We have discussed above that setting up a model includes choosing the equations that describe the physical process one wants to study. In a fluid dynamics model, these governing equations usually consist of a mass balance equation, a force balance or momentum conservation equation, and an energy conservation equation. The solution to these equations states how the values of the unknown variables such as the material velocity, pressure, and temperature (i.e. the dependent variables) change in space and how they evolve over time (i.e. when one or more of the known and/or independent variables change). Even though these governing equations are conservation equations, geodynamic models often consider them in a local rather than a global context, i.e. material and energy flow into or out of a system, or an external force is applied. Additionally, the equations only consider specific types of energy, i.e. thermal energy, and not, for example, the potential 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 property into or out of the system, and thermal energy may be generated or consumed through a conversion into other types of energy, e.g. through radioactive decay. This can be expressed using the basic principle. 1Accumulation=influx-outflux+generation-consumption. This statement suggests that accumulation, or the rate of change of a quantity within a system, is balanced by net flux through the system boundaries (influx–outflux) and net production within the system (generation–consumption). Depending on which physical processes are relevant and/or of interest, different terms may be included in the equations, which is a source of differences between geodynamic studies. Since each term describes a different phenomenon and may imply characteristic timescales and length scales, being aware of these relations is not only important for setting up the model, but also for interpreting the model results and comparing them to observations. Mathematical tools like scaling analysis or linear stability analysis are helpful to determine the relative importance of each of the terms (see also Sect. ) and for determining which terms should be included in any given model. In solid Earth geodynamic modelling, we are usually interested in problems occurring on large timescales on the order of thousands or millions of years (Fig. ). Accordingly, we usually model the Earth's interior as a viscous fluid. The corresponding equations are the Stokes equations, describing the motion of a highly viscous fluid driven in a gravitational field, and the energy balance written as a function of temperature (Fig. ). These equations are partial differential equations describing the relation between the unknowns (velocity, pressure, and temperature) and their partial derivatives with respect to space and time. Solving them requires additional relations, the so-called constitutive equations (Sect. ), that describe the material properties appearing as parameters in the Stokes equations and energy balance (such as the density) as well as their dependence on velocity, temperature, and pressure. We look at each equation in more detail in the following sections, but we do not provide their derivations and instead refer the interested reader to .

Using the equations discussed in the previous section (Sect. ) to model how Earth materials move under the influence of gravity requires some assumptions about how Earth materials respond to external forces or conditions. These relations are often called constitutive equations, and they relate the material properties to the solution variables of the conservation equations, like temperature and pressure. Which of these relations is most important for the model evolution depends on the application. In regional models that impose the plate driving forces externally, the relation between stress and deformation, i.e. the rheology (Sect. ), can be the most important parameter choice in the model and can control most of the model evolution. Since buoyancy is one of the main forces that drive the flow of Earth materials, it is often most important to take into account how the density depends on other variables and to include this dependency in the buoyancy term, i.e. the right-hand side of Eq. (). This is particularly important for mantle convection models and is described by the equation of state (Sect. ) of the material. In models of the lithosphere and crust, stress and state of deformation become more important. The constitutive equations can also include time-dependent terms, like in cases in which the strength of a rock depends on its deformation history, requiring additional equations to be solved (see also Sect. ).

The mass, momentum, and energy conservation equations are visibly coupled since velocity (derivatives) and pressure enter the stress tensor, and thermal energy transport due to advection depends on the velocity. More importantly, the previous sections have highlighted how material properties such as α, ρ, and Cp depend on pressure, temperature, and other quantities such as composition and the fact that deformation is partitioned between various mechanisms. Their pressure, temperature, and strain rate dependence makes the exact partitioning complex and renders the conservation in Eqs. (), (), and () very nonlinear with regards to the primary variables v, p, and T. Hence, the equations usually contain nonlinear terms that imply a relationship between the solution variables that is no longer linear.

In addition to temperature, pressure, and velocity there may be other conditions in the model that change over time and are important for the model evolution, but not directly related to changes in temperature, pressure, and velocity. A common example is the chemical composition of the material (which can refer to the major element composition but may also relate to the water content, for example). In this case, a transport equation is required for every additional quantity that should be tracked in the model and moves with the material flow: 15∂ci∂t+v⋅∇ci=0i=1,…nc, where ci is the to-be-tracked quantity, t is time, v is velocity, and nc is the number of compositions, fields, or materials present. This advection equation assumes that at any given location, changes in composition over time ∂ci/∂t are caused by the transport of material with the velocity v of the flow. The stronger the spatial variations in composition, expressed by the gradient of the composition ∇ci, and the faster the flow, the bigger the changes in composition over time. Another way to think about this is that this equation describes the mass conservation of each composition (compare to Eq. ). Consequently, if there are other processes that influence the composition, like chemical reactions or diffusion, the corresponding terms would have to be included in the equation.

Other physical processes may require additional terms or additional equations. Examples are the generation of the magnetic field in the outer core , two-phase or multi-phase flow , disequilibrium melting , complex magma dynamics in the crust , reactive melt transport , (de)hydration reactions and water transport , the evolution of the mineral grain size , the fluid dynamics and thermodynamics of a magma ocean , the interaction of tectonic processes with erosion and other surface processes , anisotropic fabric , phase transformation kinetics , and inertial processes and seismic cycles . Since the aim of this contribution is to introduce the general concepts of geodynamic modelling, we will not discuss these more complex effects here and will instead focus on the main conservation equations presented above.

In the end, the partial differential equations in Eqs. (), (), (), and potentially (), as well as the constitutive equations, must be supplemented by the geometry of the domain on which they are to be solved (e.g. 2-D or 3-D, Cartesian, cylindrical, spherical; Sect. ), a set of initial conditions for the time-dependent temperature and compositions, and a set of boundary conditions (Sect. ).

The solutions of the continuum equations described in Sect.  need to be computed on a finite number of moments in time and locations in space. Rewriting the equations in Sect.  for a discrete number of points is called discretisation. The three main methods of discretisation in geodynamics are the finite-element method (FEM), the finite-difference method (FDM), and the finite-volume method (FVM). The last two are equivalent in some instances, such as in the case of the commonly used staggered-grid finite-difference discretisation in geodynamic codes. Often combinations of these methods are used to deal with time and space discretisation separately. All three discretisation methods convert ordinary differential equations (ODEs) or partial differential equations (PDEs), which may be nonlinear, into a system of linear equations that can be solved by matrix algebra techniques. None is intrinsically better than the other, although there are differences that make a certain method more suitable for certain types of science questions. For instance, the finite-element and finite-volume methods are capable of dealing with non-Cartesian geometries, such as the spherical shape of the Earth, or topography at the surface of a model, while the finite-difference method is not. On the other hand, the finite-difference method is more intuitive in its design than the other two. An overview of the basic principles of these numerical methods is available in and , and an in-depth exposé of the finite-difference method in geodynamics is available in . It is worth noting that spectral methods are also encountered in mantle dynamics modelling , as is the boundary element method (BEM) and the lattice Boltzmann method . Meshless methods such as the discrete-element method DEM;, the smoothed particle hydrodynamics method SPH; , and the radial basis function method RBF; have also been used in geodynamic modelling. In what follows we focus on the most popular methods, i.e. finite differences, finite elements, and finite volumes.

