Geodynamics 101

The geodynamics of planetary habitability

The geodynamics of planetary habitability

The Geodynamics 101 series serves to showcase the diversity of research topics and/or methods in the geodynamics community in an understandable manner. In this week’s Geodynamics 101 post, Brad Foley, Assistant Professor at the Department of Geosciences, Pennsylvania State University, talks about the geodynamics of planetary habitability and in particular the key role of CO2 cycling in the mantle. 

Figure 1: Assistant professor Brad Foley at the Department of Geosciences, Penn State University.

Earth’s mantle is the planet’s engine. The loss of heat from the interior to space drives Earth’s tectonic processes, mountain building and orogeny, volcanism, and the core dynamo generating Earth’s magnetic field. But perhaps less appreciated is that the mantle also plays a critical role in shaping the state of the atmosphere. This link between surface and interior evolution is not just important for studying the Earth, but also the other rocky planets in our solar system, and rocky exoplanets. Factors that make a planet, like Earth, a suitable home for life, such as the presence of liquid water oceans, weathering processes that provide critical nutrients to the oceans, and a temperate climate are all directly influenced by deep interior processes (Foley & Driscoll, 2016). Likewise, a complex interaction between life, atmospheric chemistry, weathering, volcanism, and sediment burial led to the rise of oxygen on Earth, which is both critical for some forms of life and a signature of the presence of life (Claire et al, 2006; Kump & Barley, 2007; Lyons et al, 2014). Thus, unravelling the factors that allowed Earth to develop into a planet teaming with life, whether those same factors are likely to be present on other rocky planets, and whether potential biosignatures, like atmospheric oxygen, are likely to arise on exoplanets if life is present, all require considering the role of the mantle.


Cycling of CO2: A key factor in planetary habitability

The abundance of atmospheric gases is determined by the balance between their sources and sinks, and the mantle acts as an important source and sink for many gases: volcanism releases volatiles locked in rocks to the surface, while subduction brings volatiles from the surface back into the interior. One of the most critical for habitability is CO2, which controls the climate state. On Earth, the cycling of CObetween surface, interior, and atmosphere involve a stabilizing feedback that acts to buffer climate (Kasting & Catling, 2003). COis drawn out of the atmosphere by weathering of silicate rocks and the formation of carbonate minerals on the seafloor, which are then subducted to return carbon to the mantle (Figure 2). Critically, the rate of silicate weathering increases with increasing surface temperature or atmospheric CO2. Thus when the climate warms the weathering rate increases, acting to cool the climate down, and when the climate is cool the weathering rate decreases, allowing outgassing of COby volcanism to warm the climate up (Walker et al, 1981).


Figure 2: Schematic cartoon of the carbonate-silicate cycle on Earth. Silicate weathering on land and seafloor weathering near mid-ocean ridges remove CO2 from the atmosphere and deposit it in the ocean crust. This carbon is then subducted, where some fraction is outgassed at arc volcanoes, with the rest returning the mantle. Outgassing from mid-ocean ridges and ocean islands returns mantle carbon to the atmosphere. From Foley & Driscoll, 2016.


However, this feedback can fail in two ways: first, rates of CO2outgassing must be high enough to keep the climate from plunging into a globally glaciated or snowball state (Kadoya & Tajika, 2014); second, there must be sufficiently high rates of physical erosion to remove weathered rock and bring fresh rock into the near-surface weathering zone (Foley, 2015). The mantle plays an important role in both CO2outgassing and surface erosion rates. The CO2outgassing rate is determined by the rate of volcanism, mantle carbon content, and oxidation state, while erosion rates are controlled by rates of tectonic uplift and mountain building over geologic timescales.


