 So, to end our morning, we will talk about the interaction between tectonics and volcanoes, in particular how the passage of seismic waves or the stresses produced by an earthquake might influence the occurrence of eruptions. We'll begin by looking at whether this is in fact the case that eruptions are triggered by earthquakes, and second, we'll try and understand why this might happen if it does happen. But first, it's worth thinking about why this is worth investigating. There are two reasons why we're interested in whether or not earthquakes might affect eruptions. First, it provides a probe of some of the processes that initiate eruptions or modulate eruptions, because earthquakes generate stresses, and Paul spent the last two days explaining how stresses are coupled to deformation and eruptions. And second, we know the stresses produced by earthquakes, so we might get new insights into some of the thresholds, the critical thresholds, that influence eruptions or initiate eruptions. It has been documented, in fact, for a long time in the almost-referee literature that earthquakes induce a variety of responses in the Earth. Pliny in one of his chapters in The Natural History, his encyclopedia published 2,000 years ago, has one chapter devoted to these types of phenomena. So first topic we'd like to address today is whether earthquakes do initiate eruptions, and how do we establish this? First is we can look for correlations in space and time between the occurrence of earthquakes and eruptions. And this is not, turns out, not to be so easy for a variety of reasons that we'll explore as we look through a variety of case studies. But the easiest types of relationships to document are very short-term eruptions. Earthquake happens, and we see eruptions within days. And so here's an exercise doing this. What I'm plotting on the vertical axis is the number of eruptions that occur within five days of regional large earthquakes. In all the examples I'll be showing you, the details do matter. And so I'll have to look at my notes to try and look up some of these details. In this case, these are magnitude 8 and bigger earthquakes, and we're looking at eruptions that occur within 800 kilometers. And the magnitude of eruptions exceeds BEI2. No one has talked about this thing called the Volcano Explosivity Index, abbreviated BEI. It's some measure of how explosive an eruption is. It's based on how high the eruption column goes. It's roughly proportional to the mass that erupts. And it's appropriate for explosive eruptions. BEI2 and bigger corresponds to volcanic eruptions or violent strombolian eruptions are bigger. So eruption columns can go to heights of, say, five kilometers. And why don't we look at smaller eruptions? Our record of smaller eruptions in the geological record is not very good. And so one thing that we'll see on the next slide is how do we assess whether our information is good enough to look at this relationship? Okay, so going back to the figure then, again, big earthquakes and volcanoes that are close enough. We're looking at the time interval between the earthquake and the eruption. Notice we have negative times here. What does a negative time mean? Eruption happens before the earthquake. Okay, so there is a correlation between earthquakes and eruptions. You see roughly four times more eruptions within five days of earthquakes than you would expect by chance. I should point out this is based on a volcano eruption catalog maintained by the Smithsonian Institution called the Global Volcano Database. And it's been around for the longest, it's one of the most complete. Okay, so I think I've provided all the information you need to assess this correlation. But there are a variety of issues to think about when we look for these relationships. Foremost, I guess, is whether the data is accurate in what we call complete. So we do this with earthquakes, we can do the same thing for volcanoes. We know with earthquakes, for example, there's a relationship between the frequency of earthquakes and their magnitude. And that every time you increase the magnitude of earthquakes by one, you see roughly ten times fewer earthquakes. So if we're to plot the number of earthquakes bigger than a given size versus magnitude, we would see a straight line. And where we start seeing deviations from the straight line for smaller magnitudes, we assume that we're simply not recording those smaller earthquakes. So the same thing we expect to be true for volcanic eruptions. So what I'm plotting on the vertical axis is the number of volcanic eruptions in this catalog, bigger than a certain size versus that size. And these are subdivided into different time periods. Since 1900, for example, in green, we expect the catalog of eruptions in the last century or so to be pretty complete. We probably haven't missed too much. Although if you look in detail, in fact, during the world wars, there were fewer volcanic eruptions. But assuming it's reasonably complete, we can see that up to magnitude two or so, we see a nice power law. And that suggests that we're probably not missing in the historical record too many eruptions comparable to volcanic eruptions. And that's why in the analysis I went through on the previous slide, we only looked at eruptions with magnitudes bigger than BEI2 as reported in the catalog. So it might be first worth thinking about what we expect the answer to the question we started with to be, do earthquakes trigger eruptions? And so I'm going to build on something that Paul showed us earlier so that we can see if we can