Transcription
All right, we're gonna get going again. One of the organizers has explained to me, extremely worried about the fact that we're at slide 34, and in the file that you have, it goes up to 105 or something like that. So relax, I am NOT going to cover all of these slides. Okay? This is a very modular talk; it's planned that way. I would much rather go through 45 slides slowly, with lots of questions, than run like a lunatic across, you know, the hundred-slide presentation. So no worries about that.
So we're going to pick up, in some sense, where we left it, in the sense that we know by running these large simulations, we've—we think that we've captured something of dynamo action in the Sun and sun-like stars—at least stars that are not too different from the Sun. It seems to work as well in stars that are more different from the Sun, but the fact is that these simulations are really big: hundreds of core years of computing time to get that. You got terabytes of data that you have to move around. And if you've read, you know, the books and stuff like that, models of the solar cycles have been around for a lot more than the last five years, right?
So what we're going to do in the first part of the second lecture is go through, you know, the more conventional, simpler approaches to the problem that are still dominating the literature. So, you know, in terms of education, this is an important thing to do. Now, the basic idea is actually quite simple. I'm gonna put this there, and we are going to—I will explain to you in a very cartoon, simple fashion what is usually, still to this day, considered as the milk—the most likely mechanism for producing large-scale magnetic field out of small-scale flows. For historical reasons, it's called the Alpha effect, but even before it was called the Alpha effect, this guy Parker came up with the basic idea. So that I am going to select—huh, yeah, fire effect. I have slides on this coming up, don't worry. I'm gonna pick volunteers, and I like your haircuts, so that's—it's as good as a new criterion, right? Oh, I know all about that northward. So, so now these guys, not much for the time being, but you just want this end of the rope, and you go back a bit and hold this end of the room, so a bit more—not too much tension—yeah, that's later. Okay.
So suppose this is a magnetic field line. So this is—this could be, you know, like al—or a big magnetic loop, a large-scale field that runs along the whole Sun. So it's a field that is structured on a large scale. Convection in the Sun is influenced by rotation. What that means, you know, on the Earth, when you start to have these big upwellings, they turn into cyclones; they always burn in the same way. That's because the Coriolis force is getting a preferred sense of twists: Phillip flows and down flows. The same thing happens in rotating stars because of rotation, because of the Coriolis force. Remember that in MHD, the current density is given by the curl of the magnetic field. So that's the right-hand rule business. So yeah, maybe not—maybe in the front you can work better, get over here. So like, back to basic E&M. Okay, if this is the wire with a current, right, you take your right hand, and if the current goes this way, you have a field that wraps around like that. That's the field induced by a rectilinear current. Now there's a really funky twist in MHD—in the MHD now imagine that this is now a magnetic field line. Do you remember from first-year E&M how you make a uniform rectilinear field? Starts with S—so annoying—exactly. So a series of current loops like this will produce a nice uniform field that way. So in—so if you have a—if you have a magnetic—a uniform magnetic field, uniform on this scale here inside the Sun, it's being driven by currents that are in planes perpendicular to that magnetic field line. Now suppose that convection arrives and twists this like that, and twist—no, not too much. So I'm making a twist in—in this one direction by a quarter turn. So the field is going this way, this way. Now this is like a circular magnetic field line, right there. The associated current density runs this way. This is perpendicular to the currents that hold the uniform field. Now if these convective—from plumbing happen all the time, all over the place, you add all of these together, and what you end up with is a nice current density that is now running parallel to the magnetic field line. So out of the uniform field, using small scale that can occur cyclonic events, you build a large-scale current density that will then induce a field on much larger scales, on the same scales as that thing. So thank God—or whoever you like—for the superposition principle that holds with Maxwell's equations, right? We can add up all these little bits of induced field and build a large scale out of that, except that there are a number of catches. Suppose that my cyclonic event arrives, does this, and then twists half a turn instead of a quarter. My current density is now going this way; doesn't do me any good because I already have current going this way. If it goes 3/4 of a turn, well my current density is going this way. So if this guy goes 3/4 of a return and this one goes 1/4, I've just induced two small bits of currents that cancel each other. All right. So the only way that you can get something coherent out of that is that everybody's gotta agree to turn a quarter turn or something like that—unlikely.