Role of the mantle in CO2 cycling: Future directions

However, there are many aspects of how the mantle influences COoutgassing and weathering rates that are still poorly understood, and exciting avenues of future research. First-order constraints on rates of volcanism and outgassing, and how they change over time, are straightforward to calculate from both simple box models of planetary thermal evolution or 2- and 3-D mantle convection calculations (Noack et al, 2017; Tosi et al, 2017). As planets cool over time, volcanic outgassing rates decline and eventually become low enough for frozen, snowball climates to develop. Factors that keep a planet’s mantle warmer for longer, such as higher rates of radiogenic heat production or tidal heating, will thus act to prolong the lifetime of habitable surface conditions (Foley & Smye, 2018; Valencia et al, 2018). Yet there are still important uncertainties, in particular on how carbon is carried into, and circulates within, the mantle that are key avenues for future research. Moreover, the connection between mantle dynamics, mountain building, erosion, and weathering rate is still poorly understood. Erosion rates are high when topographic gradients are large, as in mountainous regions. Mountain building is most likely connected to surface plate speed and the vigor of mantle convection, however just what form this connection takes is not known. How mantle convection and plate tectonics leads to the formation of topography, and hence influences weathering and erosion, is a critical area of research for understanding the controls on long-term climate evolution.


How Earth-like must a planet be to be habitable?

Ultimately one of the most important questions driving future research in planetary evolution and exoplanets, and which geodynamicists should be a central part of answering, is how Earth-like a planet needs to be in order to sustain volatile cycles that allow for the development of life and for biosignatures, such as oxygen, to accumulate in the atmosphere once life has developed (Tasker et al, 2017). Exoplanets come in a wide range of sizes (see Figure 3): planets up to about 4 Earth masses are found to be rocky, while beyond this limit planets are volatile-rich like Neptune (Rogers, 2015), and likely compositions as well (Hinkel & Unterborn, 2018). These planets could display a range of different surface tectonic modes, including plate tectonics, stagnant lids, or some intermediary style of tectonics. Oxidation states could be different, influencing the type of gases outgassed by volcanism. Instead of outgassing predominantly CO2, a planet with a more reduced mantle could outgas mostly CO or CH4. Likewise, different crustal compositions could alter weathering processes and the stability of volatiles as they are recycled into the interior at subduction zones or by crustal foundering. Exploring these issues will require interdisciplinary research including geochemists, mineral physicists, and geodynamicists, as well as biogeochemists, climate scientists, and astronomers. With future space telescopes poised to image exoplanet atmospheres, research on the role of the planetary interior in shaping the surface environment and atmosphere has never been so relevant.


Figure 3: Exoplanet population as of August 2017. Image credit: NASA/Ames Research Center/Natalie Batalha/Wendy Stenzel



Claire, M. W., Catling, D. C., & Zahnle, K. J. (2006). Biogeochemical modelling of the rise in atmospheric oxygen. Geobiology4(4), 239-269.

Foley, B. J., & Driscoll, P. E. (2016). Whole planet coupling between climate, mantle, and core: Implications for rocky planet evolution. Geochemistry, Geophysics, Geosystems17(5), 1885-1914.

Foley, B. J., & Smye, A. J. (2018). Carbon Cycling and Habitability of Earth-Sized Stagnant Lid Planets. Astrobiology18(7), 873-896.

Hinkel, N. R., & Unterborn, C. T. (2018). The Star–Planet Connection. I. Using Stellar Composition to Observationally Constrain Planetary Mineralogy for the 10 Closest Stars. The Astrophysical Journal853(1), 83.

Kadoya, S., & Tajika, E. (2014). Conditions for oceans on Earth-like planets orbiting within the habitable zone: importance of volcanic CO2degassing. The Astrophysical Journal790(2), 107.

Kasting, J. F., & Catling, D. (2003). Evolution of a habitable planet. Annual Review of Astronomy and Astrophysics41(1), 429-463.

Kump, L. R., & Barley, M. E. (2007). Increased subaerial volcanism and the rise of atmospheric oxygen 2.5 billion years ago. Nature448(7157), 1033.

Lyons, T. W., Reinhard, C. T., & Planavsky, N. J. (2014). The rise of oxygen in Earth’s early ocean and atmosphere. Nature506(7488), 307.

Noack, L., Rivoldini, A., & Van Hoolst, T. (2017). Volcanism and outgassing of stagnant-lid planets: Implications for the habitable zone. Physics of the Earth and Planetary Interiors269, 40-57.

Rogers, L. A. (2015). Most 1.6 Earth-radius planets are not rocky. The Astrophysical Journal801(1), 41.

Tasker, E., Tan, J., Heng, K., Kane, S., Spiegel, D., Brasser, R., ... & Houser, C. (2017). The language of exoplanet ranking metrics needs to change. Nature astronomy1, 0042.

