Convective penetration exists and we found it*

Most stars host convection zones in which heat is transported directly by fluid motion. Parameterizations like mixing length theory adequately describe convective flows in the bulk of these regions, 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 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" 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 the extent of penetration is controlled by P. We test this theory using 3D numerical simulations which employ a simplified Boussinesq model of stellar convection. We find significant convective penetration in all simulations. Our simple theory describes the simulations well. In stellar contexts, we expect P ≈ 1 and in this regime our results suggest that convection zones may extend beyond the Schwarzschild boundary by up to ∼20-30% of a mixing length.