 Thank you for coming back from lunch on time This afternoon both in the lecture and then the lab exercise we will do later We will look at how magma ascends from places where it's stored up through volcanic conduits to the surface. I will rephrase this Question here as something that's hopefully a little bit more compelling or interesting The kind of the question we would like to answer today is why is it that volcanoes erupt in so many different styles? One specific example is shown here in 1980 Mount St. Helens erupted explosively right to create this thing We call a plenian eruption column a mixture of ash and particles that go high up into the atmosphere 14 years late 24 years later in 2004 this through the same vent presumably magma continued to erupt But instead it created this feature in the middle. We call a lava dome and that erupt eruption lasted for about four years Essentially the same magma, but erupting in two very different styles And so this is the question we'd like to address and understand How does Matt? Why does magma erupt in so many different ways? And of course to answer this question we need to understand the processes that govern the ascent and ultimate eruption of magma What I'll be showing you today is an is outlined in a review paper published a couple years ago So I will find some way to make this available and I've added a few other items In addition to what's in the review paper So we'll begin with a textbook picture of how it is that magma erupts how volcanoes work This is a cross-section through an idealized Volcano starting with your spherical magma chamber down here at the bottom called a magma reservoir and this magma Right or molten rock contains dissolved gases mostly water, but also co2 and some other gases If for some reason this magma starts to Ascend the pressure decreases the solubility of these gases and other gases decreases With decreasing pressure So as this magma rises at some depth that becomes the liquid becomes saturated with respect to these gases and at some depth called the Exolution surface here's bubbles nucleate and gas starts going from the liquid into the bubbles If this magma continues to rise more gas goes into the bubbles the gas in the bubbles can expand and the total volume Fraction of gas continues to increase Until some point is reached here term fragmentation and this is an important event in the life of a volcano Because there's a change in topology of the magma we go from a magma with a continuous phase of liquid continuous liquid phase To now having discrete pieces of potentially bubbly magma surrounded by a continuous gas phase and the rheology or the viscosity and the behavior of those Those two mixtures either gas and liquid or liquid suspended in gas are fundamentally different And presumably and it sees fragmented bits of magma of course that go into the exposed the products of the explosive eruptions And this generally assume that when we have a fuse of eruptions so a fuse of means non-explosive The differences that this fragmentation process doesn't happen or it must happen in a fundamentally different way Let me remind you once again interrupt and ask questions at any time on any topic from terminology of course to the equations or anything else Yeah, that's right So within I think exsolution is a chemical reaction, but I'm not sure Yeah, it must be a chemical reaction because there's a change in bulk chemistry But you can also get gas from outside the magma, right? Through water magma interactions But for what we normally call magmatic eruptions that are driven by gases dissolved in the liquid This is the dominant way in which you get bubbles is changes in solubility For the most part today, I'll try not to give you too many real numbers Some of you know all these numbers if you don't this is not the time to learn the range of magma viscoses and compositions Only when necessary. Well, I try and give you real numbers Okay, so to understand why does that volcanoes only sometimes erupt explosively why we see a range of eruption styles Really, we need to understand two questions When where and how does this fragmentation process occur and how does the interaction? How does the interaction between all the different processes you see illustrated schematically here? Interact with each other to explain the diversity of the eruption styles that we see So I'll begin by outlining what we'll see are the three key processes that govern the ascent of magma and eruption and Then we'll do this qualitatively first and then we'll look at these processes in a little bit more quantitative detail And then we'll try and put them all together So again what makes magma rise up to the surface is the presence of dissolved gases in the liquid that form bubbles those bubbles grow Virtually all volcanic rocks we see contain vesicles That's what we get the geobalkan illogical term for bubbles and there even styles of volcanic eruption They're probably caused by the eruption of single large bubbles at the surface Okay, so what causes bubbles to grow or two reasons first as pressure decreases the solubility of gases decreases And when you're saturated with respect to a gas you can nucleate a bubble and then once you've nucleated a bubble Gases diffuse through the liquid into the bubble The second reason these bubbles will grow is that as the magma's rising pressures decreasing and gases expand in response to a change a decrease in pressure Now presumably to keep magma's from erupting Explosively since all magma's at depth contain dissolved gases You need to find some way to get rid of those bubbles to get rid of the gases in the magma as it rises towards the surface So here after I'm going to refer to this process of losing gas as outgassing and by this we mean gas is going from the Magma into the surroundings these gases may work the way out the top of the volcano or maybe out to the side Like when I say wet here this means magma with lots of dissolved water when I say dry it means magma with less dissolved water But we know that this process of outgassing must happen because we can see magma's erupted at the surface with no dissolved water Very few bubbles, but we know for geochemical and petrological reasons. They once did have lots of water dissolved in them We can measure the initial dissolved water content by looking at little inclusions of melt or looking at the phase assemblage But magma's that erupt do start wet and some of them do erupt dry So one way in which of course gases can escape from this rising magma is that once you start making bubbles Right here's some bubbles in an SEM image with a hundred micron scale There are connections between those bubbles and that will allow gases to flow through the magma Potentially escape from the magma as it rises to the surface Okay, and the third key process is the process of fragmentation Fragmentation refers to the breakage of magma into pieces What causes fragmentation? There's there many What causes fragmentation in some ways is a little bit controversial and in part? This is because there are many different processes that can cause magma to fragment and so I'm going to talk you through two of these Okay, so remember we have bubbles that nucleate and grow inside our magma So here's our bubble in the middle surrounded by melt around it and as this bubble surrounded by melt rises towards the surface The pressure in the melt is decreasing because you're going to shallower depths The question is do we need a liquid phase to nucleate and grow bubbles? And I think the answer is yes. I don't think bubbles will form in a solid I can't imagine how because to make a bubble you have to displace This the other material out of which you're exalting So I think the answer is yes You need to have a liquid we will see later that the properties of that liquid right here in blue Matter a lot and in fact if it behaves elastically then this is when we get magma or magma to fragment So in fact what we mean by liquid Fluid and soluble vary today Yeah, certainly in the conduit the pressures are different and so the physical The question is in the conduit the liquid and solid are not what we know them at the surface So it's possible to fragment or break solids right you can take a solid piece of material and shatter it But to nucleate bubbles and have the bubbles grow I think you do have to have a liquid that behaves in a fluid like manner Okay, so I maybe this is a good time since it's coming up What do we mean by the expression liquid fluid and solid? Right so a liquid and a solid will distinguish based on a state of matter, right this the degree of order So a liquid however can behave actually I'll save this for the next slide liquids We will see can behave in a brittle manner, but they can also flow in a fluid like manner And so that's a property called rheology. How how a material deforms in response to applied stresses and Liquids can respond in very different ways to applied stresses depending on time scales and their state Okay, so let me try and explain the two different ways in which our magma Containing bubbles can fragment again our magma is rising so the pressure in the liquid is