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The Sassy Scientist – Earthquake Exoteries Nr. II

The Sassy Scientist – Earthquake Exoteries Nr. II

Every week, The Sassy Scientist answers a question on geodynamics, related topics, academic life, the universe or anything in between with a healthy dose of sarcasm. Do you have a question for The Sassy Scientist? Submit your question here or leave a comment below.

In a comment on a post about the key papers in geodynamics, the Curmudgeonly Commenter asked:


Could you please point out some exceptionally important papers in geodynamics and tell us something interesting about the history of the field?


Dear CC,

As explained in last week’s post, we will start from the very beginning: a time and place where everything was still considered elastic.

Elastic rebound theory
After millennia of attributing earthquakes to gods and mythical beasts (what a wondrous time that must’ve been), and a plethora of decades since the design of the first western seismographs, with a poor understanding of the source mechanism behind an earthquake, H.F. Reid (1910) proposed a theory on the occasion of the great 1906 San Francisco earthquake. The elastic rebound theory states that elastically stored energy is released suddenly during an earthquake on a pre-existing fault plane that resides in the solid crust. There is no need to generate a new fracture interface. Benioff (1949, 1954, 1964) — yes, the one comprising half of the name of the Wadati-Benioff zone — championed this theory and concluded that at least the shallow and intermediate-depth earthquakes could be attributed to this mechanism. Benioff considered deeper earthquakes (>300 km), posing less of a seismic hazard at the surface, to result from volumetric changes; no fracturing nor fault slip on a plane. A further refinement of the elastic rebound theory was necessary though in order to explain e.g., the low-magnitude stress changes that arise in the lithospheric medium as a result of earthquake ruptures.

Elastic dislocation theory
Whereas the nucleation of earthquakes was one line of inquiry, another was to use actual surface observations to infer fault slip at depth. There’s nothing better than guessing from afar what’s actually happening below the surface, right? Building on Volterra’s (1907) theory of distortions in elastic bodies — you know, by atomic off-sets in crystal structure — Steketee (1958) and Chinnery (1961) provided elementary steps forward in determining the surface deformation that result from motion on dislocations that reach the surface. Later, Savage and Burford (1970, 1973) and Okada (1985, 1992) elaborated and expanded elastic dislocation theory to include a complete set of analytical expressions of (surface) deformation due to slip on a (buried) fault (segment) in an elastic half-space. To this day still, these simple analytical expressions are widely used, both for fault slip inversions (e.g., Bagnardi and Hooper 2018), and elastic block models (e.g., Meade and Loveless 2009). In the elastic realm of the lithosphere, static stress changes are transferred and may trigger new earthquakes on other faults (King 1994; Stein 1999). This is already getting dangerously close to actual dynamics, instead of only finding descriptions of surface kinematics. What a waste …

Unfortunately, that’s all we have time for this week. Next week, I promise we will finally get into a bit more of the dynamics. Let’s discuss the earthquake cycle for instance.

Yours truly,

The Sassy Scientist

PS: This post was written with all things elastic in my mind. Is your mind as flexural as mine?

References:
Bagnardi, M. and Hooper, A. (2018), Inversion of surface deformation data for rapid estimates of source parameters and uncertainties: A Bayesian approach. Geochemistry, Geophysics, Geosystems, 19, 2194–2211. https://doi.org/10.1029/2018GC007585
Benioff, V.H. (1949), Seismic evidence for the fault origin of oceanic deeps. Bulletin of the Geological Society of America, 60, 1837-56
Benioff, V.H. (1954), Orogenesis and deep crustal structure—additional evidence from seismology. Bulletin of the Geological Society of America, 65, 385-400
Benioff, V.H. (1964), Earthquake source mechanisms. Science, 143, 1399-1406
Chinnery, M.A. (1961), The deformation of ground around surface faults. Bulletin of the Seismological Society of America, 51, 355-372
King, G.C., Stein, R. and Lin, J. (1994). Static stress changes and the triggering of earthquakes. Bulletin of the Seismological Society of America, 84(3), 935–953
Meade, B.J. and Loveless, J.P. (2009), Block modeling with connected fault network geometries and a linear elastic coupling estimator in spherical coordinates. Bulletin of the Seismological Society of America, 99(6), 3124–3139
Okada, Y. (1985), Surface deformation due to shear and tensile faults in a half-space. Bulletin of the Seismological Society of America, 75, 1135-1154
Okada, Y. (1992), Internal deformation due to shear and tensile faults in a half-space. Bulletin of the Seismological Society of America, 82(2), 1018–1040
Reid, H.F. (1910), The California Earthquake of April 18, 1906. Volume II. The Mechanics of the Earthquake. Washington DC: Carnegie Institution of Washington, Publication No. 87
Savage, J.C. and Burford, R.O. (1970), Accumulation of tectonic strain in California. Bulletin of the Seismological Society of America, 60, 1877–1896
Savage, J.C. and Burford, R.O. (1973), Geodetic determination of relative plate motion in central California. Journal of Geophysical Research, 78, 5, 832–845
Stein, R.S. (1999), The role of stress transfer in earthquake occurrence. Nature, 402, 605–609
Steketee, J.A. (1958), On Volterra's dislocation in a semi-infinite elastic medium. Canadian Journal of Physics, 36, 192-205
Volterra, V. (1907), Sur l'équilibre des corps élastiques multiplement connexes. Annales scientifiques de l'École Normale Supérieure, 24, 401-517

Magma dynamics

Magma dynamics
Assistant Professor Juliane Dannberg, University of Florida.

Assistant Professor Juliane Dannberg, University of Florida.

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

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

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

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

The forces at play

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

The equations

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

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

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

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

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

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

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

 

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

 

 

 

 

 

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

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

Solitary waves

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

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

Formation of solitary waves.

