What can we learn about solar activity from studying magnetogram evolution? Brian T. Welsch SSL, UC-Berkeley I will briefly review results from recent studies of magnetic evolution in sequences of magnetograms, and discuss avenues for future research suggested by these results. Further work is necessary to understand both the physics underlying a recently identified photospheric flow - flare association, and how this association is related to other flare-associated phenomena, including vertical electric currents at the photosphere and subsurface flow properties and their evolution. Efforts to understand flare activity will be aided by improvements in techniques for estimating the electric fields that govern magnetic evolution in magnetograms, e.g., incorporating constraints from Doppler shifts and multi- height magnetic measurements. Electric field estimates can also be applied to several other problems, including understanding the ultimate fate of active region flux and time-dependent coronal magnetic field modeling.

Topics I’ll begin by reviewing results from recent research. (i) flows & flares (ii) estimating magnetogram electric fields Next, I’ll discuss ideas for research with the VSM. (iii) case studies of short-timescale evolution (iv) studies involving the “core” database

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

What physical processes produce the electric currents that store energy in B cor ? 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.

“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.

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

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:”

If B cor drives flares, why study magnetograms of the photospheric and chromospheric fields, B ph and B ch ? The coronal magnetic field B cor powers flares and CMEs, but measurements of (vector) B cor are rare and uncertain. When not flaring, coronal magnetic evolution should be nearly ideal  magnetic connectivity is preserved. As the photospheric and chromospheric fields, B ph and B ch, evolve, changes in the coronal field B cor are induced. Further, following active region (AR) fields in time can provide information about their history and development.

9 Assuming B ph 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 = -c[  x E ] z = [  x (v x B) ] z = -   (u B z ) Many “optical flow” methods to estimate the u have been developed, e.g., LCT (November & Simon 1988), FLCT (Fisher & Welsch 2008), DAVE (Schuck 2006). Purely numerical “inductive” techniques have also been developed (Welsch & Fisher 2008; Fisher et al. 2010).

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 z /B z )B hor u f 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 f  0. In this figure, v hor = 0, but v z  0, so u f  0. hor z z

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.

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.

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

Autocorrelation of u x and u y suggest the 96 minutes cadence for magnetograms is not unreasonably slow. 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.

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.

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 each magnetogram and flow estimate, we computed several properties of B R and u, e.g., - average unsigned field |B R | - summed unsigned flux,  = Σ |B R | da 2 - summed flux near strong-field PILs, R (Schrijver 2007) - sum of field squared, Σ B R 2 - rates of change d  /dt and dR/dt - summed speed, Σ u. - averages and sums of divergences (  h · u), (  h · u B R ) - averages and sums of curls (  h x u), (  h x u B R ) - the summed “proxy Poynting flux,” S R = Σ u B R 2 (and many more!)

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

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 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) 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 average flare flux F. This plot is for disk-passage averages. 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).

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.

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

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

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.

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.

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.

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

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

Physically, why is the proxy Poynting flux, S R = Σ uB R 2, associated with flaring? Open questions: Why should u B R 2 – part of the horizontal Poynting flux from E h x B r – matter for flaring? – The vertical Poynting flux, due to E h x B h, is presumably primarily responsible for injecting energy into the corona. – (Another component of the horizontal Poynting flux, from E r x B h, was neglected in our analysis. Is it also significant?) – With B h available from vector magnetograms, these questions can be addressed! Do flows from flux emergence or rotating sunspots also produce large values of u B R 2 ? How is u B R 2 related to flare-associated subsurface flow properties (e.g., Komm & Hill 2009; Reinard et al. 2010)?

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.

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.

Conclusions, (i) 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. At a minimum, we can say that ARs that are both relatively large and rapidly evolving are more flare-prone. (No surprise!) This study had relatively few strong flares, and used only LOS magnetograms; further study is warranted.

Topics I’ll begin by reviewing results from recent research. (i) flows & flares (ii) estimating magnetogram electric fields Next, I’ll discuss ideas for research with the VSM. (iii) case studies of short-timescale evolution (iv) studies involving the “core” database

Conclusions (i), 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!

Shifting gears, I will now discuss methods to use vector  t B (not just  t B z ) to estimate v or E. 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 Recently, my colleagues & I have been developing methods consistent with the vector induction equation,  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.)

Why try to estimate E from magnetogram series? It can reveal information about the evolution of B cor. 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 Coupling of B cor to B beneath the corona implies estimates of E there provide boundary conditions for data-driven, time-dependent simulations of B cor.

