 Good morning. I'm sorry for this change of schedule, but I could not come beforehand due to unforeseen circumstances. So today I will give you three lectures on three different aspects which are linked to one another, and they deal with what's happening below the surface, and also what we can infer from what we can see at the surface about what's happening below the surface. So I'll begin contrary to the schedule that was printed. I'll begin with lava flows because we will investigate how lava flows are in fact reflecting conditions in the magma reservoir and how the eruption rate is linked to the pressure in the reservoir and also to the density distribution in the conduit. And then in the other two lectures we'll worry about several different effects that may change the volcanic plumbing system as a function of time. So that will involve a few calculations, some examples from nature and also some laboratory experiments, and finally a few numerical simulations because in some cases you have to go numerical to handle the geological complexity. So I will begin with lava flows and domes. I'm worried about the dynamics of spreading the morphology because I would need those equations and those formulas for most of the rest of the lectures. So domes form at the surface, and when the lava has exhausted gases, that may happen during the descent in the conduit. That may also happen because simply the lava and the magma that store that depth has no gas in solution. And they can form very thick flows. Some of them may reach several hundred meters in thickness. And they are seen everywhere. That's one example from Volcano Colima. That's from Marlinzen in Japan. This dome was about a hundred meters thick. Don't be fooled by the rubbly appearance. This is an active lava flow. It's probably because of course the lava still has some bubbles and explodes and there's cinder and ash that fall back on the flow, but it's an active flow. And sometimes because the lava is too viscous to even spread, which you have the erection of the spine, these are quite common in fact. And this was the one that erupted out of Montemple in 1902, and that rose to a height of more than 300 meters above the vent. Here these are not errors in the photograph. This is a human being. And you can see that human beings, of course, were being completely dwarfed by such a... Now this is one of very interesting flow because if you record the height of the spine as a function of time, you will see that's going to be one of our exercises this afternoon. You can derive a lot of things about pressure in the conduit and in the reservoir. And this was recorded faithfully by a marine officer who made very precise measurements of the way the spine evolved with height. And these data were forgotten until they were picked up again in 1990. Domes are seen in other planets as well, so that's on planet Venus. This is a series of so-called pancake domes. You can see some of them are cross-cutting previous ones. These are very thick bodies. They may reach a kilometer in height. And so you can see that we have information about the flow thickness. We have information about the flow morphology, and that is something we can use. And if we understand how the morphology is related to the eruption rate, for example, we can guess a lot of things about the system that actually fed those flows. To get a close-up, you see these flows extend over a radii that exceed 10 kilometers. Very thick, much thicker than what we see on Earth. And just that's the information we've got. And what can we make with that information is what I'm going to talk about now. Sometimes domes have got very smooth morphology. That's one dome from Mount Popucatepetl. It looks like a big cow dung. In fact, there's very similar features. In fact, it's not probably very exerting to look at the dynamics of cow dung. But this is exactly the same problem. So why is it important to look at lava domes? Well, lava domes build a blava over the vent, and that will change the pressure in the magma column below. And so that will reflect on the eruption conditions. And of course, we can record. If we have data, we can record out the thickness of lava over this time. And we can also record out the volume of the lava flow over this time and link the eruption right to the lava dome thickness. And this is something that's easy to do, and that will make a lot of use of. So I will go back to a few equations that you've already seen. So I'll go fast. But we start from a reservoir where the pressure is the little static pressure. The country rock is at density, we'll see. The reservoir is at big H. And the pressure there is the little static pressure. There is the equilibrium pressure if all the stresses have relaxed at that depth. Plus, there is some over pressure around the pressure that's related to the inflation, deflation, or past history of the reservoir. And of course, as we empty the reservoir, or we may fill it up again with the deeper sort of magma, this is, of course, time varying. And then we've got throughout the conduit. I will assume for simplicity that viscosity and density will not change. That's an assumption. And then we'll see how the flow is dependent upon the thickness of lava that builds up at the vent. So in these conditions, we know that we can derive what's the driving pressure without much effort, simply because we know the pressure distribution in the system. So we have a same good conduit that extends from z equal zero at the base of the conduit or at the top of the reservoir. And it extends to z equal h, which is the earth surface. Now the earth