Tosi, Nicola, Mareike Godolt, Barbara Stracke, Thomas Ruedas, John Lee Grenfell, Dennis Höning, Athanasia Nikolaou, A-C. Plesa, Doris Breuer, and Tilman Spohn. "The habitability of a stagnant-lid Earth." Astronomy & Astrophysics 605 (2017): A71.

Valencia, D., Tan, V. Y. Y., & Zajac, Z. (2018). Habitability from Tidally Induced Tectonics. The Astrophysical Journal857(2), 106.

Walker, J. C., Hays, P. B., & Kasting, J. F. (1981). A negative feedback mechanism for the long‐term stabilization of Earth's surface temperature. Journal of Geophysical Research: Oceans86(C10), 9776-9782.



Magma dynamics

Magma dynamics
Assistant Professor Juliane Dannberg, University of Florida.

Assistant Professor Juliane Dannberg, University of Florida.

In this week’s Geodynamics 101 post, Juliane Dannberg, Assistant Professor at the University of Florida, outlines the role of mantle melt generation and transport in geodynamics.

Mantle melting and magma transport are important influences on the dynamics and chemical evolution of the Earth’s interior. All of Earth’s oceanic crust and depleted oceanic lithosphere is generated through melting at mid-ocean ridges, the main process to introduce chemical heterogeneity into the mantle. Mechanically, melt and fluids play an important role in the dynamics of plate boundaries like mid-ocean ridges and subduction zones, and in the formation of oceanic islands. Specifically, partially molten regions are weaker than solid rock, and because of that, mechanisms that localise melt often also localise deformation.

Many mantle convection models include the effects of melting and melt transport to a first order. They allow for the generation of oceanic crust and lithosphere when mantle material approaches the surface. This is typically done by using a thermodynamic model to determine where melting occurs, removing melt as soon as it it forms, and distributing it further up or at the surface of the model as basaltic crust. The residuum that is left behind becomes the depleted, harzburgitic lithosphere.

But considering the addition of melt, a low-viscosity fluid phase, in models of mantle dynamics is not only important for geochemical considerations. It also introduces new length and time scales and new dynamics. Rather than talking about numerical methods, my goal for this article is to give a short overview of the physics of magma dynamics, and to build intuition on how magma generation and transport can affect geodynamic processes.

The forces at play

In mantle convection models, the driving force, buoyancy, is usually balanced by viscous stresses.
Consequently, the length scales of convective features are usually determined by how fast boundary layers can grow, by the density differences causing material to be buoyant, and by the mantle viscosity. When fluids are added to this system, new forces become important1:
(1) Darcy drag. Melt can segregate from where it forms by flowing through the pore spaces of the solid host rock. We can think of the system as a sponge-like solid rock matrix that can be squeezed and stretched, saturated with liquid melt. As melt starts to flow, the traction on the interface between the melt and the mineral grains opposes this motion. The more pore space there is, the better it is connected, and the lower the viscosity of the melt, the easier it becomes for the for the melt to flow through.
(2) Viscous compaction. When melt is generated, or when melt is about to flow into a space that had no melt before, the pore space of the solid rock has to dilate so that melt can flow in. In other words, magma is pushing the mineral grains of the rock apart. Conversely, when melt flows out of an existing pore space, it pulls the grains together and the solid rock needs to compact to fill this volume. This means that the forces that drive the flow of melt have to overcome the resistance of the solid rock to volumetric deformation, governed by the host rock’s compaction viscosity.

The equations

So instead of the Stokes equation used in mantle convection models, magma dynamics is described by a more complicated force balance with additional terms (McKenzie, 1984; Scott & Stevenson, 1984,1986; Bercovici, Ricard et al., 2001, 2003; Sramek et al, 2007):

This equation includes a number of material properties that control magma dynamics. The melt viscosity ηmelt and the permeability k express how easy it is for melt to flow through the pore space of the rock. The larger the pore space, and the better the connection between the pores, the larger the permeability. Because the rock is saturated with melt, the amount of pore space, or porosity ϕ, is equivalent to the fraction of melt. Consequently, the more melt is present, the larger the permeability. Δv is the difference between solid velocity and melt velocity and can be expressed using Darcy’s law:

In words, melt segregation relative to the motion of the solid is driven by differences in the melt or fluid pressure pf relative to the hydrostatic pressure in the melt, ρmelt g.

