Transcription
All right, everyone. We should probably get started again. If you're not in the room yet, please hurry back. Okay. So my name is Sabina Stanley. I'll be giving the next couple of 45-minute lectures this morning, getting into the early afternoon. My background: I'm someone who does planetary dynamos, focusing mainly on numerical simulations. How do we get the planets to make the magnetic fields that they do? And so this is really sort of one boundary or one input into the whole heliophysics problem. And so I hope to give you a feel for that today and to maybe give you some intuitive understanding of what goes on inside planetary dynamos, but then also do a nice survey of what we know about all the planets in our solar system. Give you some updates because there have been recent data from missions that have been giving us all sorts of new questions to answer. And then also kind of move on to the frontier or the fringe, depending on your perspective, and really talk about what might be happening in dynamos and new areas. So, for example, in small bodies, things like asteroids or planetesimals early in the solar system. And then also go far away and talk about exoplanets, because as we know there are lots of new exoplanets out there, and we could ask the question, what does it mean for their dynamos? Okay.
So here's just a quick outline of what's going to happen over the next two sessions. I'll start by giving you a warm-up intro to planetary magnetic fields and dynamos. Then we'll do a bit of dynamo theory basics, talk about the important points you need in order to understand the rest of the presentation and also some of the chapter. And then we'll go into doing a survey of what we know about planetary dynamos. You should stop me at any point in time if you have questions; just wave your hand or yell out. I'm happy to stop at any time. All right, so starting with our intro to planetary dynamos. Dynamos, essentially as you know, are a way of converting mechanical energy into electromagnetic energy, producing the magnetic fields that we observe. Okay. There are a variety of ingredients you need in order to get a dynamo acting inside a planet, as well as in any other body. First of all, you have to have fairly complex motions. Those motions have to occur in electrically conducting fluid. And if you had do that in the presence of the magnetic field, and you can generate magnetic field faster than it decays, then you have what's called a self-sustained dynamo. So this is just sort of a cutout of the type of thing you want to think of inside planets. You're gonna have some sort of spherical shell domain where you have complicated motions; that's intended to be shown with these yellow ribbon-like structures here. And those motions interact to stretch the magnetic fields around, generating new fields. This actual picture is meant to be a particular schematic for Jupiter, but the same sort of idea holds in other bodies.
So looking at these ingredients a little closer. Okay, the important things: there, electrically conducting fluid; fluid must have complex motions; and the motions must be vigorous enough. Okay. So let's start talking about the electrically conducting fluids. These are actually different in different bodies. Okay. So let's think about three different types of bodies: you've got the terrestrial planets; okay, you've got the gas giants; and then you have the ice giants. You're innocent, Upton. So in the terrestrial planets, what's the electrically conducting fluid inside the terrestrial planets where the dynamo is generated? It's iron; it's fluid iron; it's in the core, right? And this is just like a nice schematic showing the core region of the earth. What about in the in Jupiter and Saturn and the gas giants? What's the electrically conducting fluid? It's hydrogen, right? So the pressures get high enough inside Jupiter and Saturn that the hydrogen metallizes and becomes an excellent conductor. Okay. What about Uranus and Neptune? Look, this is the hard one. It yes, right? It's water. So you're innocent, Knapton. You start thinking about it. Let's start on the outside; you've got hydrogen again, okay? But hydrogen has to get it to high enough pressures before it metallizes, and that doesn't happen in Uranus and Neptune. So it's not the hydrogen regions. Maybe there's a rocky core on the inside of Uranus and Neptune; maybe the dynamo is generated in an iron layer in there. It's possible, but if it's happening deep inside Uranus and Neptune, that region is so far removed from the surface that we actually wouldn't see the magnetic fields from the other magnetic fields decay too fast with distance. Yes. Ah, ah, good question. So essentially when we think of a metal, we think of the electrons becoming able to freely move and therefore be able to be able to sort of carry currents easier. So that's what we're talking about with hydrogen is that the electrons dissociate from the from the nuclei from the protons; that's at least my mind non-experts view of what it means to metallize. Does anyone have a better explanation? Okay. Well, let's talk about water then. So in Uranus and Neptune, does the water become metallic? No. Okay. So we're not talking about a metallic water in unison, Epton. What happens in your descent, up tune, is that water under high pressure can dissociate; your ions can dissociate. So you can end up with things like OHS and H's, right? H2O, the other was right; OH, HS, and HS separated. Okay. So you can actually get an ionic conductivity in the water bodies. Okay. Now, ionic conductivity is not going to be as high as a metallic conductivity, but it's still going to be pretty good. It might be two to three orders of magnitude lower than say the conductivity of iron, but it's still going to be pretty good. And what we're interested in is it high enough in order to generate dynamo action, and that's what we're going to see is indeed the case. Okay. So the electrically conducting fluid in the bodies can be quite different. What's important is that it can that the conductivity is high enough to generate dynamo action.