The discretisation concept for all three main methods (FEM, FDM, FVM) is identical. The domain is subdivided in cells or elements, as shown in Fig. . In essence, the methods look for the solution of the equations in Sect.  in the form of combinations of polynomial expressions (also called shape functions in FEM terminology) defined on each element or cell.

To illustrate this concept, we provide a small example here for the conservation of energy using the finite-difference method, which is based on a Taylor expansion keeping only first- and second-order terms (see Appendix  for the complete example). In one dimension and under the assumptions that there is no advection or heat sources and that the coefficients are all constant in space, Eq. () becomes what is commonly called the heat equation: 16∂T∂t=κ∂2T∂x2. This is to be solved for T(x,t), for x between 0 and Lx, and t between 0 and tfinal. The time discretisation describes how to use a temperature distribution that is known at time tn to compute a new temperature at time tn+1=tn+Δt, with Δt being the time step size. The discretisation in space means that this temperature distribution is computed at a finite number np of points. The simplest way of choosing the position of these points is such that xi=ih, with h=Lx/(np-1) being the distance between points (which is often referred to as the resolution or grid size of the model) and the point indices running 0≤i≤np-1. As shown in Appendix , there are different ways of discretising Eq. (), but the so-called explicit version is written as 17Tin+1-TinΔt=κTi+1n-2Tin+Ti-1nh2, where the subscripts refer to space indices and the superscripts refer to time indices, i.e. Tin=T(xi,tn). This also means that Ti0 is the initial temperature condition needed to start the calculation. In this example, we know all temperatures at time step n and we can easily compute the temperatures at time step n+1 at all locations i inside the domain, while the temperature on the boundary T(xi=0) and T(xi=np-1) is prescribed by the boundary conditions. It also illustrates what the name of the finite-difference method means: when going from the continuum to a finite number of grid points, derivatives are approximated by differences in temperature between these points. In addition, “finite” refers to the mathematical definition of a derivative as a limit where h→0 is replaced by a formula in which h remains finite (see Appendix ).

While finite-difference codes only need to specify the order of the approximation they rely on (and the associated stencil), finite-element codes often specify the type of element they use (i.e. the specific polynomial expressions for the shape functions). This also controls how the partial differential equations are solved on the grid, with finite-difference methods solving the equations pointwise and finite-element methods averaging the equations per element. Earlier codes such as Citcom (S/CU) , ConMan , and SOPALE relied on the computationally cheap (i.e. yielding the smallest possible linear system size) Q1×P0 element. This finite-element pair is known to be unstable and this manifests itself via so-called pressure modes (see Sect. ). Stable pairs are to be preferred, and modern codes such as ASPECT and pTatin3D rely on the more accurate Q2×Q1 and Q2×P-1 (sometimes denoted Q2×P1disc) elements, respectively.

For quadrilaterals and hexahedra, the designation Qm×Qn means that each component of the velocity is approximated by a continuous piecewise polynomial of degree m in each direction on the element and likewise for pressure, except that the polynomial is of degree n. Again for the same families, Qm×P-n indicates the same velocity approximation with a pressure approximation that is a discontinuous complete piecewise polynomial of degree n (not of degree n in each direction) . Stable elements are typically characterised by m>n.

For all methods, the discretisation process results in a linear system of equations with its size being the number of unknowns, i.e. a multiple of the number of nodes and/or elements. This system of equations is written as 18A⋅X=b, where A is a large and very sparse matrix, X is the vector of unknowns, typically consisting of velocity components, pressure, and temperature on the grid, and b is the known right-hand-side vector. In each time step, the system is solved, and new solution fields are obtained, post-processed, and analysed. If the model evolves in time, the domain and therefore the mesh may evolve, compositions are transported, and a new system is formed.

After the discretisation step, a kinematical description must be chosen to define how the material is going to move in the model. There are several widely used options, i.e. the Eulerian, Lagrangian, and arbitrary Lagrangian–Eulerian (ALE) formulations (Fig. ).

Eulerian codes have a fixed mesh through which material flows. Since the evolution of the top boundary of the model is often of prime importance in geodynamical studies as it accounts for the generated topography, a feature that is directly and easily observable on Earth, the air above the crust must be modelled as well to allow for the formation of topography. This air layer has been coined “sticky air” because of its much higher viscosity (∼1018 Pa s) than real air (∼10-5 Pa s). This higher viscosity is necessary to avoid numerically difficult viscosity contrasts in the domain of more than 20 orders of magnitude. Although the sticky air viscosity seems high, it remains very low compared to the effective viscosity of the crust (∼1024 Pa s) bordering the sticky air, so negligible shear forces are exerted on the crust surface by the air layer. Hence, topography can be developed. Finite-difference codes typically use the Eulerian kinematical description and the sticky air approach. Problems associated with the use of sticky air and a possible solution are presented in .

In contrast to a Eulerian kinematical description, the mesh of Lagrangian codes deforms with the computed flow and therefore does not require the use of sticky air to model topography. This limits Lagrangian codes to small deformation. For example, subduction processes would quickly deform the mesh to such a point that it would not be suitable for accurate calculations. PyLith , a finite-element code for dynamic and quasistatic simulations of crustal deformation, uses a Lagrangian kinematical description.

Finally, as its name implies, the arbitrary Lagrangian–Eulerian (ALE) method, part of the semi-Lagrangian class of methods, is a kinematical description that combines features of both the Lagrangian and Eulerian formulations. In geodynamical codes, it often amounts to the mesh conforming vertically or radially to the free surface while retaining its structure in the horizontal or tangential direction (Fig. ). This approach forms the basis of the SOPALE , FANTOM , pTatin3D , and ASPECT codes.

The discretisation process outlined in Sect. () leads to a linear system as given in Eq. (). There are many ways to solve these large linear systems of equations, which can be split into two families: direct methods and iterative methods. Direct methods exploit the sparsity of the matrix (i.e. the ratio of non-zero terms to the number of terms in the matrix is very small) and arrive at the desired solution within a predictable number of operations. Because of the memory requirements of the direct solvers to store and process the matrix terms, these tend to be used for 2-D applications (or low-resolution 3-D problems) only. Examples of direct solver packages frequently used in geodynamics e.g. are MUMPS , Pardiso , and WSMP .

Iterative methods start with an initial solution (a guess) and incrementally improve on it until a convergence criterion is met; i.e. the remaining error (residual) is small. Common iterative methods in geodynamic codes are the conjugate gradient (CG) method and the GMRES (generalised minimal residual) method , which are used in conjunction with multigrid methods to accelerate their convergence. Note that the choice of iterative solving method is intrinsically tied to the properties of the matrix A. The most popular iterative solvers packages are PETSc used in pTatin3D, and Trilinos used in ASPECT,. Recently, iterative pseudo-transient solvers have been used to solve coupled sets of equations . These methods introduce a physics-based transient term (a time derivative) to a steady-state equation in order to iterate towards the steady-state solution. The matrix-free, finite-difference, pseudo-transient schemes of are well-suited for GPU (graphical processing unit) accelerated systems .

