1. Nonlinear force-free coronal magnetic field extrapolation scheme for solar active regions Han He, Huaning Wang, Yihua Yan National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China Hinode Workshop Beijing, China Dec. 8-10, 2007

2. Solar eruption events are connected with the coronal magnetic structures Direct observationCalculation based on a physical model OR

3. Nonlinear force-free field (NLFFF) extrapolation for solar active regions vector magnetogram observed in the photosphere magnetic field configuration in the corona Calculation based on NLFFF model Solar active region Given Bx, By, Bz at the bottom boundary

4. field line or Force-free: Divergence-free: The field can be considered force-free roughly 400km (0.55 arc second) above the photosphere (Metcalf et al., 1995, ApJ, 439, 474) Alpha is a constant along one field line Nonlinear force-free field (NLFFF) model

5. Direct Boundary Integral Equation (DBIE) method for NLFFF extrapolation ( Yan, Y., Li, Z. 2006, ApJ, 638, 1162 ) Infinite plane surface boundary

6. Boundary condition of DBIE method Infinite plane surface boundary vector magnetogram B is assumed to be zero out of the vector magnetogram

7. Parameter λ ► same dimension as the force-free factor α ► defined locally at a field point ► three components λ x, λ y, λ z corresponding to B x, B y, B z

8. Optimal method to determine λ and B locally ( Yan, Y., Li, Z. 2006, ApJ, 638, 1162 ) Best convergence property at the field points near the bottom boundary

9. The bottom boundary for applying the DBIE is moved upwardly layer by layer to achieve the best convergence property Enlarge the area for integration at higher layers Keep the original number of pixels at each layer to save computing time Upward boundary integration scheme Output region of the code

10. Global field configuration Test Case I ► Low and Lou (1990) analytical field ►

11. Case I 3D-View Analytical solutionExtrapolated field

12. Test Case II Global field configuration ►

13. Case II 3D-View Analytical solutionExtrapolated field

14. The degree of agreement between the extrapolated field and the analytical solution The extrapolated fields deviate from the analytical fields gradually with the increase in height

15. Internal consistency of the extrapolated Field (Case I) Force-free and divergence-free constraints are satisfied in the extrapolated field

16. Internal consistency of the extrapolated Field (Case II) Force-free and divergence-free constraints are satisfied in the extrapolated field

17. NLFFF extrapolation for solar active region NOAA 9077 2000.07.14 04:14UT Vector magnetogram of NOAA 9077 was observed at 04:14 UT on 14 July 2000 by Solar Magnetic Field Telescope (SMFT) at Huairou Solar Observing Station (HSOS)

18. BxByBz BxByBz The noises in the boundary data can be suppressed by DBIE through the integration over the whole boundary

19. AR 9077 2000.7.14 04:14UT Observed by SMFT at Huairou FOV: 269x269 arcsec 64x64 grid 4.2 arcsec/pixel ~ 3000km/pixel Red: field lines that leave the modeling box Blue: closed field lines

20. Internal consistency of the extrapolated field Force-free constraintDivergence-free constraint

21. U-shaped field lines above the X-point Side view 3D view

22. Compared with filament images 04:27UT Top view 04:14UT 04:12UT TRACE 195A

23. Compared with TRACE images Before the flareDuring the flare (http://trace.lmsal.com) maximum 10:10UT Onset of the flare 04:14UT 10:24UT

24. SOHO MDI NOAA 10436 2003.08.22 Vector magnetogram was observed at 2003.08.22 01:29:57UT by Solar Flare Telescope NLFFF extrapolation for solar active region NOAA 10436

25. AR 10436 2003.8.22 01:29:57UT Solar Flare Telescope Lower layers Red: field lines that leave the modeling box Blue: closed field lines

26. Compared with coronal image NOAA 10436 GOES 12 SXI 2003.08.22 01:30:37UT Solar Flare Telescope 2003.08.22 01:29:57UT 340 x 320 arcsec 64 x 64 grid 5.3 arcsec/pixel

27. Hinode (Solar-B) SOT XRT EIS Field of View EIS (576”x512”)‏ XRT (2048”x2048”)‏ SOT:NFI/SP (328”x164”)‏ SOT: BFI (205”x102”)‏ N E W S (From Hinode Team) The instruments

28. 46.8” X 162.3” (156 X 512 grids) Hinode Solar Optical Telescope – Spectro-polarimeter Fast Map: Time per position: 3.2 sec 162.3 ” x 0.3 ” Normal map Fast map Dynamics Deep magnetogram Sample data 2007-6-4 08:49:04 - 08:58:54UT

29. Hinode Solar Optical Telescope (SOT) Spectro-Polarimeter (SP)-- level-0 data Level-0 data IQUV Spectral coverage: 630.08nm - 630.32nm Fe I lines 630.15 and 630.25 nm

30. Hinode SOT SP level-1 data (by SSW code) Level-1 data IQUV

31. Milne-Eddington inversion code (High Altitude Observatory) IQ U V

32. NLFFF extrapolation for solar active region NOAA 10930 ► Flare event: 2006.12.13 0214 0240 0257UT X3.4 SOHO MDI NOAA 10930 2006.12.12 Vector magnetogram was observed at 20:30UT on 12 Dec. 2006 by Hinode satellite

33. Images of Hinode SOT Spectro-Polarimeter NOAA 10930 (630.15 nm) I Time of the observation ： 2006.12.12 20:30-21:33UT FOV ： 295.20 x 162.30 arcsec Original grid number ： 1000 x 512 Q U V Level-1 data

34. Remove 180 degree ambiguity of the direction of the transverse field component By a reference field with a force-free factor best fitting the observed fields Wang, Yan and Sakurai (2001)

35. Vector magnetogram of NOAA 10930 Hinode SOT SP data Time ： 2006.12.12 20:30-21:33UT FOV ： 295.20 x 162.30 arcsec Original grid number ： 1000 x 512 Grid number for extrapolation ： 111 x 60 (2.7arcsec/pixel) Milne-Eddington inversion code by High Altitude Observatory

36. Compared with coronal image observed by XRT NOAA 10930 Hinode XRT image 21:30:32UT 2006.12.12 Extrapolated field lines based on vector magnetogram observed by Hinode SOT SP 2006.12.12 20:30-21:33UT FOV:295.20 x 162.30 arcsec

37. Compared with coronal image observed by XRT NOAA 10930

38. Central domain of the extrapolated field NOAA 10930 During the flare Before the flare

39. Thanks