101 geodynamic modelling: how to design, interpret, and communicate numerical studies of the solid Earth
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Solid Earth, 13, 583–637, 2022
https://doi.org/10.5194/se-13-583-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.
101 geodynamic modelling: how to design, interpret, and
communicate numerical studies of the solid Earth
Iris van Zelst1,2 , Fabio Crameri3,4 , Adina E. Pusok5 , Anne Glerum6 , Juliane Dannberg7 , and Cedric Thieulot8
1 School of Earth and Environment, University of Leeds, Leeds, LS2 9JT, UK
2 Instituteof Planetary Research, German Aerospace Center (DLR), Berlin, Germany
3 Undertone Design, Bern, Switzerland
4 Centre for Earth Evolution and Dynamics (CEED), University of Oslo, Postbox 1028 Blindern, 0315 Oslo, Norway
5 Department of Earth Sciences, University of Oxford, UK
6 Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany
7 Department of Geological Sciences, University of Florida, USA
8 Department of Earth Sciences, Utrecht University, Utrecht, the Netherlands
Correspondence: Iris van Zelst (iris.vanzelst@dlr.de, iris.v.zelst@gmail.com)
Received: 8 February 2021 – Discussion started: 18 February 2021
Revised: 23 January 2022 – Accepted: 27 January 2022 – Published: 17 March 2022
Abstract. Geodynamic modelling provides a powerful tool backgrounds to (re)gain a solid understanding of geodynamic
to investigate processes in the Earth’s crust, mantle, and core modelling as a whole.
that are not directly observable. However, numerical mod-
els are inherently subject to the assumptions and simplifica-
tions on which they are based. In order to use and review
1 Introduction
numerical modelling studies appropriately, one needs to be
aware of the limitations of geodynamic modelling as well as The term geodynamics1 combines the Greek word “geo”,
its advantages. Here, we present a comprehensive yet con- meaning “Earth”, and the term “dynamics” – a discipline of
cise overview of the geodynamic modelling process applied physics that concerns itself with forces acting on a body and
to the solid Earth from the choice of governing equations the subsequent motions they produce (Forte, 2011). Hence,
to numerical methods, model setup, model interpretation, geodynamics is the study of forces acting on the Earth and
and the eventual communication of the model results. We the subsequent motion and deformation occurring in the
highlight best practices and discuss their implementations Earth. Studying geodynamics contributes to our understand-
including code verification, model validation, internal con- ing of the evolution and current state of the Earth and other
sistency checks, and software and data management. Thus, planets, as well as the natural hazards and resources on Earth.
with this perspective, we encourage high-quality modelling The broad definition of geodynamics typically results in a
studies, fair external interpretation, and sensible use of pub- subdivision of disciplines relating to one of the Earth’s spher-
lished work. We provide ample examples, from lithosphere ical shells and specific spatial and temporal scales (Fig. 1).
and mantle dynamics specifically, and point out synergies So, geodynamics considers brittle and ductile deformation
with related fields such as seismology, tectonophysics, ge- and plate tectonic processes on the crustal and lithospheric
ology, mineral physics, planetary science, and geodesy. We scale as well as convection processes in the Earth’s mantle
clarify and consolidate terminology across geodynamics and and core. Lithospheric and crustal deformation operates on
numerical modelling to set a standard for clear communi- scales of tens to thousands of kilometres and thousands to
cation of modelling studies. All in all, this paper presents millions of years to study both the folding of lithospheric
the basics of geodynamic modelling for first-time and expe-
rienced modellers, collaborators, and reviewers from diverse 1 See the glossary in the Supplement for definitions of the bold
terms that occur throughout the text.
Published by Copernicus Publications on behalf of the European Geosciences Union.
584 I. van Zelst et al.: 101 geodynamic modelling Figure 1. Spatial and temporal scales of common geodynamic processes. These processes occur over a wide range of timescales and length scales, and modellers have to take into account which of them can be included in any given model. plates and the faulting patterns accommodating them (Watts et al., 2004; Bercovici and Ricard, 2012; Keller and Katz, et al., 2013). Mantle convection encompasses individual 2016; Katz, 2008; Montési and Behn, 2007). Hence, in order flow features like mantle plumes and subducted slabs on to fully understand geodynamics, many processes on differ- scales of hundreds of kilometres and hundred of thousands ent temporal and spatial scales need to be taken into account of years as well as whole-mantle overturns on thousand- simultaneously. However, this is difficult to achieve using kilometre and billion-year scales (Androvandi et al., 2011; only observational or experimental studies. King, 2015). In the outer core, there are both small convec- Indeed, since the study of geodynamics is predominantly tion patterns on hundred-metre and hour scales and large- occupied with processes occurring below the surface of the scale convection processes taking place over tens of kilome- Earth, one of the challenges is the limited number of direct tres and tens of thousands of years (Bouffard et al., 2019). observations in space and time. Engineering limitations are Inner-core dynamics are believed to act on similar scales responsible for a lack of direct observations of processes at as mantle convection, although it is has not yet been es- depth, with, for example, the deepest borehole on Earth be- tablished whether or not the Rayleigh number allows for ing a mere 12 km (e.g. Ganchin et al., 1998), which is less convection (Deguen and Cardin, 2011). In addition to the than 0.2 % of the radius of the Earth (6371 km). In addition, large-scale dynamics of the different spherical shells of the the available rock record grows increasingly scarce when go- Earth, there are other processes that play a role and oper- ing back in time. Other disciplines, such as seismology, ge- ate on yet other different spatial and temporal scales. For ex- ology, and geodesy, relate to geodynamics as they study the ample, earthquakes – and on larger timescales, earthquake (indirectly) observable expressions of geodynamic processes cycles – facilitate the large-scale deformation of the litho- at the surface and at depth. Disciplines like mineral physics sphere, but their nucleation process can occur on scales as and rock mechanics relate to geodynamics by studying the little as milliseconds and millimetres (Stein and Wysession, physical properties and flow laws of rocks involved in geo- 2009; Beroza and Kanamori, 2015). Similarly, surface pro- dynamic processes. cesses such as erosion can play an important role in litho- To compensate for the limited amount of data in geo- sphere dynamics, although they take place on a smaller spa- dynamics, many studies have turned towards modelling. tial and temporal scale (Pelletier, 2008). For lithosphere and Roughly speaking, there are two main branches of mod- mantle dynamics, processes including magma, fluid, volatile, elling based on the tools that they use: analogue modelling, and grain dynamics have been shown to be important, which wherein real-world materials are used in a lab, and phys- all have smaller spatial and temporal scales (Solomatov and ical or mathematical modelling, wherein physical princi- Reese, 2008; Keller et al., 2013; McKenzie, 1984; Rüpke ples and computational methods are used. Analogue models Solid Earth, 13, 583–637, 2022 https://doi.org/10.5194/se-13-583-2022
I. van Zelst et al.: 101 geodynamic modelling 585
have many advantages, the most straightforward of which is can help to narrow down whether or not a geodynamic model
that they are automatically subject to all the physical laws is the best way of answering a particular research question.