Early computing architectures of the 1970s were quite limited by today's standards and predominantly relied on sequential programming whereby one task is performed after the other (Fig. ). Hence, early models in the 1970s were confined to a few thousand unknowns at most . The computer architectures and processor speeds on which calculations are performed have since vastly increased in the past decades following Moore's law , resulting in models with several hundred million unknowns now possible . Accommodating such enormous computational loads was only made possible by the use of parallel architectures (Fig. ) and their growing availability. Such architectures can be multi-core processor-based commodity-grade desktop computers, homemade (so-called Beowulf clusters), or super calculators counting up to a few million processors as listed in the TOP500 (https://www.top500.org/, last access: 24 February 2022). The majority of the state-of-the-art geodynamics codes run in parallel on distributed memory systems and rely on the MPI (Message Passing Interface) standard (Fig. ). In MPI, domain decomposition is used; i.e. each computer thread of tasks only “knows” about a subset of the mesh for which the solution is obtained. This allows for a virtually unlimited number of threads but it requires the whole code to be built with parallelism in mind to achieve maximum efficiency .

Other codes, such as the surface processes code FastScape or the geodynamics code I2/I3ELVIS , rely on a different approach based on OpenMP, which is limited to running on shared memory systems. OpenMP can be added later on to an already existing sequential code and targets areas of the code which take the most time. Although appealing at first, this level of parallelism is limited by the number of CPUs attached to the single memory (typically a few dozen at the maximum). It is worth noticing that many codes or the linear algebra libraries that they link to use a combination of both MPI and OpenMP.

When documenting the parallel performance of a code, one often talks of strong and weak scaling . Strong scaling is defined as how the solution time varies with the number of processors for a fixed problem size. Optimal strong scaling is achieved if the solution time is inversely proportional to the number of processors. Conversely, when looking at weak scaling, both the number of processors and the problem size are increased by the same factor. This also results in a constant workload per processor, and one therefore expects a constant solution time for optimal weak scaling. In practice, all geodynamic modelling codes have different limits for their strong and weak scaling, thereby limiting the size of the model they can effectively compute.

As mentioned in Sect. , the set of partial differential equations to be solved is likely to be nonlinear; i.e. the coefficients of the equations depend on the solution. For example, the viscosity may depend on the velocity via the strain rate tensor, which makes the term of the momentum equation that is the product of viscosity and strain rate a nonlinear term. Special techniques must then be employed to solve this system, and so-called nonlinear iterations are carried out on the linearised equations until a converged solution is obtained. Note that these nonlinear iterations are distinct from the iterations taking place in the iterative method employed to solve the system. One type of nonlinear iteration is a fixed-point (Picard) iteration, whereby a guess of the solution is made and used to compute the solution-dependent coefficients. The linear system is solved, and the coefficients are updated with the new solution. This process is repeated until the left-hand side and the right-hand side of the linear system match up to a given tolerance. Then, the iterative process has reached convergence. This technique is sub-optimal in its convergence rate as it often requires dozens or hundreds of iterations for strongly nonlinear problems; i.e. the linear system must be solved as many times .

State-of-the-art codes now all rely on some form of Newton iterations based the Newton–Raphson method, which guarantee quadratic convergence towards the solution in contrast to the linear convergence of the Picard iteration, provided the initial guess is close enough to the solution of the nonlinear problem . It is worth noting that the implementation of Newton's method is substantially more complex and therefore difficult than Picard iterations .

Aside from the conservation of mass, momentum, and energy, special care must be taken when solving the advection in Eq. (), which accounts for the tracking of the different fluids, chemical species, or materials in a broad sense. Many techniques have been devised in the field of computational fluid dynamics and are used in geodynamic codes. The methods differ in whether they are designed to treat smoothly varying fields such as temperature or discontinuous fields such as those that represent lithologies. Many methods have been developed to advect smooth fields while minimising artefacts such as numerical diffusion and dispersion (i.e. streamline upwind Petrov–Galerkin or SUPG, flux limiters, total variation diminishing or TVD, flux-corrected transport or FCT, and the multidimensional positive-definite advection transport algorithm or MPDATA; ). Other employed techniques for tracking discontinuous fields are the level-set method , grid-based advection methods (sometimes referred to as compositional fields methods, Fig. , ), the marker-chain method , and the volume-of-fluid method . A popular advection method, especially for discontinuities, is the particle-in-cell (sometimes called marker-in-cell) technique (Fig. ), whereby a set of markers is placed throughout the domain at the beginning of the simulation with an assigned material identity, e.g. crust, mantle lithosphere, or sediments. These properties are then projected onto the mesh on which they are needed for solving the partial differential equations. Once a new velocity field has been computed, these markers are advected with the flow velocity field, and the process is repeated at each time step. This method is found in finite-element and finite-difference codes alike .

It is important to mention that there is no single best recipe for advection, and oftentimes methods are tested against each other . Each method comes with its own strengths and weaknesses and needs to be assessed in the context of the problem at hand. For example, particle-based methods are relatively easy to implement, not subject to numerical diffusion (a form of undesirable smearing of sharp gradients), and can also be used to track properties like deformation history, water content, composition, and phase transitions. However, they take up considerable memory, introduce artificial noise, and may violate conservation laws of the advected quantities when averaged to the grid. On the other hand, field-based methods are straightforward to parallelise and much lighter in terms of memory usage, but they are more challenging to advect and require numerical stabilisation. As such, choosing a field-based method for tracking a single smooth property is generally warranted, while particle methods serve well for tracking multiple properties that vary throughout the model domain.

On Earth, the lithosphere interacts with the cryosphere, biosphere, atmosphere, hydrosphere, magnetosphere, and other systems, and the deformation of the lithosphere is related to many natural hazards such as earthquakes, volcanic eruptions, landslides, and tsunamis. These systems are often inherently multi-scale with different processes occurring on vastly different timescales, length scales (Fig. ), and/or multiphysics wherein the coupling of different physical processes is important. Such multiphysics or multidisciplinary research often sees researchers coupling a geodynamics code with another existing one, be it for surface processes , atmosphere evolution , planetary impact , plate reconstruction modelling , elastic flexure , or dynamic rupture models and tsunami models . This coupling often takes place in the boundary conditions of the geodynamics code at the surface or at the bottom of the lithosphere or mantle. This coupling can be one-sided when e.g. plate velocities are prescribed. The coupling can also be two-sided when e.g. the uplift rate and a current topography of a geodynamic model are used in a surface process model that returns an eroded surface which forms the new top boundary of a geodynamic model. Coupling codes and the clever use of boundary conditions (Sect. ) are promising avenues to incorporate geodynamic models in a host of multidisciplinary research.

