# 3D convection with an Earth-like initial condition

## Contents

# 3D convection with an Earth-like initial condition#

*This section was contributed by Jacqueline Austermann*

For any model run with we have to choose an initial condition for the temperature field. If we want to model convection in the Earth’s mantle we want to choose an initial temperature distribution that captures the Earth’s buoyancy structure. In this cookbook we present how to use temperature perturbations based on the shear wave velocity model S20RTS (Ritsema and Heijst 2000) to initialize a mantle convection calculation.

## The input shear wave model.#

The current version of can read in the shear wave velocity models S20RTS
(Ritsema and Heijst 2000) and S40RTS (Ritsema et al. 2011), which are located
in data/initial-temperature/S40RTS/. Those models provide spherical
harmonic coefficients up do degree 20 and 40, respectively, for 21 depth
layers. The interpolation with depth is done through a cubic spline
interpolation. The input files `S20RTS.sph`

and `S40RTS.sph`

were downloaded
from http://www.earth.lsa.umich.edu/~jritsema/Research.html and have the
following format (this example is S20RTS):

```
```

The first number in the first line denotes the maximum degree. This is
followed in the next line by the spherical harmonic coefficients from the
surface down to the CMB. The coefficients are arranged in the following way:
\(a_{00}\)
\(a_{10}\) \(a_{11}\) \(b_{11}\)
\(a_{20}\) \(a_{21}\) \(b_{21}\) \(a_{22}\) \(b_{22}\)
…
\(a_{yz}\) is the cosine coefficient of degree \(y\) and order \(z\); \(b_{yz}\) is
the sine coefficient of degree \(y\) and order \(z\). The depth layers are
specified in the file `Spline_knots.txt`

by a normalized depth value ranging
from the CMB (3480km, normalized to -1) to the Moho (6346km, normalized to 1).
This is the original format provided on the homepage.

Any other perturbation model in this same format can also be used, one only
has to specify the different filename in the parameter file (see next
section). For models with different depth layers one has to adjust the
`Spline_knots.txt`

file as well as the number of depth layers, which is hard
coded in the current code. A further note of caution when switching to a
different input model concerns the normalization of the spherical harmonics,
which might differ. After reading in the shear wave velocity perturbation one
has several options to scale this into temperature differences, which are then
used to initialize the temperature field. It should be noted that the shear
wave velocity perturbations in S20RTS and S40RTS are expressed in terms of
percentage deviation from PREM. Wavespeed perturbations in other models may be
referenced to other absolute values and this should be taken into account when
interpreting absolute values of temperature, density and other physical
parameters in ASPECT.

## Setting up the model.#

For this cookbook we will use the parameter file provided in cookbooks/initial-condition-S20RTS/S20RTS.prm, which uses a 3d spherical shell geometry similar to section sec:shell-simple-3d. This plugin is only sensible for a 3D spherical shell with Earth-like dimensions.

The relevant section in the input file is as follows:

```
```

For this initial condition model we need to first specify the data directory in which the input files are located as well as the initial condition file (S20RTS.sph or S40RTS.sph) and the file that contains the normalized depth layers (Spline knots depth file name). We next have the option to remove the degree 0 perturbation from the shear wave model. This might be the case if we want to make sure that the depth average temperature follows the background (adiabatic or constant) temperature.

The next input parameters describe the scaling from the shear wave velocity perturbation to the final temperature field. The shear wave velocity perturbation \(\delta v_s / v_s\) (that is provided by S20RTS) is scaled into a density perturbation \(\delta \rho / \rho\) with a constant that is specified in the initial condition section of the input parameter file as ‘Vs to density scaling.’ Here we choose a constant scaling of 0.15. This perturbation is further translated into a temperature difference \(\Delta T\) by multiplying it by the negative inverse of thermal expansion, which is also specified in this section of the parameter file as ‘Thermal expansion coefficient in initial temperature scaling.’ This temperature difference is then added to the background temperature, which is the adiabatic temperature for a compressible model or the reference temperature (as specified in this section of the parameter file) for an incompressible model. Features in the upper mantle such as cratons might be chemically buoyant and therefore isostatically compensated, in which case their shear wave perturbation would not contribute buoyancy variations. We therefore included an additional option to zero out temperature perturbations within a certain depth, however, in this example we don’t make use of this functionality. The chemical variation within the mantle might require a more sophisticated ‘Vs to density’ scaling that varies for example with depth or as a function of the perturbation itself, which is not captured in this model. The described procedure provides an absolute temperature for every point, which will only be adjusted at the boundaries if indicated in the Boundary temperature model. In this example we chose a surface and core mantle boundary temperature that differ from the reference mantle temperature in order to approximate thermal boundary layers.