decreasing The pressure inside that bubble I call P in is controlled by the side the amount of molecules in that bubble and its size So as the pressure decreases the gas inside the bubble is going to expand and to do that It has to stretch that blue film of liquid around it to stretch that blue film of liquid You have to physically deform it if the viscosity of the surrounding liquid is high And the magma is rising fast the liquid may not be able to deform fast enough To accommodate the change in pressure in which case the pressure inside the bubble can become bigger than the pressure outside Potentially significantly so and if that pressure difference exceeds some critical value that film of liquid can rupture. Yeah okay, so the question is as Pressures decreasing our bubbles are expanding but as pressure decreases solubility decreases and you can nucleate new bubbles and The nucleation of new bubbles will allow gas to diffuse into the bubbles and that should decrease the over pressure actually So it would do the opposite okay, but the key is right if if the film is if the melt is not Low enough viscosity The bubble may not be able to expand fast enough to accommodate changes in pressure the pressure inside the bubble can become large and The surrounding can rupture in a way. This is not too different from what happens in a balloon right if you blow gas into a balloon you build up stresses in the Rubber the film and if those stresses get too big the film can rupture Okay, the second way in which magma or liquids can break is if you try and deform them too quickly If you have a liquid it's made up of molecules and as you try and deform it those molecules have to move around each other To accommodate the applied deformation this the stresses in order to deform And if you try and deform it too fast the molecules can't Can't rearrange themselves or can't deform fast enough and instead of flowing like a fluid it will become brittle and it will break And so I illustrate this process here Schematically where we're plotting the deformation rate that's applied to our liquid as a function of the temperature of the liquid Okay, and let's see. I think I had animate this Okay, so let's initially deform it at some strain rate at this given temperature that's low Because the deformation rate is low the molecules can rearrange and our material is going to behave in a fluid like manner Now as the magma is rising towards the surface the bubbles are expanding the fluid is stretching faster It's deforming faster and so we can imagine as we rise towards the surface Temperature is not changing too much But the deformation rate is increasing we now increase the deformation rate to the point that the structure of the fluid Or the liquid can't adjust and our liquid becomes brittle and it breaks Ah Good question. So why would deformation rate increase as magma is rising to the surface? Let's take a conduit of a fixed radius So at the bottom we just have liquid Right and as we rise we make bubbles the bubbles expand the density goes down So the speed you have to move goes up just because you're making bubbles in the groin You will do a lab exercise a very simple one the set I shouldn't say simple Hopefully straightforward exercise this afternoon where you will solve the equations we'll see next and you will see that the velocity increases As magma rises to the surface Okay, so sometimes people call this the glass transition where you go from behaving in a liquid like manner fluid like manner To becoming brittle. Okay, the last variable that's plotted here is there's temperature on this axis as Temperature goes up. What happens to the viscosity of most liquids or fluids? It usually decreases and the lower the viscosity the easier it is to deform So the faster you have to deform it to pass through this thing called the glass transition So as temperature goes up you can deform it faster Okay, and people the actual time scale over which you Transition between liquid like behavior and brittle behavior scales like the viscosity of the liquid eta in the numerator Divided by the shear modulus g. So it's a ratio of a viscosity that characterizes a fluid behavior To an elastic modulus that characterizes an elastic behavior. So Let me ask you a question now, right? We've got magma rising through a volcanic conduit pictured on the left I'm only showing you half the conduit So on the left-hand side is the edge of the conduit in the middle is the On the right is the middle right and I haven't told you anything about boundary conditions for fluid flow But one of the boundary conditions is that if you have a solid surface the velocity of the fluid goes to zero Right the velocity of the boundary Okay, so the question is then where in this conduit are the deformation rates going to be the biggest Near the boundary right deformation rate scale like velocity gradients Right in fact, there's zero at the middle. They're biggest at the side So this process where we deform things fast and they break. Where is this going to happen inside our volcano? Along the sides to where the deformation rates are the biggest So I noticed here I reminded myself I'm going to use a couple different words So we said there are two different ways we can break fragment or magma one if the pressure inside the bubbles gets too big I will continue to call that fragmentation But you can also deform things too quickly and then it will break and here I'm going to call this brecciation to make an analogy to the texture of the rock We'll see later It breaches the name we use for a rock that's made up of broken up angular fragments And that's what you do at the side of the conduit right if you deform it too fast, so it breaks And I just remembered I brought all the way from California a piece of silly putty to do a demonstration you may all have seen right silly putty is an example of a Complex material, but at very low deformation rates like what am I doing to it? It's stretching right in the fluid like manner. So if I go back a couple slides Right. I'm deforming it at a deformation rate similar to the red dot at the bottom right at stretching in a fluid like manner And I can do the same thing if I'm strong enough, right? I pulled it fast enough right maybe near the top of the arrow that it broke into two pieces So it becomes brittle but I can put touch those two pieces back together and I keep pulling and what happens You go back to behaving in a Liquid like manner a fluid manner, right? So this glass transition is something that in principle is reversible you can go back and forth brittle Fluid like brittle fluid like depending on the strain rate that's applied. Hey, so that was meant to be an oh, yes It's called a transition because you go from one type of behavior to another so it can be All the same type of molecule It's really there's a there's a time scale over which the stresses you apply can relax that scales like the viscosity compared to the elastic modulus Yes, there is a okay, so now you're getting to the concepts. They're more sophisticated But that's right. There is in fact it's plotted just as a single curve as you approach that glass transition In fact a variety of other properties change thermodynamic properties change The viscous properties change as well So I think by analogy to phase transitions It's it's not quite the same as say a liquid solid phase transition and that material properties very continuously leading up to it But I think for our purposes to understand how volcanoes do what they do We know the conditions under which this transition happens and at least Microscopically what the process is the liquid simply can adjust its microstructure in response to the stresses Okay, so that was meant to be an overview of those three processes, right? Which was nucleating bubbles the growth of bubbles getting rid of the gas and fragmentation But already I'd like to hint at what some of the interactions between those processes will be so you can get a Sense of where we're going to go with why we see a range of eruption styles Right, so you just answered that the biggest stresses and strain rates are at the sides of the conduit This is where the magma is going to break apart Once you break apart the magma you have a liquid now that has big fractures running through it This will make it much easier for gases that we're in those bubbles to escape from the rising magma Because instead of having to flow between little tiny bubbles the gases can go into the cracks and through the cracks escape into the surroundings and Once you get rid of those crack the cracks and the gases the magma can stick back together Just like your silly putty does right and then it can continue to rise up to the surface You make new bubbles as the pressure decreases, right? And so this process of breaking getting rid of gases sticking back together and breaking may happen repeatedly as the magma rises up to the surface But here what is the feedback between process? It's the nucleation of bubbles the breakage of magma as well as the outgassing process They're all coupled to each other Okay, any questions about These concepts yeah Yeah, so the orientation of the Extensional