Well, under what condition can this happen? There are three ways I can wear this is possible. I'll do all three of them now. This time you have to work; I need tension. Okay, okay, I think I need more tension than that. Okay, this—so this is good. Now suppose that the field is really intense, and this convective event comes in, right? Now, you know, I've done a lot of weight lifting in my life, but there's no way with these two guys pulling this way that I can, you know, make this loop I was doing before. Tension resists it. So in a situation—but this is not necessarily a bad thing. If I can just tension—if I can just twist a little bit—that a little bit of twist, right, still has a component of current going this way, but their tension is making sure that I'm never gonna go beyond, you know, a quarter turn or even an eighth of a turn. So in a situation where the strength of a large-scale field is a lot larger than that of the small-scale field, this process can work, except that in the Sun there's much more energy in the small-scale field than in the large-scale one. So for Sun and stars, this doesn't work. The other way that this could possibly work is if I'm in a situation where the coupling between—so the coupling between—less tension, please—thank you—so that the tension between the flowing field is not so good. So if the coupling—not the tension—the coupling between the flow and field is not good, my magnetic Reynolds number is not a lot larger than the flux freezing is not holding very well. When that means is that whenever I try to do this, it slips. I twist a bit and it slips, but still, as long as I can get a little bit of a twist, then it works. So that process will function if the magnetic Reynolds number is a lot—while is of order unity, except that in the Sun and stars it's like ten to the eight, ten to the nine. So this one is out the window as well. So we're down to the last option now.
Well, I told you is that we get in trouble if this twists starts going, you know, many, many ways around. That's when things end up adding almost like—like a random walk in some sense. One way to avoid that is if the lifetime of the convective upflows is quite short compared to their turnover time. That's called short coherence time turbulence. If these convective updrafts and downdrafts lose their identity really quick before they had a chance to do a full turn, then what happens is that this upflow comes around—what—a little bit of a twist, goes away, this one like that, and the collective addition of all of this, because none of these guys was able to go more than, you know, not even a quarter turn, but none ever went a full turn or a half a turn, at that point it still works. So by elimination, we can think that this process might work in the convection zone if the can—if the turbulence is so vigorous that it's—it loses its coherence really quick. A cyclone in the Earth's atmosphere, you know, this thing lives for weeks and weeks, right? So if you happen hopefully enough to be in there, you're gonna spin around a couple of times before you finally dropped somewhere, right? So that is law—that that would be called a long coherence time, you know, turbulence if you want to call it that. So we want to be at the—in the other regime. We want to be in the regime where as soon as another—for—and the flow forms, it disappears and another one happens and another one happens. So under the circumstance that works—yes, if no—because you twist one way when you're going up, one way when you're going down, you need cyclicity. You need a correlation between sense of twins and upflowing, downflow. But Jeff is basically saying is that while you want to conserve mass, so every time you have an upflow you have a downflow. So if I twist the same way when I go up and down, things cancel out. But the point being that what matters is what's called a flow helicity, the correlation between the helicity and the radial flow direction, and the Coriolis force does—is the right way. Probably didn't make any sense, neither the question or the answer. There's symmetries and stuff like that, but that is basically the basic—other—you—thank you, gentlemen.
