Estimating Free Magnetic Energy from an HMI Magnetogram

1. Estimating Free Magnetic Energy from an HMI Magnetogram Several methods have been proposed to estimate coronal free magnetic energy, UF, 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. by Brian T. Welsch Space Sciences Lab, UC-Berkeley

2. 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

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

4. 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.Ais evolved via ∂A/∂t = v ×B – ηJ • guarantees∇∙B = 0; relative helicityeasy to calculate • Usesexplicit 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 cEfrom ∂B(O)/∂t(see also Fisher et al. 2011, 2012); note: this specifies ∇∙E, i.e., gauge! • Energy in model arises from Poynting flux, Sz=c(E ×B(O))/4πon bottom boundary (slide content courtesy G. Fisher et al.)

5. For AR 11158, the model field opened at the same time in the model sequence as in the observations. AR 11158 was on disk from c. 2011 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!

6. 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 Edrives model’s photosphericB(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 actualphotospheric 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

7. 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), B1(x,y), B2(x,y), … , Bi(x,y), … B(O)(x,y)} • Ethat will evolve Bi(x,y)to Bi+1(x,y) can then be estimated. • Poynting flux can then be computed from (Ex B) • This approach requires only one magnetogram! • It also does not assume the photospheric field is force-free.

8. But it doesn’t work well: In tests with a known field (Low & Lou 1990), this approach only gets 1/6th 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, andL is length scale of the coronal current system.

9. Aside:With real data, the estimated free energy is too small --- of order ∼1031 erg, too small for a big CME.

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

11. 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.

12. 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.