surface, of course, is now covered with lava. P of z will denote pressure in the conduit. The pressure at the vent is simply the weight of lava at the vent, which is related to the thickness of the dome. And the pressure in the reservoir is known because that's where we started from. That also gives you the pressure at the base of the conduit. Now we can compare that pressure, which is the pressure that will drive all the flow with the hydrostatic pressure component, which is related to the height of the lava column above. And we can make the difference between the two. And that gives us the driving pressure for the flow. If that term is zero, if the pressure in the reservoir is balanced by the weight of the lava column above it, then the flow will stop because you are in hydrostatic conditions. And that's very simple equations. It shows you several things. One is that you have two driving terms, possibly they might not always be driving terms. So one of them will be related to the difference between the magma density and the country rock density. So if the magma is buoyant with respect to the country rock, this is the driving term. The flow will be driven by buoyancy. It might also be driven by the pressure in the reservoir if it's positive, if the reservoir is over pressure. Of course, these two terms can be also negative. And a very good condition, I will see that later in the talk, that many magmas are in fact negatively buoyant with their coast rocks for very good reasons that are related to the way the magma reservoir gets in place. And of course, this pressure may vary with time. And as you empty the reservoir, this might become negative. So these two terms might be positive or negative. They have to be positive somehow if we have to start the process, but eventually it might become negative. And then of course, this term is always negative. It always acts to decrease the driving pressure and hence to slow down the flow. And this is why we can couple the four in the country to really what's happening at the exit of this term is of course related to the eruption rate. In the case of Montanpelle, there was no spreading away from the vent. So all the eruption rate was used to build up the thick lava spine. And of course, that's a very simple system. But if the system is more complicated, you have to account for the spreading of lava to figure out how the thickness of lava at the vent evolves. So that's a simple recap of what I've just said. So any Russian may stop when the driving pressure drops to zero and that might be achieved by decreasing the reservoir over pressure. That might also be achieved by increasing the thickness of lava at the vent. That might also be achieved by changing the magma because the magma reservoir may actually be made of several different magmas. The important thing is that if the lava is negatively buoyant, then you might end up with another pressure in the reservoir and another pressure in the reservoir might drive collapse and the failure of the whole system. And that has happened in the vent. And there's a second thing that we have to remember, that I've assumed that the country remains open. And that's used because I've used the pressure distribution across the system by calculating the adversarial pressure distribution. And of course that assumes a construction that is a continuous, a conduit filled with lava. However complicated it is, that's always going to be the pressure distribution in other static conditions. But of course that has to be open and we'll see that may not always happen. So we're going to calculate the Russian very rapidly because I believe that Michael has already done this. And just to show you that we're going to handle the Nadia Stokes equation, but we'll find again at the end the same result that is that the driving pressure is exactly what I've calculated before. And that's just to bring on the point that you can do a lot of things by simply considering other statics. Other statics, the first thing you have to handle when you look at fluid systems and that's a very simple set of equations. And if you do more complicated things you still have to relate to other statics eventually. So we'll deal with an increase of magma of MCD, UAM, and viscosity mu, small Reynolds numbers, that is there's no inertia. So at all times there's a balance between the forces that are applied to the flow. We'll work in a symmetrical coordinate system, velocity components u, the V-sata, the autoradial component w. We'll assume that the flow is purely vertical, that's an eruption. So there's no horizontal velocity components. And we'll also assume there's no thrilling motion or cyclonic motion in the conduit, so variables do not depend on the polar angle theta. Straight forward the assumptions. So now 96 equations in a small Reynolds number, no inertia. So the continuity, the mass conservation equation takes a very simple form. And then the radial, the conservation of radial momentum is also very simple, because we have no horizontal velocity components. And then the vertical momentum balance is quite simple too. Now you can see with the first two equations we have two big simplifications that arise. The velocity w, which is a vertical velocity component, does not depend on z. So that's going to get rid of one of these differentials here. And the pressure does not depend on the real distance. And hence dp dz only depends on z. So we can recast momentum balance this way. And that can be integrated very simply. Because of course we can integrate with respect to the radial distance across the conduit. So you integrate once and of