New Directions for Improving Electric Field Estimates Derived from Magnetograms

1. New Directions for Improving Electric Field Estimates Derived from Magnetograms Brian T. Welsch Space Sciences Lab, UC-Berkeley Via Faraday's law, sequences of photospheric vector magnetograms can be used to derive electric fields in the atmospheric layer imaged by the magnetograph. These electric field estimates have applications for space weather prediction: they determine the Poynting flux of energy across the area imaged by the magnetogam, and can be used to drive time-dependent models of the coronal magnetic field. Tests of current approaches for estimating electric fields using artificial data from MHD simulations of photospheric magnetic evolution reveal that electric field estimation methods are imperfect in several respects; most notably, the estimated electric fields underestimated the fluxes of magnetic energy and helicity in some circumstances. Here, I will outline some speculative approaches that might improve the accuracy of electric field estimates.

2. Motivations:Photospheric electric fields Eph can quantify aspects of evolution in Bcor. • The fluxes of magnetic energy & helicityacross the magnetogram surface into the corona depend upon Eph: dU/dt = ∫ dA (Eph x Bph)z /4π(1) dH/dt = 2 ∫ dA (Ephx Aph)z (2) U and H probably play central roles in flares / CMEs. • Coupling of Bcor to Bph also implies that Eph provides boundary conditions for data-driven,time-dependent simulations of Bcor.

3. Faraday’s law implies magnetogram sequences can be used to derive an electric field E from tBph. “Component methods” derive v or Eh from the normal component of the ideal induction equation, Bz/t = -c[ h x Eh ]z= [  x (v x B) ]z; and assume the ideal Ohm’s law*to getEzfromEh: E = -(v x B)/c ==>E·B = 0 The PTD method uses the vectorinduction equation, B/t = -c( x E), to determine EzandEhindependently: • tJz is used in addition to tBz to determine E • Tracking is not needed to get E, but v from tracking can better determine E! • For details of PTD, see Fisher et al. 2010 (ApJ 715, 242) *One can instead use E = -(v x B)/c + R, if some known resistivity R is assumed.

4. Problem: tBph alone does not fully constrain Eph. “Uncurling” tBph = -c( x Eph) only determines the “inductive” electric field, Eind. tB doesn’t constrain any “non-inductive” component of E--- both Eind and (Eind- ψ) are consistent with tB! Since PTD only uses tB to derive Eind, the ideal Ohm’s law (Etot·B = 0) can be enforced by solving (Eind- ψOhm)·B = 0. NB: Even this doesn’t fully constrain E: any ψfor which ψ·B= 0added to E won’t affect consistency with tB

5. Progress thus far: What approaches for better constraints on ψhave already been investigated? • NB: The non-inductive part of E is very important! • The inductive PTD field Eind, by itself, does not closely match the actual Ein test cases using MHD. • Currently, we include the following terms in Etot: The second and third right-side terms represent non-inductive contributions from Doppler shifts and pattern motions (derived from e.g. FLCT or DAVE), respectively, from which the inductive contributions have been removed. Additional constraints can be imposed, represented by the fourth term. The fifth term, imposed as a final step, enforces the condition EtotB= 0. See Fisher et al. 2012, Sol. Phys. 277, p153 for more details.

6. Goal here: Consider two ways that might better constrain ψ. • Timescale Matching: In active region fields, t in B/t inmagnetograms should match the time scale of evolution due to flows 2. Flux Emergence Constraints: Ignoring measurement errors, changes in flux in any unipolar patch constrain E around that patch’s boundary

7. (1) First, the time scales of magnetic evolution do not necessarily match that of the flow field. • Assuming Bph evolves ideally (e.g., Parker 1984), tracking methods can be used to constrain Eph. • Démoulin& Berger 2003: tracking determines the velocity u of photospheric “footpoints” of the coronal magnetic field • Many tracking (“optical flow”) methods to estimate u have been developed, e.g., • LCT (November & Simon 1988), FLCT (Fisher & Welsch 2008) • DAVE (Schuck 2006), DAVE4VM (Schuck 2008) • The flow u estimated by tracking constrains the non-inductive electric field hψ, sinceh2 ψ= [h x u Bz]z

8. In data, the fastest flows are the shortest-lived, and therefore don’t affect B as strongly as longer-lived flows. Slow flows live longer, so affect B more: Footpoint displacement x from flow of speed v0 with lifetime τ is x = v0τ τ ∝(1/v02), so x ~ 1/v0 ==> faster flows tend to contribute less Fitted lifetime (assuming exponential decay) vs. flow speed averaged over space and time for each choice of Δt, spatial binning, and σ. Lifetimes for ux and uyare plotted with +’s and x’s, respectively. Generally, higher average speeds correspond to shorter lifetimes. While a range of lifetimes exists at each average speed, there appears to be an upper limit at a given average speed, with the peak fitted lifetime scaling roughly as (average speed)−2 (dashed line; note that this line is not a fit). Points are color-coded by spatial scale of the flow (binning x σ). NB: speeds from tracking Bz are usu. smaller than those from tracking intensities. Fig. is from Welsch et al., ApJ v.747, p.130 2012

9. Here’s a simplistic illustration of the different contributions to a typical photospheric velocity vector: Faster flows are probably the most easily detected: higher S/N. This dominant component of the instantaneous v, however, is likely short-lived. The fastest flows are also probably the most turbulent, meaning their net effect over many flow turnover times is essentially produces a random walk. (NB: diffusion results in a random walk, but turbulent flux dispersal is not the same as diffusion!)

