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Geodynamics 101

On the influence of grain size in numerical modelling

On the influence of grain size in numerical modelling

The Geodynamics 101 series serves to showcase the diversity of research topics and methods in the geodynamics community in an understandable manner. We welcome all researchers – PhD students to Professors – to introduce their area of expertise in a lighthearted, entertaining manner and touch upon some of the outstanding questions and problems related to their fields. This month Juliane Dannberg from Colorado State University, discusses the influence of grain size and why it is important to consider it in numerical models. Do you want to talk about your research? Contact us!

Juliane Dannberg

When I started my PhD on geodynamic modelling, I was not aware that the size of mineral grains was something I might need to consider in my simulations. To me, grain size was something that sedimentologists need to describe rocks, and not something I had to deal with in my computations. In all the modelling papers I had read, if the mineral grain size was even mentioned, it was always assumed to be constant. However, it turns out that these tiny grains can have huge effects.

I first heard about the importance of grain size in a series of lectures given by Uli Faul when I participated in the CIDER summer program in 2014 (in case you’re interested, the lectures were recorded, and are available here and here). Primarily, I learned that for diffusion creep – the deformation mechanism people predominantly use in convection models – the viscosity does not only depend on temperature, but is also strongly controlled by the grain size, and that this grain size varies both in space and in time.

This made me think. If grain size in the mantle changes by several orders of magnitude, and the viscosity scales with the grain size to the power of 3, didn’t that mean that grain size variations could cause huge variations in viscosity that we do not account for in our models? Shouldn’t that have a major effect on mantle dynamics, and on the evolution of mantle plumes and subduction zones? How large are the errors we make by not including this effect? I was perplexed that there was such a major control on viscosity I had not thought about before, and wanted to look into this further myself.

Luckily, the multidisciplinary nature of CIDER meant that there were a number of people who could help me answer my questions. I teamed up with other participants1 interested in the topic, and from them I learned a lot about deformation in the Earth’s mantle.

How does the mantle deform?
For most of the mantle, the two important deformation mechanisms are diffusion and dislocation creep. In diffusion creep, single defects (or vacancies) – where an atom is missing in the lattice of the crystal – move through this lattice and the crystal deforms. For this type of deformation, the strain rate is generally proportional to the stress, which means that the viscosity does not depend on the strain rate. Many global mantle convection models use this kind of rheology, because it is thought to be dominant in the lower mantle, and also because it means that the problems described using this rheology are usually linear, which makes them easier to solve numerically. Furthermore, this is also the deformation mechanism that depends on the grain size. Usually, it is assumed that the viscosity scales with the grain size to the power of 3:

where ndiff is the diffusion creep viscosity, d is the grain size, m=3, T is the temperature, P is the pressure, and Adiff, E*diff, V*diff and R are constants.

However, if the stress is high (or the grain size is large, Figure 1), dislocation creep is the dominant deformation mechanism. In dislocation creep, linear defects – so-called dislocations – move through the crystal and cause deformation. In this regime, the viscosity depends on the strain rate, but not on grain size. Dislocation creep is generally assumed to be the dominant deformation mechanism in the upper mantle.

Figure 1: Deformation mechanisms in olivine

Why do grains grow or shrink?

In general, from an energy standpoint, larger crystals are more stable than smaller crystals, and so crystals tend to grow over time in a process called Ostwald ripening. The smaller the grains are, and the higher the temperature, the faster the grains grow. One the other hand, the propagation of dislocations through the grains causes so-called dynamic recrystallization, which reduces the grain size if the rock deforms due to dislocation creep. This means that there are always the competing mechanisms of grain growth and grain size reduction, and their balance depends on the dominance of either of the two deformation mechanisms described above – diffusion or dislocation creep:

The left-hand side of the equation describes the change of the grain size d over time, the first term on the right-hand side is grain growth (depending on grain size and temperature), the second term describes grain size reduction (depending the strain rate, the stress, and also on the grain size itself). The parameters Pg, kg, Eg, Vg, R, λ, c and γ are all constants.

If the flow field does not change, grains will evolve towards an equilibrium grain size, balancing grain growth and grain size reduction. In addition, the grain size may change when minerals cross a phase transition. If the mineral composition does not change upon crossing a phase transition (a polymorphic phase transition such as olivine–wadsleyite), there is almost no effect on grain size. But if the composition of the mineral that is stable after crossing the transition is different from the one before, the mineral breaks down, and the grain size is reduced, probably to the micrometer-scale [Solomatov and Reese, 2008].

And what does that mean for the dynamics of the Earth?

As there is a complex interaction between grain size evolution, mantle rheology and the deformation in the mantle, it is not straightforward to predict how an evolving grain size changes mantle dynamics. But it turned out that there had been a number of modelling studies investigating this effect. And they indeed found that grain size evolution may substantially influence the onset and dynamics of convection [Hall and Parmentier, 2003], the shape of mantle plumes [Korenaga, 2005], mixing of chemical heterogeneities [Solomatov and Reese, 2008], the seismic structure of the mantle [Behn et al., 2009], and the convection regime and thermal history of terrestrial planets [Rozel, 2012].