estimate how many eruptions we would expect to be triggered by earthquakes. And we'll imagine the very simplest possible model for a volcanic system where we have magma stored at some pressure in a reservoir. Eruption has happened, pressure drops to some pressure we'll call P0. You can think about that as the pressure of time zero. And over time, magma fills up the reservoir, the pressure goes up. And for illustrative purposes, I'm going to assume it increases linearly with respect to time. As you saw in Paul's lecture, that's not the case. We'll revisit this in just a second. There's going to be some increase in pressure. I'll call it delta P critical. Where when the pressure becomes big enough, we can create dikes, we can open up cracks, and the eruption begins. So I'll indicate the special with the dotted line. Eruption happens, pressure drops, and the cycle repeats. The time between two eruptions, we'll call T subscript V, the interval between eruptions. So how is an earthquake going to influence the time between eruptions or the occurrence of an eruption? What do earthquakes do? They stress the earth, right? They deform the earth, right? When an earthquake happens, the crust moves, the crust is stressed. It also creates seismic waves that pass through an area. They are also associated with stresses. They may change the pressure. By some amount here, I'll call delta P eq, the change in pressure associated with the earthquake. And let's imagine it's always this amount. If we've waited long enough for the pressure to be close enough to eruption, the extra pressure change from the earthquake, if it's sufficient, may initiate an eruption earlier than it would otherwise have occurred. So taking these concepts, then we can estimate what fraction of volcanic eruptions we might expect to be triggered by earthquakes. And so the arithmetic is on the board, right? The probability that a given earthquake will influence an eruption depends on the magnitude of the pressure changes generated by the earthquake, delta P eq, compared to the pressure changes needed for the eruption to occur. And the equality given on the board is based on assuming pressure increases linearly with respect to time between eruptions. We know the probability that an earthquake of sufficient magnitude will occur. It's went over the frequency of that size earthquake. So we can then compute the fraction of eruptions that will be triggered by the earthquake by multiplying those two probabilities and multiplying by the time between volcanic eruptions, TV on the board. OK, so now we have to put in real numbers to assess whether we expect how many volcanic eruptions will be triggered by earthquakes. OK, typical stress changes from earthquakes that we've been considering are quite small, say 0.01 to 0.1 megapascals. So that's 0.1 to 1 atmosphere. These are the static stress changes. I'll define static stress change in a minute if it's a new concept associated with big earthquakes. The critical stress I'm going to assume is 10 megapascals. Why 10 megapascals? That's a typical 10 cell strength for rocks. OK, we'll assume a recurrence interval for eruptions between 1 and 100 years. Big earthquakes happen every 100 to 1,000 years or so, in which case we expect a modest fraction of volcanic eruptions to be triggered by earthquakes. From the analysis I showed you on the previous slide, it turns out to be about 0.1 in the catalog. So not wildly different. OK, so given what Paul explained this morning, can you think about how you can increase the number of eruptions triggered by earthquakes based on these kinds of arguments here? Just looking at statistics. What have I left out that may matter? When Paul showed you how pressure evolves in a magma chamber, did it look like this? It was definitely going up, but it was it going up linearly? It was an exponential function, right? So it looked like this. And so for these ones, I can draw it. So you more rapidly approach failure. And so the small stress changes from an earthquake, in fact, would be more likely to trigger an eruption than we were assuming a linear, and we would obtain assuming a linear increase in pressure. OK, so the point of this going through this exercise was to see whether the observations on this slide here that a small number of volcanic eruptions are triggered by earthquakes, is it reasonable, is it something we expect? OK, so what I'm going to do now is take you through a variety of papers from the literature that have tried to establish relationships between earthquakes and eruptions. And I'll try and summarize what I got out of these papers. I know there's some of you in the room here who know as much or more about some of these papers than I do, so if I get something wrong, just jump in and fix things. OK, and I'll provide the references for all these figures and papers are on the slides. So if you look in the upper right of this figure, there's something called type 1 and type 2 volcanoes. Obviously these need to be defined. OK, so what's done in this analysis is to think more critically about the eruptions, the volcanoes that are erupting. So type 1 volcanoes are those that have not erupted within the past 50 years. Type 2 have had 1 to 4 eruptions within the past 50 years. Type 3 have had 5 to 14. And then type 4 are things that are continually erupting. So the point of categorizing things by type is simply to look at how long it's been between the last eruption. With the rationale being the longer it's been since an eruption, the more likely you are to trigger an eruption. OK, in the analysis