The viscous stresses are controlled by the shear viscosity η and the compaction viscosity ξ. Both properties depend on the melt fraction: The more melt is present (or, the larger the pore space of the rock), the easier it becomes to deform the solid rock matrix. This dependence is quite strong, and as soon as the porosity reaches 20–30% (the disaggregation threshold), the solid rock will break apart and will no longer form a connected matrix. Instead, the mineral grains will be suspended within the melt. Buoyancy is controlled by the density difference between solid and melt Δρ and the gravity g.

Depending on the importance of each of these forces in a given setting, the transport of melt can occur in different ways (Sramek et al., 2007). Buoyancy is caused by the lower density of the melt compared to the solid rock, so the more melt is present, the stronger the buoyancy forces. If viscous compaction forces are small (which corresponds to rocks that are easy to deform), buoyancy is balanced by Darcy drag (the so-called Darcy equilibrium; Sramek et al., 2007). In this case, melt ascends pervasively through the pores of the host rock, and its segregation velocity is limited by the rock’s permeability. Accordingly, the more melt is present, the faster it can ascend. This mode of melt transport may be representative of melting zones below mid-ocean ridges, where the amount of melt present in the pores of the host rock is small.

Numerical model of melt transport below MOR and in plume head.

Figure 1: Below mid-ocean ridges, melt can segregate from the solid and flow towards the ridge axis (top, modified from Dannberg et al., 2019). If there is a high-viscosity barrier to melt ascent, such as at the top of plumes, melt may circulate together with the solid in form of diapirs (bottom).


Conversely, if Darcy drag is small, then buoyancy is balanced by viscous compaction (the so-called viscogravitational equilibrium; Sramek et al., 2007). Melt can only flow if the solid rock deforms, and in this case the viscosity of the rock limits how fast melt can rise. If the compaction viscosity is comparable to or smaller than the shear viscosity, the host rock expands and melt can flow through the pores (Scott, 1988). But if the bulk viscosity is much larger than the shear viscosity, melt is forced to stay in the pocket of host rock it was created in over a long time scale. Instead of segregating, it will ascend together with the solid in form of a diapir (Scott, 1988). Because the Darcy drag becomes smaller the more melt is present (corresponding to a larger permeability), this mode of melt transport may occur near the top of plume heads (Fig. 1; Dannberg & Heister, 2016). Within the plume, high temperatures reduce the viscosity, but above the plume head, ambient mantle viscosities are larger and may limit melt segregation.






The equations also introduce a new length scale that controls the size of features that emerge in magma dynamics. This scale is controlled by the ratio of viscous forces and Darcy forces:

and is called the compaction length. It controls how far dynamic pressure differences between melt and solid, the compaction pressures, are transferred in partially molten rocks. This is important for the buoyant ascent of melt, which is hindered by the viscous resistance of the solid rock matrix to this compaction (Spiegelman 1993a, Spiegelman 1993b, Katz, 2015).

Solitary waves

Variations in porosity in the direction of gravity cause disturbances in the compaction pressure (Fig. 2): For example, if the porosity decreases in the direction of flow, the corresponding decrease in permeability makes it more difficult for melt to flow through and causes a decreasing flux of melt upwards into the low-porosity region. This negative gradient in melt flux leads to melt pressures being larger than solid pressures (a positive compaction pressure) at the location of the porosity perturbation, pushing the grains of the solid rock apart. This means that more melt can flow into this region and porosity increases. At the upper end of this perturbation, this same process continues to act, drawing more melt in.

At the lower end of the high-porosity region, the opposite happens: Because permeability increases in the direction of flow, the melt flux increases in upwards direction, causing a negative compaction pressure. Mineral grains are pulled together, and melt is expelled. Because the process continues in upwards and downwards direction, variations in the amount of melt will develop into magmatic waves (or solitary waves). At the front of the wave, the positive compaction pressure draws in more melt, and at the back of the wave, the negative compaction pressure pulls the mineral grains together as melt flows out. The length scale of these waves is on the order of the compaction length. In one dimension, this process can also be illustrated by the flow of fluid through a viscously deformable pipe (Scott, 1988). Imagine a part of the pipe that locally has a larger radius than the rest of the pipe and that moves upwards. To accommodate the arrival of more liquid, the pipe has to expand in front of the perturbation. In the same way, the pipe contracts behind the perturbation, reverting to its original radius.

