1. Solar Convection Simulations Robert Stein, David Benson - Mich. State Univ. Aake Nordlund - Niels Bohr Institute

2. Movie by Mats Carlsson

3. METHOD Solve conservation equations for: mass, momentum, internal energy & induction equation

4. Conservation Equations Mass Momentum Energy Magnetic Flux

5. Numerical Method Spatial differencing –6 th -order staggered finite difference, 3 points either side Spatial interpolation –5 th order, staggered Time advancement –3 rd order Runga-Kutta

6. Radiation Heating/Cooling LTE Non-gray, 4 bin multi-group Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface. Produces low entropy plasma whose buoyancy work drives convection

7. 5 Rays Through Each Surface Grid Point Interpolate source function to rays at each height

8. Opacity is binned, according to its magnitude, into 4 bins.

9. Solve Transfer Equation for each bin i

10. Equation of State Tabular EOS includes ionization, excitation H, He, H 2, other abundant elements

11. Boundary Conditions Current: ghost zones loaded by extrapolation –Density, top hydrostatic, bottom logarithmic –Velocity, symmetric –Energy (per unit mass), top = slowly evolving average –Magnetic (Electric field), top -> potential, bottom -> fixed value in inflows, damped in outflows Future: ghost zones loaded from characteristics normal to boundary (Poinsot & Lele, JCP, 101, 104-129, 1992) modified for real gases

12. Fluid Parcels reaching the surface Radiate away their Energy and Entropy Z S E  Q 

13. Observables

14. Granulation

15. Solar velocity spectrum MDI doppler (Hathaway) TRACE correlation tracking (Shine) MDI correlation tracking (Shine) 3-D simulations (Stein & Nordlund) v ~ k v ~ k -1/3

16. Velocity Spectrum

17. Line Profiles Line profile without velocities. Line profile with velocities. simulation observed

18. Convection produces line shifts, changes in line widths. No microturbulence, macroturbulence. Average profile is combination of lines of different shifts & widths. average profile

19. P-Mode Excitation Triangles = simulation, Squares = observations (l=0-3) Excitation decreases both at low and high frequencies

20. SUPER- GRANULATION SCALE CONVECTION

21. Initialization Start from existing 12 x 12 x 9 Mm simulation Extend adiabatically in depth to 20 Mm, no fluctuations in extended portion, relax for a solar day to develop structure in extended region Double horizontally + small fraction of stretched fluctuations to remove symmetry, relax to develop large scale structures Currently: 48x48x20 Mm 100 km horizontal, 12-75 km vertical resolution

22. Initialization Double horizontally + small fraction stretched : Uz at 0.25 Mm Snapshots of methods + composite (?)

23. Initialization Double horizontally + small fraction stretched : Uz at 17.3 Mm

27. Mean Atmosphere Temperature, Density and Pressure (10 5 dynes/cm 2 ) (10 -7 gm/cm 2 ) (K)

28. Mean Atmosphere Ionization of He, He I and He II

29. Energy Fluxes ionization energy 3X larger energy than thermal

30. Convective Flux, 48 Mm wide, after 2 hours

31. Problem

32. MAGNETO- CONVECTION

33. Unipolar Field Impose uniform vertical field on snapshot of hydrodynamic convection Boundary Conditions: B -> potential at top, B vertical at bottom B rapidly swept into intergranular lanes

34. Magnetic Field Lines - initially vertical

35. G-band images from simulation at disk center & towards limb (by Mats Carlsson) Notice: Hilly appearance of granules Striated bright walls of granules Micropore at top center Dark bands moving across granules

36. Comparison with observations Simulation, mu=0.6 Observation, mu=0.63

37. Center to Limb Movie by Mats Carlsson

38. G-Band Center to Limb Appearance

39. G-band image & magnetic field contours (-.3,1,2 kG)

40. Magnetic Field & Velocity (@ surface) Up Down

41. G-band Bright Points = large B, but some large B dark

42. G-band & Magnetic Field Contours:.5, 1, 1.5 kG (gray) 20 G (red/green)

43. Individual features

44. Magnetic field

45. Vertical velocity

46. Height where tau=1

47. Temperature structure

48. Magnetic concentrations: cool, low  low opacity. Towards limb, radiation emerges from hot granule walls behind. On optical depth scale, magnetic concentrations are hot, contrast increases with opacity

49. Temperature Gradients largest next to magnetic concentrations

50. Magnetic Field & Velocity High velocity sheets at edges of flux concentration

51. Temperature + B contours (1, 2, 3, kG)

52. Temperature & Magnetic Field (contours 1, 2 kG)

53. Temperature & Velocity

54. Magnetic Field & Velocity

55. Temperature & Velocity

56. Micropore Formation Small granule is squeezed out of existence Magnetic flux moves into location of previous granule

57. G-band images from simulation at disk center & towards limb (by Mats Carlsson) Notice: Dark bands moving across granules

58. Temperature fluctuations + Velocity

65. Boundary Conditions Magnetic structure depends on boundary conditions 1)Inflows at bottom advect horizontal field in 2)At bottom: boundary magnetic field vertical At top: B tends toward potential

66. B Swept to Cell Boundaries

67. Magnetic Field Lines - fed horizontally

68. Flux Emergence & Disappearance 12 34 Emerging flux Disappearing flux

70. The End