Stellar Dynamos #1 | Paul Charbonneau

I tend to speak quickly. I am aware of this fact, and I'm working on it. But you know how it goes when you get into your stuff; that bit rate goes up and all of that. So at some point, it gets to be too much. Just, you know, wave or do something. I should add as well: Anglish is not my mother tongue, so I'm still very vividly aware of what it means to sit in a lecture hall and, as a foreign language machine gun, that you had very, very high speed. So again, do not hesitate if this goes too quick; tell me, and I'll calm down, and we'll go—we'll take it from here.

So, stellar dynamos. This talk, show that normally logically have taken place earlier in the school, but the organizer is currently agreed to accommodate an unexpected schedule change I had to deal with. Tackle driver has already covered some observational aspects of this in Yerevan intro. I will actually have a couple of slides, just know it's been a long week, right, to remind you a bit about a few things, and then we'll talk a bit about magnetohydrodynamics. You've heard of that as well in the course of these lectures. I'll do just a little recap and make a few more points about that. I'll tell you about the numerical simulations of dynamo action, the production of magnetic field in star-like objects. That is something that is included in Chapter five of the books that you have, but it's one topic where a lot of new developments have taken place in the last—since that book was printed. So I'll spend a significant amount of time on that and then tell you a little bit about more conventional approaches to the problem. That should take us to the break.

And after the break, I will try to explain how we take all of that and make little models of the dynamo—of dynamos—for the Sun and stars. That is actually already taking us a little bit into material that will be covered in the lab session that comes at the end of the afternoon. The types of models that I will describe there are really the types of models you will be playing with when we do the lab. So as I work out through this material, I will point things out, you know, that are already in anticipation of the lab session. So that is how we're gonna go about doing it.

This is a magnetic ram of the Sun. You probably have seen one of these last week. It's basically a magnetic map where the color scale—the grayscale here—encodes the strength of the magnetic field perpendicular to the plane of the image. So the North Solar Pole is there; South Solar Pole. These big, very intense, kilogram-like field patches are associated with sunspots and active regions. But you see that there is magnetic field, you know, pretty much everywhere at the surface of the Sun. The—that stuff—all that magnetic field is produced by dynamo action. By dynamo action, as we will get into in a couple of slides, is really tapping into the kinetic energy of the flows in the Sun or stars and transferring some of that emerging energy into electromagnetic energy, making a magnetic field; pumping kinetic energy into magnetic energy. That takes place at all kinds of scales on the Sun—at least in the Sun it's like that—and that all kinds of scales. And that will turn out to be one of the major problems that we'll have to be dealing with.

Now, this is just one image taken at—on one day. If I take a thing like that and I divide it in strips of constant latitude and I then average the magnetic field in these strips, right, at that point, what I have is a function of latitude only. So on that day, I have an only averaged variation in latitude of the mean magnetic field. Sunspots and active regions tend to come, you know, in panels of opposite polarity. So if you average—or resultantly—on there, these things will cancel out. The thing that will not necessarily cancel out are places where the field is over a large area more unipolar—every single polarity. So if you play this game, you get something like this. I'm going to spend a bit of time on this diagram because the overall format of it is one that will recur in the course of this presentation. So this is time running this way, covering 2035 years from 1975 to 2010. This is latitude; equator in the centre; North Pole; South Pole. And the way this diagram was constructed is to start with magnetagrams like—like the one I was showing you before—averaging them in longitude; you get a function of latitude. And at that point, you repeat the process—they have today after day—and you stack these functions of laddies—1d function of latitudes—you stack them day after day after day, and you get this 2d function. This is—to the object—and then you code up the strength of the magnetic field, you know, using this color scheme. And what you see there is the solar cycle. These patches here—this is what's left of sunspots when you average in longitude—but you see that there is a dipole moment. This isn't positive; this is negative; that flips polarity. So what you're seeing here is the polarity reversal of the zonally averaged—as a neutrally averaged—large-scale magnetic field of the Sun. So this is the sort of stuff that we're trying to model and simulate when we make models of the solar dynamo.

