Estimating Free Magnetic Energy from an HMI Magnetogram by Brian T. Welsch Space Sciences Lab, UC-Berkeley Several methods have been proposed to estimate coronal free magnetic energy, U F, from magnetograms. Generally, each approach has significant shortcomings. Here, I present a half-baked idea to make a crude estimate, essentially using a dirty trick.

60 Sec. Review: Several methods have been used to estimate free energy, which powers flares & CMEs. [Extrap] Potential field, B (P) : actually assumes *no* free energy! – Still good for order-of-magnitude estimate (used in Emslie et al. 2012). – Viable for limb events. [Extrap] Linear, Force-Free Field (LFFF) from observed photosph. vector B (O) ; – currents extend to ∞, so energy = ∞ [Extrap] Non-Linear, Force-Free Field (NLFFF): – localized /finite free energy, but inconsistent with observed forces in photosph. Field – no data at limb; imprecise/ wrong in tests (Schrijver et al.) [Inject] Integrate Poynting flux: – initial energy unknown, so needs photosph. B (O) (t) for long ∆t; – no data at limb; imprecise/wrong in tests (Welsch et al. 2007) [Extrap+Inject] Evolve an initial “guess” for B(x,y,z,0) in time, using B (O) (x,y,0,t) – Difficult (and expensive) to do with MHD model – Can use “magnetofrictional” model, but dynamics are unphysical

Cheung & DeRosa (2012) have been running magnetogram-driven coronal models: inductive evolution mimics coronal memory.

Recently, we have been collaborating w/ Mark & Marc to supply photosph. electric fields to drive their code. Follows van Ballegooijen, Priest & Mackay (2000): vect. pot. A is evolved via ∂A/∂t = v × B – ηJ – guarantees ∇ ∙B = 0 ; relative helicity easy to calculate – Uses explicit 2nd-order time derivatives, – spatial discretization on a Yee (1966) grid By Faraday’s law, ∂B (O) /∂t at lower boundary determines ∇ × cE = - ∂A/∂t – Masha discussed deriving cE from ∂B (O) /∂t (see also Fisher et al. 2011, 2012); note: this specifies ∇ ∙ E, i.e., gauge! Energy in model arises from Poynting flux, S z =c(E × B (O) )/4π on bottom boundary (slide content courtesy G. Fisher et al.)

For AR 11158, the model field opened at the same time in the model sequence as in the observations. AR was on disk from c Feb. 10 – 19 Model ran from Feb. 13 at 00:00 to Feb. 15 at 24:00 An X2.2 flare occurred on Feb. 15 at 01:45 Coincidence in time was probably due to flare- induced effects on HMI fields -- the model field was unstable to perturbation!

Hypothetical Evolution: Drive coronal model from init. pot. B (P), using E at model base, to match observed B (O) (x,y,0). Initial field has no free energy. Electric field E drives model’s photospheric B(x,y,0,t’) toward observed B (O) (x,y,0) supplies Poynting flux – This differs from Masha’s estimate of Poynting flux, which is derived from actual photospheric evolution Evolution ceases when B at model’s bottom boundary matches B (O) (x,y) observed at photosphere. Mikic & McClymont (1994) did this with an MHD code, and called it the “Evolutionary Method” Valori, Kliem, and Fuhrmann (2007) used a “magnetofrictional” code for this

Trick: Forget the coronal model! Just sum the Poynting flux implied by E needed to evolve B (P) (x,y,0) --> B (O) (x,y). Create a fictitious sequence of magnetogram fields, { B (P) (x,y), B 1 (x,y), B 2 (x,y), …, B i (x,y), … B (O) (x,y)} E that will evolve B i (x,y) to B i+1 (x,y) can then be estimated. Poynting flux can then be computed from ( E x B ) This approach requires only one magnetogram! – It also does not assume the photospheric field is force-free.

But it doesn’t work well: In tests with a known field (Low & Lou 1990), this approach only gets 1/6 th of free energy. Problem: the coronal field will “absorb” some of the imposed twist. Hence, to actually change model photospheric field, E must be applied for longer. This implies the Poynting flux is underestimated. The underestimate probably scales as ∆x/L, where ∆x is pix. size, and L is length scale of the coronal current system.

Aside: With real data, the estimated free energy is too small --- of order ∼ erg, too small for a big CME.

Mismatch in twist between interior and corona implies twist will propagate between the two. This is what my approach does. Longcope & Welsch 2000 From sketch by Parker 1987 Corona Photosphere Recognizing this, McClymont et al. (1997) drive model in proportion to the discrepancy between model and observation. ---->

Conclusion: You (probably) can’t cheat --- you’ve actually got to do the coronal modeling. This is bad news for lazy people like me. But I still hold out (delusional?) hope that some similar “cheat” can exist.

Summary Available techniques for estimating magnetic free energy are lousy. - Assumptions that are unphysical, or in conflict with data are made. One promising approach is data-driven, time-dependent modeling of coronal fields. - This requires substantial effort by personnel and supercomputer time. A much simpler --- probably flawed! --- approach is to compute the Poynting flux for a hypothetical set of E fields. - These would evolve a potential magnetogram to the observed field. 12