Free Energies via Velocity Estimates

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Free Energies via Velocity Estimates

• B.T. Welsch G.H. Fisher,
• Space Sciences Lab, UC Berkeley

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In ideal MHD, photospheric flows move mag-netic
flux with a flux transport rate, Bnuf.
(1)
Demoulin Berger (2003) Apparent motion of flux
on a surface can arise from horizontal and/or
vertical flows. In either case, uf represents
flux transport velocity.
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Magnetic diffusivity ? also causes flux
transport, as field lines can slip through the
plasma.

• Even non-ideal transport can be represented as a
  flux transport velocity.
• Quantitatively, one can approximate 3-D
  non-ideal effects as 2-D diffusion, in Ficks Law
  form,

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The change in the actual magnetic energy is given
by the Poynting flux, c(E x B)/4?.

• In ideal MHD, E -(v x B)/c, so
• uf is the flux transport velocity from eqn. (1)
• uf is related to the induction eqns z-component,

(2)
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A Poynting-like flux can be derived for the
potential magnetic field, B(P), too.

• B evolves via the induction equation, meaning its
  connectivity is preserved (or nearly so for small
  ?).
• B(P) does not necessarily obey the induction
  equation, meaning its connectivity can change!
• Welsch (2006) derived a Poynting-like flux for
  B(P)

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The free energy flux (FEF) density is the
difference between energy fluxes into B and B(P).

(4)
Depends on photospheric (Bx, By, Bz), (ux,uy),
and (Bx(P), By(P)). Requires vector
magnetograms. Compute from Bz.
What about v or u?
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Several techniques exist to determine velocities
required to calculate the free energy flux
density.

• Time series of vector magnetograms can be used
  with
• FLCT, ILCT (Welsch et al. 2004),
• MEF (Longcope 2004),
• MSR (Georgoulis LaBonte 2006),
• DAVE (Schuck, 2006), or
• LCT (e.g., Démoulin Berger 2003)
• to find
• Proposed locations of free energy injection can
  be tested, e.g., rotating sunspots shearing
  along PILs.

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We use ILCT to modify the FLCT flows, via the
induction equation, to match ?Bz/?t.
With

• and the approximation uf ? uLCT, solving

with (vB) 0, completely specifies (vx, vy, vz).
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Tests with simulated data show that LCT
underestimates Sz more than ILCT does.
Images from Welsch et al., in prep.
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The spatially integrated free energy flux density
quantifies the flux across the magnetogram FOV.

• Large ?tU(F) gt 0 could lead to flares/CMEs.
• Small flares can dissipate U(F), but should not
  dissipate much magnetic helicity.
• Hence, tracking helicity flux is important, too!

(5)
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We used both LCT ILCT to derive flows between
pairs of boxcar- averaged mgrams.
? 15 pix thr(Bz) 100 G
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Poynting fluxes into AR 8210 from ILCT LCT both
show increasing magnetic energy, U.
ILCT shows an increase of 5 x 1031 erg FLCT
shows an increase of 1 x 1031 erg
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Poynting fluxes from ILCT LCT are correlated.
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Fluxes into the potential field, Sz(P),
calculated from ILCT FLCT flows, however,
strongly disagree.

• Recall that Sz(F) Sz - Sz(P), so the increase
  seen in ILCTs Sz(P) will cause a decrease in
  Sz(F).

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Changes in U(F) derived via ILCT are 1031erg,
and vary in both sign and magnitude.
Changes in U(F) derived via FLCT are much
smaller, and not well correlated with ILCT.
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The cumulative FEFs ( ? ?U(F)) do not match ILCT
shows decreasing U(F), LCT does not.
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Conclusions Re FEF

• Both FLCT ILCT show an increase in magnetic
  energy U, of roughly 1031 erg and 5 x 1031 erg,
  resp.
• FLCT also shows an increase in free energy U(F),
  of about 1031 erg over the 6 hr magnetogram
  sequence.
• ILCT, however, shows a decrease in U(F), of 4
  x1031 erg
• Apparently, this arises from a pathology in the
  estimation
• of the change in potential field energy, ?U(P).
  
• This shortcoming should be easily surmountable.

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References

• Démoulin Berger, 2003 Magnetic Energy and
  Helicity Fluxes at the Photospheric Level,
  Démoulin, P., and Berger, M. A. Sol. Phys., v.
  215, p. 203.
• Longcope, 2004 Inferring a Photospheric Velocity
  Field from a Sequence of Vector Magnetograms The
  Minimum Energy Fit, ApJ, v. 612, p. 1181-1192.
• Georgoulis LaBonte, 2006 Reconstruction of an
  Inductive Velocity Field Vector from Doppler
  Motions and a Pair of Solar Vector Magnetograms,
  Georgoulis, M.K. and LaBonte, B.J., ApJ, v. 636,
  p 475.
• Schuck, 2006 Tracking Magnetic Footpoints with
  the Magnetic Induction Equation, ApJ v. 646, p.
  1358.
• 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 Regniér, S.,
  ApJ, v. 610, p. 1148.
• Welsch, 2006 Magnetic Flux Cancellation and
  Coronal Magnetic Energy, Welsch, B. T, ApJ, v.
  638, p. 1101.

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Some LCT vectors flip as difference images
fluctuate!
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Derivation of Poynting-like Flux for B(P)
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Also works w/ non-ideal terms
at eqns left is valid w/any non-ideal term!