Okay. I mentioned that the fluid must have complex motions. What do I mean by complex motions? This is actually fairly difficult to quantify. Okay. We know that having lots of sort of twisting helical flows is very good for dynamo action. Okay. Rotation, and as was mentioned in the earlier talk as well, rotation; it's actually not required to have dynamo action occur, but it is a very good way to organize motions in such a way that you can produce large-scale fields. Okay. Rotation by itself, not necessarily necessary, but good for organizing fluid magnetic fields and producing large-scale things that we can see. Now, it's not enough if your motions are complex enough; they also have to be sort of strong enough. And there is a measure of the intensity of the motions; it's actually a measure of the product of three important quantities: the velocity of motions, the size of your conducting region, or the size, the general length scale of the fluid flows, and the electrical conductivity of the region. So the product of these three things essentially has to be big enough in order to generate a dynamo, and we usually quantify this by something known as the magnetic Reynolds number condition, and I'll talk about that further in a minute. Okay. All right, talking a little bit more of these complex motions, what causes them inside planets? Okay. Typically, and this is I think you could say this is true for almost all planets, this is mostly due to convection. Okay. And convection is occurring because the deep interiors of planets are hot and the surrounding layers are cold. So heat's trying to be removed. If the heat that wants to be removed, if this, so for example, if the temperature difference between the center and the outer region is big enough that you can't be done through conduction, through thermal conduction, then it's going to be done through convection in these regions in planets. Radiation is not an important factor in heat transfer. Okay. So convection will occur; convection will move heat from the center outwards, and that's going to generate these complex fluid motions. Okay. The fact that rotation tends to be an important force, so as Carl is mentioning, we have these fictitious forces, the Coriolis force in planets. Planets are all pretty much rapid rotators; the rotation acts to organize the motions through the Coriolis force and make these nice helical twisting flows that we need in order to generate dynamo action. Another important motion inside planetary cores and other places for dynamo mention is shear. Anytime you can stretch stuff, and you can do that by having a differential rotation in the radial direction, for example, any shear will just stretch a magnetic field. Okay. So shear is very important inside dynamo regions, and as I mentioned, the rotational constraint helps. Okay.
All right. So let's look at the nearest planet, Earth. Okay, where we are. If you look at sort of a starting textbook what the magnetic field of the earth looks like, you might get a picture something like this, where you have the earth and you're showing magnetic field lines, and they're typically represented as if you have a bar magnet at the center. That bar magnet is somewhat tilted with respect to the rotation axis by about 10 degrees in Earth. Okay. It can also be offset a little bit from the center with this representation. Okay. But essentially what we see at the surface is the equivalent, or to a good approximation, the equivalent of what type of field you would get from this. This is a surface map of the radial component of the magnetic field of the surface, where colors are contours of the field strength. So if you have red here, you can see that you have lots of field lines coming out of the southern hemisphere here, and they go into the northern hemisphere here. And this is what you typically see if you have what's called a dipolar dominated field, of field where most of the radio fields coming out of one hemisphere going into the other. We call it an axially dipolar dominated field if if this dipole is fairly well aligned with the rotation axis. Okay. So these are what we measure; we stand at the surface, or we're from satellite, and we take magnetic field measurements. In reality, this is at the surface. If this is an output from a numerical simulation, so the surface of the plant is somewhere far out here, if we then downward continue to where the core is in this simulation, the core-mantle boundary is about here; it's you can kind of tell where it is. And what you want to notice from this is that the magnetic fields inside the car core are much more complicated than the magnetic fields that we see out at the surface. Okay. So the dynamo problem inside planets is take data out here and infer what's happening in here, and as you can see that's going to be somewhat difficult. Okay. We really only see a small component of the field leaving the core that we take data that we can observe, and from that we have to infer a much more complex problem or a much more complex system going on on the inside. Okay. Okay. So that so the question is why is the field so much more complex in the core? Thank you, great question. So the core is the electrically conducting region inside the earth, for example, then you have the mantle, which is the rocky layer of the planet; that's a pretty good insulator. Well, pretend, pretend it's a perfect insulator. Okay. And you have to start asking questions about what types of currents can exist in insulators. And so, right. So it turns out that the magnetic field, which you can think of as consisting of two parts, we typically talk about the toroidal magnetic field in the poloidal magnetic field, and they're due to different current systems. And though the toroidal magnetic field cannot leave the conducting region, so there's an entire component of the magnetic field that cannot exist in an insulator, so we don't see it. Okay. So this field that you see is really just a potential extrapolation or a potential expansion of the field at this boundary. Okay. So because you have an insulator, currents can't do their thing, and you essentially get a Laplacian solution for the magnetic field outside of this region. Okay. So the magnetic field out here solves del squared B equals zero; the magnetic field in here is much more complicated. Okay. All right. So let's look at some. I like this picture. This is data of the Earth's magnetic field observations that we have. Okay. Here's what we know pretty much