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

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

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

The bigger picture

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


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

References:

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

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

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

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

Katz, R.F., 2015. An introduction to coupled magma/mantle dynamics. http://foalab.earth.ox.ac.uk/files/IntroMagmaLectures.pdf

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

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

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

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

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

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

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

The Sassy Scientist – Earthquake Exoteries Nr. I

The Sassy Scientist – Earthquake Exoteries Nr. I

Every week, The Sassy Scientist answers a question on geodynamics, related topics, academic life, the universe or anything in between with a healthy dose of sarcasm. Do you have a question for The Sassy Scientist? Submit your question here or leave a comment below.

In a comment on a post about the key papers in geodynamics, the Curmudgeonly Commenter asked:


Could you please point out some exceptionally important papers in geodynamics and tell us something interesting about the history of the field?


Dear CC,

I can call you CC, right? To satisfy your needs, I’ll take on a topic that is unquestionably home to the geodynamics territory, but people are not always aware of the history behind it. I’ve chosen one that resides in the (relatively) short-timescale corner of possible topics, with many other fields contributing fundamental observations and processes: the earthquake cycle. After all, I have already discussed mantle and lithosphere dynamics at length.

The major contributions to understanding earthquakes in the past originated through several lines of inquiry with seismology, rock mechanics and geodesy providing a variety of observations. The latest contributions arise from numerical modelling (and sure… some analogue modelling too) with e.g. dynamic rupture modelling and the embedding of the earthquake cycle in long-term, evolutionary geodynamic models. To accommodate your desire for some historical provenance on the topic, these posts will be somewhat more serious than you’re used to. Bear in mind I still can only cover some aspects. Apparently, some people have thought about this over the past century… I’ll insist that at least the basics in this little story should be common understanding for those involved in the field of geodynamics. With this I mean… everyone! If that’s not the case for you, you’ve got some reading to do.

Now, I had prepared a lengthy post to impress you with my knowledge on all things earthquakes, but my …lovely… Editor-in-Chief stepped in and straight-out forbid me to publish it as it is (well, was). The nerve! Apparently she thinks people wouldn’t read long blog posts. I guess you would like to prove her wrong, right? Unfortunately, my hands are tied. As a faithful, but desperately subjugated, Sassy Scientist, the answer to your question is now split up into multiple posts. With a heavy heart, you’ll get to experience my response to your question over multiple weeks. Isn’t that fantastic? I know I think it is…

So, next week, I’ll actually start going in depth and teach you all abound elastic rebound and dislocation theory. You’re welcome.

Yours truly,

The Sassy Scientist

PS: This post was written reluctantly, despite another, well-written, just a wee bit longer, literary magnificent response to your question, CC. Do you like Editor-in-Chiefs as much as I do?

PS2: As Leonardo Da Vinci once said: “Nothing strengthens authority so much as silence.” I’ll thus shut up about this in the forthcoming series of posts.

PS3: However; “The Sassy Scientist remembers”. And also; “A Sassy Scientist always pays his debts”.

PS4: …

The Sassy Scientist – Pluto Panic

The Sassy Scientist – Pluto Panic

Every week, The Sassy Scientist answers a question on geodynamics, related topics, academic life, the universe or anything in between with a healthy dose of sarcasm. Do you have a question for The Sassy Scientist? Submit your question here or leave a comment below.

After a distraught period (of more than a decade!) since the news first came out that Pluto was not considered a true planet anymore, and due to a post and comment/discussion on this very blog, Goofy finally found the courage to ask:


Do you think Pluto is a planet or a dwarf planet?


Dear Goofy,

Somethin’s wrong here! Definitely. I think Pluto is stellar in his own right, and I don’t think that the way you look at Pluto should change simply because some ‘authority’ dictates that you should change your opinion. That being said; does it matter that Pluto has been demoted to a dwarf planet status? There are a lot of great dwarfs: I know about seven who took in a beautiful lady lost in the forest and a couple who found a dragon in their pile of gold inside a mountain. Let’s take the rough with the smooth and don’t think of Pluto anymore as the last of the planets in our solar system, but rather as one of the main, and certainly the best known, dwarf planets.

Aw shucks. Pluto should have a special status. Beyond the possibility of planetary status, I’m with Laurent in that status should be imposed upon geological/geodynamical features. Why bother with all those heavenly bodies that are just swerving around? Focus on interesting features. Let’s bundle Pluto with Ceres, Titan, Europe and all those other moons, satellites, dwarf planets and create a new scale, taking one step beyond the “Geophysical Planet Definition” of Runyon et al. (2017): “Interesting Units”. So Earth, with all its tectonics and earthquakes and atmosphere scores 10 IUs. Venus, with active volcanism, tectonics and atmosphere scores 9 IUs. Mars, with signs of water presence, geology and relief, but no atmosphere, scores 8 IUs, whilst the gas giants score 7 IUs. Pluto, along with Titan and Ceres scores 5 IUs – our own, rather unimpressive, Moon only takes 3 IUs home. Not too shabby for Pluto I would say. To be fair, let’s award space rubble like Halley’s comet still 1 IU on account of possible, future space mining. For now, as a humble Earth scientist, I urge us to put our primary focus on the highly ranked space monsters, and leave the rest to the astrologers. Oops, astronomers. That’s right (in my mind all the same…).

Yours truly,

The Sassy Scientist

PS: This post was written after some distressing emails from miss. Goofy about her husbands depression due to Pluto’s demotion. I urged her to arrange some counseling.

References:
Runyon, K.D., Stern, S.A., Lauer, T.R., Grundy, W., Summers, M.E. and Singer, K.N. (2017), A Geophysical Planet Definition, Lunar and Planetary Science XLVIII, Abstract 1448