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

Yesterday afternoon, um, of course, the Jeans theorem on its own works in a fake medium which is homogeneous and uniform and doesn't exist anywhere in the world. Um, the real medium is turbulent with neutral flows and everything. Um, and the other big player, it's not really a force, but it kind of appears as a force, is angular momentum. Because everything is chaotic and things are moving against each other and they're not on axis, there's angular momentum. And of course, you all know that stars form via collapse. Typically, accretion discs or accretion rings form. They might have winds, and in order for accretion to proceed, you've got to get rid of the magnetic field. We're not going to talk about that because we haven't done MHD yet, so, so I won't avoid discussing that. Um, but gravity stratifies things. And so homogeneous media don't exist unless you're lucky enough to put just the right expansion to kill gravity, which sometimes happens, but not very frequently.

And so we have our sound wave, which in this room propagates isotropically: Omega squared equals K squared CS squared. And I apologize for the notation, I've been lazy, so I didn't feel like writing out all the bunches of equations again. So I'm going to be changing notation a little bit, but I'll tell you when that happens. Now, the sound wave, of course, if you're in a gra- in a gravitational strat- gravitationally stratified medium, changes. Okay? And so, for example, what happens? So what happens is that when you write the equation for the density perturbation of the sound wave, um, you have an extra term where the scale height, the density scale height of the medium, H, appears. Okay? So basically, you end up with a C squared over H here. So there's an intrinsic wave number, if you like, that appears in there, and you get a dispersion relation which no longer looks like so. CU squared is clearly the sound speed again. Sorry, K squared just minus i k over H. And that changes things. And what happens is that if the frequency is below a cutoff frequency, C over 2H, um, you can't propagate. So the wave doesn't manage to propagate. If it does, if the frequency is higher than this critical frequency, then it does propagate. And in fact, it steepens because the relative amplitude grows with height, delta row over row, because the scale height of delta row is twice the scale height of row. And so delta row over row goes up, and you form shock waves. And indeed, we think, we believe that in the chromosphere, we see plenty of shock waves. Luckily, if there's a strong magnetic field, the sound wave propagating upward is really the slow magnetoacoustic mode. It behaves the same way, and you get shocks. That's, um, mode number one. Yes. What? Oh, sorry, the scale height. So locally, even in this room, the density is, well, it's one over row d by dz is one over H, essentially. The scale over which the density decreases by a factor of e. That okay with everybody? Okay.

But there's another animal that propagates in atmospheres too, and this animal is much stranger, and it's called an atmospheric gravity wave. It's an incompressible or quasi-incompressible mode. And the wave equation, this is the vertical oscillation, sorry, this is the vertical oscillation fluctuation. Let's say, turn this guy on here. It is this delta is is along the z. Um, this is now propagating obliquely. So x is a horizontal axis, z is a vertical axis. And now in this quasi-incompressible case, is what's called the Brunt-Väisälä frequency, which essentially has gravity and the temperature scale height. So the scale, the typical scale over which the temperature varies by a factor of e. If you have an adiabatic atmosphere, the relationship between the previous scale height, um, the previous frequency cutoff frequency, and this cutoff frequency is a gamma minus one. Okay? So you have the cut-off, the acoustic cut-off, and the Brunt-Väisälä frequency are related by one over gamma minus one. Right. Gamma is the, sorry, you're right, adiabatic index of the gas. That's why I said adiabatic before. So typically for a monatomic, for a plasma, you can consider it to be 5/3. Got to be careful about the electrons and protons, but right. But if it's, you know, if it's here where it's mostly, you know, most, mostly the atomic molecules, and we got a seven-fifths coming in, it's not quite 5/3. Okay. So you find, you look for solutions of this thing. I'm not going to go into the details of it, but there's a dispersion relation. Unfortunately, this is Omega here, so apologize. This is Omega. And there's a wind because normally atmospheres have winds in them. So there's a horizontal wind here with U bar. Doesn't really matter. So it tells you that you get an Omega which is K dot U, which is just propagation of a horizontal wind. But then you have a propagation, K is the horizontal wave number. So you have N squared, which is the Brunt-Väisälä frequency, K squared over root of K squared plus N squared. Okay? So if you don't have any propagation vertically, you just get plus or minus N. So you, you're basically have a mode that propagates at the Brunt-Väisälä frequency horizontally. However, if one takes a look at oblique propagation, some weird things start happening. So the fluctuations are orthogonal to the wave number, of course, like so. Okay. And so these are, um, and this is the phase velocity. And what you're going to find is that the group velocity is in the direction of the fluctuations and it's orthogonal to the phase velocity. So the energy goes along the crests and troughs of the fluctuation. Um, and I, the reason I mentioned this is that, you know, this propagates energy even in the solar atmosphere. It's a mode that's hard to see because it's incompressible, just like the Alfvén wave that everybody's always going on about. So it's very difficult to measure, but it can carry a significant amount of energy even for a star. And I think there's abundant evidence of it existing in planets and propagating. You can see clouds that are essentially driven by these kinds of waves in, in the sky. Okay. All right. So, blah, blah, blah. I don't want to get too many equations, but this is the group velocity and the phase velocity. It's a very easy calculation. If you've never done these calculations by yourself once in your life, do that. Take a simple stratified atmosphere and look for the waves. Just gravity, don't add any magnetic fields because that will make a mess of things. Okay. All right. So these are very relevant both to propagation of energy in atmospheres of planets and of stars. And, um, and I'll stop there about this. So I'll just pause for a second. If you have questions about these modes, you've never heard of them, in particular, the atmospheric gravity mode is anti-intuitive. Okay. Magnetic modes are also quite anti-intuitive, but this in particular is particularly, um, um, hard. So it's basically an effect of buoyancy and gravity that balance each other. And, and the way this mode propagates. So if there are any comments or questions? But it's the Väisälä frequency. It's the gravitational piece of that. Yeah. Otherwise, you just, this, this wouldn't propagate. This mode wouldn't propagate at all. It'd just be an undamped motion by a wind. It'd be just a Doppler shift of a wind propagating Omega equals KU. If there was no gravity.

All right. So, so fluids. We treat these gases, the gas in this room, we treat it as a fluid. We have plasmas, we treat them as fluids. I'm not going to derive stuff, but I just want to say a few words on, uh, where you go and how and how how things work. So in reality, if you want a more detailed description or a kinetic description, you would use the distribution function. The distribution function, if you don't create and don't destroy, um, anything, of course, um, obeys a conservation equation in phase space, which is the top equation. U, which basically tells you that the, um, total number of particles is conserved in phase space. A is the acceleration caused by a force in phase space. You have six independent variables. If space has three, then velocity space has only three. If you're more exotic, you can have a lot more than three, but that's basically it. Now, fortunately, for the types of forces that we deal with, the acceleration does not depend explicitly on velocity. And when it does, it depends so in an incompressible way. And i.e., the divergence in velocity space of the velocity dependence of the force is zero. V cross B, for example, for the Lorentz force. Okay? If you, if you consider a viscous force, of course, that wouldn't be true. But then everything that I'm going to talk about, which lives in an ideal world, kind of breaks down. Then, if in a, in a typical gas, the forces are smooth. Gravity is a smooth force. The only force that's not smooth and that's violent are collisions, which are like hard ball collisions in, in a, in a neutral gas. Right. And so you can pull that out. You can pull that out and you can distinguish the force between a slowly variable force, an average force, which is my Italian, unfortunately, stands out here, slowly variable force, and a collisional force. Okay? And you typically pull that out and you say, okay, we're just going to call that df by dt collisional. And then you can write out this equation here, which then becomes one of your favorite equations, depending on what you do with the collision integral. If you, if you write df by dt with the Boltzmann integral, it becomes the Boltzmann equation. If you write df by dt collisional, the Landau prescription, it becomes the Landau prescription. If you write that as, um, Plunkett collision integral, that becomes the Plunkett equation.

All right. Now, when is a plasma a plasma? In a plasma, of course, the force is going to be electromagnetic. So in the plasma, we basically have to remember that we have two different types of particles. That most plasmas are neutral. Non-neutral plasmas are very interesting on their own. Gravitational field with stars is a non-neutral plasma. Basically, the, you want to study how the, the shape of the galaxy formed. It kind of is plasma physics because there's a collective potential determined by the motion of all the stars, and the collisions between stars don't really count. And so it's kind of a collisionless plasma in that sense. Um, you can, in fact, you can build galaxies using electrons essentially, or, or laboratory experiments of galaxy formation using electrons only. But in a neutral plasma, um, there is a shielding effect, which occurs on the Debye length, which is indicated here. And we call a plasma a plasma. So that collective effects are, are really important when there's a lot of particles in the Debye sphere. So when the number of particles inside a sphere of radius the Debye length is much greater than one. Now, normally, you know, we're dealing with plasmas that are not very, very high temperature. There are, of course, a lot of very exotic plasmas that are high temperatures, starting from the corona. But the photosphere is not very exotic. And can you say that the photosphere is a plasma? Can you say that the chromosphere is a plasma? Well, the interesting bit is that, for example, if you take a hydrogen gas and you use the Saha equation, what you discover is that collisions actually ionize the plasma before you would expect on the basis of pure theory of a single particle. So, for example, the ionization of hydrogen occurs at 1.58 times 10 to the 5 degrees Kelvin, technically. But at 10 to the 4, using the Saha equation, you discover that at 10 to the 4, the ionization of hydrogen is basically complete. So in the upper chromosphere, the plasma is almost completely ionized, even though you wouldn't expect it to be on the basis of just simple measurements. The other fact is, if you calculate the conductivity, you don't even have to have a high degree of ionization to get a conductivity to was almost 100%. In other words, the photosphere, where the ionization degree is fairly small, still can be considered a plasma because you can get 90% conductivity with ionization of only 8%. So the 6,000, 5,000 to 6,000 degree range of the, of the photosphere is also a plasma. So this is important.

Collisions we discussed very quickly yesterday, and this is going to take me back now to the, to the, uh, to the solar wind. So the simple way to model collisions is by saying, well, two particles of charge Z are going to come close enough until the kinetic energy is balanced by the potential energy to the repulsive. Then you call pi v squared the dimension of your target. That shows you that sigma is proportional to T to the minus 2. And when you calculate the collision frequency, it's n sigma v. And so you've got another v thermal v that kills the T to the minus 2. And so you get the T to the minus 3 halves underneath. So the collision frequency goes like T to the minus 3. The thermal conductivity, essentially caused by electrons because protons aren't that mobile, is proportional essentially vt squared over nu c. And therefore vt squared is just temperature, temperature divided by temperature to the minus three halves gives you temperature to the five halves. So the minute you start increasing the temperature of your system, the thermal conductivity goes way up. You might argue whether that's a good estimate of the conductivity itself, which brings us back to the discussion that we had of collisionality and non-collisionality. But this expression for the conductivity is gives the reasoning behind Parker's original idea of why the solar wind is supersonic. And I want to re-examine that today. Um, before I do that, I want to mention one more thing, which is that these collision frequencies depend on the masses. And therefore the equilibration time, if you have a gas with with electrons and ions with a different weight, the time for the electrons to reach equilibrium amongst themselves is much faster than the time for the ions to reach equilibrium amongst themselves. And you can see it by going back, oops, I have to go back, and looking at the, at the mass appearing in the collision. Right. So you see there's a one over m to the one half in that, in that. And so that turns this into, this fact here, that te is me over mi i half ti. And therefore electrons go to equilibrium amongst themselves before they come into, um, before the ions come into equilibrium amongst themselves on their own. And then the longest, um, intermediate time, the electrons and the ions, um, can get, can go together. And that allows you to build a fluid theory, which is kind of separate for electrons and ions. So before you go to the single fluid magnetohydrodynamic description, there's also a description in terms of an electron fluid and an ion fluid, or an electron fluid and a kinetic ion. Um, you take moments of those equations. I think you'll see more about that in the next talk. And you can derive conservation principles, which is my basic assertion as to why fluid theories work. Fluid theories tend to work because at least the lowest orders are just expressions of conservation of quantities, whether it be mass, momentum, energy, and so on. The difficulty, of course, comes in the fact that every time you take one of these moments, you get a higher order moment appearing. For example, equation for the second order moment, oops, there's Q. Q is the heat flux vector. And Q is a third order moment. So you have to find a third order moment couples into a fourth. This whole picture in which you derive this, um, set of equations, it holds in exactly the same way for a gas. Um, but in a gas, you have a small expansion parameter. Where in the plasma, it's much harder to identify a small expansion parameter which allows the higher and higher order moments to close. The same problem, of course, occurs when you're trying to study any kind of random variable or random process, like turbulence, for example. In turbulence, you have exactly the same problem where you want to find out what happens to the moments of the correlations between the fluctuating variable, for example, the velocity, the energy, and so on, and third order, fourth order moments. Okay.

