NASA's Heliophysics Summer School - August 15, 2024 - Marco Velli

Okay, the kinetic Alfvén wave is a is a is a wave that normally is of interest and is different from this wave only when the wavelength perpendicular to the magnetic field is of the same order or close to the Larmor radius, the ion Larmor radius. We're talking about microscopic objects here; we're probably animals that have huge wave numbers. Sorry, sorry, microscopic huge wavelengths, microscopic wave numbers. So k rho is very small, so so you're you're in a different regime when we're talking about kinetic Alfvén waves. Ktic have a density fluctuation associated with them as well as well as a parallel electric field, so you know it's it's slightly different animal. Okay, this is a very large-scale thing. Okay, yeah. Okay, so if we could transition to the slides now, thanks.

All right, so I'm wondering whether we should just skip ahead for a few minutes to the Parker Solar Probe. Um, so what I'm going to ask you to do is um let me just pull out my computer so I can tell you what slide to skip to because I because you know all this blah blah that I just did might seem, you know, academic, and it was academic until recently. Um, so if you could slip to skip to slide number slip to slide number 34. Wow. Okay, what you're seeing on the left is the radial component of the magnetic field as observed by Parker Solar Probe during its first perihelion. Okay, that's the raw measurement on the left. The numbers down there are what are called New York seconds. They're called New York seconds because they're shorter than a second, and everyone in New York is in a hurry. And the reason they're called New York seconds is that the spacecraft has a clock, and you do not want to have resonances between the time of observation and the signal that's periodic on your electronics because it will pollute the signal. So you try to take an irrational number which is close to a second to get your data so that you can remove the signal, the artificial signal due to the electronics on the machine. Okay, so those are New York seconds, but basically close to a second, and you can tell that we were sitting in a region of the heliosphere where the topological field was coming in. V is negative by and large. V is negative. Sorry, just B_R. Yeah, I'm told that if I keep this alone, I saw something there. Oh, there we go. So B_R is negative, and as we move closer to this on here, we're quite far away. This is our first perihelion at 35 solar radii, then we climb out again, but you're seeing the field woo, it's reversing all the time. You might think, well, you're just sitting really since we're in the ecliptic plane, we're really close to the current sheet, and we're just crossing the current sheet all the time, right? Well, it turns out that we're not. This is a magnification of this thing; we're not crossing the current sheet. We can tell from the electron pitch angle. So the angle between the electrons always have a beam in front of them called the strahl. The angle between the strahl and the magnetic field doesn't change sign. Okay, if we were moving from one side to a different side of the current sheet, the beam would still be coming out. The magnetic field is going in, then the beam is coming out, the magnetic field is going out. You'd have a pi, an angle difference of pi, so you'd have a cosine that goes from one to minus one. The fact that it doesn't means that these are folds in the field, so the magnetic field line that we don't like is bending backwards. Okay. All right, so that looks a lot like the beginning of a large amplitude Alfvén wave, right? Oh, next slide please. Sorry. Oh, I can go with this. Right, thanks.

Okay, uh, now this is an inset with all three components of the magnetic field uh this. So on the top you see the same same image. Um, the black is the magnitude of the field, and it varies a little bit, but not a lot. And if you look at an inset, the bottom graph, you see that the magnitude of the field is sitting more or less at 90 nanotesla. The radial field is going all the way from minus 90 to plus 60, but the magnetic field doesn't care; it's staying the same. So we're in that B² constant. Now the only thing we have to check on is the velocity field. Yes. Sorry, I don't know what a highly Alfvénic magnetic field means. Oh, oh, no, but you can't stop at magnetic field; you have to read the next word too. Magnetic field switchbacks. Magnetic field; there is an adjective to the word switchbacks. Highly Alfvénic magnetic field switchbacks. Switchbacks because they're oscillations that flip the magnetic field, so the magnetic field flips over, but its magnitude is constant. And I said it's highly Alfvénic, but I didn't show you it is, and to do that I have to do one more thing, which to show you the velocity field. And this is the velocity field. It's not beautiful. This is the radial velocity, but do you notice something? It's got this kind of lower level that's sort of, you know, I say a little bit, but it's all these jets coming out all in one direction, exactly like I showed you. So the first dramatic thing that Lea. Yes, no, it's not it's not it's not done for anything. This is the distance of the spacecraft. This is the distance of the spacecraft. Okay, 55 solar radii here, 35 solar radii here, and back up to 50 here. This is the flow velocity; we remove the spacecraft velocity, but that's the V_R in the absolute frame. Okay, so you see that the wind is actually kind of slowish, slowish, almost, you know, 300 km per second, but it's got these jets on it all the time. And yes, I do have that slide, so let me go back to the. And if you're worried about. Oh, sorry, no, I'm keeping going forward. Sorry, this is the correlation between V and B. So V is in blue, v is in red. Remember the average field is negative, so an outward wave has v and B positively correlated. Well, what can I say? So it's not that we hadn't seen these things before, but not as dramatic as this. So Parker really showed that the inner heliosphere is dominated by these large amplitude spherically polarized Alfvén waves coming away from the Sun. I think it was the first. I guess it was the first big kind of wow um that Parker Solar Probe gave us, and it shows you that this thing that might seem to be like this mathematical abstraction that you have to put this constant, this has to do this, this has to do that, actually happens, and in fact it's a big piece of the inner heliosphere. All right, that's that's all I wanted to say about Alfvén waves now, so I'm going to go all the way back to the [Music] beginning of course. Mm, yes.