Now, the Sun, of course, we can observe, you know, with very high temporal resolution, high spatial resolution, all of that. When you go to stars—which is really with this stars other than the Sun—then you don't—you can't afford that. You've heard this morning that we can see hints that there are star spots. We see them photometrically when you look at transits. You saw an example this morning. This is an older way to look at stellar activity, which is to look at the emission line emission in the line of the calcium—calcium H&K lines. This is a line that picks up emission—emission in its core—the core—the spectral line—in regions that have strong magnetic fields. So if you look at the Sun as if it was a star, you integrate over the whole surface, right? You don't—you pretend that you don't see a disk; that you only see a point. This is the calcium emission that you see, and the first thing that comes out is this cycle. And that is the same cycle that you were seeing on the preceding—yes; it just happens to be a spectral line whose core picks up emission in regions that are magnetized. So it's a convenient proxy and indicator of the presence of magnetic field. You can do that with magnesiums; there are other spectral lines that do that. It is a—it is a line that is—that is—that is really easy to pick up spectroscopically in stars when you do spectroscopy of stars. So historically, that's why, I guess, it's one of the lines that was being used for this. That's—yeah; you don't need to go into UV or do anything like that. There are lots of lines in the UV that are good for this as well, but this one is, of course, so anyone—that's the Sun. And if you do the same trick with stars—this is just a small sample—you see similar kind of stuff. These are—this is a small subset of a huge data set of calcium emission and stars that run—that started back in the 60s. That's a long way even for me. And what you see when you look at that is that not all, but many of these stars tend to show cycles a bit like what the Sun looks like when you look at it as a star, with periods that tend to run from—then here's the Sun with 10 years—11 years activity cycle period. The Sun is not unusual. You have stars that are Flatliners that don't seem to show many cycles; some that show rather irregular by behavior. You see all kinds of behavior.

Now, to—to actually take this measure of calcium emission and turn it into something like an average magnetic field strength is not easy. But one piece of observation that you can pull out of there that is entirely unambiguous—or almost completely unambiguous—is the cycle period. This comes out pretty directly from the data. And this is why, in trying to—people to work on stellar dynamos—trying to make models of Thera dynamos, things like the cycle period is often a very important constraint on your—on your models because the observations are rather—and it can be interpreted rather unambiguously. Now, we—these are all fairly long period cycles. We know now with Kepler data there are a lot of stars that show much shorter periods. So the range of periods that these things can cover is actually quite large. That's all I wanted to say about observations. You've heard about some of that last week, and there's a—I—you know, you could get full lectures just on that stuff, but I want to move on into magnetohydrodynamics, which is covered in a fair amount of detail in the first textbook that you have.

So magnetohydrodynamics starts at a place where everybody is comfortable, which is Maxwell's equation. Old stuff; 19th century; sailed through the quantum revolution without any problems; it was already relativity compatible even before relativity existed. So this is pretty good stuff. You've had that as undergrads already. This is Gauss's law, saying that electrical charges—that charge density—are the source of electric fields. Gauss's law is telling you that there are no such thing as magnetic monopoles. Faraday's induction law tells you that if you start screwing around with the magnetic field, you will induce an electric field. That's why these light bulbs are shining ultimately. And Ampere's law, saying that you can create a magnetic field either with electric currents or with time-varying electric fields. Some of you might think that I'm missing some 4π factors in there. Up in the great white north, we're told it was supposed to teach in the SI system of units. This is what this is, but I will still talk about Gauss and stuff like that. That's been too much time here. I'm kind of not being consistent, I'm afraid. So anyhow, this—as far as we know—applies to the production of magnetic field or the evolution of magnetic field anywhere.

