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

Outline 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 B c are rare and uncertain. 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 ARs. We found a “proxy Poynting flux” to be statistically related to flare activity. This association merits additional study.

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)

Only free magnetic energy in the coronal field is available to power flares/CMEs. For a given coronal field B C, the coronal magnetic energy is: U   dV (B C · B C )/8 . The lowest energy the field could have would match the same boundary condition B n, but would be current-free (curl-free), or “potential:” B (P) = - , with  2  = 0. U (P)   dV (B (P) · B (P) )/8  =  dA (  ·  n  )/8  Free energy is the difference U (F)  [U – U (P) ], and is stored in “non- potential structures”, i.e., electric currents. Flares & CMEs release free energy by reducing currents in B C.

It’s difficult to measure the coronal field B C, but the photospheric B P and B LOS are routinely measured. Coronal field measurements are very rare, 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.) What is the expected relationship between photospheric and coronal fields?

Active region (AR) magnetic fields produce flares & CMEs, and are anchored at the photosphere. AR fields originate in the solar interior, emerge across the photosphere and into the corona; B P is the source of B C. Credit: Hinode/SOT Team; LMSAL, NASA

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?

Large-scale, gross properties of coronal fields can be inferred by extrapolating photospheric fields. (1) Coronal holes in potential field models often compare well with images of coronal (soft X- ray, EUV) or chromospheric (He Å) emission. (This particular work focused on testing coronal heating models.) From Schrijver et al. 2004

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

Inference of detailed properties of coronal fields, however, has not been demonstrated. (1) While potential extrapolations can match higher resolution coronal observations, they often don’t. From Schrijver et al Good potential extrapolationBad potential extrapolation

Inference of detailed properties of coronal fields, however, has not been demonstrated. (2) Non-potential extrapolations can give wildly diverging magnetic energies.

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!)

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.

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

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

Aside: Magnetic shear is the discrepancy between the actual and potential photospheric fields along PILs. From Welsch & Fisher (2006)

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

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!

Schrijver (2007) found a rough maximum GOES flare flux vs. magnetic flux near strong-field polarity inversion lines (SPILs). R is the total unsigned  near strong-field PILs As expected, there are more weak flares than strong flares. AR again, and its masked PILs at right

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

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.”

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.

23 Assuming B P evolves ideally (see Parker 1984), then photospheric flow and magnetic fields are coupled. The magnetic induction equation’s z-component relates footpoint motion u to dB z /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 || “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).

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.

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

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)

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.

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

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.

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

When weighted by the estimated radial field B R, the FLCT-DAVE correlations increased to > 0.8.

To baseline the importance of field evolution, we computed intensive and extensive properties of B R. Intensive properties do not intrinsically grow with AR size: - 4 statistical moments of average unsigned field |B R |, (mean, variance, skew, kurtosis), denoted M (|B R |) - 4 moments of M ( B R 2 ) Extensive properties scale with the physical size of an AR: - total unsigned flux,  = Σ |B R | da 2 ; 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 B R 2

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) B R   ± da B R 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 B R )) & M (z ·  h  ( u B R )), and their sums The sum of “proxy” Poynting flux, S R = Σ u B R 2 Measures of shearing converging flows near PILs

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 were tracked, using both FLCT and DAVE separately. GOES catalog was used to determine source ARs for flares at and above C1.0 level.

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.

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.

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.

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 window: F = (100 S (X) + 10 S (M) S (C) )/ τ ; exponents are summed in-class GOES significands Our sample: 154 C-flares, 15 M-flares, and 2 X-flares

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). No purely intensive flow properties appear --- all contain extensive properties.

With 2-variable discriminant analysis (DA), we paired Σ u B R 2 “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 B R 2 among the two most powerful discriminators.

Conclusions, pt. 1 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 predictor: - speed u was only weakly correlated with B R ; - Σ B R 2 was also tested; - using u from either DAVE or FLCT gave the same result. This study suffers from low statistics; further study is needed. (A proposal to extend this work has been submitted!) The study of photospheric magnetic evolution is still *very much* a research topic.

Conclusions, pt. 2 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 --- GOES soft X-ray flux --- is also extensive! What “intensive” flare measures are available? Better spatial resolution of flare emission, e.g., from SSL’s FOXSI sounding rocket (Krucker et al.) should help!