Okay, so in order to do that we would have to have measurements of the same or similar solar wind parcels at one. We have done that for a couple of parcels, and the only thing I can really say is this: a switchback, they actually come in patches, but doesn't really matter; they tend to disappear within the M within the distance measured by spacecraft. And if you do some statistical correlations between Helios and um and Anna, help me, Solar Probe and what else do we use? Ulysses? No, did we use Ulysses? Helios and PSP or Wind also? So although you can't recognize the same, the number of switchbacks per unit length traversed by spacecraft for the medium size doesn't change much, right? The longer ones seem to grow, the shorter ones tend to disappear here, and the intermediate ones seem to be more or less the same. Is that right? All right, so I wanted to, we're going to talk about the real solar wind, so we already started talking about some features of the real solar wind, which is really interesting. Um, so let's go back to modeling the coronal wind, and as Genan said, you can't escape MHD, but you can go beyond it. That's really a pun on the fact that if you take the moments, I mean you got to respect those moments; they they do deserve some respect. You can do better, but basically the forces were shown today. Um, uh yeah. Okay, what else? So these equations, you know, um, mass conservation, momentum conservation, induction equation with resistivity, and there's a reason I put resistivity in there, and then there's an energy equation in which you have heating and loss losses because the corona does radiate away. And so if you want to do a solar wind model which is a little bit more serious than the one we were discussing this morning, you have to conserve mass, but you have fluctuations all the time, so you got to be careful where you're looking because you now have possible correlations between Delta rho and Delta U that act like a source term, and the same goes with div R uu. You have the classical stuff. Um, you have the gradient of total pressure, B² over 8 pi is in there. If there's a J cross B force, it's included. The only place you might have some J cross B forces is where the solar wind is ripping open magnetic field lines in the neighborhood between open and closed; there you can have currents and close to the heliospheric current sheet, where by definition you have currents because the the wind has ripped open field lines with opposite topology, and therefore all that energy in the solar wind, if you think of it, which is pushing the wind outward, some of it gets reabsorbed by the current that it has to create, so there's there's some storage of energy there, there's some free energy available to do things. Um, and then okay, gravity is always there to disrupt us, wave pressure, and then there's this tensor here um which is uh you know, if you have purely outwardly propagating Alfvén waves, this stuff all vanishes, and you just have a nice gradient of the pushing of the waves on the wind, which is the reason why their energy isn't concerned. Okay, so then you can actually do a lot um with very little if you imagine that once the field once the flow has disrupted the field, you can consider the flow to be parallel to the field everywhere, and you can just choose instead of doing a 3D solar wind model, you can basically write an equation of the solar wind along a field line. For somebody who hates field lines, that's not a good thing, but you can take a small little bundle of field lines, maybe if you're in a stationary state, you can fake your way and do that. And uh if you do that, the energy conservation turns into this thing up here. Please turn on. What's the top equation? Sorry. Oh, thank you. Cool. So here we have our mechanical energy flux, which is made up of mechanical energy, enthalpy, gravitational potential. Here you have the work of, you know, viscous stresses. Here you have heat conduction. Here's your Poynting flux, and that would be conserved except that you might have radiative losses. Now in the coronal hole you can neglect them, but in the quiet sun you can't. So you can then cut off the terms, and you're going to call F_M the transport of the wave energy, which is the Poynting flux plus from plus the mass loss times the perpendicular fluctuations, the conductive flux, which you can write if you're lucky in terms of some conductivity, that's not a good approximation far from the Sun, the integrated radiative loss from zero to R, but L_R is really localized around the base of the corona, and then the solar wind loss loss, which is right here, and that sum is conserved. And so you can basically, if you write out this conservation at the base of the corona and at a distant distance, you can write out the asymptotic wind speed, and it's two times the mechanical energy flux crossing the base minus the energy you lost from radiation divided by the mass flux minus V_Gs, which is essentially the escape speed from the Sun squared. And so you can estimate if you want a 700 km per second solar wind with a given mass flux, you know how much energy you have to put in, and so you can do this calculation and and do some energetics with it. All right. Oops, wrong direction. Um, you couple that with the famous equation that we were discussing today. Um, you put those two together, and there's an interesting point that concerns how how much this flux tube expands, the distance from the side. Where does this come in? It comes in the fact that you, in general, this flux tube will not be in radial expansion. The magnetic field has very very different expansions depending on where it comes from, so there's this areal expansion that you write here, and when you integrate the equations, you find that the base velocity depends depends on how much you expand from the coronal base to the critical point where the wind becomes supersonic. Now if you increase the base speed, which is basically in the chromosphere if you like, where the density is fixed, you're increasing the mass flux, and if you fix the energy flux in the waves and you increase the mass flux, you end up reducing the wind speed. So there's an inverse correlation between expansion factor and asymptotic wind speed. This was shown by Yiming Wang many many years ago. So there's an inverse correlation of mass flux to asymptotic solar wind speed, and there's an inverse correlation of flux tube expansion to asymptotic solar wind speed. Solar wind carries angular momentum, uh, and to find out how much angular momentum is carried out carried away by the solar wind, Weon Davis showed this. Um, you can do this in CGS; there's an there's temperature and isotropy included in the price when you do that. You find that you can estimate the angular momentum as though the solar wind was co-rotating with the Sun out to the Alfvén radius, the where the Alfvén speed is equal to the speed of the wind. So from from the point of view of the angular momentum carried away by solar wind, it's as though everything was co-rotating out to the Alfvén radius. Okay, it's as though your sprinkler started at the Alfvén radius basically, and then you can try to build more realistic models that include the waves, turbulence that you're going to hear about tomorrow, and you can produce things that look like the real solar wind in this way. Many people have done this in one dimension, and you find temperatures, and you can produce. So these Alfvén waves are important because they allow you to heat the wind to produce the asymptotic wind speeds that we see, and you know, if you take a what you know, a stellar astrophysicist point of view, we do a very good job; you can make models that really work quite well for the fast solar wind in this way. Now the wind comes from lots of different places, and I'm afraid I I misjudged this, so I hope you have a little bit of patience. What I want to take you through now is um issues about um the MHD equations and the validity of ideal MHD. So the equation written upstairs, since I've been using this as a pastime, um, it's not quite correct. Eta appears in the gradients, and Eta here is the magnetic diffusivity; it's not the resistivity, but um if you remove any non-ideal term in Ohm's law, so all the terms that I showed this morning, you chuck them away, including the resistive term that I've kept here, then you have dV/dt equals curl of v cross b, in which case you conserve magnetic flux. Um, this can be said in very many different ways; it means that you have two particles that are connected by a field line at time t equals zero, and they move around with a flow at time T; they're still going to be connected by one field line, or you can say that if you take a any closed circuit moving with the fluid, the magnetic flux threading that circuit is going to remain the same. That's only true if Eta is equal to zero, of course, and you use this to define some numbers. If you take a scale Big L and a typical velocity U, then you can define a time scale, a dynamic time scale L/U. There's a diffusion time scale which is L²/Eta. You have the Alfvén speed, and you can define an Alfvén time L/V_A. If you calculate these numbers for something like the solar cor solar corona, you have T_D is of the order of 10¹⁵ seconds, T_A is of the order of 10. The ratio, which is called the Lundquist number, L V_A/Eta, T_D/T_A is of the order 10¹⁴. It's a huge number, which might lead you to believe that you can neglect Eta completely. Problem is, you can't, and the reason that you can't is the same reason that the limit of viscosity tending to zero of Navier-Stokes is not Euler. There's stuff that happens in Navier-Stokes that doesn't happen in Euler, no matter how small Eta is, and the same thing here. There's stuff that happens in non-ideal MHD when Eta tends to zero that doesn't happen if you write the equation without, and how can you tell? It's a singular perturbation because it changes the order of the differential equations. The differential equations without this piece have fewer derivatives in space than the equations with that piece. Okay, it's called a singular perturbation. Singular perturbations introduce solutions that don't exist in the original equations. Okay, and that's where our friend reconnection comes in. You have topologies of the magnetic field where it's impossible, no matter how small the non-ideal terms are, it's impossible not for them to be important. That happens at a null point, which is something like where field lines cross each other like this, and in this particular, this is a a two-dimensional map of a 3D thing which looks more like a dome with a field line coming coming through and what's called a uh fan surface and a spine coming through it. There's a null point of the field there, and if you try to propagate waves, they accumulate in that place, and therefore they develop very intense currents, and typically things like this happen, and you form current sheets and reconnection. If you rotate around, you can inject helicity, and you can cause these things to erupt. Um, something that looks very much like an H-alpha macrospicule is sitting right here for quite a long period of time. Okay, the the film isn't perfect, but um you can build models of coronal heating that are based on the idea that even though the Lundquist number is very very high, you will form regions where you have to include it. And this is the Parker model of coronal heating. In Parker's model, he made a very strong abstraction; he said, okay, let's this is maybe what reality looks like; let's take let's take two photospheres instead of one, straighten out these field lines so that we get any get rid of any expansion forces that might push our plasma away from the Sun, and then we said, well, convection is going to start tangling these field lines together, and he proposed a theorem; he said, for an arbitrary displacement of field lines at the coronal base, there does not exist a smooth equilibrium in the corona, so any equilibrium that forms in the solar corona has to have discontinuities. Now when he made his picture, he made it look like there would be tons of knots in there. Um, in reality you can't form so many knots; um, things smooth out much faster, but oh, this is the contrast on this is horrible, so um so these are numerical simulations of this Parker problem where the model is constructed as so you take two photospheres slabs where you impose line tying; we're talking about boundary conditions earlier today. This is a simpler problem; it's called reduced MHD, so it's basically incompressible magnetohydrodynamics in which you have linear waves propagating along the field but complete nonlinearity in the planes perpendicular to the field, and you can impose the velocity field at the boundaries, and the boundaries are closed, so it's a consistent set of boundary divisions. You impose these random boundaries to see what happens in your volume. This movie that really doesn't show anything uh here, but maybe it's a little better over on those screens. What it showed you was a slice perpendicular to the field, and the evolution of the current was this picture, and the evolution of the vorticity was this picture, and this picture is actually showing you that these are current sheets that form, and these current sheets actually extend throughout the volume of the plasma, and they're reconnecting; they're reconnecting because you can see quadrupoles of vorticity correspond to the sheets; you can see blue red red blue here if you're careful, and so that means that there's an inflow outflow a little bit à la Sweet-Parker if you like, and the reason it's Sweet-Parker like is that hey, we're amateurs; the the resolution of this simulation was maybe 1000 by 1000 if it even reached that, and 1000 by 1000 means that my Lundquist number was maybe 10.

