A physically-based sandpile model for the prediction of solar flares using data assimilation

Transcription

1 A physically-based sandpile model for the prediction of solar flares using data assimilation Antoine Strugarek CEA/Saclay Observatoire Paris-Meudon Université de Montréal With R. Barnabé, A.S. Brun, P. Charbonneau, N. Vilmer

2 Detection of solar flares in X-rays 2

3 Flare classification and statistics Hα classification Importance Class Area (Sq. Deg.) Area 10-6 solar disk Radio flux at 5000 MHz in s.f.u. Soft X-ray class 1 s.f.u. = 10 4 jansky = 10-2 W m -2 Hz -1 [Bhatnagar & Livingston 2005] Importance class Peak flux in 1-8 Å w/m 2 S A 10-8 to B 10-7 to C 10-6 to M 10-5 to >24.7 > X >10-4 [Aschwanden+, 2014] 3

4 Flare classification and statistics Hα classification Importance Class Area (Sq. Deg.) Area 10-6 solar disk Radio flux at 5000 MHz in s.f.u. Soft X-ray class 1 s.f.u. = 10 4 jansky = 10-2 W m -2 Hz -1 [Bhatnagar & Livingston 2005] Importance class Peak flux in 1-8 Å w/m 2 S A 10-8 to B 10-7 to C 10-6 to M 10-5 to >24.7 > X >10-4 [Aschwanden+, 2014] 3

5 The number of flares strongly varies during the solar cycle Line of sight magnetic field in the north hemisphere of the Sun [Aschwanden & Freeland 2012] 4

6 The number of flares strongly varies during the solar cycle Line of sight magnetic field in the north hemisphere of the Sun But no correlation of the power-law slope of the peak X-ray flux during flares [Aschwanden & Freeland 2012] 4

7 Outline Sandpile model and solar flares Strugarek+ 2014, Solar Physics Predicting individual (large) events with a stochastic model? Strugarek & Charbonneau 2014, Solar Physics Data assimilation: towards the Solar Orbiter era R. Barnabé s PhD thesis (AIM) 5

8 Basics of sandpile models (I) [Wiesenfeld+ 1989; Bak+ 1987] Basic model ingredients: Driver Self-Organized Criticality [Lu & Hamilton 1993] 6

9 Basics of sandpile models (I) [Wiesenfeld+ 1989; Bak+ 1987] Basic model ingredients: Driver Self-Organized Criticality [Lu & Hamilton 1993] 6

10 Basics of sandpile models (I) [Wiesenfeld+ 1989; Bak+ 1987] Basic model ingredients: Driver Self-Organized Criticality [Lu & Hamilton 1993] 6

11 Basics of sandpile models (I) [Wiesenfeld+ 1989; Bak+ 1987] Basic model ingredients: Driver Self-Organized Criticality [Lu & Hamilton 1993] 6

12 Basics of sandpile models (I) [Wiesenfeld+ 1989; Bak+ 1987] Basic model ingredients: Driver Threshold Self-Organized Criticality [Lu & Hamilton 1993] 6

13 Basics of sandpile models (I) [Wiesenfeld+ 1989; Bak+ 1987] Basic model ingredients: Driver Threshold Redistribution rule Self-Organized Criticality [Lu & Hamilton 1993] 6

14 Basics of sandpile models (II) 7

15 Basics of sandpile models (II) 7

16 Basics of sandpile models (III) Energy released Nb av = alpha =1.31 E Power law obtained for the energy release, peak energy during the avalanche, and also duration and area covered by the avalanche 8

17 A physical interpretation of sandpiles z forcing Sandpile Coronal Loop Magnetic potential Az Turbulent twisting of loop Currents Magnetic reconnection Sandpile Height Homogenous forcing Curvature Stochastic redistribution [Strugarek+ 2014] 9

18 A physical interpretation of sandpiles z forcing Sandpile Coronal Loop Magnetic potential Az Turbulent twisting of loop Currents Magnetic reconnection Sandpile Height Homogenous forcing Curvature Stochastic redistribution [Strugarek+ 2014] 9

19 A physical interpretation of sandpiles z forcing Sandpile Coronal Loop Magnetic potential Az Turbulent twisting of loop Currents Magnetic reconnection Sandpile Height Homogenous forcing Curvature Stochastic redistribution [Strugarek+ 2014] 9

20 Deterministically-driven sandpile models Deterministic driving on all nodes (Non)-Conservative redistribution rule Random process in threshold, redistribution and/or extraction [Strugarek+ 2014] 10

21 Conservative models do not reach the SOC state Random extraction Random extraction + Random redistribution Random extraction + Random redistribution + Random threshold [Strugarek+ 2014] 11

22 Conservative models do not reach the SOC state Loading/unloading cycles: breaking of power-law statistics Random extraction Random extraction + Random redistribution Random extraction + Random redistribution + Random threshold [Strugarek+ 2014] 11