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

PTD can also be used to represent  t B, and derive an electric field E from Faraday’s Law. In this case,  t B =  x (  x  t B z) +  x  t J z, with  t B z =  h 2 (  t B ) 4 π  t J z /c =  h 2 (  t J )  h ·(  t B h ) = -  z (  t B z ) =  h 2 (  z (  t B )) Faraday told us that  t B = -c(  x E), which can be “uncurled” to derive a PTD electric field: E PTD = (  h x  t B z) +  t J z Note: E PTD plus a “gauge” electric field -   still matches  t B. Note also: No tracking was used to derive E PTD ! ^ ^^ ^

The E derived via PTD uses only  t B, so E PTD ·B ≠ 0. Hence, we must solve for  (x,y) so (E PTD -   )·B = 0. I have developed a practical iterative approach: 1. Define b = unit vector along B 2. Define   = 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  =  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   )/b h 2 and s 3 =  z  - (b h ·   ) 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   (x,y) satisfying   ·B = 0 can be added to this solution! For (many) more details about PTD, see Fisher et al ^ ^ ^ ^

How accurate is PTD? We used data from MHD simulations to compare E = E PTD -   with E MHD. Synthetic data were those used by Welsch et al. (2007) to test tracking methods. The PTD + iteration solution was more accurate than most other methods test- ed by Welsch et al. (2007). ExEx EyEy EzEz E PTD -   E MHD

While  t B provides more information about E than  t B z alone, it still does not fully determine E. 1.Faraday’s Law only relates  t B to the curl of E, not E itself; the gauge electric field   is unconstrained by  t B. (We used Ohm’s Law as an additional constraint.) 2.  t B h also depends upon vertical derivatives in E h, which single-height magnetograms do not fully constrain. Additional observational data must be used to obtain more information about both of these unknowns.

Keeping track of variables, all of these methods 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 , since  h 2  = (  h x u B z )·z 2. In particular circumstances, Doppler shifts can constrain E. 3. Multi-height magnetograms can constrain  z E h. (Given noise in the data, overdetermining E is fine!) ^

1. Tracking with “component methods” constrains  by estimating u in the source term (  h x u B z ) · z. Methods to find  via tracking include, e.g.: – Local Correlation Tracking (LCT, November & Simon 1988; ILCT, Welsch et al. 2004; FLCT Fisher & Welsch 2008) – the Differential Affine Velocity Estimator (DAVE, and DAVE4VM; Schuck 2006 & Schuck 2008) (Methods to find  via integral constraints also exist, e.g., Longcope’s [2004] Minimum Energy Fit [MEF] method.) Welsch et al. (2007) tested some of these methods using “data” from MHD simulations; MEF performed best. Further tests with more realistic data are underway. ^

2. Flows v || along B do not contribute to E = -(v x B)/c, but do “contaminate” Doppler measurements. v LOS v v v =

Aside: Dopplergrams are sometimes consistent with “siphon flows” moving along B. MDI Dopplergram at 19:12 UT on 2003 October 29 superposed with the magnetic polarity inversion line. (From Deng et al. 2006) Why should a polarity inversion line (PIL) also be a velocity inversion line (VIL)? One plausible explanation is siphon flows arching over (or ducking under) the PIL. What’s the DC Doppler shift along this PIL? Is flux emerging or submerging?

2. (cont’d) Doppler shifts along PILs of the LOS mag- netic field B LOS can constrain the ideal electric field E. Measurements of v Dopp and B trans on PILs are direct observations of the ideal E perpendicular to both. How do PTD E-fields compare with measurements of this “Doppler electric field” E Dopp ? The gradient of a scalar potential  derived from E Dopp can be added to PTD E-fields to improve consistency.

B h (t i ) v B h (t f ) v B h (t i )  t B h =  z (E h x z) - (  h x E z z) vertical shear in v hor ^ converging/diverging surface flows 3. Horizontal flows with either vertical shear or nonzero horizontal divergence (or both) alter the horizontal field B h. ^ If only vertical shear causes  t B h, then E h = 0, and there is no vertical Poynting flux!  z E h estimated from multi-height magnetograms can constrain which process is at work.

The PTD method determines an electric field E consistent with vector magnetic evolution,  t B. – Previous “component” methods (e.g., ILCT, FLCT, DAVE, DAVE4VM, MEF) were consistent with  t B z. – Faraday’s Law implies that  t B alone does not fully specify E – there is an undetermined “gauge” electric field  . Tracking can help constrain  h . Flows perpendicular to B contribute to E, and Doppler shifts along PILs constrain perp. flows, so also constrain E. E h derived from multi-height magnetograms can constrain  z E h. Conclusions, (ii)

Topics I’ll begin by reviewing results from recent research. (i) flows & flares (ii) estimating magnetogram electric fields Next, I’ll discuss ideas for research with the VSM. (iii) case studies of short-timescale evolution (iv) studies involving the “core” database

Shifting gears yet again, I present a few ideas (I have more!) for research projects with SOLIS / VSM. Many projects I envision involve VSM observations in “campaign mode.” – That is, obtaining sequences magnetograms at higher cadence than the “core data rate” of 3 per day. – HMI will also record photospheric vector magneto- grams; SOLIS data should complement (not “compete”). I would value input from NSO collaborators about prioritization. This presentation of my ideas is terse and cursory. So please interrupt me to ask questions!