3. If all goes well, it was 10 to the third. Basically, the number of points you use is essentially your list number. If, if some—by the way, somebody carries out a simulation in a paper or talks to you friendly like, he says, "Oh, I did this great simulation. I had a Lus number of 10 million," oh, and what was the resolution? Ah, 256 squared. You can tell him that he's either he's knowingly cheating or he doesn't know what he's talking about. You can find out by having a conversation, but that's true always. You cannot renormalize your way out of the fact that your resolution dictates your Lus number or your viscous number. Okay, there's no way that you can get out of that. All right, you can cheat, as we'll see, but with appropriate filters.

And if you look at the evolution in time of the current and the vorticity, you see that this is really a Parker system. The, there's very low velocity field in any of this. Current dominates. Dissipation is essentially ohmic in this system because we don't go to sufficiently small scales, and we don't have kinetic effects in there. But it's a model for coronal heating, um, based on Parker's idea. And here I'm showing you two examples. This is without cheating, which you see current sheets that extend from one—the actual loop is 10 times longer, but we compacted it for the image. Um, if you cheat by using what's called a hyper-resistivity, basically you accumulate all the dissipation to the edge, edge, edge of your computational domain in Fourier space, so that you can try to use as much—doesn't really work because there's all sorts of nonlinear interactions between the viscous—the model of the resistivity that you do, so you can't trust the small scales. But if you do it that way, the sheets are more, more fractal-like. If, if you want, um.

And then you can do the same thing with an open and closed field. You can essentially do this Parker diagram where the field is open on one side and closed on the other, and, and this is called typically interchange reconnection, and, um, I have the—I don't have the beautiful picture here, unfortunately. I put the wrong picture in, um, I think this—this was—but the—this was illustrating what happens. And what happens is that the frontier between open and closed field lines becomes extremely corrugated, so you have line tying. So on one—on one side it's perfect—this is closed on one side and it's open on the other, but the—but the—so this is at the top of the box, but the bottom boundary, at the surface, surface of the Sun—of the—of the demarcation between open and closed—becomes basically fractal-like. So it's really telling you that even with such a stupid kind of—easiest—very complicated, but easy kind of simulation from the point of view of reality, you get a very complicated boundary between the open and closed field on the Sun. So you guys have much, much more computational power—you're probably doing it in more realistic geometry, some of you at least—and you'll see even more and stronger effects of this type. Anyhow, let's come back to—oh, that's—that's a mistake. I'm sorry. So let's come back to Sweet-Parker.

So, so over the past, I would say 40 years to be honest, um, there's been a change of course in the triggering of magnetic reconnection. So first of all, why is Sweet-Parker called Sweet-Parker at all? So the reason it's called Sweet-Parker is because Parker was at a meeting where Sweet showed a model of steady-state reconnection, and Sweet was more—I don't know—I don't—you say maybe less competitive, so Parker saw—came back, wrote the paper before Sweet did, and so he published it before Sweet did, but he did recognize that he'd gotten the idea from Sweet. However, Parker realized that this thing was really slow. There's no way with a Lus number of 10 to the 13 that you're going to dissipate any reasonable amount of energy in any reasonable amount of time, especially if you want to do something like a flare. And so this is the conclusion. So the Parker—Sweet-Parker you should remember is really the negation of steady-state reconnection, not the positive application—observation of theoretically—with the apothesis of magnifi—in annihilation suggests that other alternatives for the flare must be explored. That's what Parker concludes. But of course, Parker didn't know at the time that the reason we could see Sweet-Parker in numerical simulations was precisely the reason that I was just telling you before—that we were doing 100 by 100 or 200 by 200 or 300 by 300.