direction actually whoops gotta put use the right button won't be parallel to the conduit walls It'll be at roughly 45 degrees to the conduit walls. That's the direction of maximum extension But once the magma also fragments I and I use the word bretcher right now You have a bunch of things behaving solid that are broken up They're going to be jumbled around and tossed around I'll show you some pictures later of magma that's been fragmented in this way and you can see that there's a lot of reorientation But something that is important just because you have pathways in the magma for gases to which they can flow much fast To them that could before doesn't mean those gases can escape. They have to escape to something so To escape from the rising magma the surroundings have to be permeable enough to allow them to escape Or they have to find their way out the sides of the volcanic conduit and based on observations It seems that both ways of gas escape happen So let me just remind you right the three key processes were the nucleation growth of bubbles the loss of gases that allowed that and caused that magma to ascend and The fragmentation of magma of course to create the fragments that can erupt and we can also break up magma the sides of the volcanic conduit so what we need to do now is See how these different processes interact with each other to explain the kinds of eruption phenomena we see and In the community people use a variety of different approaches to address these questions People do lab experiments on real magnets to determine when they break and how they break in the rheology Right so we can study individual processes and properties experimentally or theoretically to put everything together Relies to a large extent on computer simulations because these processes interact And so this is what I'll focus on for much of the rest of today today, and you will do in your lab exercise But of course we also can test our models with measurements. We can make on rocks We don't have time to do this today our theme is physics of volcanoes of course, too But I will show you a few examples of measurements to show you a variety of ways people Used to try and integrate predictions of these models with measurements we can make Question so without giving you I don't want to distort your she can flop back and forth too much with the slides But the question is does magma composition play a role? You've already seen one place where magma composition matters right through the viscosity the viscosity affected when things break Viscosity will affect how bubbles grow so we will see a viscosity plays a huge role in eruption style Right because viscosity affects in fact all three of these processes we talked about Any other questions? Okay So I'm going to once again show you some numerical models and the results of numerical simulations There is in fact something similar to what we'll be doing now to what we looked at earlier this morning that We were arguing that many of the key processes occur at the length scale of individual bubbles And so on the other hand we would like to simulate everything going on in a large volcanic conduit And so we are going to use a hierarchy of numerical models or models Again we need to solve the equations for conservation of mass momentum and energy, but we're going to do this at two different scales Right so the scale of the magma rising through a volcanic conduit We're going to treat our bubbly magma as it's rising to the surface as being a Homogeneous flow of a certain type of material with some set of bulk properties that is our bulk magma That contains bubbles in liquid has a certain viscosity has a variety of other properties At the same time we care about those bubbles, right? Because of the pressure on the side the bubbles matters For some of these fragmentation processes So to determine properties of the bubbles We're going to simultaneously solve a second problem how bubbles nucleate how they grow and how those the growth of those Bubbles affects properties of the bubbles and the liquid So that is there's two scales of problems We need to solve the fluid mechanics problem of what a bubble is doing and the fluid mechanics problem of how that bubbly liquid Rises through the conduit and those two scales of problems are coupled to each other Through the pressure of the magma and the temperature of the magma Temper temperature affects viscosity pressure affects solubility for example So there's a feedback between what happens at the bubble scale on the large scale So let me tell you a little bit about the large scale first We're going to treat let me go through this list solve the equations for conservation of mass momentum and energy Again where our magma has a certain set of properties. So let me explain the first equation Maybe in the interest of time I will not derive this for you, but it's a it's very similar to what Paul showed you this morning. We have buoyancy forces here pressure gradients causing flow up the conduit and They're balanced by viscous stresses stresses that arise from the viscosity of the liquid right this equation actually is Pressure gradients balanced by the divergence of the stress tensor and it's been integrated once And that's where the factor of two comes from it comes the factor of two arises because we're solved We're assuming we have a cylindrical conduit Okay, and so the expression you see on the right hand side viscose times velocity gradients is the stress arising from viscosity Okay, so this is an assumption that that's all that matters in the in the momentum equation viscous stresses are balanced by the buoyancy forces and pressure Gradients that cause the flow to happen That's also equivalent to saying the flow is laminar or not turbulent So any motion associated with the turbulence can be neglected. Okay, let's see Second the properties of the magma will treat every the magma is having a certain set of properties or a certain viscosity So viscosity will be denoted with the symbol eta here and that viscosity if you look at this equation This is a very simple equation. It just says that variations in vertical velocity across the conduit are given by a pressure gradient Divided by a viscosity. So the real complexity in what we end up seeing all goes into that viscosity and The viscosity of magmas is unfortunately complicated. It depends on the rate at which it's deforming the temperature of the melt The amount of bubbles present in the liquid know by the symbol fee R is the size of bubbles and CW is the amount of water dissolved inside the liquid And if we had crystals present, they would add an additional level of complexity So we could have a volume fraction of bubbles a volume fraction of crystals And in the review paper that I referred you to actually there's a description of how you deal with both But I left out crystals for today's presentation just to keep it simple Sort of okay, so I will show you some simulations There's a point here. I call fragmentation something happens above that point, right? And so there is a simultaneously another model That's right, okay, so above here There's we treat this as a material with the different set of properties when Brecheation or things break apart at the conduit. We don't let we don't feed that into our model for viscosity Because I have no idea how to do that And does I don't to be fair? I don't know if I mean we're being recorded. So this is always a dangerous thing to say I don't know if anyone really knows how you deal with a Brecheated material that has passed through the glass transition So when the stress is applied it goes back to behaving like a liquid and sticks back together again Okay, but in principle this is straightforward right just conservation momentum bounds viscous stresses and the pressure gradients driving the flow The last equation is an equation for conservation of energy And we care a little bit about this because viscosity turns out to be a strong function of temperature TM is the temperature of the melt. I hope I'm consistent always using M a subscript M for the melt Okay, I can't remember what I use for bubble Okay, so change in thermal energy of the temperature of the magma as it's rising towards the surface Occurs for two sets of reasons here first We can conduct heat across the volcanic conduit D is a thermal diffusivity. It depends on temperature gradients Right, so we can lose heat from the magma to the surroundings This last term might be less familiar. It's the stress times a strain rate It's actually integral over volume of the dot product between the stress tensor and the strain rate That's a measure of how much energy is dissipated through deformation Sometimes people call this viscous deformation Right, but when you do when you deform a material you're doing work that creates heat It's a heat source and you know this if you rub your hands together, right you and you push hard enough your hands warm up This isn't the analogous term And we include it we include it here in fact because it will matter So that's our model for magma rising through the conduit Questions Maybe one last point is that we're going to allow the velocity of