here, then they do something called declustering the eruptions. If you're a seismologist, you know how to decluster earthquakes. You try and get rid of aftershocks. So what they're doing here is taking out any eruptions that occur shortly after preceding eruptions, the argument being that maybe it's really one eruption with just multiple episodes. And last, they removed from the catalog unconfirmed eruptions. I'm not quite sure what unconfirmed eruptions are. I guess this means things that are questionable. And that's very difficult to do. But perhaps I should point out, as an aside, one of the most famous examples of an eruption triggered by an earthquake was reported by Charles Darwin in his book based on, I think it's probably, third-hand information. And the date was wrong. And so the paper he published was attributed to an earthquake, whereas, in fact, it happened before the earthquake. So what's shown in red then are volcanoes in this category, type 1 and 2. Volcanoes that do not erupt too often, that erupt within three years of earthquakes. And there's a fair number. And so this relationship between eruption space and time can be looked at in a little bit more detail. So remember earlier I showed you this figure looking at the time between eruption and the time of earthquakes. And so this can now be separated into widths of that time window you look at. Do the eruptions occur within a day? And that's called within three days, or within a month. And then you can also separate the distance between the earthquake and the eruption by how far away you're willing to consider eruptions. And so the first panel are distances within 250 kilometers. The next 250 kilometers. And the next 250 kilometers. So from this compilation, we can get a sense of the space-time relationships of eruptions in response to earthquakes. OK. So given this, what's your conclusion? Yeah? The epicentral distance. I think these are always epicentral distance. And these are all really big earthquakes. And so the distance to areas on the fault that slip could be substantially different from the epicentral distance, especially if there's been unilateral propagation of slip, which is the case for some of these big earthquakes. And there's a reason, actually, why epicentral distance is often used in many of these analyses, like the one I showed you earlier. Because as you go back further in time, we simply don't have the information about what's slipped in the slip distribution. Excellent question, though. Please interrupt with any questions at any time. This is kind of a funny little subject and topic. But our hope is by the end of the day we're going to learn something new about volcanic processes and volcano physics. OK, so would you conclude from the compilation of correlations between eruptions and earthquakes? Yeah, maybe. But maybe it's not a good answer to what did you learn. It's always an honest answer. Maybe I learned something. Well, I see some of the pattern. There's certainly a distance relationship. As we go further away from the epicenter, we're less likely to see eruptions, right? What do I mean by that? When we see an increase in eruptions in response to earthquakes, higher peaks mean we're seeing more eruptions. So there's a spatial relationship. As we go further away from the earthquake, not surprisingly, we see fewer eruptions in response. And it's interesting, too, that most of the eruptions appear to occur quite quickly within a day. And if we average it over a longer window of a month, of course, that peak comes down because most of the triggered eruptions are happening very quickly. So following on this analysis, then, they can look at spatial relationships globally between where they see the best correlation between earthquakes and eruptions. So obviously, a map of the earth and volcanoes are shown with the triangles. Earthquakes that they analyzed in the study are shown with the green circles. And maybe the first thing you will notice, and you shouldn't be aware of, of course, is earthquakes and volcanoes are spatially correlated because they're governed by plate tectonics. OK. And then the colors that you see here show how well correlated eruptions are with the occurrence of earthquakes. And remember, a very small number of eruptions occur in response to earthquakes. And so a high correlation here really just means that there are some examples of eruptions in response to earthquakes. Yes, I think there's probably on the order of 10 pairs. So they're not shown here. And I can't remember if they have a figure that shows them in their paper. There's only a list. OK, but anything in rather places where there's a better correlation, anything in green, there's a weaker correlation. All I got out of this global assessment of this relationship is that ocean islands are the least sensitive to earthquakes. And it's not because there aren't big earthquakes, because that's accounted for in this analysis. What's the difference between ocean islands and the other types of volcanoes in this map? Yeah, it's compositional. They tend to be effusive. Actually, Hawaii has lots of explosive eruptions, too, from magma water interaction. But in general, it seems like these basaltic systems are less sensitive to earthquakes than the more evolved and more solistic systems. OK, so again, I'm taking you through what's been done in the literature to look at this correlation. So we'll now look at South American more detail, a study by Sebastian Watt. And Chile has the advantage for this analysis of having lots of earthquakes and lots of volcanoes. Let's