course at the axis this term is zero because r equals zero and also because the vertical velocity is maximum at the axis. And then you have to integrate the second time assuming that there's no slip at the conduit walls. That's the assumption which is valid for all natural flows but it may not be valid for all conduit flows if you have a very, very smooth conduit you might have slip at the walls. And if you have also foams that rise and this happens sometimes in magma you might have slip at the walls but not in this particular case. And then you end up with this parabolic flow profile which Michael has derived before which is a Poisson flow profile. Now we can calculate the mass flux of magma which is at any depth z which is calculated this way. But of course as of now we have not solved yet for the pressure gradient and we have to solve for it. And in the exercise this afternoon we'll investigate how this pressure gradient might depend on the shape of the conduit and we'll see that. But here we've got a straight conduit and it's a much simpler result that we're going to get. Now we're going to use that the mass flux is constant mass conservation dictates that the mass flux does not depend on height in the conduit. So if you look at any height z the flow that comes from below has to be balanced by flow that's above. So q is a constant which also implies that dp dz is a constant. And because we know the pressures at the top of the reservoir and the base of the conduit we also know the pressure at the exit. That's a very straightforward calculation. And then we end up with this result which tells you what the mass flux the mass eruption rate is. And you can see that now we can relate this eruption rate to viscosity also to the conduit radius and we have here the driving pressure term which we have derived simply from very simple adostatic considerations. There's two things here of course to note one is that the adostatic argument is quite powerful if you don't want to go into the details of the flow in a conduit. The other thing of course is that the eruption rate is very sensitive to the conduit radius to the power 4 but we shall see that although it's very sensitive to the conduit radius it's not very sensitive to the shape of the conduit and that's something we'll talk about this afternoon. So a very basic equation I'll tell you again why we have to worry about the thickness of lava that builds up at the vent. Just as it takes what we're going to do this afternoon is again the Montanpelli spine 1902 and this is a beautiful set of data that was recorded by the marine officer using simply marine measurement techniques. However, it's beautiful data and you can see another thing that's always forgotten about the eruption that they may last very long Montanpelli is famous because it exploded and destroyed the whole city but in fact the explosion was only the very initial phase of that eruption and it lasted for more than a year and during a year it built up the spine the spine grows then of course it's made of bubbly lava so that's not a very robust and rigid system so it decays and can be degraded by rainfall and also other things so it crumbles down and then it picks up again and you can see it picks up again you have three phases of growth very nicely recorded here one, two, three the third one is not very useful but we're going to make use of these first two we'll blow up the data for these two phases starting at the same reference level and this is the same thing but starting from the same reference level you can see first that the spine rose above that reference level which is already way above the vent over more than almost 200 meters very thick lava thickness two phases these are the theoretical model that you can build and that you will build this afternoon and that's going to tell you a lot about what happens in the reservoir you can see between phase one and phase two that of course the eruption rate which is simply the... can be rated to the derivative of the height of the fraction of time how much growth we have on the spine decreases with time so that's telling us that the... the only thing that changes because the lava composition did not change the only thing that changed was the pressure in the reservoir that's telling us that the pressure reservoir decayed with time and we can actually measure how... how it decayed it's also telling us of course there was a reservoir which is important and then that's... because we have also this fit to the data we can get a lot of things about the size of the conduit and the magma properties and in the end you can derive that for this eruption the total pressure drop in the reservoir was larger than about 30 megapascals quite a sizable pressure drop so that's just to wet your appetite for this afternoon so dynamics are spreading so what we want to find out is how the height of lava varies with time and we're going to do it for simple flows the idea is to get an idea of how sensitive it is to the eruption rate the idea also is to get an idea of what are the time scales for the spreading because we're going to be using that to infer how the flow behaves in the very circumstances we're also going to use that for example this afternoon to find out under which conditions the compressibility of lava has to be taken into account and so we have a flow it's fed from below or it's not fed from below even if it's not fed from below it's still going to spread because once you've built up this lava volume at the surface it will spread under its own weight but of course it will also spread if it's fed from below so what you