1. The Magnetic Connection Between the Sun’s Corona and Convective Interior W.P. Abbett Space Sciences Laboratory, UC Berkeley Nov. 2007, Rice Univ. P & A Colloquium

2. Motivation All solar activity – variations in energy released by the Sun in the form of electromagnetic radiation or energetic particles – is mediated by the Sun’s magnetic field Solar activity arises as a result of the coupling of the solar magnetic field to the rotating, turbulent plasma of the Sun’s convective envelope Thus, to understand and predict solar activity, we must understand the magnetic and energetic connection between the solar interior and atmosphere Image credit: Hinode JAXA / NASA

3. Motivation A quantitative description of the magnetic connection between the convection zone and corona will greatly improve our understanding of the Image credit: Hinode JAXA / NASA and much more……. origin, evolution, and decay of active region magnetic fields physics of the solar cycle and convective surface dynamo CME initiation mechanism, and the physics of eruptive events coronal heating mechanism and irradiance variations transport of magnetic flux, energy, and helicity into the corona origin and source of the solar wind

4. Approach 1. Analyze observational data Above the visible surface, we can obtain measurements of atmospheric emission and energetic particles over an impressive range of wavelengths and energies. These data can then be used to infer, e.g., the structure and strength of the magnetic field in the solar atmosphere thermodynamic characteristics of the magnetized plasma plasma flows at and above the visible surface Where direct observations are not possible – e.g., below the visible surface and on the far side of the Sun – we must rely upon helioseismology to provide information about the sub-surface structure, flow pattern, and differential rotation rate presence of active regions on the Sun’s far side

5. Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). Movie courtesy of Marc DeRosa, LMSAL Hinode G-band images of the solar photosphere

6. Hinode LOS magnetograms of the photospheric magnetic field Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). Movie courtesy of Marc DeRosa, LMSAL

7. Hinode Ca II H images of the chromosphere Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). Movie courtesy of Marc DeRosa, LMSAL

8. Hinode XRT soft X-ray images of the low corona Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). Movie courtesy of Marc DeRosa, LMSAL

9. LASCO images of the global corona during a CME Movie courtesy of LMSAL, TRACE & LASCO consortia

10. Approach 2. First principles forward modeling Utilize idealized theoretical and numerical models to approximate the physics of the solar magnetic field in regions of the interior and atmosphere where such idealizations apply. the tachocline and convective overshoot layer the optically thick convective interior well below the visible surface the convectively unstable surface layers and the low atmosphere the optically thin, magnetically-dominated corona Historically, numerical investigations of the Sun’s magnetic field have focused separately on several physically-distinct regions: Image credit: NASA

11. Approach 3. Combine observational data with models Investigate the physics of the solar atmosphere and interior by directly incorporating observations of the magnetic field and flows into models. This coupling can be performed at varying levels of sophistication depending on the desired objective. Examples include The use of magnetograms and force-free extrapolations to study the buildup of, e.g., free magnetic energy in the corona Studies of the flux of helicity and magnetic energy into the atmosphere from below the surface using sequences of magnetograms and velocity inversion techniques The use of magnetogram sequences and velocity inversion techniques to provide the necessary boundary conditions for numerical models of the solar corona The use of 3D flow fields from helioseismic inversions and data assimilation techniques to produce models with improved forecasting capability Image credit: SOHO / MDI

12. Outline 2. First principles forward modeling 3. Combine observational data with models Most recent work modeling the magnetic connection between the solar interior and atmosphere has involved studies of the emergence of active regions. I’ll therefore briefly discuss the standard picture of the origin and evolution of active regions quantitative models of the sub-surface evolution of active regions quantitative models of active region fields in the solar corona the challenge of modeling the convection zone-to-corona system recent quantitative models of the combined system: active region emergence, ephemeral active regions, and the magnetic field of the quiet Sun 1. Analyze observational data

13. Characteristics of Active Region Magnetic Fields SOHO EIT Image of coronal plasma at ~1.3 MKMDI magnetogram from May 11, 2000Mean tilt of active regions vs. latitude (Fisher et al. 1995) Most active regions emerge as simple bipoles Leading polarities of active regions in a given hemisphere are the same, and oppose those of the opposite hemisphere Active region bipoles are oriented nearly parallel to the E-W direction (Hale’s Law 1919) On average, the leading polarity of an active region is positioned closer to the equator than the trailing polarity The mean tilt angle of active regions increases with latitude (Joy’s Law)

14. The standard cartoon picture of active region magnetic fields below the surface Since most active regions emerge as simple bipoles, these structures can be interpreted as the tops of large Omega-shaped magnetic flux tubes anchored deep in the convection zone. Similarly, active regions exhibiting non-Hale configurations (e.g., delta spots) can be interpreted as twisted, or writhed flux tubes