start with the far right. It's a plot of the total number of eruptions over time. We thought about this issue earlier. Why do we care about the number of eruptions over time? Right, we're interested whether we have a complete record of earthquakes and eruptions so that we have a biased record. And notice that for some reason, around 1900, there are more eruptions. Obviously, that's just because the documentation is much better. And so they focus their analysis on that time window when they assume that all the eruptions that have occurred have been documented. OK, so now for the figures on the left. On the vertical axis are the number of eruptions per year over time, over the last 150 years. The time of big earthquakes are shown by the vertical red bars. And the vertical height or the extent of those bars shows the length of the ruptured fault. Which of those earthquakes is the biggest? 1960, which I think is the biggest recorded earthquake. And is there a relationship between eruptions and earthquakes? I'm asking this question because that's what they're trying to answer too. You can answer it just as well as they can. Not so bad, right? After 1960, there's an increase in eruptions, more eruptions after the 1906 earthquake. There's certainly other time intervals, though, where there are more than a few eruptions per year. Unfortunately, this paper was published before the Malay earthquake, magnitude 8.8 in 2010. And as far as I know, there was not an increase in eruptions since then either. OK, so they take the analysis in the lower left. They simply look at the number of eruptions in a three-year window in case there's a longer time period that it takes for eruptions to occur. And again, you do see an increase in eruptions after these earthquakes. Maybe for 1960, maybe. And plot on the lower right is going back in time when the record of eruptions was not complete. Yeah, I don't know about the question, is there an increase before? The peak is definitely after the earthquake. The peak is after the earthquake before. Yeah, in 1906, I'm not sure. This is the data we have to work with. So those were the initiation of new eruptions. It turns out that already erupting systems seem to be much more sensitive to earthquakes. And so I'll take you through what we know about already erupting systems. And it's good to evaluate this data critically, because the conclusions that are drawn in these papers are based on exactly the data that you're seeing. OK, so this is some responses to two volcanic systems in Indonesia to the 2006 magnitude 6.3 Yogyakarta earthquake, which one of the systems is Marapi that we've heard about earlier today. And the next is Semaru. Marapi on the left is 50 kilometers away from the epicenter. Semaru on the right is 260 kilometers away. Plotted on the vertical axis, measured by satellite, is the thermal emissions measured by a spectral radiometer. And I don't know details of this, some of you may know in more detail what's being measured. But when these little points, their point measurements in time as the satellite passes over the site, when they're high, it means there's more thermal emissions from that volcanic system. So the vertical line with the stars at the time of the earthquake, it'd be nice to, I should have covered those up, and then covered up the horizontal lines and asked you when the earthquake happened. There's certainly a lot of scatter. But the argument in this paper is that there is an increase in activity at Marapi and Semaru. At Marapi, they attribute it to an increase in the generation of small pyroclastic density currents from the dome collapse on the right to an increase in gas emissions. Any comments on this data? Yeah, it's beautiful. I mean, it's tough to get data. I think Paul made this point earlier. And we have very little data from volcanoes. Here it has to be collected from space. But we'll keep going with this measurements from space, because in a subsequent paper, Deladona and colleagues looked at the global response of already erupting systems to earthquakes. So now we're looking at a cumulative amount of energy released by earthquakes, as well as a cumulative thermal emission from already erupting volcanic systems. So horizontal axis is timed. The gray curve shows the cumulative moment release from global earthquakes. And so every time you see a vertical increase here, those correspond to the largest earthquakes. And this time series ends, I guess, in 2008. But here we can see the Sumatra earthquake. We don't have Tohoku on this plot. The black curve is the total thermal emission, as measured from satellites at these volcanoes. And you can see that after these increases in moment release, we see an increase in general in thermal emissions. But it's not directly coincident. For example, here in 2003, it lags a little bit and takes some time to develop. There are time periods, for example, in Japan, where you see almost no increase in thermal emissions. The biggest earthquakes we'll see in the next very busy plot that's coming up seem to have global impacts. And there are also time periods, for example, here around 2006, where we see global increases in thermal emission without any big earthquakes globally. So it's a complicated record. But I think this is pretty compelling. Big earthquakes happen globally at already erupting systems. We see an increase in thermal emission. This could be fumaroles. It could be hydrothermal systems. OK, so this is the busy, complicated figure I was referring to. I'll take you through three of these panels, because it's a summary of