on Earth. However, it is non-trivial to upscale the models Here, we provide an overview of the whole geodynamic
to real-world applications that occur on a different temporal modelling process (Fig. 2). We discuss how to develop the
and spatial scale. Similarly, important aspects such as grav- numerical code, set up the model from a geological problem,
ity are difficult to scale in a lab setting (e.g. Ramberg, 1967; test and interpret the model, and communicate the results to
Mart et al., 2005; Noble and Dixon, 2011). In addition, it is the scientific community. We highlight the common and best
hard to determine variables such as stress and strain across practices in numerical thermo-mechanical modelling studies
the entire domain at any given time. Physical models aim of the solid Earth and the assumptions that enter such mod-
to describe processes through equations. For relatively sim- els. We will focus on the Earth’s crust and mantle, although
ple processes, analytical solutions or semi-analytical models the methods outlined here are similarly valid for other solid
can be used (e.g. Lamb, 1879; Samuel and Bercovici, 2006; planetary bodies such as terrestrial planets and icy satellites,
Montési and Behn, 2007; Turcotte and Schubert, 2014; Ribe, as well as the Earth’s solid inner core. More specifically, we
2018; McKenzie, 1969; Hager and O’Connell, 1981). How- concentrate on and provide examples of lithosphere and man-
ever, when the equations and their dependencies become in- tle dynamics. We focus on recent advances in good mod-
creasingly complex in their description of geodynamic pro- elling practices and clearly, unambiguously communicated,
cesses, numerical methods become necessary to solve the set open science that make thermo-mechanical modelling stud-
of equations. Hence, numerical modelling is the use of nu- ies reproducible and available for everyone.
merical methods to solve a set of physical equations describ-
ing a physical process. Numerical models are powerful tools 1.1 What is a model?
to study any desired geometry and time evolution with full
control over the physics and access to all variables at any We will use the word model to mean a simplified version
point in time and space in the model, but they come with of reality that can be used to isolate and investigate certain
their own set of limitations and caveats. For example, de- aspects of the real world. Therefore, by definition, a model
spite recent developments in numerical methods and mod- never equals the complexity of the real world and is never
elling studies, it remains difficult to combine diverse spatial a complete representation of the world; i.e. all models are
and temporal scales in one numerical model. This is due to inherently wrong, but some are useful (Box, 1976). We de-
the fact that numerical models are increasingly more difficult fine a physical model as the set of equations that describes a
to solve computationally when large contrasts are present in physical or chemical process occurring in the Earth.
the physical properties (e.g. in space or time). Therefore, dif- Computational geodynamics concerns itself with using
ferent modelling approaches have been developed to tackle numerical methods to solve a physical model. It is a rela-
the different natures of the processes outlined in Fig. 1. Ide- tively new discipline that took off with the discovery of plate
ally, these modelling approaches would be combined in the tectonics and the rise of computers in the 1960s (Schubert
future. In this article, we focus on the continuously growing et al., 2001; Gerya, 2019). Within computational geodynam-
branch of thermo-mechanical numerical modelling studies in ics, a numerical model refers to a specific set of physical
geodynamics which mainly deal with lithosphere deforma- equations which are discretised and solved numerically with
tion and mantle convection, but we point out the links with initial and boundary conditions. Here, we limit ourselves to
other processes wherever relevant. Other disciplines, such as thermo-mechanical numerical models of the Earth’s crust,
atmospheric dynamics and hydrology, use similar equations lithosphere, and mantle, with “thermo-mechanical” referring
and assumptions but are not discussed in detail here. to the specific set of equations the numerical code solves
Since geodynamics has much in common with other Earth (Sect. 2).
science disciplines, there is a frequent exchange of knowl- It is important to realise that a numerical model is not
edge; e.g. geodynamic studies use data from other disciplines equivalent to the numerical code. A numerical code can
to constrain their models. Vice versa, geodynamic models contain and solve one or more numerical models (i.e. differ-
can inform other disciplines on the theoretically possible mo- ent sets of discretised equations). Each numerical model can
tions in the Earth. Therefore, scientists from widely diverging have a different model setup with different dimensions, ge-
backgrounds are starting to incorporate geodynamic models ometries, and initial and boundary conditions. For a specific
into their own projects, apply the interpretation of geody- model setup, the numerical model can then be executed mul-
namic models to their own hypotheses, or be asked to review tiple times by varying different aspects of the model setup
geodynamic articles. In order to correctly use, interpret, and to constitute multiple model simulations (also often called
review geodynamic modelling studies, it is important to be model runs).
