1. 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, Oct 2012
SOC & TURBULENCE 1

2. 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, OCT, 2012

3. 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, OCT, 2012

4. 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, OCT, 2012

5. OBSERVATIONAL FACTS: FRACTALITY, MULTIFRACTALITY, TURBULENCE
SOLAE MAGNETIC FIELDS: MULTISCALING
SOC & TURBULENCE
BERN, OCT, 2012

6. 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, OCT, 2012
SOLAR MAGNETIC FIELDS: MULTISCALING
SOC & TURBULENCE

7. 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, OCT, 2012
SCALING OF FLARING VOLUMES: OBSERVATIONS
SOC & TURBULENCE

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

9. 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, OCT, 2012
SOLAR FLARE STATISTICS
SOC & TURBULENCE

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

11. 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, OCT, 2012
HISTORICAL, SELF-SIMILAR FLARE MODELS
SOC & TURBULENCE

12. 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, OCT, 2012
SELF ORGANIZATION
SOC & TURBULENCE

13. 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, OCT, 2012
PERCOLATION MODELS
SOC & TURBULENCE

14. 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, OCT, 2012
PERCOLATION MODELS
SOC & TURBULENCE

15. 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, OCT, 2012
SOC MODELS
SOC & TURBULENCE

16. 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, OCT, 2012
SOC FLARE MODELS
SOC & TURBULENCE

17. 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, OCT, 2012
SOC FLARE MODELS
SOC & TURBULENCE

18. 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, OCT, 2012
SOC FLARE MODELS
SOC & TURBULENCE

19. 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, OCT, 2012
SOC FLARE MODELS
SOC & TURBULENCE

20. 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 [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, OCT, 2012
X-CA SOC FLARE MODELS
SOC & TURBULENCE

21. 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, OCT, 2012
X-CA SOC FLARE MODELS
SOC & TURBULENCE

22. 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, OCT, 2012
X-CA SOC FLARE MODELS
SOC & TURBULENCE

23. 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
: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, OCT, 2012
DATA-DRIVEN SOC FLARE MODELS
SOC & TURBULENCE

24. 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, OCT, 2012
DATA-DRIVEN SOC FLARE MODELS
SOC & TURBULENCE

25. 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, OCT, 2012
TOP-LEVEL SUMMARY
SOC & TURBULENCE

26. 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, OCT, 2012
CRITICAL QUESTIONS FOR SOC APPLICABILITY
SOC & TURBULENCE

27. 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, OCT, 2012
CRITICAL QUESTIONS FOR SOC APPLICABILITY
SOC & TURBULENCE

28. 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, OCT, 2012
CRITICAL QUESTIONS FOR SOC APPLICABILITY
SOC & TURBULENCE

29. 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, OCT, 2012
CRITICAL QUESTIONS FOR SOC APPLICABILITY
SOC & TURBULENCE

30. 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, OCT, 2012
POTENTIAL BENEFITS FROM SOC APPLICATION
SOC & TURBULENCE

31. 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, OCT, 2012
POTENTIAL BENEFITS FROM SOC APPLICATION
SOC & TURBULENCE

32. 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, OCT, 2012
POTENTIAL BENEFITS FROM SOC APPLICATION
SOC & TURBULENCE

33. 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, OCT, 2012
POTENTIAL BENEFITS FROM SOC APPLICATION
SOC & TURBULENCE

34. 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 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, OCT, 2012
POTENTIAL BENEFITS FROM SOC APPLICATION
SOC & TURBULENCE

35. 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, OCT, 2012
POTENTIAL BENEFITS FROM SOC APPLICATION
SOC & TURBULENCE

36. 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, OCT, 2012
POTENTIAL BENEFITS FROM SOC APPLICATION
SOC & TURBULENCE

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

38. 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, OCT, 2012
POTENTIAL BENEFITS FROM SOC APPLICATION
SOC & TURBULENCE

39. 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, OCT, 2012
CONCLUSION
SOC & TURBULENCE

40. 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

41. BACKUP SLIDES

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