With a uniform or radial viscosity structure, the relationship between the pattern of viscous flow and the driving force is simple---when the variables in the equations of motion are represented in harmonic form, the equation for each harmonic contains only terms with the same harmonic degree as the driving force. Hence, the equation for one harmonic decouples from the equations for all the other harmonics of the flow. Once lateral variations in viscosity are introduced, equations from one harmonic degree, or mode, couple into equations from other modes. This greatly complicates the analysis because, simple Green's functions used in the studies in the first section no longer exist. One strategy has been to produce pseudo-Green's functions [e.g., Richards and Hager, 1989]. Another strategy has been to calculate the coupling coefficients between different modes [ Stewart, 1992], still others have left the modal formulation and solve the convective flow problem with a finite difference or finite element approach [ Ritzler and Jacobi, 1992; Zhang and Christensen, 1993; King and Hager, 1994].
There appears to be a consensus that at the long wavelengths, such as those used in the studies discussed in the first section, the effect of mode coupling are on the order of a few percent [ Richards and Hager, 1989; Stewart, 1992; Ritzler and Jacobi, 1992; King and Hager, 1994]. King and Hager find at shorter wavelengths, such as within 1000 km of the trench, that there can be strong coupling of the modes. An interesting, counter-example to the long wavelength studies is the investigation by Rivine and Phipps Morgan [1993] who use a mixed boundary condition with free-slip and no-slip on the top surface to simulate continental and oceanic plates. They find significant coupling at wavelengths equal to the length scale of the variation of the surface boundary condition. The amplitudes of the geoid variation can be large as 20% to 40% in some cases. While they use a constant viscosity formulation, one might think that the mixed boundary condition is a simple approximation to a large lateral viscosity contrast at the top of the mantle. This illustrates the potentially important role of the chemically and rheologically non-homogeneous surface of the Earth in understanding surface observables because the lateral variations in rheology may not be simply related to thermal anomalies as assumed by most studies.
The choice of equation of state used in modeling the mantle has an effect on geoid and topography profiles driven by internal density contrasts. Many studies use an incompressible flow formulation, while other studies use a compressible flow formulation. In an incompressible formulation, the density is assumed to be constant everywhere except in the driving buoyancy term. In a compressible formulation, the effect of pressure and temperature are carried through out the equations. The effect of compressibility seems to only have an effect on the degree 2 and 3 terms of the flow. In addition to the compressibility, the coefficient of thermal expansion is needed to convert from temperature to density anomalies. The coefficient of thermal expansion decreases significantly with pressure over the depth range of the mantle [cf., Chopelas and Boehler, 1989]. In addition, the coefficient of thermal expansion varies significantly with temperature [cf., Chopelas and Boehler, 1992]. Hong and Yuen [1990] find that the equation of state can change the longest wavelength geoid by as much as 20%, but again this effect falls off rapidly as the wavelength decreases. The decrease in the coefficient of thermal expansion over the depth of the mantle suggests that the magnitude of the thermal anomalies in the lower mantle, based on tomographic models, is larger than most previous estimates [ Cadek et al. , 1994].