of 250 km thickness. All of the models include time‐dependent prescribed surface plate motions, which shape the large‐scale structure of mantle flow. We adopt plate motions from a recent paleogeographic reconstruction by Matthews et al. (2016), which spans 410 Ma‐present, although some of the calculations do not include the entire plate motion history. All of the models except Case 40 (Table 1.1) impose the initial plate motions for a period of 150 Myr to spin‐up the model and initialize large‐scale structure following Zhang et al. (2010). The mechanical boundary conditions at the core‐mantle boundary are free‐slip, and the temperature boundary conditions are isothermal with a nondimensional temperature of 0 at the surface and 1 at the core–mantle boundary. We use a temperature‐ and depth‐dependent viscosity with the form η(z) = ηz(z) exp [E(0.5 − T)], where ηz(z) is a depth‐dependent viscosity prefactor and E = 9.21 is a dimensionless activation energy, which gives rise to relative viscosity variations of 104 due to temperature variations. The models are heated by a combination of basal and internal heating, with a dimensionless internal heating rate Q = 100.
We include the effects of a phase transition at 660 km depth in some of the models. Phase transitions are implemented in CitcomS using a phase function approach (Christensen and Yuen, 1985). We adopt a density increase across 660 km of 8%, a reference depth of 660 km, a reference temperature of 1573 K, and a phase change width of 40 km. We assume a Clapeyron slope of –2 MPa/K. Recent experimental work favors a range of –2 to –0.4 MPa/K (Fei et al., 2004; Katsura et al., 2003), considerably less negative than values employed in earlier geodynamical modeling studies that produced layered convection (Christensen and Yuen, 1985). The models shown in the present work are a subset of a more exhaustive suite of models from Lourenço and Rudolph (in review), which consider a broader range of convective vigor and additional viscosity structures. We list the parameters that are varied between the five models in Table 1.1 and the radial viscosity profiles used in all of the models are shown in Figure 1.4.
1.2.3 Inversions for Viscosity
We carried out inversions for the mantle viscosity profile constrained by the long‐wavelength nonhydrostatic geoid. The amplitude and sign of geoid anomalies depend on the internal mantle buoyancy structure as well as the deflection of the free surface and core‐mantle boundary, which, in turn, are sensitive to the relative viscosity variations with depth (Richards and Hager, 1984; Hager et al., 1985). Because the long‐wavelength geoid is not very sensitive to lateral viscosity variations (e.g., Richards and Hager, 1989; Ghosh et al., 2010), we neglect these, solving only for the radial viscosity profile. The geoid is not sensitive to absolute variations in viscosity, so the profiles determined here show only relative variations in viscosity, and absolute viscosities could be constrained using a joint inversion that includes additional constraints such as those offered by observations related to glacial isostatic adjustment. In order to estimate the viscosity profile, we first convert buoyancy anomalies from mantle tomographic models into density anomalies and then carry out a forward model to generate model geoid coefficients. We then compare the modeled and observed geoids using the Mahalanobis distance
(1.3)
where
We used geoid coefficients from the GRACE geoid model GGM05 (Ries et al., 2016) and the hydrostatic correction from Chambat et al. (2010). We use a transdimensional, hierarchical, Bayesian approach to the inverse problem (e.g., Sambridge et al., 2013), based on the methodology described in (Rudolph et al., 2015). We carry out forward models of the geoid using the propagator matrix code HC (Hager and O’Connell, 1981; Becker et al., 2014). Relative to our previous related work (Rudolph et al., 2015), the inversions presented here differ in their treatment of uncertainty, scaling of velocity to density variations, and parameterization of radial viscosity variations.
Table 1.1 Summary of parameters used in geodynamic models. zlm denotes the depth of the viscosity increase between the upper and lower mantle and Δηlm is the magnitude of the viscosity increase at this depth. LVC indicates whether the model includes a low‐viscosity channel below 660 km. Spinup time is the duration for which the initial plate motions are imposed prior to the start of the time‐dependent plate model. We indicate whether the model includes the endothermic phase transition, which always occurs at a depth of 660 km and with Clapeyron slope –2 MPa/K.
| Case | z lm | Δηlm | LVC? | Spinup time | Phase transition? | Start time |
|---|---|---|---|---|---|---|
| Case 8 | 660 km | 100 | No | 150 Myr | No | 400 Ma |
| Case 9 | 660 km | 30 | No | 150 Myr | No | 400 Ma |
| Case 18 | 1000 km | 100 | No | 150 Myr | Yes | 400 Ma |
| Case 32 | 660 km | 100 | No | 150 Myr | Yes | 400 Ma |
| Case 40 | 660 km | 100 | Yes | 0 Myr | Yes | 250 Ma |
Figure 1.4 (A) Viscosity profiles used in our geodynamic models. For comparison, we also show viscosity profiles obtained in a joint inversion constrained by glacial isostatic adjustment (GIA) and convection‐related observables (Mitrovica and Forte (2004)