## Visualizing 3D models.#

In this cookbook we calculate the instantaneous solution to examine the flow field. Figures 1 and 2 show some of the output for a resolution of 2 global refinement steps (1c and 2a, c, e) as used in the cookbook, as well as 4 global refinement steps (other panels in these figures). Computations with 4 global refinements are expensive, and consequently this is not the default for this cookbook. For example, as of 2017, it takes 64 cores approximately 2 hours of walltime to finish this cookbook with 4 global refinements. Figure 1a and b shows the density variation that has been obtained from scaling S20RTS in the way described above. One can see the two large low shear wave velocity provinces underneath Africa and the Pacific that lead to upwelling if they are assumed to be buoyant (as is done in this case). One can also see the subducting slabs underneath South America and the Philippine region that lead to local downwelling. Figure 1c and d shows the heat flux density at the surface for 2 refinement steps (c, colorbar ranges from 13 to 19 mW/\(m^2\)) and for 4 refinement steps (d, colorbar ranges from 35 to 95 mW/\(m^2\)). A first order correlation with upper mantle features such as high heat flow at mid ocean ridges and low heat flow at cratons is correctly initialized by the tomography model. The mantle flow and buoyancy variations produce dynamic topography on the top and bottom surface, which is shown for 2 refinement steps (2a and c, respectively) and 4 refinement steps (2b and d, respectively). One can see that subduction zones are visible as depressed surface topography due to the downward flow, while regions such as Iceland, Hawaii, or mid ocean ridges are elevated due to (deep and) shallow upward flow. The core mantle boundary topography shows that the upwelling large low shear wave velocity provinces deflect the core mantle boundary up. Figure 2e and f shows geoid perturbations for 2 and 4 global refinement steps, respectively. The geoid anomalies show a strong correlation with the surface dynamic topography. This is in part expected given that the geoid anomalies are driven by the deflection of the upper and lower surface as well as internal density variations. The relative importance of these different contributors is dictated by the Earth’s viscosity profile. Due to the isoviscous assumption in this cookbook, we don’t properly recover patterns of the observed geoid. Lastly, Figure 2g and h shows geoid perturbations for 2 and 4 global refinement steps, respectively.

As discussed in the previous cookbook, dynamic topography does not necessarily average to zero if the resolution is not high enough. While one can simply subtract the mean as a postprocessing step this should be done with caution since a non-zero mean indicates that the refinement is not sufficiently high to resolve the convective flow. In Figure 2a-d we refrained from subtracting the mean but indicated it at the bottom left of each panel. The mean dynamic topography approaches zero for increasing refinement. Furthermore, the mean bottom dynamic topography is closer to zero than the mean top dynamic topography. This is likely due to the larger magnitude of dynamic topography at the surface and the difference in resolution between the top and bottom domain (for a given refinement, the resolution at the core mantle boundary is higher than the resolution at the surface). The average geoid height and gravity anomaly is zero since the minimum degree in the geoid anomaly expansion is set to 2.

This model uses a highly simplified material model that is incompressible and
isoviscous and does therefore not represent real mantle flow. More realistic
material properties, density scaling as well as boundary conditions will
affect the magnitudes and patterns shown here. A comparison between surface
dynamic topography, the geoid, and gravity anomalies from ASPECT and a
spectral based code shows good agreement (see
`benchmarks/spectral-comparison/`

for figure and details).

Ritsema, J., A. Deuss, H. J. van Heijst, and J. H. Woodhouse. 2011.
“S40rts: A Degree-40 Shear-Velocity Model for the Mantle from New
Rayleigh Wave Dispersion, Teleseismic Traveltime and Normal-Mode Splitting
Function Measurements.” *Geophysical Journal International* 184:
1223–36.

Ritsema, J., and H. J. van Heijst. 2000. “Seismic Imaging of Structural
Heterogeneity in Earth’s Mantle: Evidence for Large-Scale Mantle
Flow.” *Sci. Progr.* 83: 243–59.