23 Non-conservative models manage to recover SOC D nc = Non-conservation parameter (D nc =1 for conservative model) [Strugarek+ 2014] 12

24 Power-law exponents in the D-models Energy Peak Duration [Strugarek+ 2014] 13

25 Lattice size: it only affects the accessible energy range Energy Peak Duration [Strugarek+ 2014] 14

26 Outline - Sandpile model and solar flares Strugarek+ 2014, Solar Physics - Predicting individual (large) events with a stochastic model? Strugarek & Charbonneau 2014, Solar Physics - Data assimilation: towards the Solar Orbiter era R. Barnabé s PhD thesis (AIM) 15

27 Motivation: predicting solar flares still gives us a hard time 16

28 Stochasticity vs memory: the issue with classical sandpile Random seed #1 Standard sandpile model Random seed #2 17

29 Stochasticity vs memory: the issue with classical sandpile Random seed #1 Standard sandpile model Random seed #2 17

30 Stochasticity vs memory: D-models Random seed #1 Non- Conservative sandpile model Random seed #2 18

31 Stochasticity vs memory: D-models Random seed #1 Non- Conservative sandpile model Random seed #2 18

32 Robustness of one large event (standard model) Energy of the largest event in the time window PDF LH Weibull Gaussian stochastic realisations [Strugarek & Charbonneau 2014] 19

33 Robustness of one large event (standard model) Energy of the largest event in the time window PDF LH Weibull Gaussian Targeted event energy 2000 stochastic realisations [Strugarek & Charbonneau 2014] 19

34 Robustness of one large event (standard model) Energy of the largest event in the time window PDF LH Weibull Gaussian Targeted event energy Statistical max in the time window 2000 stochastic realisations [Strugarek & Charbonneau 2014] 19

35 Robustness of one large event (standard model) Energy of the largest event in the time window Occurence of the largest event in the time window LH PDF Weibull Gaussian PDF Targeted event energy Statistical max in the time window 2000 stochastic realisations [Strugarek & Charbonneau 2014] 19

36 Robustness of one large event (standard model) PDF Energy of the largest event in the time window Weibull LH Gaussian Targeted event energy Statistical max in the time window PDF Occurence of the largest event in the time window Not possible to predict large events 2000 stochastic realisations [Strugarek & Charbonneau 2014] 19

37 Robustness of one large event (D model) 250 Energy of the largest event in the time window D Gaussian Weibull Released energy 2000 stochastic realisations [Strugarek & Charbonneau 2014] 20

38 Robustness of one large event (D model) Energy of the largest event in the time window D2 Gaussian Weibull Released energy Targeted event energy 2000 stochastic realisations [Strugarek & Charbonneau 2014] 20

39 Robustness of one large event (D model) Energy of the largest event in the time window D2 Gaussian Weibull Released energy Targeted event energy Statistical max in the time window 2000 stochastic realisations [Strugarek & Charbonneau 2014] 20

40 Robustness of one large event (D model) Energy of the largest event in the time window D Occurence of the large event in the time window Gaussian Weibull PDF Released energy Targeted event energy Statistical max in the time window 2000 stochastic realisations [Strugarek & Charbonneau 2014] 20

41 Robustness of one large event (D model) Energy of the largest event in the time window D2 Gaussian Weibull Released energy Targeted event energy PDF Predicting yes you can Occurence of the large event in the time window Statistical max in the time window 2000 stochastic realisations [Strugarek & Charbonneau 2014] 20

42 Predictive performance of sandpile models 10 1 LH GV D1 D2 D3 Good predictive capabilities Predicted Energy Bad predictive capabilities Predicted Time Significance s t [Strugarek & Charbonneau 2014] 21

43 Bias with occurence in the time window 10 1 LH GV Predicted Energy D1 D2 Predicted Energy t A /t w t A /t w [Strugarek & Charbonneau 2014] 22

44 Outline - Sandpile model and solar flares Strugarek+ 2014, Solar Physics - Predicting individual (large) events with a stochastic model? Strugarek & Charbonneau 2014, Solar Physics - Data assimilation: towards the Solar Orbiter era R. Barnabé s PhD thesis (AIM) 23

45 Data assimilation in sandpile models: FlarePredict Initial condition (+model parameters) Variational data assimilation 24

46 Data assimilation in sandpile models: FlarePredict Initial condition (+model parameters) A) Compare model vs observations Variational data assimilation 24

47 Data assimilation in sandpile models: FlarePredict Initial condition (+model parameters) A) Compare model vs observations Variational data assimilation B) Observations are not reproduced 24

48 Data assimilation in sandpile models: FlarePredict Initial condition (+model parameters) A) Compare model vs observations Variational data assimilation B) Observations are not reproduced Promising technique: simulated annealing 24

49 Data assimilation in sandpile models: FlarePredict Initial condition (+model parameters) A) Compare model vs observations Variational data assimilation B) Observations are not reproduced C) Observations are reproduced Promising technique: simulated annealing Determine eruptive probability (prediction) Hindcasting on past records first 24