about Earth's field. I'm gonna show you the field at the core-mantle boundary, so this sort of top of the Dynamo region. This is going to be a movie in a second. What you will notice here is a number; it's back, but you will notice a number; this is a date. Okay. So this is 1590, and what you're gonna see is a movie of what the radial component of magnetic field has looked like at the core-mantle boundary of the earth over the past 400 years or so. Okay. So let me play this. Okay. And what you will see is that it's mostly dipolar; you know, it's red in one hemisphere, blue in the other, but it's active; things are moving along. Some interesting features to notice: you will notice that these equatorial region flux spots are these strong flux spots tend to move to the west. I'll play that again. I can figure out. Okay. So you fixate on one of these; they tend to move to the west. Okay. If you think of the core as being a really good conductor, the magnetic fields are frozen into the fluids, so this is how we get information about what's happening for the fluid motions inside the core. Another interesting feature you will see is the growth of this blue spot in the southern hemisphere here. See these two things? They're growing; they're growing; they connect. Okay. This blue spot here, even other parts coming in, is much stronger today than it was say two hundred, three hundred years ago. Some people have suggested this might be the beginning of the reversal of the Earth's magnetic field. Okay. Imagine what happens during a reversal of the Earth's field; you have to change the polarity of the dipole. The way you're gonna do that essentially is you're going to collapse the dipole as it is, have a much more complex field, and then grow the dipole and the dipole in the opposite orientation. And this might be how you'd expect it to start. So as long as this keeps growing, and maybe an equivalent type of patch occurs in the northern hemisphere, we might be seeing the start of a reversal. This might also just be an anomaly, right? This might then dissipate away and go away, in which case we don't have a reversal. Okay. But there has been a lot of work trying to understand this anomaly in the magnetic field. Yes. Yes. So if we look at 1590, I don't think it's that far, but I pause it and I go back; it was more to the west, east, right? So if we go back, are you talking about essentially this kink? Yeah, it's me. So you don't have a flat, a parallel equator line, right? So there are definitely kinks. Now, keep in mind this is at the core-mantle boundary. Yes. So this is at the core-mantle boundary, and these are high-wavelength features. So smaller scales, as you move away from the core-mantle boundary, the stuff we see at the surface, you end up preferentially filtering the smaller scale components because the potential field, the power goes off, decays faster with length scale, right? So you do tend to see a much more flatter equator, let's say at the surface than you do at the core-mantle boundary. So you might not see as much of an effect of this, but it is; it does certainly move with time. Okay. Now, one question that I like to ask you this. Okay. 15. Here I've got a global map of the magnetic field. Okay. In 1590. Okay. There were no satellites in 1590, right? There were a few stations, observatories, so fixed locations where they set up a magnetometer and took data, but nowhere near enough that you could make a nice global map like this. Okay. It's anyone know where else we could get data to make maps like this from around 1590 onwards? I couldn't. People just speak louder. Okay. Okay. So rocks. We tend to use to get data much even much earlier, so from paleomagnetism you can get data of what the dipole field was like at a certain time and location, but you can't really get; it's very difficult to get a global map like this, and I and I'm I'm not really being fair with this because what I'm looking for won't give you a global map, but it will help. Are there any other sources? Not ice cores, no, but a good guess. Sorry, bird migration. You're getting closer, actually. Did anyone use magnetic fields for any sort of navigation? Yes. So ships used magnetic fields to navigate. Everyone was carrying around a compass in those days, right? That's my image of the 1600s; everyone's walking around with a compass, but that's not true anyway. So there are ship logs of people crossing the Atlantic and taking magnetic field measurements in locations, and people have actually gone through all of that data and made magnetic maps going back to this time from that. So I find that very interesting that we can go back and look at ship logs and and get my new. This is at the core-mantle. Very sorry to tell you how we made a map; we don't; we don't have anything down at the core-mantle boundary to take pictures. What you do is you make a map of what's going on at the surface; you assume that the field is a potential field, and then you do an extrapolation to what the field would look like, but at the boundary of the source region. Okay. So you have to take some of this with a grain of salt because it is an extrapolation. Yeah. So absolutely. So we so the question was can the potential feel that the surface be disrupted by other currents or other things around? If I'm understanding, and the answer is absolutely. Assuming that, first of all, assuming that the mantle is a perfect insulator is wrong, so so that's an approximation that's not necessarily true. Then, of course, you know, we've got a lovely ionosphere around us, and there are all sorts of other current systems that we have to take care of. Some of this you can do if you have sources of the field that are external to where you're measuring, then you expect it to behave differently as a function of distance than you would an internal source. So internal sources are proportional to 1 over R with something, whereas the external or proportions are the something. So you can in some ways remove them all. So the timescales of the features are important, right? Anything happening in the atmosphere has probably got de timescales, sorts of things, whereas stuff from coming from the core, we expect to have much longer timescales. Ok, other questions? Yeah, this. Yeah, that is exactly the. So the question was is this region of reversed field here that's essentially in the South Atlantic, does that have anything to it? South Atlantic anomaly; that is the cause of the South Atlantic anomaly. So this