1. 14 Feb 2006C4: Coronal Energy Inputs I. Mapping Free Energy in the Solar Atmosphere What can we learn from HMI & AIA? Brian Welsch, Space Sciences Lab, UC Berkeley What kinds of observations are required to compute and understand the creation and dissipation of free energy? How can we best make use of the joint AIA and HMI dataset? What jobs need to get done before launch to allow proper analysis and use of the SDO data? This last point, in particular, is the focus of this session; consequently, my talk is meant to engender discussion!

2. 14 Feb 2006C4: Coronal Energy Inputs I. Free energy U (F) is the actual magnetic energy minus the potential magnetic energy. 8  U (F) =  dV [ (B · B) - (B (P) · B (P) )] Both B(x 1,x 2,x 3 ) and B (P) (x 1,x 2,x 3 ) match the dist- ribution of normal flux, B n (x 1,x 2 ), at corona’s base. Nonpotential part of field is  B (x 1,x 2,x 3 ) = B - B (P) B (P) carries no currents, or, equivalently, is curl free. 1.For any B n | S, B (P) has minimal energy. 2. B (P) = - , with  2   Laplace’s eqn. gives B (P) 3.For any B n | S, B (P) is unique.

3. 14 Feb 2006C4: Coronal Energy Inputs I. The ‘Free Energy Release Paradigm’ motivates studying free magnetic energy (cf., earthquakes). 1.Slow photospheric motions (v ~ 1 km/s) add magnetic energy to the coronal field, as E x B Poynting flux. 2.The “frozen-in flux” condition prevents relaxation B  B (P), so free energy is stored in the corona – “latency.” 3.With enough free energy, the corona reaches an unstable configuration (??) and “spontaneously” relaxes toward B (P). Most CME models – flux cancellation, breakout, tether cutting, kink instability – accord with this picture. (But cf., “free energy injection” paradigm of Chen [1996]).

4. 14 Feb 2006C4: Coronal Energy Inputs I. HMI data can be used in several ways to quantify free magnetic energy. 1.Use B(x 1,x 2,0) to extrapolate B & B (P) (McTiernan, Thurs. a.m.) –can compare model fields to AIA data 2.Magnetic Virial Theorem (Wheatland & Metcalf 2005) –novel application to photospheric magnetograms 3.Free Energy Flux (FEF) through photosphere (Welsch, 2006) –gives photospheric loci of energy injection 4.Magnetic charge topology (MCT, e.g., Barnes et al. 2005). –can give coronal loci of departures from potential field

5. 14 Feb 2006C4: Coronal Energy Inputs I. 1) Using B(x 1,x 2,0) from HMI, extrapolations give B(x 1,x 2,x 3 ), allowing integration of B 2 /8 . In general, specification of B is required on all surfaces –magnetograms give photospheric B( x 1,x 2 ) – what is used on other boundaries? Strictly, NLFFF extrapolation should not be applied to non-force-free photospheric magnetograms.

6. 14 Feb 2006C4: Coronal Energy Inputs I. 2) The Magnetic Virial Theorem (MVT) gives field energy via integration over boundary surface. MVT assumes B is force-free, even on the boundaries - but photosphere is forced! So MVT is best applied to chromospheric vector magnetograms, e.g., Metcalf et al., 2003. Wheatland & Metcalf (2005) proposed extrapolating from the forced to force-free layers.

7. 14 Feb 2006C4: Coronal Energy Inputs I. 3) Knowledge of v(x 1,x 2 ) determines energy changes for B and B (P) due to boundary flows. Depends on photospheric (B x, B y, B z ), (v x, v y,v z ), and (B x (P), B y (P) ). Requires vector magnetograms. Compute from B z. How to find v?

8. 14 Feb 2006C4: Coronal Energy Inputs I. 3) ILCT (Welsch et al. 2004) & other methods can determine flows from pairs of magnetograms.

9. 14 Feb 2006C4: Coronal Energy Inputs I. 3) From B(x 1,x 2,0) and v(x 1,x 2 ), maps of the free energy flux can be computed (Welsch et al. 2006)

10. 14 Feb 2006C4: Coronal Energy Inputs I. 4) From B (P) (t 1 ), B (P) (t 2 ), MCT calculates changes in flux  ij connecting photospheric sources  i &  j to estimate U (F). Each magnetogram in a sequence is partitioned into fluxes  i.

