1. Understanding Links Between the Solar Interior and Atmosphere Brian Welsch, George Fisher*, and Bill Abbett Space Sciences Laboratory, UC Berkeley *NB: much material presented here was borrowed from George!

2. 2 Links between the interior and atmosphere?! Waves, current sheets, however --- it’s all from the convection! Are you kidding?? There are too many! Coronal heating! And CMEs! And flares! And streamer structure, and corotating interaction regions! Oh, yeah, and solar wind acceleration! Well, not driving them. Links, links, everywhere… an interior conspiracy! And maybe a trigger. What about the grassy knoll?! Call in Glenn Beck! But at least their ultimate source.

3. Restricted Focus! 1.The Usual Idea: The Interior Drives the Atmosphere Primarily a tale of magnetic energy transport. 2.Heterodoxy: The Atmosphere Can Drive the Interior! On short and long time scales, and in steady state.

4. Surface magnetism is seen as one manifestation of structures extending from the interior into the corona. Image credits: George Fisher, LMSAL/TRACE

5. 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)

6. Magnetic energy must get from the interior into the atmosphere, implying an outward energy flux. The Poynting flux of magnetic energy depends upon E, or in the ideal MHD approx., -(v x B)/c: dU/dt = ∫ dA S z = c ∫dA (E x B) z /4 π = ∫dA (B x [v x B]) z /4 π Hence, photospheric electric fields --- or flows, if the flux is frozen-in --- play a central role in the solar activity that interests most of us!

7. Aside: How good is the ideal MHD approximation? 7 Diffusion is not evident in high-resolution movies of magnetic evolution. Typical speeds are 0.5 km/s or more. Parker 1984: for ℓ= 10 km, η =10 8 cm 2 /s, flux diffusion speed u < 10 m/s; ambipolar slip vel. w/ion frac. of 10 -3 is 1 mm/s; e - conduction speed for  B = 2kG over ℓ is 0.5 m/s.

8. Digression: a thought experiment emphasizes the role of convective driving in atmospheric evolution. Q: What would happen if all photospheric flows ceased? (For this exercise, ignore the fact that the Sun needs these flows to expel the heat it produces!) Partial answers, I believe: The corona would relax, on the Alfvén time – fast! – but would then do basically nothing! Coronal heating would cease – no driver! Flares wouldn’t happen – no new energy/perturbations! Fast CMEs wouldn’t occur -- but perhaps some streamer blowouts, via slow magnetic reconnection

9. The PTD method can also be used to decompose the magnetic field, and determine E from its evolution. 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. ^^  t B =  x (  x  t B z) +  x  t J z  t B z =  h 2 (  t B ) 4 π  t J z /c =  h 2 (  t J )  h ·(  t B h ) =  h 2 (  z (  t B )) ^^

10. Faraday’s law,  B/  t = -c(  x E) =  x (v x B), can then be used -- but this does not fully determine E! Note that:  t B h also depends upon vertical derivatives in E h, which single-height magnetograms do not fully constrain. But most importantly: Faraday’s law only relates  t B to the curl of E, not E itself; a “gauge electric field”  ψ is unconstrained by  t B. ==> Even multiple-height magnetograms won’t fix this! Ohm’s law is one additional constraint. What about others?

11. Schematic illustration of flux emergence in a bipolar magnetic region, viewed in cross-section normal to the polarity inversion line (PIL). But Doppler measurements can detect vertical flows along PILs! Note the strong signature of the field change at the edges of the region, while the field change at the PIL is zero. Important magnetodynamics is not always apparent in ΔB z /Δt -- e.g., flux emergence!

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

13. 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?

14. 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.

15. How can Doppler shifts be combined with the inductive electric field E I from PTD? Near PILs of B LOS, Doppler shifts and B transverse unambiguously determine a “Doppler electric field:” We define the PIL-weighted “modulated” field E M, We can then find the curl-free component of E M, via 15

16. Next, we combine E χ and the PTD E I field, and then find another potential field to ensure total E is ideal. The total electric field is given by where we used the iterative scheme of Fisher et al. (2010) to determine the scalar potential ψ, so that E ⋅ B=0, as implied by the ideal Ohm’s law, E = -(v x B)/c. 16

17. Validation is essential before use with real data! Use MHD simulation with known magnetic field evolution, electric fields, and velocity fields: 17 Our test case is an ANMHD simulation of a bipolar magnetic region rising through a convecting medium. The simulation was performed by Bill Abbett. Welsch et al. (ApJ 2007) used this same simulation for a detailed evaluation and comparison of velocity/electric-field inversion techniques.

18. Validation is essential before use with real data! Use MHD simulation with known magnetic field evolution, electric fields, and velocity fields: 18 First three panels: White vectors show horizontal flows, red / blue contours show upward / downward flows. Lower right: Comparison of [  x (v x B) ] z with  B z /  t, demonstrating the evolution obeys the ideal induction equation.