So the deal now is to take this and apply it to a rather specific physical circumstance, which is a plasma of the type that is characteristic of solar and stellar interiors. So this is a relatively dense plasma—much a denser than what you get in the solar wind in the magnetosphere—and because that plasma is so dense—well, dense compared to all this other stuff—the collision frequency of electrons and protons and whatever is much, much, much higher than any of the plasma frequencies that a lot of you guys in your plasma courses have—you know—that'll do it in a plasma that is collision-dominated like that. It turns out that you can reduce all these equations into something that is actually fairly simple, and that's what I'll do now with you guys. The first thing we'll do is take Ampere's law and, you know, with all respect to the man, drop out Maxwell's displacement current. That turns out to be a good approximation if you're dealing with nonrelativistic fluid motions. We will also write Ohm's law like this. We write Ohm's law in the frame—cone moving with the fluid. So we have this plasma—this fluid—that's magnetized and moving around. So in the frame moving with the fluid, we write Ohm's law in this way. That's the electric—electrical current density. Yes, yeah, nonrelativistic fluid motions. If the fluid is long relativistic and no joker is turning batteries on and off here and there, you know, that this is—this is legit. Alright. Now, when you're—right—this is—this is dead already. When you're right Ohm's law in this way, you're saying that if I turn on an electric field this way, the current density is going to go this way, right? These are two vectors, and if I link them with a scalar, it means that the orientation of that vector is always the same as that one. You might recall from our earlier lectures that if you're in the magnetosphere or the solar way—in a very dilute plasma—where, for example, I have a magnetic field that points this way, well, if I—even if I turn on an electric field pointing that way—because of the influence of the magnetic field, my current density is going to go that away, right? So in a case like that, which is the most general case, this thing would be a tensor, right? The conductivity depends on the direction in space. In a plasma that is conditionally dominated, that's not the case. So that makes it easy. Now, this is through in the comoving frame, so we do a nonrelativistic transformation into the lab frame, and we pick up this piece, and from that I can write the ellipse. And by making use of the fact that in MHD this holds always instantaneously in some sense, I can write the electric field in this way. And if I take that and plug it into Faraday's law, I get this thing that you've seen a few times already this week. This is the magnetohydrodynamics—or induction equation—and it's telling you that the field—the magnetic field—varies and responds to induction by flow of a globally neutral plasma, but made up of charged constituents across a magnetic field. And this is what you get out of Ohm's law. This thing is called the magnetic diffusing, and it is inversely proportional to the conductivity. So this is what we work with.

Now you can take that equation and do some order-of-magnitude analysis. I—you guys are familiar with this—no, replacing time derivatives by one over timescale, you know, all these good stuff. So if you apply dimensional analysis to this, you know, this becomes that thing. That's a typical value for this; this is a typical value for lane scales; derivatives become 1 over line scales; this thing is linear, so the B goes away anyway. Notice that we've done away with everything geometrical about the orientation of flow in field, you know; this is really a scaling-type analysis. And if you do that and you look at the relative magnitude of this versus that, you get a thing called a magnetic Reynolds number that you've heard about in Sabine Stanley's talk, I think last week, and it is basically a dimensionless number that measures—they relate—the relative importance of induction versus dissipation—a typical Lane scale for the flow and field—typical length scale for the flow and size scale for the flow and diffusivity. So whoops, that's not supposed to be there. Another thing you can do is say, okay, suppose that this guy is a lot bigger than this one, and I drop this one. I do the same sort of analysis, and I can pull a timescale out of that, which is going to end up being the Lane scale over the flow scale. So if this term is negligible, the—this thing—will evolve on a timescale that really depends on the flow. It's called a—yes, okay, this one—this is actually pretty bright—okay, so we'll do stereo now. So this is—this is—this telling you that under this circumstance where you drop this, the field would devolve on the timescale that's really given by the flow in the—and this is called ideal MHD. Okay. And this being much bigger than that corresponds to saying that this quantity—this magnetic Reynolds number—is a lot larger than 1. Under this circumstance, you have something called flux freezing that will hold—you—I think you heard about that last week as well. What that means is that the magnetic field and the plasma are completely couple. If you move the plasma, the field will follow. And if you can find a way to move the magnetic field by grabbing the field lines and you move the magnetic field, the plasma has to follow as well. So that's what happens in ideal energy. On the other hand, if you're in the other regime and this one is a lot smaller than that one, you do the same kind of deal, and you get a timescale now that's called the diffusion timescale, which holds when this guy now is a lot smaller than one. And this is going to be telling you—in the absence of a flow—suppose I think the Sun or a star, and I turn off all the flows—u is equal to zero—okay—and there's a magnetic field in there—what is it going to do? Well, it's going to decay because of Ohm's law; resistivity is slowly gonna kill the electrical currents that are supporting the field in the timescale you get for that. If you plug it numbers in the case of the Sun would be something of the order of a few billion years—kind of depends which way in scale you pick—when that means is that if there—there were no flow inside the Sun or stars and they happen to be—to retain some magnetic field from the cloud from which they collapsed—once they settle onto the main sequence—that field is going to be there forever—effectively forever—it'll draw by a factor of two in the course of the lifetime—lifetime of the Sun. So explaining why there are magnetic fields long is not a problem; what is problematic is to explain how can it flip polarity in ten years—on much, much, much shorter timescale. That's where the catch is; that's where the—yes—that's where we have to do some work.