15. Beyond the cartoon – the “thin flux tube” description of active region magnetic fields below the surface Solve an equation of motion for a flux tube moving in a field-free background model convection zone assuming that as the tube moves, it retains its identity and does not disperse or fragment the tube’s cross-section is small relative to all other relevant length scales of the problem quasi-static pressure balance is maintained across the diameter of the tube at all times ABOVE: F B refers to the magnetic buoyancy force, F T the force due to magnetic tension, F C the Coriolis force, and F D the force resulting from aerodynamic drag (ρ e and ρ i refer to the gas density external to the tube, and in the tube’s interior respectively) Spruit 1981, Moreno-Insertis 1986, Ferriz- Mas & Schussler 1993, Caligari et al.1995

16. Successes of the thin flux tube model provided an estimate of the magnetic field strength of the toroidal layer at the base of the convection zone provided a physical description of “Joy’s Law” provided a physical basis for asymmetric spot motions explained the dispersion of tilt versus active region size described the physical basis of morphological asymmetries in active regions is currently the basis of the only viable theory of the origin of twist in active regions (the “Sigma effect” of Longcope et al. 1998) Image from Caligari et al. 1995

17. Beyond the thin flux tube model: MHD simulations of the interior Anelastic simulations of Fan et al. (1998)Boussinesq simulations of Longcope et al (1996) 2D results show that without substantial fieldline twist (far more than is, on average, observed), flux tubes fragment and are unable to reach the surface

18. 3D MHD simulations of sub-surface active region magnetic fields Relaxing the axisymmetric assumption resolves the apparent paradox. In 3D only a modest amount of twist is required for the flux rope to remain cohesive Fieldline twist is relatively unimportant: what matters is the axial field strength relative to the kinetic energy density of strong downdrafts (Fan et al. 2003): Fan et al. (2003)Abbett et al. (2000)

19. 3D MHD global simulations of active region magnetic fields below the visible surface (Fan 2007) Parameter space exploration of e.g., different initial field strengths and twists in a stratified model convection zone Jouve et al. (2007) are performing similar global simulations with the ASH code (Brun et al. 2004) in a turbulent background state

20. 3D MHD simulations of the solar transition region and corona Top row: SAIC model (from their website), Bottom row: image from Gudiksen & Nordlund 2005 (left) and Abbett & Fisher 2003 (right)

21. The surface layers and low atmosphere Left: 3D model of a flux rope interacting with surface flows from Cheung 2007 Right: 3D simulation of surface magnetoconvection from Bob Stein’s website These types of models are highly realistic, since they include the LTE radiative transfer equation in the MHD system. Like the coronal models of the previous slide that include optically thin cooling and thermal conduction in their MHD energy equation, the results of these models can be directly compared with observational data. fig_gband_moviep.mov

22. The different types of numerical models discussed so far … Sub-surface MHDSurface radiative-MHDMHD model coronae Spatial coverage: individual active regions to global models of the convective envelope Spatial coverage: surface layers (~5 Mm below the surface out into the low chromosphere) Spatial coverage: transition region (>3 x 10 5 K) to the corona. Individual active regions to the global corona Numerical techniques: TFT, anelastic MHD Fully-compressible MHD with the LTE radiative transfer equation included in the system Fully-compressible MHD with the optically thin radiative cooling and thermal conduction Applicability: from ~10 Mm below the surface to the tachocline. Applicability: “anywhere”Applicability: the optically thin low-density plasma of the corona Computationally efficientComputationally expensive“Moderately” computationally expensive

23. Putting it all together: an early attempt From Abbett & Fisher 2003

24. Putting it all together: more self-consistent, idealized simulations of flux emergence Left: Magara (2004) ideal MHD AR flux emergence simulation as shown in Abbett et al. 2005 Right: Manchester et al. (2004) BATS-R-US MHD simulation of AR flux emergence 10

25. Modeling the combined convection zone-to-corona system in a physically self-consistent way: The resistive fully-compressible MHD system of equations: Closure relation: a non-ideal equation of state obtained through an inversion of the OPAL tables (Rogers 2000),

26. Modeling the combined convection zone-to-corona system in a physically self-consistent way: The source term in the energy equation, must include the important physics believed to govern the evolution of the combined system. In the corona, this includes radiative cooling (in the optically thin limit), the divergence of the electron heat flux, a coronal heating mechanism (if necessary). In the lower atmosphere at and above the visible surface, radiative cooling (optically thick) Below the surface in the deeper layers of the convective interior radiative cooling (in the optically thick diffusion limit)

27. Numerical techniques and challenges: A dynamic numerical model extending from below the photosphere out into the corona must: span a ~ 10 - 15 order of magnitude change in gas density and a thermodynamic transition from the 1 MK corona to the optically thick, cooler layers of the low atmosphere, visible surface, and below; resolve a ~ 100 km photospheric pressure scale height while simultaneously following large-scale evolution (we use the Mikic et al. 2005 technique to mitigate the need to resolve the 1 km transition region scale height characteristic of a Spitzer-type conductivity); remain highly accurate in the turbulent sub-surface layers, while still employing an effective shock capture scheme to follow and resolve shock fronts in the upper atmosphere address the extreme temporal disparity of the combined system

28. RADMHD: Numerical techniques and challenges For the quiet Sun: we use a semi-implicit, operator-split method. Explicit sub-step: Explicit sub-step: We use a 3D extension of the semi-discrete method of Kurganov & Levy (2000) with the third order-accurate central weighted essentially non-oscillatory (CWENO) polynomial reconstruction of Levy et al. (2000). CWENO interpolation provides an efficient, accurate, simple shock capture scheme that allows us to resolve shocks in the transition region and corona without refining the mesh. The solenoidal constraint on B is enforced implicitly.

