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-Bénard 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.

中文翻译：

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湍流瑞利-贝纳德对流中惯性粒子的沉降

小惯性粒子在湍流对流中的沉降行为是从地球物理学到冶金学等多个学科的基本问题。在地球物理背景下，致密晶体的沉降控制着岩浆室和行星尺度岩浆海洋的凝固模式，而挥发性轻质气泡的上升则推动了火山岩的除气作用和原始大气的形成。受这些地球物理系统的激励，我们对矩形二维二维Rayleigh-Bénard系统中的粒子沉降速率进行了系统的数值研究，其瑞利数最大为${10}^{12}$普朗特数从10到50。在理想的雷诺数小的球形粒子的条件下，沉降速度存在两个极限行为。一方面，斯托克斯定律适用于响应时间短但有限的颗粒，从而导致稳定的沉降速率。另一方面，预期响应时间消失的粒子将以指数速率沉降。在模拟的基础上，我们提出了一个物理模型，通过将惯性颗粒的沉降描述为具有两个关键成分的随机过程来弥合上述极限行为之间的差距：（i）颗粒从剧烈对流的区域向缓慢，低速的区域迁移在系统水平边界自然发展的速度“桩”，（ii）粒子逃逸到此类低速区域而不沉降在其底部的可能性。此外，我们确定了四种不同的沉降方式，并分析了沉降颗粒的水平分布。对于这两种情况，沉降特别缓慢，并且分布非常不均匀，致密颗粒优先沉积在上升流的主要簇下面。最后，我们将结果应用于岩浆海洋的结晶。我们对特征停留时间的预测与分步结晶是一致的，即与从较轻的残余液体中有效分离致密晶体是一致的。在缺乏重新夹带沉淀颗粒的有效机制的情况下，仅对于密度对比度极低的颗粒，才可能出现平衡结晶。