have is the thickness of the lava which is going to be a function of radial distance away from the vent it's going to depend on radius and time distance away from the vent which we will call rn of t and I'm going to derive very simple scaling arguments to find out how the thickness of lava evolves as a function of the control variables and the control variables will depend on what we'll assume for example the constant eruption rate so we'll derive how the height varies as a function of the eruption rate also we'll derive a scaling of the radius how it evolves with time and also as a function of eruption rate and lava viscosity and of course the rn of t is going to be another end result these rows are again in the laminar regime small rows number, no inertia so we'll derive the scaling loads by simple scaling arguments but then we'll go back to the governing equations and we'll derive the full set of equations and we shall see how the scaling arguments are very useful to in fact arrive at a mathematical solution so the scaling arguments are a very powerful way of handling the physics but also a very powerful way of handling the mathematics so flow dimension and spreading rate constant eruption rate because I'm going to assume the incompressible lava, we'll discuss that you will discuss it this afternoon under which conditions you have to worry about compressibility and you will see that even lava is intrinsically compressible even if it's very compressible for some eruption rates these effects will not matter we'll discuss that so the control variables are the eruption rate by volume because we assume that lava is incompressible so the density is constant and the volume of flow rate is proportional to the mass flow rate plus the lava properties and so the balance equations are very straightforward first we have a global mass balance what we have is a flow that has the shape of an expanding circle and so the volume is thickness times the radius squared to some unknown proportionality constant in the scaling arguments we don't worry about proportionality constant we're just deriving the functional relationship between the variables so of course the volume has got to be linearly increasing with time because we assume that the eruption rate is constant with time the volume increases as a Q times T and that's proportional to HR squared the flow is spreading in the orbital direction so we're going to write the orbital force balance the driving term is a pressure acting on the vertical surface to the flow so pressure acting on the spherical surface with area 2 pi Rh is proportional to HR imagine that you have a flow like that and that's the same surface at the radius R it has h and radius R so the surface of that cylinder is proportional to HR and this pressure acting here and this is the pressure that's providing the horizontal force you can see that gravity plays tricks the pressure will be rated to the thickness of lava and gravity is acting vertically but here we are linking to the vertical surface so the pressure is driving a horizontal flow so that's the total force associated with that it's the pressure which is simply the weight of lava or MGH times the area and of course this resisted that's the driving term and the resisting term is the viscous shear at the base of the flow the flow is actually spreading against the rigid surface the shear stress is viscosity times the strain rate and the strain rate is the vertical gradient of velocity the new term divided by H and that's acting on the whole flow which has a radius pi R squared and so the angle force balance is that so we have this equal to that times the area and that's the simple relationship so we have three unknown variables H, R and U so of course we are going to use the fact that the velocity of the flow is in fact rated to the spreading rate such that the original velocity is dr dt and if we are interested in power laws dr dt is proportional to R divided by time and now we have all the relationships that we need we have three unknowns three equations and we can derive these spreading scales and we can see that the radius increases as a square root of time that's mostly due to the fact we have radial spreading so the flow has got to cover an ever increasing area as it goes farther from the vent and maybe the surprising thing is you can see our very poorly sensitive viscosity this is new term 3.18 so it's not a very good idea to use these flows to infer what the viscosity is and it's not that sensitive to the eruption rate q to the power 3.8 square root of q and the same thing with the lava thickness it is not very sensitive to either viscosity or q the important thing of course here is you can see that we now have the relationship that we need to figure out how the eruption rate impacts the thickness of lava at the vent the 1.4 the larger the eruption rate the thicker the lava flow or it's low for a given eruption rate the larger the viscosity the larger the thickness of lava at the vent so these are very simple arguments that give you you can see quite complicated scales these are non-linear and for those who are interested in three mechanics it's interesting that although we are in a laminar regime at low rails number we don't have the inertial terms the inertial terms in three mechanics always create problems because they are intrinsically non-linear and in this particular case we are still going to be non-linear simply because the shape and the domain over which the flow is occurring is changing with time but even though it's the simplest possible flow regime you end up with a complicated equation