1. Photospheric Flows & Flare Forecasting tentative plans for Welsch & Kazachenko

2. Topics 0. Why do should photospheric electric or velocity fields E or v matter for flares? 1.Previous work with B LOS 2.New opportunities with vector B

3. A photospheric electric field E derived from magnetogram evolution can quantify aspects of evolution in B corona. The fluxes of magnetic energy & helicity across the magnetogram surface into the corona depend upon E: dU/dt = ∫ dA (E x B) z /4 π dH/dt = 2 ∫ dA (E x A) z U and H probably play central roles in flares / CMEs. Assuming B ph evolves ideally (e.g., Parker 1984), then photospheric flow and electric fields are related: cE = -(v x B)

4. Topics 0. Why do should photospheric electric or velocity fields E or v matter for flares? 1.Previous work with B LOS 2.New opportunities with vector B

5. 5 The ideal assumption relates dB z /dt to v. The magnetic induction equation’s z-component relates the flux transport velocity u to dB z /dt (Demoulin & Berger 2003):  B z /  t = -c[  x E ] z = [  x (v x B) ] z = -   (u B z ) Many tracking (“optical flow”) methods to estimate u have been developed, e.g., LCT (November & Simon 1988), FLCT (Fisher & Welsch 2008), DAVE (Schuck 2006).

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

7. We studied flows {u} from MDI magnetograms and flares from GOES for a few dozen active region (ARs). N AR = 46 ARs from 1996-1998 were selected. > 2500 MDI full-disk, 96-minute cadence, line-of-sight magnetograms were compiled We estimated flows in these magnetograms using two separate tracking methods, FLCT and DAVE. The GOES soft X-ray flare catalog was used to determine source ARs for flares at and above C1.0 level.

8. 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). [dirty laundry!] 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.

9. Sample maps of FLCT and DAVE flows show them to be strongly correlated, but far from identical. When weighted by the estimated radial field |B R |, the FLCT-DAVE correlations of flow components were > 0.7.

10. Many parameters were studied…

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

12. flaring Given two input variables, DA finds an optimal dividing line between the flaring and quiet populations. flaring Blue circles are means of the flaring and non-flaring populations. The angle of the dividing line can indicate which variable discriminates most strongly. We paired field/ flow properties “head to head” to identify the strongest flare discriminators. (\ Standardized “proxy Poynting flux,” S R = Σ u B R 2 Standardized Strong-field PIL Flux R

13. 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.

14. 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.

15. What’s the physics behind the S R -flare association? Yan Li (UC Berkeley / SSL) has suggested that: (i) S R is a proxy for the actual Poynting flux, (ii)flares are more likely when the cumulative coronal energy is higher. From Li, Welsch, Lynch, Luhmann, & Fisher 2011

16. The “proxy Poynting flux” S R = Σ u B R 2 bears further study… With MDI, we found R and the proxy Poynting flux (PPF) to be most strongly associated with flares. Our sample size was small, so must redo this with larger N! Our results were empirical; we still need to understand the underlying processes. Also, it’ll be good to compare our parameter S R with others. For more details, see Welsch et al., ApJ v. 705 p. 821 (2009)

17. Topics 0. Why do should photospheric electric or velocity fields E or v matter for flares? 1.Previous work with B LOS 2.New opportunities with vector B

18. We have developed a way to use vector  t B (not just  t B z ) to estimate E (or v). Previous “component methods” derived v or E h from the normal component of the ideal induction equation,  B z /  t = -c[  h x E h ] z = [  x (v x B) ] z But the vector induction equation can place additional constraints on E:  B/  t = -c(  x E)=  x (v x B), where I assume the ideal Ohm’s Law,* so v E: E = -(v x B)/c ==> E·B = 0 *One can instead use E = -(v x B)/c + R, if some model resistivity R is assumed. (I assume R might be a function of B or J or ??, but is not a function of E.)

