Idealised hydrodynamic simulations of turbulent oxygen-burning shell   convection in 4{\pi} geometry
  This work investigates the properties of convection in stars with particular emphasis on entrainment across the upper convective boundary (CB). Idealised simulations of turbulent convection in the O-burning shell of a massive star are performed in $4\pi$ geometry on $768^3$ and $1536^3$ grids, driven by a representative heating rate. A heating series is also performed on the $768^3$ grid. The $1536^3$ simulation exhibits an entrainment rate at the upper CB of $1.33\times10^{-6}~M_\odot~\mathrm{s}^{-1}$. The $768^3$ simulation with the same heating rate agrees within 17 per cent. The entrainment rate at the upper convective boundary is found to scale linearly with the driving luminosity and with the cube of the shear velocity at the upper boundary, while the radial RMS fluid velocity scales with the cube root of the driving luminosity, as expected. The mixing is analysed in a 1D diffusion framework, resulting in a simple model for CB mixing. The analysis confirms previous findings that limiting the MLT mixing length to the distance to the CB in 1D simulations better represents the spherically-averaged radial velocity profiles from the 3D simulations and provides an improved determination of the reference diffusion coefficient $D_0$ for the exponential diffusion CB mixing model in 1D. From the 3D simulation data we adopt as the convective boundary the location of the maximum gradient in the horizontal velocity component which has $2\sigma$ spatial fluctuations of $\approx0.17 H_P$ . The exponentially decaying diffusion CB mixing model with $f = 0.03$ reproduces the spherically-averaged 3D abundance profiles. 
Samuel Jones (1,2), Robert Andrassy (2,3), Stou Sandalski (4,3),   Austin Davis (2), Paul Woodward (4,3) and Falk Herwig (2,...
astro-ph.SR