Stellar convective penetration: parameterized theory and dynamical simulations

Most stars host convection zones in which heat is transported directly by fluid motion, but the behavior of convective boundaries is not well understood. Here we present 3D numerical simulations which exhibit penetration zones: regions where the entire luminosity \emph{could} be carried by radiation, but where the temperature gradient is approximately adiabatic and convection is present. To parameterize this effect, we define the "penetration parameter" $\mathcal{P}$ which compares how far the radiative gradient deviates from the adiabatic gradient on either side of the Schwarzschild convective boundary. Following Roxburgh (1989) and Zahn (1991), we construct an energy-based theoretical model in which $\mathcal{P}$ controls the extent of penetration. We test this theory using 3D numerical simulations which employ a simplified Boussinesq model of stellar convection. The convection is driven by internal heating and we use a height-dependent radiative conductivity; this allows us to separately specify $\mathcal{P}$ and the stiffness $\mathcal{S}$ of the radiative-convective boundary. We find significant convective penetration in all simulations. Our simple theory describes the simulations well. Penetration zones can take thousands of overturn times to develop, so long simulations or accelerated evolutionary techniques are required. In stars, we expect $\mathcal{P} \approx 1$ and in this regime our results suggest that convection zones may extend beyond the Schwarzschild boundary by up to $\sim$20-30% of a mixing length. We present a MESA stellar model of the Sun which employs our parameterization of convective penetration as a proof of concept. We discuss prospects for extending these results to more realistic stellar contexts.