The sinking block benchmark#

This benchmark is based on the benchmark presented in (Gerya 2010) and extended in (Thieulot 2011). It consists of a two-dimensional \(512~\si{\km}\times 512~\si{\km}\) domain filled with a fluid (the “mantle”) of density \(\rho_1=3200\si{\kg\per\cubic\meter}\) and viscosity \(\eta_1=10^{21}~\si{\pascal\second}\). A square block of size \(128~\si{\km}\times 128~\si{\km}\) is placed in the domain and is centered at location \((x_c,y_c)=(256~\si{\km},384~\si{\km})\) so as to ensure that its sides align with cell boundaries at all resolutions (GMR level \(\geq 3\)). It is filled with a fluid of density \(\rho_2=\rho_1+\delta \rho\) and viscosity \(\eta_2\). The gravity vector points downwards with \(|\boldsymbol{g}|=10~\si{\meter\per\square\second}\). Boundary conditions are free slip on all sides. Only one time step is carried out and we measure the absolute velocity \(|v_z|\) in the middle of the block.

In a geodynamical context, the block could be interpreted as a detached slab or a plume head. As such its viscosity and density can vary (a cold slab has a higher effective viscosity than the surrounding mantle while it is the other way around for a plume head). The block densities can then vary from a few units to several hundreds of \(\si{\kg\per\cubic\meter}\) and the viscosities by several orders of magnitude to represent a wide array of scenarios. The velocity field obtained for \(\eta_2=10^{27}~\si{\pascal\second}\) and \(\delta\rho=32~\si{\kg\per\cubic\meter}\) is shown in Figure 1.

As shown in (Thieulot 2011) one can independently vary \(\eta_1\), \(\rho_2\), \(\eta_2\), and measure \(|v_z|\) for each combination: the quantity \(|v_z| \eta_1/\delta\rho\) is then found to be a simple function of the ratio \(\eta^\star=\eta_1/\eta_2\): at high enough mesh resolution all data points collapse onto a single line. The shell script run_benchmark in the folder runs the experiment for values \(\eta_2\in [10^{17},10^{26}]~\si{\pascal\second}\) and \(\delta\rho=8,32,128~\si{\kg\per\cubic\meter}\). Results are shown in Figure 2 and we indeed recover the expected trend with all data points forming a single smooth line.


Fig. 143 Density field with velocity arrows for \eta_2=10^{27}~\si{\pascal\second} and \delta\rho=32~\si{\kg\per\cubic\meter}#


Fig. 144 Scaled velocity measurements as a function of the viscosity contrast between surrounding medium and block for all experiments.#

Gerya, Taras. 2010. Numerical Geodynamic Modelling. Cambridge University Press.

Thieulot, C. 2011. “FANTOM: two- and three-dimensional numerical modelling of creeping flows for the solution of geological problems.” Phys. Earth. Planet. Inter. 188: 47–68.