1. How are photospheric flows related to solar flares? Brian T. Welsch 1, Yan Li 1, Peter W. Schuck 2, & George H. Fisher 1 1 SSL, UC-Berkeley 2 NASA-GSFC See also ApJ v. 705 p. 821

2. Preliminary Ideas We don’t understand processes that produce flares and CMEs, but would like to. The coronal magnetic field B C powers flares and CMEs, but measurements of (vector) B C are rare and uncertain. The instantaneous state of the photospheric field B P provides limited information about the coronal field B C. Evolution of B P can reveal additional information about the coronal field B C. We used tracking methods (and other techniques) to quantitatively analyze photospheric magnetic evolution in a few dozen active regions (ARs). We found a “proxy Poynting flux” to be statistically related to flare activity. This association merits additional study.

3. Can the photospheric magnetic field B P be used to empirically predict flares? Early idea: big & “complex” ARs are likely to produce flares. (Complex is tough to define objectively!) Kunzel 1960: δ sunspots are more likely to flare than non-δ sunspots. Hagyard et al., 1980s: sheared fields along polarity inversion lines (PILs) are associated with flare activity Falconer et al., 2000s: Both shear and flares are associated with “strong gradient” PILs

4. The instantaneous photospheric magnetic field vector B P isn’t very useful for predicting flares. Leka & Barnes (2007) studied 1200 vector magnetograms, and considered many quantitative measures of AR field structure. They summarize nicely: “[W]e conclude that the state of the photospheric magnetic field at any given time has limited bearing on whether that region will be flare productive.”

5. Can we learn anything about solar flares from the evolution of the photospheric magnetic field B P ? The coronal magnetic field B C powers flares and CMEs, but measurements of (vector) B C are rare and uncertain. When not flaring, coronal magnetic evolution should be nearly ideal  connectivity with photosphere is preserved. As B P evolves, changes in the coronal field B C are induced. Further, following active region (AR) fields in time can provide information about their history and development.

6. 6 Assuming B P evolves ideally (see Parker 1984), then photospheric flow and magnetic fields are coupled. The magnetic induction equation’s normal component relates apparent motion u to dB n /dt (Demoulin & Berger 2003).  B n /  t = [  x (v x B) ] n = -   (u B n ) Flows v || along B do not affect  B n /  t, but v || “contaminates” Doppler measurements, diminishing their utility. Many “optical flow” methods to estimate u have been developed, e.g., LCT (November & Simon 1988), FLCT (Welsch et al. 2004), DAVE (Schuck 2006).

7. 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 hor = 0, but v n  0, so u f  0.

8. Aside: Doppler shifts cannot fully determine v 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. Photospheric electric fields can affect flare- related magnetic structure in the corona. Since E = -(v x B)/c, the fluxes of magnetic energy & helicity across the photosphere depend upon v. ∂ t U = c ∫ dA (E x B) ∙ n / 4π ∂ t H = c ∫ dA (E x A) ∙ n / 4π B C  B P coupling means the surface v provides an essential boundary condition for data-driven MHD simulations of B C. (Abbett et al., in progress). Studying v could also improve evolutionary models of B P, e.g., flux transport models.

11. We studied flows {u} from MDI line-of-sight (LOS) magnetograms from N AR = 46 active regions (ARs). ARs were mostly bipolar  NOT a random sample! The radial field B R was estimated via B R = B LOS /cos(Θ) Magnetograms were reprojected to compensate for distortion by foreshortening. We tracked > 2500 MDI full-disk, 96-minute cadence magnetograms from 1996-1998, using both FLCT and DAVE separately.

12. 12 Fourier local correlation tracking (FLCT) finds v( x, y) by correlating subregions, to find local shifts. * = = =

13. FLCT and DAVE flow estimates are correlated, but differ significantly. When weighted by the estimated radial field |B R |, the FLCT-DAVE correlations of flow components were > 0.7.

14. For both FLCT and DAVE flows, speeds {u} were not strongly correlated with B R --- rank-order correlations were 0.07 and -0.02, respectively. The highest speeds were found in weak-field pixels, but a range of speeds were found at each B R.

15. 1.Total unsigned flux,  = Σ |B R | da 2 –  is strongly correlated with flaring (Leka & Barnes 2007) 2.Total unsigned flux R near strong-field polarity inversion lines (PILs), where B R changes sign – Schrijver (2007) found R to be correlated with flaring – (R ≈ to the L SG measure from Falconer et al. 2003, 2006) 1.Total of field squared, Σ B R 2 … and many, many more properties of B R ! To baseline the importance of field evolution to previous results, we also analyzed properties of B R.