However, some multiphysics problems are so closely intertwined that solving the coupled system of equations requires different numerical methods than solving the problems individually (for example, coupled magma–mantle dynamics). In these cases, coupling requires the development of a new code that tightly integrates the different physical processes. One recent approach in building numerical applications for multiphysics problems that could be used for this makes use of Application Programming Interfaces (APIs) instead of readily available community codes. APIs can be a collection of routines that are optimised to perform certain operations such as assembling vectors and matrices, solving systems of equations in parallel (i.e. PETSc, https://petsc.org, , last access: 24 February 2022; ParallelStencil, https://github.com/omlins/ParallelStencil.jl, last access: 24 February 2022 for Julia https://julialang.org/, last access: 24 February 2022), or creating and solving geodynamically oriented objects such as tensors or even partial differential equation systems (i.e. FEniCS, https://fenicsproject.org/, last access: 24 February 2022; MILAMIN, http://milamin.org/, , last access: 24 February 2022). These APIs are not necessarily stand-alone codes, but are instead building blocks that require the user to set up their model (i.e. programme their code application using available routines). Therefore, they provide the user with more flexibility in creating and coupling models with different physics and can be more robustly tested as the APIs usually come with their own suite of tests. This approach also allows scientists to build simple experimental numerical applications, which can then be scaled to large parallelised production codes in a relatively short time span.

A model is, by definition, a simplified representation of a more complex natural phenomenon. This is a simple and obvious truth that is easily forgotten when geodynamic models are interpreted, presented, and reviewed. It is the modeller's responsibility to not only constantly remind themselves, but also others, about this key underlying fact.

The complexity of the planet Earth as a whole is vast. It is therefore challenging to reconcile such true complexity with a desired simplicity. A model can easily become too complex and, just as easily, oversimplified (Fig. ). A geodynamic modeller should always strive for the appropriate middle ground between a model that is too complex and a model that is too simple. This is one of the first major tasks of a modeller when setting up a new modelling approach to address a certain hypothesis: everything should be made as simple as possible but not simpler.

So, how simple should a model optimally be? The answer to this question is not an easy one, as it strongly depends on the purpose of the model, the capabilities to diagnose and understand it, and the hypothesis that it will test. It is clear though that a more complex model does not necessarily mean a better model. In fact, a simpler model is often better than a more complex model. A simpler model is clearer to understand, clearer to communicate, and, by making fewer assumptions, more likely right Occam's razor;, although not necessarily directly applicable to the real world.

There are various ways to reduce model complexity (Fig. ). Simplifying the physical model is one of them. Any given physical complexity of the natural phenomenon in question has to be evaluated and a decision has to be made to either reproduce it, parameterise it (i.e. mimic the phenomenon with a simplified approach), or neglect it in the model. This decision is based upon the model's purpose and the spatial and temporal extent of the process under investigation.

Further model simplification is achieved through numerical adjustments. For example, all the following studies model plate tectonics, but the geometry of the model can be complex (e.g. a 3-D spherical domain like ) or simple (e.g. a 1-D or 2-D domain like ). One can choose the temporal extent of the model to be more complex (e.g. time-dependent with a long evolution as in ) or simply instantaneous as done by . For the same model, such simplifying choices make for a generally simpler model. However, a model with simpler geometry (or time evolution) can, of course, also feature more complicated processes than one with a more complex geometry. Indeed, a simpler model geometry (or time evolution) often enables the modelling of more complex physical processes.

The numerical model complexity can also be adjusted by changing the initial and boundary conditions (heterogeneous as in , or homogeneous as in ) and the imposed forcing (space- and time-dependent forcing like , or self-consistent like ). Overall, complexity should be decided based on the scientific question addressed and the focus and scope of the study.

There are two overarching geodynamic modelling philosophies: specific modelling and generic modelling (Fig. ). The first philosophy attempts to reproduce the specific state of a certain geodynamic system (e.g. based on a specific observation) with a specific model to better understand the system's specific state. In contrast, generic modelling attempts to produce different regimes of behaviour of a certain geodynamic system (e.g. based on a general observation) to better understand the system's general behaviour.

Both overarching modelling philosophies can either fulfil or reject a hypothesis. Most results published to date fulfil a hypothesis, even though positive modelling results only hint at a certain phenomenon being responsible for an observation. Modelling results that reject a hypothesis (often called “failed models”) are of course more abundant, but also much clearer as they indeed serve as proof that a certain situation does not lead to a specific observation.

Furthermore, both overarching modelling philosophies can result in instantaneous and time-dependent studies . Instantaneous models are focused on resolving a certain state and usually rely heavily on a comparison with measured geophysical, geological, and geochemical data. Time-dependent models tend to focus more on the evolution or the natural state of a system. For general modelling, time-dependent studies can be performed either focusing on the transient evolution from a known initial state or the statistical (quasi-)steady state.

After choosing the equations that will be solved and the model geometry, both initial and boundary conditions are needed to solve the numerical model. The solution of the numerical model will depend on the initial and boundary conditions used, so it is important to choose them carefully .

The construction of a specific model setup to investigate a particular problem or hypothesis can give rise to numerical issues, despite successful code verification. During the model validation process, these issues are identified and addressed. They can usually be spotted through monitoring solver convergence behaviour and visual inspection of the solution throughout the model evolution, with model breakdown (i.e. a crash of the programme) and unexpected behaviour being the most obvious red flags. In this section, we describe a number of common problems and their potential solutions.

A resolution test should be standard to check whether a certain model, or model aspect, is mesh-dependent or not. So, the modeller should check the change in model results with higher mesh resolution. In the ideal case, from a certain resolution onward the solution no longer changes significantly or the spatial discretisation error becomes smaller than other errors like that resulting from time discretisation; i.e. the numerical solution has converged. It is desirable to use a grid spacing for which, for example, the thermal boundary layers or crustal compositions are well-resolved and the resolution therefore does not affect the model evolution anymore. However, it is not always possible to completely avoid grid dependency. For example, most implementations of brittle (plastic) deformation do not include an internal length scale and are therefore grid-dependent (; Fig. c). This grid dependency causes shear zone width to keep decreasing with increased resolution. There is an active research effort to include internal length scales that can limit grid dependency . The practical solution is to have one fixed resolution for the affected domain throughout the modelling study after assessment of changes in the overall model behaviour with resolution.

When using a free surface, quickly increasing model velocities, a corresponding increase in solver iterations, and a sloshing movement of the surface are indicative of the drunken sailor effect (Fig. a). It occurs for models with a free surface wherein the time step is chosen too large to accurately reproduce changes in surface boundary elevation. The interface then overshoots slightly in one time step, which causes it to overshoot in the other direction even further the next time step. The positive feedback of this numerical instability deriving from the stark density contrasts at the surface usually leads to the programme crashing. The stark density contrasts (approximately 1.2 kg m-3 versus 2830 kg m-3) lead to much larger stress perturbations from topographic changes compared to similar topography variations at a typical density contrast inside the Earth (e.g. the density jump at the continental crust–mantle boundary is ∼280 kg m-3; ). Solutions to the drunken sailor problem involve using either smaller time steps or a stabilisation algorithm e.g. to enable the model to run stably with larger time steps.