29. RADMHD: Numerical techniques and challenges For the quiet Sun: we use a semi-implicit, operator-split method Implicit sub-step: Implicit sub-step: We use a “Jacobian-free” Newton-Krylov (JFNK) solver (see Knoll & Keyes 2003). The Krylov sub-step employs the generalized minimum residual (GMRES) technique. JFNK provides a memory-efficient means of implicitly solving a non-linear system, and frees us from the restrictive CFL stability conditions imposed by e.g., the electron thermal conductivity and radiative cooling.

30. The thermodynamic structure of the model is controlled by the energy source terms, the gravitational acceleration and the applied thermodynamic boundary conditions. No stratification is imposed a priori. RADMHD: Modeling the combined convection zone-to-corona system:

31. The quiet Sun magnetic field in the model chromosphere Magnetic field generated through the action of a convective surface dynamo. Fieldlines drawn (in both directions) from points located 700 km above the visible surface. Grayscale image represents the vertical component of the velocity field at the model photosphere. The low-chromosphere acts as a dynamic, high-β plasma except along thin rope-like structures threading the atmosphere, connecting strong photospheric structures to the transition region- corona interface. Plasma-β ~ 1 at the photosphere only in localized regions of concentrated field (near strong high-vorticity downdrafts From Abbett (2007)

32. Flux submergence in the quiet Sun and the connectivity between an initially vertical coronal field and the turbulent convection zone From Abbett (2007)

33. Average horizontal magnetic field as a function of height above the surface in the quiet Sun model atmosphere Horizontally averaged magnetic field strength at the visible surface is ~108 G, and drops to ~ 37 G in the low chromosphere. Corresponding values for the vertical component: ~71 G at the surface, falling to ~15 G in the low chromosphere. The maximum values are roughly similar, ranging from ~1 kG at the surface to ~275 G in the low atmosphere In the corona, the field becomes more vertically oriented, and drops to ~2.5 G on average

34. Reverse granulation A brightness reversal with height in the atmosphere is a common feature of Ca II H and K observations of the quiet Sun chromosphere. In the simulations, a temperature (or convective) reversal in the model chromosphere occurs as a result of the p div u work of converging and diverging flows in the lower-density layers above the photosphere where radiative cooling is less dominant.

35. Gas temperature and B z at ~ 700 km above and below the model photosphere From Abbett (2007)

36. Flux cancellation and the effects of resolution: The Quiet Sun magnetic flux threading the model photosphere over a 15 minute interval. Grid resolution ~ 117 X 117 km Average unsigned flux per pixel: 34.5 G Simulated noise-free magnetograms reduced to MDI resolution (high-resolution mode) by convolving the dataset with a 2D Gaussian with a FWHM of 0.62” or 459 km. Average unsigned flux per pixel is now: 19.9 G Simulated noise-free magnetograms reduced to Kitt Peak resolution. FWHM of the Gaussian Kernel is 1.0” or 740 km. Average unsigned flux per pixel: 15.0 G Observed unsigned flux per pixel at Kitt Peak: 5.5 G

37. How force-free is the quiet Sun atmosphere? Temperature Chromosphere Photosphere

38. log B log J log βBzBz log B

39. Characteristics of the quiet Sun model atmosphere: Note: Above movie is not a timeseries!

40. Recent progress and (near) future plans: 120 Mm Increase domain size. Below: Starting state for a 128 node run on NASA’s Discover cluster. Symmetry will be broken by a random entropy perturbation below the surface. Release beta version of Radmhd-1.0 under GPLv2 for further validation and testing by independent groups Perform AR emergence simulations once the background state is dynamically and energetically relaxed Significantly extend the coronal portion of the domain using a non-uniform mesh

41. Summary Great progress has been made throughout the past decade in our ability to model the dynamic evolution of the solar magnetic field in three separate physically-distinct regions in the Sun: the convective interior, the low atmosphere, and the upper transition region and corona Recently, substantial progress has been made in the effort to model the upper convection zone and the low-corona in a physically self-consistent way and in an efficient manner within a single computational volume New numerical techniques have been applied to the study of dynamo- generated quiet Sun magnetic fields. This study will soon be extended to active region spatial scales. It is then possible to investigate the large-scale emergence, evolution and decay of active regions the numerics of data driving and data assimilation the coupling of sub-surface, and coronal codes to RADMHD through publicly available computational frameworks