1. Surface Flows From Magnetograms Brian Welsch, George Fisher, Bill Abbett, & Yan Li Space Sciences Laboratory, UC-Berkeley M.K. Georgoulis Applied Physics Laboratory, Johns Hopkins University P. W. Schuck Naval Research Lab – D. C. D.W. Longcope & B. Ravindra Physics Department, Montana State University Marc DeRosa Lockheed-Martin Advanced Technology Center K. Kusano Earth Simulator Ctr., Japan Agency for Marine-Earth Sci. & Tech.

2. Main Ideas 1. Why should we study surface flows derived from magnetograms? Poynting/Helicity Fluxes, Data Driving, Flux Transport Models 2. What classes of flows can we derive, in principle, from magnetogram sequences? In theory, flows perpendicular to both B &  B z can be found. 3. Practically, what are the best available methods for deriving flow information? Tests suggest tracking (e.g., LCT) combined with MEF. 4. How should available methods be improved? We must develop the capability to match both ∂ t B hor & ∂ t B z.

3. Abstract Estimates of velocities from time series of photospheric and/or chromospheric vector magnetograms can be used to determine transport rates for magnetic flux (i.e., emergence and submergence), energy (Poynting flux), and helicity across the magnetogram layer, and to provide time-dependent boundary conditions for data-driven simulations of the solar atmosphere above this layer. Velocity components perpendicular to the magnetic field are necessary both to compute these transport rates and to derive model boundary conditions. Since Doppler shifts also contain contributions from flows parallel to the magnetic field, perpendicular velocities are not generally recoverable from Doppler shifts alone. Consequently, several methods have been developed to estimate the perpendicular velocity from magnetograms, and have recently been validated using synthetic magnetograms from MHD simulations. Combined with data from the next generation of magnetographs (SOLIS, SOT/Hindode, and HMI/SDO), these techniques should provide valuable tools for understanding and modeling solar variability on time scales both short (e.g., flares and CMEs) and long (e.g., the 11-year sunspot cycle).

4. Flows, v, affect magnetic evolution at both the surface and in the corona. Since E = -(v x B)/c + R, the fluxes of magnetic energy & helicity across the surface depend upon v. ∂ t U = c ∫ dA (E x B) ∙ n / 4π(1) ∂ t H = c ∫ dA (E x A) ∙ n / 4π(2) B in the corona is coupled to B at the surface, so the surface v provides an essential boundary condition for data-driven MHD simulations of the coronal B field. Studying v could improve evolutionary models of photospheric B fields, e.g., flux transport models.

5. Progress Slide #1

6. The Induction Equation relates ∂ t B to v, ∂ t B = - c (  x E) =  x [(v x B) + cR] (3) Resistive effects, R, are model-dependent. Using, e.g., R =η  2 B would require magnetograms at two heights in the atmosphere – these are rare! R=0 (ideal MHD) is usually assumed. Even ideally, only the normal component of (3) does not require magnetograms from two heights.

7. Consequently, the goal is to derive v from ∂ t B n =  ∙ (v n B hor - v hor B n ) (4) Clearly, (4) can’t determine v, which has three components! By assumtion, v ||, the component of v parallel to B, cannot affect ∂ t B n  (v x B). In fact, v || is not useful for any of the reasons I’ve given for wanting to study surface flows! So ignore v || ! With v = v  + v ||, try to find v , and use v  ∙ B = 0 (5) as an additional equation to solve for v . But the system is still not closed!

8. Aside: Doppler shifts cannot close the system! Generally, Doppler shifts cannot distinguish flows || to B (red), perp. to B (blue), or in an intermediate direction (gray). With v  estimated another way & projected onto the LOS, the Doppler shift determines v || (Georgoulis & LaBonte, 2006) Doppler shifts are only unambiguous along polarity inversion lines, where B n changes sign (Chae et al. 2004, Lites 2005). v LOS

9. Dopplergrams are sometimes consistent with “siphon flows” moving along the magnetic field. Left: MDI Dopplergram at 19:12 UT on 2003 October 29 superposed with the magnetic neutral line. Right: Evolution of the vertical shear flow speed calculated in the box region of the left panel. The two vertical dashed lines mark the beginning and end of the X10 flare. (From Deng et al. 2006)

10. Aside #2: In addition to v ||, other “null flows,” for which ∂ t B n = 0, lie in the null space of (4). Any flow obeying (v n B hor - v hor B n ) =  x f n, for some scalar function f, lies in the null space of (4). For instance, “contour flows” -- flows along contours of B n -- do not alter B n, but correspond to twisting motions. Hence, they inject energy and helicity into the corona! No method based upon ∂ t B n alone will estimate null flows well. Happily, magnetograms usually have sufficient spatial structure that the null space of (4) is expected to be small.

