# Category Archives: My Research

## Anisotropic viscosity

Nondimensionalized temperature is plotted over time. The model on the left has isotropic viscosity; the model on the right has anisotropic viscosity, oriented diagonally (45 degrees). The shear viscosity is 0.01 times the normal viscosity in the anisotropic case. Both models have no-slip boundaries on the top and bottom, and free slip on the sides. Note that in a confined box, the existence of uniform easy slip planes basically just reduces the effective viscosity of the entire box. The model on the right would look very similar with isotropically lower viscosity than the model on the left.

I’ve been working a bit lately on fluid flow problems involving materials with anisotropic viscosities. That is, deforming such a material in one direction is easier than in another direction. An example of such a material in the real world is a sequence of rock layers with varying strengths. In simple shear along the layered plane the stronger layers are able to slide past each other along the weak layers, but the rock is difficult to squish in “pure shear“, because it is supported by the strong layers.

Alternating strong and weak layers in sedimentary rock.
Sandy Hollow, MT.

Deformation in layered rock is accommodated primarily in the weak layers.
Sun River, MT.

In order to model this sort of behavior I’ve been working on extending a geodynamic modeling code, ASPECT, to incorporate more complex constitutive laws. Before I get too far in to ASPECT though I’d like to to step back a bit and define the equations I plan to solve. To that end we can start with the basic governing equations of fluid dynamics.

In general fluid flow obeys the Cauchy momentum equation,

$$\rho \frac{D\vec{u}}{Dt} = \nabla \cdot \sigma + \rho \vec{g}$$,

where $$\vec{u}$$ is the fluid velocity field, $$\frac{D}{Dt} \equiv \frac{d}{dt} + \vec{u} \cdot \nabla$$ is the material derivative, $$\sigma$$ is the stress tensor, $$\rho$$ is the fluid density, and $$\vec{g}$$ is the gravity vector. In geodynamics we deal with very low Reynolds number systems (ie, momentum is negligibly small), so we set the left hand side of Cauchy’s equation to zero. This leaves us with the governing equation:

$$-\nabla \cdot \sigma = \rho \vec{g}$$.

We often also add the constraint that mass is conserved, $$\nabla \cdot (\rho \vec{u}) = 0$$, or in some cases that the flow is entirely incompressible, $$\nabla \cdot \vec{u} = 0$$.

In general, we don’t care so much about the stress state of the fluid, but we want to solve for the velocity. Therefore we need an equation to relate applied stress to strain rate. This type of equation is called a “constitutive law,” and is dependent on the chemical and state properties of the material. The most general constitutive law can be written in index notation, $$\sigma_{ij} = -\overline P \delta_{ij} + C_{ijkl} \varepsilon_{kl}$$, where $$\overline P$$ is the dynamic pressure; $$\varepsilon$$ is the strain rate, defined as the symmetric component of the gradient of the velocity, $$\varepsilon \equiv \frac{1}{2} (\nabla \vec{u} + (\nabla \vec{u})^T)$$; and $$C_{ijkl}$$ is a fourth order tensor, related to the viscosity, which maps strain rates to stresses.

Aspect assumes that the fluid in question is isotropic, and thus reduces to only two values, bulk viscosity (which dissipates energy during compaction and dilation), and shear viscosity which dissipates energy during shear deformation. Further, it is assumed that because bulk viscosity is only important in rapid compressions and dilations, such as sound waves and shock waves, it can be ignored in slow flows like geologic applications. Therefore, in Aspect $$C_{ijkl}$$ is reduced to a single scalar state variable, $$2 \eta$$, called the shear viscosity. Generalization back to a full constitutive tensor turns out to be as straightforward as changing the variable type of $$\eta$$ from a floating point scalar to a symmetric, fourth-order tensor. This modification is easily supported in the finite element library on which Aspect is built. With such a modification it is possible to model any arbitrary constitutive law.

Note (2016/02/16): This draft is already over half a year old, and I’ve been up to a lot since I wrote it, so I think it’s time to just post it and write more in a next installment. For now I’ll leave off with some figures to preview effects of the constitutive law I’m using:

Velocity magnitude is plotted here with velocity vectors overlain. The model setups are the same as in the animated example above except the left and right are stress-free boundaries, and the bottom has a prescribed horizontal velocity to the left. We can see the development of preferred shear directions in the velocity vectors.

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## Comprehensive Exams – Part I: Abstracts

After delaying them a year, I’m finally doing comprehensive exams for my Ph.D. program. Comps is a long process, which I only really started thinking about when I got back from Kyrgyzstan, and will go until some time mid-February.

