1. Roles for Data Assimilation in Studying Solar Flares and CMEs Brian Welsch, Bill Abbett, and George Fisher, Space Sciences Laboratory, UC Berkeley

2. Only 17 minutes to talk, so here’s a quick summary of background in modeling flare/CME processes! 1. Physically, flares / CMEs are driven by the electric currents in the coronal magnetic field, B C. 2. Observationally, measurements of (vector) B C are not generally available. 3. Practically, B C must therefore be modeled. Which data should be used as input? 4. Physically, coronal magnetic energy originates in the interior, then passes across the photosphere and chromosphere, and upward into the corona. 5. Observationally, photospheric fields can be measured. 6. Hence, many coronal magnetic models are therefore derived from or driven by photospheric boundary data. Examples: A. Non-linear force-free fields (NLFFFs), e.g., Wiegelmann, Thalmann, et al.; DeRosa et al., 2009. B. Time-dependent models, e.g., Abbett & Fisher 2010; Cheung & DeRosa, in prep

3. Coronal magnetism is a manifestation of structures that extend from the interior into the corona. Image credits: George Fisher, LMSAL/TRACE

4. Magnetic energy --- from the interior! --- drives flares and CMEs, as well as coronal heating. From T.G. Forbes, “A Review on the Genesis of Coronal Mass Ejections”, JGR (2000)

5. Sequences of independent, static models (e.g., NLFFF extrapolations) can underestimate coronal energy. Evolutionary techniques used to estimate a static NLFFF typically involve relaxation. Further, most NLFFF extrapolation methods do not preserve topology. – Exception: FLUX code (DeForest & Kankelborg 2007) NLFFF models can therefore approximately satisfy FFF and boundary conditions, but have less energy than actual field. 5 Antiochos, Klimchuk, & DeVore, 1999 Hence, dynamic coronal models show the most promise for deterministic forecasting.

6. Dynamic models have been described using terms with imprecise meanings. One possible classification: Post-buildup: study eruption dynamics without focus on development of eruptive state --- e.g., start with an unstable flux rope, and simulate disruption. (e.g., CCHM eruption generator; Torok & Kliem 2005). Data – inspired: use idealization of observed physical processes to drive eruption, e.g., flux cancellation (e.g., Amari, Aly, Mikic, & Linker 2010) Data – driven: Use photospheric observations to derive time-dependent boundary condition for the coronal field. (e.g., Abbett & Fisher 2010; Cheung & DeRosa, in prep.) Data Assimilation: Incorporate additional data (e.g., coronal field measurements) into a dynamical model within the model domain. Implication: While assimilative models are consistent with data input, they arecapable of running independently of input! 6

7. A post-buildup example: Torok & Kliem (2005) start their simulation with a kink-unstable flux rope. This approach evidently reproduces aspects of an observed failed eruption, associated with a strong flare. But this approach does not illuminate how the kink- unstable magnetic field configuration arose. 7

8. Poloidal-toroidal decomposition (PTD) can be used to derive electric fields for driving coronal models. B =  x (  x B z) +  x J z B z = -  h 2 B, 4 π J z /c =  h 2 J,  h ·B h =  h 2 (  z B ) Left: the full vector field B in AR 8210. Right: the part of B h due only to J z. ^^ See Fisher et al. 2010  t B =  x (  x  t B z) +  x  t J z -cE PTD =  x  t B z +  t J z cE TOT = cE PTD +  ψ ^^ ^ ^

9. Cheung & DeRosa ran a data-driven magnetofrictional model using  t B ; but also imposed an ad-hoc vorticity of ω = 1/4 turn per day. Orange ∝ L -1 ∫j 2 dl Lavender ∝ L -1 (B/B base )(∫j 2 dl) Top view y side view x side view Problem: can’t model forces in “lift-offs” – this model is, in fact, static!

10. The RADMHD model (Abbett 2007) can model dynamics in the upper layers of the convection zone to corona in a single domain. LEFT: Magnetic field lines initiated from a set of points located in the model chromosphere. The grayscale intensity on the horizontal slice representing the photosphere denotes the magnitude of vertical velocity along this layer. RIGHT: Magnetic field lines initiated from equidistant points along a horizontal line positioned near the upper boundary of the model corona. The image illustrates how magnetic flux entrained in overturning flows and strong convective downdrafts can be pushed below the surface. The horizontal slice denotes the approximate position of the photosphere, and grayscale contours of vertically directed flows (dark shades indicate downflows, while light shades indicate upflows) are displayed along the slice. INSET IMAGES: A timeseries (over ~ 5 minutes) of the magnetic flux penetrating a small portion of the model photosphere. This sub-domain is centered on the location featured in the background image where magnetic flux is being advected below the surface. 10

11. Assuming photospheric evolution is ideal, cE = -u x B, so u derived from estimates of E can be used to drive RADMHD. F sim ≡ – ∇ ·[ ρuu + (p + B 2 /8π) I – BB/4π + ∏] + ρg F data ≡ ∂ t ( ρu est ) ∂ t ( ρu est )| phot = ξ (F data ) ⊥ + (1 - ξ) (F sim ) ⊥ + (F sim ) || Here, 0 < ξ < 1 represents a Kalman-like “confidence matrix” defined at each mesh element within the photospheric volume. – ξ = 1: forces perp. to B are determined entirely by the data. – ξ = 0: forces perp. to B are determined entirely by the radiative-MHD system – no observational forcing! – Parallel (hydrodynamic) forces are always evolved by the MHD system. 11

