Diffusive Braking of Penetrative Convection in Stably-Stratified Fluids

TL;DR

The paper investigates how diffusion influences penetrative convection at the interface between a convection zone and a stable layer, using 2D and 3D Boussinesq simulations. It extends classical entrainment theory by incorporating thermal and compositional diffusion, deriving a diffusion-aware entrainment law that predicts the interface speed $V_{ m cb}$ in the weak-diffusion limit and reveals a critical condition when $V_{ m cb}$ matches the compositional-diffusion speed $V_C=(kappa_C/t)^{1/2}$. Two regimes emerge: continuous penetrative growth and a diffusively driven stall with overshoot, with the transition governed by Ri, Le, and diffusion strengths. In astrophysical contexts where the Lewis number is large, diffusively driven stalling is unlikely, and diffusion mainly slows the growth of the convective boundary or requires additional processes such as radiative diffusion or rotation to halt expansion. The results provide a framework for predicting convective boundary evolution and heavy-element transport in stars and planets, highlighting when diffusion can meaningfully modify boundary dynamics.

Abstract

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 into 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.

Figures (6)

• Figure 1: The behavior of the suite of simulations as a function of the Lewis Number, $\mathrm{Le}$, and the Richardson Number, $\mathrm{Ri}$. All of the simulations have the same flux Rayleigh Number, $\mathrm{Ra}= 10^8$, and Prandtl Number, $\Pr = 0.1$. The black squares indicate 2D simulations that exhibit penetrative convection with a rapidly ascending convective boundary that eventually chews through the entire computational domain. Red disks correspond to 2D simulations that eventually reach a diffusive quasi-equilibrium where the convective interface stalls; these models enter a regime of overshooting convection. The large circles correspond to 3D simulations that also exhibit penetrative (black) or stalled (red) convective interfaces. The blue dotted line corresponds to parameter values where the analytically derived speed of the convective interface matches the compositional diffusion speed, see Equation (\ref{['eqn:Vcb=VC']}).
• Figure 2: Snapshots from a 3D simulation exhibiting penetrative convection. The simulation has a relatively large initial stratification and weak compositional diffusion ($\mathrm{Ri}=5$, $\mathrm{Le}=10$). Each image shows the vertical component of the velocity. Three distinct times have been illustrated: (a) An early time just after the onset of convection. (b) An intermediate time when the convection zone is fully formed and has entrained roughly half the spatial domain. (c) A late time when the convection zone has entrained about 75% of the domain. In all snapshots, the convection evinces fast upflows and downflows, but even in the overlying stable layer weak flows appear due to the presence of internal gravity waves excited by pummeling of the convective boundary by rising plumes.
• Figure 3: Vertical profiles and a time trace from a 3D simulation ($\mathrm{Ri} = 1$, $\mathrm{Le} = 10$) exhibiting penetrative convection. The vertical profiles have been measured by averaging the illustrated field horizontally and temporally. (a) Vertical profiles of density fluctuations induced by composition and temperature $\rho^\prime/\rho_0 = \beta C-\alpha T$. The dotted curves correspond to early times during the diffusive stage before the onset of convection, whereas the solid curves indicate later times during the convective stage. The thick dotted curve corresponds to the initial stratification and the thick solid curve is for a single time which we have emphasized to make the profile shapes readily apparent. (b) The thickness of the convection zone as a function of time for both the 2D and 3d models. Only times after the convection zone is formed are shown. The time emphasized in panel a is indicated with the violet circle. The black reference line indicates a power law $h\propto t^{1/2}$. (c) Profiles of the vertical heat flux normalized by the heat flux imposed at the lower boundary, $F_*$. The diffusive and turbulent fluxes are shown separately, as well as added together. (d) Profiles of the vertical composition flux normalized by the flux imposed at the upper surface, $F_C=\kappa_C (\beta g)^{-1} N_0^2$. Both the thermal and compositional fluxes are extracted from the simulation at the time indicated by the violet circle in panel b.
• Figure 4: Nondimensional convective boundary speed from the simulations (squares) and from theory (curves) as a function of the Richardson and Lewis numbers. As indicated in the caption, the colors indicate either the Lewis number or Richardson number depending on the panel. The convective boundary speed was obtained from those simulations undergoing penetrative convection by fitting the height of the convective boundary with a power law in time, $h = q^{-1} V_{\rm cb} \, (\kappa_T t)^q$. The theoretical curves come from Equation (\ref{['eqn:Vi']}), which is applicable in the weak diffusion limit.
• Figure 5: Vertical profiles and time traces from a 3D simulation ($\mathrm{Ri} = 10$, $\mathrm{Le} = 10$) exhibiting a stalled convective boundary. The curves and colors have the same meaning as in Figure \ref{['fig:fast_profs']}. (a) Profiles of density fluctuations. (b) The height of the convective boundary as a function of time for both the 3D and 2D models with $\mathrm{Ri} = 10$ and $\mathrm{Le} = 10$. (c) Profiles of the normalized vertical heat flux. (d) Profiles of the normalized vertical composition flux. Both of the fluxes are extracted from the simulation at the time indicated by the violet circle in panel b.