A Simple Model of Plate Generation from Mantle Flow

A simple model of non-Newtonian creeping flow is used to evaluate classes of rheologies which allow viscous mantle flow to become plate-like. The model describes shallow-layer lithospheric motion driven by sources and sinks. The sources represent spreading ridges while the sinks represent subduction zones; the sources and sinks thus also prescribe the poloidal component of the surface flow field. The toroidal (strike-slip) component of the flow field is found via the solution of the Stokes equation with non-Newtonian rheology. As a first basic investigation of the model, the horizontal divergence from the two-dimensional rectangular velocity field of Olson & Bercovici (1991) is used for the source-sink field. The degree to which the induced fluid flow reproduces the rectangular plate is used to measure the success of different rheologies in generating plate-like flows. Results indicate that power-law rheologies, even in the limit of very high power-law index ν, can only produce modest plate-like flow. For example, the ratio of toroidal to poloidal kinetic energy for a source-sink field derived from a square plate is at best 0.65, whereas a perfect square plate has a ratio of 1.0. Moreover, the power-law rheology appears to reach an asymptotic limit in its ability to produce plate-like behavior. This implies that plate tectonics is unlikely to arise from a power-law rheology even in the limit of very high ν. A class of rheologies that yield significantly more promising results arise from the Carreau pseudo-plastic rheology with the power-law index taken to be ν < 0. One rheology in this class is the continuum model for stick-slip, earthquake behavior of Whitehead & Gans (1974), which is essentially the Carreau equation with ν = −1. This class of rheologies, referred to as the stick-slip rheologies, induces a toroidal to poloidal kinetic energy ratio for a square plates’s source-sink function which can be as high as 0.9. The viscosity (or strength) distribution for this class of rheologies also appears more plate-like, showing fairly uniform high viscosity regions (pseudo-plates) and sharply defined low viscosity zones (pseudo-margins). In contrast, even the most nonlinear power-law rheology produces spatially varying high viscosity regions and relatively smooth low viscosity margins. The greater success of the stick-slip rheologies in producing plates is attributed to a self-lubricating mechanism in which the transfer of momentum from regions of high shear to low shear is inhibited. In contrast, even in the limit of infinite power-law index, a power-law rheology can retard but never prohibit momentum transfer. This feature is essential to the sharpening of velocity profiles into plate-like profiles, which is illustrated with a simple boundary-layer theory.