1. Inductive Local Correlation Tracking or, Getting from One Magnetogram to the Next Goal (MURI grant): Realistically simulate coronal magnetic field in eruptive events. New Idea: Evolve boundary of MHD code consistently with magnetograms. Question: How can one derive a velocity from a time series of magnetograms? Brian Welsch, Bill Abbett, and George Fisher, Space Sciences Lab, UC-Berkeley

2. Outline 1.How does the field in magnetograms evolve? 2.How can we derive flows consistent with both this evolution and MHD? 3.How do we use these flows?

3. An example of magnetic evolution in an active region NOAA AR 8210, May 1 1998 – 1 day of evolution seen by MDI

4. Local Correlation Tracking, Fourier-style Central idea of LCT schema: find proper motions of features in a pair of successive images are by maximizing a cross-correlation function (or minimizing an error function) between sub-regions of the images. The concept is generally attributed to November & Simon (1988). Useful with G-band filtergrams, H  images, or magnetograms. The FLCT method (which we developed) is similar. For each pixel, we: –mask each image with a Gaussian, of width , centered at that pixel; –crop the resulting images, keeping only signficant regions; –compute the cross-correlation function between the two cropped images, using standard Fast Fourier Transform (FFT) techniques; –use cubic-convolution interpolation to find the shifts in x and y that maximize the cross-correlation function to one of two precisions (chosen by the user), either 0.1 or 0.02 pixel; and –use the shifts in x and y and  t between images to find the intensity features' apparent motion along the solar surface.

5. Example of FLCT applied to NOAA AR 8210 (May 1 1998)

6. The Demoulin & Berger (2003) Interpretation of LCT Apparent horizontal motion, U LCT, is from combination of hori- zontal motions and vertical motions acting on non-vertical fields.

7. The Ideal MHD Induction Equation How can we ensure that LCT-determined velocities are physically consistent with the magnetic induction equation? Only the z-component of the induction equation contains no unobservable vertical derivatives: Now, substitute in the Demoulin & Berger hypothesis The ideal MHD induction equation simplifies to this form:

8. I+LCT: Use LCT to constrain solutions of the induction equation Solve for ,  with 2D divergence and 2D curl (z-comp), and the approximation that U=U LCT : Let Note that if only B z (or an approximation to it, B LOS ) is known, we can still solve for ,  !

9. Apply ILCT to IVM vector magnetogram data for AR 8210 Vector magnetic field data enables us to find 3-D flow field from ILCT via the equations shown on slide 5. Transverse flows are shown as arrows, up/down flows shown as blue/red contours.

12. Flows Consistent with Induction Equation!

13. We have v(x,y,0;t) --- now put it all together...

18. Bill Abbett has driven an MHD code with v(x,y)!

19. Conclusions From a time series of vector magnetograms, we’ve derived three-component, photospheric flows that are quantitatively consistent with the induction equation’s z-component. Qualitatively, the flows appear consistent with the observed field evolution. We’re using these flows to drive MHD simulations – please stay tuned!

20. ILCT Flows are Only Consistent with Induction Eqn’s Normal Component! Directly measured Derived by ILCT

21. Directly measured ???? ???? What about other components?Derived by ILCT From NLFFF Extrapolation/ Prev. Time Step at photosphere, z = 0above photosphere, z > 0

22. Incorporate Vertical Gradients with a Boundary Code!

23. Directly measured Calculated by Boundary Code What about other components?Derived by ILCT From NLFFF Extrapolation/ Prev. Time Step at photosphere, z = 0above photosphere, z > 0

24. Induction Equation’s Components

25. Driving Simulations from LOS Data, p.1 1.Start with a vector magnetogram. 2.Derive ,  that will move system from vector data to initial LOS data. 3.Derive ,  to move system from one LOS magnetogram to the next. 4.Evolve code, using B and ,  at each step to find v(x,y) via:

26. Driving Simulations from LOS Data, p.2

27. Driving Simulations from LOS Data, p.3

28. Driving Simulations from LOS Data, p.4

29. Driving Simulations from LOS Data, p.5

30. Driving Simulations from LOS Data, p.6

31. Driving Simulations from LOS Data, p.7