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Community Benchmarks and Examples

Below is a list of examples presented in Davies et al. (2021) -- Geoscientific Model Development -- "Automating Finite Element Methods for Geodynamics via Firedrake". For each case, we provide a brief description and point the reader towards the associated code in our repository. The key steps involved with setting up each example are highlighted. We start with the most basic problem -- isoviscous, incompressible convection, in an enclosed 2-D Cartesian box -- and systematically build complexity, initially moving into more realistic physical approximations (compressibility, viscoplastic viscosity) and, subsequently, geometries that are more representative of Earth's mantle (2D cylindrical, 3D spherical). We end with a realistic simulation that incorporates plate motion histories from GPlates, as a surface boundary condition. For further detail on each case, please refer to the paper, which can be found here: (upload link post submission).

1. 2-D Cartesian cases:

1.1. Base case (script)
1.2. Compressible case (script) (diff)
1.3. Viscoplastic case (script) (diff)

2-D Cartesian
Fig 1 - Results from 2-D benchmark case from Tosi et al. (2015), with a viscoplastic rheology, at Ra0= 102: (a) Nusselt number versus number of pressure and velocity DOF, for a series of uniform, structured meshes; (b) final steady-state temperature field, with contours spanning temperatures of 0 to 1, at 0.05 intervals; (c) RMS velocity versus number of pressure and velocity DOF; (d) final steady-state viscosity field (note logarithmic scale). In panels a and c, the range of solutions provided by different codes in the Tosi et al. (2015) benchmarkstudy are bounded by dashed red lines.

2. 2-D cylindrical case (script) (diff)

2-D Cylindrical
Fig 2 - (a)/(b) Nusselt number/RMS velocity vs. number of pressure and velocity DOF, at Ra = 1×105, for a series of uniform, structured meshes in a 2-D cylindrical domain. High-resolution, adaptive mesh, results from the Fluidity computational modelling framework are delineated by dashed red lines; (c) final steady-state temperature field, with contours spanning temperatures of 0 to 1, at intervals of 0.05.

3. 3-D Cartesian case (script) (diff)

2-D Cylindrical
Fig 3 - Final steady-state temperature field, in 2-D and 3-D, from Firedrake simulations, designed to match: (a) Case 1a from Blankenbachet al. (1989), with contours spanning temperatures of 0 to 1, at 0.05 intervals; (b) Case 1a from Busse et al. (1994), with transparent isosurfaces plotted at T=0.3, 0.5 and 0.7.

4. 3-D spherical case (script) (diff)

3-D Spherical
Fig 4 - (a)/(b) Nusselt number/RMS velocity vs. number of pressure and velocity DOF, designed to match an isoviscous 3-D spherical benchmark case at Ra= 7×103, for a series of uniform, structured meshes. The range of solutions predicted in previous studies are boundedby dashed red lines (Bercovici et al., 1989; Ratcliff et al., 1996; Yoshida and Kageyama, 2004; Stemmer et al., 2006; Choblet et al., 2007; Tackley, 2008; Zhong et al., 2008; Davies et al., 2013); (c) final steady-state temperature field highlighted through isosurfaces at temperatureanomalies (i.e. away from radial average) of T=−0.15 (blue) and T= 0.15 (orange), with the core-mantle-boundary at the base of the spherical shell marked by a red surface; (d-f) as in a-c, but for a temperature-dependent-viscosity case, with thermally induced viscosity contrasts of 100. Fewer codes have published predictions for this case, but results of Zhong et al. (2008) are marked by dashed red lines.

5. Real-Earth problem: assimilation of 230 Ma of plate motion history. (script)

Realistic
Fig 5 - Present-day thermal structure, predicted from our global mantle convection simulation where the geographic distribution of heterogeneity is dictated by 230 Myr of imposed plate motion history (Muller et al., 2016). Each image includes a radial surface at r = 1.25 (i.e. immediately above the core-mantle boundary), a cross-section, and transparent isosurfaces at temperature anomalies (i.e. away from the radial average) of T=−0.15 (blue) and T= 0.15 (red), highlighting the location of downwelling slabs and upwelling mantle plumes (below r=2.19), respectively. Continental boundaries provide geographic reference. Panel (a) provides an Africa-centered view, with panel (b) centered on the Pacific Ocean, and including glyphs at the surface highlighting the imposed plate velocities.