Anelastic and compressible simulations of stellar oxygen burning

In this paper we compare fully compressible (Meakin & Arnett 2006, 2007) and anelastic (Kuhlen et al. 2003) simulations of stellar oxygen shell burning. It is found that the two models are in agreement in terms of the velocity scale (u_c ∼ 10⁷ cm s^_1) and thermodynamic fluctuation amplitudes (e.g., ρ′/〈ρ〉 ∼2 × 10^-3) in the convective flow. Large fluctuations (∼11%) arise in the compressible model, localized to the convective boundaries, and are due to internal waves excited in stable layers. Fluctuations on the several percent level are also present in the compressible model due to composition inhomogeneities from ongoing entrainment events at the convective boundaries. Comparable fluctuations (with amplitudes greater than ∼ 1%) are absent in the anelastic simulation, because they are due to physics not included in that model. We derive an analytic estimate for the expected density fluctuation amplitudes at convective boundaries by assuming that the pressure fluctuations due to internal waves at the boundary, ρ′_w, balance the ram pressure of the convective motions, ρv_c ². The predicted amplitudes agree well with the simulation data. The good agreement between the anelastic and the compressible solution within the convection zone and the agreement between the stable layer dynamics and analytic solutions to the nonradial wave equation indicate that the compressible hydrodynamic techniques used are robust for the simulated stellar convection model, even at the low Mach numbers found, M ∼ 0.01.