GD
Geodynamics

# numerical modelling

## Rheological Laws: Atoms on the Move

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. For our first ‘101’ for 2018, we have an entry by postdoctoral researcher Elvira Mulyukova from Yale University about rheology and deformation occurring on atomic scales … it’s a fun and informative read indeed! Do you want to talk about your research? Contact us!

Elvira Mulyukova, Yale University

Most of us have an intuitive understanding that different materials resist being moved, or deformed, to different degrees. Splashing around in the mud is more energy-consuming (and fun, but never mind that) than in the water, and splashing around in the block of concrete is energy-intensive bordering on deadly. What are the physical reasons for these differences?

For Earth materials (rocks), the answer lies in the restless nature of their atoms: the little buggers constantly try to sneak out of their crystal lattice sites and relocate. Some are more successful at it than others, making those materials more easily deformable. A lattice site is really just other atoms surrounding the one that is trying to escape. You see, atoms are like a bunch of introverts: each is trying to escape from its neighbours, but doesn’t want to get near them. The ones that do escape have to overcome a temporary discomfort (or an increase in their potential energy, for those physically inclined) of getting close to their neighbours. This requires energy. The closer the neighbours – the more energy it takes to get past them. When you exert force on a material, you force some of the neighbours to be further away from our potential atomic fugitive, making it more likely for the atom to sneak in the direction of those neighbours. The fun part (well, fun for nerds like me) is that it doesn’t happen to just one atom, but to a whole bunch of them, wherever the stress field induced by the applied force is felt. A bunch of atoms escaping in some preferential direction is what we observe as material deformation. The more energy you need to supply to induce the mass migration of atoms – the stronger the material. But it’s really a question of how much energy the atom has to begin with, and how much energy is overall needed to barge through its detested neighbuors. For example, when you crank up the temperature, atoms wiggle more energetically and don’t need as much energy supplied from external forcing in order to escape – thus the material gets weaker.

“A lattice site is really just other atoms surrounding the one that is trying to escape. You see, atoms are like a bunch of introverts: each is trying to escape from its neighbours, but doesn’t want to get near them.” cartoon by Elvira Mulyukova

One more thing. Where are the atoms escaping to? Well, there happen to be sanctuaries within the crystal lattice – namely, crystalline defects such as vacancies (aka point defects, where an atom is missing from the otherwise ordered lattice), dislocations (where a whole row of atoms is missing), grain boundaries (where one crystal lattice borders another, which is tilted relative to it) and other crystalline imperfections. These regions are sanctuaries because the lattice is more disordered there, which allows for larger distances in-between the neighbors. When occupying a regular lattice site – the atom is sort of trapped by the crystalline order. Think of the lattice as an oppressive regime, and the crystalline defects as liberal countries that are welcoming refugees. I don’t know, is this not the place for political metaphors? *…whistling and looking away…*

Ok, enough anthropomorphisms, let’s get to the physics. If this is the last sentence you’ll read in this blog entry, let it be this: rocks are made up of atoms that are arranged into crystal lattices (i.e., ordered rows and columns of atoms), which are further organized into crystal grains (adjacent crystals tilted relative to each other); applying force to a material encourages atoms to move in a preferential direction of the largest atomic spacing, as determined by the direction of the applied force; the ability of the lattice sites to keep their atoms in place (call it a potential energy barrier) determines how easily a material deforms. Ok, so it was more like three sentences, but now you know why we need to get to the atomic intricacies of the matter to understand materials macroscopic behaviour.

Alright, so we’re applying a force (or stress, which is simply force per area) to a material and watch it deform (a zen-inducing activity in its own right). We say that a material behaves like a fluid when its response to the applied stress (and not just any stress, but differential stress) is to acquire a strain rate (i.e., to progressively shorten or elongate in one direction or the other at some rate). On geological time scales, rocks behave like fluids, and their continuous deformation (mass migration of atoms within a crystal lattice) under stress is called creep.

The resistance to deformation is termed viscosity (let’s denote it µ), which basically tells you how much strain rate (˙e) you get for a given applied differential stress (τ). Buckle up, here comes the math. For a given dimension (say x, and for the record – I’ll only be dealing with one dimension here to keep the math symbols simple, but bear in mind that µ, ˙e and τ are all tensors, so you’d normally either have a separate set of equations for each dimension, or some cleverly indexed symbols in a single set of equations), you have:

So if I’m holding a chunk of peridotite with a viscosity of 1020 Pa s (that’s units of Pascal-seconds, and that’s a typical upper mantle viscosity) and squeezing it in horizontal direction with a stress of 108 Pa (typical tectonic stress), it’ll shorten at a rate of 5 · 10-13 s-1 (typical tectonic rates). A lower viscosity would give me a higher strain rate, or, equivalently, with a lower viscosity I could obtain the same strain rate by applying a smaller stress. If at this point you’re not thinking “oh, cool, so what determines the viscosity then?,” I failed massively at motivating the subject of this blog entry. So I’m just gonna go ahead and assume that you are thinking that. Right, so what controls the viscosity? We already mentioned temperature (let’s call it T), and this one is a beast of an effect. Viscosity depends on temperature exponentially, which is another way of saying that viscosity depends on temperature hellavulot. To throw more math at you, here is what this dependence looks like:

where R = 8.3144598 J/K/mol (that’s Joule per Kelvin per mol) is the gas constant and E is the activation energy. Activation energy is the amount of energy that an atom needs to have in order to even start thinking about escaping from its lattice site, which of course depends on the potential energy barrier set up by its neighbours. Let’s say your activation energy is E = 530·103 J mol-1 . If I raised your temperature from 900 to 1000 K (that’s Kelvin, and those are typical mid-lithospheric temperatures), your viscosity would drop by a factor of ∼ 1000. That’s a three orders of magnitude drop.