reversed field, if you look at this near the end, like today, let's look at it over here. Okay. So again, this is at the core-mantle boundary. So here's another thing to think about: here, I have very strong blue field and very strong red field. Ok. So now you've got field lines going in in certain places and going out in certain places. Think about what that will look like if you go some distance away from it. What will happen to the blue field lines and the red field ones? They'll cancel, right? Yeah. So so they'll they'll end up cancelling and producing a much weaker field. So if I were to extrapolate this picture, picture to outer distances, you would actually just see a very weak field spot here, and that's what causes the South Atlantic anomaly. Uh, well, it might be; it so it might not be as strong four hundred years ago. Okay. And it will have moved a bit; like things seem to shift to the west; that's another thing that we noticed about the magnetic field is thirst; everything drifts westward; it's known as the westward drift. Yep. I don't have one; I can. You know what? I will find one, and I'll put it online somewhere. Something; there's an equivalent of this. The reason I like to show the core-mantle boundary one is because there's more structure because you see the smaller scales. The magnetic field at the surface, it'll be mostly dipolar; you'll see that that kink moved to the west, and you'll see sort of weaker fields in this region. Other questions? Okay. So what else do we know about the Earth's field? We know that it reverses polarity. This is from studies of paleomagnetic magnetism, magnetism in rocks, mostly the data we've gotten from the seafloor. Okay. This was actually after in the 1950s or so, after, you know, at the beginning of the Cold War and so forth, everyone was suddenly interested in the oceans, right? Because they were worried about subs and all sorts of things. So people were doing lots of magnetic measurements, and they actually discovered the magnetic
Striping pattern on the ocean floor, and that was the first hint that the Earth's magnetic field actually reverses polarity. The polarity reversals in the Earth are very different from in the Sun. Okay, in the Sun, you've got a nice regular cycle. I know it's not perfectly regular, a periodic, but compared to the Earth, it's it's, you know, it's an atomic clock, right? So in the Earth, the field reverses polarity randomly, chaotically. Okay, we can't predict the next one. If you take, to say, the last billion years ago or so, and average the number of reversals by time, you could say they occur roughly every half a million years. Okay, but the last one was 750,000 years ago, so we're kind of overdue. Okay, but it's much more random and chaotic than compared to the solar cycle. We also know that the field has existed for about the at least the last 3.5 billion years. So this comes from paleomagnetism in rocks. Okay, rocks that are 3.5 billion years ago aged are magnetized. Okay, so we have some information of the field by studying its history as well. All right.
Moving on to planets, thinking about comparing Earth to the other planets. These are now maps of the radial component of the magnetic field at the surfaces of the planets in our solar system that have active dynamos. Okay, so here's Earth, here's the radial field at the surface. Okay, if you look at the giant planets, let's start with those. Jupiter's here, aside from a sign flip, Jupiter's field has a structure that's very similar to Earth's. It's mostly dipolar, but it's a little wobbly. The tilt of the dipoles about 10 degrees, very similar to Earth's. The field strength is about 10 times stronger, but that's expected based on scalings arguments for the size of the planet. Okay.
Saturn also has a dynamo-generated magnetic field. It's also dipolar, so red in one hemisphere, blue in the other. There are anomalies with Saturn associated with how perfectly axi-symmetric it is, and I'll talk about that, but essentially there's no tilt of the dipole with respect to the rotation axes. Okay. Uranus and Neptune have very multi-polar fields, so very different from the other planets. The dipole is not a dominant structure in the magnetic field of Uranus and Neptune. Okay.
Mercury, we're getting lots of data from the Messenger mission these days from Mercury. Mercury's field is also fairly dipolar, but it has a fairly strong quadrupole as well, and that's why if you look at the magnetic equator, which is kind of up here, it's shifted upwards. So if you add a dipole and a quadrupole, you get a dipole that's offset northwards. Okay, so Mercury's field is fairly dipolar but a strong quadrupole and also fairly axisymmetric like Saturn, although we don't have as good data in this case. So I'm going to talk about these. Another body to mention is Ganymede. Ganymede is one of the moons of Jupiter. It does have an active dynamo today. We probably don't have enough data to make a nice map like this for it to tell you information about the dipole tilted and the non-dipolar components, which is why I haven't made a picture. Okay.
What's missing from here? So we started looking at this and we say, well, what's missing? Well, in terms of planets, the planet that's missing is Venus. Venus does not have an active dynamo today. Okay, somewhat puzzling if you think about the fact that dynamos are generated inside the interiors of planets, electrically conducting regions, bla bla bla. Venus is the planet most similar to Earth in almost every regard, size, location, composition, yet it has no magnetic field. If I asked you what planet is least like Earth, you would probably tell me Jupiter in terms of size, composition, and yet Jupiter's magnetic field is most like Earth's. Okay, so obviously it's more than just these ideas of size, location, so forth that go into what makes your magnetic field, and we'll talk about those sorts of things. Mmm. Mars also missing here. Okay, Mars doesn't have an active dynamo today, but there are crustal magnetic fields on the surface which tell us that it had a dynamo in the past, and I'll tell you about that. Okay. The Moon also has a crustal magnetic field due to a dynamo in the past. No other, I don't know where that's happening. No other moons in our solar system have been shown to have crustal magnetic fields or an active dynamo. Okay, so in terms of moon, we've got in terms of moons in the solar system, we've got our Moon had one in the past, and we have Ganymede has one today.
