Curvature, heat flow and normal stresses in two-dimensional models of mantle convection

Abstract

Predictions of surface heat flow and normal stresses at the boundaries from two-dimensional finite-difference models of mantle convection in curvilinear coordinates are compared with similar predictions from plane layer models. Curvature effects are parametrized in terms of f, the ratio of the radii, or f a, the ratio of the areas, of the inner and outer bounding surfaces of the model solutions. Suites of numerical solutions with values of f ranging from 0.1 to 1.0 are presented for both cylindrical shell and axisymmetric spherical shell geometry. For either shell geometry it is shown that heat flow decreases with decreasing f according to a simple geometrical scaling factor which depends only on the ratio f. Results from the two shell geometries agree for common values of f a. In cylindrical geometry it is shown that decreasing f leads to an asymmetry in the distribution of normal stresses (which produce topography) at the upper surface relative to the lower surface. In the constant-viscosity models studied here, this asymmetry results in a maximum normal stress at the lower boundary which exceeds that at the upper boundary by about 60% for values of f appropriate to the whole mantle. Thus, curvature alone contributes to producing differences in the magnitude of stresses at the two bounding surfaces, although this contribution may be small compared with those of varying material properties, such as viscosity.