We present the first detailed three-dimensional (3D) hydrodynamic implicit large eddy simulations of turbulent convection of carbon burning in massive stars. Simulations begin with radial profiles mapped from a carbon burning shell within a 15$\,\textrm{M}_\odot$ one-dimensional stellar evolution model. We consider models with $128^3$, $256^3$, $512^3$ and $1024^3$ zones. The turbulent flow properties of these carbon burning simulations are very similar to the oxygen burning case. We performed a mean field analysis of the kinetic energy budgets within the Reynolds-averaged Navier-Stokes framework. For the upper convective boundary region, we find that the numerical dissipation is insensitive to resolution for linear mesh resolutions above 512 grid points. For the stiffer, more stratified lower boundary, our highest resolution model still shows signs of decreasing sub-grid dissipation suggesting it is not yet numerically converged. We find that the widths of the upper and lower boundaries are roughly 30% and 10% of the local pressure scale heights, respectively. The shape of the boundaries is significantly different from those used in stellar evolution models. As in past oxygen-shell burning simulations, we observe entrainment at both boundaries in our carbon-shell burning simulations. In the large P��clet number regime found in the advanced phases, the entrainment rate is roughly inversely proportional to the bulk Richardson number, Ri$_{\rm B}$ ($\propto $Ri${\rm_B}^{-��}$, $0.5\lesssim ��\lesssim 1.0$). We thus suggest the use of Ri$_{\rm B}$ as a means to take into account the results of 3D hydrodynamics simulations in new 1D prescriptions of convective boundary mixing.

26 pages, 15 figures, accepted for publication in MNRAS, movie available at the following URL: http://www.astro.keele.ac.uk/shyne/321D/convection-and-convective- boundary-mixing/visualisations/very-high-resolution-movie-of-the-c-shell/view