the magma as it's rising to vary across the conduit So it's zero at the sides. It's biggest in the middle The lab exercise you will do this afternoon will also use a model for magma sent in the conduit But it will be a one-dimensional model and so it will only solve for the average velocity of everything rising in the conduit and that's mostly to keep the Little script you have much simpler so that you can read it and try and answer the suggested questions that go with the exercise So now I also said if we go back to this viscosity, right? He is a volume fraction of bubbles are as the size we need to know something about those bubbles And we also need to know the pressure inside the bubbles as well So simultaneously we're going to solve a second problem for the growth of a bubble surrounded by melt or a liquid Okay, so what we're going to do is treat our gas bubbles We're going to assume our gas bubbles spherical rather than dealing with the complicated geometry of the melt around the bubbles We'll assume it's surrounded by a shell of melt that's symmetric about the bubble So it becomes a one-dimensional problem where the shell can expand and gases can diffuse from the liquid into the bubble We need to know the solubility of gases because as pressure decreases Right solubility decreases gases go into the bubbles and this is something people measure experimentally And we also need to know the speed at which those gases can diffuse from the liquid into the bubbles and once again This is something people measure experimentally. Yeah, this is unfortunately a messy set of equations I'll just talk you through how it is a bubble grows right in words And then you can look at these equations if you care about the details because in practice This is not much different from the equations you saw before The first line is a statement of conservation of mass, right density times volume is mass so the mass of her bubble can change as gases Diffuse right with the diffusivity D based on concentration gradients into the bubble, right? And then we just multiply the flux at the boundary by the surface area, right? Paul asked you this question this morning right the surface area is 4 pi r squared Times the displacement here you replace displacement with the flux We have an equation of motion for how the pressure difference causes the melt to expand I will not derive this equation for you now But this afternoon I'd like to derive the equation for how bubbles grow for you because it's reasonably straightforward And you can think about whether the model you're solving this afternoon is a good approximation and then we have two equations for Conservation of energy the first describes how the gas temperature changes the second how the melt temperature changes Let's do the melt one first because it's most similar to what you saw right changes in thermal energy of the melt occur because we're conducting heat and Because we're dissipating energy as the liquid is deforming For the gas, it's a little bit more complicated the gas temperature can change as we conduct heat into the gas There's energy associated with the x-solution So with the thermodynamics, there's temperature changes from the gas expansion as well from the ideal gas law Okay, so again just a summary as we solve two problems how a bubble grows and then we feed that into how the Magma ascends that magma scent does change things that matter here right it affects the pressure which affects the solubility and hence the diffusion Okay, so I'm going to spend a few moments now talking about the different ways in which bubbles grow and different limits under which bubble growth occurs Because actually the three different regimes of bubble growth will translate into three different ways in which we see magmas erupted the surface But I will do this mostly qualitatively Okay, so the first way in which bubbles can grow will be at conditions that approach equilibrium And what does that mean the concentration of our dissolved gases is it roughly at equilibrium with the solubility? This requires that the magmas ascending slowly so gases can diffuse into the bubble and the bubble can grow in response to the x-solution the gas is going into the bubble and This will occur if the magma is rising slow enough P dot dots mean time derivative So the rate of pressure changes from the ascent P dot M Is small compared to P will give us a timescale for the decompression And if the timescale for gas x-solution and diffusion is short Compared to the timescale over which pressure changes. We will closely approximate equilibrium conditions So slow ascent gases can diffuse into the bubble If however the magma is rising fast enough that gases can't diffuse into the bubble quick enough to maintain equilibrium The liquid will be super saturated with respect to those dissolved gases Because pressure is decreasing fast you become super saturated Okay, so what I'm illustrating in the little panel here on the right is on the vertical axis the concentration of our dissolved gas As a function of position through the liquid R is the radius of the bubble S is the halfway point to the next bubble. So between R and S is this liquid film surrounding the bubble the equilibrium solubility is the dotted line and So now we're rising quick enough that we become super saturated in which case the gases diffused from the liquid into the bubble And so that's what the solid blue curve is at indicating Okay, and the growth of bubbles is going to be limited by diffusion of gas into the bubbles if the timescale for diffusion Is much longer than the timescale for decompression one consequence though of it being Slow to diffuse gases into the bubbles is that this is the conditions when you'll become greatly super saturated If the ascent time the decompression timescale is really short when you're super saturated This is when you nucleate new bubbles and this will change the number of bubbles per unit volume Okay, so the last limit under which bubbles grow That's second one. We're growth is limited by diffusion the third growth is going to be limited by the viscosity of the liquid surrounding the bubble So the timescale for bubbles to grow Be owing to to grow in response to the pressure changes depends on the viscosity and This timescale looks like a viscosity Divided by the pressure difference causing that expansion And if the timescale for viscous deformation is long compared to the timescale over which pressures are changing Large pressures PG can develop inside the bubble compared to the surroundings That is pressures changing so quickly Right that the melt around the bubbles not able to flow fast enough to let the bubble expand in response to the pressure changes And the importance of this limit. This is when we will get big over pressures or extra pressures inside the bubble So that's it three different ways that bubbles grow right equilibrium conditions non-equilibrium where gas growth is limited by Diffusion and of the viscosity is really high. We can get big pressures inside the bubble Okay, so you've seen several times viscosity matters for some of these processes And so what I'm going to do is talk a little bit about what governs the viscosity of magmas First temperature and the amount of water dissolved in the gas has a large influence on melt viscosity So I illustrate this here for a specific composition That doesn't really matter too much What's plotted on the vertical axis on a logarithmic scale right so each tick mark is Factor of a hundred apart from each other As a function of the temperature the melt on one horizontal axis the amount of water dissolved in the melt on the other axis is that viscosity Now for the most part I haven't given you numbers, but now I will write these magmas of this type of composition This is a rhyolite at depth will contain say 5% water dissolved in them by weight And they might have temperatures say 800 or 850 So I'll you build it near surface pressures as much lower closer to point 1% And so as this magma rises up to the surface and water comes out of solution and goes into the bubbles The viscosity is going to increase by about six orders of magnitude Point being the water as a big effect on viscosity and you can see temperature as a big effect as well Of course as we change the temperature by what is that? 300 degrees viscosity is changing by six orders of magnitude as well So second point is we also said that we're making bubbles inside the liquid the presence of bubbles and the presence of solid crystals Also affect the overall viscosity of the magma just interrupt with any questions again any at any time Okay, so a little bit about what governs the viscosity of the melt and or magma The deformation rate of the liquid matters a little bit here I'm plotting on the vertical axis the viscosity of the liquid Compared to the viscosity of very low strain rates Over on the left as a function of how fast you're deforming it You might remember if you have a great memory A to the is the viscosity G is the shear modulus eight over the shear modules controlled whether something breaks And as you approach the conditions under which the magma breaks which is right