1. Active Region Flux Transport Observational Techniques, Results, & Implications B. T. Welsch (welsch@ssl.berkeley.edu) G. H. Fisher (fisher@ssl.berkeley.edu) Space Sciences Lab, Univ. of California, Berkeley, CA 94720-7450 Fourier local correlation tracking (FLCT) was applied to time series of the normal magnetic field in vector magnetograms of NOAA AR 8210, from 17:13 - 21:29 UT on 1998 May 1, to derive photospheric flows. These flows, combined with both the observed horizontal magnetic field and an inferred horizontal potential magnetic field, can be used to derive the flux of free magnetic energy across the photosphere. Here, we present the formalism used, as well as a map of the time- averaged free energy flux. Over the time interval covered by the magnetograms, ~10 31 erg of free magnetic energy flowed upward across the photosphere, a significant fraction of the energy typically released in large flares coronal mass ejections (CMEs). A flare and CME began at 22:30 UT. We acknowledge the kind support of AFOSR's MURI program.

2. We derived flows from vector magnetograms of AR 8210, on 01 May 1998, from 17:13 – 21:29 UT. IVM spectra were summed in time to generate magnetograms We tracked the initial & final magnetograms, so  t = 4 hr., 16 min. 1 Pix ~ 1280 km ~ 1.8” M flare & CME occur- red near 22:30 UT

3. Fourier local correlation tracking (FLCT, Welsch et al. 2004) finds u(x 1,x 2 ) by correlating subregions. 1) for ea. (x i, y i ) above |B| threshold … 2) apply Gaussian mask at (x i, y i ) … 3) truncate and cross-correlate… * 4) u(x i, y i ) is inter- polated max. of correlation funct = = =

4. Demoulin & Berger (2003) argued that LCT applied to magnetograms does not necessarily give plasma velocities. u f  v n B h -v h B n is the flux transport velocity u f is the apparent velocity (2 components) v  is the actual plasma velocity (3 comps) The apparent motion of flux on the photosphere, u f, is a combination of horizontal flows and vertical flows acting on non-vertical fields.

5. Aside: Doppler shifts (  v n ) can’t distinguish between flows that are parallel to B, perpendicular to B, or neither. “Inductive flow” methods derive flows from observed normal magnetic field changes,  B n /  t. Since  B n /  t =  x (v x B), flows v || along B do not affect  B n /  t, so inductive methods can only determine v . Once v  is known, the measured Doppler shift allows determination of v ||.

6. Many techniques exist to determine velocities from time series of vector magnetograms, e.g., LCT (Démoulin & Berger 2003); ILCT, Inductive LCT (Welsch et al. 2004); MEF, Minimum Energy Fit (Longcope 2004); DLCT & DAVE, Differential LCT & Diff’l Affine Velocity Estimator (Schuck 2006). A paper comparing these methods’ accuracy, using synthetic magnetograms generated by an MHD simulation, is currently in preparation.

7. Aside: Free magnetic energy is the difference in energies of the actual field B and the potential field B (P). The actual magnetic energy is U   dV (B · B)/8 . The lowest possible magnetic energy matching the observed B n is U (P)   dV (B (P) · B (P) )/8 . –B (P) is current-free (curl-free); –Equivalently, B (P) is “potential:” B (P) = - , with  2  = 0. Free energy is the difference U (F) = U – U (P) This is the energy available to power flares and CMEs!

8. Photospheric velocities determine the flux of energy across the photosphere “into” B & B (P). The Poynting flux, S z, is the change in actual magnetic energy, equal to c(E x B)/4 , with E = -(v x B)/c. 4  S n = [(v x B) x B] n = (B h · B h )v n – (v h ·B h )B n = – (B n v h – v n B h ) · B h = – (u f B n ) · B h A “Poynting-like” flux can also be derived for the potential magnetic field, B (P) (Welsch, 2006): 4  S n (P) = – (B n v h – v n B h ) ·B h (P) = – (u f B n ) · B h (P) B evolves via the induction equation & preserves field lines’ connections, but B (P) does not – so connections in B (P) change.

9. The “free energy flux (FEF) density” is the difference between energy fluxes into B and B (P). Depends on photospheric (B x, B y, B z ), (u x,u y ), and (B x (P), B y (P) ). From vector magnetograms. Computed* from B z. Derived via FLCT. S n (F) = (S n - S n (P) )/4  = – (u f B n ) · (B h - B h (P) ) The free energy flux is U (F) =  dxdy S n (F) Large  dt (  t U (F) ) could predict flares/CMEs. * B (P) was computed using a both a Green’s function method.

10. Measured B h (red) and potential B h (P) (blue) photospheric magnetic vectors, superimposed on a grayscale image of B n. Measured values shown are averaged from two magnetograms at 17:13 & 21:29 UT.

11. Grayscale image of S n (F), the flux of free energy across the photosphere; white & black correspond to upward & downward (resp.) fluxes. Black vectors are measured B h, and white vectors are derived flows, u (LCT). Black & white contours are 100 G level curves of negative & positive (resp.) regions of B n. (For clarity, only contours up to 500 G in |B n | are shown.)

12. Using flows u f derived by FLCT, combined with the measured vector magnetic field B h and a potential extrapolation B h (P), we computed a net upward flux of S n (F) ~10 31 ergs of free magnetic energy across the photosphere in NOAA AR 8210 from 17:13 – 21:29 UT on 01 May 1998, a significant fraction of the energy released in typical flares & CMEs. Similar studies using next generation vector magnetographs (SOLIS, FPP on Solar B, HMI on SDO) could show a correlation between free energy flux and flares & CMEs. References Démoulin & Berger, 2003: Magnetic Energy and Helicity Fluxes at the Photospheric Level, Démoulin, P., and Berger, M. A. Sol. Phys., v. 215, # 2, p. 203-215. Longcope, 2004: Inferring a Photospheric Velocity Field from a Sequence of Vector Magnetograms: The Minimum Energy Fit, ApJ, v. 612, # 2, p. 1181-1192. Schuck, 2006: Tracking Magnetic Footpoints with the Magnetic Induction Equation, ApJ (submitted, 2006) Welsch et al., 2004: ILCT: Recovering Photospheric Velocities from Magnetograms by Combining the Induction Equation with Local Correlation Tracking, Welsch, B. T., Fisher, G. H., Abbett, W.P., and Regnier, S., ApJ, v. 610, #2, p. 1148-1156. Welsch, 2006: Magnetic Flux Cancellation and Coronal Magnetic Energy, ApJ, v. 638, #2, p. 1101-1109. CONCLUSIONS