Another problem that can occur when mesh deformation is allowed in finite elements (i.e. in Lagrangian and ALE methods; Fig. ) is distortion of the mesh elements. The quality of the mesh decreases when element aspect ratios change too much from 1 for triangular as well as quadrilateral elements (e.g. the ratio of the width and height of a 2-D quadrilateral element, Fig. , and similarly in 3-D), and the accuracy of the computations therefore decreases. When the element distortion is too large, the computation will crash or at the very least become extremely inaccurate. To avoid such a distortion of the elements, local or global remeshing (i.e. mesh regeneration) or mesh smoothing can be applied (e.g. ). Diffusion of the surface topography can also help to stabilise the model by smoothing high-gradient topography. (Nonlinear) diffusion can be implemented atop the model, mimicking erosional processes e.g., but remeshing of the free surface also introduces some numerical diffusion that could help remedy instabilities. However, adding diffusion could possibly change the physics included in the model.

Visual inspection of the modelling results can uncover other issues. For one, smaller features (e.g. a subduction interface) can be seen to spread out over time and disappear; this can be due to diffusion or smearing of the advected field in the grid-based advection method. On the other hand, steep gradients of advected fields can lead to oscillations of these fields normal to the gradients. Mitigating such undershooting and overshooting requires more diffusion or different stabilisation algorithms of advection e.g..

Unstable yet very popular finite-element pairs such as the Q1×P0 element (Sect. ) are prone to spurious oscillations and an element-wise chequerboard pressure pattern (Fig. b as shown in Fig. 18 of or in ). Before this pressure can be used in rheological expressions it must be post-processed and smoothed so as to remove the chequerboard mode . Stable elements, which fulfil the Ladyzhenskaya–Babuška–Brezzi compatibility condition (LBB or inf-sup condition), do not exhibit pressure artefacts and are preferable (see , for examples of such elements). Moreover, the required number of outer iterations does not increase significantly with mesh resolution compared to the Q1×P0 element .

Internal inconsistencies can arise from disagreements in the modeller's choices in terms of boundary conditions, density formulations in the different governing equations, and the equations' coefficients. Not all inconsistencies are easily detectable or manifest themselves as numerical problems. For example, when the net prescribed inflow and outflow through the model boundaries is not (close to) zero, while a model is assumed incompressible, volume is no longer conserved and the solver for the Stokes equations might crash or return a nonsensical solution. When a free surface is used, this problem might be overlooked, as the surface can rise or fall in response to a net increase or decrease in volume, respectively. This physical inconsistency is also harder to detect in compressible models. Another example is prescribed surface velocities based on, for example, plate reconstructions models, which can add unrealistic amounts of energy into the modelled system.

Care should also be taken that the assumptions made to simplify the treatment of density in the governing equations (see Sect.  and specifically Sect. ) agree with one another. The simplest accepted combination of simplifications is the Boussinesq approximation.

The thermodynamics of Earth materials are very complex, especially in multi-phase, multi-material systems; hence, they are often simplified in the numerical model. For example, in nature the thermal expansivity varies both smoothly and abruptly with depth (e.g. Fig. 7 of ), but in models it is often taken as constant or merely smoothly increasing with depth. At first, the material properties described in Sect.  may appear to be independent. However, the definition of properties like density, thermal expansivity, specific heat, and compressibility need to satisfy thermodynamic relations in order for them to be consistent. These thermodynamic relations can be derived through thermodynamic potentials. For example, the thermodynamic potential of the thermal expansivity is defined as 19α=-1ρ∂ρ∂Tp, where α is thermal expansivity. It defines the fractional increase in volume of a material per degree of temperature increase at constant pressure.

The thermodynamic potential of the isothermal compressibility is defined as 20βT=1ρ∂ρ∂pT, where βT is the isothermal compressibility and the potential describes the percentage increase in density per unit change in pressure at constant temperature. For the isobaric heat capacity, the thermodynamic potential is defined as 21Cp=∂q∂Tp, where Cp is the isobaric heat capacity and the potential is defined as the ratio of the increment of heat δq added to the material to the corresponding change in temperature δT. The thermodynamic potentials also imply that 22∂Cp∂pT=-T∂α/ρ∂Tp.

All these relations have to be fulfilled at all temperatures and pressures. Consequently, it is often not immediately apparent if a given material description is thermodynamically consistent or not, and equations of state used in the geodynamic literature do not always take this into account Sect. 6.10. For example, it would be thermodynamically inconsistent to take thermal expansivity, compressibility at constant entropy (along an adiabat), and isobaric specific heat all as constants if the temperature varies in the model because, from Eq. (), any temperature dependence in α/ρ leads to a pressure dependence of the specific heat.

After the steps described in the previous section, checking for potential numerical issues and the internal consistency of the model setup, it is time to test whether the model results are consistent with our understanding of geodynamic processes. In a broad sense, does the model evolution stay within the bounds of what we know to be possible from geological and geophysical observations? More specifically, do the velocities obtained make sense? For example, does the sinking velocity of a slab lie within the estimated bounds from reconstructions and mantle tomography e.g.? Does the plume rise at acceptable speeds e.g.? Do the plate motions agree with plate reconstruction models e.g.for global reconstructions? Does the surface heat flow lie within observed values e.g.? Does the density structure agree with 1-D average profiles like the preliminary reference Earth model (PREM) ?

Note that deviations of the model results from generic observations do not necessarily mean that the results are wrong. In fact, a model of a natural system like the geodynamic models described here can never truly be validated . This is because natural systems are never truly closed. For example, neither the rocky planetary surface nor the core–mantle boundary represents true closed boundaries devoid of any temperature, compositional, or mechanical exchange with the outside world. As such, all model results are always non-unique and the model cannot be validated even if compared to a natural observation.

Considering this lack of experimental control from both the real world and analogue models from the lab (see Sect. ), techniques such as uncertainty quantification and cross-validation of the models beyond the typical testing and predicting of hypotheses become increasingly important to assess how well models capture the real world.

Visualising the model output allows us to test, analyse, diagnose, and communicate the model results. Figures can describe and summarise the enormous amounts of data that numerical modelling can produce and highlight important features that support the initial hypothesis. Depending on the complexity of the data and the objective of the figure, visualisation methods differ widely and range from a graph of root mean square velocity (a quantitative model diagnostic, Sect. ) over time to a complete 4-D animation of a certain solution variable like the velocity (see also Sect. ).