I'll leave this and we'll come back. I just write the Ohm's law because the Ohm's law is fundamental. Some things that we're going to have to say later. Um, the Ohm's law is important because to lowest order, as you know, as you heard yesterday, in the frame of the plasma, the electric field vanishes. That's because the electrons are so mobile. And so we say that the electric field vanishes. And when the electric field in the co-moving frame vanishes, you're essentially dealing with a very good conductor. And you know that very good conductors don't like to change magnetic fields, or rather, they refuse the magnet. They move around. So the magnetic field doesn't change, right? But that's not always true. And, um, you know, we'll explore the separate, the importance of that. But for a plasma, in particular, I just want to point out that you have, of course, collisions that provide resistivity. Sigma is a conductor here. But this is really just the equation of motion of the electrons written in terms of fluid-like variables. So what you, which you basically said is, okay, this is me dv by dt, but I've written the electron velocity, average velocity, in terms of a current. And so this is called the electron inertia term. This green one. Then this yellow one comes from the fact that the velocity here is the flow, the fluid velocity, but in reality, we're talking about, so that should be the electron velocity. And the difference between the electron velocity and the average flow is really the difference between electron velocity and the ions. And that's the current. And so that's what's called the Hall term. It's the fact that you have a difference in velocity between the electrons and the ions. And then, of course, because this is the electron equation of motion, there is the second order, um, uh, second moment of the electrons, that's the pressure. And so there's a pressure gradient appearing. And in most plasmas that we deal with, whether the magnetosphere or, um, the corona, well, the corona they're slightly different, but oftentimes this is one of the first terms that become important in this equation. So in fact, most, most of the MMS reconnection events are, are, um, driven by the electron, the electron pressure gradient term. Now, you can do a dimensional analysis of this equation. It's quite fun to do. If you've never done it, you can try doing it on your, no, your own. You probably evolved taking a plasma course, but it's always, you always learn something by doing these kinds of things. So if you assume, you know, non-relativistic plasma, blah, blah, and all these terms of the same order of magnitude by definition, these purple ones, so that goes one to one. And then you try to normalize all of these, you'll find that they all boil down to ratios of characteristic frequencies. U is a typical speed of the plasma of the flow. Omega p is the plasma frequency. Nu e i is the electron-ion collision frequency, and so on. And you can write down what those things are. And as a result, you can find when it is that the terms become important. Okay. And so what you find, for example, for the Hall term is that you can only neglect the Hall term if the frequency of your motions are much smaller than the cyclotron frequency by factors of the order U over VA squared. So if you have a flow that is highly super-Alfvénic, you're probably doing okay. But if U over VA is very, very small, small, then you, you're going to hit the Hall term really quickly. And so, so on, so forth. Okay. Um, okay. There's a U over CS there. Unfortunately, that's a sound speed squared. Okay. All right.

So let's go back to the solar wind. We discussed this briefly yesterday, and I'm going to tell you about some things that I don't think are commonly known, but I, but are very relevant both to the Sun and other stars. Parker in 1958 comes up with this idea that because the pressure of the interstellar medium is so small, you have to have a supersonic flow. And we've seen these equations. I told you again today that thermal conductivity goes like T to the 5 halves. You plug that in there. Um, basically, because the thermal conductivity goes like T to the 5 halves, T of R goes something like R to the minus 2/7, that is slower than minus one. And therefore, this integral converges. And so you have to have a finite pressure at infinity to confine the solar corona. And because that's not generally not possible, you have to have a flow. And this is actually the figure from Parker's paper where he does essentially isothermal winds. So this is 500,000 degrees, 1 million, 1.5 million, 2 million, 2.5 million, 3 million, 4 million. And these are the acceleration profiles that he predicts. Let's, let's do something very simple. Do you have an idea what the thermal velocity of a hydrogen gas at a million degrees is? Do the calculation. It's not that hard. Come up with an answer. I'll give you a minute or so. Taking a page from yesterday's lectures, please do it. Sorry, I don't want it in eVs, I want it in kilometers per second. The reason is that I know the escape speed from the Sun in kilometers per second. So I want to find out what the thermal speed is compared to the escape speed from the Sun. You, you can all look up the NRL plasma parameter book if you need K and you need the mass of a proton. Should be a fairly quick calculation. You can also ask ChatGPT, which I think would have come up with an answer already. Not complaining about your speed, guys. Or maybe I am. I don't know. Sorry. That number. Yeah. Was meant to be a joke. Yeah, yeah. Sorry. Slow. Yeah, you're right. Very slow. Oh, I'm, I'm definitely confined by gravity. No problem. Okay. On an estimate, give me a number. 100 something. 100. What did you say? 50, 50, 50. That's probably more or less okay. So it's around 100, 150 kilometers per second. We're not interested. I'm not interested in the second. I'm just order of magnitudes. 100 kilometers per second. The escape speed from the Sun is 600 kilometers per second. The atmosphere is confined. There's no way. Why is there a wind? Okay. So tell me now, why is there a wind? Parker has this strange argument for a wind involving a de Laval nozzle. It's complicated. Why on Earth is this 100 kilometers per second thermal speed gas escaping? What's going on? What's the problem? Why is there a problem? It's ionized, remember? It's 2 million degrees. What's the ratio of the electron mass to the ion mass? About 2,000 in my book. So what's the thermal speed of an electron if the thermal speed of the gas is 100 kilometers per second? The thermal speed of the electrons is about 42 times, 41 point something times. It's 4,000 kilometers per second. Whoops. The electrons aren't confined. They want to take off. Now, what happens? They start taking off. The protons are left behind. Are the protons happy? What happens? An electric field is set up. The electric field pulls the protons up and calls the electrons, "Hey, get back down here!" Why? Because you don't want the Sun to charge up, right? So in the end, some equilibrium has to be reached where there's no current flowing out of the Sun. We don't want to charge up the Sun. It's not going to happen, right? We can't charge up the Sun, except when you have rubber shoes and you rub against something. But, you know, the Sun, if it tried to do that, we'd be really in trouble. Anyway. So, um, so that's what happens. How does, how is that expressed in terms of fluid? We don't, we don't have enough of the algebra done. But what happens is really, it appears in the fluid equations in terms of the pressure gradient. The pressure gradient, if you think of the electron equation of motion, there's the electric field E times E, and then there's minus grad PE. And then if you look at the protons, there's an electric field and there's a minus grad pi. And when you sum those things up, you don't care about the speed of the electrons, but you sum that electron pressure gradient, which is proportional to the electric field that gets set up. And that's how that folds in. And it's hidden in the fluid theory. And that's how Parker gets his solar wind. It's through the pressure gradient, which is, which mediates that ambipolar electric field that's getting set up. Okay. So that's Parker. Except that a very famous astrophysicist, Eugene Parker, photograph on the bottom left, has discussions with Paul Roberts, another very famous fluid dynamicist. Unfortunately, both passed away fairly recently. But Roberts was at the time, was here in Colorado, was trying to figure out what is this non-about the wind, how much is the energy, how does it work? Do you get really, so wind, the supersonic wind? Can you get a subsonic flow? And in that paper, he says, in the paper, says, well, you know, we all, you know, it's, it's late now, we're in the 70s, we've seen the supersonic solar, oops, we've seen the supersonic solar wind already. So it's kind of an academic discussion. But Eugene Parker has pointed out where the temperature at the base of the solar corona, 100,000 degrees instead of a million. Then the pressure far from the Sun, the interstellar medium has a finite pressure. It's in clouds. It would suffice to suppress the solar wind entirely. So you read that interstellar medium pressure is 10 to the minus 12 dynes per square centimeter and it confines 10 to the 5 Kelvin. So what is it? So it's not really true that the pressure of the interstellar medium is small, right? So is that a, is the solar wind a good argument? Let's take a look at this argument a little bit more in detail.

All right. So let's write, let's consider that high conductivity makes the solar wind isothermal, gamma equals 1, make things real simple. Let's consider the Sun to be a perfect sphere. Let's forget about magnetic fields. An ionized gas. Okay. Magnetic fields may be involved within the corona. We don't care. Mass flux is conserved. Row R squared. Sound speed is the isothermal sound speed. And you can write it. P is C squared times row. Okay. The equation of motion is simple. U du by dr is minus one minus one over row dp by dr minus G r squared. I've written it that way because I've normalized the radius with the solar radius. So little r there is in units of solar radius. I then decide to divide the velocity of the solar wind by C, the sound speed, and write the equation in terms of the Mach number. And what happens? Because the density and the pressure are the same, same thing, this one over r dp by dr r is actually the logarithmic derivative of density. And so you have that the logarithmic derivative of density plus the logarithmic derivative of U plus 2 over r is equal to zero. So you can replace this animal with 2 over r plus the logarithmic derivative of the velocity, which is Mach prime over Mach. Then you pull this to the left and you get M minus 1 M dm by dr r, which I've written as M prime, is equal to so 2 over r minus G over r squared. Okay. That's a number. Um, and this equation is really trivial to integrate actually, because you can do the integral on your own. And what do you find? You find that the first term gives you one half M squared. And then at the solar base, you have some Mach zero, Mach zero squared. Minus M prime over M is trivial to integrate. Log of M over M. 2 over r is trivial to integrate. It's 2 log r. And that's simple to integrate. It gives you G over C squared minus 1 plus 1 over r. That means that in r equals 1, the surface of the base of the corona, everything vanishes on the right, and Mach is equal to Mach zero. So everything vanishes on the left. Now, because of the little game that we played before, we can also use that same equation, row R squared, to write this in terms of a pressure rather than in terms of M. M. So you can also write this as you get rid of the two log r, get rid of the log M M zero, and you transform that into this plus log of P over P zero. It's just the consequence of this quantity here being constant. All right. There are two ways to integrate this equation. Fine. You find all the solutions of that ordinary differential equation. You write the phase space of it, and this is what it looks like. And you find that there are solutions where Mach tends to infinity at r equals 1. Remember, we start at the base, normalized radius r equals 1. And these solutions are always supersonic. All solutions. Then there are solutions that start with a very low Mach number, they accelerate and they reach this critical point where the right hand side goes to zero, but they didn't go to zero. So the derivative has to go to zero. So they flatten out and they decelerate. And then you have these weird animals that are double-valued solutions that aren't physical. Right? Those aren't solutions. Right? It's two values. We want to chuck them away. Actually, they're very important. We're going to use them in a minute, and I'll show you how they work. And then there's these two splendid solutions. One that goes from very high and comes descending through this ordinary critical point of the differential equation, where U is equal to VT, and one over dA by dr, that's the 2 over r, it just generalizes to arbitrary area, is equal to this. And one that rises. And the yellow is the Parker solution, the transonic solution. So that's the phase diagram mapped out. And Parker said, well, you can't have this solution static because you don't have enough pressure. But then, but then we find out that there is pressure. So what does Parker do? Parker produces a solar wind. And he says, oh, but there's pressure at infinity. So I have to put a shock in the wind. Okay. So that's really what happens. He puts a shock in the wind.

All right. Well, let's go look at the pressure as a function of distance. So remember that equation that we just, oops, that equation that we just integrated here. Turn on. Looking at the wrong. Thank you. Here it is. See this equation here? We can actually write out what the pressure is as a function of the pressure at the base. Right? This equation tells us that P over P zero is equal to e to the G R C squared minus 1 plus 1 over r. Which looks like a static. That's exactly the static profile because the stuff on the left has to do with the Mach number. But we can just pull that onto the right hand side. And it tells us that P over P zero is e to the this times e to the that as well. Okay. Okay. Actually, e to the minus that. So that the e to the minus M Mach number, since the Mach number tends to infinity as you move far away, that's how that pressure goes to zero at infinity. Okay. But you can actually go and calculate it. Now, what's going to happen? What's going to happen is that for these breezes, or these breezes, the Mach number goes to zero. It's actually trivial to show the Mach number goes. If you look at the top equation, look at the top equation and you, and you say, whoa, how does, how does M go with r if the Mach never becomes bigger than one? Ask yourself that. Okay? So that means the Mach number doesn't, never counts very much. Okay? So, so the only terms that count are log n here and log r with a minus sign. So it tells you that the Mach goes like r to the minus 2. Okay. So this equation tells you that for these subsonic solutions, Mach number goes like one over r squared at large distances. Right? But we can calculate the pressure at infinity. Now, for all these solutions, for example, for the static atmosphere, there's no Mach. P when you go to infinity is P zero times e to the minus G R C squared. So if the pressure of the intermedium is lower than that, you can't be static. But since the breezes also have a Mach number that goes to zero to infinity, the breezes also have a finite pressure. But the finite pressure of the breezes is the same pressure at the base times e to the M zero squared minus G over r over C squared. So for all these solutions down here, the pressure at infinity is finite and varies from what possibility? From the static case, e to the minus G over G R sun over C squared, all the way up. What's the critical? What's the critical breeze? The fastest possible breeze you can produce. It's a breeze that goes up the critical point and then there's a cusp there. And instead of continuing up on the transonic, comes back down. Right? You can't make a breeze better than that. It's a quasi-supersonic wind. It touches supersonic but then goes down. It's actually like an infinitesimal shock. Right? And so for any pressure of the interstellar medium between these two, you could actually get one of these subsonic solutions. So maybe Parker wasn't right after all.

