Free Energies via Velocity Estimates B.T. Welsch & G.H. Fisher, Space Sciences Lab, UC Berkeley

In ideal MHD, photospheric flows move mag- netic flux with a flux transport rate, B n u f. Demoulin & Berger (2003): Apparent motion of flux on a surface can arise from horizontal and/or vertical flows. In either case, u f represents “flux transport velocity.” (1)

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,

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: u f is the flux transport velocity from eqn. (1) u f is related to the induction eqn’s z-component, (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) :

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) ). Requires vector magnetograms. Compute from B z. What about v or u? (4)

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.

We use ILCT to modify the FLCT flows, via the induction equation, to match  B z /  t. and the approximation u f  u LCT, solving With with (v·B) = 0, completely specifies (v x, v y, v z ).

Tests with simulated data show that LCT underestimates S z 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. Large  t U (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! (5)

We used both LCT & ILCT to derive flows between pairs of boxcar- averaged m’grams.  = 15 pix thr(|B z |) = 100 G

Poynting fluxes into AR 8210 from ILCT & LCT both show increasing magnetic energy, U. ILCT shows an increase of ~5 x erg; FLCT shows an increase of ~1 x erg

Poynting fluxes from ILCT & LCT are correlated.

Fluxes into the potential field, S z (P), calculated from ILCT & FLCT flows, however, strongly disagree. Recall that S z (F) = S z - S z (P), so the increase seen in ILCT’s S z (P) will cause a decrease in S z (F).

Changes in U (F) derived via ILCT are ~10 31 erg, 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 ~10 31 erg and ~5 x erg, resp. FLCT also shows an increase in free energy U (F), of about ~10 31 erg over the ~ 6 hr magnetogram sequence. ILCT, however, shows a decrease in U (F), of ~4 x10 31 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 Longcope, 2004: Inferring a Photospheric Velocity Field from a Sequence of Vector Magnetograms: The Minimum Energy Fit, ApJ, v. 612, p 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 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 Welsch, 2006: Magnetic Flux Cancellation and Coronal Magnetic Energy, Welsch, B. T, ApJ, v. 638, p

Some LCT vectors flip as difference images fluctuate!

Derivation of Poynting-like Flux for B (P)

Also works w/ non-ideal terms… at eqn’s left is valid w/any non-ideal term!