simply because the flow domain is part of the solution so the full solution Navier-Stokes again again the cynical coordinate system we have actual symmetry with respect to vent velocity again we're not going to assume that there's no radial component of velocity there's no swirling motion you don't drive cyclonic flow patterns in a lava flow Navier-Stokes equation that's mass conservation constant density conservation of radial momentum conservation of vertical momentum now we can make several simplifications we know that the thickness of the lava flow is very much smaller than the radial extent that's not true of course for the for the mountain pulley spine but we're not handling this typical lava flows they're less than a kilometer thick and they extend over more than 10 kilometers in width so there's more than a factor of 10 difference between thickness and radius and you can use that in the concrete equation to infer that the vertical velocity has got to be much smaller than the vertical one so viscous stresses are great so that will bring a very useful simplification for that equation that for these flows the viscous terms are straight with velocity gradients along vertical are small in that equation we're going to neglect the radial derivatives because we carried over a typical distance r which is much larger than the vertical distance so that derivative is much smaller than this one and so that's a reduced set of equations you can see that the vertical now momentum balance is reduced to something which looks like a hydrostatic balance but it is not hydrostatic in the sense that because the thickness of the flow is changing radially it's not a hydrostatic pressure distribution so very straightforward simplifications you can integrate the vertical momentum equation to get at the pressure distribution and the pressure distribution again varies in the function of r because the thickness of lava changes with r and you can derive therefore what the horizontal or radial pressure gradient is you can see that it rates it to the thickness sorry that's a g mistake here sorry g so the flow is driven by thickness variations and the flow of course is in direction from big small thickness simply the flow is trying to get even so we get rid of thickness variations so we can integrate the radial momentum balance with the following boundary condition no shear stress at the top we assume that there's no wind that's able to derive definition in the flow that's not true if you had a spreading for example of a big waterproof shear stress in the ocean are able to drive waves but not for these very thick no slip at the base that's an assumption because sometimes these flows are in fact occurring over a previous ash deposit and the ash deposit might actually be possible to get some slippage but there was assumed zero slip so you derive from the equation that I had in the previous slide the distribution of horizontal velocity and that's almost a pressure profile half of a pressure profile nothing earth-shaking yet now once you have this you can write down you have to write mass conservation in this flow and you're going to write down mass conservation over a control volume in this hatched area which is the area that's between two concentric cylinders that rate EIR and R plus DR and so that's volume the volume is 2 pi HRDR and the auto mass flux on the cylinder at radius R is called pi R that's the horizontal flux and you will have the mass of volume conservation which is going to be related to that and you can calculate the mass flux using what you know about the horizontal velocity and you substitute into the mass balance equation this one then you derive an equation for the lava thickness of the ration of time you can see that's a highly non-linear equation again the point here is that because the domain over which the flow is occurring is changing at this time although we are in a very simple dynamical regime such that inertia is negligible we end up with a non-linear equation and that's true for most flows where the shape of the flow is determined by the flow itself so to solve the equation you have to add the fact that we're feeding the flow at a constant rate so you can either do this by assuming something at the inlet here but we don't want to do that because we don't want to be sensitive to congruence et cetera in large distances so we treat it as a bulk balance for the whole volume now solutions for this type of flows two variables we have non-linear equations there's not that many techniques the most powerful one is to use a similarity variable a similarity variable will combine radius and time and it's very physical because if you scale the radius by the total extent of the flow then that you're saying of the flow is spreading it is spreading so it extends over larger and larger distances but if I want to look at the flow shape I'm going to look at the flow shape at distance one over thickness one and that's just going to give me a shape and what we're going to assume and the best way to look at this is that the shape does not vary as time the flow is self-similar it just changes in its dimensions but the shape is dimensionless term is the same so the flow is self-similar and that's the similarity and of course we know what the radius of the direction of time is because we have our scaling argument so we know how this depends on time and so that's telling you that this similarity variable has this form it combines r and time but also what we've gained is that this is purely dimensionless we involve all the control variables for the flow we do the same thing for