what we know about already erupting systems. Let's start in the upper right. On the vertical axis is the distance between the earthquake's epicenter and the volcano. The horizontal axis is the earthquake magnitude. The little pluses that you see illustrated here are volcanoes that erupt in response to earthquakes. And those are volcanoes that were not already erupting. So if you remember those histograms you saw several slides ago, those are the earthquakes that are plotted here. The diamonds are already erupting earthquakes, already erupting volcanoes. And I think it's obvious what you see, right? For a given earthquake magnitude, we'll pick eight. You can see that you can go to much bigger distances and see responses at already erupting systems. Figure in the upper left shows the proportion of already erupting volcanoes that respond to these global earthquakes, or these earthquakes as a function of their magnitude. The top here, 10 to the 2, means 100%. So that means all of these volcanoes globally respond to earthquakes. And it seems like when earthquake magnitudes reach about magnitude 9, all volcanoes globally respond to the earthquake. OK, and then the last product that we'll look at is the one in the lower left, which is the duration over which they see these thermal anomalies as a function of the earthquake magnitude. And not only when you get to bigger earthquake magnitudes, you see more volcanoes respond. The duration of the response seems to be longer as well. So a couple caveats here. Since those papers were published, we've had two very large earthquakes. And the response of a number of volcanoes have been imaged or monitored by space with INSAR. And I'll show you two figures from two of these papers. This is from Pritchard and co-authors looking at Chile. The location of the Malay earthquake, the hot colors on the left show the slip on the ruptured fault. On the right are the INSAR images. And the bluish and purple colors show places where the surface subsides at the volcanoes. So rather than seeing evidence for renewed activity, in fact, it looks like the ground is moving down. Here they attribute it to degassing the loss of vapor or volatiles from the volcanic system. Not two different observations were made after the Tohoku earthquake in Japan. So the patch that ruptured here in pink, the location of the volcanoes off to the left, with the color indicating the surface displacement. Again, the ground goes down. In this paper, they argue there are almost no fumaroles at the systems. And so they attribute the subsidence to the hydrothermal system losing fluids, so not the magma directly. So one more topic I'd like to address about the observations themselves. And let me check the time is the argument behind why earthquakes influence volcanoes is that earthquakes create stresses. And those stresses influence eruptions. Volcanic eruptions also create stresses. Paul quantified the magnitude for us earlier. And so you might expect volcanoes to influence other volcanoes. So this is a similar plot to what you saw earlier. It's a time between one volcano erupts and another volcano erupts. All the same caveats from before fall here. We ensure that the eruption catalog is complete. Not the most beautiful plot. You can't see the big peak here at time interval zero to five days. But volcanic eruptions seem to be paired. When one volcano erupts, you're more likely to see another volcano erupt within 800 kilometers. Now a much busier plot looking at this for very specific eruption pairs from nature geoscience earlier this year. Let me explain the axes. And we'll go through some of what's plotted here. Horizontal axis is the change in volume of the magmatic system or the volume of the magma that erupts. The vertical axis is the distance between the volcano pairs. There are names associated with these volcanic pairs. That's Nova, Repta, and Katmai in Alaska, for example. And there's a convenient map at the top to help you identify where those volcanoes are. So now we have colors to understand. Red are volcanoes that erupt in response. I didn't write this paper, and so I'm hopefully getting it right. Eruptions in response to other eruptions. So let's see. Nova, Repta responding to Katmai. Then we have with circles places where there's unrest. And then we see an eruption at a different volcano. But those would be red circles. Blue are cases where it's thought that there's a physical lateral connection in the magmatic system directly. And then in green are examples of volcanoes close to each other where we don't see a response. And I should point out in passing that when we're trying to assess this question, I think it's often underappreciated the value in looking at the cases where we don't see a response, as opposed to just looking at responses. And so last, everything in gray are places where given the volume and the distance from what Paul showed us, we can calculate the stress from that volume change. And everything in gray are places where the stress changes are bigger than 0.1 megapascals. Everything under this yellow line represent length scales that are comparable to a typical magmatic system. They call it a mush here. It's basically the partially molten region associated with the volcano. So I think I covered everything that's there. And then in blue, they assume that all the deformation is accommodated by opening up something like a dike. And the blue area corresponds to openings that are big enough to allow dikes to propagate and not freeze. So Paul brought this up earlier this morning. There are thermal