The goal is to formulate two sufficiently different research projects which I will propose to do, and convince a committee of faculty members that they are reasonable projects and that I’m capable of doing them. There’s also a written test component, which happens in January.

Anyway, the first step is to write a one-page abstract for each project. They’re due today, and I’m happy to say that I did finally distribute them to my committee members this morning!

Abstract 1
Abstract 2

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## Kyrgyzstan – Part I: What/Why/Where?

I’ve started to develop a backlog of things to write about, and seemingly less and less time to actually write. Most significantly, probably, I need to write something (anything!) about my trip to Kyrgyzstan. I’ll start with some overview stuff here, and talk about specifics in later posts.

Q: Whaaat? You went where? Why? How? Where the heck is Kyrgyzstan, anyway?
A: Our department has a fund for sending a bunch of students to study geology in a remote place every couple of years. It’s always a place where one of our faculty has an active research project, and it switches each time. The last trip was just before I got to UO, my adviser led a trip to Spain and Morocco. This year the destination was Kyrgyzstan. Kyrgyzstan is a small central Asian country sandwiched between Kazakhstan to the North, China to the East, and Tajikistan and Uzbekistan to the South and West respectively. More intuitively, it’s here:

Q: What active research are UO workers doing in Kyrgyzstan?
A: Ray Weldon, one of the UO geology faculty members, has been working for over 20 years on a continental dynamics project with Dr. Kanat Abdrakhmatov, the director of the Institute of Seismology at the Kyrgyz National Academy of Sciences. They’re trying to understand the mechanisms of deformation that have uplifted the Tien Shan mountain range, and by proxy understand seismic hazards associated with the active tectonic deformation. There have also been countless grad students over those 20 years who have done their dissertation work on that project. Most of which has been geologic mapping of some of the major fault zones. Most recently, Ray has been trying to interest the paleontology group in helping to pin down absolute ages on some of the important rock layers by identifying fossils in them. This year my friend, Win McLaughlin, got a grant through the Fulbright Program to work towards that goal by looking for fossils in the Tien Shan mountains for 10 months next year.

Q: Why research Kyrgyzstan? What’s significant about those mountains?
A: The Tien Shan mountains are the northernmost extent of the Tethyan Orogeny, a result of the continental collision between India and Eurasia which also created the Himalaya. Both mountain ranges are the result of horizontal compression of the Asian continent: as it gets squished horizontally, there’s nowhere to go but up (and down, but these structural geologists don’t usually care about that). The Himalaya have been around for a long time now, and the Tien Shan are relatively young. The Himalaya are now increasingly inactive, and most of the modern deformation is taking place farther North.

Currently, around 50% of the total shortening of the Eurasian continent, is taking place in the Tien Shan [1]. That shortening has to be accommodated somewhere. Deep in the crust, the rock is warm enough, and under high enough pressure, that it can deform slowly over time, but near the surface that stress is accommodated through sudden ruptures, ie. earthquakes.

Just as an example of the active seismicity, according to the USGS earthquake catalog, http://earthquake.usgs.gov/, there have been 21 earthquakes greater than magnitude 4.0 in the Tien Shan in the past 30 days.

Kyrgyzstan might be the most seismically active region in Asia, but the record of historic earthquakes there is surprisingly sparse. As the population has grown, and urbanization increased since the beginning of what everyone there just calls “soviet times”, building and infrastructure regulation needs to start taking in to account the possibility of very large earthquakes, but nobody knows how often such an earthquake might happen, or how big it could be.

During our trip we learned a bit of everything about Kyrgyz geology, but for most of the trip we focused on understanding the history of uplift and deformation, to better constrain the net rate of movement on the major faults, and ultimately determine which ones could present significant seismic hazard.

Of course we also got to meet lots of very cool people, and see all kinds of beautiful places; and meet some shady people, and see some very ugly places, but more on all that in later posts!

[1] Abdrakhmatov, K. Ye, et al. “Relatively recent construction of the Tien Shan inferred from GPS measurements of present-day crustal deformation rates.” (1996): 450-453.