Now, let's consider that Parker is right and we go through the supersonic wind and we have a small pressure at infinity, but it's smaller than those that we were just considering. Well, then there's no solution we can do. We have to put a shock in. You have to put a shock in. And you notice that I put this dashed line in. I put the shock in and then I'm continuing on one of these double valued solutions. You said, Mark, what are you doing? Well, I'll tell you in a second how I derived this dashed line. But clearly, the shock has to connect something up here with something downstairs. Sorry for using upstairs and downstairs again. But the reason is that these are the only solutions of the equation. So the flow has to respect it. Right? I can put a shock in, but I still have to be on segments of the stationary flow. Is that clear to everyone? All right. How do I put a shock in? We're talking about a strange animal, an isothermal shock. Isothermal shocks, shock everyone except astrophysicists normally, because there's no such thing as a thermal shock. Except that in astrophysics, you have these exotic things that happen like radiation takes off. And so you can actually have a shock that wants to heat things up, but it can't because things radiate away so fast that it stays isothermal. Right? So we're considering this animal. So what's the isothermal shock? How do you derive the conditions? Well, you have to conserve mass. And mass conservation is row minus M minus equals row plus M plus. And then you have to conserve the momentum equation. And you're going to get a discontinuity in the pressure. And, and you have to consider that there's a shock. So the advective term, row U squared, has a discontinuity too. And so basically, you get row M plus squared plus the pressure gradient, the discontinuity in pressure has to be equal on two sides. And when you put these little two algebraic equations together, what you find is that across the shock, M plus M minus has to be equal to one. So the Mach number, when you go through a shock, becomes one over M. And since you're on a supersonic thing, you go on to one over M, which is the dashed line. The dashed line is nothing but that Parker solution, one over it. Okay. So that tells me how I can rigorously put a shock in the flow. So everything in that picture is exact. Okay. It's actually a solution. Okay.

Now, it stands to reason, stands to reason that if I now have this wind with a shock, and I increase the pressure at infinity, well, a sound wave is going to travel. It's going to reach the base of the shock and it's going to tell the shock, "I'm higher than what you want." So what's the shock going to do? It's going to retreat. Right? So it's going to jump a little less. So its pressure is a little bit higher. So it can accommodate the new higher pressure. And if you push harder, you're going to bring the shock a little bit further in. Right? And if you push harder, how far can you push it? You can only push it. So you got this little one here, this critical breeze, up, down. That's the smallest possible shock you can get, an infinitesimal shock. As I said, the algebra is really simple if you want to do the calculation. Okay. So now there's something we are going on. Since I could reach that critical breeze by pushing the shock in, there's got to be a range of pressures where we can have two solutions, a wind with a shock or a breeze. Right? But normally, when you have two potential solutions for the same boundary conditions, typically one of them isn't going to be stable. So let's take a look at the stability of the flow. Now, this is an interesting problem because Parker, who had proved that the solar wind exists, or had suggested solar wind, very quickly did a stability study and decided all the flows were stable. Except that, um, a few years later, Jokers, a scientist in Germany, who liked the kinetic model I was telling you about with electrons and protons, came up with this idea, really, we should be talking about electron-proton separately, we shouldn't be doing the fluid theory, decided to go back and see what Parker had done and realized that Parker had been a little bit in a hurry when he tried to prove the stability in the sense that he'd applied a little bit too many boundary conditions. And in a stability problem, if you apply too many boundary conditions, typically you kill the instability because you're asking too much of the solution. So let's go back and look at it. So how do you do that? Well, you want to see if you can go from one solution to another. So you want to propagate sound waves in your system and you want to make sure that the pressure doesn't change at the boundaries. Right? So basically, you study the motion of sound waves in the two directions and you impose that the pressure doesn't change at the two edges. So that's that defines your stability problem. So you define now a delta U and a delta P. And you, instead of writing it in terms of delta U and delta P, which kind of confuses the atmosphere, you write it in terms of waves going in one direction or another direction. So it turns out you, that sound wave that moves in one direction has a fluctuation velocity exactly equal to the sound speed times the fluctuation in density. And if it goes in the other way, it has a fluctuation velocity which is equal to minus the fluctuation in density. So these two things, called Y plus and minus, are actually the eigen modes of a wave going one way or the other. And you write them and you say, okay, I can't, I can't be Fourier in the radial direction because things depend on r. But I'm going to look for a solution of the form Omega plus i gamma T. And then you write the perturbation equations, and they look like this. But they're interestingly symmetric. So what you'll notice is that if you didn't have any gradients in the speed, they would be decoupled. There would be no coupling between Y plus and Y minus. The reason is that if you have a homogeneous atmosphere, a sound wave that goes this way is not linearly coupled to a sound wave that goes that way. The coupling is nonlinear. Okay? But because there's a gradient, we're propagating in an atmosphere, there's reflection. And that is provided by this animal here. What happens to waves propagating in a system with gradients? If there is no velocity field and you propagate a wave in a system with gradients, you get reflection. But the energy flux in the wave is conserved. So, for example, you shine light onto a glass. Part of the light is reflected. If you calculate the net flux of energy going forward, that is exactly conserved. It's not going to be equal to the flux that you impose at the beginning because some energy is going to be coming back. But if you go calculate the energy in the ray coming out of the glass on the other side, it's exactly equal to the difference between the energies coming back and forward behind the glass. You, you agree with that? I hope. Right? What happens if you have a flow? If you have a flow, energy flux isn't conserved anymore. Do you have any idea why that might be? I have a static medium and have waves propagating. They can get reflected. They can forward. Light does it all the time. It doesn't do work on the medium. But you have a flow. I kind of gave you a hint. What's going to happen? Well, the wave can push. Therefore, it can do work. Okay? And if the medium is a medium like a wind, or then it can actually push it to expand. Therefore, energy is no longer conserved. But there is a new invariant that takes the place of the energy, which is called the wave action. Yeah. The wave action is actually the ratio of the energy to the frequency. It's the number of photons, if you like. But in a homogeneous medium, it's the difference between the outward photons and the inward photons that's conserved. So the net number of photons moving out is preserved. That's the wave action. It's that, it's that thing that's written up there. So that's the outward photons, that's the inward photons. So if the medium is stable, wave action is conserved. But if there's an instability, well, the flow can generate waves. So you're not going to conserve this anymore. If you have an instability where the flow generates photons, you know, and so you can, you think, okay, I know that that should be conserved if I'm stable. So I can write, write the equation that corresponds to the conservation of that. And you write it down. And all this is beautifully analytical, as you can see. But now you have the possibility of instability, e to the gamma T. And so you have this piece. So no, if this thing, if gamma were zero, if I were in a stable flow, gamma were zero, then the derivative of this would be zero. And so this quantity would be conserved. S would be a conserved quantity. But now it might be unstable. But can turn this equation around to find gamma. Okay. If I know the flow that I'm talking about and I find the solution to Y, I can use this to estimate gamma. Turn this equation around. And that's what you do. And interestingly enough, you get a very simple calculation to do. It tells you the gamma is actually two times the amplitude of your fluctuation at the base minus the amplitude fluctuation at infinity divided by this integral. Now all you need is an asymptotic for for the solution. And it's trivial to do. You can do it on the back of an envelope, literally on the back of an envelope. And what happens? You find that the solutions that allow for the pressure to remain constant, i.e., Y plus equal to Y minus at infinity, have to be the ones with e to the minus gamma r if gamma is positive, or they have to be e to the plus gamma r if gamma is negative. In any case, these things go to zero. So at, at infinity, this animal on the right hand side is positive. Two Y plus zero squared. Now, let's look at the stuff that's underneath. Well, if you're subsonic, M minus 1 is always negative. If you're subsonic, the Mach number is always negative. So this integral underneath is also always positive. It's the sum of two positive numbers. So downstairs, again, sorry, is positive. Upstairs is positive. Gamma is positive. Subsonic flow is unstable. Breezes are unstable. You cannot produce a breeze. And you might ask, that's strange. You can do the thing numerically if you want to be more precise. And it's trivial. These days, ChatGPT would have done it in an instant. Calculate that dispersion relation for you. And there you have it. That's the growth rate of the instability as a function of the base Mach number. When the, when the static, you're a marginal stability. As you increase the velocity of the base of the breeze, it goes up, up, up. And then when you reach the transonic, the Parker solution goes back down to zero. Okay. Why are breezes unstable? Wait a minute. Let's look back at the asymptotic pressures. The asymptotic pressures say P infinity is P zero e to the M zero squared minus G over C squared. Suppose I have a static atmosphere. Then M zero is zero. So I have a certain pressure at infinity. Now I get an outflow. And the pressure on infinity is actually bigger than the pressure of the static atmosphere. Wait a minute. If I am in a room and there's a window and I'm in equilibrium, so I open the window and nothing happens, static. And I close the window again. And I increase the pressure outside, i.e., at infinity, and I open the window. What's going to happen? It's going to come in. It's not going to go out. Oh, in if you ask that master, increased pressure at infinity, you would expect.

flow to go in not out. Well, it turns out that a famous astrophysicist, Herman Bondi, in 1952, was working on accretion. How do you form stars? And he wrote a most beautiful paper. If you haven't read this paper, you want to be called an astrophysicist, you need to read it. It is a difficult paper, but again, it's all done by hand, no computers. And it's difficult equations because he's not isothermal. I cheated; I'm isothermal to make the equations easy. He takes a polytrope, which makes it harder, but you can still do it.

And he suggests that there's a maximum rate of accretion because he's interested in the transonic, except not the transonic of Parker, but the one for inflow, which is the other one. Okay, he's interested in flows that are zero at infinity and that accelerate towards the star. He actually drew the Parker diagram. That diagram that I just drew you in black and white, he drew it for accretion in 1952. Not only did that happen, but another genial astrophysicist, who was at Berkeley at the time, an Irishman whose name is McCrae, had actually put shocks in the flow in 1956. So when Parker invents the solar wind, actually all the solutions had already been written down.

Because if you remember in the static atmosphere, very quickly, oops, I'm going the other way around. You should notice something in this equation. What happens if you change the sign of U in this equation? You send U into minus U. Nothing happens. It's exactly the same equation. It's symmetric in the sign of U. So all the solutions that I just talked about would be equally valid for accretion.

Okay, let's see how far you guys go now. So if this equation is even in U and I linearize it for perturbations, what sign are those spatial terms going to be in U when I do the first-order perturbation? They're going to be odd, right? Because if I linearize dU/dr by putting U plus Delta U, an equation that is even, I now go odd. But now I have d/dt. So if I want the equation to remain the same and I change the sign of U, I have to say change the sign of d/dt. So gamma, my instability, goes into minus gamma. So if the breeze was unstable, an accretion breeze is stable. Ah, so breezes aren't really wind solutions at all; they're accretion solutions. It's subsonic accretion.

Okay, but that means something, something too. Let's go back to this hypothetical question. The pressure of the interstellar medium goes up. Turn on again. Sorry, this is off. Okay. The pressure of the ISM goes up, goes up, goes up. It hits the critical breeze, and the pressure goes up. So B now, what happens? I'm pushing harder. Well, until now, I had a supersonic wind with a shock, so the sun didn't really know what was going on at infinity because it was protected, right? Have a supersonic outflow, it didn't know. I push the shock in just enough to get a little cusp in my solution. And if I push a little harder, the flow's going to decelerate, right? All of a sudden, now a sound wave can tell the sun what the pressure at infinity is. And since the pressure of the breezes depends, goes up with the Mach number, the base, and I'm going to lower Mach number, that pressure is smaller. So there's no place to go. Those solutions with that such a high pressure at infinity, they don't exist. Those breeze solutions, and they're unstable anyway, so you can't go there.

So what's going to happen? Well, all what's going to happen is this: the sun's going to realize it can't take it, and that flow is going to reverse and become supersonic accretion with a shock. That's what's going to happen. So where I put, would I throw the clicker? Did you take the clicker from it? I hope the clicker part still works. The PP out. It works again. Okay, great. So here, here we go. Got it. Let's change color too. Um, so if you push it in here, you got to go to accretion. Oops, stability. That's what happens. You go to supersonic accretion with a shock. There's a, there's a catastrophe. You go to supersonic accretion with a shock. And then if you decrease the pressure at infinity again, you go up, you go up. But this time, the breezes are stable, so you can get an accretion breeze. And so there are times where the star, depending on the history, can have a shock wind or an accretion breeze at the same time with the same boundary. But once it becomes static again, then you have to blow a wind. The shock, there's nothing between. So the story is a little bit more complicated than Parker thought. And, uh, right. And it turns out that it, that that thing works. Um, so when, when, when I discussed this with Parker, he was quite interested. And you can prove that it works because it's very simple to do a numerical calculation these days. You have computers, so you can actually do this exact problem.

