Hydrodynamic simulations with a radiative surface
We solve the equations of radiation hydrodynamics to compute the time evolution toward one-dimensional equilibrium solutions using ageneralized Kramers opacity, κ=κ0 ρa Tb, with adjustable prefactor κ0 and exponents a and b on density ρ and temperature T, respectively. We choose our initial conditions to be isothermal and find that the early time evolution away from the isothermal state is fastest near the height where the optical depth is unity, and is slower both above and below it. In all cases where the quantity n=(3-b)/(1+a) is larger than -1, we find a nearly polytropic solution with ρTn in the lower part and a nearly isothermal solution in the upper part with a radiating surface in between, where the optical depth is unity. In the lower part, the radiative diffusivity is found to be approximately constant, while in the upper optically thin part it increases linearly. Interestingly, solutions with different parameter combinations a and b that result in the same value of n are rather similar, but not identical. Increasing the prefactor increases the temperature contrast and lowers the value of the effective temperature. We find that the Péclet number based on sound speed and pressure scale height exceeds numerically manageable values of around 104 when the prefactor κ0 is chosen to be approximately six orders of magnitude below the physically correct value. In the special case where a=-1 and b=3, the value of n is undetermined and the radiative diffusivity is strictly constant everywhere. In that case we find a stratification that is approximately adiabatic. Finally, exploratory two-dimensional calculations are presented where we include turbulent values of viscosity and diffusivity and find that onset of convection occurs when these values are around 31013 cm2 s-1. The addition of an imposed horizontal magnetic field suppresses small-scale convection, but has not led to instability in the cases investigated so far.
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