constraints on dike propagation in addition to the stresses. And so the argument here, well, first you'll notice as you get to bigger eruptions, there are bigger distances over which you see responses. In the same magmatic system, it's maybe not surprising you can see multiple eruptions. And last, the places where we see physical connections, it's plausible to attribute them to there being a physical opening between the magmatic system. Good. It's coming up in two slides. In two slides, I'll give you some real numbers. So let me point out a couple other complications here that it could be in some of these systems, the triggering is indirect. That an earthquake happens, it might create a landslide, for example. The landslide creates big stress changes on the underlying volcanic system, and we see eruptions. So I'll show you one example from Mike Cassidy. So we're looking at the island of Moserat in the Caribbean. And Moserat and many of these volcanic islands create large landslides, flank collapses or debris avalanches. And one of them is outlined in yellow on the right. And there was an IODP cruise to look, drill through that debris avalanche deposit to understand the timing of both the eruptions and these landslides. I will skip all the going, taking you through the stratigraphy in the interest of time. But it turns out after the big debris avalanche, there was a pulse of volcanism. It was mostly basaltic volcanism. Some of you may know Moserat, and mostly what erupts there now is andesites. And these basaltic products of the eruption came up right after the landslide. And then the magmatic system evolved to the present day andesites. And so it could be that these are indirect connections that the unloading from the landslide initiated the eruption. So just a couple of reminders of what to think about when we're trying to answer this question, do earthquakes trigger eruptions? It's good to think about the completeness, whether there might be other biases in your record because earthquakes happen, so you're just more, you pay more better attention to what's happening around you. And I put this in red because this is much harder to assess just by looking at a catalog. Okay, so to answer the question, what are the magnitude of the stresses we're dealing with because so far I've tried to establish what happened. We're going to spend the next 15 minutes or so trying to understand why, right? What is the process by which an earthquake influences eruptions? Okay, and it's going to be, of course, through the stress changes that the earthquakes produce. So listed here are the magnitude of stress changes from a variety of different processes that operate in the earth. Okay, hopefully this table's not too confusing. Let's just look at the first four lines. Okay, so stresses from solid earth tides, generated by the sun and the moon. We've got the loading from the ocean tides, changes in precipitation and climate, make water levels go up and down, glaciers come and go. And just on the right-hand side, for reference is the time scale over which we see these changes. Let me note that, which of these stress changes is the biggest? Glacier loading by far, right? Glaciers are big and they're thick. And in fact, we do know that the loss of mass from the retreat of glaciers is associated with volcanism. We see this globally, we see it in Iceland, we see it at Stratovolcanoes. And so I made this little note that deglaciation is in fact accompanied by eruptions. Okay, so now for our case, we're interested in the bottom two rows. Let me also define for you the two types of stress that I'll refer to, static stress changes and dynamic stress changes. Paul talked about the static stress changes earlier. A magma chamber expands, or an earthquake happens, the crust deforms, and I'll call that a static stress change. When seismic waves that are generated by the earthquake pass through a region, the crust is expanding and contracting and deforming. Those also generate stresses, but they're temporary short-lived stresses, right? They're only there, of course, so long as the seismic wave is passing through the region. And so I'll refer to those hereafter as dynamic stresses. Okay, so now this table gets more complicated. We have two different, we have a magnitude eight earthquake. We're going to look at the stresses at three different distances, 100, 1000, and globally. Static stress changes are small, right? Always less than 1.1 megapascal. Stresses from the passage of seismic waves are much bigger, so keep that in mind. So very briefly, a couple ideas about how static stress changes might influence eruptions, and they're listed here on the top. Maybe in the interest of time, I won't read them, but we'll start by the figure in the lower right. So we have a subduction zone earthquake. We have slip on the thrust here. Illustrated in red are the volumetric strains, which will produce stresses associated with that slip. Volcanoes are located in the arc, and typically in the location where the volcanoes are located, the crust expands, and so any process that can take advantage of expansion of the crust to initiate an eruption could be responsible for triggered eruptions. And so some of these processes are listed here, unclamping dikes, maybe making new dikes. Pressure goes down, so you might make bubbles, for example. It's worth keeping in mind though, the magnitude of these stress changes were very small, going back to the previous slide. They were less than 0.1 megapascals, and maybe I'll do that now. 0.1 megapascals is comparable to the stresses