To cover the wide range of potential visualisation products, a multitude of visualisation programmes is available. Some of the commonly used software packages are gunplot (http://www.gnuplot.info/, last access: 24 February 2022) for creating 2-D and 3-D graphs, GMT (https://www.generic-mapping-tools.org, last access: 24 February 2022) for making maps and plotting data on maps , MATLAB (https://de.mathworks.com/products/matlab.html, last access: 24 February 2022) for scripted analysis and plotting of model data, Matplotlib (https://matplotlib.org/, last access: 24 February 2022) for static and interactive plotting in Python, ParaView (https://www.paraview.org/, last access: 24 February 2022) for looking at 2-D and 3-D datasets both interactively and in batch jobs , and similarly VisIt (https://wci.llnl.gov/simulation/computer-codes/visit/, last access: 24 February 2022) . There are even fully immersive visualisation options like CAVE and virtual reality display technologies that overcome the necessity to project 3-D data to 2-D surfaces for visual inspection. However, no matter the software, a modeller has to pay careful attention to circumvent the most common visualisation pitfalls for effective graphics, as outlined in Sect. .

Mere visual inspection of the model results is not sufficient to analyse and interpret the outcome of the simulations; a quantitative analysis of the results is also required. Deciding what specific post-processing is to be done can be a time-consuming process. There are a range of non-dimensional numbers that can be calculated to characterise the flow of fluids in a range of geodynamic environments. If multiple physical processes influence the behaviour of a system, the non-dimensional numbers derived from the governing equations (Sect. ) can be used to analyse the relative importance of each of these processes. Examples are the Rayleigh number, Ra, the Knudsen number, Kn, the Mach number, M, the Argand number, Ar, the Ekman number, Ek, the Reynolds number, Re, the Péclet number, Pe, the Prandtl number, Pr, and the Nusselt number, Nu. The Rayleigh number, in particular, is a widely used non-dimensional number characterising the vigour of buoyancy-driven flow. Many of these non-dimensional numbers can be defined differently for different environments, but they can all be insightful diagnostic metrics. With increasing model complexity, newer diagnostic numbers have been defined, such as plateness .

Further analysing and diagnosing a model then varies with the modelling approach that has been taken (see Sect. ). For specific modelling (i.e. model approaches directly comparing to an observation to understand the origin of a specific state of a system), a modeller should diagnose whether the model predictions match the observations to a satisfying degree. To test the match between the model prediction and the natural observation, a modeller can perform a visual comparison, conduct statistical analysis like a misfit measure, and check for a comparable dynamic model behaviour. Secondly, it should be diagnosed whether the hypothesised mechanism is actually responsible for creating the physical complexity of interest. To this end, the model sensitivity to the parameters used (and optimally also unused) needs to be systematically tested. It should be clear whether a variation of a parameter over its uncertainty range affects a model outcome or not, and if it does, by how much. Similarly, it should be clarified what impact currently neglected parameters would have on the model outcome, for example by discussing the results of other modelling studies that did include these parameters. After such diagnostics, the model can provide insight into what causes a specific state of the system.

For generic modelling (i.e. models used to reproduce basic fluid dynamics to understand how a specific system works), a modeller should diagnose whether there are individual models that exhibit similar behaviour within the model parameter space and whether there are controlling parameters that can, when changed, cause a switch from one regime to another. In general, often-used diagnostic quantities to define regimes are the root mean square (rms) and maximum values of certain model parameters like velocity or temperature. For example, the root mean square velocity over time can be used to check whether a model has reached steady state or shows signs of periodicity e.g.. For subduction models, the slab dip, slab tip depth, slab sinking velocity, plate age at the trench, and the interaction of the slab with the mantle transition zone are often monitored e.g.. Specific isotherms, fault length and offset, rift symmetry, and the migration of fault activity are often tracked in extensional models e.g.. From general convection models, the degree of entrainment or mixing, the convective regime (e.g. stagnant lid versus mobile lid), and the degree of layering are often extracted .

While some model analyses can be done by hand, the more elaborate post-processing that is becoming increasingly popular nowadays needs to be automated using open-source, testable, and extendable algorithms and shared as user-friendly software . To achieve such next-generation post-processing, like plate boundary tracking, extraction of lithosphere thickness, or computing the dynamic or isostatic topography , the output of geodynamic codes should ideally use a standard, widely accepted format and include metadata that can also be accessed by machines. Accessing individual subsets of the data, like individual time steps or parameter fields, should be straightforward. Model output should therefore be independent from computational details, like the number of computational cores the results have been produced on.

A few software packages that allow for automated post-processing and diagnosis of geodynamic models are available to support geodynamicists with analysing their increasingly complex models and the large datasets originating from them. However, such tools are rare because while most individual researchers spend a large amount of time in coding post-processing scripts, they often do not share those scripts with the geodynamics community. Moreover, scripts that are shared in the context of repeatability and transparency are not necessarily applicable or relevant to other software output. Making their own post-processing scripts more generically applicable can also not be required of individual scientists. Contributing to post-processing tools as part of a community software project is a great step forward, reducing the duplication of work while providing author recognition. Unfortunately, not all available post-processing tools supplied with community software can be applied to results from other codes. Defining a set of interfacing functions, like the Basic Model Interface (https://bmi-spec.readthedocs.io/en/latest/, last access: 24 February 2022) established by the Community Surface Dynamics Modeling System (CSDMS), facilitates communication between different modelling and post-processing tools.

Generic, open-access geodynamic diagnostic tools are Adopt to detect and outline surface plates, StagPy (https://pypi.python.org/pypi/stagpy, last access: 24 February 2022) to diagnose plate boundaries, and StagLab (https://www.fabiocrameri.ch/staglab/, last access: 24 February 2022) to perform a large set of geodynamic diagnostics. StagLab in particular offers an extensive set of robust 2-D and 3-D diagnostics. It works across multiple model domain geometries, resolutions, and model setups and produces publication-ready, fully annotated figures and movies using exclusively scientific colour maps (Sect. ). Moreover, StagLab is versioned and extendable, and it can process output of more than one mantle convection code while also offering low-effort compatibility for any other machine-readable output.

Other recent developments include the automated comparison of observations to model predictions to find the smallest misfit between the two. Such statistical and probabilistic inversion methods help determine the model parameters, e.g. mantle viscosity or crustal density, that result in the best fit of the model solution with the observed quantity through forward geodynamic modelling e.g..

Peer-reviewed scientific papers are essential to disseminate relevant information and research findings. In particular, it is important to make results understandable and reproducible in the methods and results sections. Reviewers will criticise incomplete or incorrect method descriptions and may recommend rejection because these sections are critical in the process of making the results reproducible and replicable. In this section, we briefly review the structure of a manuscript and highlight the parts required in a geodynamic numerical modelling study .

While there are many ways of writing a paper, the main purpose of a scientific paper is to convey information. Historically, the structure of scientific papers evolved from mainly single-author letters and descriptive experimental reports to a modern-day comprehensive organisation of the manuscript known as “theory–experiment–discussion” . The formal IMRAD structure (i.e. introduction, methods, results, and discussion/conclusions) was adopted in the 1980s and, at present, is the format most widely used and encouraged by scientific journals. The IMRAD structure facilitates modular reading and locating of specific information in pre-established sections of an article . In geodynamics, the general structure of a manuscript follows the IMRAD structure (Fig. ), although journals can place different emphasis on the individual components through reordering and formatting.

