The dynamics of thin shells with variable viscosity and the origin of toroidal flow in the mantle

Abstract

SUMMARY The observed equipartition of kinetic energy between the poloidal (convergence/divergence) and toroidal (strike-slip) parts of the Earth’s surface velocity field is evidence that large lateral viscosity variations exist in the mantle. The largest such variations probably occur within the lithosphere, in the form of relatively weak boundaries between nearly rigid plates. To understand better the dynamical effects of such viscosity variations, we study an Earth model consisting of a lithospheric shell of outer radius a with laterally variable thickness h(8, q) and viscosity ti(@, q) overlying a mantle whose viscosity fj(r) is a function of depth only. Flow in the model is driven by lateral density variations beneath the shell. An asymptotic (long wave) analysis reduces the dynamics of the shell to a pair of boundary conditions, and shows that the effect of the shell is governed by a single dimensionless parameter, the ‘stiffness’ f = ~(y - l), where E = h/a and y = fj/fj(u). A perturbation analysis for the case of small lateral stiffness variations shows that the poloidal flow generated by an internal load interacts with the stiffness variations to generate additional poloidal and toroidal fields of various wavelengths according to definite selection rules. A simple model in which the shell consists of stiff (f >> 1) ‘plates’ separated by narrow weak (f<O.5) zones shows that (1) large lateral stiffness variations give rise to surface motion with a substantial toroidal component; (2) the observed surface velocity field is extremely sensitive to lateral stiffness variations, and thus provides poor constraints on the mantle viscosity; (3) lateral stiffness variations have a large (-50 per cent) effect on geoid anomalies if the mantle viscosity increases with depth; and (4) geoid anomalies predicted by models that ignore the order-unity relative change of long-wavelength normal stress across the shell are likely to be in error by 30-50 per cent. These results suggest that previous models may have overestimated the viscosity increase in the lower mantle that is required to fit the geoid data.