There is a trick you can do. If you dot the magnetic field into this and you pull out, you know, your vector algebra identities and you work on it for a few pages, you can eventually work up an equation that's called a magnetic energy equation. What you do is you—you adopt B through the induction equation; you integrate over the whole volume that you're considering, and then you just turn the crank and, you know, use vector identities. And by the time all is said and done, you obtain an equation for how the magnetic energy—this—this is simply—actually put it there—this guy is some constant v square integrated over the whole volume. All right, so that's the magnetic energy content of the plasma. You get three bits; you get a boundary term, you know, where this is the pointing flux—the flux of electromagnetic energy. Now, if you have a star embedded in vacuum, you know, and your boundary is really far away, the best that the field can do is fall like 1 over R cubed for a dipole. So this thing goes to zero if your boundary is far enough away; you're not pumping in the energy into the system. This thing—that's the current density—this is really—or make dissipation. Now, this is J square, right? So do what you want; that integral is always going to be positive because you're integrating of a positive quantity ending on that minus sign there, right? So this is bad—that Dad—this is basically destroying the field; this is dissipating magnetic energy into heat. So this is the enemy; this is the thing we're trying to fight. That one—that one—is where our hope lies. This is the flow speed, and that's the Lorentz force—J crosby—in MHD with a minus sign in the front, but that's not too worrisome because that thing can be either positive or negative, right? So that—that thing—is what expresses the fact that you can tap energy into the flow if the flow does work against the Lorentz force, right? You can pump energy; you can increase magnetic energy. Now, of course, the force—the magnetic force—could be accelerating the plasma, in which case you lose magnetic energy. So the sign of that thing really determines whether—well, really what's done—what's—what happens to—how the interaction between kinetic energy and the transfer of kinetic energy to magnetic energy or the other way around is taking place. This is where the dynamo is really sitting. A dynamo is a fluid flow then as a topological configuration such that this term will be negative; therefore, you will increase magnetic energy. So that—friend—yep—conductivity—yes—you will—you can get in the actual magnitude of electrical conductivity that will depend on the—on the temperature—on the density—on a lot of things—but that thing will always kill you—or make dissipation—always dissipates, you know, electrical current into heat. So at the rate will vary from star to star, but that is always a sink term. This is what you're fighting. So this is the equation we were dealing with. Now there is a magnetic force; that magnetic force can alter the flow. So if you really want to solve this as a complete simulation, you need to solve a set of equations that includes the induction equation but also includes other stuff. It looks kind of scary, but this is just mass conservation; this is momentum; this is F = ma if you want, you know, for a plasma; pressure gradient; gravity; magnetic force; viscous dissipative—viscous force if any; this is energy conservation; and this is field induction. So a bunch of partial differential equations; you find yourself a big computer; you put them on

There, and you let it go and see what happens. That's the brute force approach to the problem, and it's been done. It's been done in various contexts, and I will show you a little bit what all of that looks like.

I would like to distinguish between different approaches or definitions, or what we like to call the dynamo problem. There's something called the kinematic dynamo problem, and that is dealing only with the induction equations. We're not going to bother about dynamics. We're going to say, okay, suppose that I'm—I have the freedom to specify to take any flow that I like, whether it makes physical sense or not, right? I can just invent a flow. So the deal is to invent a flow that I will then put in here without considering any of that, and can I actually amplify the magnetic field in this way? So you would think if I have complete freedom to specify the flow, this should be really easy, right? But it actually isn't. Come up with any sort of flow that you can think of; most of them will not be a dynamo. You know, you will put a field in there, it will turn on your favorite flow, let it go, and after a while, everything is dead, right? So that thing is actually quite hard already, and it seems that turbulence—okay, flows with chaotic trajectories, if you won't really want to be picky about it—seems to be the trick.

Now there's an even—there's a harder version, which is to say, okay, we're not just going to pick any flow; we're going to take a flow that is dynamically consistent, right? The flow that satisfies, you know, the equations of physics, and that is what's called often the self-excited dynamo problem. Yes, some very weird time-dependent laminar flows can actually produce a magnetic field, but in the sort of astrophysical context that we're thinking about, turbulence seems to be the—the thing. Interrupt me anytime for questions, right now if you want to get even harder, you can talk about the solar or stellar dynamo problem, which is this plus the fact that what comes out has to look like the solar magnetic field or, you know, stellar magnetic field observations, which is yet another—another step up. So, and this involves likely more than just turbulence.