Like I said, helluvalot. If instead you had a lower activation energy, say E = 300 · 103 J mol-1 , the same temperature experiment would bring your viscosity down by a factor of ∼ 50, which is less dramatic, but still significant. It’s like running through peanut butter versus running through chocolate syrup (running through peanut butter is a little harder… I clearly need to work on my intuition-enhancing examples). Notice, however, that while the temperature dependence is stronger for materials with higher activation energies, it is more energy-consuming to get the creep going in those materials in the first place, since atoms have to overcome higher energy barriers. There’s more to the story. Viscosity also depends on pressure (call it P), which has a say in both the energy barrier the atoms have to overcome in order to escape their neighbours, as well as how many lattice defects (called sanctuaries earlier) the atoms have available to escape to. The higher the pressure, the higher the energy barrier and the fewer lattice sanctuaries to resort to, thus the higher the viscosity. Throwing in the pressure effect, viscosity goes as:

The exact dependence of viscosity on pressure is determined by V – the activation volume.

Alright, we’re finally getting to my favourite part – the atoms’ choice of sanctuary sites. If the atomic mass migration happens mainly via point defects, i.e., by atoms hopping from one single lattice vacancy to another, the deformation regime is called diffusion creep. As atoms hop away, vacancies accumulate in regions of compressive stress, and fewer vacancies remain in regions of tensional stress. Such redistribution of vacancies can come about by atoms migrating through the bulk of a crystal (i.e., the interior of a grain, which is really just a crystal that is tilted relative to its surrounding crystals), or atoms migrating along the boundary of a crystal (i.e., a grain boundary). In both cases, the rate at which atoms and vacancies get redistributed depends on grain size (let’s denote it r). The larger the grains – the more distance an atom has to cover to get from the part of the grain that is being compressed to the part that is under tension. More math is due. Here is what the viscosity of a material deforming by diffusion creep looks like:

Exponent m depends on whether the atoms are barging through the bulk of a grain (m = 2), or along the grain boundaries (m = 3). What’s that new symbol B in the denominator, you ask? That’s creep compliance (in this case – diffusion creep compliance), and you two have already met, sort of. Creep compliance specifies how a given creep mechanism depends on pressure and temperature:

For diffusion creep of upper mantle rocks, I typically use m = 3, B0 ∼ 13 µmm MPa-1 s-1 (which is just a material-specific prefactor), Ediff = 300 · 103 J mol-1 and Vdiff = 5 cm3 mol-1 from Karato and Wu (1993). Sometimes I go bananas and set Vdiff = 0 cm3 mol-1 , blatantly ignoring pressure dependence of viscosity, which is ok as long as I’m looking at relatively modest depth-ranges, like a few tens of kilometers.

At sufficiently high stresses, a whole row of atoms can become mobilized and move through the crystal, instead of the meagre one-by-one atomic hopping between the vacancies. This mode of deformation is called dislocation creep. Dislocations are really just a larger scale glitch in the structure of atoms (compared to vacancies). They are linear lattice defects, where a whole row of atoms can be out of order, displaced or missing. It requires more energy to displace a dislocation, because you are displacing more than one atom, but once it’s on the move, it accommodates strain much more efficiently than in the each-atom-for-itself diffusion creep regime. As the material creeps, dislocations get born (nucleated), get displaced and get dead (annihilated). Dislocations don’t care about grain size. What they do care about is stress. Stress determines the rate at which dislocations appear, move and disappear. I know you saw it coming, more math, here is the dislocation creep viscosity:

Exponent n dictates the stress dependence of viscosity. Stress dependence of dislocation creep viscosity is a real pain, making the whole thing nonlinear and difficult to use in a geodynamical model. Not impossible, but rage-inducingly difficult. Say you’re trying to increase the strain rate by some amount, so you increase the stress, but then the viscosity drops, and suddenly you have a monster of a strain rate you never asked for. Ok, maybe it’s not quite this bad, but it’s not as good as if the viscosity just stayed constant. You wouldn’t be able to have strain localization, form tectonic plate boundaries and develop life on Earth then, but maaaan would you be cracking geodynamic problems like they were peanuts! I’m derailing. Just like all the other creeps, dislocation creep has its own compliance, A, that governs its dependence on pressure and temperature:

For dislocation creep of upper mantle rocks, I typically use n = 3, A0 = 1.1 · 105 MPa-n s -1 (which is just a material-specific prefactor), Edisl = 530 · 103 J mol -1 and Vdiff = 20 cm3 mol-1 , similar to Karato and Wu (1993). Just like for diffusion creep, I sometimes just set Vdisl = 0 cm3 mol-1 to keep things simple.

