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.

New Directions for Improving Electric Field Estimates Derived from Magnetograms SHINE Session 14: Data driven MHD modeling of CME events Brian T. Welsch, Space Sciences Lab, UC-Berkeley ΔΦ LOS /Δt EhEh Estimates of photospheric electric field E ph can be used to model pre-CME energy & helicity buildup. In principle, E = -(v x B)/c can be derived from  B/  t in magnetogram sequences via Faraday’s law. This will require finding length and time scales of flows that most strongly govern  B/  t. Treating flux emergence / submergence will require explicit use of proper boundary conditions.

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

“Component methods” derive v or E h from the normal component of the ideal induction equation,  B z /  t = -c[  h x E h ] z = [  x (v x B) ] z ; and assume the ideal Ohm’s law* to get E z from E h : E = -(v x B)/c ==> E·B = 0 The PTD method uses the vector induction equation,  B/  t = -c(  x E), to determine E z and E h independently:  t J z is used in addition to  t B z 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 (ApJ 715, 242) *One can instead use E = -(v x B)/c + R, if some known resistivity R is assumed. Faraday’s law implies magnetogram sequences can be used to derive an electric field E from  t B ph.

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

Progress thus far: What approaches for better constraints on  ψ have already been investigated? NB: The non-inductive part of E is very important! – The inductive PTD field E ind, by itself, does not closely match the actual E in test cases using MHD. Currently, we include the following terms in E tot : 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 E tot  B = 0. See Fisher et al. 2012, Sol. Phys. 277, p153 for more details.

Goal here: Consider two ways that might better constrain  ψ. 1.Timescale Matching: In active region fields,  t in  B/  t in magnetograms 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

8 (1) First, the time scales of magnetic evolution do not necessarily match that of the flow field. Assuming B ph evolves ideally (e.g., Parker 1984), tracking methods can be used to constrain E ph. 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  h ψ, since  h 2 ψ = [  h x u B z ] z

The apparent motion of magnetic flux in magnetograms is the flux transport velocity, u. u is not equivalent to v; rather, u  v hor - (v z /B z )B hor u is the apparent velocity (2 components) v is the actual plasma velocity (3 components) (NB: non-ideal effects can also cause flux transport!) Démoulin & Berger (2003): In addition to horizontal flows, vertical velocities can lead to u ≠0. In this figure, v hor = 0, but v z ≠ 0, so u ≠ 0. hor z z

The apparent motion of magnetic flux in magnetograms is the flux transport velocity, u. u is not equivalent to v; rather, u  v hor - (v z /B z )B hor u is the apparent velocity (2 components) v perp is the perpendicular plasma velocity (2 comps) (NB: non-ideal effects can also cause flux transport!) Démoulin & Berger (2003) didn’t use the fact that only the components of v perpendicular to B can change B. Hence, one can ignore the comp. of v along B. hor z v perp

In data, the fastest flows are the shortest-lived, and therefore don’t affect B as strongly as longer-lived flows. Fitted lifetime (assuming exponential decay) vs. flow speed averaged over space and time for each choice of Δt, spatial binning, and σ. Lifetimes for u x and u y are 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 B z are usu. smaller than those from tracking intensities. Fig. is from Welsch et al., ApJ v.747, p Slow flows live longer, so affect B more: Footpoint displacement  x from flow of speed v 0 with lifetime τ is  x = v 0 τ τ ∝ (1/v 0 2 ), so  x ~ 1/v 0 ==> faster flows tend to contribute less

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!)

Inferred speed decreases with increasing Δt. Flow speed as a function of cadence, averaged over both space and time for each spatial resolution (binning x σ) and over spatial resolutions at each t. Error bars show standard deviation over resolutions. Least squares linear fit to the logarithms of cadence and speeds returns a power-law exponent of

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

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 v does’t govern  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.

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

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. Important magnetodynamics is not always apparent in ΔB z /Δt at PIL -- e.g., flux emergence!

