1. 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

2. 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

3. 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

4. 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 …”

5. 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

6. 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

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

8. 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

9. 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

10. 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

11.  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)

12. 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

13. Probability Distributions

14. 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

15. 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

16. Lattice Energy ~ ∑ L i (t) 2 i Lattice + perturbation NEW MODEL (2008) Threshold  =   1 +  2 angle formed by 2 fieldlines 11 22

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

18. 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

19. The lattice in action 32 x 32 64 x 64

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

21. 1.73-1.84 PP 1.63-1.71 EE New SOCClassic SOCObservations 1.54 1.40 1.7 1.79-2.11 Morales, L. & Charbonneau, P. ApJ. 682,(1), 654-666. 2008

22. 1.79-1.95 TT New SOCClassic SOCObservations 1.15 – 2.93 1.70 Morales, L. & Charbonneau, P. ApJ. 682,(1), 654-666. 2008

23. Area covered by an avalanche: a movie

24. 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

25. 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* AA Morales, L. & Charbonneau, P. GRL., 35, L04108

26. 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

27. 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

28. From a 2D lattice to a loop fold bend

29. Avalanching strands in the loop

30. Projection

31. Projections

32. Geometrical properties for the projected areas  A = 2.39 ± 0.05   A = 1.84 ± 0.07

33. 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

34. 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

35. 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

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

37. Density ] ]

38. 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…..

39. 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: