Introduction

Numerical models of convection are solutions to the equations of continuity, motion, temperature and state obtained through a variety of approaches including finite difference, finite element, finite volume, Galerkin and spectral-transform techniques. Early numerical models of mantle convection were two dimensional, either planar or axisymmetric (e.g., Richter, 1973; Moore and Weiss, 1973; Houston and De Bremaecker, 1975; Parmentier and Turcotte, 1978; Lux et al., 1979; Schubert and Zebib, 1980). As discussed in Chapter 9, two-dimensional models provide insights into mantle convection and are useful to explore complex and nonlinear effects of material behavior. However, they cannot simulate the actual style of mantle convection, which is fully three dimensional. The main features of mantle convection, linear slabs, quasi-cylindrical plumes, and toroidal motion are fundamentally three dimensional in nature. In addition, two-dimensional modes of convection are often unstable to three-dimensional disturbances even at relatively low Rayleigh number (Busse, 1967; Richter, 1978; Travis et al., 1990a). Three-dimensional numerical models of thermal convection in rectangular (Cserepes et al., 1988; Houseman, 1988; Travis et al., 1990a, b; Cserepes and Christensen, 1990; Christensen and Harder, 1991; Ogawa et al., 1991) and spherical (Baumgardner, 1985, 1988; Machetel et al., 1986; Glatzmaier, 1988; Glatzmaier et al., 1990; Bercovici et al., 1989a, b, c, 1991, 1992; Schubert et al., 1990) geometries began to appear in the late 1980s and provided a more realistic picture of the form of convection in the mantle. At present, three-dimensional numerical models of mantle convection are widely carried out and ever-increasing computational power permits inclusion of increasingly realistic material properties and behavior into the models.