I was a postdoc in 1985, 1986, 1987 in St. Andrews in Scotland, at the University, and Eric Priest was very interested in generalizations of this, um, steady-state reconnection model, and he invited a German, uh, scientist—come visiter—Biscamp. Biscamp was doing numerical simulations, and the numerical simulations were just then—1985—reaching things like a thousand squared. It was like, "Oh wow, thousand squared!" Now people are doing 68,000 squared and things like that, um, and Biscamp had started to see that he couldn't keep together this Sweet-Parker current sheet. This flow was breaking apart, and so he was saying, "Something's going on." In fact, um, currents can be unstable. They can be unstable to a mode which is called the reconnecting mode, generally speaking. Often times it's called the tearing mode now, and it was studied in the same years, believe it or not, um, because people that were doing fusion were interested in current gradient driven instabilities, and current sheets are a common animal. And so the Harris current sheet, the hyperbolic tangent current sheet was a very common thing, and they found many instabilities, but one they found particularly relevant called the tearing mode. And tearing mode, um, is essentially a, a mode where the singular perturbation of the field becomes important. So you basically have a field which switches sign, and resistivity is very small—it doesn't count very much—but it counts far away from where the field vanishes. And the reason actually is quite simple. The reason, if you remember the induction equation that we wrote, we had dB by DT is equal to B.grad U, so if you—plus other stuff—de—we don't care about—D—consider ourselves to be incompressible, and we don't care about U.grad B—the moment it's not really important. If we linearize that equation, we have a perturbed magnetic field which is equal to B0 dot the gradient of the velocity field. Now if we want to reconnect, we have to bring the velocity field in, so we take—taking the radial part—the, the part of the field—this is perpendicular to the mean field—and so basically you—you can say that the radial component of the magnetic field—so if you have a field that's going vertical here and you have a transverse field to reconnect, you have to introduce a B that goes in this direction. That change in time of that field is proportional to the B0 times the gradient of U, but that Bz vanishes at the center. Okay, so if you have a finite BR equal to B0, which vanishes times U, it means that the flow has to become infinite—there's a singularity if you remove resistivity. So if you try to suck in material into a current sheet, the flow tends to diverge—to become infinite right where the magnetic field vanishes. And because it's trying to become infinite, you have to include the non-ideal term, because you have to get rid of that singularity. And therefore, resistivity in a thin layer around where the magnetic field vanishes intervenes and slows the flow down, turns it around, and makes it—makes it come down to zero where it has to vanish because it's a flow coming in, and you have a—what's it called—a stagnation point flow. This is what the flow looks like—this is an example. So you see the accelerating flow, and then it switches over, and there's this little singular layer inside. The instability is a long wavelength instability when it's measured in terms of the thickness of the sheet, and the thickness of the sheet is called a here, and K is the wavelength along the sheet. And so in order to have instability, you have to have Ka less than one. This number Delta Prime has to be bigger than zero. Delta Prime is very important. If you look for Delta Prime and reconnection, you will see thousands of papers on Google or whatever—scholar—whatever. Okay.

And why am I saying all this? Oh, because the normalization—remember MHD is scale-free—so if you have a—the only thing you have is an infinite current sheet in one dimension—the only length with which you can normalize things is the thickness of the sheet essentially. So you find that you have this, um, long wavelength mode—Ka less than one—and then you get a growth rate, and the growth rate scales like a fractional power of the Lus number. So the Lus number, though—again, we're scale-free—the only parameter we have is a. So here the Lus number is written in terms of a, okay, the thickness a—the thickness—let me repeat that again—a, the thickness. I don't know if you were carefully listening when I was talking about the Sweet-Parker case. I didn't even say it probably, but the Lus number defined by Sweet-Parker is in terms of the length of the sheet, and that's really important for what we have to say now. Okay, so this is a very slow instability, and it didn't seem to be particularly interesting, um, it was interesting for—for other instabilities that are called the kink mode—for—for double current sheets—for plasma configurations—for the lab, but more generally you can write the dispersion relation and solve it, and what you find is, in fact, there are three regimes. There's—there's a regime where the wave—where the wave number is particularly, um, uh, low, and the scales go like S to the minus 1/3; the regime with a wave number is as high as it can be—remember Ka has to remain less than one—then it scales like S to the 3/5; but if you change the Lus number and change the wave number at which you get the maximum growth rate and take the envelope of all the possible solutions to get the—the fastest growing mode at a given number—changing K with S—then you get S to the 1/2, which sounds incredibly like Sweet-Parker from before, except that this S is defined in terms of the thickness of the sheet and not of the length. So Biscamp had indeed observed the fact that the Sweet-Parker current sheet was breaking up because of this instability.

