Free Magnetic Energy: Crude Estimates by Brian Welsch, Space Sciences Lab, UC-Berkeley

The magnetic free energy U F is the difference between U actual and U potential. U actual =  dV B 2 /8 , with V  “the volume of interest” U potential =  dV B 2 potential /8  The normal components of B and B pot match on the boundaries of V. For a given normal boundary field, B pot is unique. U F = U actual - U potential > 0

How can free energies be estimated? A non-exhaustive list: Extrapolation of non-potential field from photospheric vector magnetogram (“mgram”) – MHD: Mikic & McClymont (1994) – NLFFF: Wheatland et al. (2000) Magnetic virial theorem, from either… –chromospheric vector mgram: Metcalf et al. (1995) –“processed” photospheric vector mgram: Wheatland and Metcalf (2006)

An MHD code can be used to evolve an initial B until its bottom bound matches an observed vector mgram. Mikic, McClymont, and collaborators developed this “evolutionary method” Mikic & McClymont (1994): U F ~0.35 U pot Jiao et al. (1997): U F ~0.1 U pot

Schrijver et al. (2006, 2008) and Metcalf et al. (2008) studied NLFFF extrapolations. NLFFF = non-linear force-free field –gravity & pressure forces are neglected –electric current density J is parallel to B,  x B =  ( x ) B Recently, NLFFF methods have been tested 1. Schrijver et al. (2006): test with known, analytic B 2. Metcalf et al. (2008): test with known, numerically constructed B 3.Schrijver et al. (2008): extrapolation from Hinode SOT/SP photospheric vector magnetogram

NLFFF free energies from Hinode vector magnetogram are ~0.2 U pot From Schrijver et al. (2008)  Various Methods  Note scatter in estimates!

Virial Theorem energies probably overestimate the true energy. From Wheatland & Metcalf (2006): “McClymont, Jiao & Mikic (1997) subsequently pointed out that the virial energy estimate (1) is dimensionally of order B 2 L 3 /μ 0, where B is the average field strength and L is the horizontal extent of the active region, whereas the true energy is of order B 2 L 2 H/μ 0, where H < L is the scale height of the field.” Metcalf et al. (1995, 2002, 2005) and Wheatland & Metcalf (2006) found U free ~ (1 – few) x U pot. (1)

So: how to compute free magnetic energies for SADOSC24 events? Many events were at the limb, so no simultaneous mgrams exist. Even for events on the disk, vector mgrams are generally unavailable… –can’t do NLFFF extrapolations –can’t use virial theorem Linear FFF (LFFF) extrapolations can match SXR/EUV images --- but LFFF energies are infinite! PUNT! Compute U pot, and assume U F ~ 0.3 U pot

Get MDI full-disk mgrams with the source AR near disk center, deproject & crop, then take U F ~ 0.3 U pot. See

It turns out that U pot scales roughly linearly with total unsigned active region flux, . (This is probably not surprising to most folks.) Assuming U F  U pot implies that unsigned flux should be a good predictor of flares!

Leka & Barnes (2007) [and Welsch et al., in prep.] find that  *is* highly correlated with flaring. So the ad-hoc, hand-wavy assumption that U F scales with U pot is prob’ly a pretty good one! From Fisher et al., 1998: From Leka and Barnes (2007): “total magnetic flux and total vertical current [are] two of the most powerful predictors” of flare likelihood.