aware of the numerous assumptions that go into these models While it can be tested if a numerical model solves the
and how they affect the modelling results. Similarly, knowing equations correctly through analytical solutions, it cannot
the strengths and weaknesses of numerical modelling studies be proven that the numerical model solves the appropriate
physical model, i.e. equations. In addition, the results of sim-
https://doi.org/10.5194/se-13-583-2022 Solid Earth, 13, 583–637, 2022
586 I. van Zelst et al.: 101 geodynamic modelling
ulations with complexity beyond analytical solutions can be then needs to choose the equations that describe the physics
compared against observations and results from other codes, of the processes they are interested in, i.e. the physical model
but again these cannot be proven to be correct or, indeed, the (Sect. 2). Here, we confine ourselves to the conservation
only possibilities for producing such results. equations of mass, momentum, and energy (Sect. 2.1). In or-
The models we are concerned with here are all forward der to solve the physical model, a numerical model is needed,
models; we obtain a model result by solving equations that which solves the discretised versions of the equations using
describe the physics of a system. These results are a pre- a specific numerical scheme (Sect. 3). Before applying the
diction of how the system behaves given its physical state code to a particular problem, the code needs to be verified to
which afterwards can be compared to observations. On the ensure that it solves the equations correctly (Sect. 4). Once it
other hand, inverse models start from an existing dataset of has been established that the code works as intended, the to-
observations and aim to determine the conditions responsi- be-tackled geological problem needs to be simplified into a
ble for producing the observed dataset. A well-known ex- model setup with appropriate geometry, material properties,
ample of these kinds of models are tomographic models of initial and boundary conditions, and the correct modelling
the Earth’s mantle, which determine the 3-D seismic velocity philosophy (Sect. 5). After the model has successfully run,
structure of the mantle based on seismic datasets consisting the model results should be validated in terms of robustness
of e.g. P-wave arrival times or full waveforms (e.g. Dziewon- and against observations (Sect. 6). The results of the simula-
ski, 1984; Bijwaard and Spakman, 2000; Fichtner et al., tion then need to be interpreted, visualised, and diagnosed
2013). The possibility of incorporating data and estimating while taking into account the assumptions and limitations
the model uncertainties in inverse models has recently led to that entered the modelling study in previous steps (Sect. 7).
increasing efforts combining both inverse and forward meth- The final step of a modelling study is to communicate the
ods in geodynamic modelling. For example, adjoint methods results to peers as openly, clearly, and reproducibly as pos-
(Liu and Gurnis, 2008; Burstedde et al., 2009; Reuber et al., sible (Sects. 8, 9). It is important to note that geodynamic
2018c, 2020a), pattern recognition (Atkins et al., 2016), and modelling is not a linear process as depicted in Fig. 2. For
data assimilation (Bunge et al., 2002, 2003; Ismail-Zadeh example, modelling does not necessarily start with the for-
et al., 2004; Hier Majumder et al., 2005; Bocher et al., mulation of a hypothesis, as this can also arise from the en-
2016, 2018) have been incorporated in lithosphere and man- counter of interesting, unexpected modelling results, result-
tle dynamics models. One of the most common ways of solv- ing in the formulation of a hypothesis later on in the process.
ing the inverse problem in geodynamics is to run many for- Similarly, observations from nature feed directly into all of
ward models and subsequently test which one fits the ob- the above-mentioned steps, as illustrated by the dark grey ar-
servations best (Baumann and Kaus, 2015). This is difficult row through all modelling steps (Fig. 2). Indeed, it is im-
because individual forward models are computationally ex- portant to evaluate and adjust the numerical modelling study
pensive, which results in a limited number of forward mod- throughout the entire process.
els that can be run. Additionally, there are too many param- In the remaining parts of this paper, each of the above-
eters in the forward models to invert for. An alternative ap- mentioned steps of a modelling study correspond to individ-
proach is to incorporate automatic parameter scaling routines ual sections, making this a comprehensive guide to geody-
or use adjoint methods to test parameter sensitivities in mod- namic modelling studies.
els (Reuber et al., 2018a, c), which could considerably reduce
the number of models required. However, whether it is for-
ward or inverse models of geodynamics, a thorough under- 2 The physical model
standing of the forward model is generally required, which
we focus on in this work. The prospect of estimating uncer- From seismology, we know that on short timescales the Earth
tainties in geodynamic modelling and the increasing inclu- predominantly deforms like an elastic medium. Our own ex-
sion of data make the combination of inverse and forward perience tells us that when strong forces are applied to rocks,
modelling an exciting avenue in geodynamics. they will break (brittle failure). But from the observation
that continents have moved over the history of the Earth and
1.2 What is modelling? from the postglacial rebound of regions like Scandinavia, we
know that on long timescales the mantle strongly deforms in-
A scientific modelling study encompasses more than simply ternally. In geodynamic models, this ductile deformation of
running a model, as is illustrated in Fig. 2. First and fore- rocks is usually approximated as the flow of a highly viscous
most, the modeller needs to decide which natural process fluid (Schubert et al., 2001; Gerya, 2019). The transition
they want to study based on observations to increase the un- from short to long timescales that controls which deforma-
derstanding of the Earth’s system. From this, a hypothesis tion behaviour is dominant is marked by the Maxwell relax-
(i.e. a proposed explanation based on limited evidence) is ation time (Ricard, 2015), which is on the order of 450 years
formulated as a starting point to further investigate a knowl- for the Earth’s sublithospheric mantle (Schubert et al., 2001).
edge gap driven by the modeller’s curiosity. The modeller The rocks in the Earth’s interior are not the only material that
Solid Earth, 13, 583–637, 2022 https://doi.org/10.5194/se-13-583-2022
I. van Zelst et al.: 101 geodynamic modelling 587
Figure 2. The geodynamic modelling procedure. A modelling study encompasses everything from the assemblage of both a physical (Sect. 2)
and a numerical model (Sect. 3) based on a verified numerical code (Sect. 4), to the design of a simplified model setup based on a certain
modelling philosophy (Sect. 5), the validation of the model through careful testing (Sect. 6), the unbiased analysis of the produced model
output (Sect. 7), the oral, written, and graphical communication of the modelling approach and results (Sect. 8), and the management of both
software and data (Sect. 9). Note that constant (re-)evaluation and potential subsequent adjustments of previous steps are key, and indeed
necessary, throughout this process.