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## Kyrgyzstan: August 25 – September 15, 2014

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## Lithospheric Delamination in the Western U.S.– Part IV: Delam Styles / Type Localities

In my last post I tried to explain the fundamentals of delamination. I used a floating wooden board being weighed down by metal as an analogy for the state of the Earth’s lithosphere. In the Earth, the crust is buoyantly floating on top of the mantle and is being weighed down by cold mantle material stuck to its bottom. If the two materials were to separate, the cold mantle material would sink off in to the warmer mantle below and the buoyant crust would rebound, rapidly building mountain ranges. So, in the most general sense, delamination refers to the process of heavy material peeling of the lithosphere and sinking down in to the mantle. This can be manifested in a few different forms.

The first form I’ll mention is the one which most resembles the analogy I gave before. In this case the cold mantle lithosphere peels back from the crust and sinks away in to the mantle (Figure 1). Hotter mantle material from deeper in the Earth replaces the cold lithosphere as it recedes.

Figure 1. Mantle lithosphere can peel off from the base of the crust, forcing hotter mantle to flow in to its place. (Figure adapted from [3])

This sort of thing is most likely happening beneath the edges of the Colorado Plateau, contributing the dish-like shape to its edges [1]. Whether or not this is actually the case is a contentious question but the evidence is convincing enough to mention it here.

Secondly, it is possible for the mantle lithosphere to drip off, with similar shape to the form which cold wax at the top of a lava lamp takes as it convects back toward the bottom. This is known as a “Rayleigh-Taylor Instability.”

Figure 2. Mantle lithosphere can drip off of the crust as a Rayleigh-Taylor instability. (Figure adapted from [3])

This appears to have happened as a late stage in the formation of the Wallowa Mountains in northeastern Oregon [2].

Lastly, lithosphere can be forced to depth by horizontal tectonic forces from the plates. In this case, the lithosphere isn’t really delaminating because of its own inherent density instability, but because it is being pushed to depth by the material above it.

Figure 3 Horizontal tectonic forces can push lithosphere deep in to the mantle.

This is apparently the case for lithosphere beneath the transverse ranges of southern California, around the Los Angeles basin for instance [4].

My research is focused on understanding the mechanisms that enable each of these processes to happen. Assuming the mantle lithosphere is always buoyantly unstable, can it actually ever stably stick around? Is delamination the exception or the rule? And what controls whether the lithosphere will drip off in small Rayleigh-Taylor stabilities or in massive intact slab roll-backs.

All of these questions have impacts on, and feedback from other, associated tectonic and magmatic processes. I’m interested in untangling the causes and effects of each of these mechanisms and placing them in real-world context using the western US as a natural laboratory. All of my experiments have already been run for me, I just need to sort out what happened.

#### References

[1] Levander, A., et al. “Continuing Colorado plateau uplift by delamination-style convective lithospheric downwelling.” Nature 472.7344 (2011): 461-465.

[2] Darold, Amberlee, and Eugene Humphreys. “Upper mantle seismic structure beneath the Pacific Northwest: A plume-triggered delamination origin for the Columbia River flood basalt eruptions.” Earth and Planetary Science Letters 365 (2013): 232-242.

[3] Göğüş, Oğuz H., and Russell N. Pysklywec. “Near‐surface diagnostics of dripping or delaminating lithosphere.” Journal of Geophysical Research: Solid Earth (1978–2012) 113.B11 (2008).

[4] Houseman, Gregory A., Emily A. Neil, and Monica D. Kohler. “Lithospheric instability beneath the Transverse Ranges of California.” Journal of Geophysical Research: Solid Earth (1978–2012) 105.B7 (2000): 16237-16250.

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## Lithospheric Delamination in the Western U.S.– Part III: Delamination

In my last post I covered the basics of mantle dynamics and now I should be just about ready to dive in to the actual content of this series of posts: lithospheric delamination. I’ll start by just explaining the concept. In my next post in this series I’ll talk about type-localities in the western U.S. and some basics on the methods I use to study the process. I’ll finish the series by explaining specifics of my research and where I hope to go with it.

In geodynamics, delamination refers to a process in which something that was formerly attached to the lithosphere becomes unstable and falls off, sinking deep into the mantle. This can happen if the material is substantially more dense than the material below it. We call this a buoyant instability.

A nice way to visualize what’s happening, and an analogy that I’ll return to in a moment, is to imagine a piece of wood floating in water with a heavy chunk of metal glued to its bottom. The wood in this case represents the buoyant crust and the metal represents the cold upper mantle that’s stuck to its bottom. The water it’s floating on represents the rest of the mantle. Delamination would be the process of the glue that holds the two together coming undone and allowing the metal to sink away.

So, why would we care? This is a process that happens tens or hundreds of kilometers below the surface of the Earth so how does it affect anything else on Earth?