Now we need to make math out of that, to put it into a model, and I'm gonna do that—I'm gonna do that here relatively quickly because we are, in some sense, running out of time. And this is done very well in the chapter that Matias Temple Lucy are back—they are rolled for volume 1. Okay, he's covered that topic in great detail in Chapter 3, I believe, of Volume 1. So this is the twist I was mimicking or imitating, and the idea that this twist takes place along these magnetic field lines all over and add up, and the addition of all these processes leads to a large-scale current system that, in this case, can produce a magnetic component. Now the way you—oh, that's why—very confusing—we have to these things. Okay, that's fine. So mathematically, how you do this—and again, I realize I am going through this quickly, but again, it's done in all great details in the book—you separate your total flow in field into a large-scale component and a small-scale component, and that small scale is such that when you average it—that's what these brackets are—think of the zonal average—these things go to zero. You take that and you plug it into the induction equation. So every time you add d B/dt, becomes d/dt of that thing plus d/dt of that thing, and you average so that all these bits cancel out. What you end up with is a something that looks exactly like the induction equation you started with, except it's for mean quantities. We have the U cross B term; we have the dissipative term written differently here. This is the novelty; this is something that is related to the—the cross of the small-scale flow in the small-scale B. Now these might average to 0 individually, but they don't necessarily average out to 0 in their cross-correlation. For some type of turbulent flow they will; for others maybe they don't. And this thing is the turbulent electromotive force, and that is the thing that mathematically captures the process we were mimicking in the front here with them—with my piece of rope.
Now the way you tend to do this—the turbulent electromotive force is a vector. What I've shown you here is that I have a large-scale magnetic field—right, that piece of rope—and they have these small-scale events that produce this large-scale current. The current is like producing a large field electromotive force. So this—this electro-motive—this—this thing that I'm writing here is a function of the mean field. I'm working on a mean field to twist this thing around. The mean field is a vector quantity; the electromotive force is a vector quantity; it's oriented in the certain way. And the math—a back mathematical object that you need to link to vectors is a tensor. So we're going to link the—from all the force with the mean field through a tensor, and we're actually going to express—develop this as a sort of like a Taylor series expansion in terms of the mean field and its derivatives. And the point being that these—if you do it that way—these things should not depend on the mean field, and they can only depend on the statistics of the flow. Now Alpha, Beta, Gamma, Delta, and so on—that's an infinite series. The reason it's called the Alpha effect is that we decided to use the Greek letter alpha for the first term of this expansion back in 1966, I believe. And astronomers being creatures of habit, you know, no one went to Sarah Lake alphabet or anything like that in between. So this is now and forever known as the Alpha effect, the first term in that infinite expansion that we usually truncate after the second term. So the first term looks this way, and if you have, you know, isotropic—almost isotropic turbulence—then the tensor has to be isotropic; it ends up being a scalar. And what this is basically expressing is that for turbulence like that, the mean electromotive force is actually parallel—because this is a scalar—to the mean magnetic field. And this is precisely what we were showing with the rope in front of you guys before. And you know, you can actually—by—by various approximations that will hold well—that are expected to hold for small coherence time turbulence—you can express that scalar in terms of the flow helicity, which is that thing—that big—the correlation time of the turbulence. And what is this—this is basically telling you is that the source term—this turbulent electromotive force—is proportional to the flow helicity, and the flow helicity or cyclonicity is what is being driven by the Coriolis force. And this is why rotation is so crucial in this business. Without rotation, you don't have net flow helicity, and without net flow helicity, you don't have enough. And you can go further in—or by making assumptions about the statistics of this—and you know, all of this is in the books—we can measure the Alpha effect out of numerical simulations. There's various ways to do this, and you know, the results that you get for some simulations anyway actually fit not so badly with what these fairly simple analytical theories provide you. So there's really an alpha effect that changes sign—changes sign across the equator. That's a slice of the simulation; rotation axis is like that; equatorial plane of the plane this way. These are meters per second; that's the units of this alpha. And the fact is we can actually, in those simulations, measure a large-scale alpha effect that kind of looks like what it's supposed to look like. And if I measure the flow helicity, you know, these things are—we—this almost looks like the negative of that, which is what was predicted by this earlier expression.