We’re almost done. Allow me one last remark. A rock has an insane amount of atoms, crystal grains and defects, all subject to local and far-field conditions (stress, temperature, pressure, deformation history, etc). A typical rock is therefore heterogeneous on the atomic (nano), granular (micro) and outcrop (meter) scales. Thus, within one and the same rock, deformation will likely be accommodated by more than just one mechanism. With that in mind, and sticking to just two deformation mechanisms described above, we can mix it all together to get:

This is known as composite rheology, where we assumed that the strain rates accommodated by diffusion and dislocation creep can be simply summed up, like so:

Alright. If you got down to here, I salute you! Next time you’re squeezing a peridotite, or splashing in the mud, or running through peanut butter – give a shout out to those little atoms that enable you to do such madness. And if you want to get to the physics of it all, you can find some good introductory texts in Karato (2008); Turcotte and Schubert (2002).

```References

S. Karato. Deformation of earth materials: an introduction to the rheology of solid earth. Cambridge Univ Pr, 2008.

S. Karato and P. Wu. Rheology of the upper mantle: a synthesis. Science, 260(5109):771–778, 1993.

D.L. Turcotte and G. Schubert. Geodynamics. Cambridge Univ Pr, 2002.```

## Remarkable Regions – The India-Asia collision zone

Every 8 weeks we turn our attention to a Remarkable Region that deserves a spot in the scientific limelight. This week we zoom in on a particular part of the eastern Tethys as Adina Pusok discusses the India-Asia collision zone. She is a postdoctoral researcher at the Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, UCSD, US.

Without doubt, one of the most striking features of plate tectonics and lithospheric deformation on Earth is the India-Asia collision zone, largely comprised of the Himalayan and Karakoram mountain belts and the Tibetan plateau. What makes this collision zone so remarkable? For one, Tibet is the largest, highest and flattest plateau on Earth with an average elevation exceeding 5 km, and it includes over 80% of the world’s land surface higher than 4 km. Then, the bordering Himalayas and the Karakoram Mountains include the only peaks on Earth reaching more than 8 km above sea level.

It makes one wonder, how can such a mountain belt and high plateau form? Most of the major mountain belts and orogenic plateaus on Earth are found within the overlying plate of subduction and/or collision zones (e.g. the Alps, the Andes, the American Cordilleras etc.). When an ocean closes and two continental plates meet at a destructive (subduction) boundary, the continents themselves collide. Such collisions result in intense deformation at the edges of the colliding plates. Neither continent can be subducted into the mantle due to the buoyancy of continental crust, so the forces that drive the plate movement prior to collision are brought to act directly on the continental lithosphere itself. At this stage, further convergence of the plates must be taken up by deforming one or both of the plates of continental lithosphere over hundreds of kilometres [Figure 1]. Mountain belts can form under these circumstances.

Figure 1 Global map of surface velocities and the second invariant of strain rate (from Moresi [2015]). The surface velocities show the location and extent of plates, and the strain rate map highlights the fact that most of the deformation is concentrated at plate boundaries (high strain rates), while the continental interiors have little or no deformation (low strain rates). In some places, deformation occurs over broader regions, especially following mountain belts. These boundaries are called diffuse plate boundaries. The white rectangle roughly indicates the extent of the India-Asia collision zone.

The Himalayas and the Tibetan plateau are no different. Following the closure of the Tethys ocean (see earlier blog post), the Indian continent collided with Eurasia around 50 million years ago (e.g., Patriat and Achache [1984]), thus giving rise to this anomalously high region. This tectonic boundary is complex and changes character along its length. The Tibetan plateau is a collage of continental blocks (terranes) that were added successively to the Eurasian plate during the Paleozoic and Mesozoic [Figure 2]. The boundaries between these terranes are marked by scattered occurrences of ophiolitic material, which are rocks characteristic of oceanic lithosphere. The Himalayas represent the traditional accretionary wedge formed by folding and thrusting of sediments scraped off the subducting slab.

Figure 2 Simplified tectonic map of Tibet and surrounding region showing approximate boundaries of the major terranes, suture zones, and strike-slip faults (from Ninomiya and Bihong [2016]). Blocks and terranes: ALT-EKL-QL: Altyn Tagh–East Kunlun–Qilian terrane; BS: Baoshan terrane; HM: Himalayan terrane; IC: Indo-China block; KA: Kohistan Arc terrane; LA: Ladakh arc terrane; LC: Lincang–Sukhothai–Chanthaburi Arc terrane; LST: Lhasa terrane; NCB: North China Block; NQT: North Qiangtang terrane; QT: Qiangtang terrane; SP: South Pamir terrane; SPGZ: Songpan-Ganze terrane; SQT: South Qiangtang terrane; TC: Tengchong terrane; TSH: Tianshuihai terrane; WB: West Burma terrane; WKL: West Kunlun terrane. Suture zones: BNS: Bangong-Nujiang Suture; EKLS: East Kunlun Suture; ITS: Indus-Tsangbo Suture; JSS: Jinsha Suture; LSS: Longmu Tso–Shuanghu–Menglian–Inthanon Suture; WKLS: West Kunlun Suture. Basins: QB: Qaidam Basin; KB: Kumkol Basin. Faults: ALT: Altyn Tagh Thrust; ALTF: Altyn Tagh Fault; KKF: Karakorum Fault; LMST: Longmen Shan Thrust; MFT: Main Frontal Thrust; NQLT: North Qilian Thrust; RRF: Red River Fault; SGF: Sagaing Fault; XXF: Xianshui River–Xiaojiang Fault.

Interestingly, the India-Asia collision orogen is not just the youngest and most spectacular active continent collision belt, it is also the most studied research area on Earth. Studies on this region span a wide range of topics and methods for over more than 100 years. I am not sure if it is the fascination with the highest mountain on Earth (Mt. Everest was actually climbed for the first time as late as 1953 by Tenzing Norgay and Edmund Hillary), similar to our fascination for exploring the Moon, Mars and the other planets in our Solar System nowadays, or the hope that studying the youngest orogeny will help us decipher the older ones (soon to realize different mountain belts evolve differently).

To understand the magnitude of the work done in the past 100 years, a simple search of the keywords “India Asia collision” on Google Scholar yielded ~90k results, and a more focused geosciences search on Web of Science (where I filtered the results to those from geophysics, geochemistry, geology, geosciences multidisciplinary only) yielded >1600 results for the same keywords (other keywords: “Himalaya” > 5600 results, “Tibet” > 6500 results, “India Asia” > 2200 results). These numbers can be intimidating to a new student taking on the topic, but it is a topic worth studying and I’ll explain why below.

From a general perspective, it is important to study the India-Asia collision zone due to the interaction between tectonics and climate and the formation of the Indian monsoon [Molnar et al., 1993], but also because it is a highly populated area (>200 million people in the Hindu Kush Himalaya region) regularly shaken by natural phenomena, such as earthquakes, floods or landslides. For example, the last large earthquake in Nepal, the Gorkha earthquake (Mw 7.8) in April 2015 caused more than 9000 deaths.