I will now get into this brute force approach. What happens if we—if we actually do this? There's a chapter in volume three of the books that you have that talked about these sorts of simulations. That's back in 2009, I believe, and if you read that chapter, that's actually not going to be much about cycles and large-scale magnetic fields, because believe it or not, back in 2009, these big simulations that were being done—had been done for quite a few years—were not producing very well-organized large-scale magnetic fields, and even less fields that would undergo polarity reversals in any manner looking like the Sun. So what follows is actually sort of fairly recent stuff that has happened over the last about five years—really four or five years—but as I said, people have been working on this for quite a while. It's always dangerous to do—to have a milestone slide because you end up leaving out, you know, someone that may be in the audience back there, you know. I just wanted to have one of these instead of three, but all this to say that people have been at this for since the—since the early eighties. In fact, this place—any car—you know, is where a lot of that started with Peter Gilman and Gary Glatzmaier, and we've now finally reached a point where we can add simulations of, you know, MHD convection in something like the Sun or stars that it's starting to look like—as some aspects of it are starting to look like what we see in the Sun. So after 20 years, we finally caught up with our geodynamo friends who actually managed to make their more or less correct dynamo, you know, in the early 90s, I guess it was. So we're finally there. We caught up.

The way you do this—so the idea is to take, you know, you—you take a sphere, we cut out the interior—so maybe the 60% of the interior of the Sun—we cut it out, we fill the remaining part with this electrically conducting plasma, we set a background structure—a variation in density and temperature that looks like the Sun. We know that from helioseismology that profile is typically convectively unstable in the outer 30%. It means that the temperature gradient is such that if you displace a blob of plasma, it starts moving and—contact. We happen to a vase—you can or cannot put a stable fluid layer underneath you. There's some physical simplifications of the MHD equations that are not—you know, they're actually pretty good inside the Sun and stars that we do, and you just solve that. You basically discretize these PDEs and solve them like people do, you know, model flows around airplanes and stuff like that. The important things there is that even if we—you know, we can try to do this at various spatial resolutions, but even with the biggest computers around, we cannot reach physical parameter regimes that are those of the Sun. We're still orders of magnitude below what we should be at in terms of the intensity of turbulence and stuff like that. That's just because, you know, we work with what we have. Even then—even at these—these Reynolds numbers—you end up in a state where the convective turbulence that develops in the outer—authority percent—is strongly affected by rotation and the stratification—as this effect that it tends to pump the magnetic field that you produce downwards across these layers, and you'll see that in later slides. This is an example—this is stuff that—that we've been working on. What you're seeing here at the surface of the simulation is temperature, while we remove the latitudinal perturbation. So you're seeing warm cells—cells of warm fluid coming up. This is the radial flow near the surface, and this is magnetic field. So well, the first good news is that yes, we can produce magnetic fields. You know, this thing is strongly convecting systems; you put a tiny seed field in it, and it builds up a very strong field. That is good. However, that field picks the scale of the flow that induces it, which makes perfect sense, right? If the flow is what drives the field, you expect that the field will evolve on scales—spatial and temporal—there are those of the flow that is driving the induction. This is actually what you expect, and that is already good, but the Sun also has this large-scale component that makes a nice dipole that produces these sunspots—are well organized across hemispheres and stuff like that. So how do you take an induction process that operates on fairly small scales and make a large-scale field out of that? That's called—nowadays people call that the magnetic self-organization problem, and I'll come back to that in a short while.