How does that work? Well, let's take a look at the Parker—Sweet-Parker current sheet. In the Sweet-Parker current sheet, τA is defined in terms of the length of the sheet; S is term—is, um, is defined in terms of the length of the sheet. In the tearing instability, τA is determined by a—the thickness of the sheet—and the Lus number is determined by the thickness of the sheet. Okay, how do we go from one to the other? Well, we have to renormalize by the factor a over L. So let's do that. So τ0 is τA a star S star to the 1/12 L over VA L over VA a over L, and you put all these things together, and what do you get? You get the fastest growing mode of the tearing mode scales like S to the minus 1/2. This time it's the Sweet-Parker S times the inverse aspect ratio a over L to the minus 3/5. That's the renormalization from a to L. Okay, but in the stationary Sweet-Parker current sheet, a over L is determined by the fact that you have to have inflow outflow, and you have to dissipate the sheet, and what you find is that a over L scales like S to the 1/2. So what—put a over L equal S to the 1/2 in here—what happens? What happens is that the maximum growth rate of the tearing mode, γmax—big L over VA—now goes like S to the minus 2 elevated to the power minus 3/4—minus 1/2 is equal to plus 1/4. And you might say, "Okay, yeah, that means that the time scale of the instability—to 1 over γ—so γ to the a now scales like S to the 1/4." Now if you see this—you look at this—you start getting queasy if you think about what it means, right? Why do you start getting queasy when you think about what it means? This is now telling you that if I—if I back up and I say, "I want my resistivity to get smaller and smaller," what happens to S when the resistivity becomes smaller and smaller? S goes up. What happens to γ because S goes up? Let's make S go to infinity. What does γ do? Goes to infinity. Houston, we have a problem. Ideal MHD doesn't have reconnection, right? This is telling me that in the ideal limit, not only do I have reconnection, but I have an infinite growth rate. Oops, something goes wrong. Now there are two schools of thought here. One is, "I don't care," and you publish papers and you say, "Sweet-Parker current sheet—S to the 1/4," and you say the islands—you can form as many islands as you want; you can go as fast as you can—or you can sit back and say, "H—maybe all this is telling me is that since this sheet is so unstable that it takes zero time to destroy it—how do I even make it in the first place?" Right, if you're—if you're trying to build a building that falls faster than you can build it, it's really trying to tell you you can't make that building. Right? So what this is really telling you is that you can't form Sweet-Parker current sheets in a high Lus number plasma, so you really can't get there. So it's useless that you try using the scaling to count plasmoids because it ain't going to happen. Okay, but then how was it possible that we saw it? Because we do numerical simulations with S being a thousand. Sure, Sweet-Parker is probably more or less stable. It's just an artifact of the fact that you're at low Lus number. And at high Lus number you can't. And then you can ask yourself, "Okay, how—how thin a current sheet can I make then if I can't do that?" Well, you can argue and say, "I—I have to remain causal, right? I have to—I don't have to get back to ideal MHD, but I have to respect ideal MHD." So the—the best thing I can do is produce an instability which doesn't scale with the Lus number, because that's compatible with MHD. I could go to γ if S tends to infinity, and I find that γ is constant—that's okay because it respects ideal MHD. And you can ask yourself, "Okay, if I put in here a L proportional to S to the minus α, what α makes γ how be independent of S?" And they find that that's 1/3. So the—you never get to Sweet-Parker if you have access, of course, to any length you want, but this is really a way of thinking now. What this is telling you is that any non-ideal plasma as current sheets thin will reach a point at which their instability is compatible with any time scale which is dictated by the system. This is kind of a paradigm shift because in ideal MHD you put together equilibria, and this is really telling you, "Well, watch out, because you might always reach places where there's a dynamics going on at the fastest time scale you have in your system." So you're really now thinking about the whole plasma in a slightly different way. So it's kind of a paradigm shift, and it can influence how things go. So I've already said all this. People used to see Sweet-Parker, and the reason was blah blah. Um, you can prove with numbers that a over L S to the minus 1/3 leads to an ideal growth rate. And interestingly, the singular layer—that layer where resistivity counts—in that ideal instability scales precisely like Sweet-Parker. It has to, because in that layer you have to dissipate the field on an ideal time scale, and the thickness that will produce dissipation on—on an ideal time scale is S to the minus 1/2. So everything now makes sense. You can add viscosity, and that will make the sheet thicker again if the viscosity is efficient, and this is what happens when the instability takes off in two dimensions. Um, just enjoy the movie. And what you're seeing there is a box, and then there's a box inside the box, and then there's a box inside the box. And if you look—first look at upstairs—then look at the box—the little box that became the big box—and then you look at the little box that becomes a big box—what do you see? The whole thing looks self-similar, right? So this is what's happening. It's partly an artifact of 2D because 2D doesn't give you a lot of wiggle room. So the sheets are one-dimensional. What's really happening is this—this fast instability starts taking off on an ideal time scale. It produces an X—there's an inflow and an outflow, right? What does the alala do? It stretches that singular layer. When that singular layer is born, it's just a little square; its aspect ratio is one, and then it gets stretched out; its aspect ratio goes down. You can renormalize. Now you have a Lus number on this little scale. When this Lus number becomes unstable—S to the 1/3 on this scale—it'll reconnect again, and it'll produce another X point. This B—it'll lengthen. Now not all the islands are born alike, and so one of them becomes bigger; the other one gets crushed. It can't happen in exactly the same way, but in resistive MHD there's no limit until you get to resistivity equal to one. So now you have a process that—bang, bang, bang, bang—very quickly brings you down to the smallest possible scale you have, and this is a paper by Anna, and this is one of the most—for me—this is one of the most beautiful figures I've ever seen in a problem in plasma physics, because it wasn't predicted to be so perfect. So what am I showing you? I'm showing you the linear instability. So this is a nonlinear code doing this calculation of—of D subse—stabilization. Then the top picture shows you the reconnecting magnetic field at different times. Okay, so at the initial time you have the perfect black line here, which is the perfect eigenmode—tearing eigenmode—corresponding to the scale at which you're at. Okay, at later times, though, you see the red line—it does some wiggling in there—and then at later times—but notice the times are getting closer—18 Alin times, 19 Alin times, 19.8 times—it becomes even thinner. So we said, "Let's blow up that central part." If you blow up the central part, you'll notice that at time 19 the red curve is exactly the same as the black curve at time 18, and the blue curve at time 19.8 is like the red curve at time 19. This thing is doing the same thing at smaller and smaller scale, and so you can do it. So suppose you start at S 10 to the 13. Your threshold for instability is a over L scaling like S to the 1/3, the N, and stable layer is the diffusion region of the N minus one. So an over Ln is Sn to the 1/3; an Ln minus 1 is Sn minus 2; Sn goes like S to the 3/4 to the N, which is a hypergeometric; Ln goes—over L goes like S to the 1 plus 3/4. 10 to the 13 in four steps you get to 10 to the 4, so it doesn't take long and you're gone. You've reached whatever scale you needed to reach. This evades—so now you have a pathway via instability that gets you to the smaller scales. Now you might ask, "Okay, that's with Z resistivity—it's easy—what happens if you go to Hall MHD?" Will things change a little bit? You now have to introduce di as a small parameter, but the paradigm—the idea that things will go down to the place—will make things fast—now gives you a predictive technique to say, "Oh, how thick can I make a current sheet to make it unstable?" Even tells you the answer to why you have electron-only reconnection. Why does that happen? You can make very complicated models, or you can think in this way. Suppose you have something that limits the length of the current sheet, so now you don't have access. And so suppose that I have my non-ideal effect, and a over L is limited now because L is limited. So in order to reach whatever level I have to do—a has to become much much smaller. So suppose that di over L is not sufficiently small for reconnection to be fast at that scale, then the ions say, "Sorry, guys, we give up—we can't do it," and the current sheet precipitates to smaller scales, but de over L is much smaller, and we'll make it, and we'll transition to fast, and so you'll have electron reconnection. Imagine you have a partially ionized plasma, and a partially ionized plasma you have—you know—if you're going slow enough.

