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

Outline Equations Time Advance Spatial Derivatives Diffusion Boundary Conditions Radiation Examples

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

Equations

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

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

Time Advance 3rd order, leapfrog Predictor Corrector

Spatial Derivatives 5th order

Diffusion Numerical viscosity Hyperviscous enhancement

1D Shock Tube Test

MHD Shock Tube Test

Boundary Conditions horizontally periodic vertically open Original Top Boundary Conditions

Wave Reflection Gravity wave Acoustic Wave

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

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

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

Characteristic z-Derivatives

Boundary Equations

Local, 1D, inviscid Characteristic Equations

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

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

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

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

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

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

Characteristic Magnetic Boundary Condition -- a work in progress

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

Energy Fluxes ionization energy 3X larger energy than thermal

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

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

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

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

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

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

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

Example: 3D, Compressible Magneto-Convection

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

Stein & Nordlund, ApJL 1989

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

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

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

Turbulent downdrafts

Simulation Results: B Field lines

Magnetic Field Lines, t=0.5 min

Magnetic Field Lines, t=3.5 min

Magnetic Field Lines: t=6 min

Granulation

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

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

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

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

Magnetic Field Strength

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

Acoustic Oscillations (p-modes)

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

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

The End