deforms predominantly in this way. In fact, many other ma- 2.1 Basic equations
terials show the same behaviour: solid ice, for example, also
flows, causing the constant motion of glaciers (Ricard, 2015). We have discussed above that setting up a model includes
In the following, we will focus on problems that occur choosing the equations that describe the physical process one
on large timescales on the order of thousands or millions of wants to study. In a fluid dynamics model, these governing
years (i.e. 450 years, the Maxwell time, see Fig. 1). Ac- equations usually consist of a mass balance equation, a force
cordingly, we will treat the Earth’s interior as a highly vis- balance or momentum conservation equation, and an energy
cous fluid. We further discuss this assumption and how well conservation equation. The solution to these equations states
it approximates rocks in the Earth in Sect. 2.2.1. We will also how the values of the unknown variables such as the material
treat Earth materials as a continuum; i.e. we assume that the velocity, pressure, and temperature (i.e. the dependent vari-
material is continuously distributed and completely fills the ables) change in space and how they evolve over time (i.e.
space it occupies and that material properties are averaged when one or more of the known and/or independent variables
over any unit volume. Thus, we ignore the fact that the ma- change). Even though these governing equations are conser-
terial is made up of individual atoms (Helena, 2017). This vation equations, geodynamic models often consider them in
implies that, on a macroscopic scale, there are no mass-free a local rather than a global context, i.e. material and energy
voids or gaps (Gerya, 2019). Under these assumptions and flow into or out of a system, or an external force is applied.
keeping the large uncertainties of Earth’s materials in mind, Additionally, the equations only consider specific types of
we can apply the concepts of fluid dynamics to model the energy, i.e. thermal energy, and not, for example, the po-
Earth’s interior. tential energy related to nuclear forces. This means that for
any system considered – or, in other words, within a given
volume – a property can change due to the transport of that
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588 I. van Zelst et al.: 101 geodynamic modelling
property into or out of the system, and thermal energy may implies a change in density ρ. This usually happens as a re-
be generated or consumed through a conversion into other sponse to a change in external conditions, such as a change in
types of energy, e.g. through radioactive decay. This can be temperature (thermal expansion or contraction) or pressure,
expressed using the basic principle. and is described by the equation of state (see Sect. 2.2.5).
Other specific examples in the Earth are phase transitions or
Accumulation =influx − outflux + generation chemical reactions.
−consumption. (1) Because the first term explicitly includes a time depen-
dence, it introduces a characteristic timescale into the model
This statement suggests that accumulation, or the rate of due to viscous (Curbelo et al., 2019) and elastic forces (Pa-
change of a quantity within a system, is balanced by net točka et al., 2019). For viscous forces, this is called the vis-
flux through the system boundaries (influx–outflux) and net cous isentropic relaxation timescale and is on the order of a
production within the system (generation–consumption). De- few hundred years for the upper mantle to a few tens of thou-
pending on which physical processes are relevant and/or of sands of years for the lower mantle (Curbelo et al., 2019). For
interest, different terms may be included in the equations, visco-elastic deformation, this timescale is the Maxwell time
which is a source of differences between geodynamic stud- (see above). When we consider the Earth to be a visco-elastic
ies. Since each term describes a different phenomenon and body (see Sect. 2.2.1), the relaxation time is dominated by
may imply characteristic timescales and length scales, be- elastic forces.
ing aware of these relations is not only important for set- The timescale of viscous relaxation is usually shorter than
ting up the model, but also for interpreting the model results that of convective processes and is often shorter than the
and comparing them to observations. Mathematical tools like timescales a model is focused on. In addition, these local
scaling analysis or linear stability analysis are helpful to changes in mass are often quite small compared to the overall
determine the relative importance of each of the terms (see mass flux in the Earth. Accordingly, many geodynamic mod-
also Sect. 7.2) and for determining which terms should be in- els do not include this term (see also Jarvis and McKenzie,
cluded in any given model. In solid Earth geodynamic mod- 1980; Glatzmaier, 1988; Bercovici et al., 1992), and instead
elling, we are usually interested in problems occurring on Eq. (2) is replaced by
large timescales on the order of thousands or millions of
years (Fig. 1). Accordingly, we usually model the Earth’s in- ∇ · (ρv) = 0. (3)
terior as a viscous fluid. The corresponding equations are the
This means that the net influx and outflux of mass in
Stokes equations, describing the motion of a highly viscous
any given volume of material is zero. The density can still
fluid driven in a gravitational field, and the energy balance
change, e.g. if material is advected into a region of higher or
written as a function of temperature (Fig. 3). These equa-
lower pressure (i.e. downwards or upwards within the Earth),
tions are partial differential equations describing the relation
but these changes in density are always associated with the
between the unknowns (velocity, pressure, and temperature)
motion of material to a new location. Using this approxi-
and their partial derivatives with respect to space and time.
mation still takes into account the largest density changes.
Solving them requires additional relations, the so-called con-
For example, for the Earth’s mantle, density increases by ap-
stitutive equations (Sect. 2.2), that describe the material prop-
proximately 65 % from the Earth’s surface to the core–mantle
erties appearing as parameters in the Stokes equations and
boundary.
energy balance (such as the density) as well as their depen-
In some geodynamic models, particularly the ones that
dence on velocity, temperature, and pressure. We look at each
span only a small depth range, even this kind of density
equation in more detail in the following sections, but we do
change is small. For example, within the Earth’s upper man-
not provide their derivations and instead refer the interested
tle, the average density changes by less than 20 %. This is
reader to Schubert et al. (2001).
the basis for another approximation: assuming that material
2.1.1 Mass conservation is incompressible (i.e. its density ρ is constant). In this case,
Eq. (2) becomes
The first equation describes the conservation of mass:
∇ · v = 0. (4)
∂ρ
+ ∇ · (ρv) = 0, (2) Assuming incompressibility also has implications for an-
∂t
other material property in the model, the Poisson ratio ν,
λ
where ρ is the mass density, t is time, and v is the velocity which is defined as ν = 2(λ+G) for a homogeneous isotropic
vector. The conservation of mass in Eq. (2) states that in any linear elastic material, where λ is Lamé’s first parameter and
given volume of material, local changes in mass ∂ρ/∂t can G is the shear modulus. Poisson’s ratio is a measure fre-
only occur when they are compensated for by an equal net in- quently used in seismology to describe the volume change
flux or outflux of mass ∇ · (ρv). These local changes in mass of a material in response to loading or, in other words, how
are caused by the expansion or contraction of material, which much a material under compression expands in the direction
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I. van Zelst et al.: 101 geodynamic modelling 589
Figure 3. The governing equations: conservation of mass (Eq. 2, Sect. 2.1.1), conservation of momentum (Eq. 5, Sect. 2.1.2), and conser-
vation of energy (Eq. 6, Sect. 2.1.3) with different types of rheology (Sect. 2.2.1). In these equations, ρ is the density, t is time, v is the
velocity vector, σ is the stress tensor, g is the gravitational acceleration vector, Cp is isobaric the heat capacity, T is the temperature, k is
the thermal conductivity, H is a volumetric heat production term (e.g. due to radioactive decay), and the term S = S1 + S2 + S3 accounts for
viscous dissipation, adiabatic heating, and the release or consumption of latent heat (e.g. associated with phase changes), respectively. Many
geodynamic models include additional constitutive and evolution equations. Note that the brittle rheology depicted here is approached by
plasticity in geodynamic models, as depicted by the plastic slider in dark grey underneath the breaking bar. See Sect. 2.2.4 for more details.