from things like weather and tide. So it's for this reason that many people have also explored the possibility that somehow the passage of seismic waves, the dynamic stresses also can trigger eruptions because the magnitude of the stresses are much bigger. The challenge here, however, is that those stresses are not permanent. They're only there so long as the seismic waves are passing through the region. So what you really need to do then, if you want to attribute an eruption to seismic waves, is you need to figure out a mechanism by which you can take the passage of seismic waves, the time-dependent strains, and make something permanent. So I list here for you five different proposals in the literature, going from some of the earliest ideas to the most recent. And let me explain the first two very quickly on the blackboard, because the first one came up yesterday in our discussions. So here's our volcano, and we have a bubble in it, located at some depth H below the surface. The magma has density rho. What's the pressure inside that bubble? The pressure should be approximately rho G H, right? We're neglecting surface tension, so. This idea was originally proposed by, we're doing number two, by Steinberg and rediscovered by Sahagin and Prusovich. So here's the idea. Seismic waves pass through the area. For some reason, this bubble is kind of stuck, and the bubble is liberated, and it moves, and it goes up. So now you ask the question, what is the pressure change? So we're going to, the bubble radius is A. We're going to let the bubble go all the way to the top with some radius A. The argument in both of these papers is that if the radius does not change, and gas doesn't go from the bubble into the surroundings or vice versa, the pressure inside here has to be the same. So we're effectively taking high pressure rho G H, bringing up to the surface, and so we've changed the pressure by an amount rho G H. So if you can take a bubble over a kilometer, that's a huge pressure change. Good question, okay, so let's make a list of the key assumptions that have to be satisfied for this to work. Right, so radiuses assume not to change, and what does that require? Yes, short time scales and... Right, so the bubble's going up so fast, gas is not diffusing into the bubble or out of the bubble, and there's something mechanical that also has to happen. The liquid has to be incompressible. It can't change volume for the bubble not to expand as the pressure changes. And not only does the liquid have to be incompressible, what else has to be incompressible? The rock surrounding the magma, too. And those turn out to be not the best approximations. And so if you wanna look through the analysis of this, I mentioned the two papers right here, and you can read them. But this is one of the earliest ideas about how you might trigger eruptions with bubbles. I should point out that almost everything here appeals to bubbles through one process or another. And part of the rationale for invoking a mechanism that involves bubbles is we know the bubble's drive volcanic eruptions, right? So if that's what causes eruptions, you might as well explore mechanisms that involve what causes eruptions. Let me explain qualitatively rectified diffusion. We have a bubble sitting here, gases are diffusing into the bubble. There's a thin region around the bubble where concentration gradients are the largest. We're solving a diffusion equation. Passage of seismic waves oscillates pressure. The bubble's expanding and contracting. That's stretching and thinning the boundary layer over which diffusion is happening. When the boundary layer gets stretched to be thinner, diffusion can happen faster. And because boundary layer thicknesses are non-linear, you end up with a net input of gas during this process. And this is the process called rectified diffusion. The most recent analyses that have tried to actually quantify this process have shown it's not so important. Maybe that we're nucleating brand new bubbles. The passage of seismic waves are oscillating pressure. And it takes a certain over pressure or super saturation to nucleate bubbles as well. And it's been demonstrated experimentally that the passage of seismic waves can nucleate bubbles. And I guess the most recent paper that I'm aware of that I didn't write down here is Jackson Cruz and co-authors in JGR 2015. Another possibility is that somehow you mechanically disrupt crystal rich magmas. And so something that was strong, sometimes people use the word rheologically locked, meaning the rheology is such that it behaves like a solid, becomes mobilized and fluid and behaves more like a fluid, and then it can start moving and ascending. And the last idea is that there's a resonance that occurs in the magma bodies underground. And the passage of seismic waves, if it can excite this resonance, causes the magma to oscillate back and forth or slosh, that can allow it, through the motion, a variety of processes that ultimately culminate in the eruption. And so I went through that list quickly. I'm going to try and decide what to show you. I think I'll just show you two pictures. So one example of this idea that you might initiate convection is a kin or analogous to the process of liquefaction. So let me describe qualitatively what liquefaction is. It's something that happens in loose, unconsolidated materials like soils if you shake them. And you've experienced this actually when you've been to the beach. If you stand on wet sand and you shake your toes or wiggle your feet, you know that you can liquefy the sand. Instead