Stellar Convective Penetration: Parameterized Theory and Dynamical Simulations

Abstract

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 that 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”  , which compares how far the radiative gradient deviates from the adiabatic gradient on either side of the Schwarzschild convective boundary. Following Roxburgh and Zahn, we construct an energy-based theoretical model in which  controls the extent of penetration. We test this theory using 3D numerical simulations that 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  and the stiffness  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  ≈ 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. We present a MESA stellar model of the Sun that employs our parameterization of convective penetration as a proof of concept. Finally, we discuss prospects for extending these results to more realistic stellar contexts.

Schwarzschild and Ledoux are equivalent on evolutionary timescales

“…These implementations should include time-dependent entrainment models to properly advance convective boundaries (e.g., Turner 1968;Fuentes & Cumming 2020). Anders et al (2022a) showed convective motions can extend significantly into the radiative zones of stars via "penetrative convection." In this work, we used parameters which do not have significant penetration.…”

Convective Penetration in Early-type Stars

“…There are several uncertainties that require further study to provide a complete theory of convective penetration. For instance, Anders et al (2022) only examined Cartesian domains in the Boussinesq limit. For stellar cores a spherical geometry is more appropriate, and may change the obtained values of f and ξ in Equation (8), though we do not expect this to change the orders of magnitude or the trends we see.…”

“…Building on Zahn (1991) and Roxburgh (1989), Anders et al (2022) developed a theory of convective penetration to predict the extent of the PZ and calibrated that theory to 3D Cartesian hydrodynamical simulations of Boussinesq convection. 5 This theory is based on an exact rewriting of the time-and-spatially averaged energy equation, accounting for all energy fluxes, sources, and sinks.…”

“…While Anders et al (2022) did not include the effects of composition gradients, we do not expect these to change the equilibrium extent of the PZ. Overshooting motions still occur when there is a composition gradient at the convective boundary.…”

“…Recently, Anders et al (2022) proposed a theoretical model of convective penetration and calibrated this to 3D hydrodynamical simulations of Boussinesq convection. Here we study the implications of the resulting calibrated model in early-type stars.…”