sinking_block.md

The sinking block benchmark

This benchmark is based on the benchmark presented in {cite:t}gerya:2010 and extended in {cite:t}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 {numref}fig:sinking_block1.

As shown in {cite:t}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 {numref}fig:sinking_block2 and we indeed recover the expected trend with all data points forming a single smooth line.

Description of benchmark files

<img src="dens_vel.png" style="width:60.0%" />

 Density field with velocity arrows for $\eta_2=10^{27}~\text{ Pa s}$ and $\delta\rho=32~\text{ kg m}^{-3}$

<img src="plot.svg" style="width:80.0%" />

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

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