Near-polytropic simulations with a radiative surface

Context. Studies of solar and stellar convection often employ simple polytropic setups using the diffusion approximation instead of solving the proper radiative transfer equation. This allows one to control separately the polytropic index of the hydrostatic reference solution, the temperature contrast between top and bottom, and the Rayleigh and Péclet numbers. Aims. Here we extend such studies by including radiative transfer in the gray approximation using a Kramers-like opacity with freely adjustable coefficients. We study the properties of such models and compare with results from the diffusion approximation. Methods. We use the PENCIL CODE, which is a high-order finite difference code where radiation is treated using the method of long characteristics. The source function is given by the Planck function. The opacity is written as κ = κ0ρaT , where a = 1 in most cases, b is varied from −3.5 to +5, and κ0 is varied by four orders of magnitude. We adopt a perfect monatomic gas. We consider sets of one-dimensional models and perform a comparison with the diffusion approximation in a two-dimensional model. Results. Except for the case with b = 5, we find one-dimensional hydrostatic equilibria with a nearly polytropic stratification and a polytropic index close to n = (3− b)/(1+ a), covering both convectively stable (n > 3/2) and unstable (n < 3/2) cases. For b = 3 and a = −1, the value of n is undefined, but the final equilibrium stratification turns out to be nearly isentropic. For large values of κ0, the thermal adjustment time becomes long, the Péclet and Rayleigh numbers become large, and the temperature contrast increases and is thus no longer an independent input parameter, unless the Stefan–Boltzmann constant is considered adjustable. Conclusions. Proper radiative transfer with Kramers-like opacities provides a useful tool for studying stratified layers with a radiative surface in ways that are more physical than what is possible with polytropic models using the diffusion approximation.