# 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.

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.