perpendicular to the direction of compression. The assump- on the geodynamic problem, additional terms become rele-
tion of incompressibility implies a Poisson ratio of ν = 0.5. vant in the force balance.
When converting material properties from geodynamic mod- The surface forces are expressed as the spatial derivatives
els to wave speeds for seismological applications, incom- of the stress ∇ · σ . Stresses can be perpendicular to the sur-
pressible models result in unrealistic infinite P-wave speeds face of a given parcel of material like the pressure; they can
unless a different value for the Poisson ratio is assumed dur- act parallel to the surface, or they can point in any other ar-
ing the conversion (van Zelst et al., 2019). bitrary direction. In addition, the forces acting on a surface
may be different for each surface of a parcel of material. Ac-
2.1.2 Momentum conservation cordingly, the stress can be expressed as a 3 × 3 tensor in
three dimensions (3-D), with each entry corresponding to the
Equation (5) describes the conservation of momentum or, in component of the force acting in one of the three coordinate
the way we use it here, a force balance: directions (x, y, z) on a surface oriented in any one of the
∇ · σ + ρg = 0, (5) three coordinate directions, giving a total of 3×3 = 9 entries.
One of the most complex choices in the design of a geody-
where σ is the stress tensor, ρ is the density, and g is gravi- namic model is the relation between these stresses and the
tational acceleration vector. The conservation of momentum deformation (rate) of rocks, i.e. the rheology (Sect. 2.2.1).
states that the sum of all forces acting on any parcel of ma- Under the assumption that deformation is dominantly vis-
terial is zero. Specifically, the first term ∇ · σ represents the cous, Eqs. (2) and (5) are often referred to as the Stokes
net surface forces and the second term the net body forces. equations. In their more general form, called the Navier–
In mantle convection and long-term tectonics models, the Stokes equations, Eq. (5) would contain an additional term
gravity force ρg is generally the only body force consid- ρ Dv/Dt that governs inertia effects describing the acceler-
ered, and the gravitational acceleration g is often taken as ation of the material. In the Earth’s mantle, material moves
a constant of 10 m s−2 for global studies because it varies so slowly that inertia (and, consequently, the momentum of
by less than 10 % over the whole mantle depth (Dziewonski the material) does not have a significant influence on the mo-
and Anderson, 1981), whereas for lithospheric-scale studies tion of the material. Consequently, this term is very small
g is taken to be 9.8 m s−2 . Other body forces like electro- and usually neglected in geodynamic models. In other words,
magnetic or Coriolis forces are negligibly small compared to we look at the behaviour of the material on such a long
gravity. Conversely, these forces are very important for mod- timescale that from our point of view, material accelerates or
elling the magnetic field in the outer core. Hence, depending
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590 I. van Zelst et al.: 101 geodynamic modelling
decelerates almost immediately to its steady-state velocity. concentration of heat-producing elements is high, such as the
Conversely, for models of seismic wave propagation (which continental crust. Viscous dissipation, also called shear heat-
cover a much shorter timescale and consequently use a dif- ing, describes the amount of energy that is released as heat
ferent form of the momentum equation because deformation when material is deformed viscously and/or plastically. The
is predominantly elastic; Stein and Wysession, 2009), it is more stress required to deform the material and the larger
crucial to include the inertia term because it introduces the the amount of deformation, the more shear heating is pro-
physical behaviour that allows for the formation of seismic duced. This can be expressed as S1 = σ 0 : ε̇0 , where σ 0 is
waves. the deviatoric stress, i.e. the part of the stress not related
Dropping the inertia term means that the Stokes equations to the hydrostatic pressure, ε̇0 is the non-recoverable, i.e.
(Eqs. 2 and 5) describe an instantaneous problem if the mass non-elastic, deviatoric strain rate (see also Eqs. 8 and 9 in
conservation is solved using one of the common approxima- Sect. 2.2.1), and the colon (:) is the matrix inner product, so
0 . Adiabatic heating describes how the
σ 0 : ε̇0 = i j σij0 ε̇ij
PP
tions in Eqs. (3) or (4). The time does not appear explicitly in
these equations, so the solution for velocity and pressure at temperature of the material changes when it is compressed
any point in time is independent of the history unless this is or when it expands due to changes in pressure, assuming that
explicitly incorporated in the material properties (Sect. 2.2). no energy is exchanged with the surrounding material dur-
The model evolution in time is incorporated through Eq. (6), ing this process. Hence, the work being done to compress the
which describes the conservation of energy. For more details material is released as heat. This temperature change depends
on the Stokes equations and a complete derivation, we refer on the thermal expansivity α, which expresses how much the
the reader to Schubert et al. (2001). material expands for a given increase in temperature and on
the changes in pressure over time: S2 = αT Dp/Dt. Here,
2.1.3 Energy conservation Dp/Dt = ∂p/∂t +v·∇p is the material (Lagrangian) deriva-
tive of the pressure. Since the dominant pressure variation
Equation (6) describes the conservation of thermal energy, in the Earth’s interior is the effect of the lithostatic pres-
expressed as ρCp T : sure increase with depth, the usual assumption is that pres-
∂T
sure only changes due to the vertical motion of material with
ρCp + v · ∇T − ∇ · (k∇T ) = ρH + S, (6) velocity vz along a (lithostatic) pressure gradient ∇p = ρg ẑ
∂t
(with ẑ the unit vector in the vertical direction), resulting in
where ρ is density, Cp is the isobaric heat capacity (i.e. p is S2 = αT ρgvz .
