The Crameri et al. benchmarks#

This section was contributed by Ian Rose.

This section follows the two free surface benchmarks described by Crameri et al. [2012].

Case 1: Relaxation of topography.#

The first benchmark involves a high viscosity lid sitting on top of a lower viscosity mantle. There is an initial sinusoidal topography which is then allowed to relax. This benchmark has a semi-analytical solution (which is exact for infinitesimally small topography). Details for the benchmark setup are in Fig. 173.


Fig. 173 Setup for the topography relaxation benchmark. The box is 2800 km wide and 700 km high, with a 100 km lid on top. The lid has a viscosity of \(10^{23} \, {Pa\,s}\), while the mantle has a viscosity of \(10^{21} \, {Pa\,s}\). The sides are free slip, the bottom is no slip, and the top is a free surface. Both the lid and the mantle have a density of \(3300 \,{kg/m^3}\), and gravity is \(10 \, {m/s^2}\). There is a \(7 \, {km}\) sinusoidal initial topography on the free surface.#

The complete parameter file for this benchmark can be found in benchmarks/crameri_et_al/case_1/crameri_benchmark_1.prm, the most relevant parts of which are excerpted here:

set CFL number                             = 0.01

set Additional shared libraries = ./

subsection Geometry model
  set Model name = rebound box
  subsection Rebound Box
    set Order = 1
    set Amplitude = 7.e3
  subsection Box
    set X extent = 28.e5
    set Y extent = 7.e5
    set X repetitions = 300
    set Y repetitions = 75

In particular, this benchmark uses a custom geometry model to set the initial geometry. This geometry model, called “ReboundBox,” is based on the Box geometry model. It generates a domain in using the same parameters as Box, but then displaces all the nodes vertically with a sinusoidal perturbation, where the magnitude and order of that perturbation are specified in the ReboundBox subsection.

The characteristic timescales of topography relaxation are significantly smaller than those of mantle convection. Taking timesteps larger than this relaxation timescale tends to cause sloshing instabilities, which are described further in Free surface calculations. Some sort of stabilization is required to take large timesteps. In this benchmark, however, we are interested in the relaxation timescale, so we are free to take very small timesteps (in this case, 0.01 times the CFL number). As can be seen in Fig. 174, the results of all the codes which are included in this comparison are basically indistinguishable.


Fig. 174 Results for the topography relaxation benchmark, showing maximum topography versus time. Over about 100 ka the topography completely disappears. The results of four free surface codes, as well as the semi-analytic solution, are nearly identical.#

Case 2: Dynamic topography.#

Case two is more complicated. Unlike the case one, it occurs over mantle convection timescales. In this benchmark there is the same high viscosity lid over a lower viscosity mantle. However, now there is a blob of buoyant material rising in the center of the domain, causing dynamic topography at the surface. The details for the setup are in the caption of Fig. 175.


Fig. 175 Setup for the dynamic topography benchmark. Again, the domain is 2800 km wide and 700 km high. A 100 km thick lid with viscosity \(10^{23}\) overlies a mantle with viscosity \(10^{21}\). Both the lid and the mantle have a density of \(3300\,kg/m^3\). A blob with diameter 100 km lies 300 km from the bottom of the domain. The blob has a density of \(3200\,kg/m^3\) and a viscosity of \(10^{20}\) Pa s.#

Case two requires higher resolution and longer time integrations than case one. The benchmark is over 20 million years and builds dynamic topography of \(\sim 800\) meters.


Fig. 176 Evolution of topography for the dynamic topography benchmark. The maximum topography is shown as a function of time, for as well as for several other codes participating in the benchmark. This benchmark shows considerably more scatter between the codes.#

Again, we excerpt the most relevant parts of the parameter file for this benchmark, with the full thing available in benchmarks/crameri_et_al/case_2/crameri_benchmark_2.prm. Here we use the “Multicomponent” material model, which allows us to easily set up a number of compositional fields with different material properties. The first compositional field corresponds to background mantle, the second corresponds to the rising blob, and the third corresponds to the viscous lid.

Furthermore, the results of this benchmark are sensitive to the mesh refinement and timestepping parameters. Here we have nine refinement levels, and refine according to density and the compositional fields.

set CFL number                   = 0.1

subsection Material model
  set Model name = multicomponent
  subsection Multicomponent
    set Densities = 3300, 3200, 3300
    set Viscosities = 1.e21, 1.e20, 1.e23
    set Viscosity averaging scheme = harmonic

subsection Mesh refinement
  set Additional refinement times        =
  set Initial adaptive refinement        = 4
  set Initial global refinement          = 5
  set Refinement fraction                = 0.3
  set Coarsening fraction                = 0.0
  set Strategy                           = density,composition
  set Refinement criteria merge operation = plus
  set Time steps between mesh refinement = 5

Unlike the first benchmark, for case two there is no (semi) analytical solution to compare against. Furthermore, the time integration for this benchmark is much longer, allowing for errors to accumulate. As such, there is considerably more scatter between the participating codes. does, however, fall within the range of the other results, and the curve is somewhat less wiggly. The results for maximum topography versus time are shown Fig. 176.

The precise values for topography at a given time are quite dependent on the resolution and timestepping parameters. Following Crameri et al. [2012] we investigate the convergence of the maximum topography at 3 Ma as a function of CFL number and mesh resolution. The results are shown in Fig. 177.


Fig. 177 Convergence for case two. Left: Logarithm of the error with decreasing CFL number. As the CFL number decreases, the error gets smaller. However, once it reaches a value of  ∼ 0.1, there stops being much improvement in accuracy. Right: Logarithm of the error with increasing maximum mesh resolution. As the resolution increases, so does the accuracy.#

We find that at 3 Ma converges to a maximum topography of \(\sim 396\) meters. This is slightly different from what MILAMIN_VEP reported as its convergent value in Crameri et al. [2012], but still well within the range of variation of the codes. Additionally, we note that is able to achieve good results with relatively less mesh resolution due to the ability to adaptively refine in the regions of interest (namely, the blob and the high viscosity lid).

Accuracy improves roughly linearly with decreasing CFL number, though stops improving at CFL \(\sim 0.1\). Accuracy also improves with increasing mesh resolution, though its convergence order does not seem to be excellent. It is possible that other mesh refinement parameters than we tried in this benchmark could improve the convergence. The primary challenge in accuracy is limiting numerical diffusion of the rising blob. If the blob becomes too diffuse, its ability to lift topography is diminished. It would be instructive to compare the results of this benchmark using particles with the results using compositional fields.