12. Unfortunately, while  t B provides constrains E, it does not fully determine E – any  ψ can be added to E. Faraday’s law only relates  t B to the curl of E, not E itself; the gauge electric field  ψ is unconstrained by  t B. Ohm’s law is one additional constraint. Doppler shifts provide another constraint: where B los = 0, the Doppler velocity and transverse magnetic field determine a Doppler electric field, cE Dopp = – (v Dopp x B trans ) Key issue: only true where B LOS = 0 – i.e., polarity inversion lines (PILS).

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

14. Aside: Parallel flows are observed! Dopplergrams show patterns consistent with “siphon flows.” 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?

15. Away from disk center, the PTD+Doppler approach can be still be used, but does not properly capture emergence. Off disk center, Doppler shifts along PILs of the line-of- sight (LOS) field constrains E --- but both the radial (normal) and tangential (horizontal) components are constrained. Consistency of E h with the change in radial magnetic flux therefore imposes an independent constraint.

16. Fisher et al. (2011) tested a method of incorporating Doppler electric fields into estimates of total E. Top row: The three components of the electric field E and the vertical Poynting flux S z from the MHD reference simulation of emerging magnetic flux in a turbulent convection zone. 2nd row: The inductive components of E and S z determined using the PTD method. 3rd row: E and S z derived by incorporating Doppler flows around PILs into the PTD solutions. Note the dramatic improvement in the estimate of S z. 4th row: E and S z derived by incorporating only non-inductive FLCT derived flows into the PTD solutions. Note the poorer recovery of E x, E y and S z relative to the case that included only Doppler flows. 5th row: E and S z derived by including both Doppler flows and non-inductive FLCT flows into the PTD solutions. Note the good recovery of E x, E y, and S z, and the reduction in artifacts in the low-field regions for E y. 16

17. Tests with synthetic data from MHD simulations show good recovery of the simulation’s E-field and Poynting flux S z. Upper left: A comparison of the vertical component of the Poynting flux derived from the PTD method alone with the actual Poynting flux of the MHD reference simulation. Upper right: A comparison between the simulated results and the improved technique that incorporates information about the vertical flow field around PILs into the PTD solutions. Lower left: Comparison of the vertical Poynting flux when non-inductive FLCT- derived flows are incorporated into the PTD solutions. Lower right: Comparison of the vertical Poynting flux when both Doppler flow information and non-inductive FLCT-derived flows are incorporated into the PTD solutions. Poynting flux units are in [10 5 G 2 km s −1 ] 17

18. 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”

19. Because magnetic fields suppress convection, there are pseudo-redshifts in magnetized regions, as on these PILs. Here, an automated method (Welsch & Li 2008) identified PILs in AR 11117, color-coded by orbit-corrected Doppler shift.

20. But Doppler measurements are typically biased: there are pseudo-redshifts in magnetized regions. This effect is present in HMI measurements of Doppler velocities along PILs in active regions.

21. 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. 21

22. 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. 22

23. Scatter plots of Doppler shift vs. line depth show the pseudo-redshift, clear evidence of bias from the convective blueshift. 23 Dark regions correspond to low DN/s in maps of line depth. PIL pixels (shown here in blue) for the most part appear redshifted.

24. 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)

25. 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). ΔΦ LOS /Δt ΔΦ PIL /Δt NB: The analysis here applies only near disk center!

26. 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!

27. 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. 27

28. In sample HMI Data, we solved for v 0 using dozens of PILs from several successive magnetograms in AR 11117. 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

29. The inferred offset velocity v 0 can be used to correct Doppler shifts along PILs.

30. 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). 30 In addtion, there are large systematic errors in identifying PILs.

31. 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! 31 - 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

32. 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. 32 - 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

33. 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.

34. 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. 34

35. 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!) 35 For instance, what’s up with these PILs?

36. 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.”

37. 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! 37 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!

38. Conclusions Until measurements of the vector magnetic field B C in the corona can be made (efforts are underway!), B C can only be modeled. Static models (e.g., extrapolations) likely underestimate coronal energies. Magnetofrictional models cannot model eruptions, since they do not realistically treat the momentum equation. We have developed assimilative methods to: 1.Derive electric fields, or flows, from magnetogram sequences using PTD; 2.Drive the RADMHD model by imposing flows derived via PTD E fields; 3.Incorporate Doppler data into our E field estimates (“PTD +Doppler”); 4.Correct offsets in HMI Doppler velocities from convective blueshifts.

39. Challenges: Modeling High & Low High: Investigate use of coronal field measurements, e.g., field strengths from radio observations. – “8 th wave” treatment can handle  ·B = 0 – But updating B in the hyperbolic MHD system implies waves! Low: Develop treatment of subsurface B and v – How can we incorporate observed flux emergence into the model in a self-consistent manner? Global scales: models need global magnetic context – RADMHD is being converted to spherical coordinates -- “first NaN” expected soon, if not already achieved! – Large simulations are computationally expensive = difficult to run in real time!