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

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).

Conservation Equations Mass Momentum Energy Magnetic Flux

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

Variables

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

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

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

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

Actually solve for q = P - B

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

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.

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

Solve Transfer Equation for each bin i

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)

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

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

Radiative Heating/Cooling

Energy Fluxes ionization energy 3X larger energy than thermal

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

Diffusion stabilizes scheme Spreads shocks Damps small scale wiggles

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

Observables

Granulation

Emergent Intensity Distribution

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

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

Simulation Oscillation modes

Oscillation modes Simulation MDI Observations

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

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)

P-Mode Excitation

Solar Magneto-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 (?)

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

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

Never See Hot Gas

Granule ~ Fountain

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

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

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

B Swept to Cell Boundaries

Magnetic Field Lines - fed horizontally

Flux Emergence & Disappearance Emerging flux Disappearing flux

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

Magnetic Field Lines - initially vertical

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

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

Magnetic Field & Velocity surface) Up Down

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

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

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

Center to Limb Movie by Mats Carlsson

G-Band Center to Limb Appearance

Individual features

Magnetic field

Vertical velocity

Temperature structure

Height where tau=1

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

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

Temperature Gradients largest next to magnetic concentrations

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

The End