Free convective boundary layers in variable-viscosity fluids by the method of local nonsimilarity: Application to plumes in the Earth's mantle

Abstract

The effects due to departures from local similarity in steady-state boundary layers ascending through a fluid with strongly variable viscosity are examined with the local-nonsimilarity method. Both the absolute temperature and the hydrostatic pressure appear in the argument of an exponential in the viscosity function. The fluid-dynamical system studied here is that which characterizes plume structures in the Earth's mantle. By means of an iterative approach, two successive nonlinear boundary value problems are solved simultaneously and the errors incurred in the locally similar solutions are then assessed from a comparison between the first (locally similar) and the second level of a system of truncated equations. Three different sources of nonsimilarity have been considered: 1) localized radiogenic heating within the plume, 2) ambient thermal stratification, 3) pressure dependence of mantle rheology. Of particular interest is an appraisal of the degree of accuracy of the locally similar solutions as a function of viscosity contrast within the boundary layer. For the range of viscosity contrasts examined, up to 108, the velocity and temperature fields between the first- and second-level solutions differ at most by 20 to 30%, for the rheological parameter values relevant to the Earth's mantle.