From a geophysics point of view, understanding mountain-building processes and the driving forces of plate tectonics has been one of the long-term goals of solid Earth sciences community. The India-Asia collision zone is one of the best examples in which subduction, continental collision and mountain building can be studied in a global plate tectonics perspective. Prior to plate tectonics theory, Argand [1924] and Holmes [1965] thought that the Himalayan Mountains and Tibetan Plateau had been raised due to the northern edge of the Indian craton underthrusting the entire region, causing shortening and thickening of the crust to ∼80 km. This perspective remains widely accepted, but recent ideas suggest that other processes are equally important (more below).

Today, the challenge lies in refining our understanding of the dynamics of India-Asia collision by elucidating the connections between the wealth of observations available and the underlying processes occurring at depth. Decades of study have produced data sets across various disciplines, including: active tectonics, Cenozoic geology, seismicity, global positioning system (GPS) measurements, seismic profiles, tomography, gravity anomalies, mantle-crustal anisotropy, paleomagnetism, geochemistry or magnetotelluric studies. Of these, the GPS data stands out as it clearly shows the distributed deformation across the entire collision zone and suggests that this is a highly dynamic area [Figure 3].

Figure 3 Horizontal GPS velocities of crustal motion around the Tibetan Plateau relative to stable Eurasia from Liang et al. [2013].

Collectively, all these observation data sets stand as a different piece in the puzzle of the India-Asia collision. However, the same data sets can support a number of competing and sometimes mutually exclusive mechanisms for the uplift of the Tibetan Plateau. For example, the mantle lithosphere beneath Tibet has been proposed to be cold, hot, thickened by shortening, or thinned by viscous instability. Other controversies include the degree of mechanical coupling between the crust and deeper lithosphere and the nature of large-scale deformation. It is no surprise then, that several hypotheses emerged over time trying to explain the anomalous rise of the Himalayas and Tibetan Plateau [Figure 4]:

1. Figure 4 Schematic cartoons of tectonic models proposed to explain the thickening and uplift of the Himalayas and the Tibetan Plateau. (Source: personal institutional web page of A. Ozacar).

Wholescale underthrusting of the Indian plate below the Asian continent [e.g. Argand, 1924].

2. The thin-sheet model or distributed homogeneous shortening [e.g. England and McKenzie, 1982].
3. Homogeneous thickening of a weak, hot Asian crust, involving a large amount of magmatism [e.g. Dewey and Burke, 1973].
4. Slip-line field model to account for the brittle deformation in and around the Tibetan Plateau and to explain extrusion of SE Tibet away from Indian continent [e.g. Molnar and Tapponnier, 1975]. The same group also proposes a time-dependent model for the growth of Tibetan plateau [e.g. Tapponnier et al., 2001], in which successive intracontinental subduction zones maintain the stepwise growth and rise of the plateau.
5. Lower crustal flow models for the exhumation of the Himalayan units and lateral spreading of the Tibetan plateau [e.g. Royden et al., 1997, Beaumont et al., 2001].
6. Delamination or convective removal of the lithospheric mantle that induced isostatic movement, lifting the Tibetan Plateau [e.g. Molnar, 1988].

These models were applied either to the Tibetan Plateau or the Himalayan mountain belt and were able to explain the formation of specific tectonic and geological features. However, there is no conclusive answer on which of the hypotheses works best for the entire orogen, and instead, more questions arise:

• Which forces are at work during continental collision and mountain building?
• What is the deformation history and evolution of this plate boundary?
• How was the subduction accommodated in the Neo-Tethys?
• How does subduction evolve during continental collision?
• What drives the present-day fast convergence (~4-5 cm/yr) between India and Eurasia?
• Which forces propagated India northwards between 70-50 million years at anomalously high speeds (up to 16 – 20cm/yr)?
• How can you form such large elevations over such extended areas?
• What is the effect of surface processes on uplift?
• What is the structure at depth beneath the Himalayas and Tibetan plateau?
• How do the Indian and Eurasia plates deform during collision?
• How is the deformation accommodated during continental collision?
• How do mountain belts form and why not all mountain belts look the same?
• How did the crust beneath Himalaya and Tibet reach double-crustal thickness (normal continental crust is 35-40 km thick, whereas the crust beneath the Himalaya and Tibet is 70-100 km thick)?
• Which mechanisms help sustain the high topographic amplitudes?
• Why should an area as broad as the Tibetan Plateau be uplifted so high compared to other mountain belts following collision?
• Did the Tibetan Plateau and Himalayan mountain belt rise continuously or diachronously?
• Which the proposed models [Figure 4] can be applied, and where?
• How do lithospheric heterogeneities and rheology affect the deformation pattern?
• What is the degree of mechanical coupling between the crust and deeper lithosphere? Is it the “jelly sandwich” model (e.g., Burov and Watts [2006]) or the “creme-brulee” model (e.g., Jackson [2002], see earlier blog post)?
• Why do the Himalayas have a convex curvature?
• What about the high deformation of the prominent Himalayan syntaxes (the inflection points of the Himalayan belt): Nanga Parbat in the west and Namche Barwa in the east?
• What is the effect of the India-Asia collision on climate? Do the Himalayas affect the Indian monsoon or is it the other way around? A chicken-and-egg question.

Seriously, can I even stop asking questions? The question that fascinated me the most during my graduate studies was “Why is the Himalayan-Tibet region so high and broad compared to other mountain belts?”. If we tune our models to Earth parameters, can we build such large elevations in computer simulations? Which factors and forces are at play? Using 3-D numerical models to address this question [Pusok and Kaus, 2015], we were able to obtain distinct topographic modes (different types of mountain belts) [Figure 5] and to show that building topography is an interplay between providing the energy to the system and the ability of that system to store it over longer periods of time. We also suggest that the reason why Himalaya-Tibet is different from the Alps, for example, is because the shape and elevation of mountain ranges can vary depending on the boundary conditions (plate driving forces that control convergence velocity and lithospheric heterogeneities such as the Tarim Basin) and internal factors (rheology), but also on the evolution stage they are in.

To sum up, it is clear that many of the above questions remain unanswered. But I think this is good news, meaning that in the future, exciting new results will shape our understanding of this remarkable region.

Figure 5 3-D Simulation results showing different modes of surface expressions in continental collision models. Modified from Pusok and Kaus [2015].