Speaking of crustal fields, here are some maps of crustal magnetic fields. I just want to give you a flavor of this. So first of all, in Earth, we have crustal magnetic fields. What you notice if you compare this type of figure to the figure I just showed you for the dynamo-generated magnetic field, the fields are much smaller scale, cross the wise, right? There they're roughly of the size of tectonic objects on the surface, the scale of the tectonics or even on the rock scale if you want to get really fine results. Okay, so you can kind of tell us something's crustal, dominant crustal source or a dynamo source, first of all, just by its length scale. Okay. Here are some magnetic signatures on the Moon. Okay. Here's Mars's crustal magnetic field. I'll talk about that more in in a few minutes, and then there's also some evidence that there were crustal magnetic fields on planetesimals early in the solar system and possibly even on asteroids, and I'll talk about those as well. Okay, so there is evidence of past dynamo action that we can go out and explore if you have rocky surfaces on these bodies that sustain magnetic fields in the rocks. All right.
So how do we study magnetic fields? Go through some various mechanisms to do this. The first one obviously is to get magnetic observations from spacecraft. I put up some upcoming ones or presently active ones going on today. Swarm is a constellation of three satellites, that's why it's called Swarm because like bees hovering around us and that kind of thing. I was launched in I think November of this past year, and it's gonna take magnetic field measurements at Earth and tell us a lot about the temporal and spatial structure of the Earth's magnetic field. Juno is on its way to Jupiter and it's going to give us probably the best magnetic data of any planet due to a dynamo. Okay, and that includes or okay, we are going to have better magnetic data for the dynamo in Jupiter than we have for Earth today. Okay, so you might say, well, that's that's impossible. We've got, you know, we've got observatories on Earth. We have satellites hovering around, three of them hovering around Earth and Swarm. We've had mags that we've had or stat. How could we have better magnetic field data from one mission to Juno? Okay, that's gonna go around Jupiter. Admittedly, it's gonna be in a polar, but it's gonna give us a nice orbit, but how could we possibly get better magnetic data for Jupiter than we have for Earth? Any ideas? I'll pose this as a question. I mean, I'm crazy. Can't possibly. Yes. Okay, so it's definitely it's somewhat related to altitudes, but not so much that about the spacecraft's altitude. Let me ask you, and maybe this isn't a fair question because I haven't told you a lot about Jupiter's magnetic field or dynamo region yet, but the Earth dynamo is generated in the core, right? That's meant to be shown by this reddish region in here, I guess. How far is the core from the surface in due order of magnitude? Yeah, it's a few thousand kilometers away. It's about the core of the Earth is roughly half the radius of the planet. Okay. In Jupiter, the metallic hydrogen region okay extends to very close to the surface, roughly about eighty-five percent of the planetary surface. So we're actually much closer to the dynamo region when you're near Jupiter than you are on Earth. Okay, so the the Juno mission, which is going to orbit Jupiter fairly close to the planet, with the equivalent of it was if we could fly a spacecraft on Earth roughly two or three hundred kilometers above the core-mantle boundary, right? So in the middle of the mantle kind of thing or at the base in the mantle, and we can't do that. Okay, so one reason is we're much closer to the dynamo source region. Another reason goes back to this picture from before. Okay, the Earth has a crust. The crust has magnetic fields. If you may measure magnetic fields, you measure magnetic fields. You don't measure crustal magnetic fields or dynamo Jeremy Mayfield's. You just measure magnetic fields, and the problem is that the crustal magnetic fields start becoming important and start dominating the the spectral or the power we get from the magnetic fields at around a length scale of, let me say, in spectrally it around in a spherical harmonic degree 13. Okay, so that means that we can only really see spatial structures that are coming from the dynamo up until about that spectral degree. So it's hard to see small-scale things from the dynamo because there are small-scale things in the crust. Okay, and we can't tell the difference between them very easily. All right. Jupiter doesn't have a crust, so there are no crustal magnets on Jupiter, so we also are not burdened by that issue. So in Jupiter, we're actually maybe going to be able to see much smaller scale features in the magnetic field that are due to a dynamo than we wouldn't Earth. Okay. All right.