the thickness that's the scale for the flow thickness there are factors of threes here that simplify the equations if you remember there were some factor three lying about so that's simply we get rid of threes by doing these here and now the h of a is a dimensionless function that just describes the shape of the flow it's the same as the flow spread now if you cannot integrate this analytically so what you'll get is this solution where the constant now has been obtained by numerical integration and that's again we find that the flow is spreading at the square root of time you can derive an approximate solution for those of you interested I will be distributing a bibliography so those of you interested can get at the original articles you can derive a simple approximation to the flow shapes this way otherwise the flow shape has got to be solved for numerically and you can see it's a simple flow shape and you expand as a function of variable one of psi and psi is the dimensionless radial distance which goes from zero to one so what we've gained is these very nice solutions how does they compare with experiments so these are experiments that were done by her that happened a long time ago so you use different oils and you can see that you had to let them spread for quite a while in seconds you're talking about several days here and there's several oils but if you make everything dimensionless you scale the radius by this combination here and that's left this time and you expect this to be very nice function time to the ratio of power one half this is not a curve fitting exercise the solid line is the theoretical solution so it has the right slope but it also goes through all the measurements you can see how beautiful the theory works now to see how these relationships are sensitive to the eruption rate now we assume that the eruption has stopped what you have is a thick lava pile the water vent and this lava pile will spread under its own weight but it's not fed from below anymore so we just going to go through the same arguments I'm going to go fast because the only thing that changes is the mass conservation instead of having a volume of the flow that increases linearly with time now this volume is constant doesn't change because the eruption has stopped and there's no more a new lava coming in at the surface we use the same orbital balance same ratio between spreading velocity and radius and then this is what you get for the scales for the flow radius and thickness you can see how radical the change in the power loads are now the radius increases as time to the one eighth power of course this is a much smaller flow because this flow is actually thinning all the time because it's thinning all the time it's spreading at an ever decreasing rate you can see again how poorly sensitive to viscosity this is again viscosity to the rate to the power one eighth very very poor dependence on viscosity the thickness of the lava flow decreases with time of course the lava is spreading so it's thinning again you can see how poorly sensitive to viscosity this is so these are very useful relationship they illustrate very nicely the way the eruption rate influences the dimensions of the flow and if you go to laboratory experiments again you see experiments that were done over in this case almost a month that's the radius scaled using the relationships I've shown you before again there's absolutely no fitting here there's an analytical solution here so no curve fitting and you can see how beautiful the fit is and same thing for the lava thickness the lava thickness is there's more errors here simply because these are very thin flows and the measurements are in fact are not easy to make but there's another point that's important here you can see that the departures are largest for small times and this reflects the fact that we assume that the flow is self-similar but these flows were started from given initial conditions so you pour there are constant volume flows so you pour some liquid over a table and then you're going to record our express with time and the initial shape of the flow there's nothing to do with the self-similar shape once it's spread for some time so this reflects the fact that the flow in fact is initially not necessarily in a self-similar flow condition and that explains the departure here so beautiful theory and beautiful fit in the data and these scaling relationships are important and we use that type of argument several times in these lectures the power of these scaling relationships they give you very simple to derive the mathematics then if you want to do the full solution is more important but you get at a very simple exercise you get how the characteristic of the flow depends on the control variables and the control variables are either the volume but they also may be the eruption rate and the viscosity and other things you can see how fully sensitive to viscosity all these things are now we understand a little bit about how this flow spreads these were very simple flows constant viscosity constant density in nature of course things are more complicated and so the other thing that you can measure in the field you can measure flow thickness, radius, etc but you can also observe the flow morphology and of course when you're looking at Venus you can't measure flow rate eruption rate, etc the only thing you've got is the morphology of the flow and their basic dimensions the dimensions will tell you something about hydrostatics with the thicker the flow the larger the pressure of the vent and the morphology is going to tell you something about the spreading