So what do we do here? We start with a static sol. So this is distance. This is distance. This is time. And this is Mach number. So we start with a static atmosphere. So we can't go to infinity, so we go to 10. And we decrease the pressure at infinity by epsilon minus 0.1. What happens? A wind takes off. You see sound waves propagating back and forth. That's what those oscillations are, right? But you get a supersonic wind with a shock. Then what do you do? You bring the pressure back up. Epsilon equals plus 0.13. So you bring it back not only to the static solution but a little bit more. What happens? Sound waves propagate, and the shock moves in. And you see the shock is now lower amplitude than the original shock. Moves in. But they still have a supersonic. Now, even though the pressure at infinity is higher than the pressure for a static atmosphere. I push some more so much that I go through the catastrophe and I collapse from a supersonic wind with a shock to a supersonic accretion flow with a shock. And then I go back to the initial condition, and now I have an accretion breeze instead of having a supersonic wind with a shock. So I can have both. So you can actually show that that's what happens.

All right. Okay. So, so you've got to be careful with flows because it's really counter-intuitive how they work. And that's what I mean when I say that when you're dealing with a plasma, don't take anything for granted. Okay? Um, it's happened to me more than once that a problem that I thought I knew forward and backwards, when I started thinking about it again, I realized I was completely wrong. And so it might happen to you soon. Um, and hopefully it will, because normally you learn things when that happens. Okay. So I don't know how much time I have now, but I think I have a little bit of time still. Okay, fantastic. So let's talk about the real wind now, rather than this toy wind, which is interesting from a pedagogical point of view, but not. Oh, and by the way, um, you might find in the literature, because Parker Solar Probe has been launched, so somehow this, this thing, wind Parker, attracted interest. And so all, all of a sudden, people are trying to solve, solve problems once again. It's a typical thing physicists do that all the time. You know, when, when cold fusion was discovered, there were like 25 theories appearing to, to describe cold fusion. So now we have Parker Solar Probe, and, uh, there are new theories coming out as to why the solar wind is supersonic. But they don't really say anything new. Um, it's the same old stuff. All right. You've seen it. You've basically, you've seen the whole story.

But this is what Ulysses showed us in the '90s. So this is the solar wind. These are polar plots of the solar wind speed. You probably have seen these before, maybe in some talk or something. And Ulysses went all the way to Jupiter and was kicked out of the ecliptic plane. And it's the only spacecraft that we've ever sent pretty close to the poles of the sun. And that's the first orbit. And what do we notice when we see that first orbit that happened to go over the poles, basically around solar minimum? We see that in the ecliptic plane, the solar wind is, uh, not so fast. This is a polar plot of the speed, and the numbers are up there, 1,500. So you're in the ecliptic. You see this jetting, you know, from below 500, close to 250, all the way up to 750 and back. And you'll notice how crispy it is on the, how, how zig-zaggy it is on the left, and how not zig-zaggy it is on the right. Does anybody have an idea why that is? See if you think about it a second, why on Earth would it be different on the left, on the right? Think about the orbit. It's an elliptical orbit. Had to go out to Jupiter, had to go over the poles of the sun. And it's, so it's aphelion distance from, from the sun was five astronomical units. Came into perihelion at 1.3 astronomical units. So as it was climbing out of the, it was going really slow. And so that warped heliospheric current sheet, as the months go by, would go around it many, many, many times. So that's the left-hand side. But I went over the poles, it came zipping down really fast. And so it only crossed the warp sheet a few times. That's why it looks, there's a bit much less jackets on that side. It's telling you the orbit's elliptical. Okay. So that shows you that there is this warp sheet with slow wind. And you can see that there's a band at solar minimum, I would say it's almost 30 degrees, uh, 25, 24, 25 degrees, um, around the equator, around the ecliptic, where you have this alternating slow, fast wind. But once you climb out, you're up there, you have this beautiful wind, almost sitting on a circle. It's noisy, it's got fluctuations on it, but it's sitting up there at around 750 kilometers per second. So we have a 750 kilometers per second wind. That's the region where the magnetic field can't confine the gas, right? Because it's open. So the sun basically produces a 750 kilometers per second wind. Ah, I would call that turbulence. That's kind of a very vague thing. But this thing's always oscillating. And so there's a plus or minus 50 kilometers per second, maybe about 80 total, plus or minus 40 on top of it. It's always there. So there's some, some oscillation. We don't know really if it's turbulence or not, but it's called oscillation. And same on the other side. And the blue and the red are the polarity of the magnetic field. So it does show that we have a fairly beautiful dipole. Okay. Um, red on the top, outward IMF. Blue on the bottom. The next time it went around, it was a solar maximum. Anything goes. It's crossing fast and slow wind coming from all over the place. We can't even see a generic structure there. It's really hard to tell. And the wind is going up to speeds that are significantly faster. And it's going to speeds that are, you know, pretty close to the ones that we saw before.

So we do this all the time. We always do a disservice to ourselves in some sense. And if Dave Mamas heard me, he would kill me on the stage. Um, it's a disservice because this is an instantaneous photo. Here we're talking about essentially, um, five, well, it's basically five years. What's, what's the Jupiter frame? What's the Jupiter orbit time? It's a little bit less than that. But was it five years? Well, Jupiter, how long does it take for, come on, a cube over T squared, guys? So a cube is, a cube is 125. Square root of 125 is, sorry. Thank you. Oh, I was trying to do a calculation. You want me to do it like this? Uh, okay. So solar minimum was from here to here. Let's call that seven years. Probably closer to eight. Seven, eight. It kind of makes sense. Yeah. All right. Um, and that's a picture. That's an instantaneous picture. So the corona is changing all the time. So that picture was chosen because it fit that picture better. And so again, this picture was chosen because it kind of fit, but it's arbitrary, really. There's thousands of pictures that were done by SOHO over that period that could have, you know, equally well.

