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. Outline We don’t understand processes that produce flares and coronal mass ejections (CMEs), but would like to. Electric currents in the coronal magnetic field B c power flares and CMEs, but measurements of (vector) B c are rare and subject to large uncertainties. The instantaneous state of the photospheric field B P provides limited information about the coronal field B c. Properties of photospheric field evolution can reveal additional information about the coronal field. 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. Flares are transient enhancements of the Sun’s radiative output over a wide wavelength range, from radio to X-rays. This movie shows flare emission in Ca II ob- served by the Solar Optical Telescope (SOT) aboard the Hinode satellite. (Image credit: SOT Team/NASA/JAXA) Flares produce bursts of emission in spaceborne X-ray monitors. Note “two ribbon” structure

4. Flares arise from the release of energy stored in electric currents in the coronal magnetic field. Movie credit: EIT team McKenzie 2002 the “standard model” an EUV movie of ~1MK thermal emission

5. Flares and CMEs are powered by energy in the coronal magnetic field. From T.G. Forbes, “A Review on the Genesis of Coronal Mass Ejections”, JGR (2000)

6. The hypothetical coronal magnetic field with lowest energy is current-free, or “potential.” For a given coronal field B C, the coronal magnetic energy is: U   dV (B C · B C )/8 . The lowest energy coronal field would have current J = 0, and Ampére says 4 π J/c =  x B, so  x B min = 0. A curl-free vector field can be expressed as the gradient of a scalar potential, B min = - . (since  2  = 0, use electrostatics to solve!) U min   dV (B min · B min )/8  The difference U (F) = [U – U min ] is “free” energy stored in the corona, which can be suddenly released in flares or CMEs.

7. Active region (AR) magnetic fields produce flares and CMEs. Magnetograms are maps of the photo- spheric magnetic field. White/black show areas of positive/negative magnetic flux. Magnetograms are derived from spectropolarimetric measurements.

8. AR fields originate in the interior, and emerge across the photosphere into the corona. The coronal magnetic field is anchored at the photosphere; the two regions are magnetically coupled. Credit: Hinode/SOT Team; LMSAL, NASA

9. What physical processes produce coronal electric currents? Two options are: Currents could form in the interior, then emerge into the corona. – Current-carrying magnetic fields have been observed to emerge (e.g., Leka et al. 1996, Okamoto et al. 2008) Photospheric evolution could induce currents in already-emerged coronal magnetic fields. – From simple scalings, McClymont & Fisher (1989) argued induced currents would be too weak to power large flares – Detailed studies by Longcope et al. (2007) and Kazachenko et al. (2009) suggest strong enough currents can be induced General picture: slow buildup, sudden release.

10. Measuring the coronal vector field B C is difficult, but the photospheric B P and B LOS are routinely measured. Coronal field measurements are scarce, and subject to large uncertainties (e.g., Lin, Kuhn, & Coulter 2004). While photospheric magnetograms are relatively common, only B LOS, the line-of-sight (LOS) component of the vector B P has been routinely measured. (This should change soon, with NSO’s SOLIS and NASA’s HMI.) How are photospheric fields related to flare activity?

11. From Welsch & Fisher (2006) We do see non-potential fields (evidence of currents) in vector magnetograms, which measure B P.

12. What can B P tell us about the likelihood B C will produce a flare? One approach is to extrapolate a model coronal field B C (M) from B P, and study the model field.  Q: How “good” are the extrapolated fields? Another approach is to empirically relate properties of B P to flare activity.  Q: How “good” are the empirical predictions?

13. Large-scale, gross properties of coronal fields can be inferred by extrapolating photospheric fields. Coronal holes in potential field models often compare well with images of coronal (soft X-ray, EUV) or chromospheric (He 10830 Å) emission. (This particular work focused on testing coronal heating models.) From Schrijver et al. 2004

14. Large-scale, gross properties of coronal fields can be inferred by extrapolating photospheric fields, #2. Solar wind speeds can be estimated from coronal hole properties in potential extrapolations. Credit: Arge & Odstrcil

15. Inference of detailed properties of coronal fields, however, has not been demonstrated. While potential extrapolations can match higher resolution coronal observations, they often don’t. From Schrijver et al. 2005 Good match to potential extrapol. Bad match to potential extrapol. Currents must be present!

16. Accurate estimation of current-carrying coronal fields has not been demonstrated. For instance, non-potential field extrapolations can give wildly diverging magnetic energies.

17. Failing in extrapolating B C from B P, can 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.

18. Aside: δ sunspots have positive and negative flux within the same umbra. These MDI synoptic magnetic and intensity maps of Carrington Rotation 2025 show AR 10720.

