Centerline temperature of mantle plumes in various geometries: Incompressible flow

We investigate the temperature at the center of the vertically rising and sinking limbs of convection cells as a function of geometry. We carried out finite element numerical simulations of steady state convection in three different two-dimensional domains, a cartesian box, an axisymmetric cylinder, and an axisymmetric spherical shell, and looked at the centerline temperature as a function of height for the sheets and/or axial plumes at cell boundaries. We found that there is a significant change in centerline temperature near the base of the sheets due to conduction of heat in the horizontal-direction. This drop does not occur for the axial plumes, so that generally temperature loss from plumes is significantly less. Numerical results were compared with the predictions of simplified mathematical models, and key parameters controlling temperature loss were identified in terms of boundary layer velocities, Rayleigh number Ra, and cell shape. Centerline temperature loss decreases strongly as the aspect ratio (length/height of the cell) increases for both sheets and plumes, but while the loss is nearly independent of Ra for sheets, for plumes it decreases approximately as Ra^-1/3 For a bottom heated cell of aspect ratio 1, centerline temperature loss is about 60-80% for sheets and about 8-10% for plumes at Ra = 10⁶. We also find that with the same catchment area, plumes are about 2-3 times wider than sheets. These results may be applied to two-dimensional simulations of steady convection and can address questions regarding the extent that temperature is preserved in mantle plumes in their upward passage.