11. 14 Feb 2006C4: Coronal Energy Inputs I. Role(s) of Current Sheets 0 W fce W pot W DWDWDWDW Energy RELEASE: D W accumulates prior to reconn’ burst: latency Rapidly released via local field via local E field 4) [Lifted from Longope’s talk, TRACE-RHESSI-SOHO meeting, Dec. 2004]

12. 14 Feb 2006C4: Coronal Energy Inputs I. Here are a few random (and arguable!) thoughts that didn’t fit anywhere else. Mapping free energy using AIA data will require new techniques – not so with HMI. Techniques that can be automated would be good – AIA will generate a lot of data! AIA will tell us about non-potentiality from emergence – something HMI probably won’t do so well.

13. 14 Feb 2006C4: Coronal Energy Inputs I. How can we use coronal observations to determine how much and where B differs from B (P) ? Qualitative differences? –Canfield et al. (1999) – X-ray sigmoids –Schrijver, Title, & De Rosa (2005)

14. 14 Feb 2006C4: Coronal Energy Inputs I. Canfield, Hudson, & McKenzie (1999) argued that sigmoidal coronal morphologies correlate with eruptions. They also showed that spot areas also correlate with eruptive activity.

15. 14 Feb 2006C4: Coronal Energy Inputs I. Schrijver, Title, & DeRosa (2005) found that free energy can be detected qualitatively. Comparisons of TRACE EUV observations with B(P) revealed similarities & differences. similarities differences

16. 14 Feb 2006C4: Coronal Energy Inputs I. How can we use coronal observations to determine how much and where B differs from B (P) ? Qualitative differences? –Canfield et al. (1999) – X-ray sigmoids –Schrijver, Title, & De Rosa (2005) Quantitative differences? –Can we infer B directly?

17. 14 Feb 2006C4: Coronal Energy Inputs I. Can we infer B directly from coronal morphology? 1.Gary & Alexander (1999) distorted of a model B to match coronal observations. – assumed an initial topology in model B –distortions were non-force-free (but perhaps this is OK) 2.De Rosa (2004, unpublished?) investigated automated loop identification algorithms. Punchline: This is not easy to do!

18. 14 Feb 2006C4: Coronal Energy Inputs I. How can we use coronal observations to determine how much and where B differs from B (P) ? Qualitative differences? –Canfield et al. (1999) – X-ray sigmoids –Schrijver, Title, & De Rosa (2005) Quantitative differences? –Can we infer B directly? –If we cannot infer B, then what? Can we quantify departures from B (P) ?

19. 14 Feb 2006C4: Coronal Energy Inputs I. How can AIA observations be used to quantify departures from B (P) ? Aside - The corona exhibits ~two modes of emission: a) steady state – perhaps averaged over weak fluctuations b) highly intermittent – “impulsive,” stronger fluctuations What gives rise to EUV/SXR emissivity? I.) Local emissivity  steady heating?  B? or  ? independent of B - B (P) (at large scales)? II.) Local emissivity  intermittent magnetic reconnection?   B = B - B (P). Can we distinguish between these?

20. 14 Feb 2006C4: Coronal Energy Inputs I. If steady emissivity is a function of B (or  ), then what can AIA tell us about magnetic connections? Can Pevtsov’s Law (2003), relating photospheric magnetic flux to coronal SXR emission, be extended to EUV observations? Does each EUV loop correspond, on average, to a certain amount of coronal (or photospheric) flux? Study Idea: Quantify how many EUV/SXR loops connect photospheric sources (Voronoi regions?) of with varying flux. Applicable to MCT, which estimates free energy by estimating flux  ij linking photopheric sources  i and  j.

21. 14 Feb 2006C4: Coronal Energy Inputs I. From Pevtsov et al. (2003): X-ray spectral radiance L X vs. total unsigned magnetic flux for solar and stellar objects. Dots: Quiet Sun. Squares: X-ray bright points. Diamonds: Solar active regions. Pluses: Solar disk averages. Crosses: G, K, and M dwarfs. Circles: T Tauri stars. Solid line: Power-law approximation L X   1.15 of combined data set.

22. 14 Feb 2006C4: Coronal Energy Inputs I. If emissivity   B = B - B (P), can models predict loci of reconnection-driven emission in AIA? Several models predict (to varying degrees) reconnection sites: Extrapolations (NLFFF & potential) MCT – separators FEF – corona above free energy injection sites Longcope et al. 2005: 4 x 10 18 Mx per reconnection

23. 14 Feb 2006C4: Coronal Energy Inputs I. Which observations might we pursue? A “starter” list: 1.Avg. reconnected flux, , per reconnection event? –avg. reconnected flux per DN? –avg. reconnected flux per coronal loop? (vs. ?) 2.Avg. reconnection rate,  /  t? 3.Avg. “latency time” vs. spatial scale? –Separatrices/ QSLs, during emergence are thin.  Does this mean reconnection happens quickly? –How about during cancellations? –How about shear flows ? 4.What can we learn from “simulated emission” forward models? (Lundquist/Schrijver/Mok et al.’s)