here Viscosity seems to decrease a little bit if you add bubbles to the magma or the liquid the viscosity will also change and To this is a little bit complicated figure, but I'll talk you through it because This turns out to be a process that influences the results. We'll see in a few moments okay, so If we put an object inside a liquid So I put a nice spherical object inside our liquid and we're deforming our liquid like this If it's a solid object, what does the fluid have to do as we're trying to deform it? It has to flow around right? So there are a variety of ways to think how this might affect the average So now we were interested in what is the average viscosity of the blackboard with these little spherical particles in it? And by adding these spherical particles, we're going to change what that average viscosity is Because we've distorted what the flow is and there are a variety of ways to think about this But you know I always like to think what you're enhancing the energy dissipation around the particle by making the fluid Deformed more than it would otherwise and in a famous paper written in a good year for mr. Einstein He showed that the viscosity of the suspension Which I think I've been calling Ada. I Can't remember I'll call it E F F for the viscosity of the suspension Looks like the viscosity of the liquid times one plus two point five the Where fee is the volume fraction Solids and Einstein derived this in 1906 as a what as part of the process of trying to understand Brownian motion If we put a bubble in here Maybe I should clarify this is also for small volume fractions of particles Fee is a fraction of space filled with these particles If I put a bubble in here, what's the difference between a bubble and a rigid sphere? The bubble can deform. Let's start first by assuming for some reason this bubble doesn't deform It stays perfectly spherical. Is it the same as a rigid sphere? So for the rigid sphere, we have a there's a boundary condition that applies here, right? For a rigid sphere, the velocity of the fluid is equal to the velocity of the solid If we put a bubble in here boundary conditions different It's that there's no shear stresses on that boundary right because the low viscosity gas inside the bubble can't exert any shear on the object And so in 1932 GI Taylor showed that for bubbles This is the equation. It's one plus fee. So bubbles have a proportionally smaller effect on viscosity then Solid particles because it's easier. I guess you can think about this way It's easier for the surrounding liquid to flow past a free-slip surface on the bubble than it is a solid surface Okay, so the second effect for bubbles However, is that bubbles can deform in response to that deformation, right? And so I may start with this initially spherical bubble We put in a flow that's deforming it and that bubble will get stretched out and it may end up looking something like that. I suspect in your everyday life, you've probably encountered cases where bubbles can get the form like this Do you expect that deformation to increase or decrease the viscosity of the suspension? Compared to the case when that bubble is spherical. So we've we're applying only simple shear So and in response to a stress and so we only care about one viscosity here But in fact, of course the relationship between the stress and strain tensor is more complicated than a single number But let's ask a simple question, right? The shear viscosity of the suspension if we allow the bubble to deform Will it be bigger than or smaller than when if we force the bubble to be spherical? Well It's if you just look at the these flow lines, right? It looks a lot easier for the fluid to go past that stretched out bubble, right than having to go around The original bubble and so once these bubbles get stretched out it becomes a lot easier for the liquid to flow around the bubble So what am I plotting here? Oh So now we need to know right what governs whether the bubbles are going to be deformed or not What's causing the bubble to deform? It's shear stresses shear stresses look like Viscosity times the deformation rate the strain rate. Those are the shear stresses Yes, so in real magma's at the scale of individual bubbles inertial force is almost And I think I can say they will never matter because the length scales are small. Viscosities are pretty high At the end today, I will make sure to give you a list of dimensionless numbers explain what they mean and how they affect eruptions Okay, so what keeps bubbles spherical? It's surface tension surface tension forces. I haven't given you number a radius will call a Is the radius of bubbles surface tension forces look like surface tension divided by a radius of curvature So the ratio of shear stresses to surface tension stresses is a dimensionless number The people call the capillary number and so it's on the board. I'm not going to read it down because I just saw the clock But as the capillary number gets big it means shear stresses go up Relative to surface tension stresses and the bubble should get more deformed And so what I'm plotting on the vertical axis is the viscosity of our suspension Compared to that of the liquid as a function this capillary number for big capillary numbers the viscosity goes down, right? it's less than the liquid and for Small capillary numbers where bubbles remain spherical the viscosity goes up and again. This is another effect That's on the order of an order of magnitude So the second process was the loss of gases from the rising magma, right? We said if magma is permeable gases can escape from the magma Good thing is people go out there. They collect rocks. They measure this permeability They now do it in the lab in situ at high temperature. So we have models for permeability It does depend on the volume fraction of gas to some power as well as the size of the bubbles and Then the last set of processes were how magma fragments, right? We said either we deform it too fast people measure this experimentally and they do this by decompressing by stretching materials and looking at when they break and Experimentally people also measure how big the pressure inside a bubble needs to be relative to the surroundings for it to break as well I'll just show you one example of Experimental results for bubble bursting on the vertical axis is the pressure difference needed to burst bubbles as a function the volume fraction of bubbles and experimental data for a wide range of magma types and compositions and We have an experiment Experimentally determined model for when magma fragments. So that's everything about the processes any questions about any of those processes or the model Yeah, so the question is when bubbles first nucleate is this is the rate is the same There is a nucleation the initial radius of the bubble has to be big enough to overcome the surface tension stresses, right? That's right. So the surface tension stresses are surface tension sigma divided by a and that needs to overcome the super saturation So the initial nucleation size is probably not wildly variable But it does depend on surface tension and surface tension of magmas as far as I can tell from all the literature Is are not that variable, but it's actually a property. That's very difficult to measure And even though we're being recorded. I shouldn't say this there is a little bit of circularity in how it's measured that people often use theory to interpret measurements Bubbles can coalesce as well. And so this is something I'm neglecting for now in part because if we at least limit ourselves to the high viscosity magmas We don't see huge Consequences of our signatures of that coalescence. It'll appear later when we talk about the Celtic magmas Okay, so what I'm going to do is I'm going to talk you through one example in a bit of detail To give you a sense of what these processes are as magma sends and then we'll try and generalize and summarize Okay, and I picked for this particular example conditions that would have characterized the explosive eruption of Mount St. Helens in 1980 The vertical axis on all these plots will be depth And what I'm plotting on the horizontal axis here is pressure And so not surprisingly right as magma rises towards the surface the pressure decreases, right? Because we're getting closer to the surface as pressure decreases the solubility of gases decreases So we nucleate bubbles gases go from the liquid into the bubbles Fee once again as a volume fraction of bubbles going from zero no bubbles to one all bubbles And you can see that as pressure goes down the volume fraction of bubbles goes up Next I'm plotting the viscosity of our liquid compared And as water goes from being dissolved into liquid into the bubbles the viscosity is going to go up Remember water dissolved water is a huge effect on viscosity right in fact viscosity is increasing here by about four orders of magnitude At this point now the viscosity is becoming so large that it becomes hard for the bubbles to expand in response to the decrease in pressure so or I circle that right viscosity is limiting the ability bubbles to grow So what I'm plotting the final plot in the blue curve is