Tests and Comparisons of Photospheric Velocity Estimation Techniques Brian Welsch, George Fisher, Bill Abbett, & Yan Li Space Sciences Laboratory, UC-Berkeley Marc DeRosa Lockheed-Martin Advanced Technology Center M.K. Georgoulis Applied Physics Laboratory, Johns Hopkins University K. Kusano Earth Simulator Ctr., Japan Agency for Marine-Earth Sci. & Tech. D.W. Longcope & B. Ravindra Physics Department, Montana State University P. W. Schuck Naval Research Lab – D. C.

Abstract Estimates of velocities from time series of photospheric vector magnetograms can be used to determine fluxes of magnetic energy (the Poynting flux) and helicity across the magnetogram layer, which have important consequences for understanding solar flares, coronal mass ejections, and the solar dynamo, and to provide time-dependent boundary conditions for data-driven simulations of the coronal magnetic field. Velocity components perpendicular to the magnetic field are necessary both to compute these transport rates and to derive useful boundary conditions. Since Doppler shifts contain contributions from flows parallel to the magnetic field, perpendicular velocities are not generally recoverable from Doppler shifts alone. The magnetic induction equation, however, relates evolution in the magnetic field to the perpendicular flow field, and several methods to estimate perpendicular velocities from magnetic field evolution have recently been developed. These methods have not previously been tested against realistic, simulated data sets, in which both the magnetic and flow fields are exactly known. Here, I present results of such tests, using several velocity inversion techniques applied to synthetic magnetogram data sets, generated from anelastic MHD simulations of the upper convection zone with the ANMHD code. Statistically, most of the methods' estimated flows reproduced properties of the actual flows comparably well, although no technique reproduced the actual flows without significant errors. Estimates of the fluxes of magnetic energy and helicity by the "minimum energy fit" (MEF) method were, however, far more accurate than any other method's. Overall, therefore, the MEF algorithm performed best in our tests. We note that our test data simulate fully relaxed convection in a very high beta plasma (β > 1e+4), and therefore do not realistically model evolution at the β ~ 1 photosphere.

Main Ideas 1. Why should we study surface flows derived from magnetograms? Poynting/Helicity Fluxes, Data Driving, Flux Transport Models 2. What classes of flows can we derive, in principle, from magnetogram sequences? In theory, flows perpendicular to both B &  B z can be found. 3. Practically, what are the best available methods for deriving flow information? Tests suggest tracking (e.g., LCT) combined with MEF. 4. How should available methods be improved? We must develop the capability to match both ∂ t B hor & ∂ t B z.

Photospheric field evolution is often “boring.” AR 8210 magnetograms from MDI over ~24 hr. on 01 May 1998 show the field changing slowly; proper motions are ~1 km/s.

Meanwhile, in the corona… … an M-class flare & halo CME occurred. Typical speeds of CMEs are ~1000 km/s.

This seeming incongruity led to the “storage & release” paradigm. The force-free (J x B = 0) coronal magnetic field is “line- tied” to the photospheric B. The photospheric B field is not force-free. Photospheric flows slowly inject free energy into the coronal field (via, e.g., convective motions & flux emergence), which is stored in currents, J. The high temperature and long length scales of the coronal plasma prevent dissipation of J. At some point, reconnection occurs, allowing dissipation of induced currents and coronal relaxation.

This slow forcing  sudden release process in flares & CMEs closely resembles that in earthquakes. Earthquake terms  Flare terms earthquakes  flares/CMEs fault  (quasi-) separator (elastic) strain energy  free magnetic energy tectonic stress loading  photospheric evolution deformation  magnetic diffusion rupture  fast reconnection mainshock earthquake rupture  flare reconnection fault healing and re-strengthening  diffusivity quenching

“Recurrence Process of a Large Earthquake,” from the Univ. of Tokyo’s Earthquake Prediction Research Center: Large earthquakes repeatedly occur along a large-scale fault, and the entire recurrence process includes the following stages: (I) fault healing and re-strengthening just after the previous earthquake occurred, (II) accumulation of the elastic strain energy with tectonic stress loading, (III) local concentration of deformation and rupture nucleation at the final stage of tectonic stress buildup in which an enough amount of the strain energy has been stored, (IV) mainshock earthquake rupture, and (V) rupture arrest and its aftereffect.

Large earthquakes flares repeatedly occur along a large-scale fault (quasi-) separator, and the entire recurrence process includes the following stages: (I) fault healing and re-strengthening diffusivity quenching just after the previous earthquake flare occurred, (II) accumulation of the elastic strain energy free magnetic energy with tectonic stress loading photospheric evolution, (III) local concentration of deformation magnetic diffusion and rupture fast reconnection nucleation at the final stage of tectonic stress buildup photospheric displacement in which an enough amount of the strain free magnetic energy has been stored, (IV) mainshock earthquake rupture flare reconnection, and (V) rupture reconnection arrest and its aftereffect. “Recurrence Process of Large Solar Flares:”

“Non-potentiality” should imply non-zero free energy, and increased likelihood of flaring. Schrijver et al. (2005) found ARs that matched potential fields (J = 0) were not likely to flare.