The long way to a working model…

But even knowing all of these things, it was still a long way to implement these mechanisms in a geodynamic modelling code, testing and debugging the implementation, and applying it to convection in the Earth. There were several reasons for that:

Firstly, large viscosity contrasts are already a problem for most solvers we use in our codes, and the strong dependence of viscosity on grain size means that viscosity varies by several orders of magnitude over a very small length scale in the model.

Secondly, considering an evolving grain size makes the problem we want to solve strongly nonlinear: Already in models with a diffusion–dislocation composite rheology and a constant grain size, the viscosity – which is needed to calculate the solution for the velocity – depends on the strain rate, making the momentum conservation equation nonlinear. But an evolving grain size introduces an additional nonlinearity: The viscosity now also depends (nonlinearly) on the grain size, whose evolution in turn depends on the velocity field. In terms of dynamics, this means that there is now another mechanism that can localize deformation. If the strain rate is large, the grain size is reduced due to dynamic recrystallization (as described above). A smaller grain size means a lower viscosity, which again enables a larger strain rate. Due to this feedback loop, velocities can become very high, up to several meters per year, which severely limits the time step size of a numerical model.

Finally, the equation (2) that describes grain size evolution is an ordinary differential equation in itself, and the time scales of grain growth and grain size reduction can be much smaller than changes in the flow field in the mantle. So, in order to model grain size evolution and mantle convection together, one has to come up with a way to separate these scales, and use a different (and probably much smaller) time step to compute how the grain size evolves. I remember that at one point, our models generated mineral grains the size of kilometers (whereas the grain sizes we expect in the mantle are on the order of millimeters), because we had not chosen the time step size properly. And on countless occasions, the code would just crash, because the problem was so nonlinear that a small change in just one parameter or a solution variable had such a large impact that material properties, velocities and pressures went outside of the range of what was physically reasonable.

However, after a lot of debugging, we could finally investigate how an evolving grain size would influence mantle dynamics. But see for yourself below. In an example from our models, plumes become much thinner when reaching the upper mantle, and cause much more vigorous small-scale convection when they interact with the lithosphere. Slabs bend rather than thicken, and accumulate as piles at the core-mantle boundary.

Figure 2: Comparison of plumes and slabs in models with and without grain size evolution. Modified from Dannberg et al., 2017

Of course, there are also many other areas where grain size evolution is important, and many recent studies are concerned with the influence of grain size on the Earth’s dynamic evolution. Dave Bercovici and his collaborators found that grain evolution and damage mechanisms may be a key factor for the onset of plate tectonics [e.g. Bercovici and Ricard, 2014, 2016]: Grain size reduction in shear zones could make them weak enough to for subduction initiation. The evolution of grain size may also be a major factor for focusing of melt to mid-ocean ridges [Turner et al., 2017], as it influences how fast the solid matrix can dilate and compact to let melt flow in and out. And if the Large Low Shear Velocity Provinces at the core-mantle boundary are indeed piles of hot material that are stable on long time scales, mineral grains would have a long time to grow and may play a crucial role for pile stability [Schierjott et al., 2017].

So if you do not include grain size evolution in your geodynamic models – which in many cases is just not feasible to do – I hope that you now have a better feeling for how that may affect your model results.

1The other researchers in my CIDER group were Zach Eilon, Ulrich Faul, Rene Gassmöller, Raj Moulik and Bob Myhill. I learned a lot about grain size in the mantle in particular from Bob Myhill and Ulrich Faul; I developed the geodynamic models together with Rene Gassmöller, and Zach Eilon and Raj Moulik investigated how the evolving grain size predicted by our models would influence seismic observations.

References:

Solomatov, V. S., & Reese, C. C. (2008). Grain size variations in the Earth's mantle and the evolution of primordial chemical heterogeneities. Journal of Geophysical Research: Solid Earth, 113(B7).

Hall, C. E., & Parmentier, E. M. (2003). Influence of grain size evolution on convective instability. Geochemistry, Geophysics, Geosystems, 4(3).

Korenaga, J. (2005). Firm mantle plumes and the nature of the core–mantle boundary region. Earth and Planetary Science Letters, 232(1), 29-37.

Behn, M. D., Hirth, G., & Elsenbeck, J. R. (2009). Implications of grain size evolution on the seismic structure of the oceanic upper mantle. Earth and Planetary Science Letters, 282(1), 178-189.

Rozel, A. (2012). Impact of grain size on the convection of terrestrial planets. Geochemistry, Geophysics, Geosystems, 13(10).

Dannberg, J., Eilon, Z., Faul, U., Gassmöller, R., Moulik, P., & Myhill, R. (2017). The importance of grain size to mantle dynamics and seismological observations. Geochemistry, Geophysics, Geosystems.

Bercovici, D., & Ricard, Y. (2014). Plate tectonics, damage and inheritance. Nature, 508(7497), 513-516.

Bercovici, D., & Ricard, Y. (2016). Grain-damage hysteresis and plate tectonic states. Physics of the Earth and Planetary Interiors, 253, 31-47.

Turner, A. J., Katz, R. F., Behn, M. D., & Keller, T. (2017). Magmatic focusing to mid-ocean ridges: the role of grain size variability and non-Newtonian viscosity. arXiv preprint arXiv:1706.00609.

