Free Energies via Velocity Estimates
B.T. Welsch & G.H. Fisher,
Space Sciences Lab, UC Berkeley
In ideal MHD, photospheric flows move magnetic flux with a flux transport rate, Bnuf.
Bnuf  Bn v h  vn Bh
(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.”
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 Fick’s
Law form,
Bn u non-ideal ~ ηBn
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:
4 Sz  [( v x B) x B]z  v z (B h  B h ) - Bz ( v h  B h )
4 Sz  B h  (v z B h - Bz v h )
4 Sz  - B h  (B zu f )
• uf is the flux transport velocity from eqn. (1)
• uf is related to the induction eqn’s z-component,
Bz
  h  (B z u f )  0
t
(2)
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):
4 S  B h  (v z Bh - Bz v h )  - B h  (Bz uf ) (3)
(P)
z
( P)
( P)
The “free energy flux” (FEF) density is the
difference between energy fluxes into B and B(P).
S  Sz - S
(F)
z
(P)
z
 (B h  B h )  (v z B h - Bz v h ) /4
(P)
 - (B h  B h )  (B z u f ) /4
(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?
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
(v z Bh - Bz v h ) , or (- Bzuf )
• Proposed locations of free energy injection can be
tested, e.g., rotating sunspots & shearing along PILs.
We use ILCT to modify the FLCT flows, via the
induction equation, to match Bz/t.
With Bzuf  vz Bh  Bz v h       ẑ,
and the approximation uf  uLCT, solving
B z
2
  
t
  ( Bz u
LCT
f
)   
2

with (v·B) = 0, completely specifies (vx, vy, vz).
Tests with simulated data show that LCT
underestimates Sz more than ILCT does.
Images from Welsch et al., in prep.
The spatially integrated free energy flux density
quantifies the flux across the magnetogram FOV.
 t U  dx dy S  dx dy (Sz - S ) (5)
(F)
(F)
z
(P)
z
• Large tU(F) > 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!
We used both LCT & ILCT to derive flows
between pairs of boxcar- averaged m’grams.
 = 15 pix
thr(|Bz|) = 100 G
Poynting fluxes into AR 8210 from ILCT & LCT both
show increasing energy.
Poynting fluxes from ILCT & LCT are correlated.
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
ILCT’s Sz(P) will cause a decrease in Sz(F).
Changes in U(F) derived via ILCT are ~1032erg, and
vary in both sign and magnitude.
Changes in U(F) derived via FLCT are much smaller, and not well
correlated with ILCT.
The cumulative FEFs (  U(F)) do not match;
ILCT shows decreasing U(F), LCT does not.
Conclusions Re: FEF
• Both FLCT & ILCT show an increase in magnetic energy
U, of roughly ~1032 erg and ~5 x 1032 erg, resp.
• FLCT also shows an increase in free energy U(F), of about
~1032 erg over the ~ 6 hr magnetogram sequence.
• ILCT, however, shows a decrease in U(F), of ~4 x1032 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.
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.
Some LCT vectors flip as difference images fluctuate!
Derivation of Poynting-like Flux for B(P)
U  2 dV (B  B ) / 8
(P)
(P)
(P)
 x B  0   x B  B  -  , B   '
(P)
(P)
(P)
(P)
  B (P)  0    B (P)   2   0,  2  '  0
 '
U   dA ( 
) / 4   dA (  B z ) / 4
z
(P)
 t  dA (   h  ( v z B h  v h Bz )) / 4 
U
  dA  h   ( v z B h  v h Bz ) / 4
t
(P)
  dA B h  ( v z B h  v h Bz ) / 4
(P)
Also works w/ non-ideal terms…
NI
U (P)

dA
(


B
z ) / 4

NI
 t  dA (   h x (η J h ) ) / 4 c 
U (P)
NI
t
  dA  h  x (η J h ) / 4 c
  dA B h x (η J h ) / 4 c
(P)
so S
(P),NI
z
Bh  B
 B h x (η J h ); cf ., S  B h x (η J h )
(P)
h
(P)
NI
z
at eqn’s left is valid w/any non-ideal term!