# The Tosi et al. benchmarks

*This section was contributed by Anne Glerum.*

This section discusses the viscoplastic thermal convection benchmarks
described by {cite:t}`tosi:etal:2015`. The five benchmarks extend
those of {cite:t}`BBC89` with temperature-,
pressure- and strain rate-dependent rheology. As the results are published in
the original paper, we limit ourselves to a brief description of the setup and
results of the first 2 benchmark cases.

All five benchmarks solve for Boussinesq convection in a box of $1 \times 1$
dimensions with free slip boundary conditions. The initial temperature
distribution considers a linear depth profile with a slight perturbation to
start convection. Top and bottom boundaries are set to a fixed temperature
value. The parameters shared between the benchmark cases can be found in
[benchmarks/tosi_et_al_2015_gcubed/Tosi_base.prm](https://www.github.com/geodynamics/aspect/blob/main/benchmarks/tosi_et_al_2015_gcubed/Tosi_base.prm). The other input files
describe the variations on this base model, which pertain to the rheological
description. The specific rheologies used are implemented in
[benchmarks/tosi_et_al_2015_gcubed/tosi.cc](https://www.github.com/geodynamics/aspect/blob/main/benchmarks/tosi_et_al_2015_gcubed/tosi.cc) and describe a linear and a
plastic component of the viscosity:
```{math}
:label: eq:tosi-benchmark-lin-visc
  \eta_\text{linear}(T,z) = \exp(-\ln(\eta_T) T + \ln(\eta_Z) z)
```
```{math}
:label: eq:tosi-benchmark-plast-visc
  \eta_\text{plastic}(\dot\epsilon) = \eta^\ast + \frac{\sigma_y}{\sqrt{\dot\epsilon:\dot\epsilon}}
```
where $\eta^\ast$ is the
constant effective viscosity at high stresses and $\sigma_y$ the yield stress.

## Case 1: Temperature-dependent convection.

The first benchmark considers a viscosity that only depends on temperature
(Eq. {math:numref}`eq:tosi-benchmark-lin-visc`, with $\gamma_Z=0$). When run to
steady-state, this produces one convection cell with a high viscosity,
stagnant lid insulating the fluid below (see Fig. 1 of {cite:t}`tosi:etal:2015`). In
{cite:t}`tosi:etal:2015`, results of different codes are compared by looking at the
average temperature, the Nusselt number at the top and bottom of the domain,
the RMS velocity at the top boundary and in the whole domain, and the maximum
velocity at the surface. These quantities can be queried by using several of
the postprocessors, but the additional postprocessor in
[benchmarks/tosi_et_al_2015_gcubed/tosi.cc](https://www.github.com/geodynamics/aspect/blob/main/benchmarks/tosi_et_al_2015_gcubed/tosi.cc) is needed to compute the average
rate of work done against gravity, the average rate of viscous dissipation,
and the error between them. Differences between these diagnostic quantities of
the 11 codes that participated in the benchmark effort are smaller than 5% for
their preferred mesh resolution.

## Case 2: Viscoplastic convection.

Case 2 includes a strain rate-dependent component in the viscosity, which is
harmonically averaged with the linear component (see also the code snippet
below):

```{math}
:label: eq:tosi-benchmark-ave-visc
\eta(T,\epsilon,z) = 2 \left(\frac{1}{\eta_\text{linear}}+\frac{1}{\eta_\text{plastic}}\right)^{-1}
```

```{literalinclude} tosi_benchmark_2.prm
```

This rheology leads to mobile-lid convection, with the descending cold lid
cooling the cell's interior (Fig. 2 of {cite:t}`tosi:etal:2015`).

By changing the input parameters shown in the code snippet, we obtain the
other benchmark cases. Case 3 includes a depth-dependent component for the
viscosity, but no strain rate-dependence, i.e. it uses Eq. {math:numref}`eq:tosi-benchmark-lin-visc`. Case 4 considers a full temperature-,
depth- and strain rate-dependent viscosity, while in case 5 the yield stress
is varied to investigate the transitions from mobile-lid to periodic to
stagnant-lid convection regimes. The input files referenced above implement
these specific cases. As mentioned before, the results are presented in {cite:t}`tosi:etal:2015` together with the results of several other finite element, finite
volume, and spectral codes. {numref}`fig:tosi-benchmark-results1` to {numref}`fig:tosi-benchmark-results3` shows one example of the
resolved temperature and viscosity fields for case 1.


```{figure-md} fig:tosi-benchmark-results1
<img src="Case1_6_T.*" style="width:31.0%" />

 Temperature field in steady-state for case 1 of {cite}`tosi:etal:2015`.
```

```{figure-md} fig:tosi-benchmark-results2
<img src="Case1_6_visc.*" style="width:31.0%" />

 Viscosity field in steady-state for case 1 of {cite}`tosi:etal:2015`.
```

```{figure-md} fig:tosi-benchmark-results3
<img src="Case1_6_vel.*" style="width:31.0%" />

 Velocity field in steady-state for case 1 of {cite}`tosi:etal:2015`.
```