the difference in pressure between the pressure inside the bubble and the Surroundings and when the viscosity becomes large we start getting big pressures inside the bubbles because the gas is not expanding So last this red curve on the right is that experimentally determined over pressure extra pressure it takes to cause the magma to fragment and When those two curves meet and cross is when we might expect the pressure inside the bubbles to cause the magma to fragment Once you fragment the magma, then you can create the ash particles of course that give rise to the explosive eruption questions about this example so these were the two questions we started with where when and how does magma fragment and How do these processes interact with each other to explain the diversity of eruption styles that we see So what I'm going to try and do is summarize when these different processes matter or dominate and this is a busy and complicated plot So let me first explain the axes and then if you get confused you can ask questions So what I'm going to plot on the horizontal axis is the rate at which magma is rising up to the surface On the horizontal vertical axis I'm plotting depth Right, but rather than plotting depth below the surface I'm going to plot pressure because the translation between pressure and depth is kind of it can be quite non-linear And so this plot looks very different if I plot depth But surface at the top deeper down at the bottom and then at the top with these words I plot estimates of this mass flow rate for some very well studied or characterized historical eruption and So we're going to start on the left and look at eruptions that have moved to progressively bigger centrates So remember the first process was To keep magma from erupting explosively we have to get rid of gases. We call that outgassing This red curve shows the point as magma is rising to the surface once we cross this red curve The magma is permeable enough that gases escape from the magma as fast as they're coming out of solution And so if that happens then you get rid of the gases before the magma can erupt and you can erupt For example to make a lava dome that lost much of its gas as it came up to the surface Notice as we move to bigger mass flow rates, you have to get closer to the surface You need higher amounts of gas exalt to have high permeability And as we go to higher ascent rates magma is rising too fast to let the gases escape fast enough So there's really two timescales the timescale for gas to escape compared to the timescale over which you're rising Okay, the second process we introduced Was breakage of magma we call this brecciation when the magma is deformed like you're silly putty or some other way fast too fast for it that it can break Okay, so the red curve now shows the point as magma is rising to the surface where it's deforming fast enough That the stresses on the sides of the volcanic conduit may allow it to break the vertical arrow here shows an estimate of the mass eruption rate for this Lava dome in northern California. It's about a one and a half cubic kilometers of ryo light It erupted effusively of course to make this lava dome And so presumably what happened as the magma rose to the surface a broke apart into fragments that allowed the gases to escape And then it can erupt effusively like this So we're predicting of course the magma is breaking apart into pieces right allowing the gas to escape Is there any evidence that this happens? I'm confused about time on my computer says we've talked for an hour and five Minutes and are we supposed to be ending at two thirty? What time are we supposed to end? at three, okay I'm sorry for you. It's taking so long. Okay, so question though is is there any evidence that the stuff the story we're telling is Reality that magma is breaking apart letting gases escape Here's some close-ups of what that magma looks like Right if we look at it at all scales we see structure we see bands and by bands I mean these alternating layers of light dark light dark light dark Right, that's a hand sample one centimeter scale bar with a microscope hundred micron scale bar a bigger scale bar in the field and What defines these bands actually is not a big difference in composition But little tiny crystals you might be able to pick some of them out here The bands that are white have more crystals the bands that are dark have fewer crystals So the question is whether maybe these bands we see here are recording this breakage process We can take these kinds of images and look at structure, right? We can for example Stack a huge number of these images and look at what is the relationship between the size of the bands and The color of the bands we can take a power spectrum to look at the variability of color as a function of wave number One example is plotted here. If you don't know what power spectra are this is probably not the time to explain it But what we see is a one over wave number scaling over a pretty good range of scales By time we get down to very short wavelengths here on the right. That's the size of the individual crystals There are other things we can do with these kinds of images right to look at the spatial relationships between bands If we look carefully at these bands though, we don't only see nicely parallel layers We see a range of textures that go from angular fragments here on the right to nice bands on the left and On the bottom I have another example where you can see things that look like fragments some of which have been stretched and So the qualitative argument is these bands that we see right at all scales once were fragments that formed The loss of gas is what actually causes the crystallization And then the bands stick together and they get stretched out as the magma rises to the surface question So let me illus do I meant I'll illustrate this graphically right remember We said of the conduit wall the magnets deforming it's breaking apart Right and so the magma breaks apart We make fragments the fragments stick back together they get stretched out and maybe this process happens repeatedly So I'm going to skip the picture on the right But one thing we can do is we can create toy models for this process You can imagine you take something you break it into pieces you let the color change you stick them back together You jumble it up and repeat it and you can test different types of models for making these kinds of bands Right the relationship between the size of the fragments and the color change And you can try and test different conceptual models for how we make these structures to see if they're consistent with data And I'll simply say in words that One of the interesting feature of these banded rocks is that they're multi fractal if you know what that means In addition to seeing that one-over-wave number scaling and this is all indicated indicative of a process that acts repeatedly upon itself So again, we're taking it through the story right increasing ascent rates and looking at some of the consequences So now we're going to go to higher ascent rates conditions that are thought to characterize the eruptions of a sequence of Lava domes within the past thousand years in eastern, California What you're looking in the picture on the right are these bumps here? Those are lava domes the white stuff on the top is pumice from small explosive eruptions and These eruptions like many of these types of eruptions begin with explosive eruptions And then they transition to a few sub eruptions that make the lava domes And so here's the idea how this might happen the magma is rising towards the surface Right, it's breaking apart at the sides of the volcanic conduit allowing gases to escape But remember I said the conduit things very radially right the middle is not the same as the edges The black curve the top which I haven't explained yet shows Where as magmas rising towards the surface the pressure inside the bubbles can become big enough for the magma to fragment That was like our balloon bursting and so maybe what's going on at these eruptions is The magma is rising slow enough at the sides to allow gases to escape but in the middle of the eruption or maybe at the beginning It's still rising fast enough that we get big pressures and bubbles and it can fragment And that happens early on in these eruptions and actually what I should have said is the white stuff You see here is pumice that's rock with lots of bubbles and glass rock without any bubbles So we see a mixture of magmas erupting But then these eruptions do transition to just a fuse of lava domes and presumably what's happening over time and Clogio pal undoubtedly will tell you about this on Wednesday is Over time the eruption rate goes down and so we move from the right of this diagram to the left We can still break apart the magma, but maybe we don't get big over pressures and so last At the very beginning I said of course we have a variety of ways to understand the physics that's going on one of Them is we can try and interpret observations. We make in the field When we do these simulations, right? We're modeling real processes and we're keeping track of things that in principle