Neither PTD nor tracking methods “automatically” incorpo- rate the constraint on E h at the PIL due to flux emergence. 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 E ph in this case, either. Consider three hypothetical magnetogram sequences : titftitf

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

Summary Estimating photospheric electric fields from  t B ph 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

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

Aside: Flows v || along B do not contribute to E = -(v x B)/c, but do “contaminate” Doppler measurements. v LOS v v v =

Aside: Dopplergrams are sometimes consistent with “siphon flows” moving along B. MDI Dopplergram at 19:12 UT on 2003 October 29 superposed with the magnetic polarity inversion line. (From Deng et al. 2006) Why should a polarity inversion line (PIL) also be a velocity inversion line (VIL)? One plausible explanation is siphon flows arching over (or ducking under) the PIL. What’s the DC Doppler shift along this PIL? Is flux emerging or submerging?

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.) From Dravins et al. (1981) Line “bisector”

Because magnetic fields suppress convection, there are pseudo-redshifts in magnetized regions. Will this effect bias HMI measurements of Doppler velocities along PILs in active regions?

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.

Ideally, the change in LOS flux ΔΦ LOS /Δt should equal twice the flux change ΔΦ PIL /Δt from vertical flows transporting B h across the PIL (black dashed line). NB: The analysis here applies only near disk center! ΔΦ LOS /Δt ΔΦ PIL /Δt

The pseudo-redshift bias is evident in scatter plots of Doppler shift vs. |B LOS |. I find pseudo-redshifts of ~0.15 m/s/G. Schuck (2010) reported a similar trend in MDI data. 28

Schuck (2010) also found the pseudo-redshift bias in MDI data. Schuck’s trend of redshift with|B LOS |is also roughly ~0.2 m/s/G. 29

Scatter plots of Doppler shift vs. line depth show the pseudo-redshift, clear evidence of bias from the convective blueshift. 30 Dark regions correspond to low DN/s in maps of line depth. PIL pixels (shown here in blue) for the most part appear redshifted.

Changes in LOS flux are quantitatively related to PIL Doppler shifts multiplied by transverse field strengths. From Faraday’s law, Since flux can only emerge or submerge at a PIL, From LOS m’gram: Summed Dopplergram and transverse field along PIL pixels. (Eqn. 2) In the absence of errors, ΔΦ LOS /Δt = 2ΔΦ PIL /Δt. (Eqn. 1)

We can use this constraint to calibrate the bias in the velocity zero point, v 0, in observed Doppler shifts! A bias velocity v 0 implies := “magnetic length” of PIL But ΔΦ LOS /2 should match ΔΦ PIL, so we can solve for v 0 : (Eqn. 3) NB: v 0 should be the SAME for ALL PILs ==> solve statistically!

Aside: How long do Doppler flows persist? Some flow structures persist for days, e.g. the Evershed flow (outflow around sunspots). Generally, however, the spatial structure of Doppler flows decorrelates over about two 12-minute HMI sampling intervals. 33

In sample HMI Data, we solved for v 0 using dozens of PILs from several successive magnetograms in AR Error bars on v 0 were computed assuming uncertainties of ± 20 G on B LOS, ± 70G on B trs, and ± 20 m/s on v Dopp. v 0 ± σ = 266 ± 46 m/s v 0 ± σ = 293 ± 41 m/s v 0 ± σ = 320 ± 44 m/s

ASIDE p.4: Welsch & Fisher (2012) estimated this offset velocity v 0, enabling correction of HMI Doppler velocities. See Welsch & Fisher (2012), arXiv: for details. 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 v 0 : The bias is estimated by equating ΔΦ LOS /Δt = 2ΔΦ PIL /Δt.

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

Why is there a range of bias velocities? Noise! Upper left: Histogram of B LOS, consistent with noise of ~20G. Upper right: Hist. of B trans, consistent with noise of ~20G over a mean field of ~70G. Lower left: Histogram of azimuths: flat = OK! Lower right: Hists. of v LOS from filtergrams (red) and fit to ME inversion of line profile (aqua). 37 In addtion, there are large systematic errors in identifying PILs.