19. The E derived via PTD uses only  t B, so E PTD ·B ≠ 0. Hence, we must solve for ψ (x,y) so (E PTD -  ψ )·B = 0. We have developed a practical iterative approach: 1. Define b = unit vector along B 2. Define  ψ = s 1 (x, y) b + s 2 (x, y)(z x b) + s 3 (x, y) b x (z x b) 3. Set s 1 (x, y) = E PTD · b 4. Solve  h 2 ψ =  h · [s 1 (x,y)b h + s 2 (x, y)(z x b) − s 3 (x, y)b z b h ] 5. Update s 2 = z·(b h x  ψ )/b h 2 and s 3 =  z ψ - (b h ·  ψ ) b z /b h 2 6. Repeat steps 4 & 5 until convergence. This approach quickly yields a solution. However, uniqueness is still a problem: any  ψ (x,y) satisfying  ψ ·B = 0 can be added to this solution! For (many) more details about PTD, see Fisher et al. 2010. ^ ^ ^ ^

20. The “PTD” method employs a poloidal-toroidal decomposition of B into two scalar potentials. B =  x (  x B z) +  x J z B z = -  h 2 B, 4 π J z /c =  h 2 J,  h ·B h =  h 2 (  z B ) Left: the full vector field B in AR 8210. Right: the part of B h due only to J z. ^^  t B =  x (  x  t B z) +  x  t J z  t B z =  h 2 (  t B ) 4 π  t J z /c =  h 2 (  t J )  h ·(  t B h ) =  h 2 (  z (  t B )) ^^

21. Faraday’s Law implies that PTD can be used to derive an electric field E from  t B. “Uncurling”  t B = -c(  x E) gives E PTD = (  h x  t B z) +  t J z Note:  t B doesn’t constrain the “gauge” E-field -  ψ ! So: E tot = E PTD -  ψ Since PTD uses only  t B to derive E, (E PTD -  ψ )·B = 0 can be solved to enforce Ohm’s Law (E tot ·B = 0). (But applying Ohm’s Law still does not fully constrain E tot.) ^ ^

22. PTD has two advantages over previous methods for estimating E (or v): In addition to  t B z, information from  t J z is used in derivation of E. No tracking is used to derive E, but tracking methods (ILCT, DAVE4VM) can provide extra info! For more about PTD, see Fisher et al. 2010, in ApJ 715 242

23. Doppler shifts and tranverse fields B trs on LOS PILs can improve estimates of E. Will we get v LOS, Graham? Near a Polarity Inversion Line (PIL) of B LOS, B is purely transverse to v LOS We can thus measure a Doppler electric field E Dopp = -(v LOS x B trs )/c related to flux emergence at PILs. See Fisher et al. 2012, Sol. Phys. 277, 153 We can combine this info with the PTD B to improve our estimate of E

24. In tests with simulated MHD data, our reconstructed Poynting flux compared well with the true S z. Qualitative and quantitative comparisons show good recovery of the simulation’s E-field and vertical Poynting flux S z.

25. We haven’t tested the Poynting flux from the PTD approach as a flare predictor, but hope it works! Also, it’ll be good to compare our parameter S z with others.

26. Both vector and component methods of finding E are underdetermined: unknowns exceed knowns by one! MethodUnknownsKnowns Component MethodsE x, E y, E z  t B z, E·B = 0 PTD E x, E y, E z,  z E x,  z E y  t B x,  t B y,  t B z, E·B = 0 Hence, extra information about E provides useful constraints! 1. The flow u estimated by tracking can constrain the gauge electric field ψ, since  h 2 ψ = (  h x u B z )·z 2. Where B LOS = 0, Doppler shifts can constrain E. 3. Magnetograms from multiple heights can constrain  z E h. (Given noise in the data, overdetermining E is fine!) ^

27. Topics 0. Why do should photospheric electric or velocity fields E or v matter for flares? 1.Previous work with B LOS 2.New opportunities with vector B