Now these are time series. This is the same thing; this is just the beginning in the simulation. This is—this is a 2000-month simulation, so it's a fairly long simulation—covers about 150 years or something like that. So when you start with this ball of magnetized fluid that is in equilibrium in an unstable stratification, and you put a little bit of magnetic field and kick it to get it going, the first thing that happens is convection kicks up—this is the energy of the turbulence point—and then it flattens. So turbulence sets in and stays below that—stabilizes fairly quick. As soon as convection has stabilized, magnetic energy starts to go up. That's log-linear; it goes up on a straight line. So this is exponential growth of the magnetic field. This is classical dynamo stuff, okay? So this field cranks up until the magnetic force starts to resist the flow that is trying to do the induction, and at that point, you think that you're stabilizing, and in fact, this image—a bit of simulation you just saw—is around here somewhere for this same very simulation. You start waiting long enough, what—something else is happening after you think this thing has stabilized—magnetic energy starts to go up again. Now this is now the full time series. This little box here was that one. So this is on a linear scale now. So this is the exponential growth phase when you think you're saturated, and then you get into this—slowly, on a much longer time scale, you start to get this oscillation. Same simulation—we're looking at the surface—the Equator is like that—we're looking at this thing—Equator on—that's temperature again. Rotation is like that. You can see the differential rotation—the plasma at the Equator is preferentially flowing to the right. Now we're going to zoom in, right? We're at the base of the convective layer. This color coding is now the magnetic field—the zonal component of the magnetic field. It is quite turbulent, naturally; it's living in a very turbulent environment, but you can see that there's a—so we're rolling this thing—so there is this—this structure that tends to be aligned with the zonal direction—latitude—longitude. We're going to animate this over 100 years, and this green band up there—this yellow band there—will go away and reverse polarity. When I'll have this very turbulent system where you have a positive toroidal field—zonal field—and a negative zonal field this way, and if we keep going, this thing will go away, and you will see green appearing on top again and yellow appearing back there. So this thing builds a—a magnetic field that at all—at any time—it's very turbulent, and yet you can tell that it manages to build—superimposed on this very turbulent component—something that is—that knows about latitudes and longitudes. It knows it's in the sphere; it knows there's a rotation axis somewhere. That's the large-scale component. Sorry.

So what I will do now is I will average this in longitude this way—that's North Pole—South Pole—Equator, right? So I approach that thing that way at this depth, and I do that day after day after day, and I build, you know, this latitude-time diagram. This would be the simulation's equivalent to the very first diagram I showed you where these things would be the place where sunspots would be produced—the regions of strongest little field. Now the keen-eyed among you will notice that the cycle period of this thing is around 40 years instead of 11. Well, that's the way it goes. It's actually not that—if that cycle period had been, you know, 50,000 years or had been five days, it would have been more recent, but at this level, to be off by a factor of four on the cycle period is actually not bad. This is a simulation; I can cut it and slice it and dice it whichever way I want, right? So the bottom diagram is the same thing except that now I'm fixing the longitude and I'm cutting out as a function of depth and time what the field looks like. So this is the top of the simulation; this is the base of the simulation. This line here is the base of the convective layer, and what you see is the magnetic field is actually being pushed down and accumulates beneath the turbulent layers. So this thing pumps down the magnetic field, and the field piles up down there, and this is the—and that diagram is taken at this depth, and so this is—this is the magnetic field that ultimately we think would produce sunspots. We have a couple of questions. You first. Good. This one. Yes. I mean, I can put it wherever I want. No, absolutely. You'll see—you'll see some surface diagrams as well. Yeah, the question was that this diagram—this diagram is being extracted at the base of the convective layer; it's not what we would see if we would, you know, look at the top of the simulation. There was another question there. Excuse me. The units here? Oh, yeah, this is Teslas. So this is basically 4000 Gauss. It's close—it's within a factor of three—equipartition at that depth. Well, if I—if I compute my mean turbulent flow speed and I get an energy density that way, and I do that with the field—see, this is below the convective layer, and it accumulates there, so it's really approaching equipartition down there. If I go in the convection zone and about a tenth of equipartition, so it's—it—so it's a fairly strong field. It's fairly strong fields. Does this answer your question? Yeah. Okay. It was—yes. I mean—oh, yeah, that's—is another problem with the simulation. I mean, it's—it's not the Sun, right? This is a simulation, and we would like these—we would like these things to be at much lower—at much lower latitudes if this was really the Sun. Yeah, difficult. I mean, the idea is that these things will form flux ropes, and once these things destabilize and rise buoyantly—their eyes radially—so you would need very strong flows to be able to—the rise time is short—like a month. No, so you would really need strong flows to push this to the Equator. I don't think I can wiggle my way out of it that way, unfortunately. It wouldn't work.