The whole neutral ion thing moves like the same. You start going a little bit faster when you start to decouple the ions and the electrons from the neutral fluid. You get some viscosity coming in because they can kind of dissipate against each other. And then you go to smaller scales; the electrons and the ions, they go on their own in a similar way. In a current sheet, if you start reconnecting, you're probably going to start reconnecting with everything, but then it's going to form a singularity on that scale. Ooh, you're going to start feeling the slippage between the ions, and when they reconnect, the scale is going to be even smaller. Oh, now they're not going to even notice the neutrals anymore, and they're going to take off. And so now you have a scheme that even in a partially ionized plasma might lead you to acceleration of charged particles in a partially ionized plasma where you would expect things to be quasi-neutral-like. Right? Hall effect is intriguing because it opens so what happens when di is zero is that you have this multiple thing, and this multiple thing, by interacting different islands, opens things up to increase the speed of reconnection. But the Hall effect opens its arms faster. And so if you reach and you have resistivity so that you limit yourself to the scales of Hall reconnection, then this self-similar collapse is stopped, and you get single X-point reconnection. All right, why was I saying all this? Because I needed to know things about what's going to happen in the heliosphere where there are regions where reconnection has to take place.

I should say this is a lecture, so I haven't given names to people, but all of this work, the ideas, come from a number of people. And we have to mention, of course, um, Bisham, who realized, even though he has a beautiful book on reconnection, magnetic reconnection, and he didn't quite understand how the flows in Sweet-Parker were involved in the stabilization of the thing, but he realized that Sweet-Parker was unstable. In fact, even before that, in the Russian school, there were people that realized that reconnection was very fast, and there was something wrong in the way they were modeling it. Um, but after that, Shibata, son in Japan, the student of Uchida, uh, and Tajima, realized that growth rate would be very huge. This was rediscovered by Lero and uh, by um, CI Lero, KY CI, and Biskamp, who talked about the plasmoid instability as the instability of Sweet-Parker. Um, we got involved in the discussion because I pointed out some aspects of that, and so our own group did some work on that. Um, but it's been a really interesting evolution of thinking on magnetic reconnection. Baley and Lero and Alfred Mallet have done work on the effect of this kind of destabilization on turbulent cascade that you're going to hear about tomorrow.

Let's come down to Parker and the structure of the real solar wind. So this is where Parker is today; it's at perihelion essentially, and it's getting ready to do its last passage at 11 solar radii um, in about a month and a half. And then in December, Christmas Eve, we're going to have our first perihelion at 9.8 solar radii. So we're well deep into the mission now. I already showed you this; I showed you that these large-scale Alfvén waves are a dominant feature. What we didn't realize though, as you can see, the wind is really slow. It manages to become fast wind up here, but here it's really slow. So, Solar Probe is seeing a peculiar type of slow wind, which is filled with Alfvén waves. And I haven't really discussed Alfvén waves because turbulence is tomorrow, so but the point is that most of the time when you see the slow solar wind, you don't see all these Alfvén waves in the wind. It's typically non-Alfvénic, as they say; you don't see all these Alfvén waves coming out. But this has been observed before, um, Alfvénic slow wind, and Parker seems to be immersed in it most of the time, actually. This is in encounter nine. In encounter nine, you can still see the switchbacks. We call these things switchbacks when the radial net field is so you see them here. Here, this is the crossing of the heliospheric current sheet. And I like to point out, if you look at the switchbacks carefully, you can see that they don't, they're not uniform; they come like in little patches, little groups, right? And here they line up really close to the heliospheric current sheet. Often times, there's an asymmetry between the inward and outward um, uh, solar wind measured by the spacecraft, and uh, it hasn't been discussed in detail. But if you think of it, there are reasons why there's an asymmetry, and I think one of the main reasons is the following: when you're moving, the spacecraft is going really fast, and so the aberration of the spacecraft is running in the rain; in other words, the aberration sums with the speed of the wind. The wind's coming at you, and you're going in, and so the aberration sums. When it's coming out, you're running away from the rain, and so the aberration gets subtracted, and so everything seems more radial. And so your instruments that are sitting in the back of the spacecraft have a harder time catching all the particle populations, and this may affect your measurements in a significant way. Yes, that's the magnetic field boom is in the back as well, so that interest is it's I'm not I don't have an answer to that question, but I'm just wondering there is a wake, right? And the nature of the wake does depend, I think, on how things how the flow goes around the obstacle. I can't say more; I don't know.

This is in encounter 12. You can see these beautiful patches of switchbacks. This is the heliospheric current sheet crossing. I've shown you that the switchbacks are Alfvénic. This is why this is a high, really high-tech quality diagram shows you why the jet is always going forward. We've already discussed that, and these are the switchback patches, and you can see these patches. And if you measure their size by using the orbit of the spacecraft and the Carrington longitude and stuff, you can see that they're maximum at various scales, but in particular, there's a maximum supergranulation scale. So it looks as though, wow, supergranulation. Yeah, I'm worried because we were supposed to talk about the edge of the heliosphere, and well, that's supergranulation, okay? Um, and so it looks like these things scale with supergranulation. Sorry, I'm not going to go fast; I just want to show you what supergranulation looks like. Supergranulation on the sun looks like this. This is actually, it's interrupted by sunspots here, but supergranulation plays a major role in the relaxation of what happens in the corona because it's a scale at which the mo the longer-term motions are. And so if you imagine, so you can see the magnetic network; it's actually visible here. You can see the network; you can see kind of the cellular pattern of the magnetic field, right? That it follows, you know, these these kind of cells. And so the field emerges and then opens up. And so it stands to reason that as this thing expands um, and then a neighboring supergranule expands, they're going to occupy the corona in cells which have essentially the expanded version of one supergranule. And so if you then expand this out, so Parker, you expand, you expand by a factor, say 10 solar radii, and you'd expect that scale to appear, and indeed it does appear. But we don't know exactly what forms those switchback patches.

