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Laura F. Morales Canadian Space Agency / Agence Spatiale Canadienne Paul Charbonneau Département de Physique, Université de Montréal Markus Aschwanden Lockheed Martin, Adv. Tec. Center, Solar and Astrophysics Lab. Anisotropic braiding avalanche model for solar flares: A new 2D application

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Outline Solar Flares: Observations + Classical Th. Models SOC paradigm: The sandpile model SOC & Solar Flares: Lu & Hamilton's classic model New SOC model for solar flares: * Cellular Automaton * Statistical results & Spreading exponents * Expanding the model capabilities: Temperature Density

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Sun's Atmosphere PHOTOSPHERE CHROMOSPHERE SOLAR CORONA Sunspots Granules Super-granules Spicules Filaments Active regions Loops Solar Flares Etc…. http:// www-istp.gsfc.nasa.gov/istp/outreach/images/Solar/Educate/atmos.gif

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M-Class Flare - STEREO (March, 25 2008) – EUV http://stereo.gsfc.nasa.gov/img/stereoim ages/movies/Mflare2008.mpg X-Class Flare - SOHO (November, 4 2003) http://sohowww.nascom.nasa.gov/gallery/ Movies/EITX27/StormEIT195sm.mpg “...a solar flare is a process associated with a rapid temporary release of energy in the solar corona triggered by an instability of the underlying magnetic field configuration …”

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Magnetic Reconnection t onset ~ 1-2s - t thermalization ~ 100s t diffusion ~ 10 16-18 s in the solar corona another mechanism http://www.sflorg.com/spacenews/images/imsn051906_01_04.gif

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Parker's Model for solar flares B 0 uniform High conductivity Photospheric motions shuffle the footpoints of magnetic coronal loops Spontaneous Current Sheets in Magnetic Fields: With Applications to Stellar X-rays (Oxford U. Press 1) – Figure 11.2 http://helio.cfa.harvard.edu/REU /images/TRACE171_991106_023044.gif

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Photosphere Injection of kinetic Energy Solar Corona Storage of Magnetic Energy Very small Solar Flares Flares Energy Liberation Magnetic reconnection

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TURBULENCE OR SELF ORGANIZED CRITICALITY? (Dennis 1985, Solar Phys., 100, 465) Power law self similar behavior Energy is released in a wide range of scales ~10 24 -10 33 ergs

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SOC + Solar Corona Intermitent release of energy: Magnetic Reconnection Statistically stationary state: the solar corona is an statistically stationary state Slowly driven open system Photospheric motions instability threshold: Critical Angle

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t flare ~ seconds L B ~ 10 10 cm t photosphere ~ hs How can we obtain predictions by using this model? Integrate MHD aquations Cellular automaton-like simulations

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Each node is a measure of the B B(0)=0 Driving mechanism: add perturbations at some randomly selected interior nodes Stability criterion: associated to the curvature of B Classic SOC Models (Charbonneau et al. SolPhys, 203:321-353, 2001)

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Time series of lattice energy & energy released for the avalanches produced by 48 X 48 lattice (Charbonneau et al. SolPhys, 203:321-353, 2001) soc

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Probability Distributions

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Classic SOC Models: Ups Successfully reproduced statistical properties observed in solar flares: pdf’s exhibiting power law form good predictions for exponents: E, P, T

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Classic SOC Models: Downs 1. No magnetic reconnection 2. Link between CA elements & MHD If B k ↔ B .B ≠ 0 If B k ↔ A .B ≠ 0 solved & A interpreted as a twist in the magnetic field B k 2 is no longer a measure of the lattice energy 3. No good predictions for A

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Lattice Energy ~ ∑ L i (t) 2 i Lattice + perturbation NEW MODEL (2008) Threshold = 1 + 2 angle formed by 2 fieldlines 11 22

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E=1.25E 0 Reconnect + @ (1,3) Perturbation starts again One-step redistribution E = 1.22 E 0 Elim/reduce angle

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Two-step redistribution Reconnect (3,2) unstable E = 1.32E 0 E=1.4E 0 E=1.19E 0 Perturbation starts again (3,1) E = 1.19E 0

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The lattice in action 32 x 32 64 x 64

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Lattice Energy & Released Energy Morales, L. & Charbonneau, P. ApJ. 682,(1), 654-666. 2008 SOC P TT E T

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1.73-1.84 PP 1.63-1.71 EE New SOCClassic SOCObservations 1.54 1.40 1.7 1.79-2.11 Morales, L. & Charbonneau, P. ApJ. 682,(1), 654-666. 2008

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1.79-1.95 TT New SOCClassic SOCObservations 1.15 – 2.93 1.70 Morales, L. & Charbonneau, P. ApJ. 682,(1), 654-666. 2008

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Area covered by an avalanche: a movie

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Area covered by Avalanches unstable (12,2) unstable (10,1) t0t0 t 0 +30 t f = t 0 +332 t 0 +116 = t max t 0 +150 Time integrated Area Peak Area

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Geometric Properties New SOC Classic SOC EUV – TRACE 0.55 ± 0.021.02 ± 0.06 1.83 – 2.45 1.93 ± 0.072.45 ± 0.11 A*A* AA Morales, L. & Charbonneau, P. GRL., 35, L04108

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Spreading Exponents Number of unstable nodes at time t Probability of existence at t Size of an avalanche ‘death’ by t Probability of an avalanche to reach a size S

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128 x 128 c =2.5 0.09±0.02 1.1 ± 0.1 1.83±0.25 1.70±0.2 th =1+ 2.19±0.1 th =(1+ +2 )/ th 1.48 ±0.01 Just an example … Morales, L. & Charbonneau, P. GRL., 35, L04108

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From a 2D lattice to a loop fold bend

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Avalanching strands in the loop

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Projection

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Projections

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Geometrical properties for the projected areas A = 2.39 ± 0.05 A = 1.84 ± 0.07

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ND (stretch=1)D (stretch=10) 321.26 ± 0.041.21 ± 0.04 641.21 ± 0.041.23 ± 0.04 1281.20 ± 0.031.25 ± 0.05 Observations 1 – 1.93 N=64 N=32

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Another way of looking at the simulations Near vertical current sheet that extends from the coronal reconnection regions to the photospheric flare ribbons mapped into

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Temperature & Density Evolution The maximum loop temperature based on the maximum heating rate and the loop length for uniform heating case: Pressure Density k = 9.210 -7 erg s -1 K 7/2 (Spitzer conductivity) E max

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Temperatures Avalanche duration: 106 it. Avalanche duration: 138 it. N=64 THR=2 51013 avalanches in 4e5 iterations Max duration ~ 700 it

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Density ] ]

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With the temperature T(t) and density evolution n(t) of each avalanche we can compute the resulting peak fluxes and time durations for a given wavelength filter in EUV or SXR, because for optically thin emission we just have: I(t) = ∫ n(t) 2 w R(T) dT w is the loop width R(T) is the instrumental response function. We can plot the frequency distributions of energies: W =E_Hmax * duration peak fluxes (I_EUV, I_SXR) Coming up…..

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Conclusions Every element in the model can be directly mapped to Parker's model for solar flares thus solving the major problems of interpretation posed by classical SOC models. For the first time a SOC model for solar flares succeeded in reproducing observational results for all the typical magnitudes that characterize a SOC model: E, P, T, T & the time integrated A and the peak A *. The new cellular automaton we introduced and fully analyzed represents a major breakthrough in the field of self-organized critical models for solar flares since:

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