How do bias velocities vary in time, and with parameter choices? - Frame-to-frame correlation implies consistency in the presence of noise. - Agreement w/varying parameter choices implies robustness in method. - Longer-term variation implies a wandering zero-point! 38 - The two main params are PIL “dilation” d and threshold |B LOS |. - black: d=5, |B LOS |= 60G; red: d=3, |B LOS |= 60G; blue: d=5, |B LOS |= 100G

How do bias velocities vary in time, and with parameter choices? The radial component of SDO’s orbital velocity (dashed line) varies on a similar time scale The two main params are PIL “dilation” d and threshold |B LOS |. - black: d=5, |B LOS |= 60G; red: d=3, |B LOS |= 60G; blue: d=5, |B LOS |= 100G

The values we find for the convective blueshift agree with expectations from line bisector studies. Asplund & Collet (2003) used radiative MHD simulations to investigate bisectors in Fe I lines similar to HMI’s 6173 Å line, and found convective blueshifts of a few hundred m/s. From Gray (2009): Solar lines formed deeper in the atmosphere, where convective upflows are present, are blue-shifted. Dots indicate the lowest point on the bisectors.

What if PIL electric fields don’t match LOS flux loss? Possible evidence for non-ideal evolution. Kubo, Low, & Lites (2010) find some cancellations without horizontal field as in top row. “Normal” cancellation is more like bottom row. 41

If electric fields along some PILs are non-ideal, can we estimate an effective magnetic diffusivity? Linker et al. (2003) and Amari et al. (2003a,b, 2010) use non-ideal cancellation to form erupting flux ropes. Also, Pariat et al. (2004) argue that flux emergence is non-ideal. (But it’s probably just that my error bars are too small!) 42 For instance, what’s up with these PILs ?

Pariat et al. (2004), Resistive Emergence of Undulatory Flux Tubes: “These findings suggest that arch filament systems and coronal loops do not result from the smooth emergence of large-scale Ω -loops from below the photosphere, but rather from the rise of undulatory flux tubes whose upper parts emerge because of the Parker instability and whose dipped lower parts emerge because of magnetic reconnection. Ellerman Bombs are then the signature of this resistive emergence of undulatory flux tubes.”

Aside: Doppler velocities probably can’t be calibrated by fitting the center-to-limb variation. Snodgrass (1984), Hathaway (1992, 2002), and Schuck (2010) fitted center-to-limb Doppler velocities. But such fits only yield the difference in Doppler shift between the center and the limb; they don’t fit any “DC” bias! 44 Toward the limb, horizontal components of granular flows contribute to Doppler shifts. But the shape and optical thickness of granules imply receding flows will be obscured. Hence, it’s likely that there’s also a blueshift toward the limb!

Conclusions, #2 We have demonstrated a method to correct the bias velocity v 0 in HMI’s Doppler velocities from convective-blueshifts. Discrepancies between ΔΦ LOS /Δt & ΔΦ PIL /Δt in blueshift-corrected data can arise from departures from ideality --- e.g., Pariat et al Hence, the method can also be used to identify the effects of magnetic diffusivity.

Scales matter: the fluxes of energy & helicity that underlie CMEs & large flares are probably long-lived and large-scale. CMEs/large flares do not recur rapidly in a given region – Typical frequencies are c. 1/day (and usu. much less) – Small-scale (granular) flows only live for a few minutes ==> Small-scale flows probably don’t play a major role in CMEs and large flares CMEs/large flares are active-region-scale phenomena – Typical length scales are a few 10s of Mm – Granular phenomena have a scale length of ~1 Mm ==> Granular phenomena probably don’t play a major role in CMEs / large flares

The ideal induction equation is:  t B x = (-  y E z +  z E y )c =  y (v x B y - v y B x ) -  z (v z B x - v x B y )  t B y = (-  z E x +  x E z )c =  z (v y B z - v z B y ) -  x (v x B y - v y B x )  t B z = (-  x E y +  y E x )c =  x (v z B x - v x B z ) -  y (v y B z - v z B y )