  Planetary atmospheres are commonly thought to result from the efficient
outgassing of cooling magma oceans. During this stage, vigorous convective
motions in the molten interior are believed to rapidly transport the dissolved
volatiles to shallow depths where they exsolve and burst at the surface. This
assumption of efficient degassing and atmosphere formation has important
implications for planetary evolution, but has never been tested against fluid
dynamics considerations. Yet, during a convective cycle, only a finite fraction
of the magma ocean can reach the shallow depths where volatiles exsolution can
occur, and a large-scale circulation may prevent a substantial magma ocean
volume from rapidly reaching the planetary surface. Therefore, we conducted
computational fluid dynamics experiments of vigorous 2D and 3D
Rayleigh-B\'enard convection at Prandtl number of unity to characterize the
ability of the convecting fluid to reach shallow depths at which volatiles are
exsolved and extracted to the atmosphere. Outgassing efficiency is essentially
a function of the magnitude of the convective velocities. This allows deriving
simple expressions to predict the time evolution of the amount of outgassed
volatiles as a function of the magma ocean governing parameters. For plausible
cases, the time required to exsolve all oversaturated water can exceed the
magma ocean lifetime in a given highly vigorous transient stage, leading to
incomplete or even negligible outgassing. Furthermore, the planet size and the
initial magma ocean water content, through the convective vigor and the
exsolution depth, respectively, strongly affect magma oceans degassing
efficiency, possibly leading to divergent planetary evolution paths and
resulting surface conditions. Overall, despite vigorous convection, for a
significant range of parameters, convective degassing appears not as efficient
as previously thought.
