1. Numerical Simulations of Solar Magneto-Convection
Bob Stein
Michigan State University
East Lansing, MI, USA

2. Outline Equations Time Advance Spatial Derivatives Diffusion
Boundary Conditions
Radiation
Examples

3. Computation Solve 3D, Compressible Realistic EOS includes ionization
Conservation equations
mass, momentum & internal energy
Induction equation
Radiative transfer equation
3D, Compressible
Realistic EOS includes ionization
Open boundaries
Fix entropy of inflowing plasma at bottom

4. Equations

5. Centering
Ez, Jz
uz, Bz
uy, By r
, e, P
Ex, Jx

6. Time Advance
3rd order, 2N storage, Runge-Kutta

7. Time Advance 3rd order, leapfrog
Predictor
Corrector

8. Spatial Derivatives 5th order

9. Diffusion
Numerical viscosity
Hyperviscous enhancement

10. 1D Shock Tube Test

11. MHD Shock Tube Test

12. Boundary Conditions horizontally periodic vertically open
Original Top Boundary Conditions

13. Wave Reflection
Gravity wave
Acoustic Wave

14. Original Bottom Boundary Conditions: Everywhere --
Evolve towards uniform Pressure

15. Original Bottom Boundary Conditions: Inflows --
Evolve toward given entropy
Evolve velocity towards uniform vertical & zero net mass flux
Evolve magnetic field toward given horizontal field

16. Characteristic Boundary Conditions Physical Conditions
No Pressure drift
Zero net mass flux
Minimal reflected waves
Entropy of inflowing material at bottom given

17. Characteristic z-Derivatives

18. Boundary Equations

19. Local, 1D, inviscid Characteristic Equations

20. Outgoing Characteristics: - Calculate di from their definitions using 1-sided derivatives Incoming Characteristics: - Impose Physical Boundary Conditions

21. Incoming Characteristic - No Reflected Waves
Characteristic equation for incoming waves at bottom & top is
So boundary condition for no reflected waves is

22. Incoming Characteristic - No Pressure Drift
Impose condition on incoming characteristic to make
Impose condition at bottom, with

23. Incoming characteristic - Zero net mass flux
Impose condition on incoming characteristic to make So
Impose the condition at the top

24. Incoming characteristic - Specified Entropy
Impose a term on the entropy characteristic equation to make

25. Incoming characteristics - Horizontal velocities
Impose condition horizontal velocities tend toward zero

26. Characteristic Magnetic Boundary Condition -- a work in progress

27. Physics is the time consuming part
Equation of state - includes ionization and molecule formation
Radiative heating and cooling - LTE, non-gray, multi-group

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

29. Tabular Equation of State includes ionization, excitation & H2 molecule formation Lookup as function of log density & internal energy per unit mass for
Log Pressure
Temperature
Log Opacity
Source Function

30. Radiative Cooling & Heating
Produces low entropy plasma whose buoyancy work drives convection
Determines (with convection and waves) mean atmospheric structure
Provides diagnostics of velocity, temperature and magnetic field
Reverses p-mode intensity vs. velocity asymmetry

31. Energy Conservation Radiative Heating/Cooling
J is average over angles of integrals along rays through entire domain

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

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

34. Opacity is rapidly varying function of wavelength
Opacity is rapidly varying function of wavelength. Reduce number by binning like magnitudes

35. Simplifications
Only 5 rays
4 Multi-group opacity bins
Assume kL a kC

36. Example: 3D, Compressible Magneto-Convection

38. Stratified convective flow: diverging upflows, turbulent downflows
Velocity arrows, temperature fluctuation image (red hot, blue cool)

39. Stein & Nordlund, ApJL 1989

40. t Z
Fluid Parcels reaching the surface Radiate away their Energy and Entropy r Q E S

41. Entropy
Green & blue are low entropy downflows, red is high entropy upflows
Low entropy plasma rains down from the surface

42. Downflows are turbulent, upflows are more laminar.
Vorticity
Downflows are turbulent, upflows are more laminar.

43. Turbulent downdrafts

44. Simulation Results: B Field lines

45. Magnetic Field Lines, t=0.5 min

46. Magnetic Field Lines, t=3.5 min

47. Magnetic Field Lines: t=6 min

48. Granulation

49. Spectrum of granulation
Simulated intensity spectrum and distribution agree with observations
after smoothing with telescope+seeing point spread function.

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

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

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

53. Magnetic Field Strength

54. Both simulated and observed distributions are stretched exponentials.
Field Distribution
simulation
observed
Both simulated and observed distributions are stretched exponentials.

55. Acoustic Oscillations (p-modes)

56. Tests: Comparison with Solar Observations!
Granulation Intensity Distribution
Horizontal Velocity Spectrum
Line Profiles
Magnetic Field Distribution
P-Modes

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

58. The End