1. Nonlinear Force Free Field Extrapolation for AR 10486 J.McTiernan 7-Nov-2005

2. Some Jargon: Force-free field: JxB = 0, current J is parallel to B Alpha: constant of proportionality, J = αB Linear Force-free field: α is a spatial constant Non-linear Force-free field: α is not constant Potential field: α = 0, J = 0, also called “current-free”

4. Start with a VECTOR magnetogram With 180° amiguity resolved: IVM data from T.Metcalf. 29-Oct-2003 18:46:00 UT Chromospheric (field is force- free)

5. The starting point is a potential field, extrapolated from Bz on the surface: Note that the potential field lines do not show shear in the field. Colors? The brighter lines have stronger Bz at the surface.

6. Angled view of potential field.

7. Given the initial field, the next step is to calculate F, then, set B = B + F*dt dt = small (1.0e-5) The objective function, L is guaranteed to decrease. Then iterate: eventually L stops decreasing, generally not to 0, due to noise. iter dt L 0 1.000000e-05 18.79674049 10 1.104622e-05 13.91906588 20 1.220190e-05 10.78302648 30 1.347848e-05 8.72998846 40 1.488863e-05 7.37316102 50 1.644631e-05 6.44224807 60 1.816696e-05 5.77895070 70 2.006762e-05 5.27828721 80 2.216713e-05 4.89977869 90 2.448631e-05 4.60944443 100 2.704811e-05 4.38145939. 700 2.525218e-09 3.53352468 710 1.394706e-09 3.53352137 720 1.540623e-09 3.53351864 730 1.701806e-09 3.53351563 740 1.879852e-09 3.53351230 750 2.076526e-09 3.53350863 760 2.293776e-09 3.53350457 770 2.533756e-09 3.53350009 780 2.798842e-09 3.53349514 790 3.091663e-09 3.53348967 800 1.707559e-09 3.53348412

8. Here is the final field: These are the same field line starting points shown earlier. There are some differences.

9. It’s easier to see differences by angling the view. The left side is the potential field, the right is the NLFFF extrapolation. The NLFFF extrapolation shows sheared field lines. These are close to the lower boundary.

10. Quantitative Comparison Metrics for Field Vectors Cauchy-Schwarz: Good = 1.0 Field direction Mean Vector Error: Good = 0.0 Field Magnitude

11. All of the action is in the 30000 km closest to the surface. At least some of this is due to the noise.

12. For a model with no noise, the difference between NLFFF and potential varies more gradually.

13. Given a field, can you relate it to what you see in an image? This is hard, e.g., Lundquist, Fisher, McTiernan, Regneir (2004) SOHO 15 Workshop Proceedings. All I would like is to see if I can see if there are any field lines that look like the visible loops. This is an EIT 195 image, 5 mins before the IVM magnetogram. The arrows point at loop-like structures.

14. The left plot are field lines, the right is the EIT image, with MDI contours. Arrows denote possible matches. (The alignment in all of these comparisons is suspect, maybe only good to 10 arcsec.)

15. Add some more lines, maybe we see something that looks like the loops on the top, but they do not seem to close back into the surface in the same way.

16. Still more lines. Not much help on the top…

17. Even more lines,

18. More lines, some of the very dark (weak Bz) lines may match. So maybe we have had some success. But not very quantitative.

20. What about the flare? Our IVM magnetogram is 2 hrs before the X flare. (Do not yet have a post-flare mgram.) So the loops seen here may or may not correspond to our B field. TRACE image with RHESSI contours

21. IVM Bz, with RHESSI 50 – 100 keV footpoints, at different times: A: 20:43:40 to 20:44:40UT B: 20:51:12 to 20:52:12UT C: 20:58:40 to 20:59:40UT The footpoints move outwards in both directions: (Säm has movie, I lost the link)

22. Most of the lines are not in the same direction as you might expext from the footpoint position. Closer to the bottom boundary, there are lines with the same sort of directionality as the footpoints show.

23. This is easier to see if you angle the field lines. The lines in the footpoint direction are very low, and show the shear of the field seen in the mgram.

24. Those field lines do have the direction shown in the TRACE loops during the flare.

25. Conclusions? If you look very hard, you may see something you like; some field lines look sort of similar to the loops that you see in the preflare images. More quantitative work is needed … Also there are some field lines that look similar to what you might expect from the flare footpoints. Future work? Better alignment, and some quantitative estimates. Also we are trying to get some post-flare magnetograms. The NLFFF code is now part of SSW, try: IDL>ssw_upgrade, /nlfff, /spawn See http://sprg.ssl.berkeley.edu/~jimm/fff/optimization_fff.htmlhttp://sprg.ssl.berkeley.edu/~jimm/fff/optimization_fff.html for documentation.