1. Reducing the Divergence of Optimization-Generated Magnetic Fields J.M. McTiernan, B.T. Welsch, G.H. Fisher, D.J. Bercik, W.P. Abbett Space Sciences Lab. University of California, Berkeley

2. Abstract: Optimization methods (OMs) are often used to extrapolate non-linear, force-free (NLFF) coronal magnetic fields from measurements of the vector magnetic field at the Sun's photosphere. Unfortunately, OM algorithms typically generate a magnetic field that is not as divergence-free as methods that are explicitly divergence-free (for example, methods that describe the field using a vector potential). Here we describe multiple approaches to deriving a correction that can be added to an arbitrary vector field with a non-zero divergence to cancel that divergence. We then apply one technique to OM magnetic fields extrapolated from magnetograms, within the framework of a particular OM implementation.

3. Optimization method: Wheatland, Roumeliotis & Sturrock, Apj, 540, 1150 The idea: minimize the “Objective Function” Take the derivative, and manipulate. Then we can write: If we vary B, so that dB/dt = F, and require dB/dt = 0 on all boundaries, then L decreases, and the divergence and force must decrease. (Note that “t” is not time; it’s just a variable.)

4. Start with a Chromospheric Vector magnetogram With 180° ambiguity resolved: IVM data from Tom Metcalf. AR10486: 29-Oct-2003 18:46:00 UT

5. The starting point is a potential field, extrapolated from Bz on the surface: The potential field lines do not show shear in the field. Note that for purposes of this test, the IVM data have been rebinned from 512x512 to 65x65 (4.4 arcsec pixels).

6. Next, the magnetogram replaces the bottom boundary. This results in a configuration That is not force-free or divergence-free, but the optimization algorithm will attempt to minimize both. If we can “clean” some of the divergence from the initial field, this may help reduce the divergence in the final result.

7. How do you “clean” divergence? The initial field has divergence: We seek a “corrective” magnetic field that can be added to the input field to render it divergence-free.

8. How do you “clean” divergence? (Continued….) Next, require that B z and J z be kept fixed; the correction field is horizontal and curl free, and can be derived from the horizontal gradient of a scalar. This results in Poisson’s equation in two dimensions: The boundary conditions are difficult, due to the requirement that B not change on the boundaries, but B.T. Welsch has developed an IDL routine that will try it. Ask Brian to explain it….

9. Testing the method: We used 3 different procedures. The first was to optimize with no divergence cleaning. The 2 nd was to divergence-clean the initial state, and then optimize. The 3 rd was to do divergence-clean during the optimization process, once every 100 iterations. Here is a plot of the objective function L for each trial: The 3 rd method (blue line) had issues; the solution oscillated, and took much longer to converge.

10. Testing the method (cont): Here is a table of the L values for each model, for the potential field, and the initial fields. The total value of L can be split into two contributions: one from the force and one from the divergence. FieldL valueJxB termdiv B term Potential1.450.700.75 Potl+IVM22.319.82.50 Potl+IVM+divb _cleaning 27.527.10.42 (!) Final: No divb_cleaning 3.092.130.96 Final: One divb_cleaning 3.192.250.94 Final: Multi divb_cleaning 2.992.190.80

11. Testing the method (cont): Divergence cleaning works – sort of... Here is a plot of the average value of divB (normalized by B) as a function of z. The red line shows the initial field, immediately after divergence cleaning. Except for the 2 nd level above the bottom, (bug?) the divergence is smaller. But the optimization process recovers most of the lost divergence.

12. Testing the method (cont): The process increases the JxB force, away from the boundary, but the optimization recovers from this also. In fact, there is very little difference in the final fields for the three cases. The divergence cleaning results in only a slight improvement, as seen in the L values in the table.

13. Fieldlines? They are similar, but not identical (Only two models are shown here, but the third is very similar.)

14. Conclusions: The divergence cleaning works well near the lower boundary of the extrapolation volume. The optimization procedure seems to recover the missing divergence and converges to a similar solution regardless of whether the divergence was cleaned. This may point to a unique solution for a given set of boundary conditions. This is the first test that has been run. We will continue testing with other active region magnetograms and also with simulated data.