Formation of solitary waves.

Figure 2: [Updated] Formation of solitary waves. After Spiegelman 1993b.

In the Earth’s mantle, the compaction length is on the order of a few to a few tens of kilometres. So in geodynamic models on tectonic scales, these waves are so small that they may first look like pressure or porosity oscillations! But knowing where this behaviour comes from is important for understanding what these waves mean, and if pressure waves in a numerical model are numerical artefacts or part of the actual physical behaviour of the system.

There are many more mechanisms that can cause melt to localise. When partially molten rock is sheared, melt is drawn into thin, melt-rich bands in between larger melt-poor regions. Channels of high porosity can form when reactive melting is driven by the flux of magma along a solubility gradient. A good overview over these processes and sources of further information is given in the lectures notes by Katz, 2015.

The bigger picture

All of these processes may be important for the flow of melt in subduction zones, at mid-ocean ridges, and for hotspot volcanism. Therefore I believe it is essential to better understand magma dynamics if we want to answer questions such as: How much of the mantle melt reaches the surface? What is the reason for the location and spacing of volcanoes? How are plate boundaries generated and maintained? I hope this article helps to understand this topic better and inspires you to consider melt generation and transport in this bigger context.

1These are the two most important forces in the mantle, but there are other forces that may impact deformation of partially molten rock. I have made a number of assumptions here, for example that the melt viscosity is much smaller than the solid velocity, that surface tension is negligible, and that deformation of the solid rock is predominantly viscous.


Bercovici, D. et al., 2001. A two‐phase model for compaction and damage: 1. General theory. J. Geophys. Res. Solid Earth, 106(B5), pp.8887-8906.

Bercovici, D. and Ricard, Y., 2003. Energetics of a two-phase model of lithospheric damage, shear localization and plate-boundary formation. Geophys. J. Int., 152(3), pp.581-596.

Dannberg, J. and Heister, T., 2016. Compressible magma/mantle dynamics: 3-D, adaptive simulations in ASPECT. Geophys. J. Int., 207(3), pp.1343-1366.

Dannberg, J. et al., 2019. A new formulation for coupled magma/mantle dynamics. Geophys. J. Int., 219(1), pp.94-107.

Katz, R.F., 2015. An introduction to coupled magma/mantle dynamics.

McKenzie, D., 1984. The generation and compaction of partially molten rock. J. Petrol., 25(3), pp.713-765.

Scott, D.R. and Stevenson, D.J., 1984. Magma solitons. Geophys. Res. Lett., 11(11), pp.1161-1164.

Scott, D.R. and Stevenson, D.J., 1986. Magma ascent by porous flow. J. Geophys. Res. Solid Earth, 91(B9), pp.9283-9296.

Scott, D.R., 1988. The competition between percolation and circulation in a deformable porous medium. J. Geophys. Res. Solid Earth, 93(B6), pp.6451-6462.

Spiegelman, M., 1993a. Flow in deformable porous media. Part 1 Simple analysis. J. Fluid Mech., 247, pp.17-38.

Spiegelman, M., 1993b. Flow in deformable porous media. Part 2 numerical analysis–the relationship between shock waves and solitary waves. J. Fluid Mech., 247, pp.39-63.

Šrámek, O. et al., 2007. Simultaneous melting and compaction in deformable two-phase media. Geophys. J. Int., 168(3), pp.964-982.

EGU GA 2020 call-for-sessions deadline

EGU GA 2020 call-for-sessions deadline

The deadline for session (and short course!) proposals for EGU 2020 is tomorrow on September 5, 2019! So, if you have a great idea for a session or a short course you still have a little bit of time to write a smashing proposal, find a nice co-convener and submit it to ensure that you will be able to access the convener’s party next year without a fuss.

Why not share your knowledge on correct code management and version control with the community? Maybe you could spark a discussion about disabilities or gender equality in geosciences. Or maybe you and your colleagues can devise the perfect session for your niche research topic, such that finding a session won’t be a problem any more! Don’t hesitate: anyone can do it!

What? You don’t believe me? Well, we have been hosting the Geodynamics 101 short courses (inspired by our blog posts) for 2 years now and they have been a great success! And trust me, I didn’t know anything about organising short courses at EGU either before I embarked on this journey. Turns out: it’s pretty straightforward.