Third orbit that it unfortunately didn't complete. Spacecraft, they said, basically failed at the end. As you can see here, it's pretty much the same. The slow wind is much, occupies a bigger slice. This was the very famous, very, very weak minimum. And I don't, you guys are young enough that you probably don't remember what the newspaper articles were saying at that time. There were people saying that sunspots were going to disappear. And because people know that when there's magnetic activity, we get more heat, like some, they were claiming we were going to have a new Glaz, Glaz, what's called glaciation? What's, what's it called when you get a glacier? For me, we're going to have a new ice age. There were actual articles in journals saying that we were going to have a new ice age coming. Yeah, didn't, didn't come out. Sunspots didn't disappear. No high age appeared here. But it was a very deep minimum, and, and people were worried about it. Yeah. Do you want to have a question? Yeah. Yes. Yes. Yes. Yeah. So this is, this is perihelion is about 1.3 astronomical units, and this is around five astronomical units. So the wind isn't really accelerating very much beyond one AU. It's kind of coasting along there. A, a little bit of acceleration in some regions. The, the left is further away, yes, and the right. But once you get out, you see that it's pretty much the same. There's some slight decrease in speed, but not much. So at, at, at Jupiter's orbit, the speed is, well, I mean, hard to tell, but there's no big radial acceleration for sure, right? You might infer that there's some here. You see how it's curling in much more strongly than here? Yeah. So here there could be some actual average effect, perhaps. You're right. It's an interesting point. Has nothing to do with sides of the sun. Sun's rotating so much faster than, right? You're right. But anyway, so I'm glad we clarified the picture for you. Anyway, that's the, that's the number of sunspots below it. This, this picture was made trying to imagine where Parker was going to be. And so I did a very stupid prediction of the next solar cycle, which is just redrawing the same one but shifted. But it got it pretty much right. So I claimed that Parker was going to be doing, going through the ascending phase and going through solar maximum. So that's what we're doing, essentially. Um, okay. So, um, so that's the solar wind. So now we're in trouble. Next. Why, why are we in trouble? Because the speed is, is pretty high. It's 750. It's not 300. If you remember the Parker plot of the isothermal wind, a million degrees, 2 million, it gave you 200 kilometers per second at Earth orbit, not much more than that. Um, and we're in trouble because of this as well. So what is this plot? This is again Ulysses, but this time what we're looking at is the speed of the wind, um, the speed of the wind, which has got to be this plot here, dashed line. And then we're looking at the ratio of ions of different, uh, ionization of the same species, oxygen 7 over oxygen 6, and magnesium to oxygen. Okay. So what is it telling you? Well, once you're, the wind is very, very, very, very, very no density at all, so that ratio is basically fixed in the corona. And it depends, how much oxygen 7 you're going to have compared to oxygen 6 depends essentially on the electron temperature. The hotter the electrons, the more oxygen 7 you'll get with respect to oxygen 6, right? In the same way, you'll get more magnesium than oxygen. Okay. So that's telling you what's called the freezing-in temperature of the corona. I.E., it's giving you an indication of the electron temperature of the corona where that wind is coming from. And so what's that telling you? It's telling you that the fast solar wind is coming from the cold corona, which kind of fits with what I was telling you yesterday where I showed you the polar coronal hole is cold in electron temperature, and the slow solar wind is coming from the hot corona. And, and that is not the Parker solar wind. Parker solar wind depends in a positive correlation between the electron temperature and the speed. Okay. So there's something else going on. There's an indirect proof of the same thing coming from the ultraviolet coronal spectrometer on SOHO, where they use a very nifty idea developed by Janosi called Doppler dimming. And basically they can do, there's a, there's a line of oxygen which is sufficiently close to Lyman alpha. If oxygen is moving right exactly at the right speed, it absorbs the Lyman alpha line, and so you see this dip, dimming due to the Doppler effect. And using that, um, with some modeling, you can derive things like temperature and flow speed. And this shows you the temperature of minor ions and electrons in a coronal hole. You see that the electron temperature in the coronal hole, dotted line, scarcely reaches a million degrees. So a coronal hole is really cold from the point of view of the electrons. On the other hand, the minor ions, look at that. That's O6. This is neutral hydrogen. It's going up to a few million degrees. But that matches with what's seen in C2. In C2, the strongest correlation in the solar wind that exists is between the proton temperature and the measured speed at the same place. So there's a positive correlation between proton temperature and wind speed, which is the strongest, this is the best correlation that exists, actually. Solar. Okay. So this is showing us the differences now in temperature and protons and fast and slow wind. It's kind of a summary here. And fast and slow wind. So as astrophysicists do normally, but since we're solar physicists, we do that all the time, it's got to happen. Magnetic field's got to play a role here, right? One, you used, there's, there's this old joke, um, that I'll repeat here, hope you don't kill me for it. Um, I don't know who said it once. I think it was a Dutch astrophysicist who said that the magnetic fields are to astrophysics what sex is to psychoanalysis. Basically, because, you know, whenever there's a psychoanalytic problem, they always claim some sexual trauma in the past. The astrophysics didn't know what the hell's going on, they say, ah, it's got to be the magnetic field, right? And it's basically that's what's happening here too. It's the magnetic field. Mag field provides confinement. Why is the hot wind slow? It's because it has to, it has to drive its way through the magnetic field. It's closed. Um, but why is the open field cold for electrons? Well, probably because there's no time to reach any kind of thermodynamic equilibrium. There is no, there's something that clearly prefers the ions in terms of depositing energy, and the electrons can't participate because there's no, very few collisions, so they never manage. But that also means that this old picture of the electrons running away and the protons being lifted out, you know, going for the ride, isn't quite right. There's something going on for the protons as well. And if one looks at the distributions, this is from the, uh, Helios data, and then I'll stop, or I might, there's another slide available, I'll go for it. But, um, so it shows essentially the proton distribution function. Now, these were selected. So this is Earth March and back in the 1980s. So, well, I didn't realize it's 42 years old now. Interesting. Pretty shocking, actually. But anyway, the fast solar wind, remember the, the, the distribution functions I invented when I was showing you drawings the other day, saying, well, maybe in a plasma, if you can accelerate along the field, well, all sorts of weird things are going to happen. Well, you're going to get these bicycle things. People now do this in 3D, and so they're fancy, so they call them avocados instead of bicycle seats, but, you know, it's pretty much the same thing. So that's the fast wind for you. Slow wind is really a mess. But I'll show you that it's actually much more messy as you go close to the sun. But I don't want to ruin the right, so I'll stop here. To be continued. Um, and so this afternoon, we'll delve into geography. How geography plays a role in accelerating solar wind with some ideas, no solutions really, but there's all the data that's coming now from Parker Solar Probe and Solar Orbiter. We have a much better idea of all the activity that's going on close to the sun. All the things that, you know, the connection between what happens at the sun and the heliosphere. Actually, used to be two separate fields. When I got into the field, there were the people that were doing solar wind, and there were the solar people, and they were, they hardly talked to each other. It was really funny. They, it's like, oh, we see this stuff. And, and people looking at the sun said, well, but it's got to be the same stuff, right? But it was very difficult. I mean, in the corona, you see these plumes going out, but no one has identified, to this day, no one has identified a coronal plume in the solar wind. No one has actually said, that thing that I see there is this thing. We do it with CMEs, of course. The CMEs are big, fat animals. Um, we attempt to do it. But is there a safe? Well, thanks to Parker's really close, so it's kind of cheating, right? Because Parker is sitting at the edge of a coronagraph, so it's actually measuring and seeing stuff that you can actually see. So it's helping to make that connection. But the whole goal of missions like Solar Probe and Solar Orbiter is to actually not only understand the acceleration process, but figure out what happens with the magnetic field and the plasma and the star as it propagates out and make that connection once and for all. I'll stop there and let you guys ask questions. Oh, yeah. It's, it's, that's actually not very difficult to do that. Um, sure, sure. The, the, the question is, could you explain why a lower Mach number shock leads to a higher pressure far away? And it's basically, um, well, from the physical point of view, what's happening is that the, the higher the Mach number, the lower the pressure in the flow of the supersonic flow. Okay. So, so the, the jump in the pressure is higher for a high Mach number of slow. When you get to lower and lower Mach numbers, the jump is smaller in the pressure. And so essentially, now you have to compensate that with the actual distance at which the shock occurs, right? So that pressure is essentially a monotonically increasing function as you move in. Okay. So you have a smaller jump, but you're much further in. So that when you put the two together, you get a higher pressure. Basically, that's, that's it. The algebra is very simple, but that's what it's telling you. I'm saying with words what the algebra tells you. Okay. So, so wind speed is, is, uh, anti-correlated with, uh, temperature at its source. You said electron temperature. Electron temperature. So that gets rid of the idea, this ambipolar field. The ambipolar field is there, okay? It's not that it's not there, because the electrons, even if they're older by a factor of two, remember their thermal speed is still higher by a factor of square root of two, you know, square root of two, 42 divided by two, so it's still a factor of 30 higher than the protons. Okay. That was more or less my question. Is that, you know, basically negates the whole Parker? No, it doesn't negate it entirely. It's still providing like a baseline. It still provides you a baseline. In fact, I will show you this afternoon, I think, a calculation. You can use the electron distribution function to estimate the ambipolar electric field as a function of position. And what you'll see is that, well, not the fast wind, but for the sum of the slower wind, the ambipolar electric field is just right. It just provides the right acceleration for the wind. But it doesn't work for the faster wind. So you've got to ask, what's in the, what makes the faster wind fast? We have ideas. It's, of course, it's not that there aren't, I mean, you, you can probably figure them out on your own. Clearly waves and play. I just, I just spent time telling you that waves push a medium, right? So if you have a lot of waves, you can push. So there's this whole gain. So if you, if you dissipate a wave really close to the sun, provides heat, but it doesn't push. But if you heat, and we'll see this this afternoon, you basically increase the scale height, so you're pulling up more matter in the corona. But if you only have a finite amount of energy and you've got more matter to lift out, your speed is going to be less in the end. So heating close produces a larger mass flux but smaller velocity. If, on the other hand, you can't dissipate the waves, so they propagate out, they push really hard, but the scale height remains small, you get a higher velocity but a lower, slightly lower mass flux. The mass flux doesn't change much between fast and slow, it's only about a factor of two. Can we, uh, go back to the Ulysses orbits? I wanted to ask a question regarding how does it, the change in the solar cycle affect the, uh, region where we see the slow wind? So you mentioned that in the third cycle, it was the deep minima, so the slow wind is much, uh, at much higher latitudes compared to the first, or it's much closer to the equator? So what's causing that? Okay, so the, the fast answer to that question is, I don't know. But we can speculate. So let's speculate a little bit. So there are some non-trivial things that happen in the corona and with magnetic fields. Of course, plasma beta in the corona is small. It's small whether you're at minimum or at maximum. So magnetic fields kind of dictate what's going on. The fact that you're solar minimum does not mean that you have a better dipole. It means that you have a smaller field, but it doesn't necessarily mean that the dipole is clearer than in a minimum with a stronger magnetic field. So it could well be that, in fact, you have a less well-defined dipole in a deep minimum than you have at a strong minimum because of weaker fields. Because of the weaker fields, and so that might justify the fact that you have more quiet sun and therefore more regions that are actually closed and therefore are smaller regions. Or on the other hand, you may have coronal holes that are at the poles occupy less volume, and then you might have very tiny coronal holes that expand a lot closer to the equator. And those, so that, that's a discussion about where does the slow wind come from. And that's the discussion we're going to have this afternoon with magnetic, the magnetic geometry is crucial in determining, you know, what happens, the ultimate speed of the solar wind. Everything I told you about in that Parker thing was spherical, row u r squared is constant, so it was just spherical geometry. There was no magnetic stuff. A little bit about yourself that we do. Okay. Thank you very much. Nice to see so, uh, many youthful faces, and you all look awake. That's fantastic. As, uh, Max said, I come from University of Illinois, Urbana-Champaign. Okay, some of us are lazy to, uh, fill in the entire, um, name of the institution, like myself. But there's a distinction. My research is, uh, primarily on the terrestrial magnetosphere. I like to study the polar wind, how it connects with the, um, um, ring current region or the inner magnetosphere. And most of my work has to do with numerical simulations, including the VR work. So if any of you are interested in education and are good at, uh, writing code and interested also in immersive technologies and how to use them for education, feel free to contact me. We're not going to do any of that today, but I'm going to give you an introduction into magnetohydrodynamics. Okay, this is going to be the 101 level. I just want to kind of get an idea of how many of you are familiar with modeling. Okay, that's fantastic. Doing new software development. Okay, good. That's good news for all of you. Some of you will be bored, please forgive me, but some of you would be, uh, would learn a couple of new things. So I like to start my presentation by showing this, uh, a movie. It comes from, okay, it looks like it doesn't go. Have to play it, I guess. Okay. Uh, uh, it's part of a show that's called Journey Through Space. It was shown at the American, um, um, Museum of Natural History in New York. It's narrated by Opie Goldberg. So I'm not doing that of a job for you, but what's, uh, interesting, you can see that it follows the evolution of plasma or coronal mass injection in this, uh, situation. It starts with a, with a quiescent, and it tracks the evolution of this blob of plasma through the, uh, interstellar, uh, medium. And it, uh, also visualizes, shows a beautiful visualization of its interaction with the, uh, terrestrial, uh, um, environment. What I find really neat about this visualization is that every single pixel in it, it's a solution to a set of equations that's called magnetohydrodynamic equations. Okay, that's the beauty of it. Um, now, how do we describe this environment that we all actually study from a very different perspective? Now, in order to simulate it and have an accurate solution, would be nice to know what every single particle in the system does, right? So we could look at the guiding center theory. That's, uh, predicts or provides a prediction of the particle trajectory as is a proposition of the, uh, fundamental modes of motion, like of a charged particle in, in an electromagnetic field that is superposed to the gyration to a bounce and a curvature, uh, drift. If I am in, in a magnetic field that's, uh, nonuniform, well, if we have a large number of particles, and I think you learned yesterday that the number of particles, I think within the, the, of the heliosphere is about 10 to the 57. Okay, that sounds like not doable, okay, impossible. So if we have a large number of particles, then the, uh, guiding center theory might not work for us, providing a, a solution in the appropriate amount of time. So one could look at kinetic theory. This allows us to derive transport coefficients from fundamental properties of gas molecules, right? So, uh, um, Marco mentioned earlier today, the Boltzmann equation that tells you, describes the evolution of a, a distribution function, right? If now particles can be described in terms of collective behavior, right, the distribution function and how that evolves under the actions of various external, uh, forces. And, and because the, uh, distribution function, it's six-dimensional now, every particle has an associated velocity and, uh, position, both of those, okay, six of those being, uh, independent variables, and adding time in that, then you have a six, seven-dimensional, uh, U partial integral differential equation that's somewhat challenging to solve. In many cases, we reduce that by, let's say, one dimension if we were to average, if we were to average, let's say, uh, a couple of those motions, the gyration and the bounce, then we get rid of a couple of, uh, dimensions in space. Okay. Now, if we look at compressible fluid dynamics, they, uh, this theory describes the fundamental conservation laws for a continuous medium that, um, um, is composed of individual, uh, uh, particles. So the particle density, of course, is very large, that the continuum description is warranted. So the challenge comes into how do we connect these microscopic quantities to the macroscopic quantities? So I'm going to describe to you a little bit our path from particle to fluid, going from kinetic theory to fluid, fluid dynamics, and from fluid dynamics, added the, the contribution of electromagnetic field. But I'm also going to turn into the fluid dynamics because it provides a little bit of, uh, more clear intuition about, we're all familiar with fluids, right? We've seen waterfalls, we understand how that, uh, uh, happens with more ease than, uh, um, when we, we think in terms of particle distributions and how distributions evolve in time, more of a mathematical description. But let me get to the, remind you a little bit about velocity moments. Okay, the velocity moment. Do we remember what that is from a class maybe we took earlier? We remember. Do you mind telling us what it is? It's kind of written in there. It's an integral over the velocity space of the distribution function. So we say we have, let's say, a six-order, six-order velocity moment in which we have described, let's say, the sum of the powers of the velocity components in the moment integral. So F of time, position, and velocity, integrated over the velocity space, times a velocity to the power six. Okay. Now, the zeroth moment is, by definition, the density. Okay. So we see I have density by definition is the integral of F times, I don't write it in here, velocity to the power zero, that's what it is. Okay. Uh, next one, the particle flux. Particle flux, it's the moment integral that has velocity to power one. So we call it first moment. Uh, and so on. So the, the, the first three moments should be something that we will get ourselves familiar with. The zeroth moment that talks about the, provides the density. The first moment that it's, uh, the flux, and the second moment, the, the pressure tensor. Now, the Navier-Stokes equations are actually the zero, first, and second moments of the Boltzmann equation. The moment integral refers to the distribution function, that's an integration of the distribution function. But the, uh, Navier-Stokes equation, zero, first, and second moments of the Boltzmann equation. What does it mean? It means I take the Boltzmann equation to get the zero moment and integrate it over the velocity space. Okay. To get the first order, I take the velocity moment. I take the Boltzmann equation, multiply every single term in, uh, by velocity to power one, and integrate over the, uh, phase space. Now, if I were to neglect the, uh, viscosity and the heat conduction, I get something that's called Euler's equation. I don't have any expectation that you will remember every term of these equations while I talk about it right now, but I do have an expectation that by the end of this lecture, you will remember or at least have an idea what MHD equations are. Okay. So I get a, what's called Euler equations. So I should have three because I told you it's zero, first, third order of the, uh, um, first, uh, moment of the Boltzmann equation. And ideal MHD equations are actually the Euler equations that describe flow of fluid that has, uh, response to, let's say, gravity, external fields, electric and magnetic fields. Okay. And Maxwell's equations. It considers a self-consistent description of a conducting fluid under the influence of electromagnetic fields. We neglect any heat conduction, we neglect, uh, viscosity, we assume quasi-neutrality of the plasma, and we also neglect, and I get to that in a little bit, the, uh, um, um, displacement current in, uh, Ampere's law. And low and Maxwell's equations are now simplified by showing that Gauss's law has zero terms. So divergence of B is zero. Why would they say that? Causing neutrality. Divergence of B is zero. Until some of you will find a magnetic monopole, that's unlikely. Uh, then the next equation is just Faraday's law, something that we're familiar from our undergraduate forces. And, uh, the fourth equation right there is just Ampere's law where we drop the displacement field. That would be my DT, right? So, uh, uh, time-changing electric fields. And it's what's beautiful about this part makes our life a little bit easier. We see that the equations for electric field, uh, I will not get there yet. Okay. So, um, um, but maybe we can just mention that now I have a formalism for the electric field and magnetic field for which I know what the divergence and the curl of the fields are. And according to our Helmholtz theorem, that hopefully you remember, knowing the behavior at, uh, infinity, if the fields are, uh, well-behaved and vanishing, then I will have a, a good solution for the electromagnetic fields. On the other hand, we have Ohm's law in which we can think about, um, um, um, we have, sorry, we have Ohm's law in which the second part of, uh, um, the additional electric like term comes from what we call a Lorentz transformation of the magnetic field. Then if we were to manipulate the current, so I have the curl of the current, I plug in here the curl, the curl, the del operator operates then on the cross product. I play a little bit with this one, so I get an expression for the curl of curl of B. I use some, uh, vector calculus identity, curl of curl of B should be something that looked like divergence of B minus Laplacian of the magnetic field, vector, of course. Curl of divergence of B stays zero. So I get an equation that describes the change of magnetic field. Now it looks like a curl of velocity cross B, okay, plus a term that relates the magnetic resistivity, okay, that tells us something like resistivity, electric resistivity of a medium. This tells us about the environment or medium's ability to allow the flow of an electric, uh, current. So this is what we call the induction equation, okay, where eta is just a magnetic resistivity. And, um, let's see. So in the MHD, in the set of MHD equations, it's better me for me to sit here because I can't read there. In the, for the set of, uh, MHD equations, we said there are three Euler equations and one that, uh, describes the evolution of the, uh, magnetic field, um, within the fluid. So that's the, um, induction equation. And I will remind some of you that, um, I'll take an interlude here and go over the continuity equation or conservation law. So the most general way to write one, right, such a, uh, uh, conservation law is the time derivative of the conserved quantity. And I like to call that the chi density, okay, just to be fancy with my Greek letters, the density of chi, where chi can be whatever you want, plus the divergence of the flux of chi, whatever that is, equal sources of things in the system. So we say chi is then my conserved quantity, the flux of chi is the transport agent, and sources are things that are on the right-hand side of the equation. And you'll see this. I have several of these throughout this lecture. Let's talk means that you will have an opportunity to turn to your neighbor and will take a good three minutes to kind of identify now in this form that I showed you the, uh, MHD equation, and I glance over pretty, uh, fast, what is the conserved quantity in every equation and what is the transport agent in each and every one of them? And let's do this exercise. The first equation, maybe we've seen it before. We have the rate of change of mass density plus the flux of that, it's zero. There are no sources or sinks in this case. So the conserved quantity is the mass density. How about in the second equation? In the second equation, we said the conserved quantity is the part that's in the time derivative that tells us the rate of change of the thing that's conserved, plus the divergence of the thing that moves it along, momentum. Uh, how about in the fourth equation? It looks like an energy, right? Uh, do we recognize this term, B squared over 2 mu, magnetic energy? Where have we seen that before? Maybe in some EM conservation law for EM energy. Do you remember that? Say it louder. I heard some pointing. Poynting theorem. That is correct. Okay. So now we should know to identify the conserved quantity in each of those. So the first, the first equation, uh, in our MHD set would be conservation of mass. Second one, it's conservation of momentum. The third one, we call induction equation. It's an elevated kind of Faraday's law with an additional term. Okay, but if we were to think about it, and we're going to get to it in a little bit, I want you to keep your mind on what is being conserved in that, uh, um, induction equation, but not answer it right away. Okay. Uh, and the third one is conservation of energy. So as a recap, MHD is actually a fluid approximation regarded that the lowest approximation. Okay, we're describing plasma self-consistently. Lowest approximation. What does that mean? Oh, sure. Taking first moments. Other thoughts there? Say it louder. Okay, that's a way to explain the English. Very good. It's the least complicated one, the most macroscopic. Okay. Can we hold on that thought and come back to it? We'll come back to it. Okay. Now, the MHD, um, um, the, the MHD equations, we can, we can, uh, derive via fluid dynamics or show it in a little bit, or via kinetic, uh, uh, um, theory. They should be consistent with each other. And, uh, um, you should be able to do both. Now, what's important is that it applies to large scales. Okay, it applies to, uh, uh, problems that have large, uh, temporal and spatial scales. And that allows us to ignore all the single particle, uh, motion, to ignore, uh, displacement currents. And the important part, the important point that I would like to make here is that the fluid behavior now does not stand for the bi-collision type of, uh, um, scenario where we're thinking that you have a distribution function, you have enough collisions, then it, uh, uh, uh, um, stabilizes into something that's a Maxwellian. But from the collective interaction at the distance due to electromagnetic, uh, uh, forces, um, between the particles, um, it has several limitations. Okay. One inherent, uh, difficulty is that it has more variables than equations. That's not a good thing in most cases when you're trying to solve a set of equations. So you've seen that the nth velocity moment also depends on components of higher degrees, like n plus one, uh, moment. So, for instance, if you look at the zero moment, right, that, that depends, so, for instance, the mass density, right? Now, the, in, I have a solution for, for the continuity of mass density that now depends on the flux, that's the, uh, uh, first moment. Now, in the first moment equation, I can see that the flux depends on the second moment of, uh, uh, uh, a second velocity moment, that's the pressure. So that's problematic. So therefore, in order to close the system of equations, you have to make some assumptions about the fluid, and you have to assume what's the nature of the fluid, how can you restrain the, the flow, and whatnot. So you can make the assumption that plasma is in some local, uh, um, thermal equilibrium, assume that the distribution function, it's a Maxwellian, which is the intrinsic assumption in ideal MHD, and the plasma then behaves like an ideal gas with some, uh, equation of, uh, state. Now, we also assume that the fluid is infinitely conducting. Okay, all charges are free to conduct. Uh, there's no electric field, you heard this like many times, in the rest frame, and the heat conduction is neglected. But there are many non-ideal descriptions that allow for resistivity, heat conduction, NE, special, uh, charge separation, and they call this like, uh, uh, extended, oh, sorry, higher-order MHDs or, uh, modified. Nevertheless, it's still fluid. So I would like to illustrate these concepts, kind of going from the fluid perspective, because I thought it is much more intuitive than the kinetic perspective, right? Let's say we start with a, a volume that contains a large number of particles. Okay, so this is any small volume in a fluid element that has an immense number of particles, so it allows that the MHD fluid can to be treated as a continuum. So if I were to be given a, a, um, the, the, system variables, uh, for this, uh, situation, I would have a charge, oh, sorry, mass density. Okay, mass charge density, unless you see it in the math of the equation, it's going to be a mass density that moves the sun velocity, U, and for we say if the fluid is displaced in a time delta T, and then it moves a distance that's velocity times delta T, then the mass of the fluid that crosses the surface element per unit, uh, time is nothing else but its flux, that time. Okay, so Ru dotted with a, uh, unit surface vector. So we can come up with a very simple expression for the total mass, assuming that your system.