First of all, in Earth sciences even the processes which seem most esoteric and intangible usually have consequences to other processes and systems which directly affect life on Earth. Delamination most likely has a big part to play in the development of continental lithosphere and mountain-building events which affect paleo and modern ecology, tectonic history, distribution of useful minerals, etc. It’s also just cool to imagine processes on these time and length scales because it stretches the imagination past our small bubble of human experiences.

In this case however there is a direct, tangible consequence that’s fairly interesting in its own right. Let’s go back to the floating board being weighed down by the chunk of metal analogy for a moment. Basic physics tells us that the buoyant force pushing the board up equals the weight of the displaced water and that the system will find equilibrium when the weight of the floating stuff equals this buoyant force. Since the metal is dense, when the two are attached the whole thing will float relatively low (assuming it’s not so heavy that it just sinks all together). When the metal comes loose and sinks away, the board is much less dense and will float higher in the water. It’s similar to a boat weighed down with cargo floating super low until the cargo is unloaded.

So, what does this look like in geophysical terms? If there’s heavy material attached to the bottom of the crust, it’s being weighed down and will float relatively low, but if the heavy material falls off the crust will float up higher. This is usually expressed on the surface as (relatively) rapid uplift. In other words, it can build mountains.

We suspect that this has happened in the southern Sierra Nevada (adding to an already impressive mountain range), the Wallowa Mountains in eastern Oregon (building a mountain range from scratch), and around the edges of the Colorado Plateau (responsible for its dish-like topographic expression).

I think I’ll cut this post short here and talk about the different forms of delamination in my next post. As illustrations, I’ll give specifics on the three type-localities I just mentioned: the southern Sierra Nevada, Wallowas and Colorado Plateau.

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## Lithospheric Delamination in the Western U.S. – Part II: Mantle Dynamics

In my last post I tried to do a quick overview of seismic tomography. The takeaway message being that it is possible, through seismology, to “see” structures inside the Earth. It’s how we know that the Earth has layers, for example.

Other significant structures that we observe with seismic imaging are seismically fast (cold) regions beneath subduction zones where we can see tectonic plates sinking deep into the mantle, and seismically slow (hot) regions beneath mid-ocean ridges. All of these examples are to be expected from the basics of plate tectonic theory and the fact that we see them seismically nicely supports both plate tectonic theory and the methods behind seismic tomograpy.

More interestingly, in some areas we see hot and cold blobs in places that aren’t obviously associated with plate tectonics. Some examples from the western U.S. are circled in the seismic image from the last post.

This would be a nice point to lead me in to my topic, lithospheric delamination, but I feel like I should first take a tangent to explain something that I think is non-obvious about the way the Earth operates, and come back to my main point in my next post.

The Earth’s mantle makes up a huge percentage of the mass of the Earth. It’s a chemically distinct layer between the crust and the core, encompassing almost 85% of Earth’s total volume. It’s very hot but is also under enormous amounts of pressure, the deeper you go, the more pressure the mantle is under. This has weird mineralogical and rheological consequences. The mantle can be divided in to several distinct layers in which pressure and temperature trade off differently. From the top down:

At the top of the mantle there is a cold region which is, for the most part, frozen on to the bottom of the crust. The crust floats at the surface because it is chemically buoyant (made of much less dense minerals) and the top of the mantle hangs on to the bottom of it like a barnacle. If it weren’t being held up by the buoyant crust it would prefer to sink deep in to the mantle because it is cold and dense. This frozen mantle along with the crust is collectively called the lithosphere. The part of the lithosphere which is made of mantle rocks and not crustal rocks is referred to (not very imaginatively) as the mantle lithosphere.

Relative geometries of the compositional layers of the Earth, drawn to scale.

Below the lithosphere, the mantle is hotter but it isn’t under too much pressure so it’s able to flow relatively easily. It’s not melted (at least not entirely, this is an area of some debate) because the pressure forces it to be a solid. It is hot enough and under sufficiently low pressure that it can “flow” by allowing crystal grains to slide past each other and deform. This layer is called the asthenosphere and it allows plate motion by decoupling the lithosphere from the rest of the mantle..

Below the asthenosphere, the mantle is very hot but is also under much more pressure. It can still flow but much more slowly. Flow in this part of the mantle is dominated by whole-mantle scale convection. Much like the way water flows around in a pot to equalize temperature, the mantle is heated by radiogenic decay throughout, and cooled from above, so there is a constant flux of material rising and falling throughout.

