# 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](../README.md)**

```{figure-md} fig:sinking_block1
<img src="dens_vel.png" style="width:60.0%" />

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

```{figure-md} fig:sinking_block2
<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.
```

:::{toctree}
../README.md
:::