Schierjott, J., Rozel, A., & Tackley, P. (2017, April). Toward unraveling a secret of the lower mantle: Detecting and characterizing piles using a grain size-dependent, composite rheology. In EGU General Assembly Conference Abstracts (Vol. 19, p. 17433).

The jelly sandwich lithosphere: elastic bread, the jelly, and gummy bears

The jelly sandwich lithosphere: elastic bread, the jelly, and gummy bears

The Geodynamics 101 series serves to showcase the diversity of research topics and methods in the geodynamics community in an understandable manner. We welcome all researchers – PhD students to Professors – to introduce their area of expertise in a lighthearted, entertaining manner and touch upon some of the outstanding questions and problems related to their fields. This month Vojtěch Patočka from the Charles University in Prague, Czech Republic, discusses the rheology of the lithosphere and its food analogies. Do you want to talk about your research? Contact us!

It is becoming increasingly obvious that geodynamics and cooking are closely related, especially to the participants of a recent workshop in the Netherlands. Food analogies are helping students to get a physical grasp of continuum mechanics and to forget their lunches at university canteens. Here we take a basic look at a field where talking about fine cuisine has long been established: the rheology of the lithosphere. But first we step back a little.

Elastic solids deform when a force is applied and return to their original shape when the force is removed. Viscous fluids eventually take the shape of a container that is applied to them: they spontaneously flow to a state of zero shear stress. One can hardly imagine more different materials than these two, and yet the Earth’s mantle is sometimes modelled as an elastic shell and sometimes considered to be a viscous fluid.

You may be thinking:

Sure, it is a matter of what time scale is relevant for the process at hand

– just as J.C. Maxwell already thought: “Hence, a block of pitch may be so hard that you cannot make a dent in it by striking it with your knuckles; and yet it will in the course of time flatten itself by its weight, and glide downhill like a stream of water.” To visualise Maxwell’s dreams, the University of Queensland has been continuously running a pitch drop experiment for the past 90 years.

It is for the reason above that seismologists vibrate a Hookean Earth in their computers and geodynamicists play with viscous fluids. Are both of them right? Surprisingly, only the seismologists are. There is direct evidence that the outer parts of planetary mantles have an important elastic component even on geological time scales, which implies that treating the entire mantle as viscous is wrong. A textbook example can be found near the deepest part of the world’s oceans. Fig. 1, adopted from the bible by Turcotte & Schubert, shows how one can fit the bathymetric profile across the Mariana trench to the shape of a bent elastic plate. Note that the slab is subducting at the rate of a few centimeters per year, meaning that each segment of the slab is loaded for tens of millions of years before it disappears into the mantle and it still retains the elastic strain.

Figure 1: Comparison of a bathymetric profile across the Mariana trench (solid line) with the deflection of a thin elastic plate subject to end loading (dashed line). Distance xbx0 is the half-width of the forebulge from which the thickness of the plate can be inferred. Adopted from section 3 of Turcotte & Schubert, 2002.

Flexure studies and effective elastic thickness

Other similar examples, referred to as flexure studies, are summarised nicely by Watts et al., 2013. They include deflections under seamounts that litter oceanic floors and also some much longer lasting structures, such as continental foreland basins with the Ganges basin representing a particularly popular case. The free parameter that plays first fiddle in flexure studies is the thickness of elastic plate that matches observation. What does the resulting value, known as the effective elastic thickness, actually tell us?

I will use a dirty trick of bad journalism and quote an innocent geology blog slightly out of context:

The Indian crust is cold and rigid. Clever folk can do the maths on the shape of the crust as it bends down. This confirms that the pattern matches the model for rigid, elastic deformation. It also allows us to calculate the plate’s flexural rigidity, which is a measure of its strength. This means quantifying the rheology of real bits of the earth, which is a very useful trick

The math referred to involves purely elastic deformation and the flexural rigidity in its definition depends only on Young’s modulus, thickness, and Poisson ratio of the plate. Are the clever folk trying to fool us into believing that the lithosphere is an elastic plate? From lab measurements and geology we know for sure that it is not. Brittle failures are an abundant feature in both the crust and upper mantle, and various solid state creep mechanisms must be active in the deeper parts of the lithosphere.

The best fit for the Indo-Australian plate subducted below the Ganges basin is obtained with an elastic plate of circa 90 km thickness and for the Pacific plate at the Mariana trench it is circa 30 km. In both cases the plates are actually much thicker. What the computed values of 90 and 30 km tell us, is how much of elastic energy is present within the plate over the process of subduction, despite the brittle failures and despite the ductile creep activated in response to shear stresses within the plate.

To move away from sinking slabs, think about the story of a seamount that popped out on an ocean floor. Imagine a fresh, unstressed segment of oceanic plate that gets suddenly loaded by the uninvited underwater volcano. Small intra-plate fractures may immediately form, depending on the size of the load, and ductile creep will begin to continuously deform the plate’s deeper parts. The elastic energy present in the material upon loading gets partially released, either immediately through cracking or gradually via ductile creep. Measuring effective (or equivalent) elastic thickness merely tells us the amount of elastic energy left at the time of measurement.

