The Kinematics and Dynamics of Poloidal–Toroidal Coupling in Mantle Flow: The Importance of Surface Plates and Lateral Viscosity Variations

Alessandro M. Forte*    Department of Earth and Planetary Science, Harvard University, Cambridge, Massachusetts 02138
* Present address: Institut de Physique du Globe, Département de Sismologie, 75252 Paris, France

W. Richard Peltier    Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S IAI

1 INTRODUCTION

The ability of the mantle and lithosphere to creep over geologic time scales is due to the presence of naturally occurring atomic-scale defects in the lattice of crystal grains (e.g., Weertman and Weertman, 1975; Carter, 1976; Nicolas and Poirier, 1976; Weertman, 1978). The imposition of deviatoric stresses causes these defects to propagate and thus allows mantle material to creep or “flow.” If the ambient temperature is sufficiently high, this solid-state flow will persist so long as the stress is maintained and the deformation process may achieve steady state. The steady-state creep of crystalline substances may be characterized by a single parameter, the effective viscosity (e.g., Stocker and Ashby, 1973; Weertman and Weertman, 1975).

This effective viscosity η provides the link between imposed deviatoric stress and resulting deviatoric strain rate as follows:

ij=2ηEij,

in which Eij is the deviatoric strain-rate tensor and τij is the deviatoric stress tensor. The deviatoric strain-rate tensor is defined in terms of the material flow velocity u as follows:

ij=12∂iuj+∂jui−23∂kukδij,

where i ≡ /xi. Solid-state creep in crystalline media generally occurs by two independent mechanisms: dislocation glide and climb (e.g., Weertman, 1968) and diffusion of point defects through crystal grains and/or along grain boundaries (e.g., Herring, 1950; Coble, 1963; Green, 1970). The theoretical expression for the effective viscosity, derived from detailed consideration of these creep mechanisms, is

=Admτ1−nkTexpΔE+PΔVkT,

where A is a dimensional constant that depends on the details of the creep mechanism, d is the effective size of crystal grains, τ = [τij τij]1/2 is the square root of the second stress-tensor invariant (Stocker and Ashby, 1973), k is Boltzmann’s constant, T is the absolute temperature, ∆E is the diffusion activation energy, ∆V is the diffusion activation volume, and P is the total pressure.

The theoretical expression (3) for the effective viscosity of crystalline media is useful for understanding the importance of viscosity variations in the Earth’s mantle. If mantle creep occurred dominantly through diffusion of point defects, the effective viscosity in (3) would be independent of stress (i.e., n = 1). The grain-size dependence of diffusion creep is quite pronounced and m = 2 for Herring (bulk) diffusion and m = 3 for Coble (grain-boundary) diffusion. If mantle creep instead occurred through propagation of dislocations, the effective viscosity will be insensitive to grain size (i.e., m = 0) and sensitive to the ambient deviatoric stress, with the stress exponent n ≈ 3 being typical (e.g., Weertman, 1968; Carter, 1976). In either creep mechanism, steady-state creep is ultimately dependent on the creation and diffusion of point defects and is therefore thermally activated. This is manifested by the exponential temperature dependence in (3). The effective viscosity of the mantle is therefore expected to be most sensitive to variations in temperature. Over the past decade numerous studies have also indicated the importance of chemical environment (e.g., the presence of H2O and CO2) on the effective viscosity of mantle rocks (e.g., Kohlstedt and Hornack, 1981; Ricoult and Kohlstedt, 1985; Karato et al., 1986; Borch and Green, 1987). The dependence of effective viscosity on grain size, stress, pressure, temperature, and chemical environment implies that the viscosity of the mantle is expected to be very heterogeneous, owing to the lateral and depth variation of these thermodynamic state variables.

The significant mathematical difficulties arising from the treatment of arbitrary three-dimensional (3D) viscosity variations have led to an overwhelming focus on mantle flow models in which the viscosity is assumed to be constant, or to vary with depth only. Such simplifications have nonetheless led to a deep understanding of the basic physics underlying the thermal convection process responsible for the “drift” of the Earth’s tectonic plates and the global variation of surface heat flux (e.g., Turcotte and Oxburgh, 1967; McKenzie et al., 1974; Peltier, 1972, 1985; Jarvis and McKenzie, 1980; Jarvis and Peltier, 1982; Solheim and Peltier, 1990, 1993, 1994; Peltier and Solheim, 1992).

Theoretical modeling of mantle flow, based on the simplifying approximation that the viscosity depends only on depth, culminated with the development of models that are used to predict the 3D mantle circulation expected on the basis of seismically inferred lateral density heterogeneity (e.g., Richards and Hager, 1984; Ricard et al., 1984; Forte and Peltier, 1987, 1991a). Such flow modeling has demonstrated that the observed long-wavelength nonhydrostatic geoid may be successfully described in terms of the seismically inferred global heterogeneity in the mantle (e.g., Hager et al., 1985; Forte and Peltier, 1987, 1991a; Hager and Clayton, 1989; Forte et al., 1992). There therefore appears to be no evidence in the long-wavelength geoid data, or indeed the dynamic surface topography data (Forte et al., 1993a), for the presence of a significant effect due to lateral variations of mantle viscosity. Since such lateral heterogeneity must exist if the crystalline mantle is in a state of motion determined by the thermal convection process, the success of these simple models would appear to indicate that the dynamics of flow in a laterally heterogeneous mantle are such that the existence of lateral rheology variations does not significantly impact surface observables such as dynamic topography and nonhydrostatic geoid anomalies.

The inadequacy of flow models which assume a spherically symmetric viscosity distribution becomes truly apparent only by considering the observed motions of the tectonic plates (e.g., Hager and O’Connell, 1981; Forte and Peltier, 1987). In a fluid shell with spherically symmetric viscosity, buoyancy forces excite only poloidal flow (which produces a pattern of purely converging or diverging flow at the surface) and thus fails completely to account for the strong toroidal (i.e. strike–slip) component of actual plate motions (e.g., Hager and O’Connell, 1981; Forte and Peltier, 1987). The observed equipartitioning of kinetic energy between poloidal and toroidal plate motions is a direct consequence of the plate-like mechanical structure of the lithosphere. This equipartitioning has also been investigated by O’Connell et al. (1991), who suggest that the present-day ratio of toroidal to poloidal energy in the plate motions appears to be nearly a minimum. A detailed consideration of the relationship between toroidal energy and the strike–slip motion of plates at transform faults has been presented by Olson and Bercovici (1991).

The mere existence of plates, with their “weak” boundaries and relatively “strong” interiors, implies that the effective viscosity of the lithosphere exhibits extreme lateral variations. The mathematical difficulties of dealing explicitly with such extreme variations of rheology have motivated several studies that attempt to overcome these difficulties by directly employing the observed plate motions (e.g., Hager and O’Connell, 1981) or by employing the geometry of the plates as a surface boundary condition (e.g., Ricard and Vigny, 1989; Forte and Peltier, 1991a,b; Gable et al., 1991). Such treatments of the plates are essentially kinematic. To model the plates in a dynamically consistent manner, and to understand the rheologic coupling of poloidal and toroidal surface flow, requires an explicit treatment of lateral viscosity variations.

The mathematical modeling of lateral viscosity variations in numerical simulations of thermal convection has been almost exclusively carried out in two-dimensional (2D) Cartesian geometry. Perhaps the most complete of such studies, in the detailed investigation of stress-, temperature-, and pressure-dependent viscosity, is that by Christensen (1984). An investigation of...