Fabio Crameri

101 Numerical modelling

Geodynamic modelling provides a powerful tool to investigate processes in the Earth’s crust, mantle and core. To ensure high-quality modelling studies, fair external interpretation, and sensible use of published work, a basic understanding of numerical modelling is necessary.


The massive scales and inaccessibility

Numerical modelling


The Earth's interior spans massive scales, both in time and space. Most of these scales are inaccessible to direct observation. Indeed, these scales are hardly comprehensible.

Numerical modelling allows us to reproduce the evolution of the Earth's interior both during micro seconds or billions of years and over micro meters or thousands of kilometres.

Numerical modelling is the geoscientist's time machine, microscope and binocular in once.

Numerical modelling

What is it?

A modelling study encompasses everything from the assemblage of both a physical and a numerical model based on a verified numerical code, to the design of a (simplified) model setup based on a certain modelling philosophy, the validation of the model via careful testing, the unbiased analysis of the produced model output, the oral, written, and graphical communication of the modelling approach and results, and the management of both software and data.

Numerical modelling

The basic equations

The governing equations are the conservation of mass, the conservation of momentum, and conservation of energy with different types of rheology.

Here, ρ is the density, t is time, v the velocity vector, σ the stress tensor, g the gravitational acceleration vector, Cp the heat capacity, T the temperature, k the thermal conductivity, H a volumetric heat production term (e.g., for radioactive decay), and the term S can account for friction heating, adiabatic heating, and the release or consumption of latent heat (e.g., associated with phase changes), respectively. Note that the plastic rheology depicted here represents the geodynamic approximation of brittle failure.

Numerical modelling

Numerical discretisation

In order to model nature, one has to discretise time and space.

The 2-D domain discretisation on the bottom left illustrates different mesh types. The top half mesh is build on a quadtree with several levels of mesh refinement (top right) to better capture the interface of the circular feature. The bottom left quarter is based on an unstructured triangular mesh adjusted to align its element edges with the interface of the circular feature. Different methods of material tracking are available based on either the particle-in-cell method (top) or grid-based advection (bottom) .

Model simplification

A simpler model can be more useful: the basic shape of the heart is likely the most successful model to-date, indeed a true icon: it is neither too complex (it can be reproduced easily), nor too simple (its characteristic shape is still recognisable). Finding the right level of complexity is challenging and must repeatedly be evaluated by the modeller for each new modelling task using the potential options for geodynamic model simplification.

Modelling philosophies

Specific modelling and generic modelling are the two overarching modelling philosophies. Each one has a different scientific goal and needs to be used, communicated, and reviewed differently from the other.

Based on, and figures adopted from:

van Zelst, I., F. Crameri, A.E. Pusok, A.C. Glerum, J. Dannberg, C. Thieulot (2022), 101 geodynamic modelling: how to design, interpret, and communicate numerical studies of the solid Earth, Solid Earth, doi:10.5194/se-13-583-2022