Is being closed in which the mass is nothing but the uh divergence, the of view of of the volume integral of uh um divergence of the flux. Oops, something happened. I didn't do it.

Okay, so the mass continuity equation is something that you've seen many, many times. Rate of change, rate of change of mass density plus divergence of r u is zero. It is valid for all fluids. It's independent of the fluid nature. If it's adiabatic, if it's isothermal, if it's compressional, if it's uh turbulent, whatever it is, it's valid for all fluids.

Now, let's try to understand what it means. Right, you said conserve quantity. The mass density, the transporting agent is the flux of such thing. So I would like to play this simulation. It's two-dimensional, uh, U simulation for used to uh um uh describe it. I have a blob of some dense plasma that moves into the X direction with some uniform speed. Okay, so it means that that equation, it tells tells me how the blob evolves. I make a note of something while I look at the simulation. Can we play it again? Yes. Yeah, blob is the form. Let's stay with that thought. Okay, blob, it's a little deformed. That's kind of not what we want. Um, we'll come back to it.

Now, let's think about the momentum equation. Okay, momentum equation was which moment of the distribution function? You remember that? Okay. Uh, momentum equation. Let's think about the equation of motion or equation of motion for a single particle that moves in an electromagnetic uh into an. Oh, this is great. Into an electromagnetic field. So what I wrote there is nothing but Newton's second law. And on the right-hand side, I have uh Lorentz force. And if I were to assume that I have no thermal motion, no collisions, all particles move together with some velocity, uh, uh, u, then I can write my uh, the collective equation of motion for all these particles in which now I just, I have a multiplication by the number of particles. Now assuming everybody moves with the same velocity. But now if I have the thermal motion at uh uh play, that will give rise to gradients in pressure. Right, that will give PR gradient in pressure. So assuming that again, uh, the the charge particle fluids, okay, all the charge particles that make up this fluid act as an ideal gas, then I can add the correction, right, in which the gradient in pressure, it's the additional force that's stabilizing the system. Right, and also I'm making the assumption that the pressure is isotropic. The cow, it's better to be spherical. Um, so considering now ions and electrons that make up our plasma, I can write two uh, equations of motion for each of the species. Nothing fancy over there. And I'm now assuming that I'm assuming charge neutrality. I have as many electrons as ions. Therefore, the total pressure is the sum or the superposition of the pressure of electrons and ions. And the current density follows kind of the same setup.

So putting all of this together, I get something that looks like the momentum equation. Right, the momentum equation. Now we have to pay attention here that uh, this is a total derivative or convective derivative that has the the uh information about, um, uh, the propagation. So I have R is J cross B. Minus, let's try to understand what this one means because there are a bunch of letters. Have we seen this before? Yes. Okay. Um, and the always go to the, how did they call it, the lowest approximation. Okay, of course, would be the trivial fact that there's row is zero and everything, it's zero. So Z equals zero, that's less interesting. But let's assume we are in an environment that I have no ambient magnetic field. Okay, so now I have rud is just minus gradient of pressure. And I'm creating the simulation again, kind of the same setup. I'm showing X Z two-dimensional simulation. X, uh, uh, dimension. I have no magnetic field, no I'm being fi. I have a a uniform, a blob of plasma of uniform density, and I allow it to exist according to this equation. What do you think will happen? Well, maybe we can just play the movie and see what happens, and then we'll check if your intuition is correct. Um, we've seen something. It tends to expand. Okay, under the influence of pressure, it tends to expand. It's, oh, the expansion is symmetrical. It's beautiful. Our brains are correct. Whatever we learned in underground, that still holds here.

Now, let's look at the situation in which I allow myself to be in an environment that the magnetic field now is non-zero. So we created a simulation for which, let's say I have BY, BZ there zero, but I allow for a field to be uh aligned to the X direction. Right, so kind of the same thing. Uh, the same blob. Now it's in ambient magnetic field. What do we think happens? Now, let's go step back. Are we expecting a change? Yeah. Okay, better change something. At least the numerical error should be different. Uh, what would be the change? What's your intuition telling you? Maybe not much yet. If you didn't start at this stuff before. Okay. No, it's a valid answer. It's the best place to be when you don't know because you get to learn. If you already know, you're wasting your time being here. Okay, so as the blob expands, I note something. As the blob expands under the influence of magnetic field, I see that it no longer expands symmetrically, radially, but there's a flow that's inhibited that's across the field. So can we play it again? The expansion looks aligned with a. We don't know what that. The expansion looks aligned with the magnetic field. So how do we understand this?

I like to think about it in terms of currents. Right, we can understand momentum equation in terms of current. So now I am plotting the BZ component. Recall that before I showed you that the color was density. Now the color is the magnitude of BZ component. My ambient magnetic field was X-aligned. It didn't have any Z component. Z and Y were zero. And I now I'm plotting the the magnitude of B component. I also want you to take a note of the, um, order. Okay, my ambient field is about 0.1 Tesla. B, it's, let's say, four orders of magnitude smaller. So what I see that when this, when the expansion is happening, it tends to distort the magnetic field, the ambient magnetic field. And the distortion introduces, or, or the distortion is associated with a slight change in the field, which the field now has a Z component. Okay, so we call this like some sort of a heel-shaped pill lines that we can't see them in the previous simulation, or we can't see the pill line right here. Well, that would be still the B expert in a 3D view, uh, having a dent that will be a little bit hard to see. Um, and the reason for that is that now the curling magnetic field, right, now I have a B field that has two components. One is in X, one it's in Z. And the curling magnetic field, it is associated with a current that now points in the JY direction. Okay, so if I were to look at the, uh, current, now I'm showing you the associated current, basically just a curve of that, uh, uh, B field. I see it's coming out of the page on, uh, uh, on positive Z and going into the page. Actually, the other way around. I start with zero for Z negative. So the J cross B field now creates a force that would point in what direction, or set up a force that is the force. Even if you're left-handed, it's still the right-hand rule, by the way. Stick with F. There's going to be one direction above for positive Z, one direction for negative Z. So it's going down on the positive and up on the negative. It means I have this force now that's pushing that does not allow for the blob to expand across the field.

So now we look at this equation set and said, okay, uh, I tend to understand it a little better, right, rather than just letters and symbols. Uh, the expansion, okay, the the natural expansion of the fluid as it was a gas, now it is inhibited by the presence of the magnetic field in such a way that the field itself is being altered, sets up the current, and the J cross B force stabilizes the system. So we have a J cross B or a Lorentz force that it's opposing that expansion. So we say the J cross B term arises from the coupling of the current density, okay, to the magnetic field. This is the force that makes the EM EM fluid very different than the the, uh, than the neutral gas, the ordinary gas. The thing that you've seen in the first simulation, that term goes away. We don't care. It's exactly what we learned in thermodynamics if you've seen that in your earlier classes. And the J cross B term is the electromagnetic dress. Now, the minus sign in the momentum equation tells us, well, the fluid is being driven in the opposite direction from the direction of the gradient of pressure. So I hope this is somewhat clear for most of us, what the the, uh, momentum equation actually, uh, uh, means or how to interpret it.

Now, one could rearrange this, okay, with a little bit of hairy algebra, not too terrible, and we can express the momentum equation in this form. Something that arises much faster if we go from the kinetic path in which we might be able to recover a term that has to do with magnetic pressure and the magnetic tension. That actually describe those are the terms that are reshaping the the, uh, expansion of the blob.

So continuity and momentum equations apply to all fluids, independent of their nature. Problem is that they contain 15 independent variables. I have three for electric field, three for magnetic field, three for currents, taking back components, okay? And then we have the scalar fields that are the mass density, charge density, and pressure. Now we can obtain, we include additional equations, making the assumption of the nature of the fluid. So let's say if it acts as a perfect conductor, we can use Ohm's law that gives us three more equations. Okay, um, we can assign assign a thermodynamic equation of state for one fluid. It gives us one more equation. So now we got to the point that we have the number of equations equals the number of unknowns.

