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

2. Abstract Estimates of velocities from time series of photospheric 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/Hinode, 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).

3. 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 normal-field magnetogram sequences? Flows perpendicular B, that lie outside the null space of  t B n. 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? In addition to using ∂ t B n,,, full vector ∂ t B hor should be used.

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. 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 -  x 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.

6. Consequently, the goal is to derive v from ∂ t B n =  ∙ (v n B hor - v hor B n ) (4) Clearly, the single ∂ t B n equation 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!

7. 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

8. 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.

9. Several approaches have been used with ∂ t B n and v  ∙ B = 0 to solve for v . Kusano et al. (2002), Welsch et al. (FLCT, ILCT; 2004), and Schuck (DAVE; 2006) used tracking methods to close the system. Longcope (MEF; 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.

10. 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.

11. - 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.

12. 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)

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

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

15. 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

16. Using v ∙ B = 0 and u f  v hor - (v n /B n )B hor, we converted u f ’s to v’s. Estimated v’s are highly correlated with ANMHD’s v. DAVE MEF FLCT ILCT

17. 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.

18. 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):

19. How well do we estimate the electric field E in the fluxes of magnetic energy and helicity? DAVE MEF FLCT ILCT

20. 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.

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

22. 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.

23. 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, keeping B n fixed. A combined approach, using tracking and VMF, might allow incorporating  t B hor into estimates of v.

24. Future efforts: Coming soon to a review panel near you! 1) Similar tests with RADMHD sim- ulations (Abbett 2007), which model photospheric evolution more realistically. 1a) Also, test susceptiblity to noise in magnetograms. 2) In addition, run coordinated velocity studies with SOT/Hinode and HMI/SDO magnetograms.