This is another example. And so, so of course, when you see this, and I mentioned Ulysses, what I didn't mention this morning, remember those fluctuations in the velocity? They were always there. If you go and look at them a little bit more carefully, they seem to have a scale which also corresponds to supergranulation, and it lasts about a day, a day and a half. So it seems like it could be, at least intuitively, that these patches of switchbacks might kind of dissipate as you move away and leave this general oscillation in velocity, which are called micro-streams. And so this is a paper by Stuart Bale, first author, in which he kind of calls them supergranulation-scale micro-streams, and he shows that there's energetic particles that are correlated with it. I think reconnection may play a role in generating the fluctuations of the magnetic field that then lead to switchbacks. Some people believe that reconnection at the sun produces the kink in the field line directly. I think that's not possible, but um, yes, yes. If you want numbers, I can't give them to you, but I think it's the evidence is overwhelming, really overwhelming; it's they're there all the time. Oh yeah, every encounter. Yeah, Nikos, where's Nikos? He's not here; he's out there. Nikos can show you all the encounters if you ask him tonight. Go flap, flap, flap, flap, flap, flap, and you'll see the switchback lashes all the time, and they're always there. Yeah. Um, yes, uh, so that's the velocity there, there. So, so that one there is the velocity; you see the radial jets coming out, and this is the magnetic field, and this is perihelion. And of course, you'd have to try to identify streams on the sun, which leads us to the next. Oh, and then if you put, just make a histogram of all the velocity fields measured by Parker as a function of distance, this is what you get, and it's really nice because if you look at the inf of this, you get a nice profile here, and this profile corresponds to the slowest wind possible, and that's the wind that this morning I was mentioning you can use to measure the ambipolar electric field, and it seems to work like a dream as the right ambipolar electric field to produce the Parker solar wind. Okay, so it looks as though the baseline average solar wind is really a thermal wind ala Parker, but everything else is on top of it, and it needs the extra push of the Alfvén waves or the switchbacks, as as as you might call them now. Yes, right. Yeah, yeah. Well, the you know as well as I do that these numbers, but you used to measure this stuff; things haven't changed. The average field is always the average field; it's always a few gauss at the poles, three gauss, something, not more than that. Yeah, we can go into more detail. Yeah, so there's evidence from the beautiful thing about Parker is that coming and out, sometimes it crosses the same stream more than once. This is an older paper; I'm sure that I I'm trying to keep up with the number of papers coming out with Parker data; it's just incredible. So I may have missed a few; I'm sure this can be done more and better now, but we can actually measure an acceleration profile um, and see that the acceler the solar wind measured by Parker, this kind of slower wind, is still accelerating all the way out to 50, 60 solar radii. You can measure the acceleration profile.

And now we come to the current sheet, which is seen to be reconnecting at Parker Solar Probe, and you can see beautiful reconnection jets here. This is the radial velocity, and you can distinguish the reconnection jet from the um switchback jet because the magnetic signature is completely different. The magnitude of the magnetic field here is not constant at all; it just bang, it goes almost down to zero and comes back up. So that's where the current sheet, the radial field, has a beautiful bifurcated sheet; it's one sign, then it drops to close to zero, sits there, and then it drops again, again, and then you have this beautiful jet coming out. The fact that the jet is positive means we're on this side of the current sheet, and you have some signatures of leakage of proton beams being accelerated at the sheet that come out from a paper by Tjhin and the distributions. I showed you distributions today that were bicycle-like, and in the neighborhood of the current sheets, you see incredibly deformed distributions such as this one here, where there's evidence of scattering along constant magnetic field arcs, and they they've been called hammerheads, and they're papers describing theoretical models for them, but this is all all in its infancy, I would say.

Where did the solar wind that Parker saw in first perihelion come from? Well, this isn't perihelion; we're close to solar minimum. This isn't perihelion either; perihelion is here. The solar wind came from a very small coronal hole; it was expanding dramatically. And this comes to the kind of the final top, well, final two topics in the last five minutes. Well, there's the heliosphere, and well, gosh, I don't know; we'll see. The thing is that the way the solar wind expands and the way the magnetic field expands from the sun depends on the topology of the field at the sun. And you can have situations where a very small slice of sun produces a very large slice of heliosphere, and it happens especially when the field can be weak or there's some strange configurations on the sun, multipolar configurations. We had in that period, sunspots from the old cycle and the new cycle appearing at the same time, and when that happens, you have pluses and minuses occurring close to each other, which usually don't happen in opposite hemispheres, producing topologies that are different than usual, um, such as what are called unipolar streamers or or pseudo-streamers. Pseudo-streamers look like helmet streamers; they don't, they're a little bit; you can see them in the corona, in the corona, you can see that they're different because the compact part at the bottom is more tight, and the stalk in the middle is more apparent in the helmet streamer. It looks like a wisp, like a plume on the helmet of something, but in the pseudo-streamer, you see this compact thing and this very thin thing coming out, like almost like the cup of a sword. Um, and so that's really the configuration that Parker was dealing with um, and at perihelion. So so the question, so there's been a long discussion in our field as to where does the fast wind and where does the slow wind come from? And if you look at a traditional textbook, what they'll say is that the coronal hole produces the fast wind, the edges of the coronal hole produce slow wind, and then there's this in there's the helmet streamer stalks also produce slow wind, and basically that's it. But it was noticed, take this picture, sorry, this picture, it was noticed by many people that the volume occupied by the slow wind was actually quite large compared to the boundary of the coronal holes. And um, Spiro Antiochos and others proposed that in fact the slow wind was not only at the boundary of coronal holes but appeared everywhere the magnetic field mapping to the sun became complicated. What is that mean? It means that there's a lot of shear or a lot of structure in the field; the mapping from the sun to the corona and to the heliosphere is complicated in very many different ways. It's complicated very low where you get the canopy coming out of the chromosphere, but even when you move up, you have fields that close back down, fields that have longitudinal component because you have filaments, and so they meander around somewhere and then come out. And so you can have mappings that are complex. And how do you measure that mapping? Well, you take a little circle of flux in the lower corona, and you see how it deforms as you trace the magnetic field outwards. You can think of a circle; it turns into some kind of deformed pattern. And if you take the major axis of the deformed pattern divided by the minor axis, how how big that is, it's called the squashing factor; it tells you how complex the mapping is. And this is a map of the squashing factor. And so Antiochos suggested, well, wait a minute, the squashing factor tells you where the wind is slow, so forget about coronal hole expansion; it's the squashing factor that produces the slow; that's their idea. The boundary of, sorry, the boundary of coronal holes looks like an interesting proposition, but it was proposed on the basis of one-dimensional models of the solar wind. So let me explain what I mean by that. Suppose you have, suppose a magnetic field like so, this is a potential field; it was first proposed by um, Belitz and Ian Axford as a model for the field of the sun. You give, if you give yourself the geometry of the field and you study the one-dimensional solar wind model like I just did previously, you find that there's a singularity in the field lines. And so the critical point for all of these field lines lies close to the sun, and then all of a sudden there's a field line. Bing, the critical point jumps, and it was proposed that that's what the slow wind does. So basically you have fast wind, fast, fast, and then there's a singularity, and poop, the critical point moves out; therefore, the wind has to accelerate more slowly, and therefore you get slow wind in all that area. But if you think about it, it's not, and and if you do models, it kind of says, oh well, maybe maybe that's how you can get the fast-to-slow transition. There's a problem with that, and the problem with that is that when you have a configuration like that, the actual flow starts to accelerate as though it could go through the first singular point; it can't, poor thing, decelerates and then reaccelerates. So you get these weird profiles of solar wind like this, and when that happens and the solar wind decelerates, imagine you're on a highway and you decelerate; what happens? There are a lot of cars that kind of bunch up together. So all of a sudden you're accumulating mass in those dips, and therefore you're accumulating pressure. So although you solve the equations along the field, you forgot about the fact that these field lines are open, and the minute you change the pressure somewhere, they're going to wiggle themselves around. And so that's not stable; so that configuration is not going to work. So in fact, a coronal hole really essentially only produces either fast or slow; it can't do both, except in a very narrow singular layer where it moves from being closed to open. I.E., in this picture here, this field line is open, but then next to it there's a closed one here; you can get slow wind that last field line, but it's a very thin layer, not a thick layer. Okay. On the other hand, the squashing factor works, but there is another source of slow wind, which is a small little coronal hole that expands dramatically; that produces slow wind as well.