A good introduction should answer the following questions: what is the problem to be solved? What previous work has been done? What is its main limitation? What do you hope to achieve? How do you set up your investigation? One major mistake is to attempt to do an extensive literature review in the introduction, which often goes off topic. The introduction serves as the stage to lay out the motivation for the study, and any background reading should focus on the question being addressed.

The methods section is an important part of any scientific manuscript . A good methods section allows other scientists to verify results and conclusions, understand whether the design of the experiment is relevant for the scientific question (validity), and build on the work presented (reproducibility and replicability) by assessing alternative methods that might produce differing results. Thus, the major goals of the methods section are to verify the experiment layout and allow others to reproduce the results. Here, we outline standards for reporting the methods in numerical geodynamic modelling.

First, the methods should be plain and simple, objective, logically described, and thought of as a report of what was done in the study. Unstructured and incomplete methods can make the manuscript cumbersome to read or even lead the reader to question the validity of the research. Generally, journals have guidelines on how the methods should be formatted, but not necessarily what they should contain because they vary from field to field. The “who, what, when, where, how, and why” order proposed by breaks the methods section down into the following questions: who performed the experiment (not directly applicable to geodynamics, although one might mention here the specific cluster or supercomputer on which the simulations were run)? What was done to answer the research question? When and where was the experiment undertaken (i.e. what computational resources and which software versions were used?)? How was the experiment done, and how were the results analysed? Why were specific procedures chosen? The answers to these questions should be adapted to every field (i.e. in geodynamics, “results were obtained using code X on cluster Y”). Here, we focus on methods that have primarily theoretical (mathematical and physical) and numerical (computational) components, but geodynamic studies may have other aspects such as a data component (e.g. collection and post-processing of data) or an analogue component (e.g. laboratory experiments).

Figure  shows a breakdown of the most important elements of a manuscript and the methods section in particular. The methods should start with a brief outline (one paragraph) describing the study design and the main steps taken to answer the scientific question posed in the introduction. The outline should be logical and go from theoretical elements, to numerical aspects, to analysis and post-processing. First to be described is the theoretical framework. This includes the mathematical and physical concepts used in the study including the governing equations, constitutive equations, initial and boundary conditions, and any other relevant theory describing the model setup.

This is followed by a section on the computational approach explaining how the theory and the model are translated into computer language (Sect. ). This includes details on numerical methods (discretisation and other numerical techniques, code details, solvers and software libraries, etc.) and model setup. The model setup subsection should include details on the current experiment such as model geometry, resolution (numerical and physical), parameter values, and initial and boundary conditions. Any necessary figures of model geometry and tables of parameter values should be provided. More importantly, the choice of parameters should be motivated to explain their relevance to addressing the scientific question.

After the model setup has been explained, the methods should contain a section describing the design or layout of the study in detail. What is being tested or varied? How many simulations were performed in terms of model and parameter space? For example, one can use different model setups (i.e. lithosphere-scale and mantle-scale subduction models) with varying parameters in the same study. Why perform those simulations and vary those parameters? A summary table is handy to indicate all simulations that were run and which parameter was varied in which run. Additionally, it is important to include information on all input parameters and their values and units, as well as possible conversions within the code to enable reproducibility of the study (see Sect. ) and to foster transparency. This information ideally takes the form of an extensive table including the description, symbol, value, and SI units of the parameters. Errors may still be introduced during manuscript writing, and published values of input parameters may differ from values actually used in the numerical model. Automatic routines to print input parameters in publishable format directly from the code can avoid these mistakes in models and increase transparency and replicability. As an example, laboratory-derived creep laws are often listed haphazardly in tables for publications. However, published laboratory data may need conversion due to unit change (i.e. MPa-n s-1 to Pa-n s-1 for n>1 in creep laws) or correction due to the type of experiment (uniaxial, simple shear) in order to be directly usable in geodynamic models (see ). This example also demonstrates the importance of consistently listing the units of all parameters used in a study, preferably in SI units.

Analysis, visualisation, and post-processing techniques of numerical data should also be described in the methods section. This is a step generally ignored, but it is important to be open about it; e.g. “visualisation was performed in ParaView/MATLAB, and post-processing scripts were developed in Python/MATLAB/Unicorn language by the author”. If the post-processing methods are more complex, the author can provide more details (i.e. statistical methods used for data analysis). It is also good practice to provide these post-processing scripts for peer reviewing and community engagement (Sect. ).

Information should also be given on code and data availability. This was originally part of the methods section, but recently journals have introduced data management requirements (Sect. ), and this information may have a designated location in the manuscript template. However, it is good practice to write this information in the methods section. The authors should indicate code availability, code verification, and input files or other data necessary to reproduce the simulation results (e.g. installation guides). Additional questions to be answered in the methods section are the following: where were the simulations performed and on how many cores? What was the average runtime of a simulation? Can the model results be reproduced on a laptop or desktop computer, or is access to a cluster required?

Before moving to other sections, model assumptions need to be stated clearly in either the description of the theory or the numerical approach. Geodynamics is a field in which we take a complex system like the Earth or another planetary body and simplify it to a level from which we can extract some understanding (Sect. ). In doing so, we rely on a physically consistent set of assumptions. It is important to bear in mind that this set of assumptions may not always be obvious to the reader. As long as assumptions are explicit and consistent (i.e. through clear and honest communication), the reviewers and readers will find fewer flaws in the study. It is good practice to write a complete methods section for every manuscript, such as the one described here. However, some journals will ask for a short version to be included in the main manuscript and have the complete methods section in a separate resource (i.e. in the Appendix, Supplement, online repository).

Complementary to the methods section, the results section should be a report of the results obtained. The main goal of the results section is to present quantitative arguments for the initial hypothesis. However, any interpretation of the results or reference to other studies should be reserved for the discussion. For example, results in a mantle convection model might show that dense material accumulates at the bottom of the domain (i.e. core–mantle boundary). The interpretation of these results is that they provide a mechanism to explain how LLSVPs (large low-shear-velocity provinces) have formed. Illustrations, including figures and tables, are the most efficient way to present results (Sect. ). However, authors should only include material and information relevant to demonstrate the scientific arguments discussed in the next section. Therefore, to avoid distraction, writers should present additional data as Supplements, e.g. movies of the whole simulation, whereas only a few snapshots are provided in the main body of the paper.

The discussion section relates all the questions in the manuscript together: how do these results relate to the original questions or objectives outlined in the introduction section? Do the results support the hypothesis? Are the results consistent with observations and what other studies have reported? The modeller should discuss any simplifying assumptions, shortcomings of numerical methods and results, and their implications for the study. For example, the discussion in a specific modelling study (Sect. ) should address how applicable the model results are to the specific problem or region, whereas the discussion in a generic modelling study should aim to understand the underlying factors in a given system. If the results are unexpected, the authors should try to explain why. Are there other ways to interpret the results? What further research would be necessary to answer the questions raised by the results? What does it all mean? Many manuscripts are rejected because the discussion section is weak and the authors do not clearly understand the existing literature. Writers should also put their results into a global context to demonstrate what makes those results significant or original.

