# 2D compressible convection with a reference profile and material properties from BurnMan

## Contents

# 2D compressible convection with a reference profile and material properties from BurnMan#

*This section was contributed by Juliane Dannberg and René Gassmöller*

In this cookbook we will set up a compressible mantle convection model that uses the (truncated) anelastic liquid approximation (see The anelastic liquid approximation (ALA) and The truncated anelastic liquid approximation (TALA)), together with a reference profile read in from an ASCII data file.
The data we use here is generated with the open source mineral physics toolkit BurnMan (https://geodynamics.github.io/burnman/) using the python example program `simple_adiabat.py`

.
This file is available as a part of BurnMan, and provides a tutorial for how to generate ASCII data files that can be used together with ASPECT.
The computation is based on the Birch-Murnaghan equation of state, and uses a harzburgitic composition.
However, in principle, other compositions or equations of state can be used, as long as the reference profile contains data for the reference temperature, pressure, density, gravity, thermal expansivity, specific heat capacity and compressibility.
Using BurnMan to generate the reference profile has the advantage that all the material property data are consistent, for example, the gravity profile is computed using the reference density.

The reference profile is shown in Fig. 84, and the corresponding data file is located at `data/adiabatic-conditions/ascii-data/isentrope_properties.txt`

.

## Setting up the ASPECT model#

In order to use this profile, we have to import and use the data in the adiabatic conditions model, in the gravity model and in the material model, which is done using the corresponding ASCII data plugins.
The input file is provided in `cookbooks/burnman/burnman.prm`

, and it uses the 2d shell geometry previously discussed in Simple convection in a quarter of a 2d annulus and surface velocities imported from GPlates as explained in sec:cookbooks:gplates.

To use the BurnMan data in the material model, we have to specify that we want to use the `ascii reference profile`

model.
This material model makes use of the functionality provided by the `AsciiData`

classes in ASPECT, which allow plugins such as material models, boundary or initial conditions models to read in ASCII data files (see for example sec:cookbooks:geomio).
Hence, we have to provide the directory and file name of the data to be used in the separate subsection `Ascii data model`

, and the same functionality and syntax will also be used for the adiabatic conditions and gravity model.

The viscosity in this model is computed as the product of a profile \(\eta_r(z)\), where \(z\) corresponds to the depth direction of the chosen geometry model, and a term that describes the dependence on temperature:

where \(A\) and \(\eta_0\) are constants determined in the input file via the parameters `Viscosity`

and `Thermal viscosity exponent`

, and \(\eta_r(z)\) is a stepwise constant function that determines the viscosity profile.
This function can be specified by providing a list of `Viscosity prefactors`

and a list of depths that describe in which depth range each prefactor should be applied, in other words, at which depth the viscosity changes. By default, it is set to viscosity jumps at 150 km depth, between upper mantle and transition zone, and between transition zone and lower mantle). The prefactors used here lead to a low-viscosity asthenosphere, and high viscosities in the lower mantle.
To make sure that these viscosity jumps do not lead to numerical problems in our computation (see Averaging material properties), we also use harmonic averaging of the material properties.

```
subsection Material model
set Model name = ascii reference profile
subsection Ascii data model
set Data file name = isentrope_properties.txt
set Data directory = $ASPECT_SOURCE_DIR/data/adiabatic-conditions/ascii-data/
end
subsection Ascii reference profile
set Thermal viscosity exponent = 10.0
set Viscosity prefactors = 1.0, 0.1, 1.0, 10.0
end
set Material averaging = harmonic average
end
```

As the reference profile has a depth dependent density and also contains data for the compressibility, this material model supports compressible convection models.

For the adiabatic conditions and the gravity model, we also specify that we want to use the respective `ascii data`

plugin, and provide the data directory in the same way as for the material model.
The gravity model automatically uses the same file as the adiabatic conditions model.

```
subsection Adiabatic conditions model
set Model name = ascii data
subsection Ascii data model
set Data directory = $ASPECT_SOURCE_DIR/data/adiabatic-conditions/ascii-data/
set Data file name = isentrope_properties.txt
end
end
```

```
subsection Gravity model
set Model name = ascii data
end
```

To make use of the reference state we just imported from BurnMan, we choose a formulation of the equations that employs a reference state and compressible convection, in this case the anelastic liquid approximation (see The anelastic liquid approximation (ALA)).

```
subsection Formulation
set Formulation = anelastic liquid approximation
end
```

This means that the reference profiles are used for all material properties in the model, except for the density in the buoyancy term (on the right-hand side of the force balance equation (1), which in the limit of the anelastic liquid approximation becomes Equation (20)). In addition, the density derivative in the mass conservation equation (see sec:mass-conservation-approximation) is taken from the adiabatic conditions, where it is computed as the depth derivative of the provided reference density profile (see also sec:combined_formulations).

## Visualizing the model output#

If we look at the output of our model (for example in `ParaView`

), we can see how cold, highly viscous slabs are subducted and hot plumes rise from the core-mantle boundary.
The final time step of the model is shown in Fig. 85, and the full model evolution can be found at https://youtu.be/nRBOpw5kp-4.
Visualizing material properties such as density, thermal expansivity or specific heat shows how they change with depth, and reveals abrupt jumps at the phase transitions, where properties change from one mineral phase to the next.
We can also visualize the gravity and the adiabatic profile, to ensure that the data we provided in the `data/adiabatic-conditions/ascii-data/isentrope_properties.txt`

file is used in our model.

## Comparing different model approximations#

For the model described above, we have used the anelastic liquid approximation.
However, one might want to use different approximations that employ a reference state, such as the truncated anelastic liquid approximation (TALA, see The truncated anelastic liquid approximation (TALA)), which is also supported by the `ascii reference profile`

material model.
In this case, the only change compared to ALA is in the density used in the buoyancy term, the only place where the temperature-dependent density instead of the reference density is used.
For the TALA, this density only depends on the temperature (and not on the dynamic pressure, as in the ALA).
Hence, we have to make this change in the appropriate place in the material model (while keeping the formulation of the equations set to `anelastic liquid approximation`

):

```
subsection Material model
subsection Ascii reference profile
set Use TALA = true
end
end
```

We now want to compare these commonly used approximations to the “isothermal compression approximation” (see The isothermal/isentropic compression approximation (ICA)) that is unique to ASPECT.
It does not require a reference state and uses the full density everywhere in the equations except for the right-hand side mass conservation, where the compressibility is used to compute the density derivative with regard to pressure.
Nevertheless, this formulation can make use of the reference profile computed by BurnMan and compute the dependence of material properties on temperature and pressure in addition to that by taking into account deviations from the reference profile in both temperature and pressure. As this requires a modification of the equations outside of the material model, we have to specify this change in the `Formulation`

(and remove the lines for the use of TALA discussed above).

```
subsection Formulation
set Formulation = isentropic compression
end
```

As the “isothermal compression approximation” is also ASPECT’s default for compressible models, the same model setup can also be achieved by just removing the lines that specify which `Formulation`

should be used.

Fig. 86 and Fig. 87 show a comparison between the different models. They demonstrate that upwellings and downwellings may occur in slightly different places and at slightly different times when using a different approximation, but averaged model properties describing the state of the model – such as the root mean square velocity – are similar between the models.