Georgoulis, Velli, & Einaudi (1998) --> Some evidence of avalanching in a reduced MHD (RMHD), incompressible turbulent system --> However, incompressibility implies lack of magnetic energy storage Can we achieve avalanches in a fully compressible turbulent system? Event scaling laws BERN, 15 -19 OCT, 2012 CRITICAL QUESTIONS FOR SOC APPLICABILITY SOC & TURBULENCE

DOES CORONA EXHIBIT TURBULENCE AND SOC? Uritsky et al. (2007) Threshold-dependent EUV coronal emission and evidence for avalanching reminiscent of SOC behavior --> However, this type of behavior is also exhibited by a well-known intermittent turbulent model, even in 1D (Watkins et al. 2009) Question remains: Can turbulent systems also exhibit SOC? BERN, 15 -19 OCT, 2012 CRITICAL QUESTIONS FOR SOC APPLICABILITY SOC & TURBULENCE

FLARE WAITING-TIME DISTRIBUTIONS: WHAT IS THEIR FORM AND WHAT IS EXPECTED? Flares are essentially random events (Crosby et al. 1998) Power-law PDF of waiting times (Boffetta et al. 1999; Lepreti et al. 2001) Flare occurrence a non-stationary Poisson process (Wheatland 2000; 2001) --> Isotropic LH and the Statistical Flare models favor exponential waiting-times distributions, indicating lack of memory and random flare occurrence --> However, there are indications that different driving mechanisms can give rise to different waiting-time distributions in a SOC system (e.g., Charbonneau et al. 2001) Perhaps the specifics of waiting-time distributions is not universal - generalized SOC models may account for multiple cases! BERN, 15 -19 OCT, 2012 CRITICAL QUESTIONS FOR SOC APPLICABILITY SOC & TURBULENCE

What are the implications, or ramifications, of SOC validity? IF SOC IS AT WORK IN THE SOLAR ATMOSPHERE, WHAT IS THE GAIN FROM ITS APPLICATION? What are the implications, or ramifications, of SOC validity? Deeper physical insight: how can cellular automata be further refined or generalized to account for more observed properties? Implications for coronal heating: a “soft” nanoflare population? Loss-of-equilibrium models of solar eruptions: a tell-tale SOC sign? SOC ramifications for solar flare forecasting: can flares be truly predicted? BERN, 15 -19 OCT, 2012 POTENTIAL BENEFITS FROM SOC APPLICATION SOC & TURBULENCE

with physical units and even predictive power, with the computational GENERALIZATION OF CELLULAR AUTOMATA MODELS --> What would be the optimal course of action in terms of cellular automata? More physics-based, concept-oriented CAs (free energy, helicity, stress, tension, shear, twist, etc? - Morales & Charbonneau, Chou, Longope & Noonan Discrete, MHD-coupled cellular automata? - Vassiliadis, Isliker, et al. Data-driven cellular automata? - Dimitropoulou et al. Can the emphasis of cellular automata be shifted from statistics to physics, with physical units and even predictive power, with the computational convenience that automata traditionally have compared to MHD models? BERN, 15 -19 OCT, 2012 POTENTIAL BENEFITS FROM SOC APPLICATION SOC & TURBULENCE

A “SOFT” NANOFLARE POPULATION? GRANAT/WATCH flare PDFs Statistical Flare model reproduction Georgoulis et al. (2001) Besides the reproduced sizable flares, SOC anisotropic relaxation criteria predict a soft small-event population. Does it really exist? Recent statistical studies have yet to identify this population for “small” flares, “microflares”, or “subflares” However, how small is “small”? Can the soft population be hidden underneath the big events? BERN, 15 -19 OCT, 2012 POTENTIAL BENEFITS FROM SOC APPLICATION SOC & TURBULENCE

A HIDDEN “SOFT” NANOFLARE POPULATION? Georgoulis et al. (2001) Shorter events in the WATCH/GRANAT database obey steeper power laws And this is also reproduced by the Statistical Flare model, albeit more pronounced It is possible that “nanoflares”, or even “picoflares”, are there, partially hidden and/or unobserved, with scaling indices steeper than -2 to sustain a significant thermal heating hypothesis (Hudson 2001) BERN, 15 -19 OCT, 2012 POTENTIAL BENEFITS FROM SOC APPLICATION SOC & TURBULENCE