Project #1: Estimate Poynting fluxes, and compare with both the proxy Poynting flux and flare activity. Welsch et al. (2009) had only LOS magnetograms, so could only estimate a “proxy Poynting flux”, (E h x B R ) With vector magnetograms, the vertical Poynting flux can be estimated, (E h x B h ) – the full horizontal flux can also be estimated How are these quantities correlated in space and time? – For a sample of active region magnetogram sequences, how are these energy fluxes related to flare activity?

Project #2a: Study the relationship between electric field estimates and Doppler shifts. Estimate the Doppler shift of magnetized plasma along LOS PILs (cf., Chae et al. 2004) – average to remove helioseismic “noise,” get “DC” shift – compensate for intensity – blueshift correlation – survey a few to several active regions Compare E-fields inferred along PILs by Doppler shifts with E-fields from other methods. Develop and improve methods to incorporate Doppler data into electric field estimation.

Project #2a: Using the same data, check for evidence of non-ideal electric fields along PILs. If there is a component of B along the PIL (shear), then E·B≠0 at the PIL, implying the E-field is non-ideal there. – Flux cancellation CME models invoke non-ideal E-fields on PILs. – How large is the non-ideal E-field at the PIL compared to (presumed) ideal E-fields away from the PIL? If flux cancellation occurs, does the rate of flux loss match |B h | x v z (Doppler)? If ideal, these should match. Tracking sometimes shows evidence of converging flows toward PILs, implying an E-field parallel to the PIL. Left: FLCT flows from MDI LOS magnetograms of AR 8038.

Project #3: Study electric fields in multi-height magnetogram sequences. Estimate E in sequences magnetograms in spectral lines with different formation heights – chromosph. LOS & vector photosph., SOLIS 854.2nm + HMI – (also worthwhile to baseline SOLIS & HMI photospheric) Compare electric fields / flows. – Where are they correlated in space? Where are they not? (e.g., in strong field vs. weak field regions?) – Do flow vorticities / divergences appear in common loci? – Do flows decorrelate on similar timescales? Inversions of the chromospheric vector field would certainly be useful, but also certainly not necessary.

Are changes in vertical currents associated with flow vorticity? Do sunspot rotations tend to increase or decrease currents? Are shearing or converging motions statistically stronger where increases in magnetic shear occur? Project #4: What is the relationship between flows and evolution of magnetic structure? The component of B h that arises purely from currents (derived using PTD), from a SOLIS/ VSM magnetogram of AR How do flows affect these structures?

Topics I’ll begin by reviewing results from recent research. (i) flows & flares (ii) estimating magnetogram electric fields Next, I’ll discuss ideas for research with the VSM. (iii) case studies of short-timescale evolution (iv) studies involving the “core” database

Project #5: Compare chromospheric magnetic prop- erties with RHESSI hard X-ray (HXR) flare emission. HXR emission is bremsstrahlung from non-thermal particles precipitating onto the upper chromosphere. Magnetic mirroring is thought to govern the precipitation of flare particles. – If so, in magnetic fields that converge more strongly, more particles should mirror, and less HXR emission should be seen. Models of the potential magnetic field at the chromo- sphere (from SOLIS data) quantify field convergence. HXR intensities & indices of power-law fits to emission spectra can be compared to aspects of chromospheric B. (In collaboration with SSL’s Jim McTiernan & Pascal Saint-Hilaire.)

Using LOS magnetograms, we found a “proxy Poynting flux,” S R = Σ uB R 2 to be related to flare activity. – It will be interesting to compare the “proxy” Poynting flux with the Poynting flux from vector magnetogram sequences! We developed the poloidal-toroidal decomposition (PTD) method to determine an electric field E consistent with vector magnetic evolution,  t B. – The method is in its infancy, and I look forward to applying it to more vector magnetogram sequences! I presented several research ideas for SOLIS/VSM data. – I hope you don’t think they’re too crackpot! Conclusions, final

The PTD method determines an electric field E consistent with vector magnetic evolution,  t B. The method is in its infancy, and I look forward to applying it to vector magnetogram sequences. – Previous “component” methods (e.g., ILCT, FLCT, DAVE, DAVE4VM, MEF) were consistent with  t B z. – Faraday’s Law implies that  t B alone does not fully specify E – there is an undetermined “gauge” electric field  . Tracking can help constrain  h . Flows perpendicular to B contribute to E, and Doppler shifts along PILs constrain perp. flows, so also constrain E. E h derived from multi-height magnetograms can constrain  z E h. Conclusions, (ii)

The ideal induction equation is:  t B x = (-  y E z +  z E y )c =  y (v x B y - v y B x ) -  z (v z B x - v x B y )  t B y = (-  z E x +  x E z )c =  z (v y B z - v z B y ) -  x (v x B y - v y B x )  t B z = (-  x E y +  y E x )c =  x (v z B x - v x B z ) -  y (v y B z - v z B y )