19. Failing in extrapolating B C from B P, can 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

20. Failing in extrapolating B C from B P, can 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

21. Falconer found strong shear and strong gradients in B LOS along PILs to be correlated with both each other and flares. Strong shearStrong gradient Schrijver (2007) found the flux R near “strong field” PILs --- hence, strong gradient --- to be correlated with flare activity. Strong gradients are just what you’d expect in a δ spot! From Falconer et al. 2006

22. Schrijver (2007) associated large flares with the amount of magnetic flux near strong-field polarity inversion lines (PILs). R is the total unsigned flux near strong-field PILs AR 10720, and its masked PILs at right

23. Discriminant analysis can test the capability of a magnetic parameter to predict flares. 1) For a time window  t, estimate distribution functions for the parameter in the flaring (green) and nonflaring (black) populations in a “training dataset.” 2) Given an observed value, predict a flare within the next  t if: P flare > P non-flare (vertical blue line) From Barnes and Leka 2008

24. Barnes & Leka (2008) tested R against , and found them to be equally bad flare predictors!

25. Large flares are rare, so it’s a good bet that no flare will occur in a forecast window of a day or less.  “Success rates” > 90% are possible by “just saying no”

26. Barnes & Leka (2008) tested R against , and found them to be equally bad flare predictors! Large flares are rare, so it’s a good bet that no flare will occur in a forecast window of a day or less.  “Success rates” > 90% are possible by “just saying no” “Skill scores” are normalized to expected rate – 1 = perfect forecast; 0 merely matches expectation – Heidke = “just say no”; “Climate” = historical rate

27. It turns out that a snapshot of the photospheric vector field 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.”

28. Can we learn anything about flares from the evolution of B P ? When not flaring, coronal magnetic evolution should be nearly ideal  photospheric connectivity is preserved. As B P evolves, changes in B C are induced. Further, following AR fields in time can provide information about their history and development.

29. Here’s a typical sequence of 96 minute cadence active region magnetograms, from AR 10923. Q: How can we quantitatively analyze this evolution? Generally, photospheric evolution induces currents (stored energy!) in the coronal field.

30. 30 Assuming B P evolves ideally (e.g., Parker 1984), then photospheric flow and magnetic fields are coupled. The magnetic induction equation’s z-component relates the flux transport velocity u to dB z /dt (Demoulin & Berger 2003).  B z /  t = [  x (v x B) ] n = -   (u B n ) Many “optical flow” methods to estimate u have been developed, e.g., LCT (November & Simon 1988), FLCT (Welsch et al. 2004), DAVE (Schuck 2006).

31. 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.

32. Aside: Flows v || along B do not affect  B n /  t, but “contaminate” Doppler measurements. v LOS

33. Dopplergrams are sometimes consistent with “siphon flows” moving along the magnetic field. MDI Dopplergram at 19:12 UT on 2003 October 29 superposed with the magnetic polarity inversion line. (From Deng et al. 2006)

34. Photospheric electric fields can affect flare- related magnetic structure in the corona. If magnetic evolution is ideal, then E = -(v x B)/c, and the Poynting flux of magnetic energy across the photosphere depends upon v: ∂ t U = c ∫ dA (E x B) ∙ n / 4π = ∫ dA (B x (v x B)) ∙ 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.

35. 35 Fourier local correlation tracking (FLCT) finds u( x, y) by correlating subregions, to find local shifts. * = = =

36. We studied flows {u} from MDI magnetograms and flares from GOES for a few dozen active region (ARs). N AR = 46 ARs were selected. – ARs were selected for easy tracking – usu. not complex, mostly bipolar -- NOT a random sample! > 2500 MDI full-disk, 96-minute cadence magnetograms from 1996-1998 were tracked, using both FLCT and DAVE separately. GOES catalog was used to determine source ARs for flares at and above C1.0 level.

37. Magnetogram Data Handling Pixels > 45 o from disk center were not tracked. To estimate the radial field, cosine corrections were used, B R = B LOS /cos(Θ) Mercator projections were used to conformally map the irregularly gridded B R (θ,φ) to a regularly gridded B R (x,y). Corrections for scale distortion were applied.

38. 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.

39. 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.

40. To baseline the importance of field evolution, we analyzed properties of B R, including: - 4 statistical moments of average unsigned field |B R | (mean, variance, skew, kurtosis), denoted M (|B R |) - 4 moments M ( B R 2 ) - total unsigned flux,  = Σ |B R | da 2 - total unsigned flux near strong-field PILs, R (Schrijver 2007) - sum of field squared, Σ B R 2

41. We then quantified field evolution in many ways, e.g.: Un- and signed changes in flux, |d  /dt|, d  /dt. Change in R with time, dR/dt We also computed many flow properties: Moments of speed M (u), and summed speed, Σ u. Moments and sums of divergences (  h · u), (  h · u B R ) Moments and sums of curls (  h x u), (  h x u B R ) The sum of “proxy” Poynting flux, S R = Σ u B R 2

43. To find a typical flow timescale, we autocorrelated 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.

44. 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 centered each flow estimate; – 6N/24N: the “next” 6 & 24 hr windows after 6C/24C (6N is 3-9 hours in the future; 24N is 12-36 hours in the future) Following Abramenko (2005), we computed an average GOES flare flux [μW/m 2 /day] for each window: F = (100 S (X) + 10 S (M) + 1.0 S (C) )/ τ ; exponents are summed in-class GOES significands Our sample: 154 C-flares, 15 M-flares, and 2 X-flares

45. 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).