So observations, many make observations. More of these we have the better. Other things we do, we do paleomagnetism. So we go to rocks and we investigate their magnetic fields. This is just a nice picture. This is from Ben Weiss's lab. This is a meteorite. It's ALH84001. It's from Mars. This is the famous one that had potentially got a little worm-like creatures in it, right? But it's not they're not worms, but anyway, you can make them. Nowadays, you can actually put this thing in an instrument, make magnetic maps of a small, so this is 2 millimeters, right? And this is a magnetic map of this body. Okay, so you can actually get magnetic fields from meteorites and from rocks all over the Earth. Okay, and from that we can do things. You can't see this. This is today. This is in the past. This is a record of the intensity of the Earth's magnetic field essentially over time. So you can see that we can measure the intensity of the field, and you can also do for longer times. This is about a billion years ago, I think, on this scale. You can also see polarity reversals. So paleomagnetism helps if you have rocks. Other things you can do is you can try and build your own dynamo. Okay, we like, you know, as physicists, or we like to do experiments. Experiments would mean, okay, take a container, fill it with an electrically conducting fluid, make it move very fast, and see if it can grow a magnetic field. Okay, we're going to see that this is an incredibly challenging thing to do in the experiment. That doesn't mean people aren't trying to do it. Okay. These are some of the first dynamos, experimental-wise. This is the Karlsruhe dynamo. What they did, they got very clever. So the hard thing to do if you're trying to build a dynamo experiment is to get the complex fluid motions in this nice homogeneous spherical container, which is what we think of as these planetary dynamo regions. So they said, well, we won't do it in a nice spherical container. We'll say we know what we know. We want complex motions, so let's build a pipe system that has all this nice twisting helical stuff and then run the fluids through the pipes, and this will naturally give you all the complexity you need and the fluid flows in order to generate the tunnel, but it's not really a dynamo core geometry. So nowadays, people are trying to do this thing where you take a sphere, you fill it with a fluid, rotate it really fast, as fast as you can for this type of fluid, you try and get motions going on in there and see if you can generate a dynamo. Okay, so we do experiments, and I'm going to show you later that the hard thing here is getting that velocity times length scale times conductivity criterion, the magnetic Reynolds number criterion for a dynamo is incredibly hard to get working inside an experiment. Okay.
So the other thing we do is we do numerical simulations. So we tell computers the equations governing the system, and we say you solve them, right? And from that we can get maps. So here's one from one of my dynamo simulations of the radial magnetic field at the surface of one of the simulations, and you can see it's fairly dipolar. Things drift westwards. Okay, things that we saw in the Earth's magnetic field. You really need the largest supercomputers out there to perform these simulations because of the the number of length scales and time scales in the problem. You really need a ton of grid points to resolve the geometry and get everything right. Okay. All right, so that was the intro flavor. I don't know why that keeps happening. What I want to do now is just cover up a couple of sort of theory basics in order to understand dynamos. There's going to be more information on these slides than I then I'm going to cover here, just so that you can go back and look at them. Okay, but the basic idea here for studying dynamo action, we're studying electromagnetism. We're gonna need Maxwell's equations. Okay, so you start with Maxwell's equations. Here, I'm not going to go into too much detail here. You take into account the Lorentz force because you need to know how these things affect the fluid particles. So you take Maxwell's equations, you take the Lorentz force, and you take Ohm's law, which I've written in some simplified form that works nicely inside planetary cores. Here, those are the fundamentals of the yeah. Okay.
Then we start making approximations for and keeping what's important. Okay. The first thing we do is we realize that the fluid motions inside planetary bodies are really slow. They're much lower than the speed of light, and if you take that into account, you can make what's known as the MHD approximation where you assume that you oversee is very small, and that allows you to get rid of a few terms in equations. Okay. What you get left with is so here are Maxwell's equations again. The only major difference is here is you'll notice the the displacement current is missing. Okay, but otherwise Maxwell's equations hold, and in the Lorentz force, you lose the Sigma e. Okay, so you lose those guys, but otherwise they're pretty much the same. I won't get into the details here. Okay, but what we're really interested in, right, that'll give you if I go back, sorry, that'll give you a bunch of equations for magnetic fields, for currents, for electric fields. Okay, we don't want to deal with all of these if we don't have to. Okay. What you can do is you can kind of combine all four of these equations along with Ohm's law to write an equation that only has magnetic fields. So we're gonna find a way to remove electric fields and remove currents from the equations, and it's just sort of a way of starting with, say, Ohm's law. You plug in Ampere's law. You work with all the other equations. The details are here. I'm gonna leave it to you to try this yourself, but I take you step through step through this. Okay, but what you eventually end up with is an equation that gets rid of all the electric fields and currents and just involves magnetic field B. This is known as the magnetic induction equation. So this says the rate of change in time of the magnetic field B is given by this first term, which is the curl of the velocity cross the magnetic field, plus the second term. Now, the second term, if you look at it, okay, pretend the first term wasn't there. Pretend we got rid of this. This would just be d by DT of something is lambda del squared of that something. That's a diffusion equation. All right. If I asked you for a thermal diffusion equation or a conduction equation, it would look a lot like that if you accept you would have temperature here. Okay, so this is diffusion acting in here. This lambda is known as the magnetic diffusivity. It's inversely proportional to the electrical conductivity. Okay, so higher electrical conductivity, smaller magnetic diffusivity. So this is your diffusion term that makes this, if you look at it, your source term. Okay, so this is a diffusion term, but there's actually a growth potential because there's a source term for magnetic fields, and this says if you have velocity fields interacting with the magnetic field in some complicated way, right, it's the curl of a cross-product. That's terrible, right? Then you might actually be able to grow your okay. So this magnetic induction equation is really at the heart of the magnetic processes in the dynamo problem, and we can play with this equation a little bit, try and get a feel for it. Okay.