At this point in preparing the manuscript, the authors have all the necessary elements to write the abstract and conclusions and come up with a descriptive title. Both the abstract and conclusion summarise the entire publication, but in a different way: one as a preview and one as an epilogue, respectively. It is crucial to focus a paper on a key message, intended for both specialist and non-specialist readership, which is communicated in the abstract and conclusions. Some journals also include a plain language summary and/or graphical abstracts as alternative ways to engage a broader audience.

In the end, every scientific manuscript has additional components such as the references, acknowledgements, Supplement, software and data availability, and author contributions (Fig. ) that contain further information about how the study was funded, conducted, and shared with the community. Acknowledging the often substantial contributions of reviewers is a common courtesy.

In this section, we have primarily referred to scientific articles, but scientific manuscripts can also be reviews, editorials, and commentaries. The structure and contents of these manuscripts differ for each type. Each publisher and journal have their own style guidelines and preferences, so it is good practice to consult the publisher's guide for authors. Finally, even though scientific manuscripts may have a rigidly defined structure due to journal guidelines, there is still plenty of flexibility. In fact, the best manuscripts are creative, tell a story that communicates the science clearly, and encourage future work.

There are many different ways to visualise geodynamic models, and it is challenging to figure out how to do so most effectively. However, avoiding the most common visualisation pitfalls is the best start for any modeller looking into visually communicating important results across the research community and possibly beyond. The key aspects to remember when creating figures, thereby preventing misleading visual impressions, are the following: (1) scales, like graph axes and colour bars, must always be included to allow quantification of data values. (2) Bar plots must always have a zero baseline (or in the logarithmic case, have a baseline at 1), to not mislead the reader with altered relative bar heights. (3) Pie diagrams should be avoided as angles and areas are not easily quantifiable by the human brain and are therefore not directly comparable to each other. These problems are exaggerated when pie charts are displayed as 3-D structures, which causes the values underlying the pieces closest to the viewer to appear artificially larger than the others. (4) Heat maps (i.e. plots with differently coloured tiles) should have numbered tiles that include the data value, as surrounding colours heavily distort the impression of a given colour, which can mislead the viewer's perception of the underlying data values significantly . (5) Colours must be applied in such a way that data are reflected correctly and are inclusive to all readers . Scientifically derived colour maps exist, like Colorbrewer, MPL, Cividis, CMOcean, CET, and scientific colour maps , and must be chosen over unscientific default colour maps like rainbow, jet, seis, and others (Fig. ). (6) Visualisation should be subject to the same scientific scrutiny as other research methods to communicate data and concepts truthfully, accessibly, and clearly.

All aspects of a figure need to be explained and understandable. While filtering, resampling, or otherwise post-processing model results instead of plotting raw data can improve the message purveyed by the figure, such actions should be mentioned in the figure caption. Some numerical codes work, for example, in dimensionless numbers (Sect. ) and require scaling of the model output before they can be related to observations. However, too much information jammed in a figure can easily render the figure unusable to the reader. Again, the modeller has to simplify enough to reach the sweet spot, without oversimplifying the figure (compare with Sect. ). Everything that can be removed without losing key information should be removed. Unnecessary and/or duplicated axes labels, e.g. those repeated across multiple panels, should be removed. The same applies to other figure aspects like colour bars. To make a figure intuitive to readers, colour bars in multiple figure panels applied to the same parameter should optimally maintain the same range (i.e. map the same colour to the same data value), and if they do, displaying just one colour bar is sufficient.

Displaying 3-D models effectively is challenging and somewhat arbitrary, as the third dimension is often difficult to convey in a still image. Given the current dominant occurrence of non-interactive, two-dimensional canvases (e.g. the pdf format), 2-D slices of parameter fields often represent the model more effectively than 3-D volumes. The combination of various datasets, like flow characteristics on top of a temperature field, can be effective but is also challenging. Velocity arrows, for example, should not overlap or distract from the remaining important content of the figure. If the velocity in a 3-D visualisation is displayed using arrows, they should be coloured according to their magnitude because their lengths are distorted by the 3-D perspective. Stream lines and a coloured contour plot of the velocity field often provide a more suitable solution to display the flow direction and patterns, as well as its velocity magnitudes, respectively.

An uninformed, unscientific use of colours not only excludes a large portion of the readership, for example through hardly distinguishable colour combinations for readers with a colour-vision deficiency (like the most common red–green colour blindness), but also significantly distorts the underlying model data visually e.g.and Fig. . In fact, the distortion can be more than 7 % of the displayed data range (e.g. a temperature variation of 20∘ could look like 24∘ in one part along a colour axis that ranges between 0 and 100∘, while it looks like 17∘ in another part), which in most cases easily amounts to the biggest error in a modelling study. Therefore, modellers should avoid using default colour maps blindly, but instead look for scientific colour maps . If software does not offer them, or at least allow them to be imported, users should contact the software developers. Hence, scientific colour maps should be the default in common geodynamic codes and visualisation programmes.

Scientific colour maps are perceptually uniform to prevent data distortion, and they are perceptually ordered to make data intuitively readable, colour-vision deficiency friendly, and optimally readable in black and white to include all readers. Suitable scientific colour maps e.g. that offer fair data representation and are fully inclusive have become readily usable (either as built-in options or after importing them) with all major modelling, visualisation, and graphics software, like MATLAB, Python, R, GMT, QGIS, ParaView, VisIt, Mathematica, gnuplot, GIMP, and Inkscape. Indeed, the scientific colour maps suite e.g. offers custom colour map classes for any data type. The sequential class (e.g. batlow) is best suited for monotonically increasing data like a temperature field. The diverging class (e.g. vik) is best suited for zero-centred data like a horizontal velocity field. A multi-sequential class (e.g. oleron) should be used with zero-centred, non-diverging data like a combination of ocean bathymetry and land topography, and the cyclic class (e.g. romaO) is tuned for circular data like azimuthal direction. Moreover, these scientific colour maps are offered as a continuous type (e.g. batlow) to visualise small-scale data variation, a discrete type (e.g. batlow10) to highlight similar data values more clearly, and a categorical type (e.g. batlowS) to clearly differentiate unordered data for line or scatter plots. In general, colour maps are an integral part of the colour axis (or colour bar) and their key purpose is to represent – and not interpret – data. As such, the local colour gradient, like the tick spacing along an x axis, should be the same all along the entire colour bar (or vary constantly in the case of a logarithmic axis). To interpret or highlight certain parts of the rendered data, other graphic tools like superimposed contour lines or visual indicators such as arrows should be used. More in-depth information, a clear user guide to using colour and colour maps, and a concise list of the major currently available scientific colour maps are provided in .