Further evidence used to argue for non-homogeneous mantle viscosity is the nearly equal energy in the poloidal and toroidal components of global plate motions within each harmonic degree [ Hager and O'Connell, 1978]. The observed plate velocities can be represented as the sum of a toroidal and poloidal component. The convergence of two plates at a trench (perpendicular to the trench) and the divergence of two plates at a mid-ocean ridge (perpendicular to the spreading axis) are examples of the poloidal component. Transform faults and any oblique components of motion at convergent or divergent plate boundaries are expressions of the toroidal component of the velocity field. In a convecting fluid with a constant viscosity, the velocity fields can be entirely expressed as poloidal fields. Velocity fields with a non-zero toroidal component are generated by a fluid with a laterally-varying viscosity. With a temperature-dependent rheology, only a small amount of toroidal flow appears to be generated [ Christensen and Harder, 1991]; however, rigid plates with weak boundaries can generate approximately equal toroidal and poloidal components of motion [ Gable et al. , 1991].
Viscous flow in a radially stratified mantle with a thin shell of variable thickness was studied by Ribe [1992]. For the case of small lateral stiffness variations, Ribe shows that a poloidal flow generated by a simple internal load generates additional poloidal and toroidal flow. For the case with stiff plates separated by narrow weak zones, Ribe finds it is possible to generate substantial toroidal flow. The surface velocity field is sensitive to the lateral stiffness variations and less sensitive to the radial viscosity structure. In addition, Ribe finds that lateral viscosity variations on the surface can have a profound effect on the geoid, in agreement with Rivine and Phipps Morgan [1993].