This is all well known to the authors of flexure studies. In fact, they often re-draw the Christmas trees (see Fig. 2) and provide constraints on rheological zonation of the Earth. Pioneers in the field are E.B. Burov and A.B. Watts, whose papers are often accidentaly googled by chefs searching for latest trends in the dessert industry due to the extensive use of the words jelly sandwich and crème brûlée ([4] and [5], for breakfast see also [6]). The main point of the discussion is to determine the brittle/ductile transition and the active creep mechanism in a realistic, compositionally stratified lithosphere. This is usually complicated by the fact that lithologies measured in lab experiments strongly depend on composition, water content and temperature – and these are not well constrained in the real Earth.

Figure 2: The total force per unit width necessary to break or viscously deform a lithospheric section at a given strain rate. Plots like these are known as the Christmas tree plots, here adopted from Basin Analysis by P.A. Allen and J.R. Allen (without the food).

A geodynamical paradox

It is a paradox that in geodynamical modeling we often use the constraints from flexure studies and at the same time forget about elasticity, without which there would be no such studies at all. Recall that the primary result of shape fitting, for instance of the one depicted in Fig. 1, is that some elastic support is present. As I warned above, one can only hardly distinguish between a purely elastic plate of a given thickness and some other, thicker elasto-brittle or visco-elastic plate when looking at its surface flexure. However, there must be elasticity involved in some form: purely viscous or visco-plastic plates would not form the observed forebulge. Forebulges are related to the way elastic rods and plates transfer bending moment throughout the medium.

The tendency to disregard elasticity may be related to the way we use the word ‘rigid’ in the context of plate tectonics. In physics, rigid means ‘not deforming’. For plate tectonics to work the plates do not have to be rigid. They may not flow apart on geological time scales (they must remain plates), but some relatively small reversible deformation is well compliant with the concept. So the main point is: Be the lithosphere a sandwich or a crème brûlée, it is also flexible, even on geological time scales.

Batman and gummy bears

If you are in numerical modelling then there is good news for you. In the past two decades, several Robins, including myself, have enhanced the Batman codes to account for elasticity. Maybe all you have to do is to switch on the right button. The implementation of visco-elasticity is usually based on a method developed by L. Moresi, who also, according to a previous Geodynamics 101 blog post, coined the hero terminology I just borrowed. Visco-elasticity works in quite a simple way, just like gummy bears. I am currently running an experiment with them. They quickly bend and squeeze when tortured, their resistance governed by their shear modulus, and recover when let go. At the same time they can flow, the resistance being controlled by their viscosity. Let’s put a book on top of one and see if we can find its nose after a week…

The elasticity button

Don’t be afraid to push the elasticity button, if your code has it. Usually it won’t do anything dramatic to your simulations, but exceptions exist [7]. In thermal convection models without plate tectonics there is not much feedback between the lid and the underlying mantle, and so only the build-up of dynamic topography is affected [8]. In subduction modeling your slabs may obtain different dipping angles. In continental extensions the total amount of extension will become more important than the divergence rate [9]. And in the simulation of continental shortening mentioned above [7], the elastic energy accumulated in the entire model gets partially released upon the onset of a shear zone. In such cases, i.e. when large scale elastic strains suddenly influence a much smaller region, one can expect some earthquakes to shake the conventional view of elasticity in geodynamical modeling. And if you still do not care about elasticity but yet you made it all the way here, then you deserve a bonus: the convergence of Stokes solvers is way better for visco-elastic rheologies than for the viscous ones – if you are numerically troubled by large viscosity contrasts of your model, elasticity is the way to go for you.

References
(1) D.L. Turcotte and G. Schubert (2002), Geodynamics
(2) A.B. Watts, S.J. Zhong, and J. Hunter (2013), The Behavior of the Lithosphere on Seismic to Geologic Timescales, doi: 10.1146/annurev-earth-042711-105457
(3) P.A. Allen and J.R. Allen (2005), Basin analysis : principles and applications
(4) E.B. Burov and A.B. Watts (2006), The long-term strength of continental lithosphere: 'jelly sandwich' or 'creme brulée'?, doi: 10.1130/1052-5173(2006)016
(5) E.B. Burov (2009), Time to burn out creme brulee?, doi: 10.1016/j.tecto.2009.06.013
(6) E.H. Hartz, Y.Y. Podladchikov (2008), Toasting the jelly sandwich: The effect of shear heating on lithospheric geotherms and strength, doi: 10.1130/G24424A.1
(7) Y. Jaquet, T. Duretz, and S.M. Schmalholz (2016), Dramatic effect of elasticity on thermal softening and strain localization during lithospheric shortening, doi: 10.1093/gji/ggv464
(8) V. Patocka, O. Cadek, P.J. Tackley, and H. Cizkova (2017), Stress memory effect in viscoelastic stagnant lid convection, doi: 10.1093/gji/ggx102
(9) J.A. Olive, M.D. Behn, E. Mittelstaedt, G. Ito, and B.Z. Klein, The role of elasticity in simulating long-term tectonic extension, doi: 10.1093/gji/ggw044

Planting seeds of deformation in numerical models

Planting seeds of deformation in numerical models

The Geodynamics 101 series serves to showcase the diversity of research topics and methods in the geodynamics community in an understandable manner. We welcome all researchers – PhD students to Professors – to introduce their area of expertise in a lighthearted, entertaining manner and touch upon some of the outstanding questions and problems related to their fields. This month we continue the conversation that was started at NetherMod 2017 by discussing how we initiate deformation in numerical models of the lithosphere and more importantly, does it matter how we start such models? Do you want to talk about your research? Contact us!