Let me skip through this. Uh, it's just a dimensional analysis that tells us about, uh, uh, um, what it means to how the neglecting the displacement, uh, current in Maxwell's equations, um, and we said that in most cases, when the velocity, non-relativistic cases, when the velocity of transport of the fluid is, uh, much less than the speed of light, then we can, uh, um, um, ignore safely the displacement fluid. And, uh, that means that the time scale, uh, of the, um, okay, the temporal scale at which the electromagnetic fields, uh, vary should be larger than the length scale or spatial scales in which they vary over the sea. And what I said here is it's no, nothing new, right? You, you over C much less than one, you can ignore the displacement current. And actually, this is how everybody ignored it before Maxwell. Right, Maxwell figured out this inconsistency in Ampere's law purely based on mathematics. Nobody seen the displacement current before. It's also kind of a misnomer, just because it adds to the current term. Uh, I don't know if necessarily it's current in the way the convective current and conductive currents are, um, thought of. Uh, but it, uh, it arises, um, for cases in which, um, um, we deal with velocity larger, um, or approaching, um, speed of light. Okay.

Now, we heard or we looked at when we defined the MHD equations. He said we can ignore the charge density. Okay, I mean, Maxwell's equations are still okay. Why, why are we ignoring it? Why are we allowed to do that? So let's look at the conservation of charge. And I'm going to write the conservation of charge for positive charges and negative charges. Nothing over here, right? Um, um, and if I were to add them, I get something that's called, you've seen also the, uh, conservation of electrical charge that connects with the displacement field. That's, uh, that's that was the missing piece in, uh, um, Maxwell's equation that led to, um, um, the discovery of the displacement, uh, current, where the total current density is given by the superposition of the electron and ion, um, currents. And using Ohm's law and assuming we have some constant conductivity, we get an equation that looks like that, right? So I just replaced, I just replaced, where am I? Divergence of J with sigma divergence of E. That also, of course, we already said. We already said that the conductivity, it's, uh, uh, um, constant, independent of space. And plugging that in, I can have an equation PD that looks like that. Now, if I were to calculate what kind of solution it, it means, what would be a good expression for the charge density? And this is our time to think and talk. Can I like quickly derive it? Oh, yeah. So what we look at, I have a, I have the time derivative of charge, electrical charge, plus a constant times electrical charge equals zero. We learned that in our third grade. I call that third grade. Okay, okay. We don't know what grade it was. Maybe it was fourth grade. Okay. Um, with confidence, our most, our most common method to figure out the solution is guess. It's an exponential. Is it? I don't know. Let them give them time to think. They will come to picture. Maybe no, you can have it a constant on the other side. Yeah, if you would like, in the limiting. You're not completely wrong. But somebody here said an exponential, right? So if I plug it in, you know, I could guess this. I would know that equation of those kinds. It meets solutions that are exponential, right? The time derivative, you can think about if you're integrating an exponential, what do you get? An exponential. Okay, good. Um, so looking at this solution, what do you see? That the charge density, it's actually exponentially decaying. So after some enough time, it will get to zero. So in the limiting, uh, uh, um, and the limiting, uh, the limiting behavior is that it will, it could be neglected.

Okay, now there's one equation, maybe two, that we we talked about. Uh, mass continuity and we talked about momentum equations. And the energy equation could be, uh, um, could be, um, derived from combining the momentum equation, the continuity equation, assuming, uh, the fluid to be adiabatic. So assuming adiabatic equation of state, conductivity goes to zero. So it gets something like that. I showed you before, okay, where you can do the, uh, take the moments of the, uh, second moment of the, uh, Boltzmann equation. Now, the first term in the derivative tells us about the kinetic energy, right, the kinetic energy of the, uh, uh, fluid motion, the thermal energy, and magnetic energy. But the, the quantity that's conserved is the total energy of the field, right? And the second part is the rate at which this, uh, energies are flowing. Um, now the energy equation can give additional information, but it's not necessarily needed for closure in, uh, um, uh, many, um, the cases. So our goal for today is that if you are in an emergency and there's a fire outside and the password to get out of your building is, identify which Maxwell's equation is which. Hopefully, everybody could survive. Uh, we should know that there are three conservation laws for mass, momentum, and energy, and something that tells us about the evolution of the magnetic field. Okay, and we'll discuss that in a little bit. So our goals are, name each of them. Are we comfortable with that? Okay. Um, no, which, and we can tell that by checking which is the conserved quantity. The conserved quantity always appears in the time derivative. And then describe each of the terms in the equation. That's, uh, uh, if it's a tsunami, okay, maybe that's like, uh, so you should be able to explain kind of how each of the term works, how we describe it in our simulation, for instance, for the momentum equation. Hopefully, that will give you'll have a clear understanding of that.

Um, so now let's look at the induction equation. It tells us about the evolution of magnetic field due to the fluid motion, butol velocity of v. And here I, you see that I neglected the term that had to do with magnetic resistivity. Okay, just let's put that into the zero ideal MHD. U does that. So we can think about an exercise. I have a plasma of some uniform density, uniform pressure that moves into the X direction of some speed that's in the X direction. Okay, we said the initial magnetic field, it's confined in the in the some region. X goes from minus one to one. It also aligns with the X direction, exactly like the speed. Okay, and it's zero everywhere else. Can you use the inductive equation to calculate the magnetic field as a function of position after some number of seconds? And this is our time again to think about and maybe do that. Can you at least think? Don't what you would get. Work through it. And maybe let's just do it this time because most of the time I talk to you. So I wrote it in the two forms. Many times. I like to see it in the tensorial form. There's no error in here that I have U minus BU. There's no that both of those are tensors. So the notation is correct. U B, it's a tensor. B, it's a tensor. That's why it allows me to take the divergence of that. But take a look at the top equation. That's the one thing that you've seen before. It's Faraday's law in which now your U cross B, it's an electric field, right? So I have a U in X direction, B confined in the X direction. The magnetic field, it's in the X direction. Oh, somebody made a mistake. I wonder who that is. Uh, sorry. I didn't even say trust the figure. How would you think about that? Yes, okay. What does that mean? What is unchanged? So let's look at this equation. It says something about the time evolution of the magnetic field. You're very close. Yes, say it again. Yeah, there's an error in how B it's specified. The plot is correct. The text is always, uh, easier to make mistakes. Well, you can calculate this for a uniform density. Just follow, follow the, uh, example that I have. B defined in some region that, and the region is minus. It doesn't even matter between minus one and one, okay? And it's only pointing in one direction. Velocity is X. Then what can you tell about the magnetic field as the fluid flows? Peter, do you want to say that it can, it's frozen in? Have we heard about that? The field is frozen in. That means it moves. I see that everybody is doing this on in their in their hands, and I appreciate that. That's fantastic. I can't do it in my head. If I were to take just the cross product, it tells me in which direction that baby goes. Point my U cross B, it's in the Z direction. Is it in the Z? No, we said B it's in, uh, B it's in, but it's a function of X. That was correct. So U it's in the X direction, B it's in the Z direction, but the component in Z varies with X. Yeah, yes. B in this problem, forget about this. B it's in the, uh, B it's, we'll get to that. I can flip the axis. That's fine. Uh, B it's in the Z direction, right? So I have a speed that's in X, B it's in Z, but the component in Z varies with X. Uh, the lines on the plot are, I think I just, one, did I make it? Yeah, there are philosopy. I flipped it. My mistake. I didn't look at this in like four years. But what do we expect to happen? Yes, based of the text, because now I see it, it's in the correct direction. Z. What do we expect happens here? Nobody writes anything down. E, what do we expect happens? Can we just calculate it and figure it out?

[Music]

Okay, change. Okay, it becomes. Go ahead. The living edge kind of falls and failing edge. Is that your intuition or something that your equation helps you? Okay. Oh, can you say it a little louder? That's right. I don't know if I got the last part, Alex. So please forgive me if I didn't get it. What happens is that the magnetic field is frozen with the flow. So it will move with the flow. Is that what you said? So you can imagine, you can just draw something that, oh, let's say this is my X-axis, the Y-axis, and I have a magnetic field confined in one direction. It points to the Z direction, right? And now I have the flow velocity that's, uh, uh, in the X direction. Calculate based on the induction equation, and you see that the solution that you get, it's a field that moves with the flow. And I think it's likely that this one, it's a, uh, not the right simulation. So let me, and be of, okay, because of the infinite conductivity, you feel this frozen into the plasma, moves with the flow. Um, here's now an interesting exercise that I always give it as a reveal this. Um, it's a qual question. Imagine we are in a world that electrical charges have only one polarity. Describe that world. Yep. Not it from whatever perspective you want. You can think about Maxwell's equations. How would that change? What implication would we have? Well, the rock postulated magnetic monopoles can exist. It's just that we didn't find them in nature yet. Yeah, let's take five minutes and we can think. Some of us might feel more comfortable thinking out loud. Yeah, go for it. Yeah. What are they saying? Are we ready to share? For well, no electrical magnetic fields, sources of fields, right? Sources of electric fields. What are sources of electric fields? Charges, okay? So you can have static distribution of charges, right? Uh, rows. We don't need to have positive or negative. That's a source. Um, what other source for electric fields? What other source? Just look at Maxwell's equations. Let's write them down. Let's all write them down. Whatever we remember. The simplest one, no magnetic monopoles. Divergence of B equals zero. Let's write that down. Then the next one, Gauss's law. Because maybe that helps our brain a little bit more. Gauss's law. Divergence of, let's, let's stay in vacuum. Divergence of electric field is row over Epsilon naught. Okay. Next one, Faraday's. We just wrote it down. Faraday's curl of E is minus DB/DT. Okay. And Ampere's curl of B, if we stay with H, the curl of H is J plus dD/DT, right? So sources of magnetic fields are moving charges, right? Or current. Sources of electric fields, uh, are static, but also something else. Variations in magnetic field, right? That's the induction one. So now we're saying we remove half the charges, only one polarity exists. How are Maxwell's equations changing? Okay, let's write something else down. Maybe we can think about, uh, the electrical force between, or Lorentz force. Let's go that way. Lorentz force between two charge objects. It will always be repulsive. H, what does that mean? Or will have would have only one, uh, uh, will be similar to, let's say, gravity. Gravity is only attractive, but you have a single type of nature, right? Now the electrical force is repulsive or attractive, right? Uh, now you will only be either attractive, oh, no, you will be, sorry, it will be only repulsive. Okay, so it'll be similar with gravity. Um, what would that mean? Yeah, okay. If we're repulsive, matter will pulse. We will ourselves, our bodies, we are electrically neutral. Right, uh, if we are electrical neutral, what happens? Uh, if we are not electrical neutral anymore, that's what you've done initially, okay? But now we have sources of fields are also times very in field. So let's say we are the beginning of the world. The fields are set up by some charge distribution. They vary with time. Could you have waves? With the power vested in me, I made the universe. I have all these time-changing fields and then I remove, you know, charges of one polarity. I decide one. How is that work looking like? Okay, how about we keep talking? Keep thinking about it. I have like how many minutes left? I don't know. We keep on going. Okay, I want to talk. Please keep thinking about it. Okay. Um, not half an hour. Okay. Uh, let's talk a little bit about how we solve, uh, the ideal MHD equations. So mathematically, there are this set of hyperbolic partial differential conservative laws. Okay, what is, what does it mean? The partial differential equation. If we need to specify initial boundary conditions, every time you have a time derivative, we need the initial condition. Every time you have a spatial derivative like divergence and gradient, we need a boundary condition. They're hyperbolic, that means given an initial boundary, uh, um, problem, the solution can be found for any other time and location. But this is very important for solving them numerically. Information tends to propagate with the characteristic speed. And the fact that they are conservation laws, there's a quantity that is conserved as it is transported. Something we talked about that. So this is back to the mass continuity equation. We looked at this blob that is being deformed as being as it is being, uh, um, uh, transported. And this fluid, and now let's go over how did we simulate that? So we can simplify this in one dimension. So, uh, things become much, much easier to, uh, grasp. And they say, well, then I have the time change in row should be just some UX, the the, uh, um, um, space derivative of the, uh, density. And if I were to discretize it over a grid, now I select three grid points. I can define a spatial derivative, uh, as the, uh, difference, right? So let's say at some point, uh, X, it will yield the difference between, uh, um, its value at some position X, either the one prior, divided by the cell size. Nothing fancy. I'm telling you right now. And the temporal derivative, it will be a discretization of this form. And which we look at the, the comparative stance. So if I were to put them all together, now I can describe my continuity equation, mass continuity equation, saying, well, that variable at, uh, advanced in time has to do with the speed at which I'm moving, the time step that I'm taking, and the cell size. So the solution at some given location in my case, like at location J, right there, on a specific time depends on information on a limited set of points from the previous time. So I said this is important when we solve it, uh, numerically, because the propagation, um, sorry, because the information propagates at the some characteristic speed. So we are limited in how we choose the cell sizes and the, uh, time steps. So let's say if I have a blob that's moving with one kilometer per hour, and I to the, uh, cell size that's 100 kilometers, uh, and a time step that's 10 seconds, after one time step, I only traveled about 10 kilometers. So I don't reach the next, the next cell. Whatever discretization of that, it will produce a solution that's not physical. Okay, so we have to pretty much care, pay attention to, uh, what we're doing, how we're doing it. So updating the solution this J, this way would give non-physical results because the information didn't reach the, the information didn't reach the next cell. So to trust or not to trust the, the physical solution, the numerical solution. So we think about that's the solution that we discretize reliably mimics the, uh, um, the, uh, expression in the continuum. So this is the question: how to do that? We can simplify and try to compare, right, how the space derivative looks like compared to the actual one, the discretized one, with, uh, in the continuum. So to do that, we do something that's easy, right? Started the Taylor expansion and expand, uh, find the values of the next grid point. So I express set that, let's say the next grid point, I advance by some Delta X. And if I were to write the, uh, Taylor expansion, it would go as, uh, the value at J plus its derivative at that location times DX plus, uh, the second derivative at that location, the X squared, and of course, higher order terms that we don't care about them that much anymore. We're going to drop them. So now, now I want the difference in the two states, right? That means I move this, uh, on on the right-hand side, and I get, and after I divide by, uh, DX, I get an expression of this form. So let's put it back together. So the discretization is something that looks like what we want plus a gift of not nature. Okay, this is the error. Well, what I like to say now, in theory, I mean, not me, with I, according to Google, this came from Brewster in '81, not Yogi. Uh, in theory, there's no practice, there's no difference between practice and theory, but in practice, there is. So now, if we go back to our baby blob right here, we've seen that our discretization actually included some level of error. And this is on the right-hand side. I'm showing the, um, the error compared to the some analytical solution because in this case, it was a simple calculation to me. So this is the reason that my blob is being, uh, deformed. Now, this kind of error analysis that it's, uh, really back of and the V, the envelope calculation, uh, highly simplistic and not very realistic. It's also, uh, very hard to do with other, uh, MHD equations. But you can think about every perturbation, uh, break it into like free components, and then they will propagate as MHD waves. Um, so when we derive the image disp and, uh, um, maybe somebody will talk about this eventually in this MHD waves, um, okay, then I'm happy to point you to the right direction. Okay, we find, so we linearize the equations, plug in some wave solution, figure out the dispersion relation. So you get a Jacobian matrix. Okay, this is the Jacobian of the, uh, equation set. And the, uh, differential equations are characterized by, characterized by this matrix that has real eigenvalues. And those are the wave speeds. Now, in the most general sense, every set of the five minutes. Oh, I have plenty. I thought it's like only four. Okay. Uh, every set of the couple equations has a characteristic wave speed. And this is actually how the physical information is being, uh, um, propagated from, uh, cell to the other. So example, Euler equations or the fluid dynamics, information propagates with the acoustic speed. In, uh, electromagnetic waves and back, in propagation, information propagates with, uh, speed of light. In the, uh, equation, we have the Alfven and fast and slow magnetosonic waves. So when we actually, [Music] implementing, um, um, time steps or making decisions about how to discretize your equations, that you have to take into account all the speeds at which propagate information propagates. The way to check yourself, there are always as many speeds as there are equations. Okay, have three equations for F equation. One speed is always the flow speed. And, uh, whatever structure you have, it will be transported with that, uh, speed. Let me go fast over that. Uh, now implications, you'll have when you're discretizing, you'll introduce something that's called numerical diffusion. Okay, and this is would be the second term in our expansion. How does it look like? Well, your solution to CH is the way, right? Um, how to fix it? How would we fix it? Okay, but simpler than that, you want your discrete method to approach continuum. What do you need to do? Yes, make the step very, very tiny, right? Well, you can decrease the X, for instance, but it's very expensive. There are better methods to do that. Um, other implications. Uh, you've seen that, um, probably you don't remember that, but, uh, the, the, um, in the induction equation, we could write, uh, we could write actually Maxwell's equations in two forms, conservative form and non-conservative forms. Okay, now we have an Maxwell's equation. They also need to be solved and enforce that the divergence of B remains zero. That's a difficult problem to solve because machines don't like zero. Zero is a very special place in mathematics. Doesn't exist, the same as, uh, uh, infinity. So the discretization error will generally violate this equality. So you'll have to remove this residual of divergence of Fe. There are numerous schemes that take care of that, some, uh, faster, some better, some more accurate than others. So word of caution, in the real world, um, of computational physics and numerical simulation, general, we don't have analytical solutions to compare with. And that becomes a little bit problematic in trying to understand if your solution is actually physical. So of course, if we did, we wouldn't need simulations.