50 Data assimilation in sandpile models: FlarePredict Initial condition (+model parameters) A) Compare model vs observations Variational data assimilation B) Observations are not reproduced C) Observations are reproduced Promising technique: simulated annealing Determine eruptive probability (prediction) Hindcasting on past records first The different blocks are included in a general framework called FlarePredict (under development) 24

51 A) Comparing observational data with our model (I) a.u [hours] GOES nm, 06/02/ /02/2010 [AR 1045] C8 X M C B A [Aschwanden & Freeland 2012] Model iterations 25

52 A) Comparing observational data with our model (II) Least squares not efficient because of the discrete character of the events Defining a versatile cost function: Flare energy Observations time 26

53 A) Comparing observational data with our model (II) Least squares not efficient because of the discrete character of the events Defining a versatile cost function: Flare energy Model Observations time 26

54 A) Comparing observational data with our model (II) Least squares not efficient because of the discrete character of the events Defining a versatile cost function: Match Flare energy Model Observations time 26

55 A) Comparing observational data with our model (II) Least squares not efficient because of the discrete character of the events Defining a versatile cost function: Match Miss Flare energy Model Observations time 26

56 A) Comparing observational data with our model (II) Least squares not efficient because of the discrete character of the events Defining a versatile cost function: Match Miss Flare energy Model Observations False Alarm time 26

57 B) Finding a new initial condition reproducing the data (I) Using the gradient of the cost function Adjoint method 27

58 B) Finding a new initial condition reproducing the data (I) Using the gradient of the cost function Too many local minima Adjoint method 27

59 B) Finding a new initial condition reproducing the data (I) Using the gradient of the cost function Using more robust but more expensive methods (w/o explicit gradient calculation) Too many local minima Cost (x) Perturbation Solution x Adjoint method Simulated annealing 27

60 B) Finding a new initial condition reproducing the data (II) Simulated annealing is not very efficient: N 2 realizations of the model are used to minimize the cost function Work in progress: reduce this number by projecting the sandpile model on its eigenvectors In this case, N 2 = 2304 eigenvectors (N degrees of freedom) 28

61 C) Towards a predictive tool 10 7 [hours] GOES nm, 01/04/ /04/ X 3 M-flares 10 5 M 3 X-flares 10 7 [hours] GOES nm, 25/10/ /10/2014 [AR 12192] a.u C B 10 6 X 10 2 A 10 5 M a.u C Model Iterations 10 3 B 10 7 [hours] GOES nm, 25/10/ /10/ Model Iterations A X M 2 X-flares + 4 M-flares a.u C 10 3 B 10 2 A Model Iterations 29

62 Summary Develop a physics-inspired SOC model that reproduces the solar flare statistics [Strugarek+ 2014] Demonstrate the predictive capabilities of particular sandpile model(s) [Strugarek & Charbonneau, 2014] Reduce solar X-ray data to be used as an input of the data assimilation pipe-line Implement robust data assimilation techniques (minimization algorithm) Under investigation: first results are very encouraging Test the prediction package with a statistically significant sample of real observations Future: compare sandpile models with MHD models Stay tuned! 30

63 Thank you for your attention! 31

64 Time window definitions [Strugarek & Charbonneau 2014] 32

65 Simulated annealing with the simplex method Festival de théorie 2015, Aix-en-Provence A. Strugarek

66 The largest flares are extremely hard to predict Festival de théorie 2015, Aix-en-Provence A. Strugarek [Schrijver 2007] [Barnes & Leka, ApJ 2008]

67 The largest flares are extremely hard to predict Improvement factor compared to an assumed statistics of events All the estimates perform quite poorly for the large events [Schrijver 2007] Festival de théorie 2015, Aix-en-Provence A. Strugarek [Barnes & Leka, ApJ 2008]

68 The largest flares are extremely hard to predict Improvement factor compared to an assumed statistics of events All the estimates perform quite poorly for the large events Estimates using the statistical distribution of flares [Wheatland 2005] give slightly better results [Schrijver 2007] Festival de théorie 2015, Aix-en-Provence A. Strugarek [Barnes & Leka, ApJ 2008]

69 Physical interpretation of the nodal variable [Charbonneau 2013] Festival de théorie 2015, Aix-en-Provence A. Strugarek

70 Stochasticity in the d models Where to put the random process? Random extraction (Z i,j >Z c )! Bi,j = 4 B r B i±1,j±1 += B r Random threshold (Z i,j >Z r c )! Bi,j = 4 B B i±1,j±1 += B Random redistribution (Z i,j >Z c )! Bi,j = 4 B B i±1,j±1 += r k R B r k random deviate 2 [0, 1] (k 2 {1, 4}) X r k = R k Festival de théorie 2015, Aix-en-Provence A. Strugarek