Diffusive braking of penetrative convection in stably-stratified fluids

Mixing at the interface between a convection zone and an adjacent, stably-stratified layer plays a crucial role in shaping the structure and evolution of stars and planets. In this work, we present a suite of 2D and 3D Boussinesq simulations that explore how bottom-driven convection penetrates a compositionally stratified region. Our results reveal two distinct regimes: a penetrative regime, where the convection zone steadily grows by entraining fluid from above, and a stalled regime, where growth halts and transitions to overshooting convection. We extend classical entrainment theory by incorporating thermal and compositional diffusion and by deriving a modified entrainment law that predicts interface speeds in the weak-diffusion limit. We show that convection stalls when the interface speed becomes comparable to the compositional diffusion speed and validate the transition between behaviors across a wide parameter space of Richardson and Lewis numbers. Such diffusively controlled stalling is unlikely to occur in stellar and planetary interiors, where the Lewis number is typically large and compositional diffusion is extremely slow. In these environments, compositional diffusion will merely slow the growth of the convection zone, and convective boundaries are expected to stall only in the presence of other curtailing mechanisms such as strong radiative diffusion or rapid rotation.