```References:
Argand, E. (1924). La tectonique de l’Asie. Proc. 13th Int. Geol. Cong., 7:171–372.

Beaumont, C., Jamieson, R. A., Nguyen, M. H., and Lee, B. (2001). Himalayan Tectonics Explained by Extrusion of a Low-Viscosity Crustal Channel Coupled to Focused Surface Denudation. Nature, 414:738–742.

Burov, E. B. and Watts, A. B. (2006). The long-term strength of continental lithosphere: “jelly sandwich” or “crème brûlée”? GSA Today, 16(1):4.

Dewey, J. F. and Burke, K. (1973). Tibetan, Variscan, and Precambrian Basement Reactivation: Products of Continental Collision. The Journal of Geology, 81(6):683–692.

England, P. and McKenzie, D. (1982). A Thin Viscous Sheet Model for Continental Deformation. Geophys. J. R. astr. Soc., 70:295–321.

Holmes, A. (1965). Principles of Physical Geology. The Ronald Press Company, New York, second edition.

Jackson, J. (2002). Strength of the Continental Lithosphere: Time to Abandon the Jelly Sandwich? GSA Today, 4–9.

Liang, S., Gan, W., Shen, C., Xiao, G., Liu, J., Chen, W., Ding, X., and Zhou, D. (2013). Three-dimensional velocity field of present-day crustal motion of the Tibetan Plateau derived from GPS measurements. Journal of Geophysical Research: Solid Earth, 118:1–11.

Molnar, P. and Tapponnier, P. (1975). Cenozoic Tectonics of Asia: Effects of a Continental Collision. Science, 189:419–426.

Molnar, P. (1988). A Review of Geophysical Constraints on the Deep Structure of the Tibetan Plateau, the Himalaya and the Karakoram, and their Tectonic Implications. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 326(1589):33–88.

Molnar, P., England, P., and Martinod, J. (1993). Mantle Dynamics, Uplift of the Tibetan Plateau, and the Indian Monsoon. Reviews of Geophysics, 31:357–396.

Moresi, L. (2015). Computational Plate Tectonics and the Geological Record in the Continents. SIAM News, 48:1–6.

Ninomiya, Y. and Bihong Fu, B. (2016). Regional Lithological Mapping Using ASTER-TIR Data: Case Study for the Tibetan Plateau and the Surrounding Area. Geosciences 2016, 6(3), 39; doi:10.3390/geosciences6030039.

Patriat, P. and Achache, J. (1984). India-Eurasia collision chronology has implications for crustal shortening and driving mechanism of plates. Nature, 311:615–621.

Pusok, A. E. and Kaus, B. J. P. (2015). Development of topography in 3-D continental-collision models. Geochemistry, Geophysics, Geosystems, 16(5):1378–1400.

Royden, L. H., Burch el, B. C., King, R., Wang, E., Chen, Z., Shen, F., and Liu, Y. (1997). Surface Deformation and Lower Crustal Flow in Eastern Tibet. Science, 276(5313):788–790.

Tapponnier, P., Zhiqin, X., Roger, F., Meyer, B., Arnaud, N., Wittlinger, G., and Jingsui, Y. (2001). Oblique Stepwise Rise and Growth of the Tibet Plateau. Science, 294(5547):1671–1677.```

## 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).```

## Alaska: a gold rush of along strike variations

Every 8 weeks we turn our attention to a Remarkable Region that deserves a spot in the scientific limelight. After exploring the Mediterranean and the ancient Tethys realm, we now move further north and across the Pacific to the Aleutian-Alaska subduction zone. This post was contributed by Kirstie Haynie who is a PhD candidate at the department of geology at the University at Buffalo, State University of New York, in the United States of America.

The Aleutian-Alaska subducton zone marks a North American-Pacific plate boundary where subduction varies greatly along strike (Figure 1). At the western end of the subduction zone, the Aleutian volcanic islands are the result of oceanic-oceanic subduction while in the eastern part of the subduction zone there is oceanic-continental collision where the Pacific plate descends beneath the North American plate. The age of the subducting sea floor increases laterally from around 30 Ma in the eastern subduction corner to 80 Ma at the end of the Aleutian volcanic arc (Müller et al., 2008). Slab dip changes drastically from 50° to 60° in the west and central Aleutians to flat slab subduction under south-central Alaska (Ratchkovski and Hansen, 2002a; Lallemand et al., 2005; Jadamec and Billen, 2010). This leads to a variation in the slab pull force, which is a main driving force of subduction caused by the weight of dense slabs sinking into the mantle (Morra et al., 2006).

