Estimating Free Magnetic Energy from an HMI Magnetogram

by Brian T. WelschSpace Sciences Lab, UC-Berkeley

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.

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,

Sz=c(E × B(O))/4π on bottom boundary (slid

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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!

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), B1(x,y), B2(x,y), … , Bi(x,y), … B(O)(x,y)}

• E that will evolve Bi(x,y) to Bi+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 1/6th 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, and L is length scale of the coronal current system.

Aside: With real data, the estimated free energy is too small --- of order ∼1031 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.

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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