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 \times 10 ^ { -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 \approx 0.17 H _ { P } .
The exponentially decaying diffusion CB mixing model with f = 0.03 reproduces the spherically-averaged 3D abundance profiles .