Transcript Self-Organized Criticality in the Solar Atmosphere

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
RCAAM of the Academy of Athens
SOC & TURBULENCE
Bern, CH, 15 - 19 Oct 2012
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OUTLINE
What is the extent of SOC validity in solar magnetic structures?
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Observational facts in the solar activeregion atmosphere
Courtesy: TRACE
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?
Falling grains of sand
If SOC is at work, what can we gain from its
application?
Conclusions
OUTLINE
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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 everincreasing intermittency in the observed spatial
structures
SOLAR MAGNETIC FIELDS: COMPLEXITY
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SOLAR MAGNETIC FIELDS: COMPLEXITY AT
WORK
Ever-increasing temporal resolution leads to everincreasing intermittency in the observed dynamical
response
SOLAR MAGNETIC FIELDS: COMPLEXITY
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OBSERVATIONAL FACTS: FRACTALITY,
MULTIFRACTALITY, TURBULENCE
SOLAE MAGNETIC FIELDS: MULTISCALING
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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)
SOLAR MAGNETIC FIELDS: MULTISCALING
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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)
SCALING OF FLARING VOLUMES: OBSERVATIONS
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MODELED BEHAVIOR OF FLARING VOLUMES
MacIntosh & Charbonneau (2001)
Clearly fractal flaring volumes in 3D
Aschwanden & Aschwanden (2008)
SCALING OF FLARING VOLUMES: MODELS
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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
SOLAR FLARE STATISTICS
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Compilation of various statistical studies
(Hannah et al. 2011)
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WHAT LIES BEHIND THESE OBSERVABLES?
-- Observables:
-- Interpretations:
RECAP & INTERPRETATIONS
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SELF-SIMILAR FLARE MODELS
Rosner & Vaiana (1978)
• From a Poissonian flare-occurrence
probability:
P(t) = n e
-n t
• To a power-law probability of flares with
energy between E and E+dE, if E >> E0 :
n æ
Eö
P(E) =
1+ ÷
ç
a E0 è E 0 ø
-
n +a
a
• Dependence of the PDF index on mean flaring raten 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!
HISTORICAL, SELF-SIMILAR FLARE MODELS
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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
SELF ORGANIZATION
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PERCOLATION MODELS OF SOLAR
ACTIVE-REGION EMERGENCE & EVOLUTION
Simulated magnetic flux emergence and activeregion 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)
PERCOLATION MODELS
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FLARE STATISTICS BY PERCOLATION MODELS?
Plausible
timeseries of
magnetic energy
release
Power-law PDFs in
event energy with
index ~ -1.55
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)
PERCOLATION MODELS
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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
SOC MODELS
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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
Bi ® Bi + d B
• Local gradient calculated & compared to |Bc|
1
dBi ® Bi - å Bnn
6 nn
• Isotropic redistribution rules if |dBi| > |Bc|
6
1
Bi ® Bi - dBi ; Bnn ® Bnn + dBi
7
7
• 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.
( )
2
6
Ec = 7 dBi
• Flux conservation, avalanches, and a spectral
power of the form S(f) ~ f-2
SOC FLARE MODELS
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REVISED SOC SOLAR-FLARE MODELS
-- The second “Statistical Flare” model (Vlahos, Georgoulis, et al.)
• Constant driver δΒ << critical threshold Bc
• Randomly chosen point i perturbed
Bi ® Bi + d B
SOC survived in this
“unorthodox” model, too!
• Local gradient calculated & compared to Bc, both isotropically, and anisotropically
dBiisotropic
1
® Bi - å Bnn
6 nn
dBi,nnanisotropic ® Bi - Bnn
• Isotropic redistribution rules if dBi (isotropic) > Bc and anisotropic rules in case dBi (anisotropic) > Bc
6
1
Bi ® Bi - dBi ; Bnn ® Bnn + dBi
7
7
• Elementary energy release per instability:
6 ö
æ
Ec = ç Bi - Bc ÷
è
7 ø
2
6
Bi ® Bi - dBi ; Bnn ® Bnn + d Bnn
7
åd B
i,nn
nn
d Bi,nn
( )
= 6 7 Bc
( )
dBi,nn
6
= 7 Bc
å dBi,nn
nn
SOC FLARE MODELS
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DIFFERENT FACES OF SOC
Vlahos et al. (1995)
--> Different avalanche attributes
--> Single power-law PDF (Lu et al. 1993) --> Double power-law PDF (Georgoulis & Vlahos
SOC FLARE MODELS
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1996)
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FURTHER SOC CA REVISIONS: VARIABLE DRIVER
• Constant Variable driver δΒ with PDF
P(d B) ~ d B
-a
with α --> free parameter
•For α ∈ [1.0, 2.5], SOC manages to survive
Georgoulis & Vlahos (1998)
Output power-law indices variable, and controlled by α!