46. 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 discriminant analysis (DA) to find variables most strongly associated with flaring.

47. 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 non- flaring 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

48. 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.

49. Physically, why is the proxy Poynting flux, S R = Σ uB R 2, associated with flaring? Further study is needed. u B R 2 corresponds to the part of the horizontal Poynting flux, from E h x B r – The vertical Poynting flux, due to E h x B h, is probably most relevant to flaring. – Another component of the horizontal Poynting flux, from E r x B h, is neglected in our analysis. Are horizontal & vertical Poynting fluxes similar in magnitude? Why would this be? Do flows from flux emergence or rotating sunspots also produce large values of S R ?

50. Distinct regions contribute to the sums for R and S R, implying different underlying physical processes. White regions show strong contributions to R and S R in AR 8100; white/black contours show +/- B R at 100G, 500G.

51. The distributions of flaring & non-flaring observations of R and S R differ, suggesting different underlying physics. Histograms show non-flaring (black) and flaring (red) observations for R and S R in +/-12 hr time windows.

52. 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. S R underpredicts in low and high probability ranges. Proxy Poynting flux, S R = Σ u B R 2

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

54. Again, total unsigned AR flux  is correlated with flare SXR flux. Some studies relating magnetic properties with flares have not taken this underlying correlation into account.

55. Is rapid magnetic evolution, by itself, correlated with flare activity? We computed the current- to- initial frame autocorrelation coefficients for all ARs in our sample.

56. We found that rapid magnetic evolution is anti- correlated with  --- but  is correlated with flares! Hence, rapid magnetic evolution, by itself, is anticorrelated with flaring: small ARs don’t flare, but evolve most rapidly.

57. Conclusions We found Σ u B R 2 and R to be strongly associated with average flare soft X-ray flux and flare occurrence. Σ u B R 2 seems to be a robust flare 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. It appears that ARs that are both relatively large and rapidly evolving are more flare-prone. This study suffers from low statistics; further study is needed. (A proposal to extend this work has been submitted!)

58. Conclusions, cont’d The strongest flare predictors are extensive: , R, Σ u B R 2 Does this imply that “the flare mechanism” is also extensive? This would accord with the “avalanche” model of Lu & Hamilton (1991): large flares are “built” of many small flares. BUT: our flare measure --- integrated GOES soft X-ray flux --- is also extensive! What “intensive” flare measures are available? Better spatial resolution of flare hard X-ray emission, e.g., from SSL’s FOXSI sounding rocket (Krucker et al.) should help!

59. “Recurrence Process of a Large Earthquake,” from the Univ. of Tokyo’s Earthquake Prediction Research Center: Large earthquakes repeatedly occur along a large-scale fault, and the entire recurrence process includes the following stages: (I) fault healing and re-strengthening just after the previous earthquake occurred, (II) accumulation of the elastic strain energy with tectonic stress loading, (III) local concentration of deformation and rupture nucleation at the final stage of tectonic stress buildup in which an enough amount of the strain energy has been stored, (IV) mainshock earthquake rupture, and (V) rupture arrest and its aftereffect. http://wwweprc.eri.u-tokyo.ac.jp/ENG_HP/recurren/main.html

60. This slow forcing  sudden release process in flares & CMEs closely resembles that in earthquakes. Earthquake terms  Flare terms earthquakes  flares/CMEs fault  (quasi-) separator (elastic) strain energy  free magnetic energy tectonic stress loading  photospheric evolution deformation  magnetic diffusion rupture  fast reconnection mainshock earthquake rupture  flare reconnection fault healing and re-strengthening  diffusivity quenching

61. Large earthquakes flares repeatedly occur along a large-scale fault (quasi-) separator, and the entire recurrence process includes the following stages: (I) fault healing and re-strengthening diffusivity quenching just after the previous earthquake flare occurred, (II) accumulation of the elastic strain energy free magnetic energy with tectonic stress loading photospheric evolution, (III) local concentration of deformation magnetic diffusion and rupture fast reconnection nucleation at the final stage of tectonic stress buildup photospheric evolution in which an enough amount of the strain free magnetic energy has been stored, (IV) mainshock earthquake rupture flare reconnection, and (V) rupture reconnection arrest and its aftereffect. “Recurrence Process of Large Solar Flares:”