pressure) T is the temperature, t is time, v is the velocity vec- When material undergoes phase changes, thermal energy
tor, k is the thermal conductivity, H is a volumetric heat pro- is consumed (endothermic reaction) or released (exothermic
duction term, and S = S1 +S2 +S3 represents other sources of reaction) as latent heat. This happens both for solid-state
heat. Changes in thermal energy over time ρCp ∂T /∂t can phase transitions, such as from olivine to wadsleyite, and
be caused by several different processes. The term ρCp v·∇T for melting of mantle rocks. For phase transitions that oc-
corresponds to advective heat transport, i.e. the transport of cur over a narrow pressure and/or temperature range, this can
thermal energy with the motion of material. Consequently, lead to abrupt changes in mantle temperature where phase
it depends on the velocity v the material moves with and transitions are located, such as around 410 km depth. The
on how much the temperature, and with that the thermal en- amount of energy released or consumed is proportional to the
ergy, changes in space, which can be expressed through the density ρ, the temperature T , and the change in the thermo-
temperature gradient ∇T . Heat is also transported by con- dynamic quantity entropy across the phase transitions 1S,
duction, which is represented by the term ∇ · (k∇T ). When i.e. S3 = ρT 1SDX/Dt (Schubert et al., 2001, Chapter 4.1).
the thermal conductivity k is larger or the temperature varia- Here, X is the fraction of the material that has already un-
tion as expressed by the gradient ∇T becomes steeper, more dergone the transition, sometimes also called the phase func-
heat is diffused. If the model includes no additional heating tion. For mantle melting, X would be the melt fraction. The
processes, Eq. (6) can be simplified by dividing it by ρCp , terms discussed above are generally the most important heat-
assuming that they are constants: ing processes in the Earth’s interior. While there may be other
sources of heat or heat transport, they are assumed to be so
∂T
+ v · ∇T = ∇ · (κ∇T ) , (7) small that they can be neglected in most cases, such as heat
∂t transport by radiation or the energy required for reducing the
which introduces the thermal diffusivity κ = k/(ρCp ). mineral grain size and creating new boundaries between the
The terms on the right-hand side of Eq. (6) describe other grains.
heating processes that can be significant in a geodynamic
model. Internal heating can be expressed as the product of 2.1.4 Approximations of the governing equations
the density ρ and a volumetric heat production rate H . Its
most common source is the decay of radiogenic elements. In the previous sections on the mass, momentum, and en-
Accordingly, this process is important in regions where the ergy equations, we have already seen that there are different
Solid Earth, 13, 583–637, 2022 https://doi.org/10.5194/se-13-583-2022
I. van Zelst et al.: 101 geodynamic modelling 591 ways these equations can be simplified and that there is a the same assumptions as the ALA but further assumes that choice of which physical processes to include. Based on an the variation of the density due to pressure variations can be analysis of the equations, there are a number of different ap- neglected in the buoyancy term as well. In the Earth’s man- proximations that are commonly used in geodynamics (see tle, these density variations are typically an order of magni- also Gassmöller et al., 2020; Schubert et al., 2001). They all tude smaller than the density changes caused by temperature. make choices about how the density is approximated, so it is The TALA has similar applications as ALA but introduces important to understand the magnitude of density variations an imbalance between energy dissipation calculated from the in a model (see also Sect. 2.2.5). In the Earth’s mantle, the Stokes equation and heat dissipation in the energy equation. primary density variations that drive flow are usually caused Consequently, it should be avoided when energy dissipation by temperature and are on the order of 1 % of the total density is important in the model (Leng and Zhong, 2008; Albous- (assuming a thermal expansivity of 10−5 K−1 and tempera- sière and Ricard, 2013). ture variations of about 1000 K). Density variations due to The Boussinesq approximation (BA; Oberbeck, 1879; chemical heterogeneities have a similar magnitude. Density Boussinesq, 1903; Rayleigh, 1916) assumes that density jumps at phase transitions can reach 500 kg m−3 , which is on variations are so small that they can be neglected everywhere the order of 10 % of the total density. The increase in litho- except in the buoyancy term in the momentum equation in static pressure from the surface to the core–mantle boundary Eq. (5), which is equivalent to using a constant reference causes a density increase of 65 % (Schubert et al., 2001), but density profile. This implies incompressibility, i.e. the use of these density changes do not drive the flow. Density varia- Eq. (4) for mass conservation. In addition, adiabatic heating tions caused by the dynamic pressure are much smaller, on and shear heating are not considered in the energy equation the order of 0.1 % of the total density (assuming dynamic in Eq. (6). This approximation is valid as long as density vari- pressures of up to 500 MPa). Note that these are estimates ations are small (see also Sect. 2.1.1) and the modelled pro- for the Earth’s mantle, and these values may change for other cesses would cause no substantial shear or adiabatic heating. applications. For example, a model of salt domes has a much The Boussinesq approximation is often used in lithosphere- larger compositional density contrast between the salt and scale models. Due to its simplicity, the approximation of in- the rock, and certain dehydration reactions can lead to vol- compressibility is sometimes also adopted for whole-mantle ume changes larger than 10 %. Before deciding to use any convection models, wherein it is only approximately valid, of the following approximations in a geodynamic model, it and it has been shown that compressibility can have a large is therefore important to make sure that the underlying as- effect on the pattern of convective flow (e.g. Tackley, 1996). sumptions about the density variations are justified for that The extended Boussinesq approximation (EBA; Chris- particular application. tensen and Yuen, 1985; Oxburgh and Turcotte, 1978) is based The anelastic liquid approximation (ALA; Jarvis and on the same assumptions as the BA but does consider adia- McKenzie, 1980) assumes that lateral density variations are batic and shear heating. Since it includes adiabatic heating, small relative to a reference density profile and can be ne- but not the associated volume and density changes, it can glected in the mass equation in Eq. (3) and energy conserva- lead to artificial changes of energy in the model, i.e. mate- tion equation in Eq. (6), which instead use the density of the rial is being heated or cooled based on the assumption