LOSS-OF-EQUILIBRIUM SOLAR ERUPTION MODELS If the loss-of-equilibrium, or “catastrophe” models of solar flares (Forbes & Isenberg 1991) are of any validity, can this be a tell-tale signature of a SOC system? Lin & Forbes (2000) Hinode/SOT Ca 3968.5 A --> Loss of equilibrium reminds us of marginal stability --> If so, what is the critical parameter? --> Can we justify and record the course of an eruptive active region to marginal stability? --> Many candidates: electric currents, resistivity, non-potential magnetic energy, magnetic (or current) helicity, etc. BERN, 15 -19 OCT, 2012 POTENTIAL BENEFITS FROM SOC APPLICATION SOC & TURBULENCE

IN SUPPORT OF THE MARGINAL STABILITY CONCEPT Tziotziou et al. (2012) The free magnetic energy - relative magnetic helicity diagram of solar active regions --> Eruptive active regions tend to exceed well-defined thresholds in both free magnetic energy and relative magnetic helicity --> Can these thresholds be shown to bring the system into marginal stability under SOC? --> If so, what is the driver? - FYI, Georgoulis, Titov, & Mikic, 2012, ApJ, in press BERN, 15 -19 OCT, 2012 POTENTIAL BENEFITS FROM SOC APPLICATION SOC & TURBULENCE

SOC RAMIFICATIONS FOR SOLAR FLARE FORECASTING --> The classical SOC concept implies spontaneity in the system’s dynamical response --> Therefore, if solar active regions, manifesting multiscaling behavior, are in a SOC state, can flares/eruptions be predicted? In a recent work (Georgoulis, 2012) it has been shown that flaring and non-flaring active regions show similar measures of fractality / multifractality As a result, multiscale methods cannot be used for flare prediction This conclusion, however, is subject to the outcome of the debate on waiting time distributions - notice also the work of Dimitropoulou et al. (2009) showing lack of correlation between photospheric and coronal fractal properties Flare prediction may remain inherently probabilistic! BERN, 15 -19 OCT, 2012 POTENTIAL BENEFITS FROM SOC APPLICATION SOC & TURBULENCE

PREDICTIVE ABILITY OF MULTISCALE METHODS Georgoulis (2012) 17733 SoHO/MDI magnetograms 370 AR timeseries BERN, 15 -19 OCT, 2012 POTENTIAL BENEFITS FROM SOC APPLICATION SOC & TURBULENCE

PREDICTIVE ABILITY OF MULTISCALE METHODS Georgoulis (2012) The unsigned magnetic flux, a conventional predictor used as reference, works better than multiscale parameters - these parameters, therefore, cannot be used for flare prediction Multiscaling behavior is widespread in flaring and non-flaring ARs alike Does this mean that flaring and non-flaring active regions might be in a similar - indistinguishable - SOC state (Vlahos & Georgoulis 2004) ? BERN, 15 -19 OCT, 2012 POTENTIAL BENEFITS FROM SOC APPLICATION SOC & TURBULENCE

Overarching question / food for thought: CONCLUSIONS Overarching question / food for thought: at what extent, if any, is SOC valid in solar active-region magnetic fields? BERN, 15 -19 OCT, 2012 CONCLUSION SOC & TURBULENCE

Victor Hugo, Les Misérables How do we know that the creations of worlds are not determined by falling grains of sand? Who can understand the reciprocal ebb and flow of the infinitely great and the infinitely small, the echoing of causes in the abyss of being and the avalanches of creation? Victor Hugo, Les Misérables

BACKUP SLIDES

STATISTICS OF OTHER SOLAR DYNAMICAL PHENOMENA -- Active-region sizes -- Quiet Sun fields -- Ellerman bombs -- ARTBs -- etc. BERN, 15 -19 OCT, 2012 SOLAR FLARE STATISTICS SOC & TURBULENCE