Geodynamic modelling often concerns itself with the study of localized deformation in the crust and lithosphere. The models generally start with an initial geometry, boundary conditions and a prescribed set of initial conditions that can be either thermal or mechanical. This initial setup is usually a standard representation of the lithosphere and asthenosphere, as inferred from geological and geophysical observations and laboratory rock experiments. The boundary conditions usually drive the deformation in the system. However, as a synthetic (computer) model is pristine at the start, the localization of deformation can take a long model time of up to millions of years, as numerical disturbances need to accumulate. Besides that, the deformation will likely localise near or at the boundary of the system because of the boundary conditions. In order to avoid this long starting phase, and to exert some control on the location of the initial deformation, several approaches are widely used to initiate and localize deformation. Examples of these are the S-point velocity discontinuity at the bottom of the system (e.g. Braun and Beaumont, 1995; Ellis et al., 1995; Willett, 1999; Beaumont et al., 2000; Buiter et al., 2006; Thieulot et al., 2008; Braun and Yamato, 2009) and the use of a weak seed. The latter are usually small zones that are weaker than the surrounding crust or lithosphere. As the lithosphere and crust are never homogeneous, using weak seeds in a model can be easily justified. They could represent regions of different material properties (e.g., heat production), inherited faults, inherited crustal thickness changes and/or plumes impacting the lithosphere. The popular NetherMod 2017 potatoes (yep, still referring back to that one. If you don’t know what this is about, check out this post) are an extreme case in which multiple weak seeds are used to reflect local geology. Traditionally, simpler, single weak seeds are used.

Here, I will give a brief (and by no means comprehensive) overview of the different weak seed methods used to initiate deformation in models of continental extension, and I will conclude with a discussion on how these different initial conditions affect the model evolution.

Modified figure from Burg & Podladchikov (1999): a thermal perturbation is used in the middle of their model to localise deformation there.

Seeding through thermal effects

The weak zone could be implemented as a temperature anomaly which could reflect a region in the crust of higher radiogenic heat production or a region in the lithosphere of locally reduced viscosity. They can be implemented by elevating the temperature or basal heat flux at the crust-lithosphere boundary or lithosphere-asthenosphere boundary with a certain amplitude over a finite region. Examples of studies using thermal weak seeds are Burg and Podladchikov (1999), Frederiksen and Braun (2001), Hansen and Nielsen (2003), and Brune and Autin (2013).

Seeding by mechanical inhomogeneity

Modified figure from Kaus (2009): Model setup with a rectangular, viscous inclusion in the middle of the domain to initiate and localise deformation. Dimensions of the weak zone were varied, but maintained the aspect ratio 2:1.

A weak seed could be composed of a material with a lower rheological strength than its surroundings. There are multiple ways of achieving this, but often used methods include:
• A weak seed with a lower viscosity (e.g., Gray and Pysklywec, 2010; Gray and Pysklywec, 2012b; Gray and Pysklywec, 2012a; Kaus, 2009, and Mishin, 2011)
• A weak seed with a different angle of internal friction (e.g., Pysklywec et al., 2002; Kaus and Podlachikov, 2006; Thieulot, 2011; Gray and Pysklywec, 2013; Chenin and Beaumont, 2013)
• A weak seed with a different density (e.g., Tirel et al., 2008)
• A weak seed consisting of a different material (e.g., Pysklywec et al., 2000; Huismans and Beaumont, 2007)
• A weak seed with more accumulated strain than its surroundings (e.g., Lavier et al., 2000; Huismans et al., 2005; Warren et al., 2008a; Warren et al., 2008b; Petrunin and Sobolev, 2008; Beaumont et al., 2009; Allken et al., 2011; Kneller et al., 2013; Allken et al., 2013)

Apart from these different ways of making a seed weak, you can also find weak seeds of many different shapes and sizes in the literature. Most commonly, you will find
• Square weak seeds (e.g., Gray and Pysklywec, 2013)
• Rectangular weak seeds (e.g., Huismans et al., 2005) with different aspect ratios
• Fault-shaped weak seeds (e.g., Currie et al., 2007, and Currie and Beaumont, 2011)

Modified from Gray & Pysklywec, 2013: Model setup with a square, frictionally weak zone of dimensions 10×10 km.

Seeding through geometrical discontinuity
Another method to create a zone of different rheological strength is by varying the thickness of the crust or lithosphere. A thickened lithosphere could represent a remnant of a previous mountain building phase, whereas a thinned lithosphere would represent a remnant of a previous rifting phase. Studies using this method of weakening include Gac et al. (2014) and Burg and Schmalholz (2008).

Modified from Gac et al. (2014): Model setup with an inherited thin crust.