Now this is showing four cycles. We've run these simulations for thousands of years, and what's kind of amusing—if you do it that way—we got like 40 cycles—40 activity cycles here. This is the surface field. Someone was asking about the surface field before, so we do get a nice dipole—no—concentrated at high layers, but you know—this what you see here is not really a reflection of that because we do not have sunspots rising at the surface in this simulation. We don't even have a surface; we stop at 95% of the solar radius, but if you look at the magnetic energy time series, you know, we got a fairly regular cycle. This is total magnetic energy in the whole domain, and you notice that it never goes down to zero or close to zero, and that it—and this baseline level here is really the overall level of smaller-scale magnetism that is sustained in the simulation. Now once you have such long time series, you know, you can do all kinds of things like statistics and, you know, define proxies of what should be sunspot number—measure that, you know, in various atmospheres. This is—this is again something related to magnetic energy, but in this base layer—their northern hemisphere—southern hemisphere—you number your cycles like the observers do it. Sounds better. I can—my cycle 24 is here. It's a weak cycle as well, but that isn't a complete coincidence. It has nothing to do with solar cycle 24. And then you can measure their amplitude—their—their duration and stuff like that. You can try to correlate, you know, various characteristics of your cycles like their period and their amplitude, and the reason you do these silly games is that we can do that with sunspot data as well, and most of these correlations, you know, they go—at least they seem to go—at least in the right direction in the sense that whenever sunspot data tells me that these things should be positively correlated, usually the simulation gives me positive correlations—not always. Now hard-core statisticians would look at correlation coefficients of 0.3 and they would do—yeah, no good. But the fact is that the sort of correlations you pull out of sunspot data are also at that level. So that's—that's just the best we can do. So anyway, if you kind of look at all of that—these simulations—know—they're again—they're not the Sun—there's just this simulation, but there's a certain number of things that they manage to reproduce relatively well, and others that—where they fail, but still—we have enough—whatever the limitations—this thing is dynamically consistent on all the scales that we resolve in the system. So we can actually treat that basically as experimental data and go back into the simulations and measure forces and measure things, you know, and this is—this—we can do physics with this, and this is—this is the fun part about it. You can do that. Then let's see—you can do that in stars other than the Sun as well. So this is work that's been done here—not by our group—but by a group that's based here, and that's CU-Boulder. They run a different simulation that does more or less the same thing as what I was showing you, and these are two simulations for a star that is Sun-like but that rotates at different speeds. So if they're on their version of the simulation and the solar rotation period, they do not get a large-scale field at all. We're trying to sort out still why we get it and they don't. Then we're all working together on this. If they go at three times the solar rate—so they spin their things at three times the solar rate—then they do get—this is again a time-latitude diagram of the zonal field—they get these filled bands. So they do get a well-organized large-scale field, but they don't get polarity reversals. To get polarity reversals, they need to crank it up to five times the solar rate—the solar rotation rate. So the reason I'm showing you this is that rotation is obviously something that matters. I've shown you data that illustrated that different stars that have somewhat different luminosity or different rotation rates have different sort of cycles—different levels of magnetism. So when I showed you a flat line or calcium emission, it had no cycle, but it had calcium emissions, so it had magnetic field. Is this a star like this that does build up a large magnetic field that simply does not undergo polarity reversals? I've showed you time series of calcium that were kind of irregular. Is it something like this where we do get polarity reversals, but they're out of sync between hemispheres and stuff like that, you know? So we can actually get these questions with simulation data like that. Now when you go to stars other than the Sun, and I'll run another two minutes and then we'll take a break, okay, because that—that's going to be a good stopping point. When you go—if you start with the Sun where you're basically convectively stable in the inner 70%, and you're convecting in the outer 30%, and you start looking—and the sort of stars that you've been hearing about, you know, solar-type stars that are somewhat—that's less massive—the convection zone gets deeper and deeper until things become fully convective. So what happens to the dynamo? Convection is essential. So well, you think that you get more and more convection. However, these stars are less and less luminous, so convective speeds are weaker; they get colder and colder. So at some point, flux freezing in the—in the outer layers of the star will not hold anymore. So there's a lot of potential action happening as you go to cooler and cooler—if you go to more massive—more luminous stars, and we're still only talking about main-sequence stars here, right? Stars that burn hydrogen in their core. Well, if you start making stars bigger and bigger, it turns out that around spectral—the mid-F spectral types—or the early F spectral types, I should say—the temperature—the temperature keeps increasing in the center. By the time you get here, you hit the early F stars—that temperature in the center of the star has reached a point where hydrogen burning switches from the—what's called the PP chain—to another set of nuclear reactions called the CNO cycle, and these are far more temperature sensitive. The consequence is that the—the core where the nuclear burning takes place builds up a temperature gradient that is convectively unstable. So you start to get convection in the core at the same time the star is so hot at the surface that you lose surface convection. So you end up in a situation like the Sun that had a radiative core in a convective envelope—to eventually a star with a convective core and a radiative envelope. So what happens to magnetic field production—in dynamo action—in such stars? Again, with simulations, we can get a handle on this. This is a simulation of convection in the core—that's not the whole star—that's just the inner 40%, I think—of magnetic field generation in the convective core of an early-type star, and you do get a lot of magnetic field produced in there, but it remains confined to this core. This is now going the other way—in a fully convective star—that's a simulation by Matt Browning—this—so this is now the stellar surface, and again, I'm not sure why it cuts out the core there, but you—in a fully convective case, you do get a lot of magnetic field with sometimes some hint of spatial organization. These are contours of the zonal—toroidal—azimuthal field, but again, there you get magnetic fields; you get sometimes large-scale organizations—sometimes not—but the end point of is that, you know, this thing seems to work. Convection plus rotation equals dynamo action—in magnetic field. This is, in some sense, the anticlimactic conclusion of this part of the—of the lectures. Now one last word before we break on this magnetic self-organization problem. So I mentioned the problem already. So this is basically the equation that drives the production of the magnetic field, but again, remember…