Now you can separate—I go quickly to this because it's really getting kind of technical—you can separate the symmetric and antisymmetric part of this alpha tensor. The antisymmetric part you can write as a pseudo flow. So this is not a real flow. Okay? This thing is called turbulent pumping. If you put a drop of ink in the Sun, you know, turbulent flow is not going to move the ink around. It's basically a part of the electromotive force that corresponds to displacing the large-scaled field—mathematically—from the point of view of the mean field—in the mean field only—it looks like a large-scale flow, but it's really a turbulent effect. And that can be measured as well. I'll spare you the detail, but I'll just say that that thing—that flow speed—that uses a turbulent flow speed that you can calculate out of the simulation is actually quite significant; it's measured in meters per second, which is a comparable to other large-scale flows that develop naturally in those simulations from thermal driving. So it's something that might play—that might play a role. Now you go to the next term—the term in that series expansion—the term that depends on the derivatives of the mean field. And again, under the same approximation, you can calculate that this thing ends up being proportional to the square of the turbulent intensity. This is the famous turbulent diffusion that some of you might have heard about. Yes, this is not good; this is actually destroying the magnetic field; it is creating small-lane scale that enhance dissipation. There is no free lunch in this business. The same turbulence that gives you the alpha effect that you need is also giving you enhanced dissipation, right? That's well—so you wanna—you only be in a situation where the source term that you get out of the Alpha effect somehow does not get overpowered by the enhanced dissipation you get from the—sent—from the next term in the development. So it's not—it's—it's rather tricky. So that disturb—a little ikram—of the force that we mimicked, you know, quite easily there with this piece of rope is actually a complicated beast. It induces field—the large-scale field—it transports the large field—field by turbulent punting—and it destroys the large-scale field by turbulent diffusion. So it's actually a pretty tricky business. I've shown you this before. There is a—another way that you can create—but not create is not quite the right word—there is another way where you can take small-scale field and build up a large-scale component. This famous, you know, self-organization I was talking about—that's the surface magnetogrand we were looking at before. Notice these streaks that start on active regional attitudes and extend all the way to the poles, right? These are sunspots and active regions that are decaying. Sunspots are strong concentrations of magnetic field, and as they decay, they release magnetic field into the solar photosphere, and surface convection and poleward large-scale flows carry the sealed high latitudes, you know, and this magnetic field accumulates in polar regions. So this is a process that takes stuff on very small scales and, through this process of transport, diffusion, and accumulation, builds up a large-scale dipole moment. This is yet another way to go from small scales to large scales, and it certainly seems to be happening in the Sun. What's it—what's interesting if you put numbers into that—I mean, just some observations—the total flux that emerges in one activity cycle is of order ten to the seventeen Weber. That's the SI equivalent of Maxwell's, up to a factor of ten to the five, I think. The peak polar cap flux is about ten to the fourteen Weber—three orders of magnitude smaller. So what that means is that if you can somehow tap into that small-scale stuff and only manage to convert point one percent of it into something coherent, you have enough flux—you reverse the dipole—of the surface dipole of the Sun—to create a large-scale component that is, you know, comparable to what we see. So that is yet another form of—of going from small scales—large scales. That's a surface simulation of this process. So we're showing you—you know, this is supposed to be positive and negative magnetic field—so this is a pair of sunspots, and we let this stuff, you know, be carried by surface differential rotation—original flow—diffusion. So the stuff diffuses away or wraps around, and slowly but surely you build up a large-scale dipole moment. And if you take observed sunspots emergences and put them in the model like that, you can produce, you know, surface magnetogrand synoptic that look an awful lot like what you see. So what this is basically saying is that this process is very much—is very likely—it is taking place at the solar surface. We understand it; we can model it. And the outstanding question is: is this dipole just a side effect of the fact that we have sunspots emerging, or is it feeding back into the dynamo loop? And the answer to that is, at this point, we don't know. There are people who would swear, you know, that this is a key part of the dynamo and even consider it—it's a very sobering thought—the thing that even with the Sun that we can observe in such a great glory detail, we still don't really understand what the dynamo process is.