are testable with measurements So I said earlier right we're modeling how gases go from being in the liquid into the bubbles, right? They're diffusing from the liquid into the bubbles in that calculation We're keeping track of how much gas is left in the liquid We should be able to pick up a rock and measure those gases and see does the model look like observations and maybe the key Featured ignore the graphic is that there are two gases dominant gases in these erections carbon dioxide and water and One of those molecules Diffuses much faster than the other Which one Water or co2? Let's try that again. Which one does the actually let's imagine that the data is on the right You actually made these measurements. Which of those gases is diffusing faster? Water is diffusing faster, right? You see the diffusion having propagated further away from the bubble H h diffuses much faster than co2 By about an order of magnitude so going back to the three different ways in which bubbles grow Which one of these gases is most likely to come out of solution at equilibrium or under equilibrium conditions? It'll be water because it can diffuse faster Okay, and so we can take advantage of the fact that these gases diffuse at different rates to get information of time scales and so what I'm plotting here on the vertical axis is the amount of co2 dissolved in the liquid on the vertical axis the amount of water dissolved in the liquid as The magma is ascending and pressures decreasing and so let's start with the inset which simply shows the same axes But a much bigger scales magma starts at high pressure over here where the green dot is and follows the black line as it ascends So the solubility co2 and water decreases and goes into the bubble I should let me define what these words are open and closed open means that the gases that come out of solution disappear from the magma Vanish close means the you form a bubble the gases go into the bubble and so when the gases come out of solution they're going into a space that contains a certain amount of water in co2 and so if things were coming out of solution under equilibrium conditions you would follow one of these two curves labeled open or closed and Measurements on those rocks on the glass are shown with the dots. What's the difference between the dots and? equilibrium degassing They're not on the this curves are they they're not on the curves for the liquid being at equilibrium with the gas that's come out of solution you see for a given water content more co2 left in the melt and Presum we can attribute this to co2 diffusing slower and just taking more time to get into the bubbles Okay, and so this black curve that you see here says model non-equilibrium Where that black curve falls depends on how quickly the magma is rising because that's setting the timescale over which diffusion is allowed to happen and I guess I should all I can just say Qualitatively that the speed at which the magma needs to rise to explain these kinds of observations is consistent with What makes sense in other observations, but it's not a unique interpretation of this data So getting close to the end Does magma always have to break apart at the sides of volcanic conduits? Because we're progressively increasing the strain rates magma is going faster and faster right and the faster you go the more likely you are to break So what what could you do to keep magma from breaking apart? I guess I asked the question the answers on the slide unintentionally Whether it breaks depends on the viscosity right and so if you want to keep magma from breaking apart You want to lower the viscosity? I see some of you are rubbing your hands together if you rub your hands together what happens They get warm Right, and you do that through friction through deformation and you already answered the question Where is the deformation rate the highest right at the edge of the volcanic conduit? So you may be able to generate enough heat at the sides of the volcanic conduit to lower the viscosity enough that it can no longer break So that is magma as it's rising towards the surface and the strain rates going up rather than following the blue curve and becoming brittle It may warm up and move off to the right and not Become brittle or break. I'll skip the dimensionless number What's illustrated on the right though our examples of in these numerical simulations of what's happening as magma is rising towards the surface So let me explain the top one figure on the right first in the bottom Initially as the magma is rising towards the surface right the speed is biggest in the middle of the conduit on the left drops a zero on the edge as Once the magma's risen far enough in the volcanic conduit that it's flowing Faster we're making heat at the sides of the volcanic conduit We're lowering viscosity at the side and that's where all the deformation is accommodated And so we actually end up with a much more uniform distribution of velocity So I should keep that in mind when we do the lab exercise this afternoon And what's simply plotted on the bottom is the rate at which temperature is changing because of this process right? internal deformation creating heat and most of the heat is concentrated near the sides of the volcanic conduit and So what I'm plotting in blue here now are Ascent rates or mass flow rates that are big enough that you make enough heat the magma can't break apart If you don't break apart the magma all the gases stay inside the bubbles and eventually we reach the black curve that Tells us when the pressure inside the bubbles becomes big enough for the magma to fragment and I think that's the end of the story Oh, right Okay, so any quick questions about what happens for these magmas as they rise and what the processes are I Guess the thing to keep in mind. There's many processes that operate They do interact with each other, but it's presumably the origin of much of the diversity. We see an eruption style Okay, so I should say to find these axes more carefully zero is the center of the conduit in this particular case 25 is the edge in the middle. There's no deformation right velocity gradients are zero So we're not creating any heat through the deformation The heat generation rates proportional to velocity gradient squared So but you might wonder well, why doesn't this curve keep increasing all the way to the edge? Right, that's where the deformation rates the biggest So what's keeping the deformation rate or the heating rate being the biggest at the very edge? Yeah, we're losing heat by conduction into the surroundings And so there is a dimensionless number here that characterizes the importance of dissipating heat It's the ratio of heat dissipation through that deformation compared to heat conduction and At the very edge of the conduit heat conduction dominates Okay, very briefly there are In this story we just went through we assume the bubbles move with the liquid And that's a fine approximation for viscous magmas things that we call andesites up to rhyolites But very low viscosity magmas like the salts the bubbles can move with respect to the liquid Yes The reality is the question is what viscosity Separates when bubbles can move versus not it's it's a continuous process Bubbles are always moving and so it's really a question of how fast the magmas rise and compared to how fast the bubbles rise And so I'm going to summarize everything on a single plot for you were to capture that continuum But let me just highlight what some of these processes are that happen in low viscosity liquids that we haven't talked about So what have how bubbles move in a low viscosity liquid depends on their size and their concentration if The size is low and the concentration is low and the liquids moving pretty fast the bubbles simply move with the liquid and that's what we've been assuming so far if The liquid fluxes low and there's a lot of gas and the gas is moving with respect to the liquid The bubbles can coalesce with each other they can merge and they can maybe make these big slugs of gas Right that can burst at the surface and if there's a lot of gas flowing relative to the liquid The gas may end up forming a continuous phase with suspended droplets of liquid But again the key here is how fast are the bubbles moving compared to the liquid and Does that allow the bubbles to merge and make bigger bubbles or not? Okay, so what I am going to try and do here summarize at least these physical Processes and their consequences for eruption now spanning the full spectrum of magma compositions This is a very busy plot to So I'll try and talk you through the axis axes the vertical axis is What used to be the horizontal axis on the previous figures? It's the rate at which mass is rising towards the surface You can think about the speed of which magma is rising so I called it ascent velocity on a log scale The horizontal axis is the viscosity of the magma at depth when it started It's a scent and so previously what we did is we picked a high viscosity magma somewhere over here on the right And we increase the ascent rate to look at Where we go from a few sub eruptions to make lava domes because the gases can