Figure 1: Tectonic map of Alaska modified from Haynie and Jadamec (2017). Topography/bathymetry is from Smith and Sandwell (1997) and Seafloor (SF) ages are from Müller et al. (2008). Blue lines are the slab contours of Jadamec and Billen (2010) in 40 km intervals; the thick black line is the plate boundary from Bird (2003); and the thinner black lines are faults from Plafker et al. (1994a). The location of Denali is marked by the orange hexagon. Holocene volcanoes are given by the pink triangles (Alaska Volcano Observatory). The purple polygon is the outline of the Yakutat oceanic plateau (Haynie and Jadamec, 2017). WB – Wrangell block fore-arc sliver; JdFR – Juan de Fuca Ridge.

There is also a distinct change in margin curvature from convex in the west to concave in the east. At the end of the eastern bend, the Alaska part of the subduction zone is truncated by a large transform boundary, the Fairweather-Queen Charolette fault, which gives rise to a corner-shaped subduction-transform plate boundary (Jadamec et al., 2013; Haynie and Jadamec, 2017). Here, convergence is oblique with an average velocity of 5.2 cm/year northwest (DeMets and Dixon, 1999). Seismic studies (Page et al., 1989; Ferris et al., 2003; Eberhart-Phillips et al., 2006; Fuis et al., 2008) show that thicker than normal oceanic crust lies off-shore in the subduction corner. This thick oceanic material has been identified as the Yakutat oceanic plateau (Plafker et al., 1994a; Brocher et al., 1994; Bruns, 1983; Worthington et al., 2008; Christeson et al., 2010; Worthington et al., 2012). Even though oceanic plateaus tend to resist subduction (Cloos, 1993; Kerr , 2003), the Yakutat plateau is currently subducting beneath the Central Alaska Range to depths of 150 km (Ferris et al., 2003; Eberhart-Phillips et al., 2006; Wang and Tape, 2014). It is also colliding into south-east Alaska (Mazzotti and Hyndman, 2002; Elliott et al., 2013; Marechal et al., 2015) where the largest coastal mountain range on Earth, the Saint Elias Mountains, are located (Enkelmann et al., 2015).

With regards to surface deformation, in addition to Denali (the tallest mountain in North America), other notable along strike variations reside within the broad deformation zone of south-central Alaska. For example, a normal volcanic arc occurs over the Aleutian part of the subduction zone and above the Alaska Peninsula. However, above the flat slab there is a gap in volcanism followed by the presence of the enigmatic Wrangell volcanoes (Rondenay et al., 2010; Jadamec and Billen, 2012; Martin-Short et al., 2016; Chuang et al., 2017). These volcanoes are marked by a range of morphologies as well as adakitic geochemical signatures (Richter et al., 1990; Preece and Hart , 2004), which have a petrogenesis that may be attributed to slab melting (Defant and Drummond , 1990; Peacock et al., 1994; Castillo, 2006, 2012; Ribeiro et al., 2016). Analogue (Schellart , 2004; Strak and Schellart , 2014) and 3D numerical models (Stegman et al., 2006; Piromallo et al., 2006; Jadamec and Billen, 2010, 2012) predict that toroidal flow can produce upwellings around the edge of a slab that may have implications for melting of the slab and the formation of adakites. However, the formation of the Wrangell volcanoes is still debated.

