Three-dimensional numerical experiments on thermal convection in a very viscous fluid: Implications for the dynamics of a thermal boundary layer at high Rayleigh number

Abstract

In this study we present results from three-dimensional numerical experiments on thermal convection in a volumetrically heated, infinite Prandtl number fluid cooled from above. At high Rayleigh number, a thin thermal boundary layer forms adjacent to the cold top boundary. On the basis of our numerical results we study the thermal structure and dynamics of this boundary layer and the population of plumes that it creates. Cold thermal plumes that develop by boundary layer instability form continuous nearly vertical columns that migrate horizontally sweeping off the unstable boundary layer. A plume usually persists until it coalesces with another plume. The average spacing of plumes, inferred from the variation of the observed number of plumes with Rayleigh number, is proportional to (δd)1/2, where δ and d are the thermal boundary layer and fluid layer thicknesses, respectively. Based on a “kinetic theory” of plume populations, we show that this is consistent with an equilibrium plume population in which the creation of plumes by boundary layer instability and their disappearance by coalescing with other plumes are balanced. This scaling of average plume spacing is a consequence of the width of velocity plumes in a very viscous (infinite Prandtl number) fluid comparable to the fluid layer depth. For finite Prandtl number, the same analysis but with temperature and velocity plumes of comparable width predicts a plume spacing proportional to the boundary layer thickness.