11. Progress Slide #2

12. Several approaches have been used with ∂ t B n and v  ∙ B = 0 to solve for v . Kusano et al. (2002), Welsch et al. (2004), and Schuck (2006) used tracking methods to close the system. Longcope (2004) minimized ∫ dA (v n 2 + v hor 2 ). Georgoulis & LaBonte (2006) assumed v n = 0. Tracking methods, which follow the apparent motions of magnetogram features, have the longest heritage.

13. The apparent motion of magnetic flux in magnetograms is the flux transport velocity, u f. u f is not equivalent to v; rather, u f  v hor - (v n /B n )B hor u f is the apparent velocity (2 components) v  is the actual plasma velocity (3 comps) (NB: non-ideal effects can also cause flux transport!) Démoulin & Berger (2003): In addition to horizontal flows, vertical velocities can lead to u f  0. In this figure, v  = 0, but v n  0, so u f  0.

14. Aside #3: Conformal mappings should be used to project magnetograms for tracking! Apparent motions in the plane-of-sky do not accurately represent surface flows. If spherical data are to be projected onto a plane prior to tracking, conformal (shape-preserving) mappings should be used. Equal-area (authalic) projections distort shapes, and will therefore distort the directions of estimated flows. We currently use Mercator projections, which distort length scales, but in a way that is easy to correct.

15. Does the flux transport velocity agree with white light velocities? Yes & no! Tracking MDI full- disk white light images & mag- netograms gives different results. Data are from AR 8210 on 01 May ’98; Δt = 96 min.

16. - We created “synthetic magnetograms” from ANMHD simulations of an emerging flux rope. - In these data, both v & B are known exactly. Recently, we conducted quantitative tests & comparisons of several available methods.

17. Via several methods, we estimated v from N = 7 pairs of magnetograms, with increasing Δt’s. We verified that the ANMHD data were consistent with ∂ t B n =  ∙ (v n B hor - v hor B n ). Here, I show representative results from just a few of the methods tested: 1. Fourier LCT (FLCT, Welsch et al. 2004) 2. Inductive LCT (ILCT, Welsch et al. 2004) 3. Minimum Energy Fit (MEF, Longcope 2004) 4. Differential Affine Velocity Estimator (DAVE, Schuck 2006)

19. Here are MEF’s estimated v’s plotted over ANMHD’s v. Like MEF, methods have problems at the edges of magnetic flux.

20. We verified that the estimated v’s also obeyed ∂ t B n =  ∙ (v n B hor – v hor rB n ). FLCT MEF DAVE ILCT

21. We tested Démoulin & Berger’s relation of u f to v. Estimated u f ’s are highly correlated with ANMHD’s u f. DAVE MEF FLCT ILCT

22. Not surprisingly, the methods’ performance worsened as the time between magnetograms increased. % vector errors (direction & magnitude) were at least 50% (!!!). % speed errors (magnitude) were smaller, but biases were seen.

23. Some methods estimated the direction of v to within ~30º, on average. C VEC and C CS were as defined by Schrijver et al. (2005):

24. How well do we do at estimating the E in the fluxes of magnetic energy and helicity? DAVE MEF FLCT ILCT

25. While most methods’ E is highly correlated with ANMHD’s E, only MEF gets the energy flux right. DAVE MEF FLCT ILCT Unfortunately, even MEF exhibits a large amount of scatter.

26. As with the energy flux, only MEF accurately estimates the helicity flux.

27. Conclusions from Tests Tests using MHD magnetograms, in which B is error-free, show that estimated v’s are correlated with true v’s, but are inaccurate. Only MEF reproduced energy and helicity fluxes well; other methods erred by > 50%. Further tests (Ravindra & Longcope, 2007, in prep.) show that combining MEF with tracking does enhances performance.

28. What lies ahead? All of the methods we tested were developed to find v’s that match  t B n. But we also have observations of  t B hor ! The Vector Magnetogram Fitting (VMF) technique developed by McClymont, Jiao, & Mikic (1997) was developed to drive changes ΔB hor. A combined approach, using tracking and VMF, might allow incorporating  t B hor into estimates of v.

29. Progress Slide #3

30. Rudi’s Invite Text We would like to invite you to give a review talk at the 2007 NSO Summer Workshop to be held at Sunspot, NM on 16 - 20 April, 2007. The workshop is called "Subsurface and Atmospheric Influences on Solar Activity" and consists of six half-day sessions. The topic of your talk would be "Surface flows from magnetograms” in Session 1 "Flows around active regions: surface and below". The length of the talk is approximately 25 minutes plus 10 minutesfor discussion.