So here are some things you can do with it. You can ask the question, well, do planets need dynamos? Maybe when planets formed, we know there were magnetic fields in the solar system. Okay, in the disk, maybe when planets formed, there was some magnetic field at that time. They just kind of froze it in, and that's still the magnetic field that from that time. Maybe there isn't an act of dynamo going on inside planets, and if you want to test that sort of question, what you need to do is you need to ask the question, what is the time scale that it takes for a magnetic field to decay away if it's not being actively regenerated? Okay, so if we take the magnetic induction equation and we say that there are no active generations, so let me go back, then what I'm essentially saying is that there's no velocities going on here. Okay, so we get rid of this equation, this term, and we're left just with this. Okay, so now we have the diffusion equation for the magnetic field. We ask the question, how long will it take for the field to decay? Okay, and we can just do an order of magnitude approximation. You can you can pull out more fancy mathematics, solve it for Bessel functions and stuff like that, but if you just want to do an order of magnitude calculation, say I have a characteristic magnetic field strength, which I'll just call B, and I'll ask how long does it take for that to decay. So this side of the equation I would say goes roughly as that magnetic field intensity over some timescale. So if I'm approximating a time derivative as B over the time, okay, the right-hand side, keep my magnetic diffusivity, B is approximately B, and this del squared is each gradient is inversely proportional to a length scale, so I end up getting two L's for length scale on the bottom. So this equation, if I solve now for the time, I get that it's proportional to l squared over lambda. Okay, so now you can start figuring out what length scale you want to use. Okay, you could for planetary cores, for example, take the length scale of the core as being your largest length scale possible. Okay. If you do fancier math, you end up finding that the what you really want to do is use the length scale, the slowest decaying eigenmode, which is just the length scale of a core divided by pi. Okay, so it doesn't make that big of a difference, factor of 10 because of the squared. Okay, you plug in the numbers, and you get that the magnetic decay time for the Earth is roughly around 15,000 years. Okay, so what that means is if the dynamo shut off today, okay, it would take about fifteen thousand years for the field to decay to one ovary of its current value. Okay, and if you think of the fact that solar system formation occurred much longer than fifteen thousand years ago, you realize that any sort of initial field out there would be gone by today. So you do need an active mechanism in order to explain the intensity of the field in planets today. Okay, so those are the types of games you can play with the magnetic induction equation. Another thing you can do is you can try and visualize the actual induction processes. Okay, I like to do that by doing the following. So I don't find it very easy to intuit what del cross u cross B is. Okay, I don't I don't get it, but you can use vector identities to rewrite del cross u cross B in terms of dot products. Okay, so del cross u cross B is B dot grad u minus u dot grad B minus B del dot u plus u del dot B. Okay, four terms. One of these terms is 0 automatically. Which one? You del dot B, right? Cows is a lot. Boom. Lose one. Okay. The other thing we can do is we can realize that u dot grad B, well, if I stick that on this side of the equation, then I have D by DT plus u dot grad B, and that's just the Lagrangian derivative. That's a derivative in the reference frame of the particle, right? So so we move
That over here, and we end up with, from these four terms that we got from the cross product here, the curl there, we end up with two terms. We end up with B dot grad U and B del dot u. Okay, this is just the same diffusion term we had before. These I kind of understand. Okay, the first one says that I can generate a magnetic field if I have gradients in the velocity in the same direction as the magnetic field.
Okay, visually what does that mean? Let's say I have a magnetic field like this, okay, and I have gradients in the velocity, let's say in such a way that I could stretch—I have to be careful with my gear—stretch magnetic fields in that way, then I can stretch the field and get B dot U. So this is stretching due to differential shearing motions. Okay, this term involves L dot u. Okay, this has to do with dilatation or compression of the velocity field. So this says you can also generate a magnetic field if you kind of expand a region or contract it. Okay, which again makes a sort of more physical sense to me. Okay, so these are really the mechanisms involved in generating magnetic fields, and the third term is diffusion. All right.
The last thing we'll talk about in terms of magnetic Reynolds number, in terms of the magnetic induction equation, is to derive this property that's going to tell us a criterion that needs to be met in order to get dynamo action. Okay, in order to get magnetic fields that grow, so dV by dT not going to 0, this term better be bigger than this term. You must create fields faster than you diffuse it. Okay, so the magnetic Reynolds number is essentially a measure of the ratio of this force to this place. Okay, and if you use characteristic scales for everything, you end up finding that this goes as the velocity times the length scale divided by the magnetic diffusivity. Okay, so this is the magnetic Reynolds number, and if you do some theoretical analysis, you find out that this number has to be bigger than at least roughly 10. Simulations tell us probably 20 to 50-ish. Okay, in order to get dynamo action going.