Figure 2: Top: map of south-central Alaska (zoomed in from Figure 1) with model predicted velocities (blue arrows) from Haynie and Jadamec (2017) plotted on top. Bottom: percent of slab contribution from Haynie and Jadamec (2017) models to observed Denali fault slip rates (modified from Haynie and Jadamec (2017)). Results from Haynie and Jadamec (2017) show that the slab drives northwest and counter-clockwise motion of the Wrangell block fore-arc sliver and contributes to an average of 20-28% of motion along the Denali fault. The flat slab exerts the largest contribution to motion along the eastern segment of the fault, where surface motion parallels the fault, and also along the central segment of the fault, where the slab is driving the Wrangell block into the North American backstop and subducting obliquely to the fault.

```References
Bemis, S. P., and W. K. Wallace (2007), Neotectonic framework of the north-central Alaska Range foothills, Geological Society of America Special Papers, 431, 549–572.
Bemis, S. P., R. J. Weldon, and G. A. Carver (2015), Slip partitioning along a continuously curved fault: Quaternary geologic controls on Denali fault system slip partitioning, growth of the Alaska Range, and the tectonics of south-central Alaska, Lithosphere, 7 (3), 235–246.
Bird, P. (1988), Formation of the Rocky Mountains, Western United States: A continuum computer model, Science, 239 (4847), 1501–1507.
Bird, P. (2003), An updated digital model of plate boundaries, Geochemistry, Geophysics, Geosystems, 4 (3).
Brocher, T. M., G. S. Fuis, M. A. Fisher, G. Plafker, Moses, M. J., J. J. Taber, and N. I. Christensen (1994), Mapping the megathrust beneath the northern gulf of alaska using wideangle seismic data, Journal of Geophysical Research: Solid Earth, 99 (B6), 11,663– 11,985.
Bruns, T. R. (1983), Model for the origin of the Yakutat block, an accreting terrane in the northern Gulf of Alaska, Geology, 11 (12), 718–721.
Castillo, P. R. (2006), An overview of adakite petrogenesis, Chinese Science Bulletin, 51 (3), 257–268.
Castillo, P. R. (2012), Adakite petrogenesis, Lithos, 134, 304–316.
Chapman, J. B., T. L. Pavlis, S. Gulick, A. Berger, L. Lowe, J. Spotila, R. Bruhn, M. Vorkink, P. Koons, A. Barker, et al. (2008), Neotectonics of the Yakutat collision: Changes in defor- mation driven by mass redistribution, Active Tectonics and Seismic Potential of Alaska, Geophys. Monogr. Ser, 179, 65–81.
Christeson, G. L., H. J. A. Gulick, P. S. ad Van Avendonk, L. L. Worthington, R. S. Reece, and T. L. Pavlis (2010), The Yakutat terrane: Dramatic change in crustal thickness across the Transition fault, Alaska, Geology, 38 (10), 895–898.
Chuang, L., M. Bostock, A. Wech, and A. Plourde (2017), Plateau subduction, intraslab seismicity, and the Denali (Alaska) volcanic gap, Geology, pp. G38,867–1.
Cloos, M. (1993), Lithospheric bouyancy and collisional orogenesis: Subduction of oceanic plateaus, continental margins, island arcs, spreading ridges, and seamounts, Geological Society of America Bulletin, 105 (715-737).
Cross, R. S., and J. T. Freymueller (2008), Plate coupling variation and block translation in the Andreanof segment of the Aleutian arc determined by subduction zone modeling using GPS data, Geophysical Research Letters, 34 (6).
Defant, M. J., and M. S. Drummond (1990), Derivation of some modern arc magmas by melting of young subducted lithosphere, Nature, 347 (6294), 662–665.
DeMets, C., and T. H. Dixon (1999), New kinematic models for Pacific-North America motion from 3 Ma to present, I: Evidence for steady motion and biases in the NUVEL-1A Model, Geophysical Research Letters, 26 (13), 1921–1924.
Eberhart-Phillips, D., D. H. Christensen, T. M. Brocher, R. Hansen, N. A. Ruppert, P. J. Haeussler, and G. A. Abers (2006), Imaging the transition from Aleutian subduction to Yakutat collision in central Alaska, with local earthquakes and active source data, Journal of Geophysical Research: Solid Earth, 111 (B11).
Elliott, J., J. T. Freymueller, and C. F. Larsen (2013), Active tectonics of the St. Elias orogen, Alaska, observed with GPS measurements, Journal of Geophysical Research: Solid Earth, 118 (10), 5625–5642.
Enkelmann, E., P. O. Koons, T. L. Pavlis, B. Hallet, A. Barker, J. Elliott, J. I. Garver, S. P. Gulick, R. M. Headley, G. L. Pavlis, et al. (2015), Cooperation among tectonic and surface processes in the St. Elias Range, Earth’s highest coastal mountains, Geophysical Research Letters, 42 (14), 5838–5846.
Ferris, A., G. A. Abers, D. H. Christensen, and E. Veenstra (2003), High resolution image of the subducted Pacific (?) plate beneath central Alaska, 50–150 km depth, Earth and Planetary Science Letters, 214 (3), 575–588.
Fitzgerald, P. G., R. B. Sorkhabi, T. F. Redfield, and E. Stump (1995), Uplift and denuda- tion of the central Alaska Range: A case study in the use of apatite fission track ther- mochronology to determine absolute uplift parameters, Journal of Geophysical Research: Solid Earth, 100 (B10), 20,175–20,191.
Freymueller, J. T., H. Woodard, S. C. Cohen, R. Cross, J. Elliott, C. F. Larsen, S. Hreins- dottir, and C. Zweck (2008), Active deformation processes in Alaska, based on 15 years of GPS measurements, Active tectonics and seismic potential of Alaska, 179, 1–42.
Fuis, G. S., T. E. Moore, G. Plafker, T. M. Brocher, M. A. Fisher, W. D. Mooney, W. Nodle- berg, R. Page, B. Beaudoin, N. I. Christensen, A. Levander, W. Lutter, R. Saltus, and
N. A. Ruppert (2008), Trans-Alaska Crustal Transect and continental evolution involving subduction underplating and synchronous foreland thrusting, Geology, 36 (3), 267–270.
Haeussler, P. J. (2008), An Overview of the Neotectonics of Interior Alaska: Far-Field Defor- mation From the Yakutat Microplate Collision, American Geophysical Union, pp. 86–108.
Haeussler, P. J., A. Matmon, D. P. Schwartz, and G. G. Seitz (2017), Neotectonics of interior alaska and the late quaternary slip rate along the denali fault system, Geosphere.