“Non-potentiality” should imply non-zero free energy, and increased likelihood of flaring. Schrijver et al. (2005) found that non-potenital ARs were 2.4 times more likely to flare.

Another prediction: the coronal field should evolve toward the potential state. Pevtsov et al. (1996) saw this “sigmoid- to- arcade” evolution. Sigmoids are now widely viewed as signs of non-potentiality (Canfield et al., 1999).

The Punchline: Surface flows, v, affect magnetic evolution at both the Sun’s surface and in the corona. Since E = -(v x B)/c + R, the fluxes of magnetic energy & helicity across the surface depend upon v. ∂ t U = c ∫ dA (E x B) ∙ n / 4π(1) ∂ t H = c ∫ dA (E x A) ∙ n / 4π(2) B in the corona is coupled to B at the surface, so the surface v provides an essential boundary condition for data-driven MHD simulations of the coronal B field. Studying v could improve evolutionary models of photospheric B fields, e.g., flux transport models.

Main Ideas 1. Why should we study surface flows derived from magnetograms? Photospheric evolution affects the corona: Poynting/Helicity Fluxes, Data Driving, Flux Transport Models 2. What classes of flows can we derive, in principle, from normal-field magnetogram sequences? In theory, flows perpendicular to both B &  B z can be found. 3. Practically, what are the best available methods for deriving flow information? Tests suggest tracking (e.g., LCT) combined with MEF. 4. How should available methods be improved? We must develop the capability to match both ∂ t B hor & ∂ t B z.

The Induction Equation relates ∂ t B to v, ∂ t B = - c (  x E) =  x [(v x B) - cR] (3) Resistive effects, R, are model-dependent. Using -  x R  η  2 B would require magnetograms at two heights in the atmosphere – these are rare! R=0 (ideal MHD) is usually assumed. Even ideally, only the normal component of (3) does not require magnetograms from two heights.

Consequently, the goal is to derive v from ∂ t B n =  ∙ (v n B hor - v hor B n ) (4) Clearly, this single ∂ t B n equation can’t determine v, which has three components! By assumtion, v ||, the component of v parallel to B, cannot affect ∂ t B n  (v x B). In fact, v || is not useful for any of the reasons I’ve given for wanting to study surface flows! So ignore v || ! With v = v  + v ||, try to find v , and use v  ∙ B = 0 (5) as an additional equation to solve for v . But the system is still not closed!

Aside: Doppler shifts cannot close the system! Generally, Doppler shifts cannot distinguish flows || to B (red), perp. to B (blue), or in an intermediate direction (gray). With v  estimated another way & projected onto the LOS, the Doppler shift determines v || (Georgoulis & LaBonte, 2006) Doppler shifts are unambiguous where B n changes sign near disk center (Chae et al. 2004, Lites 2005). v LOS

Dopplergrams are sometimes consistent with “siphon flows” moving along the magnetic field. Left: MDI Dopplergram at 19:12 UT on 2003 October 29 superposed with the magnetic neutral line. Right: Evolution of the vertical shear flow speed calculated in the box region of the left panel. The two vertical dashed lines mark the beginning and end of the X10 flare. (From Deng et al. 2006)

Aside #2: In addition to v ||, other “null flows,” for which ∂ t B n = 0, lie in the null space of (4). Any flow obeying (v n B hor - v hor B n ) =  x f n, for some scalar function f, lies in the null space of (4). For instance, “contour flows” -- flows along contours of B n -- do not alter B n, but correspond to twisting motions. Hence, they inject energy and helicity into the corona! No method based upon ∂ t B n alone will estimate null flows well. Happily, magnetograms usually have sufficient spatial structure that the null space of (4) is expected to be small.

Main Ideas 1. Why should we study surface flows derived from magnetograms? Poynting/Helicity Fluxes, Data Driving, Flux Transport Models 2. What classes of flows can we derive, in principle, from normal-field magnetogram sequences? Flows perpendicular B, that lie outside the null space of  t B n. 3. Practically, what are the best available methods for deriving flow information? Tests suggest tracking (e.g., LCT) combined with MEF. 4. How should available methods be improved? We must develop the capability to match both ∂ t B hor & ∂ t B z.

Several approaches have been used with ∂ t B n and v  ∙ B = 0 to solve for v . Kusano et al. (2002), Welsch et al. (2004), and Schuck (2006) used tracking methods to close the system. Longcope (2004) minimized ∫ dA (v n 2 + v hor 2 ). Georgoulis & LaBonte (2006) assumed v n = 0. Tracking methods, which follow the apparent motions of magnetogram features, have the longest heritage.

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

Aside #3: Conformal mappings should be used when projecting magnetograms for tracking! Apparent motions in the plane-of-sky do not accurately represent surface flows. If spherical data are to be projected onto a plane prior to tracking, conformal (shape-preserving) mappings should be used. Equal-area (authalic) projections distort shapes, and will therefore distort the directions of estimated flows. We currently use Mercator projections, which distort length scales, but in a way that is easy to correct.

