Convective Cooling of a Initially Stably Stratified Fluid with Temperature Dependent Viscosity

Convective cooling of a stably stratified fluid with a strongly temperature dependent viscosity is important in understanding the thermal evolution of the oceanic upper mantle, cratonic lithosphere, and planets. Potential causes of compositional stratification are partial melting due to adiabatic decompression and magmatic differentiation. In the absence of compositional stratification, we explore the relationship between temperature dependent viscosity and the onset time of convection and subsequent thermal evolution using 2D numerical experiments with secular cooling. For secular cooling after the initiation of convection, the viscosity ratio across the convecting thermal boundary layer (ctbl) evolves to nearly constant value independent of the temperature dependence of the viscosity comparable to that for volumetric heating. However in the initial stages of convection, this ratio is significantly larger and varies inversely with the temperature dependence of the viscosity. In the presence of an initial stable compositional stratification, the convective motions in isoviscous fluids are initially restricted to a vertical scale proportional to (α Δ Tδ /γ ) ^1/2 where γ =1/ρ (dρ /dz) is the conductive lid thickness at the onset of convection and Δ T is the temperature across the convecting region. The onset of convection can be determined by a critical Rayleigh number, where relevant length scales are δ and the depth of the initial convective motions. The convective motions either decay in time leaving a relatively unmixed fluid or amplify to form a convecting mixed layer that grows in time. Thickening of the convecting region occurs by the penetration of subsequent downwellings (plumes) further into the underlying stratified fluid. Following buoyancy arguments, the amount of cooling is proportional to the convecting region thickness. Initial numerical experiments with both temperature dependent viscosity and compositional stratification show behavior similar to the isoviscous experiments. There again exists a finite amplitude oscillatory mode of convection which transfers more heat than purely conductive cooling but introduces little to no chemical mixing. Mixed layer growth is expected to be controlled by the thickness and temperature difference across the ctbl.