of being solid, it behaves like a liquid and your feet sink into the sand. And if you haven't experienced this, you've had an unsatisfying life. And earthquakes do the same thing, right? Earthquakes shake up loose, wet material. And physically what's happening is you're taking the sand particles, right? You're moving them into a tighter packing. And that decreases the volume. And there's a liquid there, right? If you're decreasing the volume and there's a liquid there, the pressure in the liquid has to go up. And that pressure may be enough to support the weight of all the particles. And so you go from some of the weight of supported by the solids touching each other to being supported by the fluid pressure. And it behaves in a liquid-like manner. And this is how buildings collapse and all kinds of damage happens after earthquakes. There's absolutely no reason why this should not happen in magma, if you deform them. And the slide I skipped over was some experimental measurements at conditions similar to what would happen in a magma. The last idea in that slide was a paper that our host was involved with by Namiki and co-authors where they did some lab experiments with model magma chambers. So there's some green magma and bubbly magma at the top to make it oscillate. And look at the types of response to the system. And if the frequency of the oscillation is similar to the resonance frequency of the system, it can start to combat. And these kinds of processes that operate in magma chambers, as you'll learn about in more detail tomorrow from Claude-Joe Powell, can culminate an eruption. So let me end by just showing you a few other examples of systems on the earth that respond to earthquakes. Because these may also provide additional insight into what happens. One of the most sensitive systems in the earthquake seems to be geysers. Geysers are places where water erupts at the boiling temperature. Here are three geysers. They correspond to the names here, except for pink geyser. On the vertical axis is the time between two eruptions, called the integral between eruptions. So you just count when one eruption starts, wait for the time of the next eruption. The time of the magnitude 7.9 Denali earthquake in Alaska, 3,100 kilometers away from Yellowstone National Park is indicated by the vertical line. And what happens after the earthquake? Well, two of these geysers erupt with a shorter interval between their eruptions. One of these geysers does nothing. This particular geyser lone pine increases its eruption interval. I'll skip this, except to point out that earthquakes trigger other earthquakes. Mud volcanoes, we haven't talked about mud volcanoes, but they're like magnetic volcanoes in that a mixture of gas, liquid, and solids erupts to the surface. Big mud volcanoes can erupt material from depths of several kilometers. They create edifices at the surface that kind of look like volcanoes. They can even have flames from methane that drives the eruption burning at the surface. In fact, we have more examples of mud volcanoes erupting in response to earthquakes than we do magnetic volcanoes. And so second last slide, so there's time for questions. What I've done here to end is compiled all these other systems that have changed in response to earthquakes. You have actually seen this figure before the paper by Dela Dona and co-authors, where on the horizontal axis, we're plotting the magnitude of earthquakes. On the vertical axis, we're plotting the distance between the epicenter of the earthquake and the response that we see. And so the magnetic volcanoes we've been talking about are these red circles. And what do we see? As we go to bigger earthquakes, we see magnetic volcanoes erupting in response to the earthquakes at ever-increasing distances. And just for reference, a variety of other responses to earthquakes are plotted here. The occurrence of liquefaction, we're soils liquefying green. In blue are rivers that increase their flow after the earthquake. Let's see, there's the geyser that we just saw in the previous slide. In blue squares are the water level and wells changing significantly after the earthquake. For all these responses to earthquakes that we see, we do see that as we get to bigger earthquakes, we see an increasing distance. What's intriguing about this set of observations to me anyways is that mud volcanoes in yellow, magnetic volcanoes in red, changes in stream flow in green, and blue circles, and the occurrence of liquefaction in green, they all share roughly the same maximum distance over which people see these responses. That doesn't mean the process has to be the same. But nevertheless, it remains intriguing that all these very different types of earth systems seem to be equally sensitive to earthquakes. So I'm going to end with reminding you why we bothered with this topic in a week talking about volcano physics. And the point of looking at the relationship between earthquakes and volcanoes that may or may not erupt in response is we might get some new insights into some of the processes that initiate eruptions, or at least determine some of the critical thresholds that lead to eruption. A couple of things to always think about with this topic that we've explored today. First, are the relationships that we're establishing from observations real? Are there systematic biases that we're not accounting for? And the last question, I think, is still an open question. What are the mechanism? What is the mechanism, or the mechanisms that lead to these responses? Thanks so much. Time for questions, I think.