Mapping Free Energy in the Solar Atmosphere

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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!
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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(x1,x2,x3) and B(P) (x1,x2,x3) match the
  dist-ribution of normal flux, Bn(x1,x2), at
  coronas base.
• Nonpotential part of field is ?B (x1,x2,x3) B -
  B(P)
• B(P) carries no currents, or, equivalently, is
  curl free.
• For any BnS , B(P) has minimal energy.
• B(P) - ??, with ?2? ? Laplaces eqn. gives
  B(P)
• For any BnS , B(P) is unique.

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The Free Energy Release Paradigm motivates
studying free magnetic energy (cf., earthquakes).

• Slow photospheric motions (v 1 km/s) add
  magnetic energy to the coronal field, as E x B
  Poynting flux.
• The frozen-in flux condition prevents
  relaxation B ? B(P), so free energy is stored
  in the corona latency.
• 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).

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HMI data can be used in several ways to quantify
free magnetic energy.

• Use B(x1,x2,0) to extrapolate B B(P)
  (McTiernan, Thurs. a.m.)
• can compare model fields to AIA data
• Magnetic Virial Theorem (Wheatland Metcalf
  2005)
• novel application to photospheric magnetograms
• Free Energy Flux (FEF) through photosphere
  (Welsch, 2006)
• gives photospheric loci of energy injection
• Magnetic charge topology (MCT, e.g., Barnes et
  al. 2005).
• can give coronal loci of departures from
  potential field

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1) Using B(x1,x2,0) from HMI, extrapolations give
B(x1,x2,x3), allowing integration of B2/8?.

• In general, specification of B is required on all
  surfaces
• magnetograms give photospheric B(x1,x2)
• what is used on other boundaries?
• Strictly, NLFFF extrapolation should not be
  applied to non-force-free photospheric
  magnetograms.

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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.

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3) Knowledge of v(x1,x2) determines energy
changes for B and B(P) due to boundary flows.

Depends on photospheric (Bx, By, Bz), (vx
,vy,vz), and (Bx(P), By(P)). Requires vector
magnetograms. Compute from Bz.
How to find v?
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3) ILCT (Welsch et al. 2004) other methods can
determine flows from pairs of magnetograms.
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3) From B(x1,x2,0) and v(x1,x2), maps of the free
energy flux can be computed (Welsch et al. 2006)
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4) From B(P)(t1), B(P)(t2), 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.

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Role(s) of Current Sheets
4) Lifted from Longopes talk, TRACE-RHESSI-SOHO
meeting, Dec. 2004
W
Energy RELEASE DW accumulates prior to
reconn burst latency
Wfce
DW
Wpot
Rapidly released via local E field
0
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Here are a few random (and arguable!) thoughts
that didnt 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 wont do so
  well.

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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)

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Canfield, Hudson, McKenzie (1999) argued that
sigmoidal coronal morphologies correlate with
eruptions.

• They also showed that spot areas also correlate
  with eruptive activity.

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Schrijver, Title, DeRosa (2005) found that free
energy can be detected qualitatively.
similarities
differences

• Comparisons of TRACE EUV observations
• with B(P) revealed similarities differences.

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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?

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Can we infer B directly from coronal morphology?

• 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)
• De Rosa (2004, unpublished?) investigated
  automated loop identification algorithms.
• Punchline This is not easy to do!

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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)?

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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?

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If steady emissivity is a function of B (or ?),
then what can AIA tell us about magnetic
connections?

• Can Pevtsovs 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.

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From Pevtsov et al. (2003)

• X-ray spectral radiance LX 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 LX ? ?1.15 of combined data set.

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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 1018 Mx per reconnection

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Which observations might we pursue? A starter
list

• Avg. reconnected flux, ??, per reconnection
  event?
• avg. reconnected flux per DN?
• avg. reconnected flux per coronal loop? (vs. ??)
• Avg. reconnection rate, ??/ ?t?
• 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 ?
• What can we learn from simulated emission
  forward models? (Lundquist/Schrijver/Mok et al.s)

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Emissivity appears correlated with reconnection
rate in different spectral ranges.

• HXR (RHESSI) UV (TRACE)

Fletcher et al. 2004
Courtesy S. Krucker
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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 1018 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?

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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.
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Longcope et al. (2005) compiled statistics of EUV
loops properties.
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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

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References, contd

• 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