Fundamentals of laminar free convection in internally heated fluids at values of the Rayleigh–Roberts number up to $10^{9}$

Motions in the solid mantle of silicate planets are predominantly driven by internal heat sources and occur in laminar regimes that have not been systematically investigated. Using high-resolution numerical simulations conducted in three dimensions for a large range of Rayleigh-Roberts numbers (5 x 10(3) <= Ra-H <= 10(9)), we have determined the characteristics of flow in internally heated fluid layers with both rigid and free slip boundaries. Superficial planforms evolve with increasing Ra-H from a steady-state tessellation of hexagonal cells with axial downwellings to time-dependent clusters of thin linear downwellings within large areas of nearly isothermal fluid. The transition between the two types of planforms occurs as a remarkable flow polarity reversal over a small Ra-H range, such that downwellings go from isolated cylindrical structures encircled by upwellings to thin interconnected linear segments outlining polygonal cells. In time-dependent regimes at large values of Ra-H, linear downwellings dominate the flow field at shallow depth but split and merge at intermediate depths into nearly cylindrical plume-like structures that go through the whole layer. With increasing Ra-H, the number of plumes per unit area and their velocities increase whilst the amplitude of thermal anomalies decreases. Scaling laws for the main flow characteristics are derived for Ra-H values in a 10(6)-10(9) range. For given Ra-H, plumes are significantly colder, narrower and wider apart beneath free boundaries than beneath rigid ones. From the perspective of planetary studies, these results alert to the dramatic changes of convective planform that can occur along secular cooling.