SOC FLARE MODELS
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Event PDFs for α=1.6
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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
X-CA SOC FLARE MODELS
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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
X-CA SOC FLARE MODELS
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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
X-CA SOC FLARE MODELS
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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)
Haleakala IVM
NOAA AR 10247
2003-01-13 18:26 UT
--> 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
DATA-DRIVEN SOC FLARE MODELS
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Avalanche relaxation in NOAA AR 10247
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DATA-DRIVEN CA FLARE MODELS
--> SOC survived evolving via a timeseries of SOC-state magnetic field solutions
Dimitropoulou et al. (2012), A&A, submitted
“Dynamic” integrated flare model (D-IFM)
--> 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
DATA-DRIVEN SOC FLARE MODELS
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SUMMARY AND OVERARCHING QUESTIONS
Therefore, is SOC at work in solar active regions?
ToCritical
find out,
we should
consider
fundamental
1.
threshold:
what
is/are the
the following
critical threshold(s)
for questions:
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?
TOP-LEVEL SUMMARY
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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
CRITICAL QUESTIONS FOR SOC APPLICABILITY
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TURBULENCE & SOC:
COMPATIBLE OR CONFLICTING CONCEPTS?
Timeseries of mean Joule
dissipation rate
Georgoulis, Velli, & Einaudi (1998)
Event scaling laws
2.5D reduced MHD (RMHD
- incompressible) system
--> 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?
CRITICAL QUESTIONS FOR SOC APPLICABILITY
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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?
CRITICAL QUESTIONS FOR SOC APPLICABILITY
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FLARE WAITING-TIME DISTRIBUTIONS:
WHAT IS THEIR FORM AND WHAT IS EXPECTED?
Flares are essentially random eventsPower-law PDF of waiting times Flare occurrence a non(Boffetta et al. 1999; Lepreti et al.stationary Poisson process
(Crosby et al. 1998)
(Wheatland 2000; 2001)
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!
CRITICAL QUESTIONS FOR SOC APPLICABILITY
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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?
POTENTIAL BENEFITS FROM SOC APPLICATION
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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?
POTENTIAL BENEFITS FROM SOC APPLICATION
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A “SOFT” NANOFLARE POPULATION?
Besides the reproduced sizable
flares, SOC anisotropic
relaxation criteria predict a soft
small-event population. Does it
really exist?
Georgoulis et al. (2001)
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?
GRANAT/WATCH flare PDFs
Statistical Flare model
reproduction
POTENTIAL BENEFITS FROM SOC APPLICATION
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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)
POTENTIAL BENEFITS FROM SOC APPLICATION
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LOSS-OF-EQUILIBRIUM SOLAR ERUPTION MODELS
Lin & Forbes (2000)
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?
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.
POTENTIAL BENEFITS FROM SOC APPLICATION
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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
POTENTIAL BENEFITS FROM SOC APPLICATION
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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!
POTENTIAL BENEFITS FROM SOC APPLICATION
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PREDICTIVE ABILITY OF MULTISCALE METHODS
Georgoulis (2012)
• 17733
POTENTIAL BENEFITS FROM SOC APPLICATION
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SoHO/MDI
magnetograms
• 370 AR
timeseries
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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 nonflaring ARs alike
Does this mean that flaring
and non-flaring active
regions might be in a similar
- indistinguishable - SOC
state (Vlahos & Georgoulis
2004) ?
POTENTIAL BENEFITS FROM SOC APPLICATION
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CONCLUSIONS
Overarching question / food for thought:
at what extent, if any, is SOC valid in solar active-region magnetic fields?
CONCLUSION
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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
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BACKUP SLIDES
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STATISTICS OF OTHER SOLAR DYNAMICAL
PHENOMENA
-- Active-region sizes
-- Quiet Sun fields
-- Ellerman bombs
-- ARTBs
-- etc.
SOLAR FLARE STATISTICS
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