so I'll give you a very simple run-through some of the work that we've done in this before I do that do you have any questions for what I've done so far everybody's happy with Nadia Stokes just to clarify in the first part we have been sensitive so we have H not bearing this time the first part no because there's no inertia at all times there's a false balance there's no inertia in your system so time must be considered as a parameter this is an equilibrium situation which will evolve very slowly with time so you can carry things look at your flow at any given time in the first part and so you know the thickness you know the pressure at the vent so you know what the flow rate is going to be of course as the flow will spread this situation will change but because there's no inertia the force balance will be modified but this will be a force balance there's not going to be any inertia in the system so it's simply an evolving steady state any other conditions? no inertia so there's no acceleration inertia is not zero it will be a mistake to say my flow rate does not change with time that would be a mistake but it's the acceleration that is negligible through all the other terms so it's not saying that there's no time change but the time changes contribute nothing to the momentum balance so we'll have a flow morphology for observations so we'll look at this lava mountainous lava dome 1980 it built up over about 150 meters another stick at Montaigne-Pelet so the French spine is better than the American spine but anyway so not a sick lava flow but a very interesting morphological characteristic it should move but it doesn't so that's a PDF and in this animation you would have seen that this thing developed lobes so for example here that's the flow and this develops a small lobe on the other side so these flows will not spread as nicely as what we've seen so far from a purely viscous liquid spreading horizontal surface they will develop lobes so you have something that springs out from one of the edges and forms lobes for example in submarine lava flows where you form pillow lavas the flow front isn't stable there's some crust because it breaks and there's a surge of lava away from that breakage and it forms a nicely formed pillow in this case it's more than a thousand pillow it's output lobes and that's a summary of the various shapes and there was also two different kinds of surfaces there was a scoria smooth surface which is very regular and raggedy and also smooth surface we saw a very smooth surface for the Popocatripet dome for example but this dome was also smooth in part and this is a summary of what happened with the other function of time and you can see that as a function of time there was quite a change in the morphology of the flow there's information there we can't read all of that yet but there's information in these things but there's also information in the shape of the flow when you have a flow like that you're quite sure there's been some lobes sticking out of the original flow and that's a very simple information that we can use another beautiful characteristic is that you have resting structures that's the flow it's stretching of course its carapace stretching its carapace and the lava that's inside which is still liquid will stretch the rigid crust and will form this spreading center so to speak this is called resting structure when the crust is not able to withstand the extension that's applied by the flow to the crust that of course is something that you can observe in the field there's going to be a trace of that left even when the flow has stopped and you might be able to document this for example and flows in other planets than Earth so how can we endow that well that's the problem we are we have a cold and brittle crust and that's and now the flow is viscous lava encased in the brittle crust and that crust will break and will be stretched in many different regimes and so of course what happens is the kind of the magnitude of stresses that are applied to this crust and that's going to depend on the way the lava tends to spread so it's going to depend on the flow rate the magma viscosity and the density but also the stresses are going to be dependent on how thick the lava flow is and it depends also on cooling cooling is occurring with time so as time increases if the flow doesn't move for example the crust will become thicker and thicker and it will be too strong to deform and to do anything so you're going to have to upset what the flow is trying to do with the way with the way cooling is proceeding the simplest way you can try to figure out what happens is that the view of the flow depends on crust resistance and the crust resistance depends on the flow there's a couple between the two and the best way to add that is to look at the time scale for these two different phenomena one is the time scale for spreading which I'm going to call toe A another one is the time scale for solidification toe S now you can see that if toe A is much larger than the solidification time scale the crust is forming very rapidly much more rapidly than the flow is trying to spread so essentially what you've got is a very thick crust that develops even before the flow can really develop so crust formation has a big influence on the flow on the other hand if the flow time scale is much less than that of solidification then the flow is spreading very fast it's adding more area and area without crust because it's new lava that's coming out then the flow is much faster than crust formation and therefore the crust will not have a big effect on the spreading so these