16. We then quantified field evolution in many ways, e.g.: Changes in flux, d  /dt. Change in R with time, dR/dt Means & variances of speed u; summed speed, Σ u. Mean and total (  h · u ) & ( z ·  h  u) A “proxy” Poynting flux, S R = Σ u B R 2 Measures of shearing & converging flows near PILs

18. For some ARs in our sample, we auto-correlated u x, u y, and B R, for both FLCT and DAVE flows. BLACK shows autocorrelation for B R ; thick is current-to-previous, thin is current-to-initial. BLUE shows autocorrelation for u x ; thick is current-to-previous, thin is current-to-initial. RED shows autocorrelation for u y ; thick is current-to-previous, thin is current-to-initial.

19. Parametrization of Flare Productivity We binned flares in five time intervals, τ: – time to cross the region within 45 o of disk center (few days); – 6C/24C: the 6 & 24 hr windows (Longcope et al. 2005, Schrijver et al. 2005) centered each flow estimate; – 6N/24N: the “next” 6 & 24 hr windows after 6C/24C Following Abramenko (2005), we computed an average GOES flare flux [μW/m 2 /day] for each time interval: F = (100 S (X) + 10 S (M) + 1.0 S (C) )/ τ ; exponents are summed in-class GOES significands Our flare sample: 154 C-flares, 15 M-flares, and 2 X-flares

20. Correlation analysis showed several variables associated with flare flux F. This plot is for disk-passage averaged properties. Field and flow properties are ranked by distance from (0,0), the point of complete lack of correlation. Only the highest-ranked properties tested are shown. The more FLCT and DAVE correlations agree, the closer they lie to the diagonal line (not a fit).

21. We also used discriminant analysis (DA) to identify the strongest predictors of GOES flares > C1.0 class. flaring Given one input variable, DA finds the “optimal” point between the flaring and quiet populations. (Linear DA assumes Gaussian distributions and dichotomous pop- ulations --- not very accurate here!) Standardized “proxy Poynting flux,” S R = Σ u B R 2

22. The distributions of other variables and their correponding discriminant results look different! The Gaussian assumption seems even less applicable here! We are exploring non- parametric methods for quantifying correlations between magnetic variables and flare activity. Standardized Strong-field PIL Flux R

23. Reliability plots characterize the accuracy of forecasts based upon a discriminant function. Such plots compare the predicted and observed event frequencies. A good model will follow the 45 o line. Underpredictions (failed “all clear” predictions) lie above the dotted line. This underpredicts in low and high probability ranges. Proxy Poynting flux, S R = Σ u B R 2

24. Reliability plots for discrimination using the strong- field PIL flux R also show imperfect forecasting. This model under- predicts (makes failed at “all clear” forecasts) in the low-probability range, and over predicts in the high- probability range. Strong-field PIL Flux, R

25. flaring Given two input variables, DA finds an optimal dividing line between the flaring and quiet populations. The angle of the dividing line can indicate which variable discriminates most strongly. flaring Blue circles are means of the flaring and quiet populations. (With N input variables, DA finds an N-1 dim- ensional surface to partition the N-dimen- sional space.) Standardized “proxy Poynting flux,” S R = Σ u B R 2 Standardized Strong-field PIL Flux R

26. Many of the variables correlated with average flare SXR flux were correlated with each other. Such correlations had already been found by many authors. Leka & Barnes (2003a,b) used linear discriminant analysis to find variables most strongly associated with flaring.

27. We used discriminant analysis to pair field/ flow properties “head to head” to identify the strongest flare associations. For all time windows, regardless of whether FLCT or DAVE flows were used, DA consistently ranked Σ u B R 2 among the two most powerful discriminators.

28. Total unsigned AR flux  is correlated with average flare SXR flux. Some studies relating magnetic properties with flares have not taken this underlying correlation into account.

29. Is rapid magnetic evolution correlated with flare activity? We computed the current- to- initial frame autocorrelation coefficients for all ARs in our sample, and fit their decay rates.

30. We found that rapid magnetic evolution is anti- correlated with  --- but  is correlated with flares! Hence, rapid magnetic evolution, by itself, is anti- correlated with flare activity.

31. We made “head to head” comparisons of S R =Σ u B R 2 with other variables using two-variable discriminant functions. The strong-field PIL flux R was also a powerful predictor; it was the strongest in a few time windows. The areas that contributed most strongly to the sum Σ u B R 2 are away from PILs – suggesting u B R 2 and R are physically distinct. For most time windows, S R =Σ u B R 2 had the greatest power to discriminate between flaring and quiet populations. Grayscale: u B R 2. Contours: B R

32. Conclusions We found S R =Σ u B R 2 and R to be strongly associated with average flare soft X-ray flux and flare occurrence. - both can discriminate flaring/ quiet regions independently; - their spatial distributions differ. Σ u B R 2 seems to be a robust predictor: - speed u was only weakly correlated with B R ; - Σ B R 2 was independently tested; - using u from either DAVE or FLCT gave similar results. The association with Σ u B R 2 suggests that ARs that are both rapidly evolving and large are flare-prone. This study suffers from low statistics; further study is needed. (You might soon review a proposal to extend this work!)