SH53A-2151: Relationships Between Photospheric Flows and Solar Flares by Brian T. Welsch & Yan Li Space Sciences Laboratory, UC-Berkeley Fourier Local Correlation Tracking (FLCT) has been applied to the entire database of 96-minute cadence line-of-sight (LOS) magnetograms from the SOHO/MDI mission, to derive photospheric transverse velocities (u x,u y ). In a previous study, we applied FLCT to a few dozen active regions (ARs), and found that the "proxy Poynting flux” (PPF) --- the product |u|B 2, where |u| is the FLCT flow speed and B is the LOS field divided by the cosine of viewing angle, integrated over each AR --- was statistically related to flare activity. We will present preliminary results of our investigation of the relationship between PPF and flare activity from NOAA's GOES catalog for several hundred ARs identified in NOAA's daily Solar Region Summaries.

The ideal induction equation relates v to  t B,  t B = -c(  x E)=  x (v x B) assuming the ideal Ohm’s law applies,* relating v to E via E = -(v x B)/c Hence:  B z /  t = [  x (v x B) ] z = -  h  (v h B z - v z B h ) *One could instead use E = -(v x B)/c + R, if some known resistivity R is assumed. Background: Faraday’s & Ohm’s laws imply that v is related to field evolution  t B in magnetogram sequences.

Why do we care about photospheric flows? Flows (or electric fields) can quantify aspects of evolution in B corona. The fluxes of magnetic energy & helicity across the photosphere into the corona depend upon E ph : dU/dt = ∫ dA (B ph x [v ph x B ph ]) z /4 π dH/dt = 2 ∫ dA (A ph x [v ph x B ph ]) z U and H probably play central roles in coronal heating, flares, and CMEs. Coupling of B corona to B ph also implies that v ph can provide boundary conditions for data-driven, time-dependent simulations of B cor (e.g., Cheung & DeRosa 2012).

4 Method: Fourier local correlation tracking (FLCT) estimates u( x, y) by correlating evolution in regions to find local shifts. * = = = Windowing implies spatial averaging of the underlying flow field.

How is the apparent movement of magnetic flux, u, in magnetograms related to the plasma velocity, v? u is not equivalent to v: u is the apparent horizontal velocity (2 components) v is the actual plasma velocity (3 comps) (Note: non-ideal effects can also cause flux transport!) Démoulin & Berger (2003): u = v hor - (v n /B n )B hor Schuck (2008): u = a biased estimate of v hor

Approach 1.Apply FLCT to all “not bad” pairs of full-disk, 96-minute magnetograms in the MDI database from pix. = 1.4 Mm, sigma = 8 pixels, dt between magnetograms = 96 min. 2.From each daily NOAA Solar Region Summary (SRS), find all active regions (ARs). 3.Quantitatively characterize magnetic and flow fields in each AR once per day, at t 0 nearest the SRS (00:30 UT). 4.Using NOAA’s GOES flare catalog, quantify subsequent flare activity in each AR during t 0 +  t --- here we use  t = 24 hr. 1.Investigate relationship(s) --- if any! --- of properties of magnetic and flow fields to flare activity.

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 irregularly gridded B R ( θ,φ ) on the sphere to a regularly gridded B R (x,y) prior to tracking. Corrections for scale distortion from projection were applied to estimated flows.

Here is a sample NOAA Solar Region Summary, for 2001 Mar. 27: SRS files are online at: ftp://ftp.ngdc.noaa.gov/STP/swpc_products/daily_reports/solar_region_summaries/ Descriptions of fields are online at: Errors (though rare) in the SRS-derived NAR database in SSWIDL motivated using the SRS reports directly. Files were automatically parsed. In the process, several minor inconsistencies and errors were identified and manually corrected.

For each SRS, neighborhoods of all ARs within 45 o of disk center were found in the corresponding magnetogram. Field outside 45 o from disk center is zeroed. Red asterisks show (longitude, latitude) of each NOAA AR location. Colored lines show 10 o zones around each AR within 45 o of disk center. Pixels within 10 o of multiple ARs are assigned to the closest AR. Properties of magnetic and flow fields within each zone were computed (see below), to be associated with flaring.

