On the Purpose of Toroidal Motion in a Convecting Mantle

The purpose of toroidal flow, i.e., strike-slip motion and plate spin, in the plate-tectonic style of mantle convection is enigmatic. It is a purely horizontal, dissipative flow field that makes no apparent contribution to the release of heat. However, when plate-like toroidal motion is allowed to arise as a phenomenon of non-Newtonian mantle dynamics, it in fact acts to reduce the net amount of viscous dissipation. We show that for a non-Newtonian flow driven by an existing poloidal field (a source-sink field), the generation of toroidal motion interacts with the nonlinear rheology to cause less viscous dissipation than if there were no toroidal motion. With power-law rheologies, the generation of toroidal motion causes up to a 25% reduction in viscous dissipation. With a self-lubricating rheology, which has been shown to induce the most plate-like behavior, toroidal motion causes as much as an 80% reduction in viscous dissipation. Thus, basic non-Newtonian fluid dynamical theory of the formation of plate tectonics shows that toroidal motion is far from superfluous but in fact facilitates the efficiency of convective flow at the surface. Introduction One of the most fundamental goals of geodynamics is to understand how plate tectonics arises as a convective selforganizing structure from the lithosphere-mantle system. A defining feature of the plate-tectonic style of mantle convection is the existence of toroidal flow, i.e., motion along transform faults, oblique subduction and in the bulk spin of plates. Such motion at the Earth’s surface has almost as much kinetic energy as the poloidal part which is manifest at the surface in convergent and divergent motion [Hager and O’Connell, 1978, 1979, 1981; O’Connell et al., 1991; Cadek and Ricard, 1992; Lithgow-Bertelloni et al., 1993]. Poloidal motion throughout the mantle is associated with upwellings and downwellings; it is linked to heat transport and directly driven by buoyancy forces. Toroidal motion, on the other hand, is apparently superfluous. Since it derives and dissipates energy from the poloidal flow [see also Forte and Peltier, 1987; Gable et al., 1991; Ribe, 1992], and only involves horizontal motion, it would not seem to facilitate efficient release of gravitational potential energy or heat. Thus, the purpose of such motion in a convecting medium such as the plate-mantle system is enigmatic. Non-Newtonian fluid dynamical theory of plate generation [e.g., Ribe, 1992; Weinstein and Olson, 1992; Bercovici 1993, 1995], however, can be used to show that toroidal motion in fact serves a very important purpose. In this paper we demonstrate that the generation of the toroidal flow field acts to reduce the net viscous dissipation of the convecting system, in the top thermal boundary layer or lithosphere, in particular. If toroidal motion arises from the interaction of a non-Newtonian or nonlinear rheology with the convective (i.e., poloidal) flow, then the localization of strike-slip shear into narrow zones with low viscosity can greatly reduce the dissipation of work done to move material from a source (a spreading center) to a sink (a convergent or subduction zone). Thus, although the toroidal field is added onto a preexisting poloidal field, it acts in concert with the nonlinear plate-mantle rheology to lubricate the mass transfer of material from divergent to convergent zones. The non-Newtonian source-sink model One of the simplest theories describing how toroidal motion arises as a fluid dynamical phenomenon from the interaction of nonlinear rheology and convective flow was proposed by Bercovici [1993, 1995]. In this model, shallowlayer lithospheric motion is driven by sources and sinks which are a proxy for convective motion in that they represent spreading centers and subduction zones, respectively.