Photospheric Flows and Solar Flares

1. Photospheric Flows and Solar Flares Brian T. Welsch1, Yan Li1, Peter W. Schuck2, & George H. Fisher1 1Space Sciences Lab, UC-Berkeley 2Naval Research Lab, Washington, D.C.

2. Magnetic evolution at the photosphere is apparently steady. Magnetograms of AR 8210 from MDI over ~24 hr. on 01 May 1998 show no drastic field changes. Proper motions are ~1 km/s.

3. Meanwhile, in the corona… … an M-class flare & halo CME occurred. Q: What can we learn from photospheric evolution?

4. The evolution of the photospheric field is expected to be correlated with flares/CMEs. Observations + theory suggest converging and/ or shearing flows along polarity inversion lines (PILs) are relevant to flares/CMEs. Flux emergence is also likely to dramatically affect coronal evlution. Falconer et al. (2006) and Schrijver (2007) argue that the presence of strong-field PILs (SPILs) is related to flares/CMEs. Does flare/CME likelihood increase as SPILs form?

5. Also, photospheric flows can be used to drive time-dependent models of the coronal field. • The magnetic induction equation’s z-component relates footpoint motion u to dBz/dt (Demoulin & Berger 2003). Bz/t = [  x (v x B) ]z = -  (u Bz) • Flows v|| along B do not affect Bn/t, but v|| “contam-inates” 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).

6. Fourier local correlation tracking (FLCT) finds v( x, y) by correlating subregions, to find local shifts. = * = = 4) Δx(xi, yi) is inter- polated max. of correlation funct 1) for ea. (xi, yi) above |B|threshold 2) apply Gaussian mask at (xi, yi) 3) truncate and cross-correlate

7. We studied flows {u} from MDI magnetograms and flares from GOES for a few dozen active region (ARs). • NAR = 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.

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

9. FLCT and DAVE flow estimates were correlated, but differed substantially.

10. FLCT and DAVE flow estimates were correlated, but differed substantially.

11. To baseline the importance of field evolution, we computed intensive and extensive properties of BR. Intensive properties do not intrinsically grow with AR size: - 4 statistical moments of average unsigned field |BR|, (mean, variance, skew, kurtosis), denoted M(|BR|) - 4 moments of M( BR2 ) Extensive properties scale with the physical size of an AR: - total unsigned flux,  = Σ |BR|da2 ; this scales as area A (Fisher et al. 1998) - total unsigned flux near strong-field PILs, R (Schrijver 2007), should scale as length L - sum of field squared, Σ BR2

12. 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 • Changes in center-of-flux separation, d(x±)/dt, with x±  x+-x-, and x±  ±da (x) BR  ±da BR We computed intensive and extensive flow properties, too: • Moments of speed M(u), and summed speed, Σ u. • M(h · u ) & M( z·h u), and their sums • M(h · (u BR)) & M(z·h (u BR)), and their sums • The sum of “proxy” Poynting flux, Σ uBR2 • Measures of shearing converging flows near PILs

13. For both FLCT and DAVE flows, speeds {u} were not strongly correlated with BR--- 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 BR.

14. For some ARs in our sample, we auto-correlated ux, uy, and BR, for both FLCT and DAVE flows. BLACK shows autocorrelation for BR; thick is current-to-previous, thin is current-to-initial. BLUE shows autocorrelation for ux; thick is current-to-previous, thin is current-to-initial. RED shows autocorrelation for uy; thick is current-to-previous, thin is current-to-initial.

15. For some ARs in our sample, we auto-correlated ux, uy, and BR, for both FLCT and DAVE flows. tcorr ~ 6 hr. BLACK shows autocorrelation for BR; thick is current-to-previous, thin is current-to-initial. BLUE shows autocorrelation for ux; thick is current-to-previous, thin is current-to-initial. RED shows autocorrelation for uy; thick is current-to-previous, thin is current-to-initial.

16. Parametrization of Flare Productivity • We binned flares in five time intervals, τ: • time to cross the region within 45o of disk center; • 6C/24C: the 6 & 24 hr windows 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/m2/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

17. 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), 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). No purely intensive quantities appear --- all contain extensive properties.

18. With 2-variable discriminant analysis (DA), we paired Σ u BR2 “head to head” with each other field/ flow property. For all time windows, regardless of whether FLCT or DAVE flows were used, DA consistently ranked Σ u BR2 among the two most powerful discriminators.

19. Conclusions We found Σ u BR2 and R to be strongly associated with avg. flare flux and flare occurrence. Σ u BR2 seems to be a robust predictor: - speed u was only weakly correlated with BR; - Σ BR2 was also tested; - using u from either DAVE or FLCT gave the same result. This study suffers from low statistics, so further study is needed. (A proposal to extend this work is being written!) The study of photospheric magnetic evolution is still *very much* a research topic. FLCT however…

20. HMI magnetograms have Npix ~ 40962 (π/4) pix. • It takes ~3 min. to track all 12 Mpix, skipping every fourth pixel, with windowing parameter σ = 15 pix • If we only track pixels within 60o of disk center and • |Br| > Bthresh ~ 20 G, then tracking should take ~1 min. • We are working with the HMI Team at Stanford to port the FLCT code’s C-version to the HMI / JSOC pipeline. • http://solarmuri.ssl.berkeley.edu/~fisher/public/software/FLCT/ v. 1.01 of FLCT (Fisher & Welsch 2008) is capable of matching HMI’s ~10-minute magnetogram cadence.