# 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 $$512~\text{ km}\times 512~\text{ km}$$ domain filled with a fluid (the “mantle”) of density $$\rho_1=3200\text{ kg m}^{-3}$$ and viscosity $$\eta_1=10^{21}~\text{ Pa s}$$. A square block of size $$128~\text{ km}\times 128~\text{ km}$$ is placed in the domain and is centered at location $$(x_c,y_c)=(256~\text{ km},384~\text{ km})$$ 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 $$|\boldsymbol{g}|=10~\text{ m s}^{-2}$$. 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 $$\text{ kg m}^{-3}$$ and the viscosities by several orders of magnitude to represent a wide array of scenarios. The velocity field obtained for $$\eta_2=10^{27}~\text{ Pa s}$$ and $$\delta\rho=32~\text{ kg m}^{-3}$$ is shown in Fig. 145.

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 $$\eta_2\in [10^{17},10^{26}]~\text{ Pa s}$$ and $$\delta\rho=8,32,128~\text{ kg m}^{-3}$$. Results are shown in Fig. 146 and we indeed recover the expected trend with all data points forming a single smooth line.

Description of benchmark files