Haynie, K., and M. A. Jadamec (2017), Tectonic drivers of the Wrangell block: Insights on forearc sliver processes from 3D geodynamic models of Alaska, Tectonics, 36 (7), 1180– 1206.
Jadamec, M., and M. Billen (2010), Reconciling surface plate motions with rapid three- dimensional mantle flow around a slab edge, Nature, 465 (7296), 338–341.
Jadamec, M. A., and M. I. Billen (2012), The role of rheology and slab shape on rapid mantle flow: Three-dimensional numerical models of the Alaska slab edge, Journal of Geophysical Research: Solid Earth, 117 (B2).
Jadamec, M. A., M. I. Billen, and S. M. Roeske (2013), Three-dimensional numerical models of flat slab subduction and the Denali fault driving deformation in south-central Alaska, Earth and Planetary Science Letters, 376, 29–42.
Kerr, A. C. (2003), Oceanic plateaus, The Crust, Treaste on Geochemistry, 3, 537–565. Lallemand, S., A. Heuret, and D. Boutelier (2005), On the relationships between slab dip, back-arc stress, upper plate absolute motion, and crustal nature in subduction zones, Geochemistry Geophysics Geosystems, 6 (9).
Lease, R. O., P. J. Haeussler, and P. O'Sullivan (2016), Changing exhumation patterns during Cenozoic growth and glaciation of the Alaska Range: Insight from detrital geo-and thermo-chronology, Tectonics.
Marechal, A., S. Mazzotti, J. L. Elliot, J. T. Freymueller, and M. Schmidt (2015), Indentor- corner tectonics in the Yakutat-St. Elias collision constrained by GPS, Journal of Geo- physical Research: Solid Earth.
Martin-Short, R., R. M. Allen, and I. D. Bastow (2016), Subduction geometry beneath south-central Alaska and its relationship to volcanism, Geophysical Research Letters, doi: 10.1002/2016GL070580, 2016GL070580.
Mazzotti, S., and R. Hyndman (2002), Yakutat collision and strain transfer across the north- ern Canadian Cordillera, Geology, 30 (6), 495–498.
Morra, G., K. Regenauer-Lieb, and D. Giardini (2006), Curvature of oceanic arcs, Geology, 34 (10), 877–880.
Müller, R. D., M. Sdrolias, C. Gaina, and W. R. Roest (2008), Age, spreading rates, and spreading asymmetry of the world’s ocean crust, Geochemistry, Geophysics, Geosystems, 9 (4).
Page, R., C. Stephens, and J. Lahr (1989), Seismicity of the Wrangell and Aleutian Wadati- Benioff zones and the north American plate along the Trans-Alaska Crustal Transect, Chugach Mountains and Copper River Basin, Southern Alaska, Journal of Geophysical Research, 94 (B11), 16,059–16,082.
Peacock, S. M., T. Rushmer, and A. B. Thompson (1994), Partial melting of subducting oceanic crust, Earth and planetary science letters, 121 (1), 227–244.
Piromallo, C., T. Becker, F. Funiciello, and C. Faccenna (2006), Three-dimensional instan- taneous mantle flow induced by subduction, Geophysical Research Letters, 33 (8).
Plafker, G., J. C. Moore, and G. R. Winkler (1994a), Geology of the southern Alaska margin, The Geology of North America, The Geology of Alaska, G-1.
Plafker, G., L. M. Gilpin, and J. C. Lahr (1994b), Neotectonic map of Alaska, The Geology of North America, 1.
Preece, S. J., and W. K. Hart (2004), Geochemical variations in the <5 Ma Wrangell Volcanic Field, Alaska: implications for the magmatic and tectonic development of a complex continental arc system, Tectonophysics, 392 (1), 165–191.
Ratchkovski, N., and R. Hansen (2002a), New evidence for segmentation of the Alaska subduction zone, Bulletin Of The Seismological Society Of America, 92 (5), 1754–1765.
Ratchkovski, N. A., and R. A. Hansen (2002b), New Constraints on Tectonics of Interior Alaska: Earthquake Locations, Source Mechanisms, and Stress Regime, Bulletin Of The Seismological Society Of America, 92 (3), 998–1014.
Ribeiro, J. M., R. C. Maury, and M. Gr´egoire (2016), Are Adakites Slab Melts or High-pressure Fractionated Mantle Melts?, Journal of Petrology, 57 (5), 839, doi: 10.1093/petrology/egw023.
Richter, D., J. G. Smith, M. Lanphere, G. Dalrymple, B. Reed, and N. Shew (1990), Age and progression of volcanism, wrangell volcanic field, alaska, Bulletin of Volcanology, 53 (1), 29–44.
Rondenay, S., L. G. Mont´esi, and G. A. Abers (2010), New geophysical insight into the origin of the Denali volcanic gap, Geophysical Journal International, 182 (2), 613–630.
Schellart, W. (2004), Kinematics of subduction and subduction-induced flow in the upper mantle, Journal of Geophysical Research: Solid Earth, 109 (B7).
Smith, W. H., and D. T. Sandwell (1997), Global sea floor topography from satellite altimetry and ship depth soundings, Science, 277 (5334), 1956–1962.
Stegman, D., J. Freeman, W. Schellart, L. Moresi, and D. May (2006), Influence of trench width on subduction hinge retreat rates in 3-D models of slab rollback, Geochemistry, Geophysics, Geosystems, 7 (3).
Strak, V., and W. P. Schellart (2014), Evolution of 3-D subduction-induced mantle flow around lateral slab edges in analogue models of free subduction analysed by stereoscopic particle image velocimetry technique, Earth and Planetary Science Letters, 403, 368–379.
Waldien, T., S. M. Roeske, J. A. Benowitz, W. K. Allen, and K. D. Ridgway (2015), Neogene exhumation in the eastern Alaska Range and its relationship to splay fault activity in the Denali fault system, AGU Fall Meeting Abstracts.
Wang, Y., and C. Tape (2014), Seismic velocity structure and ansotropy of the Alaska subduction zone based on surface wave tomography, Journal of Geophysical Research: Solid Earth, 119, 8845–8865.
Worthington, L. L., S. P. Gulick, and T. L. Pavlis (2008), Identifying active structures in the Kayak Island and Pamplona zones: Implications for offshore tectonics of the Yakutat Microplate, Gulf of Alaska, Active tectonics and seismic potential of Alaska: American Geophysical Union Geophysical Monograph, 179, 257–268.
Worthington, L. L., H. Van Avendonk, S. Gulick, G. L. Christeson, and T. L. Pavlis (2012), Crustal structure of the Yakutat terrane and the evolution of subduction and collision in southern Alaska, Journal of Geophysical Research: Solid Earth, 117 (B1).```