For a little transparency and insight into organising the courses, here is a little summary of the feedback from the two courses we hosted at EGU 2019:
• Geodynamics 101A: Numerical methods
• Geodynamics 101B: Large-scale dynamic processes
And we’ll also discuss our plans for the short course next year!

We decided to have 2 short courses at EGU 2019, because of the feedback we got at EGU 2018 about people desiring more info on the applications of numerical methods in geodynamics. Your wish is our command, so we delivered. Both courses were attended well, but the first course (about numerical methods) was more popular by far with people queueing and standing even outside of the room. So, we will ask for a bigger room at EGU 2020.

People from many different disciplines attended the courses. Surprisingly a large amount of geodynamicists were present! Most of them (you?) attended to get a quick refresher in (the broad area) of geodynamics and to see how you can do geodynamics outreach. Our course was particularly popular with geologists, and there were also some hydrologists, meteorologists, and computer scientists attending!

The scientific background of the participants in the two Geodynamics 101 short courses at EGU GA 2019. Did you join us?

People gave the course a high rating with an average grade of 4.4/5 for Geodynamics 101A and Geodynamics 101B. The highest ratings were given by the geodynamicists attending the short courses. Hm… maybe because you are our friends?

The rating of the Geodynamics 101 short courses by our participants (divided by background).

The main feedback was that people would like to know how to actually run codes and see some more hands-on examples. So, next year, we will be back with a new and improved version of the “Geodynamics 101: Numerical methods” course which will include some demos of running a code.

Maybe I will see you there and who knows…? Maybe I will be at your session or short course!

The past is the key

The past is the key

Lorenzo Colli

“The present is the key to the past” is a oft-used phrase in the context of understanding our planet’s complex evolution. But this perspective can also be flipped, reflected, and reframed. In this Geodynamics 101 post, Lorenzo Colli, Research Assistant Professor at the University of Houston, USA, showcases some of the recent advances in modelling mantle convection.  


Mantle convection is the fundamental process that drives a large part of the geologic activity at the Earth’s surface. Indeed, mantle convection can be framed as a dynamical theory that complements and expands the kinematic theory of plate tectonics: on the one hand it aims to describe and quantify the forces that cause tectonic processes; on the other, it provides an explanation for features – such as hotspot volcanism, chains of seamounts, large igneous provinces and anomalous non-isostatic topography – that aren’t accounted for by plate tectonics.

Mantle convection is both very simple and very complicated. In its essence, it is simply thermal convection: hot (and lighter) material goes up, cold (and denser) material goes down. We can describe thermal convection using classical equations of fluid dynamics, which are based on well-founded physical principles: the continuity equation enforces conservation of mass; the Navier-Stokes equation deals with conservation of momentum; and the heat equation embodies conservation of energy. Moreover, given the extremely large viscosity of the Earth’s mantle and the low rates of deformation, inertia and turbulence are utterly negligible and the Navier-Stokes equation can be simplified accordingly. One incredible consequence is that the flow field only depends on an instantaneous force balance, not on its past states, and it is thus time reversible. And when I say incredible, I really mean it: it looks like a magic trick. Check it out yourself.

With four parameters I can fit an elephant, and with five I can make him wiggle his trunk

This is as simple as it gets, in the sense that from here onward every additional aspect of mantle convection results in a more complex system: 3D variations in rheology and composition; phase transitions, melting and, more generally, the thermodynamics of mantle minerals; the feedbacks between deep Earth dynamics and surface processes. Each of these additional aspects results in a system that is harder and costlier to solve numerically, so much so that numerical models need to compromise, including some but excluding others, or giving up dimensionality, domain size or the ability to advance in time. More importantly, most of these aspects are so-called subgrid-scale processes: they deal with the macroscopic effect of some microscopic process that cannot be modelled at the same scale as the macroscopic flow and is too costly to model at the appropriate scale. Consequently, it needs to be parametrized. To make matters worse, some of these microscopic processes are not understood sufficiently well to begin with: the parametrizations are not formally derived from first-principle physics but are long-range extrapolations of semi-empirical laws. The end result is that it is possible to generate more complex – thus, in this regard, more Earth-like – models of mantle convection at the cost of an increase in tunable parameters. But what parameters give a truly better model? How can we test it?

