Efficient volatile exchange between atmosphere and magma ocean

The formation and earliest evolution of a secondary atmosphere is tightly linked to its underlying magma ocean. Our current understanding of this coupled evolution is mainly built on thermal evolution coupled to chemical equilibrium models, which inherently assumes instant chemical exchange between the atmosphere and magma ocean. However, some recent numerical models [1,2] have challenged this assumption.  In this work, we address the issue both theoretically and numerically.

Volatile transport within the bulk of the magma ocean can, to a certain extent, be approximated as a passive particle diffusion process. Even when the buoyancy of volatiles is neglected, we demonstrate through two complementary approaches that the bulk transport is rapid. First, we extend a theoretical model for turbulent diffusion whose predictions align well with numerical simulations, which enables to replace empirical constants with more fundamental parameters. When extrapolated to magma ocean conditions, the characteristic diffusion timescale is found to be significantly shorter than the expected lifetime of the magma ocean. Second, we perform numerical experiments by initializing a passive scalar field at mid-depth in a statistically steady-state turbulent convection simulation. The evolution of its distribution, governed by an advection-diffusion equation, shows that the initial central peak flattens within just a few free-fall time units, which is a direct indicator of vigorous turbulent mixing.

The seemingly inefficient transport observed in some recent studies may be attributed to the behavior of a compositional boundary layer, which forms in conjunction with a laminar velocity boundary layer near the top surface. We analytically derive the composition flux across a no-slip boundary layer, which is supposed to scale with the chemical diffusivity and the square root of a characteristic Reynolds number. Numerical simulations show good agreement with this prediction. Nonetheless, this boundary-layer bottleneck is unlikely to significantly limit vertical volatile transport under realistic magma ocean conditions, for several reasons:
- Volatile parcels could grow in size as they approach the boundary layer, when buoyancy becomes significant and  the "passive particles" assumption no longer holds
- Even a no-slip boundary layer can be turbulent at the relevant extremely high Rayleigh number, where vertical transport is much more efficient than in a low-Ra laminar boundary layer
- The atmosphere-magma ocean interface is a free-surface, instead of a no-slip or free-slip wall

Building on recent findings that rotation significantly alters magma ocean dynamics (e.g., [3]), our future research will incorporate rotational effects to develop a more comprehensive understanding of volatile transport efficiency.

References:
[1] Salvador, A. & Samuel, H.  Icarus 390, 115265 (2023).
[2] Walbecq, A., Samuel, H. & Limare, A. Icarus 434, 116513 (2025).
[3] Maas, C. & Hansen, U. EPSL 513, 81–94 (2019).