Influence of weak seeds on the model evolution

As mentioned at NetherMod 2017, you would ideally either have a generic model for which you should determine the influence of different initial weak seeds to check how robust your model is, or you would have a region-specific model for which you find the optimal initial conditions to get your desired model output. Only few studies have investigated the former in detail. Dyksterhuis et al. (2013) found that a single weak seed typically produces symmetric narrow rifts; multiple seeds produce a wide rift; and an initial fault-shaped weak zone produces an asymmetric rift.

It also raises the question about whether or not there are differences between similar codes when the same initiation method is used.

To shine a very preliminary light on this problem, I ran some models of continental extension using the SULEC and ELEFANT codes with different initial conditions. As both codes are based on the same physics, and have similar implementation, they should show a high degree of similarity when using the same deformation initiation method. The results show that different initiation methods indeed result in different results, particularly with respect to the timing of the deformation (see figure below). Besides that, SULEC tends to show more asymmetric behaviour than ELEFANT. For a more complete overview of the results and the model setup, please look here (my old (first!) poster for GeoMod 2014).

Models of continental extension after 10 Myr of extension for SULEC (top) and ELEFANT (bottom) for different initial, frictionally weak, weak seeds with aspect ratios of 6×6 elements, 12×3 elements, and 3×12 elements (i.e., the weak seed consists of the same amount of elements in each model).

In conclusion, I hope to add to the discussion of ‘how we start our models’ with this post by affirming that the model evolution is affected by our choice of weak seed (if only by the amount of waiting until deformation starts) and its effect can differ slightly between codes, even if the codes are very similar. Taking into account the vast variability of methods to initiate deformation, one really should be careful when assessing model results.

References
Allken, V., Huismans, R., Fossen, H., and Thieulot, C. (2013). 3D numerical modelling of graben interaction and linkage: a case study of the Canyonlands grabens, Utah. Basin Research.

Allken, V., Huismans, R., and Thieulot, C. (2011). Three-dimensional numerical modeling of upper crustal extensional systems. Journal of Geophysical Research, page doi:10.1029/2011JB008319.

Beaumont, C., Jamieson, R., Butler, J., and Warren, C. (2009). Crustal structure: A key constraint on the mechanism of ultra-high-pressure rock exhumation. EPSL, 287:116-129.

Beaumont, C., Munoz, J., Hamilton, J., and Fullsack, P. (2000). Factors controlling the alpine evolution of the central pyrenees inferred from a comparison of observations and geodynamical models. Journal of Geophysical Research, 105:8121-8145.

Braun, J. and Beaumont, C. (1995). Three-dimensional numerical experiments of strain partitioning at oblique plate boundaries: Implications for contrasting tectonic styles in the southern Coast Ranges, California, and central South Island, New Zealand. Journal of Geophysical Research, 100(B9):18,059-18,074.

Braun, J. and Yamato, P. (2009). Structural evolution of a three-dimensional, finite-width crustal wedge.Tectonophysics, 484:181-192.

Brune, S. and Autin, J. (2013). The rift to break-up evolution of the Gulf of Aden: Insights from 3D numerical lithospheric-scale modelling. Tectonophysics, 607(0):65-79. The Gulf of Aden rifted margins system: Special Issue dedicated to the YOCMAL project (Young Conjugate Margins Laboratory in the Gulf of Aden).

Buiter, S., Babeyko, A., Ellis, S., Gerya, T., Kaus, B., Kellner, A., Schreurs, G., and Yamada, Y. (2006). The numerical sandbox: comparison of model results for a shortening and an extension experiment. Analogue and Numerical Modelling of Crustal-Scale Processes. Geological Society, London. Special Publications, 253:29-64.

Burg, J.-P. and Podladchikov, Y. (1999). Lithospheric scale folding: numerical modelling and application to the Himalayan syntaxes. International Journal of Earth Sciences, 88(2):190-200.

Burg, J.-P. and Schmalholz, S. (2008). Viscous heating allows thrusting to overcome crustal-scale buckling: Numerical investigation with application to the Himalayan syntaxes. Earth and Planetary Science Letters, 274(1):189-203.

Chenin, P. and Beaumont, C. (2013). Influence of offset weak zones on the development of rift basins: Activation and abandonment during continental extension and breakup. JGR, 118:1-23.

Currie, C. and Beaumont, C. (2011). Are diamond-bearing Cretaceous kimberlites related to low-angle subduction beneath western North America. EPSL, 303:59-70.

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How good were the old forecasts of sea level rise?

How good were the old forecasts of sea level rise?

Professor Clint Conrad

The Geodynamics 101 series serves to showcase the diversity of research topics and methods in the geodynamics community in an understandable manner. We welcome all researchers – PhD students to Professors – to introduce their area of expertise in a lighthearted, entertaining manner and touch upon some of the outstanding questions and problems related to their fields. Our latest entry for the series is by Clinton P. Conrad, Professor of Geodynamics at the Centre for Earth Evolution and Dynamics (CEED), University of Oslo. Clint’s post reflects on the predictions of sea level rise since the first Intergovernmental Panel on Climate Change (IPCC) report in 1990 and the near three decades of observations and IPCC projections that have been made since then. Do you want to talk about your research area? Contact us!