So now what we can do is we can take this magnetic Reynolds number and we can say, well, what are those four typical objects that we think of as dynamos? Okay, so in Earth's core, if we come up with estimates for the magnetic diffusivity, for the length scales, for the velocity fields—we get these from those magnetic maps of time, how fast things were drifting—we end up with a magnetic Reynolds number around 4500 or so. Okay, I shouldn't mention that this is somewhat of an old slide, and I need to update this number, and I'll tell you about that in the next session. So those numbers actually probably closer to a half than two, but it won't affect the magnetic field somewhat too much, make it a bit bigger. So the Earth, this is roughly around 500; that's fairly bigger than 10, so we're pretty safe that dynamo action occurred in the Earth. But look at a star. Okay, a star—a star wins because of two things: first of all, the length scales in stars are much bigger than planets, and the velocities are probably much faster. So you get a ginormous magnetic Reynolds number inside stars. Okay.
Then you think as an experimentalist, you're going to go build a dynamo in your own lab. Okay, you probably want to work with as conducting a material as possible, which means it's low magnetic diffusivity as possible. So let's say we work with copper; that's probably as low as you can get. Okay, a length scale—you're going to have to fill this tub right with a lot of this material—today the length scale that's feasible in terms of the fact you're going to also have to move this stuff, and moving it requires power, and power is proportional to the length scale to some horrible power, cubed or something like that, means we're probably confined to length scales around a meter. Okay, and power requirements tell us we're probably confined to velocities around one, and you get a magnetic Reynolds number of six. Okay, so it's just horrible. This is probably lower than the critical value. Okay, typically we want to make it much bigger. So doing experiments is very hard because it's very hard to get the magnetic Reynolds number criteria met. Okay. Okay, are we close to the break time? Okay, if I was at five minutes.
Okay, so the last thing I want to do in this session is talk about the actual equations that we're going to use to study dynamos. Okay, so the magnetic conduction equation is the one that's going to give us how the magnetic fields evolve in time, but typically when we model these things or when we think about these processes, there are fluid motions involved. So we need the equations governing the fluids. Okay, typically for fluids, you're talking about a combination of a conservation of mass equation, which looks like this, where Rho is density, okay, and a conservation of momentum equation. Okay, so your standard conservation laws. In the conservation of momentum equation, you have—this is essentially the acceleration in the in the Lagrangian frame—okay, because we're rotating, and we typically deal with things in the rotating reference frame, we end up with a Coriolis force, so omega's rotation rate here, u is velocity, okay, and then you have all the forces acting in the problem. Okay, there are contact forces, and these are essentially pressure gradient forces and viscous forces, by these two terms here. So nu is kinematic viscosity here; there's gravitational forces, and if you have any differences in density of materials, you'll end up with buoyancy forces from this. This is what's going to drive convection in our problem. Okay, and then you have the Lorenz force, where here I've written J as one over mu naught del cross B. Okay, so this is our momentum equation. Okay, and whenever you're doing a problem like this and you're trying to figure out, while doing, have enough equations, is everything fine here, you need to figure out your number of unknowns; then you have to count your number of equations and make sure everything matches. The problem at this point is Delta Rho; we don't really know what governs Delta Rho. If we assume that the changes in density are going to be due to thermal properties, so things that are hotter or less dense expand, then we need further equations; we need to know how Rho depends on temperature, and then we need an equation for temperature. Okay, so the other things we get, for example, using equations of state, you can see how the density depends on things like temperature; it might depend on pressure. Okay, and planets, usually we just assume a very simple linear temperature dependence. Okay, and then we need an equation for the temperature, and that comes from conservation of energy. Okay, and conservation of energy will typically give you an equation that looks something like this. Again, this is just coming from entropy conservation, essentially. This is ohmic dissipation; this is viscous dissipation; this is internal heating; and this is thermal conduction or thermal diffusion. Okay, so these are the equations that you need in order to generate, in order to study dynamos. But what we're going to do a lot of in this—and I'll end on this slide for now—is we don't like working with dimensional equations; we like making everything non-dimensional so that we can figure out what are the important combinations of properties that are important, and if you non-dimensionalize your equations with characteristic scales, you end up with four non-dimensional numbers that govern the problem: the Rayleigh number, the Prandtl number, the Ekman number, and the magnetic Prandtl number. So when we start again after the break, I'm going to show you what these actually are in some planets and what they are when we do models of these certain things. Okay, so maybe now is a good time to stop, take a break, and what time should we reconvene? Oh, okay. Okay, so there's homework going around, okay, that I think there's a session tomorrow to discuss the homework; is that correct? Okay, are there any questions about what I've gone over so far? Sorry? Yes, so I'll talk about these numbers and what they represent and also what their values are in the next session. Okay, are there questions? Okay, then also Sabine. Okay, I noticed it's on the—but I was able to access it via the website. Yeah, the Brink might see it instead. I was like, okay, I didn't realize it was there. Okay, then any other questions? So 10 minutes, so at 10:40 we'll reconvene. Okay.