Settling of inertial particles in turbulent Rayleigh-Benard convection

The settling behavior of small inertial particles in turbulent convection is a fundamental problem across several disciplines, from geophysics to metallurgy. In a geophysical context, the settling of dense crystals controls the mode of solidification of magma chambers and planetary scale magma oceans, while rising of light bubbles of volatiles drives volcanic outgassing and the formation of primordial atmospheres. Motivated by these geophysical systems, we perform a systematic numerical study on the settling rate of particles in a rectangular two-dimensional Rayleigh-Benard system with Rayleigh number up to 10(12) and Prandtl number from 10 to 50. Under the idealized condition of spherically shaped particles with small Reynolds number, two limiting behaviors exist for the settling velocity. On the one hand, Stokes law applies to particles with small but finite response time, leading to a constant settling rate. On the other hand, particles with a vanishing response time are expected to settle at an exponential rate. Based on our simulations, we present a physical model that bridges the gap between the above limiting behaviors by describing the sedimentation of inertial particles as a random process with two key components: (i) the transport of particles from vigorously convecting regions into sluggish, low-velocity "piles" that naturally develop at the horizontal boundaries of the system, and (ii) the probability that particles escape such low-velocity regions without settling at their base. In addition, we identify four distinct settling regimes and analyze the horizontal distribution of sedimented particles. For two of these regimes settling is particularly slow and the distribution is strongly nonuniform, with dense particles being deposited preferentially below major clusters of upwellings. Finally, we apply our results to the crystallization of a magma ocean. Our prediction of the characteristic residence times is consistent with fractional crystallization, i.e., with the efficient separation of dense crystals from the residual lighter fluid. In absence of an efficient mechanism to reentrain settled particles, equilibrium crystallization appears possible only for particles with extremely small density contrasts.