  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.