are very simple arguments and of course we need to find out we can estimate the solidification time scale because it's over such a thick viscous body we know that cooling has got to occur by diffusion so we know the basics of that but the flow time scale well we have to find out how it is and we know what the flow time scale is because we've done the arguments that have just developed before so the spreading time scale is going to be simply Q is the eruption rate typical Russianist that's volume of a time the volume is about a square s cube of a time that's an average thickness that's the one length scale you get out of that we use the thickness that we've derived before and that gives us the spreading time scale if Q divided by Q very simple now you can see that the spreading time scale depends on the eruption rate the larger the eruption rate the smaller the eruption time scale of course the flow is faster as the eruption rate increases you can also see that the larger the viscosity the larger the time scale for spreading not surprising density of course is important as we know so it's the less important variable here and gravity might be important because if you deal with flow on other planetary surfaces gravity is going to be different and there are large differences between the gravity of Mars for example Venus is about the same gravity as others but Mars is about one-third that of Earth and the Moon is of course even smaller so you cannot compare what happens on the planet if you don't account for gravity but again you can see the power of these relationships is that they allow you at the first plants to evaluate the influence of all the important variables including the gravity on the planet so that's my spreading time scale and time scale depends on the cooling mechanism for diffusion this is going to be about that if you want to solve the problem exactly you have to solve for the heat exchange between your flow and the atmosphere and Venus of course has got a very hot atmosphere so you cannot assume that Venus's flows are going to be cooling exactly as Earth's flows but still there's going to be some diffusion time scale because the cooling has got to propagate through the flow and so there were what we're going to see that we now have a dimensionless number which is the ratio between these two scales of time it's going to be the time scale for so education over the time scale for spreading so if there's a big phi crust has no effect on the flow so education is too slow to change things so spreading is very fast so that's what you see for these very large ellipses what you get is the solution that has arrived before lava just spreads out it's just cross warming but the crust is thin and the flow is always increasing its area so nothing happens that remarkable and as you go down inside the crust has got a larger and larger effect on the flow dynamics if it's at this particular value you're in a folding regime you can see how contorted the flow surface is but it's still governed by the dynamics of the viscous interior the crust is thin and just follows the flow it does influence the flow but you can see the flow is not as simple as it used to be the size then becomes even more limiting the crust is now thicker or less being equal and for the flow to expand you have to stretch the crust and to reduce rifting and then if you go even to smaller values then you have a rifting and pillow load formation this is when the crust has no influence you still get these very complicated things these experiments were done with wax different waxes and if you have rifting you can see how complicated the flow shapes become and you cannot compare this very readily to true flows on earth because there are some effects involved but the basic patterns are there and you can form pillows, loads of rifting structures rifting structures here and that's where I'll stop so to summarize what I've done I've shown you three important facts one is you can get a lot of information by invoking hydrostatics hydrostatics is a very straightforward physical principle that you can produce we derived for example the driving pressure for flow without any complicated calculation and we were able to find out exactly what were the controls of the flow then we derived a simple scaling relationship for the spreading of lava these relationships are very useful to investigate a lot of complicated patterns because you will always have to balance spreading with other effects here mentioned the effect of the crust morphology we used our spreading relationship to figure out when crust formation was important crust formation is not important in all cases if the flow is spreading extremely fast if the Russian is extremely large then crust formation has no effect so crust formation is always there but it may not have any effect and for this you have to compare it with the effects of the spreading we know how spreading occurs and the crust formation is a complicated business but we can find out under which circumstances the crust formation is important and we will do that this afternoon from compressibility and the last thing that I want to show you is the scaling arguments I have used you can see how simple the scaling are and you can use them if you don't want to do the largest dose solutions to the full but even though you might want to do this if you want to go into the full scale solution use these scales because these scales will tell you what are the similarity variables that allow you to find solutions to the flow so scaling arguments are extremely powerful and we will use them time and time again