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

Movie by Mats Carlsson

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

Conservation Equations Mass Momentum Energy Magnetic Flux

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

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

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

Solve Transfer Equation for each bin i

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

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, , 1992) modified for real gases

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

Observables

Granulation

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

Velocity Spectrum

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

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

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

SUPER- GRANULATION SCALE CONVECTION

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, km vertical resolution

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

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

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

Mean Atmosphere Ionization of He, He I and He II

Energy Fluxes ionization energy 3X larger energy than thermal

Convective Flux, 48 Mm wide, after 2 hours

Problem

MAGNETO- CONVECTION

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

Magnetic Field Lines - initially vertical

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

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

Center to Limb Movie by Mats Carlsson

G-Band Center to Limb Appearance

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

Magnetic Field & Velocity surface) Up Down

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

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

Individual features

Magnetic field

Vertical velocity

Height where tau=1

Temperature structure

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

Temperature Gradients largest next to magnetic concentrations

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

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

Temperature & Magnetic Field (contours 1, 2 kG)

Temperature & Velocity

Magnetic Field & Velocity

Temperature & Velocity

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

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

Temperature fluctuations + Velocity

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

B Swept to Cell Boundaries

Magnetic Field Lines - fed horizontally

Flux Emergence & Disappearance Emerging flux Disappearing flux

The End