1. Simulating Solar Convection Bob Stein - MSU David Benson - MSU Aake Nordlund - Copenhagen Univ. Mats Carlsson - Oslo Univ. Simulated Emergent Intensity

2. METHOD Solve conservation equations for: mass, momentum, internal energy & induction equation LTE non-gray radiation transfer Realistic tabular EOS and opacities No free parameters (except for resolution & diffusion model).

3. Conservation Equations Mass Momentum Energy Magnetic Flux

4. Simulation Domain 48 Mm 20 Mm 500 x 500 x 500 -> 2000 x 2000 x 500

5. Variables

6. Spatial Derivatives Spatial differencing –6th-order finite difference, non-uniform mesh c = (-1.+(3.**5-3.)/(3.**3-3.))/(5.**5-5.-5.*(3.**5-3)) b = (-1.-120.*c)/24., a = (1.-3.*b-5.*c)

7. Time Advance Time advancement –3rd order Runga-Kutta For i=1,3 do

8. 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

9. Solve Feautrier equations along rays through each grid point at the surface

10. Actually solve for q = P - B

11. Simplifications Only 5 rays 4 Multi-group opacity bins Assume  L  C

12. 5 Rays Through Each Surface Grid Point Interpolate source function to rays at each height

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

14. Line opacities are assumed proportional to the continuum opacity Weight = number of wavelengths in bin

15. Solve Transfer Equation for each bin i

16. Finite Difference Equation Problem: at small optical depth the 1 is lost re 1/  2  in B Solution: store the value - 1, (the sum of the elements in a row) and calculate B = - (1+A+B)

17. Advantage Wavelengths with same  (z) are grouped together, so integral over  and sum over commute

18. Interpolate q=P-B from slanted grid back to Cartesian grid

19. Radiative Heating/Cooling

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

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

22. Diffusion stabilizes scheme Spreads shocks Damps small scale wiggles

23. 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

24. Observables

25. Granulation

26. Emergent Intensity Distribution

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

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

29. Velocity spectrum, (kP(k)) 1/2 * * * * * * MDI doppler (Hathaway) TRACE correlation tracking (Shine) MDI correlation tracking (Shine) 3-D simulations (Stein & Nordlund)

30. Simulation Oscillation modes

31. Oscillation modes Simulation MDI Observations

32. Local Helioseismology uses wave travel times through the atmosphere (by former grad. Student Dali Georgobiani) Dark line is theoretical wave travel time.

33. P-Modes Excited by PdV work Triangles = simulation, Squares = observations (l=0-3) Excitation decreases at low frequencies because oscillation mode inertia increases and compressibility (dV) decreases. Excitation decreases at high frequencies because convective pressure fluctuations have long periods. (by former grad. students Dali Georgobiani & Regner Trampedach)

34. P-Mode Excitation

35. Solar Magneto-Convection

36. 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

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

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

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

40. Inhomogeneous T (see only cool gas), & P turb Raise atmosphere One scale height 3D atmosphere not same as 1D atmosphere

41. Never See Hot Gas

42. Granule ~ Fountain

43. Granules: diverging warm upflow at center, converging cool, turbulent downflows at edges Red=diverging flow Blue =converging flow Green=vorticity

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

46. Magnetic Boundary Conditions Magnetic structure depends on boundary conditions Bottom either: 1)Inflows advect in horizontal field or 2)Magnetic field vertical Top: B tends toward potential

47. B Swept to Cell Boundaries

48. Magnetic Field Lines - fed horizontally

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

51. Magnetic Flux Emergence Magnetic field lines rise up through the atmosphere and open out to space

52. Magnetic Field Lines - initially vertical

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

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

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

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

57. G-band images from simulation at disk center & towards limb (by Norwegian collaborator Mats Carlsson) Notice: Hilly appearance of granules Bright points, where magnetic field is strong Striated bright walls of granules, when looking through magnetic field Dark micropore, where especially large magnetic flux

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

59. Center to Limb Movie by Mats Carlsson

60. G-Band Center to Limb Appearance

61. Individual features

62. Magnetic field

63. Vertical velocity

64. Temperature structure

65. Height where tau=1

66. 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

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

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

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

70. Temperature & Velocity

71. Magnetic Field & Velocity

72. Temperature & Velocity

73. Temperature Gradients largest next to magnetic concentrations

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

75. The End