Is and what are things that we can observe are just a side effect or an important aspect of the dynamo? Now again, at that level, numerical simulations are on the way of helping us. It's a fairly recent development that out of these, the types of numerical simulations that I was showing you, we are reaching spatial resolutions in turbulent intensity levels such that this is this is a cutout; the sphere is like that. This is the inside of the simulation, and these are basically magnetic field lines within convecting layers. The rotation axis runs this way, so this is basically a large-scale structure that is being built by convection and very turbulent, as you can see. And that large-scale structures develop these loops; these loops that start, you know, rising to the surface. And even though there's not really a surface in the simulation, these things are exactly what we think would eventually produce sunspots. So we are slowly getting to the point where even with miracle simulations, we might be able to capture that process and answer that question: whether it's a side effect of the dynamo or an important part of the dynamo.
Now what I want to spend the remaining 15-20 minutes doing—that's the time we have left—is to try to look at how we take now these various simplifications of these dynamo processes and build them into a model that can actually be run, you know, without waiting months to get results. The types of simple dynamo models that I will describe here are precisely the ones that you will be working with in the lab part this afternoon. So we're kind of, you might see we're already, you know, in in the lab session here, we're starting to do look into what you'll be doing in the lab. So remember, we started there. So what we will do: we'll go back to using only this equation, and what will—is for you the thing that does all the work—will do two things: we will input are steady artificial flows that will—this will design them to look like the fluid, the large-scale flows we observe in the Sun. And for the small-scale component of that thing, we will use this alpha effect and this mean field theory to specify source terms that will capture the effect of these small scales. Okay, so this is a this is a kinematic model; I specify the flows, and I just solve this equation and look at what happens. We will do it; we will do a geometrical simplification as well. We will take the total magnetic field, will, and will decide that we will only solve for the axisymmetric component. We're not going to care about all the small-scale field; what we want to get at is the dipole; it's the Detroit flux system that gives us sunspots—nothing that depends on the azimuthal angle. So my magnetic field, which is normally a function of all three space coordinates and time, is now a function only of depth and of time. Mathematically, you can express a field like that as the curl of a vector potential that only has a Phi component plus the Phi component of the magnetic field. You can do something similar with the large-scale flow, and if you do that, that's a—you can actually, by substituting that separation into the induction equation in doing a fair amount of vector algebra, you can end up separating your induction equation into two evolution equations for that poloidal vector, toroidal vector potential, internal magnetic field. The solutions that you will be working with in the lab are solutions of these equations. This is what you'll be working with.
Now what is missing here that that does not come out of this procedure is a source term in the poloidal field equation. This is where this alpha effect business comes in. This does not—write that this twisting—now if I decide that nothing depends on azimuth, you know, this twisting, I cannot represent in there because the fact of taking a field line that is aligned in the zonal direction and twisting it that introduces a zonal variation, right? By the very design of this model, I'm throwing this out from the beginning, so I need to come back and add this stuff artificially there. Now I do have the shear of differential rotation that is a very important component in these models. This basically takes the poloidal field—so that's magnetic field lines that are in Meridian planes—and it stretches them in the zonal direction. This is one important part of the Dynamo loop in these models. So for that, this is a parametric expression that basically produces contours of angular velocity that looks qualitatively like the Sun: rapidly rotating equator, slowly rotating Pole, and a few layers straddling the base of the convection zone. So this is a parameterization that is used in the models that you will be working with. And if you take a dipole and you let it shear by such a differential rotation, you get a toroidal field that is antisymmetric about the equator, which is what observations tend to tend to indicate. So this is basically an important part of the Dynamo process in the model that you'll be using.
Now we also have this large-scale flow, UP, is a flow in Meridian all plane. This is also observed in the Sun, and we will just plug in a big nice single-cell flow. They are recent—Helios make observations that indicate that it's most likely more complicated than that, but believe me, the model you're working with is complicated enough already that we're going to stick to this just to see what happens. Now that flow will carry magnetic fields at the surface; it runs like this anticlockwise here; it will carry field to the high latitudes. We actually see that at the surface of the Sun, and if you want to conserve mass, you need this flow that will bring things back, you know, back down along the base of the convection zone. That flow in the Sun is driven by the turbulence; it's so it's again a product of turbulence. Now these source terms we're going to consider too; there's actually many possibilities that they've been worked out. There's this alpha effect business we talked about, and there's this active region decay mechanism, which is called a Babcock-Leighton mechanism for historical reasons—the two guys who worked on this developed this originally—and these are basically the only two source terms that we will put into the model that we'll be using.