That that feeds back onto this via the momentum equation. So this guy, we know from observations and for simulations, that it's something that has fairly small lane scales. The convective flow, as lane scales are much smaller than the star's radius and coherence time—you know, the turnover time that goes from, you know, days to months—so that's fairly small length and time scales. The solar cycle magnetic field is structured on the scale of the star and oscillates on a period of ten years. So there's a huge disparity between the length and time scales characterizing the guy that does the job, this one here, with its end product, the large-scale magnetic field.

So this magnet, this business of self-organization is: how can we set things up so that we can take all this small-scale process and somehow have it act coherently, you know, to build up a structure that develops much larger spatial and temporal scales? And there are a couple of things that we know already. Rotation is really, really important in all kinds of ways. This is the surface of the simulation again, the simulation of convection. There's no magnetic field in this one; it's purely hydrodynamical, where we have no rotation. So you—so the bright stuff is warm upwelling fluid; these little lanes are cold down-welling fluid. This is completely typical of convection; we see that in the Sun; we see that in all the simulations. And this is the exact same simulation where I've turned rotation at the solar rate, and you can see that the convective pattern changes quite a bit. You get this tendency for convective cells to elongate in latitude and align themselves with their rotation axis. You've heard about that in the context of geodynamo last week. These are the Taylor columns that build up parallel to the rotation axis; they're in here as well. So a rotation structures convection.

The other thing rotation does, once it introduces an anisotropy in the convection, is that it drives large-scale flows. From face-mix sounding of the Sun, by looking at acoustic waves bouncing off the Sun's surface, we can reconstruct what the rotation rate of the Sun looks like. We knew from tracking sunspots, you know, four centuries ago, that the equator of the Sun is rotating faster than the higher latitudes. With helioseismology, we can extend that. In words, this is the rapidly rotating equator; this is a slowly rotating pole; this is the base of the convective layer; and that big latitudinal shear—real angular velocity vanishes in this thin layer—we call it a tachocline. This is a differential rotation we get out of the hydro simulation. So you—we tend to get with, you know, reasonably solar parameter regimes—so we get again a rapidly rotating equator, slowly rotating pole. Eighteen shilling—or it's not perfect. Okay, with this rotation, it's squeezed in the convection zone more than it is here. We don't get much of a tack aligned in the equator in the simulation; we don't have a surface layer, so that's why we—presumably why—we don't get the surface shear. But if you run that with a magnetic field, the field tends to erase a lot of that differential rotation, but you still retain differential rotation. And differential rotation is good because it is something that builds up on the large scales of the system; it's a contribution to the total inductive flow that is structured on the large scales, and therefore you think you have a hope of tapping into that to impose that large scale on the total magnetic field.

And I think we'll pause here; that's a good place to stop. And when we—when we come back, we'll talk a little bit about another aspect of the influence of rotation on magnetic field that is quite important for dynamo action. Alright, take a break. [Applause]