Figure 1: The mantle convection model on the left runs in ten minutes on your laptop. It is not the Earth. The one on the right takes two days on a supercomputer. It is fancier, but it is still not the real Earth.

Meteorologists face similar issues with their models of atmospheric circulation. For example, processes related to turbulence, clouds and rainfall need to be parametrized. Early weather forecast models were… less than ideal. But meteorologists can compare every day their model predictions with what actually occurs, thus objectively and quantitatively assessing what works and what doesn’t. As a result, during the last 40 years weather predictions have improved steadily (Bauer et al., 2015). Current models are better at using available information (what is technically called data assimilation; more on this later) and have parametrizations that better represent the physics of the underlying processes.

If time travel is possible, where are the geophysicists from the future?

We could do the same, in theory. We can initialize a mantle convection model with some best estimate for the present-day state of the Earth’s mantle and let it run forward into the future, with the explicit aim of forecasting its future evolution. But mantle convection evolves over millions of years instead of days, thus making future predictions impractical. Another option would be to initialize a mantle convection model in the distant past and run it forward, thus making predictions-in-the-past. But in this case we really don’t know the state of the mantle in the past. And as mantle convection is a chaotic process, even a small error in the initial condition quickly grows into a completely different model trajectory (Bello et al., 2014). One can mitigate this chaotic divergence by using data assimilation and imposing surface velocities as reconstructed by a kinematic model of past plate motions (Bunge et al., 1998), which indeed tends to bring the modelled evolution closer to the true one (Colli et al., 2015). But it would take hundreds of millions of years of error-free plate motions to eliminate the influence of the unknown initial condition.

As I mentioned before, the flow field is time reversible, so one can try to start from the present-day state and integrate the governing equations backward in time. But while the flow field is time reversible, the temperature field is not. Heat diffusion is physically irreversible and mathematically unstable when solved back in time. Plainly said, the temperature field blows up. Heat diffusion needs to be turned off [1], thus keeping only heat advection. This approach, aptly called backward advection (Steinberger and O’Connell, 1997), is limited to only a few tens of millions of years in the past (Conrad and Gurnis, 2003; Moucha and Forte, 2011): the errors induced by neglecting heat diffusion add up and the recovered “initial condition”, when integrated forward in time (or should I say, back to the future), doesn’t land back at the desired present-day state, following instead a divergent trajectory.

Per aspera ad astra

As all the simple approaches turn out to be either unfeasible or unsatisfactory, we need to turn our attention to more sophisticated ones. One option is to be more clever about data assimilation, for example using a Kalman filter (Bocher et al., 2016; 2018). This methodology allow for the combining of the physics of the system, as embodied by the numerical model, with observational data, while at the same time taking into account their relative uncertainties. A different approach is given by posing a formal inverse problem aimed at finding the “optimal” initial condition that evolves into the known (best-estimate) present-day state of the mantle. This inverse problem can be solved using the adjoint method (Bunge et al., 2003; Liu and Gurnis, 2008), a rather elegant mathematical technique that exploits the physics of the system to compute the sensitivity of the final condition to variations in the initial condition. Both methodologies are computationally very expensive. Like, many millions of CPU-hours expensive. But they allow for explicit predictions of the past history of mantle flow (Spasojevic & Gurnis, 2012; Colli et al., 2018), which can then be compared with evidence of past flow states as preserved by the geologic record, for example in the form of regional- and continental-scale unconformities (Friedrich et al., 2018) and planation surfaces (Guillocheau et al., 2018). The past history of the Earth thus holds the key to significantly advance our understanding of mantle dynamics by allowing us to test and improve our models of mantle convection.

Figure 2: A schematic illustration of a reconstruction of past mantle flow obtained via the adjoint method. Symbols represent model states at discrete times. They are connected by lines representing model evolution over time. The procedure starts from a first guess of the state of the mantle in the distant past (orange circle). When evolved in time (red triangles) it will not reproduce the present-day state of the real Earth (purple cross). The adjoint method tells you in which direction the initial condition needs to be shifted in order to move the modeled present-day state closer to the real Earth. By iteratively correcting the first guess an optimized evolution (green stars) can be obtained, which matches the present-day state of the Earth.

1.Or even to be reversed in sign, to make the time-reversed heat equation unconditionally stable.