Now 15 minutes, okay. So just to give you a bit of a preview of what you will see: we'll explain this later, but we'll put you in groups, and you and you will all work on different versions of this model. Different versions means that some of you will use this alpha effect as a source term; others will use this surface decay as a source term; some of you will do models with meridional flows, and we'll do all those models without. And at the end, we'll come together and try to see what comes out of all this. That's the idea. So I wonder rapidly in my remaining 10-15 minutes, gives you a few examples of some of the solutions you'll be working with. I'll show you what they look like so that you have a bit of an idea what to look for when you start. The first thing that we'll do: these are driving equations inside the numerical code that that has produced solutions you'll be using. We are basically using non-dimensional units, and there are these various dimensionless quantities that will appear. The detail of this is not important, but these are the knobs that govern the behavior of the solution. This guy measures the strength of the Alpha effect; this one measures the strength of the differential rotation; this guy measures the strength of the meridional flow; and in a few models like that, there's actually a fourth one that will measure the strength of that third bowl of turbulent pumping, so I was telling you about. Okay, so you will you will see these these guys appear. Now I'll skip that. This is one model that that's about half the models that you will be using in the database are basically based on retaining the Alpha effect only in this equation and retaining differential rotation in this. The Alpha effect can make contributions to this equation as well; we've drawn it out; details won't matter. Now if you make a model like that, you can run it in different ways. I will just show you an example of what happens when you write the solutions as eigen solutions. Now if the the flow is time independent, everything is time independent in the coefficients of the PDEs, and therefore you can look for eigen solutions where this thing, you know, is the eigenvalue; that's a growth rate; that's a frequency. So for that model, if you if you solve this as an eigenvalue problem and you increase the strength of the Alpha effect—that this thing—that's this thing here—what you find is that if the growth—if the Alpha effect becomes too weak—the growth rate that Sigma is negative; if you put a negative in there, it means that this thing decays exponentially. So this is a star that is not convecting hard enough or rotating quick enough to be a dynamo. There's a certain threshold you have to reach before this thing can be a dynamo, before induction can be dissipation. That's actually characteristic of all these models. The other thing that you see here is that the frequency keeps increasing with with that that dynamo number for these linear models. On which—what that means is that if you get cyclic solutions—so you have you have a real frequency here—if you get a cyclic solution, the harder you can affect, you know, the faster this thing will cycle. This is kind of cool because again we see stars of different rotation rates that they have different dynamo periods. It looks like this. Their solution database that you have contains not only the numerical solution that you will analyze but also animations of these solutions, so you get a bit of a feel for what these things look like. And that's the animations that are in there. So this is the rotation axis; this is the equatorial plane; above the dashed line is the convection zone; this is a stable layer; there's a color—the solid colors back there—measure the intensity of the zonal field, of the toroidal field, and these lines—clockwise like this, know clockwise like this in orange, counterclockwise in blue—that's the poloidal field; that's the part that's given by the authorial vector potential, and that is just a toroidal component. So this is a snapshot; it doesn't look like much, but as we look at the time evolution of these things, it becomes more interesting. Not only is the field reversing polarity, but then as they—it is moving—you know, these these solutions—it's not just like the Earth's dipole that stays more or less—with all due respect to people who model this—it's of course a lot more complicated, but to a first approximation the Earth's dipole is almost aligned with the rotation axis, and then every once in a while—loop within a few hundred years—it flips. This stuff not only changes polarity but propagates spatially, and you will see things like that in the solutions that you will be looking at. And this is good because we do see some spots appearing closer and closer to the equator in the course of the solar cycle. So these dynamo waves, as they are called, were the big success in the late 50s of the first attempts to model the turbulence electro-motive force because they could explain the observed equatorial migration of the sunspots. That's a butterfly diagram. What we've done here is we've taken images of the Sun, and in strips of latitude we computed the percent of the surface area covered by spots. So this is basically tracing over 140 years where sunspots have emerged in the course of the various cycles covered by this diagram. And early in the cycle, sunspots emerge at high latitudes, and as you advance in the cycle, they get closer and closer to the equator until they show up again at high latitude. That behavior can be at least qualitatively reproduced by these mean field dynamos.