This past week I flew over the North Atlantic with a direct flight to California from Europe. From the plane we had a beautiful view of glaciers on the western edge of the Greenland ice sheet, where the ice seems to be disintegrating into the ocean. We’ve been hearing lately that the ice sheets are slowly disintegrating – is this what that looks like? Using my mobile phone’s camera, I took a photo of the glacier that happened to be visible from my seat and compared it to images of the same glacier saved in Google Earth (Figure 1). This is an interesting exercise if you like looking at glaciers, but I can’t tell about the overall dynamics of the ice sheet this way.

Figure 1. A glacier on the west coast of Greenland on September 2, 2017 (left) taken with my iPhone. From my plane’s in-flight entertainment system, it seems that this glacier is between the villages of Upernavik and Niaqornat. For comparison, the image on the right is a screenshot of the same glacier from Google Maps.

Actually, we’ve been worried about ice sheet melting – and the sea level rise with it – for decades. I re-realized this during this past summer, as I finally started unpacking the boxes that we shipped to Oslo one year ago from Hawaii. Some of these boxes probably didn’t need to be unpacked, like the one labeled “High School Junk”, but it turns out there is interesting stuff in there! Here was my diploma, a baseball glove, some varsity letters, and a pile of old schoolwork – most of which I have no recollection of creating. But I did remember one of the items – a report on global warming that I wrote for Social Science class in 1989. In particular, I remember being fascinated by the prediction that human activity would eventually cause enough sea level rise to flood land areas around the world. For years, I have been personally crediting that particular high school report as being my first real introduction to the geosciences – but until this past summer I had never revisited that report to see what I actually wrote at the time. Now here it is – twelve yellowed pages of dot-matrix type, with side perforations still remaining from the printer feed strips that I tore off 28 years ago.

My report is entitled “Global Warming – What Must Government Do?” and now I can see that it is mostly a rehashing of reporting from a bunch of newspaper articles written in 1989. It was a bit disappointing that my younger self wasn’t more creative or inspirational, but the content of the report – really the content of the newspaper articles from 1989 – is fascinating because much of the material could have been written today. There is discussion of how the warmest years in recorded history have happened only recently, that climate skeptics were unwilling to attribute recent changes to human activity, and that a few obstinate countries (then, it was Japan, the USSR, and the USA) were standing in the way of international agreements to curb CO2 emissions. Another statement is also familiar: that “oceans could rise from 1.5 to 6.5 feet”. For those of you not familiar with that measurement system, that is about 0.5 to 2.0 meters! I know that recent predictions are not quite as dire as 2 m of rise (at least in the 2100 timeframe), although sea level acceleration has been getting more attention lately. Did people in 1989 consider 2 m of sea level rise a possibility? I checked the cited New York Times article from 1989, and indeed it seems that I dutifully reported the estimate correctly. The article says that 1.5 to 6.5 feet of sea level rise is expected “to occur gradually over the next century affecting coastal areas where a billion people, a quarter of the world’s population, now live”.

Figure 2. Projections of sea level in 2100 (relative to 1990 sea level) for the five IPCC reports between 1990 and 2013, plotted as a function of IPCC report date. Shown are the minimum and maximum projections (range of red bars), and the mean of estimates (black circles).

I have contributed a little to sea level research in the intervening years, and am somewhat familiar with the current predictions. I know that the most recent (2013) report of the Intergovernmental Panel on Climate Change (IPCC) predicts up to about a meter of sea level rise by 2100, which was a large increase over the 2007 report that predicted up to about 0.6 meters. Thus, meter-scale sea level rise predictions seemed like a relatively recent development, and yet here was a prediction just as large from nearly 30 years ago. What did the IPCC have to say about sea level at the time?

I plotted the sea level projections of the five reports that the IPCC has released between 1990 and 2013 (Figure 2). Indeed, the 1990 report predicted slightly higher sea level for the year 2100 (31-110 cm higher) than did the most recent report from 2013 (28-98 cm higher). In fact, the IPCC projections for 2100 sea level declined from 1990 through 2007, until they increased again in the most recent report in 2013 (Figure 2). Why is this? Well, we have nearly 3 decades of observations that could help us to answer this question!

 

Figure 3. Sea level projection from the IPCC’s first assessment report (1990), showing that report’s low, best, and high estimates (blue lines) and predicted rates in mm/yr. Also shown is the University of Colorado sea level time series (red line), which is based on satellite altimetry observations from 1992-2016 and records a sea level rise rate of 3.4 ± 0.4 mm/yr.

First, let’s evaluate the initial predictions of the first IPCC report from 1990. Since 27 years have passed since the publication of that report, we can actually compare a sizeable fraction of those 1990 predictions to actual sea level observations. Left, I have plotted (Figure 3) the 1990 report’s sea level projection from 1990-2100 (Fig. 9.6 of that report) along with actual sea level observations made using satellite altimetry between 1992 and 2016, which have been nicely compiled by the University of Colorado’s Sea Level Research Group. The comparison shows (Figure 3) that the actual sea level change for the past 24 years has fallen slightly below the “best” estimate of the 1990 report, and well above the “low” estimate.