Where am I? And so, conclusion of this discussion, it seems to be that there's a way to recognize these different types of winds because suppose the squashing factor is very high; the magnetic field is complex. Now you're you're a wave, and you're trying to propagate; if the field is too complex, your wavefront is going to become incredibly corrugated; what's going to happen? It's going to dissipate, so that wind shouldn't have too many Alfvén waves in it. On the other hand, if you're a nicely expanding coronal hole, everything kind of equilibrates itself out; well, the waves can propagate, so that slow wind should be Alfvénic. So there should be a way to distinguish one; one should be non-Alfvénic, and the other should be Alfvénic. So there's a way to test this hypothesis. And then, of course, there is the intermittent solar wind coming from the helmet streamer tips. It used to be thought that you could have a smooth solar wind coming out of the helmet from your tips, but measurements from STEREO and Blasi and others show that you have these blobs coming out, and these blobs are fairly extended in longitude, and they accelerate outwards; they have periodicities, and that's kind of natural if you think about it because coronal heating takes some time. So if you accumulate matter and you push a blob out, it's going to take some time before this thing becomes unstable again. So it's, you know, periodicities should happen. And this is Parker, the white light instrument on Parker, and this is a really beautiful movie; see it where the contrast is higher; look at it for a long time, and what you see is plasmoids being ejected and growing as they move out, and it's a really beautiful movie. And of course, yes, so Pet Levier studied this um, in particular, and there's a paper out now um with this, but we thought that this was an example of the heliospheric current sheet instability, and we considered that ideal tearing was a good was a good uh possibility to understand how this worked. And um, so we have a couple of papers with Victor Reville showing how this might happen, and you actually do a pretty good job also of explaining the longitudinal structure and the extent of these the instability is actually a combination of what would be called in three dimensions of a ballooning mode and then a fast fast reconnection mode. All right. It's the final topic, which is, of course, fundamental to space weather and to other stars, space weather before we get to the end of the heliosphere, what actually

What happens to the current sheet when you go from solar minimum to solar maximum? Um, not many people have have thought about this, but it has implications for space weather. Of course, you know, if we think of a dipole sun with a warped current sheet, then our coronal mass ejections tend to deflect away from coronal holes; they tend to deflect away from the current sheet. And you can imagine what happens when you go to solar maximum—quadripole and octopole components start dominating over the dipole. If you had a pure quadripole on the sun, you wouldn't have a single exospheric current sheet; you'd have two cones. So now imagine putting a dipole and a quad—what happens to these current sheets? What happens when the quadripole intensifies? When is the current—the secondary current sheet—born? When does it die? This is an open field right now, but um, there's some indication that there may be this bifurcation of current sheets occurring. This is from a paper by Yiming Wang, who's always a little bit ahead of everybody else, um, and it's very interesting. And I think we've now found further evidence for this, and we're kind of working on it, but I think this really is important for other stars, especially that have a more complex magnetic field geometry, more prominence, and so we need to think about how this works, how this um pumping of uh magnetic energy might occur in a more complicated system.

All right, okay. How does the heliosphere end? So we mentioned this yesterday. I'm going to spend five minutes on this. I'm sorry. Um, so this is where we are. This is from a paper by Dave McComas from a few years ago, and uh, if you magnify percus arm, Sagittarius arm, this is Orion Spur, local Interstellar Cloud, direction of Sun's motion. This is a more—have a termination shock here, have the heliosheath, and we have this bow wave. So this bow wave we were discussing yesterday—it's not—it's not to me—it's not obvious that that I don't think this—I don't think it's settled whether there's a shock or not yet actually, but people now tend to think of it in terms of a bow wave, and um, IBEX has been measuring the energetic neutral atoms. And what IBEX has seen is that there is a there is a ribbon of energetic neutral atoms that basically defines the draping and the direction of the uh Interstellar magnetic field around the sun.

Um, this is from Gary Zank's 2013 paper. Basically, it's saying that because the the difference in speeds is very small, it may be slightly super-fast magnetosonic, but not dramatically—depends on the intensity of the interstellar magnetic field—and therefore charge exchange might play the role of dissipation and allowing the smooth kind of bow wave rather than a bow shock. This is the this is the the IBEX ribbon, and the idea here is that these energetic neutral atoms are caused by essentially Interstellar atoms—the charge exchange—so sorry, solar wind atoms charge exchange creating neutral atoms that come in, get reionized, go back out, become neutral, turn around the interstellar magnetic field, and come back in as highly energized. And so if that's the case, they just have to rope around the interstellar magnetic field and come back, and therefore that thing sits in the essentially the plane perpendicular to the uh the intersection of the plane perpendicular to the magnetic field and the uh um I'm looking for words uh anyway, it's the intersection of two planes. Okay, and and this is the shape as described by MAVE of her simulations um of the crescent shape. So this is the draping of the interstellar magnetic field around the heliosheet, and what happens is basically these—so the solar wind wraps these kind of slinkies—the solar magnetic fields wrapped into slinkies—and these two slinkies essentially get pulled open; they just get lifted around and deform backwards, and you have all all sorts of neat Kelvin-Helmholtz instabilities in the back to kind of make this thing um have this kind of either interestingly gourmet or somewhat um disgusting cocoon uh depending on your imagination. Uh, anyway, yeah, and so this is very different from the image of the sort of the comet type uh um heliosphere that we saw before, and I haven't, you know, I think this is extremely interesting, and of course, in the future we're going to have IMAP flying soon, and one of its goals is of course to measure the neutral atoms, so we'll know more about this ribbon and the interstellar magnetic field uh in the near future. Voila, I think I'm done. You are welcome.