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

2. METHOD Integrate conservation equations for: mass, momentum, internal energy + induction equation for magnetic field

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

4. 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

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

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

7. Boundary Conditions Ghost zones loaded Density: top hydrostatic, bottom logarithmic Velocity: symmetric (normal derivative =0) Energy (per unit mass): top = slowly evolving average, bottom fixed entropy in inflows Magnetic (Electric field): top -> potential, bottom -> fixed value in inflows, damped in outflows

8. 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

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

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

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

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

16. Problem

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

18. The End