escape at low ascent rates To explosive eruptions that make eruption columns that we call plenium or subplenium eruptions for high ascent rates Okay, and Let me okay, so this black line When we go from below it to above it Represents the conditions under which you generate enough heat in the magma that it doesn't break apart at the sides of the conduit We can get big pressures in the bubble and then we can fragment the magma And so this was what we called Mount St. Hallens and on pin the tubo and notice as we get to lower and lower Viscosities you have to go faster and faster to fragment the magma this way and that's because as you get to lower Viscosities, it's easier for the bubbles to grow and to expand By time we get down to shoot I animated this to show you what real volcanoes look like okay, so Mount St. Hallens plots up here That's an over rupt of the biggest eruption of the last century in Alaska down here Okay, now when we get to You go back when we get to low viscosity liquids down here You have to go impossibly fast in order to get over pressures in bubbles big enough to fragment the magma So that's not going to cause explosive eruptions because the bubbles can always expand in response to the change in pressure So what I'm going to plot next this black curve if we're below it For low ascent rates the bubbles can rise with respect to the liquid that allows the bubbles to coalesce and start making really big bubbles and Maybe this is the conditions under which we're going to see things like strombolian eruptions if we increase the ascent rate the Bubbles don't separate move with respect to each other. They erupt with the liquid. Maybe that's when we see lava flows and Last right there we go lava flows for higher ascent rates where the bubbles can't separate from the liquid There are types of explosive eruptions with low viscosity liquids, however, and these are typically characterized by very fast ascent rates And this black curve here shows notice as we go to lower viscosity lower ascent rates What the Reynolds when the Reynolds number is big enough that if liquid is ejected from the surface it can stretch Because it has enough inertia that it can stretch and break up into little droplets or fibers So it's really when is the inertia of the ejected liquid big enough for it to continue to stretch and make droplets And in fact if you look at the products of these Hawaiian style eruptions, these are fire fountains basically liquid Magma rising and breaking apart The textures you see bear witness to a liquid disruption not brittle fragmentation. They're little droplets and fibers Exactly. Yeah, so let me go back to that regime diagram Okay, so low ascent rates would be on the bottom of this figure right low ascent rates high viscosity So we get lava domes Here right there are nice rhyolite domes because the ascent rate is low the gases can escape from the rising magma Either because the magma is permeable or as it's ascending it's breaking apart creating pathways for the magma to ascend Whether or not any of the details I told you are quantitatively right is Questionable, but certainly I think the processes and the interaction between the processes or have to be right Because there's nothing wrong with what we've been assuming it, but the details would matter for any specific one thing for example, I didn't say is that in the Simulations we looked at we assume flow is steady For convenience Obviously an eruption cannot be studied by definition. It starts and it ends so there's a time period where it's not steady But the steady assumption is a good approximation if the duration of the eruption is long compared to the time scale it takes for magma to ascend to the surface and So the lab exercise you'll do in about 40 minutes. We will also be considering a steady case So let me summarize What we just talked about in a more physics kind of Perspective with a set of dimensionless numbers We've sort of talked about dimensionless numbers a little bit But I'll remind you about what they are if you haven't heard of them, right? They're essentially ratios of different forces or ratios of different timescales that allow you to compare Two different processes two different timescales when we talked about our three ways of bubble growth, right? We had a dimensionless number that characterized to competing processes for the capillary number in the board It's the ratio of viscous stresses that acted to form bubbles compared to stresses that act to make them spherical So what I'm doing here on the vertical axis is listing a variety of different dimensionless numbers that characterize this process of magma Ascent from the subsurface to the surface the process that's being captured in the middle and The consequences for the eruption phenomenon that we see Now if I talk here it comes out in the microphones, but I can't read the slides Okay, so Reynolds numbers the ratio of inertial forces The change in velocity with respect to time Compared to viscous forces and there are two different Reynolds numbers that matter There's one for the magma ascent in the conduit Right, and it's going to be much bigger than the one that's relevant for the growth of bubbles The Reynolds number looks like a velocity times a length divided by a viscosity and So for our magma ascent the length is big the velocity is big the Reynolds numbers big But almost always it's low enough that we don't need to worry about the effects of inertia or turbulence and for the bubble growth problem it's always small this dimensionless number people call a Peckley number is the ratio of diffusion timescales to it's the ratio of advection to diffusion and we've seen it in the context of bubble growth right whether or not gases can diffuse into bubbles fast enough to accommodate changes in pressure and If gases can't diffuse into the bubble fast enough compared to the change in pressure We get large degrees of super saturation. This is when we nucleate new bubbles That when you nucleate new bubbles the bubbles are closer diffusion happens faster So it allows you to more closely approximate equilibrium conditions We saw another Peckley number, which was the ratio of a viscous timescale right a viscous flow timescale to a decompression timescale and this is a Dimensionless number that characterizes whether bubbles can expand in response to pressure changes And we said that if it's big enough If the viscosity is high enough, this is when we get over pressures inside the bubbles and the magma fragments We saw a dimensionless number that I skipped over there was a dimensionless number called the Brinkman number It's the ratio of generation of heat by viscous deformation Right by the flow compared to the heat loss by conduction And if this number becomes big we dissipate energy at the size of the conduit the magma warms up and it doesn't break apart Lots of dimensionless numbers that characterize the viscosity of our suspension. We saw the capillary number on the board There's another dimensionless number, which was how fast we're deforming magma compared to the maximum deformation Radicals withstand or that tells you whether it's going to fragment and So the mechanical behavior of our magma is controlled by a variety of different Dementalist numbers all based on how fast it's being deformed and Then last for a low viscosity magmas There is another dimensionless number, which was the speed of bubble rise the bubble rise speed compared to the magma rise speed And if the bubble rise speed was big enough compared to the magma rise speed the bubbles can coalesce and separate And we don't have a homogeneous bubbly fluid erupting at the surface Okay, so just to wrap up I was going to summarize with this cartoon we started with right This was the textbook picture of how magma ascends and erupts at the surface to review what the different key processes are Again, it's the dissolved gases that ultimately create the buoyancy that allow magma to rise To keep magma from erupting explosively we need to get rid of those gases and that gas can Escape if the magma is permeable enough if not and the ascent rates are big enough The magma can break apart at the sides of the conduit to create additional pathways for that escape If the magma rises too fast however the viscosity drops enough that it can no longer break That keeps all the gases in the magma and eventually you lose enough water from the magma becomes too stiff You get big pressures in the bubble and you get an explosive eruption So we still have five minutes left for questions. We'll take questions But let me remind you what we'll do at 330 is we will take a simpler version of a conduit flow model written by Helga Garnerman and I want you to explore what's in the script For maybe 15 minutes. I'll write down these instructions when we start. There's a set of questions It involves Deriving and understanding two equations and so we will do that together is derive those equations Hopefully that won't be too tedious and Then there's a set of exercises you can go through but in the end I want you to think about what assumptions and approximations go into obtaining the answers that you will Thanks a lot