Controls on entrainment of a dense chemical layer by thermal plumes

Results from two-dimensional axisymmetric numerical models of a chemically dense layer entrained by thermal plumes, featuring a plume temperature structure fixed a priori as a Gaussian function and strongly temperature-dependent viscosity, show that the entrainment rate, Q, is controlled by the radius of the thermal plume, r_T, the ratio of excess chemical to thermal buoyancy, R_b, and viscosity contrast across the two layers, γ. Fitting of the numerical results shows that Q ∼ γ^0.65 r_T^3.90 R_b^{- 2.89 γ0.17}. A smaller r_T and/or a larger R_b results in a smaller entrainment rate, as expected. However, the dependence of Q on R_b is much stronger than previously suggested if the fluid viscosity is strongly temperature dependent. The effects of γ on Q depend on R_b because the scale of Q on R_b is dependent on γ, and when R_b is around 1.8, the dependence of Q on γ is weak. The results also show that the dependences of the radius, r_c, and the vertical velocity, V_zc, of the chemical plume on R_b are also much stronger than previously suggested if the fluid viscosity is strongly temperature dependent. Scaling the numerical results to the Earth's mantle suggests that the chemically dense layer in the mantle may survive through the age of the Earth with insignificant entrainment, e.g. in a mantle with a reference viscosity of 5 × 10²¹ Pa s and a reference density of 4000 kg/m³, a thermal plume that has a radius of 80 km, a temperature of 600 K hotter and a viscosity of 100 times lower at the center of the plume than in the ambient mantle can only entrain less than 5% of the bottom chemical layer that is 300 km thick, 0.9% denser than the above mantle (R_b = 1.5) and spread over an area of size ∼0.5D² (D = 2900 km, thickness of the mantle) through the age of the Earth.