So these are the two solutions we were looking at before, with one that ran out of gas for some reason. And what I've done here is I've again taken that solute—oh, there we go—okay, magic laser! Magic laser here. So we've made—we've extracted the toroidal field at this depth and made the usual time-latitude diagram, and you see the secret or—word propagation of the magnetic field; it's at two high latitudes, okay, but it's at least going the right way. This one is not only stuck at high latitudes—north pole, equator here—but it's propagating the wrong way, okay. So this can make dynamo waves, but the waves don't always go the same way; it depends on how you set up the parameters in the model. Let's get that—let's get that—we'll skip that as well. This is now changing the latitude, you know, dependency of this famous source term. So it's the same sort of thing we're looking at before, except obviously here you see something entirely different. You no longer have an ice dynamo wave, but you have a much more complex type of behavior where you seem to have—especially on this one—this was supposed to run in the loop—you seem to have two cycles: one starting here going this way, one starting there and going that way, right? If you look at this thing carefully enough, that is a consequence of the shape of the differential rotation inside the Sun. There are many shear regions; you have a rotation decreasing radially outwards here, increasing radially outwards there, and you have a latitudinal shear across the convection zone. Now when you run these models, the shears all contribute to the production of magnetic field, but they do so in a manner that is almost independent from one shear region to the other. So you can get these strange—you can get these strange modulations; you can get multiple dynamo modes that will run simultaneously. This is now taking that same model that—add the magnetic activity concentrated at high latitude, right—and now we've added that this large-scale flow that turns like this. And if you do that, you get an entirely different type of dynamo behavior, which is what this was supposed to show you. You can tell that the field is being brought here and dragged by the flow. This is no longer a dynamo wave; this is this treadmill, this conveyor belt, very little flow that drags the field around. Some of you will be working with models of that variety, you know, and when we do the lab. So that's yet another dynamo model that produces polarity inversions, you know, and at that level of complexity is a decent model of the solar cycle, except it's not always that easy. This is a whole sequence of these time-latitude diagrams where I start with this solution that that's too high latitude and I crank up the speed—that parameter, you know, that drives the flow—first thing that happens: I kill everything; the dynamo dies. But if I keep turning it, it reappears again, looking entirely different. This solution, which has a different source term for the Alpha effect, turns into a steady field—the field that doesn't reverse polarity—that one goes totally bonkers. So what happens to these dynamo modes when you increase or decrease that flow is actually quite complex, and this is again something that some of you will explore. Laughs. You'll take simple models like these and change various parameters, add various kinds of flows in there, and see what happens. This is a surface—this is what we would observe at the surface—so again, time-latitude—it's you—north pole, equator—this is you know, in one solution—I think this is what the surface field evolution looks like when I don't have any radial flow, and as I crank it up, you can tell that you can get entirely different behavior. This accumulation at high latitude is solar-like; this is not solar-like. So that's another thing you'll be looking for, you know, what happens at the surface. Now you can also—I can go quick on that—you can run similar models where you concentrate the source term in the very outer layers of the model instead of being distributed across the convection zone. So that these are—this is the equivalent in that framework of these models that rely on the decay of active regions to drive—drive the dynamo. And that just means a different source term; again, details are not do important; all you need to know is that mean, you know, I have a source term that is strongly concentrated in surface layers, and you can still make a model that looks—that looks pretty good—that reverses polarity, then has a period of something numbered in five, ten, fifteen years. So again, the flow, the poloidal field, the field lines are being driven; they appear here; they'll get dragged to the pole; to get pulled down there; you mean differential rotation induces this large-scale field, and you get again a treadmill kind of dynamo where this time the source—the source term—is is completely different from