Our sample consists of 7164 “active-region days,” associated with 2264 unique NOAA ARs. Each AR is typically observed multiple times; observations 24 hr. apart are treated as “independent.” It is plausible that some AR properties relevant to flare activity vary on time scales > 24 hr. If so, treating observations as independent would be inappropriate.

To start, we computed 10 quantities from each estimated radial magnetic field, B R (x,y), and flow field, u(x,y). 1.  = Σ |B R | da 2 ; this scales as area A (Fisher et al. 1998) 2.Schrijver's (2007) R, for |B R | > 150 Mx / cm 2 – extensive param: should scale as length L – σ R = 15 Mm FWHM 3.Schrijver's R, for |B R | > 50 Mx / cm 2 – σ R = 4 Mm FWHM 4. Σ |B R | 2 5. Σ |B R | 3 6. Σ |B R | 4 7. Σ |u| 8. Σ |u| 2 9. Σ |u| 2 |B R | 10. Σ |u||B R | 2 Nonlinearity weights regions of strong field / strong flow more or less heavily. These differences should be mostly irrelevant for correlation analysis, but nonlinearity might affect parameters’ discrimination capability. Meant to also capture small-scale, weaker fields.

Given B R (x,y) and estimates of the apparent motion of flux u(x,y), how can flare activity be predicted? Extensive Params Matter Most: Welsch et al. (2009) found extensive parameters were better flare predictors than intensive - extensives grow with region size, e.g., integrated quantities; - intensives do not increase with system size, e.g., average properties. Baseline Params: Barnes & Leka (2008) report that total unsigned flux Φ and flux near polarity inversion lines R are among the best known predictors of flare activity. A Promising Parameter: In their study of 46 ARs, Welsch et al. (2009) found the“proxy” Poynting flux (“PPF”), Σ u B R 2, to be as or more strongly correlated with flaring than Φ.

Distributions of Φ, R, and PPF in flaring and non-flaring AR populations are similar. (Best would be separate peaks.) Barnes et al. 2007: Using Bayes’s theorem, the probability that a region belongs to the flaring population when it is observed to have properties x is: So: where the red curve lies above the dashed line, the parameter accurately predicts a greater likelihood of flaring. This reasoning implies Schrijver’s R, with a thresh. of 50 Mx/cm 2, is a better predictor.

This result differs from Welsch et al. (2009)! Welsch et al. (2009) found the distribution of PPF in the flaring population (right panel, red curve) differed significantly from that of R (red curve at left). Is this a sample effect? The 46 ARs in the Welsch et al. (2009) sample was not objective: flare/CME active and flare/CME quiet regions were manually selected.

Discriminant analysis (DA) compares the power of one or more variables to predict population membership. In both plots, green is flaring population; means are circles. The blue line is the discriminant boundary. At upper left, values of Φ (T.U.S. flux) & PPF above it imply flares are more likely than not. In both plots, the line is more nearly horizontal than vertical implying the vertical coordinate’s parameter has more discriminatory power. This implies PPF has more discriminatory power than Φ, and R-50 has more than R-150.

Reliability plots indicate under- or over-prediction of flare activity as a function of forecast probability. At low forecast probability (“all clear”), the combo of Φ & PPF underpredicts – i.e., misses flares. At high forecast probability (“red light”), the combo of Φ & PPF overpredicts – i.e., cries wolf. These failures are reflected in limited skill scores.

Outputs of DA -- (1) coefficients of linear fits, and (2) skill scores -- be used to compare predictive powers. ParamSolo Skill, Random Solo Skill, Climatology DA Coeff. ratio to  DA Coeff. ratio to R-50 DA Coeff. ra- tio to R-150 Best 2-var. Clim. Skill  R-50 R-150 PPF / / / / / / , R , R-150 “, R , R-50 the climatological skill score [e.g., Murphy and Epstein, 1989], [is] defined by Wheatland 2005: the joint probability distribution for forecasts (denoted f ) and observations (denoted x ) may be constructed… Averages over all days are denoted by. For example, is the average of the forecast probability over all days.

Summary First, results shown here are preliminary! Everything --- from the flow fields themselves to the AR masks to the flare tabulation in each prediction window --- has not been checked! Second, our results differ from Welsch et al. (2009)! They found that PPF slightly outperformed Φ and R, but we find that R-50 works best in our sample.