19. How accurate are our methods? We tested them with MHD simulations of emerging flux from Welsch et al. (2007). 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. 19 See Fisher et al., Sol. Phys, in press, and http://arxiv.org/abs/1101.4086

20. Qualitative and quantitative comparisons show good recovery of the simulation’s E-field and Poynting flux S z. 20 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. 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. Poynting flux units are in [10 5 G 2 km s −1 ] See Fisher et al., Sol. Phys, in press, and http://arxiv.org/abs/1101.4086

21. Poynting fluxes estimated by other techniques, e.g., DAVE or DAVE4VM (Schuck 2008) don’t do as well. 21

22. Conclusions, pt. 1 Vector magnetogram sequences can, by themselves, be used to estimate electric fields E and vertical Poynting fluxes S z. These techniques are constrained to obey all three components of Faraday’s Law. We then presented a new method to incorporate Doppler shifts observed along polarity inversion lines (PILs) to improve accuracy of estimates for E and S z. We tested the technique with MHD simulation data, where the electric field and Poynting flux are known. 22

23. Future Work, pt. 1 We plan to test our methods further with more realistic simulations of photospheric evolution, including sensitivity to magnetogram noise. We plan to apply the method to HMI vector magnetogram and Doppler flow observations, to investigate relationships between estimated Poynting fluxes and flares, CMEs, and coronal heating. 23

24. Open question: Can the PTD electric field method work away from disk center? 24 George thinks so --- but this is a topic of heated discussion within our group! The usage of the ẑ vector in the definition of the Poloidal and Toroidal functions can be in any direction --- so one can take ẑ to be in the LOS direction, with the horizontal directions then residing in the plane-of-the-sky, or “transverse” directions. The PTD and Doppler Poisson equations can be solved in this coordinate system and then decomposed back into radial/horizontal coordinates on the Sun. PTD is useful in this context because it derives all three components of the electric field independently.

25. Restricted Focus! 1.The Usual Idea: The Interior Drives the Atmosphere Primarily a tale of magnetic energy transport. 2.Heterodoxy: The Atmosphere Can Drive the Interior! On short and long time scales, and in steady state.

26. Restricted Focus! 1.The Usual Idea: The Interior Drives the Atmosphere Primarily a tale of magnetic energy transport. 2.Heterodoxy: The Atmosphere Can Drive the Interior! On short and long time scales, and in steady state.

27. On short time scales: Lorentz forces during flares might cause sunquakes! See Fisher et al., Sol. Phys., in revision, http://arxiv.org/abs/1006.5247 Kosovichev & Zharkova, 1998 Hudson (2000): coronal fields should “implode” in flares and CMEs. Wang & Liu (2010) report that photospheric fields often become “more horizontal” during flares. A sudden field change can produce a Lorentz “jerk” on the interior:

28. In steady state: Quiet-sun surface layers are regions of diverging Poynting flux! At the surface, strong downflows in strong-field concentrations (turbulent pumping!) imply a downward Poynting flux. Abbett & Fisher “find a… positive… Poynting flux… along the edges of overturning granules above the surface where the field is being compressed.” The surface is a special place: flows do work on the magnetic field! Steiner et al. (2008) refer to the visible surface as “a separatrix for the vertically-directed Poynting Flux” Poynting Flux See Abbett & Fisher, Sol. Phys., in press, http://arxiv.org/abs/1102.1035 Abbett & Fisher (2011)

29. Every solar cycle, ~3000 ARs emerge, each with ~10 22 Mx of unsigned flux. And every cycle it must be removed from the photosphere --- somehow! 29 Long-term: What process removes all the flux from active regions over a solar cycle? Babcock (1961)

30. HMI’s measurements of Doppler shifts & transverse fields along PILs can constrain flux removal. Which model more accurately describes the Sun? Low (2001) Spruit et al. (1987) Van Ballegooijen (2008) Kubo et al. (2010) Several models of cancellation have been proposed, including emergence of U-loops, and submergence of Ω loops.

31. 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”

32. 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?

33. 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 a subregion of AR 11117, color-coded by Doppler shift.

34. 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. 34

35. 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. 35

36. Scatter plots of Doppler shift vs. line depth show the pseudo-redshift, clear evidence of bias from the convective blueshift. 36 Dark regions correspond to low DN/s in maps of line depth. PIL pixels (shown here in blue) for the most part appear redshifted.

37. 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)

38. 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!

39. 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!

40. 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. 40

41. 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

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

43. 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). 43 In addtion, there are large systematic errors in identifying PILs.

44. 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! 44 - 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

45. 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. 45 - 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

46. 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.

47. 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. 47

48. 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!) 48 For instance, what’s up with these PILs?

49. 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.”

50. 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! 50 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!

51. 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. 2004. Hence, the method can also be used to identify the effects of magnetic diffusivity.

52. Future Work, #2 Do a better job estimating uncertainties in both ΔΦ/Δt and magnetic lengths, to estimate v 0 via total least squares (TLS). (errors are present in both magnetic lengths & ΔΦ/Δt ) Investigate cases with large discrepancies between ΔΦ LOS /Δt & ΔΦ PIL /Δt for any evidence of non-ideal dynamics.

53. Summary The Interior Can Drive Evolution In the Atmosphere! – Duh, we knew that… – But by estimating the photospheric Poynting flux, we can try to quantify this driving! Processes in the Atmosphere Can Affect the Interior! – Changes in magnetic fields above the photosphere can cause a Lorentz jerk on the interior --- perhaps causing sunquakes. – Generic properties of convection do work on magnetic fields at the surface, and lead to a Poynting-flux divergence in the Quiet Sun. – Doppler shifts along PILs --- properly calibrated! --- can constrain how much active region flux cancels by submergence, and by emergence.