the turbulent alpha effect, and you can still make a reasonable dynamo out of that. So this is the unfortunate situation we're in; let's just say basically that's our butterfly diagram of sunspots. So again, you see a nice tendency to move towards the equator, except now it has nothing to do with dynamo waves; it's simply the—it's simply the passing—there may be—there may be a little flow that drags this—and this is time—time down there—latitude—north pole, equator—this is pretty solar-like in terms that you accumulate the surface field, you know, in the polar caps. So these are the kinds of models that you will be playing with, and there's quite a debate right now in the dynamo—solar dynamo modeling community—about which of these two classes of model is best. You could argue that they're both equally too simplistic anyway because they don't really do any dynamics, but in terms of, you know, reproducing solar features, they both have, you know, pluses and minuses; some do certain things quite well, and others do other things not so well. I will skip that. I think by now I would rather stop. We'll take a good 15-minute break, go to the bathroom, get some coffee, and when we come back, I will I will show you just a few more things on the screen, just explain you really logistical aspects of using the various dynamo codes that you will be using for the lab. That'll take maybe 10 minutes, and after that, Nick and I will go around and try to make sure that you guys can run the IDL analysis, cause that—yeah, I will give you a little demo of what this will look like, and then we'll set you up to do it, and then it will distribute the tasks, and then we'll be busy for an hour or so. Yes. Yes. No. No. I mean, it's it's a very good point. The geodynamo you heard about last week and what I've presented you—I'm not in the first—in the first lecture—the simulations—we are solving the same equations, the same partial differential equations; our geometry is the same; we're both in the sphere, okay? I mean, there's a way in which these things really look—really look the same, but what is really different, I would say, are the parameter regime; is the intensity of turbulence that we're dealing with; the impact that the Coriolis—Coriolis force—has on the turbulence is very different than the Sun than it is in the Earth's interior. It's really any—I would say it's really an issue of parameter regime. I don't know if you want to add something to that. Scale heights? Yeah. Stratification? Jewel dynamos? Probably no. Yeah. Yeah, that's—well, that's—yeah, that's a parameter in some sense. Yeah, there's compositional driven convection in planetary interiors which do not play a role here. They know that there are some physical differences, but that I think it's really the parameter regime of the rotation and the convection that that really are the the key thing. We could try to set something up for the discussion session tomorrow afternoon, maybe. Okay. Good. Yeah, Jeff, right. That is actually something that these guys will be exploring. Two of the tasks that will be assigned to two of the teams is to take this model and run it with different depths of the convection zone. So if—as long as you have a tachocline—you get changes in cycle period and magnetic field intensity. If you start to have to go really, really deep down and you do not—no longer—a tachocline, then at that point you might switch to a more purely turbulent dynamo, which then might become a steady dynamo rather than the cycling dynamo that is what mean field theory used to predict, but that's not really borne out by my simulations at this point. You do get irregular polarity reversals in simulations of fully convective stars. I mean, maybe not reversals, but oscillations and things like that. So the answer is we don't know. Yes, exactly. If if star spot production takes place more or less like in the Sun, I think there are observations of photometric variability where it looks like the rotation period of the star would be changing, which doesn't really make sense, and the interpretation is that the photometric variability you're seeing…
Is due to spots. And if spots appear at different latitudes, when you catch them, you'll see a spot at low latitude, so you'll measure a rotation period that corresponds to this latitude. And if it's at higher latitude, you'll see something else. So people have actually claimed to have measured differential rotations in stars by exploiting that, that difference in that apparent difference in rotation rate that you get by doing photometric tracking of star spots. They've got like that in half a dozen stars, I think, by now.
Yeah. Yep. We have a whole afternoon of question period tomorrow, so maybe we should—I don't know about you guys—I need their coffee badly. But anyway, I'll be around afterwards as well, after the lab, so anybody who has questions, we can continue this.