24. 14 Feb 2006C4: Coronal Energy Inputs I. Emissivity appears correlated with reconnection rate in different spectral ranges. HXR (RHESSI)UV (TRACE) Courtesy S. Krucker Fletcher et al. 2004

25. 14 Feb 2006C4: Coronal Energy Inputs I. Longcope et al. (2005) combined TRACE data with MCT to study an emerging AR & reconnection. In this case, “ flux [was] transferred as discrete bundles of 4 x 10 18 Mx each.” “The sum of cross sections of all observed loops accounts for only one-fifth of the transferred magnetic flux predicted by the model.” Could their technique be standardized?

26. 14 Feb 2006C4: Coronal Energy Inputs I. Longcope et al. (2005) quantified EUV loops’ spatial and topological properties. Left: TRACE 171 Å image of ARs 9570 & 9574. Right: Cross-sections of loops intersecting slice in left image.

27. 14 Feb 2006C4: Coronal Energy Inputs I. Longcope et al. (2005) compiled statistics of EUV loops’ properties.

28. 14 Feb 2006C4: Coronal Energy Inputs I. References Barnes et al., 2005: Implementing a Magnetic Charge Topology Model for Solar Active Regions, Barnes, G., Longcope, D.W., & Leka, K.D., ApJ, v. 629, 561. Canfield et al. 1999: Sigmoidal morphology and eruptive solar activity, Canfield, R. C., Hudson, H.S., & McKenzie, D.E., GRL, v. 26, 627 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. Fletcher et al., 2003: Tracking of TRACE Ultraviolet Flare Footpoints, Fletcher, L., Pollock, J.A.., & Potts, H.E. Sol Phys, v. 222, 279 Gary & Alexander, 1999: Constructing the Coronal Magnetic Field By Correlating Parameterized Magnetic Field Lines With Observed Coronal Plasma Structures, Gary, G.A., & Alexander, D., Sol Phys., v. 186, 123 Longcope et al., 2005: Observations of Separator Reconnection to an Emerging Active Region, Longcope, D. W.; McKenzie, D. E.; Cirtain, J.; Scott, J. ApJ, v. 630, # 1, p. 596. Lundquist et al., 2005: Predicting Coronal Emissions with Multiple Heating Rates, Lundquist, L.L., Fisher, G.H., Leka, K.D., Metcalf, T.R., & McTiernan, J.M., AGU Spring Meeting Abstracts, A2 Metcalf et al., 2005: Magnetic Free Energy in NOAA Active Region 10486 on 2003 October 29, Metcalf, T. R., Leka, K. D., Mickey, D. L., ApJ, 623, # 1, pp. L53-L56. Mok et al., 2005: Calculating the Thermal Structure of Solar Active Regions in Three Dimensions, Mok, Y., Miki\'c, Z., Lionello, R., & Linker, J.A., ApJ, v. 621, 1098

29. 14 Feb 2006C4: Coronal Energy Inputs I. Metcalf et al., 1995: Is the solar chromospheric magnetic field force-free? Metcalf, T. R., Jiao, L., McClymont, A. N., Canfield, R. C., Uitenbroek, H., ApJ, v. 439, #1, p. 474- 481. Pevtsov et al., 2003:: The Relationship Between X-Ray Radiance and Magnetic Flux, Pevtsov, A.A., Fisher, G.H., Acton, L.W., Longcope, D.W., Johns-Krull, C.M., Kankelborg, C.C., & Metcalf, T.R., ApJ, v. 598, 1387. Schrijver et al., 2005: The Nonpotentiality of Active-Region Coronae and the Dynamics of the Photospheric Magnetic Field, Schrijver, C. J, DeRosa, M. L., Title, A. M., and Metcalf, T. R., ApJ, v. 628, #1, p. 501. Schrijver et al., 2004: The Coronal Heating Mechanism as Identified by Full-Sun Visualizations, Schrijver, C. J, Sandman, A. W.; Aschwanden, M. J., DeRosa, M. L., ApJ, v. 615, #1, p. 512. 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, in press. Wheatland et al., 2000: An Optimization Approach to Reconstructing Force-free Fields, Wheatland, M. S., Sturrock, P. A., Roumeliotis, ApJ, v. 540, #2, p. 1150-1155. Wheatland & Metcalf, 2005: An improved virial estimate of solar active region energy, Wheatland, M.S. and Metcalf, T.R., ApJ, in press. (v. 636, #2, 10 Jan. 2006) [on astro-ph] References, cont’d