In retrospect, the 1990 predictions of future sea level change seem rather bold, because the 1990 IPCC report also concludes that “the average rate of rise over the last 100 years has been 1.0-2.0 mm/yr” and that “there is no firm evidence of accelerations in sea level rise during this century”. Yet, the 1990 report’s projection of 2.0-7.3 mm/yr of average sea level rise from 1990-2030 (Figure 2), represents a prediction that sea level rise would accelerate almost immediately – and this acceleration actually happened! Indeed, three recent studies (Hay et al., 2015; Dangendorf et al., 2017; Chen et al., 2017) have confirmed sea level acceleration after about 1990.

Thus, the IPCC’s 1990 sea level projection did a remarkably good job for the first three decades of its prediction timetable, and the next 8 decades don’t seem so unreasonable as a result. What did the 1990 report do right? Here the 1990 IPCC report helps us again, by breaking down its projection into contributions from four factors: thermal expansion of the seawater due to warming, the melting of mountain glaciers, and changes in the mass of the great ice sheets in Greenland and Antarctica. The 1990 report makes predictions for the changes in sea level caused by these factors for a 45-year timeframe of 1985-2030, and I have plotted these predictions as a rate (in mm/yr) in Figure 4. Thermal expansion and deglaciation in mountainous areas were predicted to be the largest contributors. Greenland was predicted to contribute only slightly, and Antarctica was predicted to gain ice, resulting in a drop in sea level.

Figure 4. Comparison of projections and observations of the various factors contributing to global mean sea level rise (GMSL, in mm/yr). Red bars show predictions that were made in 1990 (table 9.10 of the 1990 IPCC report) for the 45-year period 1985-2030 (range is given by red bars and best estimate is shown with a dark line). Blue bars show the actual contribution from each factor for the 17-year period 1993-2010, as detailed in table 13.1 of the 2013 IPCC report. Note both the sum of observed contributions and the direct observation of sea level change from satellite altimetry (bottom two blue bars) are consistent with recent analyses of tide gauge data (Hay et al., 2015; Dangendorf et al., 2017), within uncertainty.

Now 27 years later, we have actual observations of the world’s oceans, glaciers, and ice sheets that we can use to evaluate the predictions of 1990 report. Many of these observations are based on measurements made using satellites, which can now remotely measure ocean temperatures, changes in the mass of land ice (mountain glaciers and ice sheets) and even changes in groundwater volumes, over time. The IPCC report from 2013 (the most recent report) shows these contributions in the timeframe of 1993-2010, which are 17 years during the 45-year outlook predicted by the IPCC’s 1990 report. I have plotted these observations in Figure 4, and we can see how the 1990 predictions compare so far – remembering that the prediction and observation timescales do not exactly align.

First, we see that 1990 report overpredicted the contribution from thermal expansion, and slightly overpredicted the contribution from mountain glaciers. Of course, there is still time before 2030 for these factors to increase some more toward the predictions made in 1990. However, we also see that Greenland melting has already matched the 1990 report’s prediction for 2030, and that the prediction of a sea level drop from Antarctica did not materialize – Antarctica contributed almost as much sea level rise as Greenland did by 2010 (Figure 4). Furthermore, there is another significant contributor to sea level rise – land water, which represents the transfer of liquid water from the continents into the oceans. This occurs because groundwater that is mined for human activities eventually ends up in the ocean. According to the 2013 report, land water caused more sea level rise than ice sheet melting from Antarctica.

Thus, in 2010 the predicted rates of sea level rise from two factors (thermal expansion and mountain glaciers) had not yet reached the 2030 predictions of the 1990 report, but the contributions from Greenland, Antarctica, and land water loss have already nearly met or exceeded the predictions of 1990. Indeed, recent satellite observations between 2002 and 2014 show an acceleration of melting in Antarctica (Harig et al., 2015) and especially in Greenland (Harig et al., 2016). The recognition that Antarctica and Greenland may contribute significantly more to sea level rise in the future compared to earlier estimates is reflected in the 2013 IPCC report (Figure 2).

Figure 5. A dike near the town of Putten in the Netherlands, where the recent EGU-sponsored “Nethermod” meeting was held in late August 2017. This dike is one of many in the Netherlands that protect negative-elevation land (left) from a higher water level (right).

So far, it seems that the IPCC’s 1990 sea level projection has stood the test of 27 years remarkably well (Figure 3). It is rather disheartening to realize that we are on track for the ~60 cm of sea level rise that the 1990 report predicted for the year 2100, or more if the early underestimates of ice sheet contributions prove to be more significant than any overestimates of thermal expansion (Figure 4). Looking at my own high school report from the same time, it is also disappointing that to realize that the warmest years in recorded history have again happened only recently, that climate change skeptics are still unwilling to attribute recent changes to human activity, and that there are still obstinate countries (well, one country) standing in the way of international agreements to curb CO2 emissions. On the other hand, high school students writing reports on this topic today will likely find discussions of dropping beachfront real estate prices, governmental planning for future sea level rise, and engineering techniques for managing future sea level rise (Figure 5). I hope that these students save copies of their reports in a format that they can examine decades later, because it is interesting to consider how predictions of future sea level rise have changed over time, and how society has been responding to the challenges of this geodynamic phenomenon that is operating on the timescale of a human lifetime. One day in the 2040s these students may want to scrutinize another quarter century of data against the projections of the next IPCC report, to be completed by 2022. I wonder what they will find?