Local correlation tracking (LCT) finds v(x 1,x 2 ) by correlating subregions; it assumes advection. 1) for ea. (x i, y i ) above |B| threshold … 2) apply Gaussian mask at (x i, y i ) … 3) truncate and cross-correlate… * 4) v(x i, y i ) is esti- mated max. of correlation funct = = =

Can white light data provide information about the flux transport velocity? Unclear… Tracking MDI full- disk white light images & mag- netograms gives different results. Data are from AR 8210 on 01 May ’98; Δt = 96 min.

We created “synthetic magnetograms” from ANMHD simulations of rising flux rope. In these data, both v & B are known exactly. Recently, we conducted quantitative tests & comparisons of several available methods.

Via several methods, we estimated v from N = 7 pairs of magnetograms, with increasing Δt’s. I will show representative results from the shortest Δt, from a subset of the methods tested: 1. Fourier LCT (FLCT, Welsch et al. 2004) 2. Inductive LCT (ILCT, Welsch et al. 2004) 3. Minimum Energy Fit (MEF, Longcope 2004) 4. Differential Affine Velocity Estimator (DAVE, Schuck 2006)

First, we verified that the ANMHD data were consistent with ∂ t B n =  ∙ (v n B hor - v hor B n ).

Here are MEF’s estimated v’s plotted over ANMHD’s v. Like MEF, many methods have problems near edges of flux B n.

We verified that the estimated v’s also obeyed ∂ t B n =  ∙ (v n B hor – v hor B n ). FLCT MEF DAVE ILCT

We tested Démoulin & Berger’s relation of u f to v. Estimated u f ’s are highly correlated with ANMHD’s u f. DAVE MEF FLCT ILCT

Then ∂ t B n = -  ∙ (u f B n ) =  2 φ, so u f (I) B n = -  φ Any given ∂ t B n determines the “inductive” part (!) of u f B n. We can use a Helmholtz decomposition to write u f B n = v hor B n - v n B hor = -  φ +  x ψ n. (5)

Using v  ∙ B = 0 and u f  v hor - (v n /B n )B hor, we converted u f ’s to v’s. Estimated v’s are highly correlated with ANMHD’s v. DAVE MEF FLCT ILCT

Not surprisingly, the methods’ performance worsened as the time between magnetograms increased. % vector errors (direction & magnitude) were at least 50% (!!!). % speed errors (magnitude) were smaller, but biases were seen.

Some methods estimated the direction of v to within ~30º, on average. C VEC and C CS were as defined by Schrijver et al. (2005):

How well do we estimate the electric field E in the fluxes of magnetic energy and helicity? DAVE MEF FLCT ILCT

While most methods’ E is highly correlated with ANMHD’s E, only MEF gets the energy flux right. DAVE MEF FLCT ILCT Unfortunately, even MEF exhibits a large amount of scatter.

As with the energy flux, only MEF accurately estimates the helicity flux.

Conclusions from Tests Tests using MHD magnetograms, in which B is error-free, show that estimated v’s are correlated with true v’s, but are inaccurate. Only MEF reproduced energy and helicity fluxes well; other methods erred by > 50%. Other tests (Ravindra & Longcope, 2007, in prep.) show that combining MEF with tracking does enhances performance.

Main Ideas 1. Why should we study surface flows derived from magnetograms? Poynting/Helicity Fluxes, Data Driving, Flux Transport Models 2. What classes of flows can we derive, in principle, from normal-field magnetogram sequences? Flows perpendicular B, that lie outside the null space of  t B n. 3. Practically, what are the best available methods for deriving flow information? Tests suggest tracking (e.g., LCT) combined with MEF. 4. How should available methods be improved? We must develop the capability to match both ∂ t B hor & ∂ t B z.

Extant methods find v’s that match  t B n, but we should also use the observed  t B hor ! The Vector Magnetogram Fitting (VMF) technique developed by McClymont, Jiao, & Mikic (1997) derives electric fields consistent with changes  t B hor, but with  t B n = 0. A combined approach, using tracking & VMF, would incorporate  t B hor into estimates of v.

Main Ideas 1. Why should we study surface flows derived from magnetograms? Poynting/Helicity Fluxes, Data Driving, Flux Transport Models 2. What classes of flows can we derive, in principle, from normal-field magnetogram sequences? Flows perpendicular B, that lie outside the null space of  t B n. 3. Practically, what are the best available methods for deriving flow information? Tests suggest tracking (e.g., LCT) combined with MEF. 4. How should available methods be improved? Both ∂ t B hor & ∂ t B n should be used, to increase accuracy.

Future efforts: Coming soon to a review panel near you! 1) Similar tests with RADMHD sim- ulations (Abbett 2007), which model photospheric evolution more realistically. 1a) Also, test susceptiblity to noise in magnetograms. 2) In addition, run coordinated velocity studies with SOT/Hinode and HMI/SDO magnetograms.