that depth-dependent reference profile. Only the buoyancy term it is compressed or it expands, but the mechanical work that in the momentum equation in Eq. (5) uses a temperature- causes compression or expansion is not done. Consequently, and pressure-dependent density, which is approximated us- the extended Boussinesq approximation should only be used ing a Taylor expansion (Appendix A1). This approxima- in models without substantial adiabatic temperature changes. tion is commonly used in whole-mantle convection models, For a comparison between some of these approximations wherein the density changes substantially from the surface to using benchmark models, see e.g. Steinbach et al. (1989), the core–mantle boundary, and compressibility is an impor- Leng and Zhong (2008), King et al. (2010), and Gassmöller tant effect. Its use is appropriate as long as deviations from et al. (2020). In addition, the choice of approximation may the reference profile remain small or are not important for also be limited by the numerical methods being employed the interpretation of the model. For example, substantial ex- (for example, the accuracy of the solution for the variables pansion or contraction caused by temperature changes that that affect the density). Also note that, technically, these ap- are not on the reference profile, such as cooling or heating in proximations are all internally inconsistent to varying de- the thermal boundary layers caused by conduction, and the grees (Sect. 6.2), since they do not fulfil the definitions of stresses caused by this process cannot be modelled with this thermodynamic variables but use linearised versions instead, approximation. Phase transitions of different types of materi- and they use different density formulations in the differ- als can also cause large density or temperature changes, since ent equations. Nevertheless, many of them are generally ac- only one type of material can be represented by the reference cepted and widely used in geodynamic modelling studies, as profile (see also Gassmöller et al., 2020). they allow for simpler equations and more easily obtained The truncated anelastic liquid approximation (TALA; solutions. Jarvis and McKenzie, 1980; Ita and King, 1994) is based on https://doi.org/10.5194/se-13-583-2022 Solid Earth, 13, 583–637, 2022
592 I. van Zelst et al.: 101 geodynamic modelling
2.2 The constitutive equations: material properties and the flow usually does not depend on the deformation his-
tory and has no material memory. Geodynamic models still
Using the equations discussed in the previous section aim to reproduce such processes using complex rheologies
(Sect. 2.1) to model how Earth materials move under the that go beyond the basic assumption of a viscous fluid and in-
influence of gravity requires some assumptions about how clude plastic yielding, strain or strain rate weakening or hard-
Earth materials respond to external forces or conditions. ening, and other friction laws (see below). Hence, resolving
These relations are often called constitutive equations, and how to go forward with the assumption of a viscous fluid in
they relate the material properties to the solution variables the face of complex deformation behaviour of rocks remains
of the conservation equations, like temperature and pressure. an important challenge for the geodynamic modelling com-
Which of these relations is most important for the model evo- munity.
lution depends on the application. In regional models that im- In the following, we will discuss some common rheologi-
pose the plate driving forces externally, the relation between cal behaviours being used in geodynamic models.
stress and deformation, i.e. the rheology (Sect. 2.2.1), can be
the most important parameter choice in the model and can 2.2.2 Viscous deformation
control most of the model evolution. Since buoyancy is one
of the main forces that drive the flow of Earth materials, it We start by discussing a very simple type of rheology with
is often most important to take into account how the density the following features: (i) only viscous (rather than elastic
depends on other variables and to include this dependency in or brittle) deformation, (ii) deformation behaviour that does
the buoyancy term, i.e. the right-hand side of Eq. (5). This not depend on the direction of deformation (corresponding
is particularly important for mantle convection models and is to an isotropic material), (iii) a linear relation between the
described by the equation of state (Sect. 2.2.5) of the mate- stress and the rate of deformation (corresponding to a New-
rial. In models of the lithosphere and crust, stress and state of tonian fluid), and (iv) a low amount of energy dissipation
deformation become more important. The constitutive equa- under compaction or expansion (corresponding to a negligi-
tions can also include time-dependent terms, like in cases ble bulk viscosity). In this case, the (symmetric) stress tensor
in which the strength of a rock depends on its deformation σ can be written as
history, requiring additional equations to be solved (see also σ = −pI + σ 0 = −pI + 2η ε̇0 (v), (8)
Sect. 2.3).
where p is the pressure, I is the unit tensor, σ 0 is the de-
2.2.1 Rheology of Earth materials viatoric stress tensor, η is the (dynamic) shear viscosity, and
ε̇0 (v) is the deviatoric rate of deformation tensor (often called
The rheology describes how materials deform when they are the deviatoric strain rate tensor), which is directly related to
exposed to stresses. This relation between stress and defor- the velocity gradients in the fluid. The total strain rate tensor
mation enters the momentum equation in Eq. (5) in the term is defined as
∇ · σ , which represents the surface forces, and is also used to
1
compute the amount of shear heating σ 0 : ε̇0 in the energy bal- ε̇(v) = (∇v + ∇v T ). (9)
ance. In many cases, the material reacts differently to shear 2
stresses (acting parallel to the surface) and normal stresses It describes both volume changes of the material (vol-
(acting normal to the surface and leading to volumetric de- umetric) and changes in shape (deviatoric), and it can be
formation). Correspondingly, the resistance to these different written as the sum of these two deformation components:
types of deformation is expressed by different material pa- ε̇(v) = 31 (∇ · v) I + ε̇0 (v). The prime denotes the deviatoric
rameters: the shear viscosity or shear modulus and the bulk part of the tensor, i.e. the part that refers to the change in
viscosity or bulk modulus. shape. It follows that the case of incompressible flows, with
Rocks deform in different ways, necessitating different the density assumed to be constant and ∇ ·v = 0 (Eq. 4), also
rheologies. For example, rocks can deform elastically or by implies ε̇0 (v) = ε̇(v). Hence, incompressibility implies that
brittle failure. On long timescales their inelastic deformation there is no volume change of the material, so this part of the
is usually modelled as that of a highly viscous fluid. Based strain rate tensor is not taken into account.