The mantle is able to convect similarly to a pot of water in the microwave, heated throughout and cooled from above. [Figure generated with ASPECT mantle convection code]

The convecting interior of the mantle is all made of the same elements but is actually divided in to layers of its own because higher pressure at depth can force the minerals to reorganize their structure (think graphite -> diamond type transition). There are mineralogical transitions at ~410km and ~660km depth, as well as a possible transition right near the core-mantle-boundary at around 3000km. For the purpose of my story though, we’ll assume that the mantle is homogeneous below the asthenosphere.

Now that I’ve given a quick overview of the way the mantle acts and moves, I think this is a reasonable place to stop for now. Next time I’ll get to the actual content of my research and talk about the ways in which the cold mantle lithosphere can come detached from the crust to sink back down in to the mantle.

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## Lithospheric Delamination in the Western U.S. – Part I: Seismic Imaging

I recently put together a poster about lithospheric delamination in the western U.S. for the CIDER workshop that’s coming up in Berkeley and I thought I’d share the content here too.

I’d like to start with the basics to tell the whole story but it’s a long one so I’m gonna split this in to several installments. This first one is just going to fill in some background on methods we use in Geophysics, starting with teleseismic imaging of the Western U.S.

Seismic Imaging: What Is It And Why Do We Care?

As a geodynamicist I’m interested in deep-Earth processes. Most of my work is focused on understanding what’s inside the Earth, where it came from and where it’s going. More specifically, I’m interested in modeling deep-Earth processes, i.e. using analogy of a simplified system to explain the behavior of a complex one like the Earth’s mantle. Of course, just making up analogies between simple systems and complex ones isn’t really science. To relate our simplified system to the real world we need an idea of what the real system looks like and how it behaves.

Geophysicists have relatively few tools at their disposal for seeing the deep structure of the Earth: geochemists can tell us about the origin of rocks at the surface, and geomorphologists can tell us about the evolution of landscapes which can both give hints at the processes happening at depth (or at least processes that have happened at depth in the past) but neither gets at the current state of the Earth below a few kilometers.

That’s where geophysical imaging comes in.  I’ll be specifically talking about teleseismic imaging. The concept is relatively simple: seismic waves from earthquakes all over the planet propagate through the Earth’s interior and interact with the different structures they encounter along the way. Namely, they travel faster through some materials than through others. When the waves arrive somewhere far away we can measure the relative delays among arrival times at different seismometers and use that information to reconstruct an image of the structures through which they passed.

Cross section of the whole Earth, showing the complexity of paths of earthquake waves. The paths curve because the different rock types found at different depths change the speed at which the waves travel. Solid lines marked P are compressional waves; dashed lines marked S are shear waves. S waves do not travel through the core but may be converted to compressional waves (marked K) on entering the core (PKP, SKS). Waves may be reflected at the surface (PP, PPP, SS). Seismographs detect the various types of waves. Analysis of such records reveals structures within the Earth.

Techniques for seismic imaging have been meticulously developed over the past century or more, but in the past decade advancements in, and access to, computational hardware as well as amazing data from the EarthScope project have enabled unprecedented progress in imaging structures beneath the U.S.

Horizontal slice of seismic wave velocity through the upper mantle in the western U.S. (Schmandt, 2010)

Things that affect the wave speed include chemical properties, pressure, and directional strain but the one we’re most interested in studying is usually the temperature. The other effects on seismic velocity are either easy enough to correct for (like pressure) or small enough effects over large distances that we usually ignore them and assume all seismic velocity perturbations are caused by temperature perturbations. Typically, a rock at low temperature conducts seismic waves more rapidly than the same rock at higher temperature. Temperature is an interesting property because it can tell a lot about where a piece of the Earth came from and where it’s headed.

Since most materials, including rock, are less dense when they’re hot, temperature is typically a good indicator of relative buoyancy. For example, when we see a cold (seismically fast) object we can guess it’s heavy and therefore actively sinking deeper in to the Earth, or it at least wants to. There are other methods of using seismic waves to measure different properties about the Earth but temperature and buoyancy of objects in the mantle are the most interesting to me.

In my next installment I’ll talk about the significance of some of the objects we see in the most recent seismic images, such as the figure above.

References

Schmandt, Brandon, and Eugene Humphreys. “Complex subduction and small-scale convection revealed by body-wave tomography of the western United States upper mantle.” Earth and Planetary Science Letters 297.3 (2010): 435-445.

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