1. Introduction

     Simulations are Parameter Free - Physics Full

     Agree with Observations

2. Numerical Method

     Integrate Conservation Laws

     Equation of State includes ionization

     Radiative Transfer crucial

     Diffusion - hyperviscosity

     Boundary Conditions

3. Simulations vs. Observations

     Convection Zone Depth

     P-Mode Frequencies

     P-Mode Excitation Rate

     Granulation

     Line Profiles

4. Solar Convection

     Driving: radiative cooling in thin surface thermal boundary layer.

     Topology: controlled by mass conservation

     Topology: turbulent Downdrafts, smooth upflows

     Horizontal scale increases with increasing depth

5. Summary

The simulations are essentially parameter free:

numerical resolution

viscosity coefficients

Equation of State

Radiative energy transfer, including lines

Requires correct physics

Validates the simulations

Provides confidence to explore solar convection

Integrate the conservation laws: mass, momentum and energy, but not in conservative form:

Essential physics:

  Equation of State includes ionization.
Ionization energy dominates internal energy near surface.

  Radiative Transfer crucial

• Produces low entropy gas whose buoyancy work drives the convection.
• Controls what we observe.
• Boundary of CZ occurs near tau=1
• Line blocking increases surface temperature

  Diffusion

• stabilize numerics, damp short wavelength fluctuations.
• use k⁴ like hyperviscosity
  Test: shock tubes

  Boundary Conditions

• Horizontal: periodic
• Top: Transmitting
  Test: acoustic and gravity wave reflections
• Bottom: set incoming fluid properties
• Bottom: node for vertical motions

1. Depth of the convection zone (helioseismology)

   The convection zone depth is directly influenced by

    The Equation of State in the CZ

   • rather well known

    The entropy jump near the surface

   • convection details

    Spectral line blocking

   • about 12% for the Sun
     (would increase T_eff by 3% for T⁴ source function)
   • sensitive to chemical abundances and (T, P)

   Changing T_eff by +- 100 K alters CZ depth by +-7 Mm

   

2. Mode Frequencies (helioseismology)

   Standard, 1-D, Spherically Symmetric, Mixing Length Model

   
   3D Convection Simulation + Mixing Length Envelope Extension
   
   High frequency modes' cavity is enlarged by
   1. turbulent pressure support (convergence to correct value ensured by line widths)

      

   2. 3D radiative transfer effects
      Don't see hot gas.   Average temperature higher for a given effective temperature

      

   Contribute equally to elevating photosphere by 150 km.

   

   High frequency modes' frequency is reduced.

     Remaining discrepancies in mode frequencies can now be used to investigate details of the mode physics.

3. P-mode spectrum (helioseismology).

   P-mode spectrum is asymmetric. Velocity power is larger toward low frequencies and Intenstiy power is larger toward high frequencies.

   
   

   Intensity-Velocity phase jumps at modes.

   

4. P-mode excitation rate (helioseismology).

   Stochastic, non-adiabatic, gas pressure fluctuations excite the p-modes via PdV work,

   

   Mode excitation rate is,

   
   
   Driving decreases at low frequencies because the mode compression decreases and the mode mass increases.

   

   Driving decreases at high frequency because the pressure fluctuations decrease.

   

   Driving is due primarily to turbulent pressure. Non-adiabatic gas pressure fluctuations only are important at the surface.

   Driving occurs close to the surface. The higher the frequency, the closer to the surface.

5. Granulation (visible continuum).
   

   Size Spectrum

   

   But beware: any image with sharp edged features (such as ) produces such a distribution.

   Emergent intensity distribution (visible continuum).

   

6. Photospheric line profiles (visible)

   Line Formation: Fe I+II lines

   • LTE (use majority species, weak lines)
   • Accurate wavelengths and gf-values
   • No free parameters (no micro- or macroturbulence, damping enhancement, etc)

   Line widths, shifts, and shapes provide constraints

   • Width <-> flow velocity (thermal speed is small)
   • Shift <-> temperature-velocity correlations
   • Shape <-> details of convective overshoot, cf bisectors

   Excellent agreement exists between simulated and observed profiles of weak and intermediate strength FeI and FeII lines.

   Without the convective and wave velocities, the line profiles would differ drastically from the observed profiles:
   

   Including velocities and sufficient resolution, produces close agreement between simulated and observed profiles:
   2D simulations do not give observed profiles
   

   The average profile is a combination of profiles with very different shifts, widths and shapes.  (Thick red line is average profile)
   

   Line shapes depend on the details of the convection overshooting and are revealed in the line bisectors.   Note: gravitational redshift is removed.

   

   Sufficient resolution is necessary,

   

     Small differences exist between synthetic and observed profiles for strong FeI lines. These can be used to improve the physics of the upper photosphere (and chromosphere).

How do we know the simulations converge to the real thing?

  We are solving the right equations

• verify in benchmark cases

  The results satisfy a number of observational constraints

• Convection topology and flow
• Helioseismological constraints
• Emergent flux distribution and limb- darkening
• Intensity brightness contrast
• Spectral line shapes and shifts
• errors in general do not conspire to cancel

  Remaining differences are signature of missing physics

• path to improved understanding

Agreement with observations gives confidence in model of convection

• Radiative cooling in thin surface thermal boundary layer produces low entropy gas that form cores of downdrafts.
• Low entropy downdrafts generate most buoyancy work.
  

  movie (127 Mb)

  
  

• Topology controlled by mass conservation.
  What comes up must flow out sides and turnover within a scale height.

• Convection characterized by turbulent downdrafts and smooth upflows, not hierarchy of eddies.

  

  entropy fluctuations in a downdraft (31 Mb movie)
  (1.2 Mm)² x 2.5 Mm deep

• Vorticity primarily in intergranular lanes and downdrafts.
  

  

  

  vorticity in same downdraft (31 Mb movie)
  (1.2 Mm)² x 2.5 Mm deep

  vorticity movie showing ring vortex and trailing vortex tubes (127 Mb)
  

• Horizontal scale increases with increasing depth.
  No distinct mesogranular cell size.
  

  Vorticity slices

  

  Velocity slices