on this latter assumption, we use the Stokes equations to de- In the Earth’s interior, viscous deformation is the domi-
scribe the deformation. This physical model adequately ex- nant rock deformation process on long timescales if temper-
plains many processes in the Earth’s interior related to man- atures are not too far from the rocks’ melting temperature.
tle convection and observations. However, some observations Under these conditions, imperfections in the crystal lattice
such as plate tectonics, which involve strain localisation and move through mineral grains and contribute to large-scale
strong hysteresis, and the existence of ancient geochemical deformation over time. The physical process that is thought
heterogeneity revealed by isotopic studies are behaviours not to most closely resemble this idealised case of a Newtonian
commonly associated with fluids. Indeed, in fluids, different fluid is diffusion creep, whereby single atoms or vacancies,
chemical components are well-mixed in a convective flow, i.e. the places in the crystal lattice where an atom is missing,
Solid Earth, 13, 583–637, 2022 https://doi.org/10.5194/se-13-583-2022I. van Zelst et al.: 101 geodynamic modelling 593
migrate through the crystal. Diffusion creep is assumed to be dividual strain rates add up to the total “effective” rate of
the dominant deformation mechanism in the lower mantle. deformation.
Other creep processes exhibit different behaviour. Disloca-
tion creep, which is considered to be important in the up- 2.2.3 Elastic deformation
per mantle, is enabled by defects in the crystal lattice that
are not just a single atom, but an entire line of atoms. Dis- On shorter timescales, the elastic behaviour of rocks be-
location creep is highly sensitive to the stress applied to a comes important in addition to viscous flow. In the case
rock, which means that the relation between stress and strain of elasticity, the stress tensor is related to the strain tensor
rate is no longer linear but a power law. Peierls creep, which through the generalised Hooke’s law σ = C : ε(u), where C
has an exponential relation between stress and rate of defor- is the fourth-order elastic tensor and ε is the strain tensor,
mation, becomes important at even higher stresses (Gerya, which depends on the displacement vector u as follows:
2019). Grain boundary sliding describes deformation due 1
to grains sliding against each other along grain boundaries, ε = (∇u + ∇uT ). (11)
2
which has to be accommodated by another mechanism that
allows the grains themselves to deform (e.g. diffusion or dis- Hence, for elastic deformation, the stress is proportional
location creep). It becomes important at high temperatures to the amount of deformation rather than the rate of defor-
and high stresses (Hansen et al., 2011; Ohuchi et al., 2015). mation, as is the case for viscous processes. Due to the in-
Consequently, the viscosity of rocks strongly depends on herent symmetries of σ , ε, and C, the tensor C is reduced
e.g. temperature, pressure, stress, size of the mineral grains, to two numbers for homogeneous isotropic media, and the
deformation history, and the presence of melt or water, and it stress–strain relation becomes
varies by several orders of magnitude, often over small length
scales. These experimentally obtained flow laws can be ex- σ = λ(∇ · u)I + 2G, (12)
pressed in a generalised form using the relation (Hirth and where λ is the first Lamé parameter and G is the second
Kohlstedt, 2003) Lamé parameter (also referred to as shear modulus), which
describes how much the material deforms elastically under
n −q r ∗ E + pV
ε̇ = Aσd d fH2 O exp(α φ) exp − , (10) a given shear stress. These two moduli together define Pois-
RT
son’s ratio ν (also see Sect. 2.1.1).
where ε̇ is the strain rate, σd is the applied differential stress, Elastic deformation is often included in geodynamic codes
d is the grain size, f H2 O is the water fugacity and relates that solve the Stokes equations by taking the time derivative
to the presence of water, φ is the melt fraction, and A, n, of Eq. (12), which introduces the velocity and strain rate. The
q, r, α ∗ , E, V , and R are constants. In geodynamic mod- term σ̇ is then approximated by a first-order Taylor expansion
elling, this flow law is usually recast in terms of the square (see Appendix A), which ultimately amounts to adding terms
root of the second invariant of the deviatoric strain rate or to the right-hand side of the momentum equation (Eq. 5; see,
stress tensors to obtain the effective viscosity (see Chapter for example, von Tscharner and Schmalholz, 2015).
6 of Gerya, 2019). Each of these parameters (except for R,
which is the gas constant) varies depending on the mineral 2.2.4 Brittle deformation
and creep mechanism. Which of these dependencies is most
important depends on the type of model. For example, diffu- For strong deformation and large stresses on short timescales,
sion creep exhibits a linear relation between stress and strain such as occurring in the crust and lithosphere, brittle defor-
rate, so n = 1, whereas dislocation creep strongly depends mation becomes the controlling mechanism. In this case, a
on stress (typically n ∼ 3 − 4), but there is no dependence on fault or a network of faults (which can range from the atomic
grain size (i.e. q = 0). Accordingly, temperature is probably to the kilometre scale) accommodates deformation. The rela-
the most important control in the lower mantle, while in the tive motion of the two discrete sides of the fault is limited by
upper mantle, the stress dependence plays a dominant role. the friction on the interface. However, in a continuum formu-
Grain size d is another potentially strong influence (e.g. Hirth lation, discontinuous faults cannot be represented, and hence
and Kohlstedt, 2003; King, 2016; Jain et al., 2018), but since the deformation in geodynamics models typically localises
the grain size in the mantle is not well-constrained it is hard in so-called shear bands of finite width, which can be inter-
to include this effect in geodynamic models (Dannberg et al., preted as faults on crustal and lithospheric scales (van Zelst
2017). While the presence of melt can also lead to substantial et al., 2019). We approximate the brittle behaviour by so-
weakening of rocks, the exact dependence on melt fraction called plasticity, characterised by a yield criterion that de-
is less well-constrained than other dependencies, and addi- termines the yield stress (also called yield strength) that the
tional equations for the motion of melt are usually required to maximum stress must satisfy. When locally built-up stresses
model a realistic melt distribution (see also Sect. 2.3). Gen- reach this yield stress, the rock is permanently deformed and
erally, several deformation mechanisms with different flow the plastic strain rate is no longer zero. Note that it is difficult
laws (Eq. 10) can be active at the same time, and their in- to compare a tensor (the stress) and a scalar (the yield stress).
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