10. In realistic simulations, B/t =  x (v x B) is only approximately obeyed. In MURaM simulations (by M. Cheung), with t = 53 seconds, the code’s actual Bz/t is at best only approximately consistent with [  x (v x B) ]z, using the code’s (known) v. (At the code’s native time step, t = 0.53 seconds, Bz/t =[  x (v x B) ]z is satisfied.) Evidently, flows are only inductive on the shortest spatial scales --- much shorter than realistic magnetogram cadences.

11. A few implications arise from these considerations, which generally imply knowing the instantaneous v isn’t very useful. (i) B/t doesn’t reflect instantaneous v, so can’t be used to infer v. • Essentially, flow evolution leads to entropy, so not all info about v can be reconstructed from B/t. (ii) Conversely, the instantaneous vdoes’tgovern B/t. • So don’t sweat it if you don’t know the instantaneous v! What’s probably needed are ways to relate ⟨B/t⟩ to ⟨v⟩ or ⟨E⟩, where ⟨…⟩ represent averaging over space + time --- à la mean-field electrodynamics! From this, ⟨dU/dt⟩ and ⟨dH/dt⟩ can then be estimated. Approaches to this problem are currently being investigated.

12. 2. Radial Flux Constraints: changes in radial flux require a horizontal Ealong the polarity inversion line (PIL), which can constrain Eph. Vertical transport of horizontal flux at PIL. Changes in radial flux of each polarity, ΔΦ/Δt ≠ 0 • Note: This does not directly use Doppler velocities.

13. Important magnetodynamics is not always apparent in ΔBz/Δtat PIL -- e.g., flux emergence! This sketch of flux emergence in a bipolar magnetic region shows the emerging field viewed in cross-section normal to the polarity inversion line (PIL). Note the strong signature of the field change at the edges of the region, while the change in photospheric field at the PIL is zero.

14. Neither PTD nor tracking methods “automatically” incorpo-rate the constraint on Eh at the PIL due to flux emergence. Consider three hypothetical magnetogram sequences : The left two cases, what do tracking or PTD methods infer? - Presumably, a diverging flow in each case. - But these two cases violate conservation of flux! The right-most case is a superposition; linearity suggests tracking / PTD won’t get the correct Eph in this case, either. ti tf

15. Incorporate information about Eh from changes in flux ΔΦ/Δtis not straightforward, however. • Feature tracking (see, e.g., Welsch et al. 2011) can be used to define unipolar regions, and quantify changes in flux in each • PILs at the periphery of such features can be identified (see, e.g., Welsch & Fisher 2012) • Eh along each PIL is unknown, but an Ansatz of uniform Eh is plausible, which sets a Dirichlet boundary condition along the PIL • From this, scalar potential ξ(x,y) can be computed, and a non-inductive electric field Eξ(x,y) = -ξ can be derived • Eξcan be added to the inductive electric field in the neighborhood of the radial-field PIL Approaches to this problem are currently being investigated.

16. Summary • Estimatingphotospheric electric fields from tBph shows promise for modeling the buildup of energy & helicity prior to large flares and CMEs. Techniques to do this are still being developed and tested. We describe two aspects of this problem that merit further investigation: • (1) Matching timescales of magnetic evolution with velocity / electric fields. • (2) Using changes in radial flux to impose constraints on electric fields

17. ASIDE p.1: A distinct concept is to use Doppler shifts along PILs of the LOS (cf., radial) field BLOSto constrain Eph. Measurements of vDopp and Btrans on LOS PILs are direct observations of the ideal E perpendicular to both. This concept was introduced by Fisher et al. (2012), Sol. Phys. 277 153. It’s been further developed by Kazachenko et al. --- see Masha’s poster at this meeting!

18. ASIDE p.2: But there’s a problem with using HMI data for this technique: the convective blueshift! Because rising plasma is (1) brighter (it’s hotter), and (2) occupies more area, there’s an intensity-blueshift correlation (talk to P. Scherrer!) S. Couvidat: line center for HMI is derived from the median of Doppler velocities in the central 90% of the solar disk --- hence, this bias is present! Punchline: HMI Doppler shifts are not absolutely calibrated! (Helioseismology uses time evolution of Doppler shifts, doesn’t need calibration.) Line “bisector” From Dravins et al. (1981)

19. ASIDE p.3: Because magnetic fields suppress convection, magnetized regions have pseudo-redshifts, as on these PILs. Here, an automated method (Welsch & Li 2008) identified PILs in a subregion of AR 11117, color-coded by Doppler shift.

20. ASIDE p.4: Welsch & Fisher (2012) estimated this offset velocity v0, enabling correction of HMI Doppler velocities. • The method enforces consistency between changes in LOS flux and the rate of transport of transverse magnetic field inferred from Doppler velocities on LOS PILs. • Rate of change of LOS flux: • Rate of transport of transverse flux, w/bias v0: • The bias is estimated by equating ΔΦLOS/Δt = 2ΔΦPIL/Δt. • See Welsch & Fisher (2012), arXiv:1201.2451 for details.

21. ASIDE p.5: Note that radial-field and LOS PILs are not co-spatial --- more so as one moves off disk center. • Hence, constraints from vDoppler and Btransverse on LOS PILs are independent of constraints from vradial and Bhorizontal on radial-field PILs! • Magnetograms courtesy K. D. Leka • Left: White line shows LOS PIL. Right: White line shows radial-field PIL.