1. Non-linear FFF Model: AR8210 J.McTiernan 1-Mar-2004

2. Optimization method: Wheatland, Sturrock, Roumeliotis, 2000 Apj, 540, 1150

3. Optimization method (cont): Iterative process, start with Potential or linear FF field, extrapolated from magnetogram. (We prefer the potential field, since we expect the field to be potential far from the active region.) Be sure that B does not change on the boundary Calculate F, set new B = B + F*dt “Objective function”, L, is guaranteed to decrease, but the change in L (dL) becomes smaller as you go. Keep going until dL approaches 0.

4. Optimization method (cont): Wheatland, et al, tested the method using a known FFF model of Low and Lou (1990) and got good agreement. L approaches 0. It’s harder with real data: the boundary conditions require the field to be potential on the outer boundaries, and whatever the vector magnetogram is on the lower boundary. L asymptotically approaches some (small) constant value. There are other issues, finite grid size doesn’t help, and real magnetograms have noise. We try anyway For AR8210, we have a vector magnetogram (courtesy of S.Regnier) 121x121 1.77 arcsec pixels. (actually padded to make square) IDL routine, took 221 minutes to converge.

5. Bz 8210

6. 8210

7. From Abbett, et al, 2004.

9. Yes they do sort of look alike, don’t they..

10. Conclusions: The Optimization method gives a result that doesn’t look so bad when comparing to images. But …. The field does not seem to reduce down to a force-free state in the case with “real’’ boundary conditions. Need to do a more quantitative comparison, preferably using a bunch of different (100s) magnetograms. A spherical coordinates version is in development, this could be used for larger slices of the sun.