I'll briefly go over this. That there's a hierarchy of, uh, discretization of plasma, going from describing individual particles and their behavior to fluid. Okay, so fluid on the bottom, the lowest approximation one could make in starting with particle in cells. That's actually, uh, tracks the trajectory of, uh, particles. Then we have kinetic, six D or reduced, where we neglect certain motions of particles and, uh, we reduce, uh, some of the, um, dimensionality of the problem. And that solves for the distribution of a particle and the evolution of the distribution function instead of particles, uh, themselves. And the other end of the spectrum is the ideal MHD. U and there's things in between, right? Hybrid fluid and PIC codes. What do they, what do they do? They actually treat electrons as fluid, but then solve an equation of motion for for the heavier particles like, uh, uh, protons and their higher moment, uh, fluid equations. Um, and extended MHD that, um, um, have more complex, uh, physics, but still a fluid description. And I will stop here. Questions for the morning lectures. So this is just like more of a thought for Marco, actually. Do you, uh, you said something that kind of made me think that you can't really see these structures from like helmet streamers or like just the structure of the solar wind sort of out in the heliosphere? I thought I remember seeing there's this like heliospheric imager on board one of the STEREO spacecraft. Can that show anything? Okay, it depends what you mean by see. So of course, um, the helmet streamer normally we think contains the birth of the heliospheric current sheet. And so we do see the, we do cross the heliospheric current sheet in situ. Therefore, in that sense, we do see the helmet streamer. But now if you're, and sometimes when we're lucky, you see a CME take off with STEREO, for example. And if you're lucky, you might catch some cold material in situ. So it's not that there aren't examples, but I'm, I'm referring to structures that are long-living and that are observed in the corona and trying to match them up with structures in the solar wind. And and that's really hard because, because the solar wind has, as you will learn tomorrow, turbulence in it. And there are fluctuations at all scales, basically. So with one spacecraft, you never know whether you're seeing something which is just being advected, whether it's time-dependent, whether it's oriented in some direction with respect to the spacecraft or another. It's very, I think it's very, very difficult. Although there have been attempts. I think T and Marsh and others claimed that they'd observed a pressure balance structure. They corresponded to something like a coronal plume. I mean, coronal plumes are are weird animals, and there's some intrinsic contradictions even in the observations. So, um, it's hard to really discuss carefully. Yeah, so that's what I meant. I mean, there are big structures that you do observe for sure, both in situ and remote sensing. There was a magnetic tension term in the energy conservation equation. So how we get imagination of that parameter, magnetic tension, and how does it affect the fluid flow? Um, sure, yeah, that might help. Um, yes, uh, how we get imagination of that magnetic tension, how it, what does it describe, and how it affects the flow of fluid. M fluid e. So, yes, second last term in this, uh, energy conservation. What is the imagination of this magnetic tension? Last one, yes. Our M, yes, yes. Pressure is that, that, then, and magnetic tension is something. Field lines are getting curved or tension? Yes, yes, yes. Okay, I can imagine in this, uh, magnetic pressure in a similar way that you can imagine kinetic pressure, right? That has to do because it has a B squared term that has to do with how much, uh, the magnitude, right, the magnitude of the field itself, right, that sets up a pressure. The tension, we can think about it in terms of, um, let's say in mechanics, in another, uh, we understand tension like the shear tension in a material, right? Am I explaining this correctly? Am I paring this question? Maybe one way to understand it is to, um, think about it in terms of the field line itself. So if you have a field line and in every point it has a certain curvature, then the, this tension component is actually just B squared over 4 pi times the radius of curvature, underneath the radius of curvature, and it goes in the direction of the curvature. So if the field line is bent like this, it's a force like that, which is proportional to the intensity of the field and inversely proportional to the radius of curvature. So if the field is kinked very, very much, it becomes very, very, very large. Thank you. I guess good discussion. Well, I, I don't like field lines because they don't exist. Sorry. Yeah, so not necessarily. I mean, it depends. So you to look at that, right? That's that's a tensorial product. That's a vector, right? There, BB. There's no dot to make it a a scalar field, right? So that tells you something about components in which the tension, magnetic tension, like the mechanical tension, if you want. And, right, um, it's changing the, oh, I don't know. Let me think about this a little bit more. Not from magnetic. I don't like the F. Not like it, but it's a vector field, right? So you, any vector field, you can always define a tangent vector to the field, right? Sure, sure, sure. So for any line, there's always a tangent, there's a curvature vector, and there's a torsion vector. And the BB over mu Z is in the direction of the curvature. There's nothing you can do about it, even if you don't like the field line. Sure, sure, sure. But we're talking about what particles do from that perspective. How do I imagine? Okay, individual particles is different. I was thinking about the fluid, right, which I think can also be like, you got to be careful about doing doing that because this is a fluid interpretation. And if you start like trying to think about what the, it's like thinking about water, right? If water flowing in a pipe, I don't want to think about what the individual particles are doing, what the individual molecules of water doing. I want to think about the, the parcel of water that's Don't mix up your models. Build kinks. If it's a, right, the, what's the problem with the field line interpretation? Like you cannot account for superposition of fields because all the time, right, the field line is not superposed. I know this is a very, this is a very hairy question in some sense because because of this interpretation of field lines, which is, um, normally used, often times I would say used, H D, as a as a convenient, uh, yeah, so, yeah, it's a convenient way to see things, um, given that in lots of places, the conservation of flux theorem holds and and kind of that's what justifies the use of this thing as a line, even though it's not. But I mean, if you, if you had to, for example, explain the Alfven wave, that we haven't yet discussed at all, but you to describe what the restoring force in the Alfven wave is, the fastest shortcut for explaining that is to use the tension in the field line. And if you don't use that, you have to kind of use many more words to describe what the restoring force in the Alfven wave is. I imagine just like a line. I mean, I'm fortunately, I think of them as lines as strings in plasma. Oh, that takes us to a nice location to ask. We've seen Alfven speed repeatedly used in a lot of these derivations, discussions. What is intuitively an Alfven wave? What is an Alfven speed? What do, what does this mean in the simplest representation? Um, are the waves that are propagating along the field lines for the field lines? Now, waves that arise in an MHD formalism. And one way to derive them, right, or think about them, is you assume some perturbation and fields and, um, uh, work out through the, uh, linearizing the equations, what would be the dispersion relationships. And from there, you can derive the speeds at which I mean, you recover an equation of motion, sorry, wave equation in which you can identify the, the, uh, three fundamental speeds of MHD. And that would be the fast, go, and, um, um, if you can go maybe either, I don't know whether forward or is there one, one slide where you have all the MHD equations on at the same time? I just wanted to mention something up. Yeah, I think right, one of those should work. Okay, so there's, um, if you look at the momentum equation and you look at the induction equation at the same time, okay, now let's kill gravity for simplicity. Oh, no gravity. And assume the pressure is uniform for simplicity. And the, then the momentum equation and assume the density is uniform too. It bothers us incredibly. So it's trying to make some big simplifications. Then the momentum equation turns into DU by DT plus DU U. P is uniform. So whether it's inside, we can kill it. We don't care. B squared, we're going to say it's constant. You mind, but just for me. Um, and we're going to get DU by DT plus div UU minus BB equals zero. All right. Now, let's take DB by DT plus div U minus BU equals zero. Okay, now we have some very, very symmetric equations. And if we put in there U equals B or U equals minus B, they're solutions. DU by DT equals zero, DB by DT equals zero. Those are Alfven waves. But they're not linearized Alfven waves. They have a speciality, B squared is constant. They have a speciality, P is constant. They have a speciality, row is uniform. And I should have, I said something a little wrong. It's row U squared that has to be equal B squared over mu. And so U is equal to plus or minus B over the square root of M row zero. Okay, those are exact solutions of that equation. So they're nonlinear. These are weird animals that don't belong to the categorization the erotic was talking about just now of the three linearized modes. They're a combination of these modes. Um, but they're, they can be any amplitude you want as long as U is equal to plus or minus B over the square root. And, um, why am I mentioning this? Because people usually think of the Alfven wave as the shear Alfven wave. But once we sent a spacecraft up into space, it's filled with these other animals, these large amplitude Alfven waves. You see them all the time. I will show you a plot later in the afternoon of the magnitude of the field, and it's just sitting there, but the components of the field are going all over the place, and the velocity field is perfectly correlated with the magnetic field. So somehow this solution that appears to be completely idealized of these equations is enjoyed or liked by nature. And it's pretty impressive. Um, I'll show you an example later. But scarily, um, existing, put it that way. Um, not well, I could, if we have a panel, the math is actually, it's, you know, it's not, it doesn't really require much thought. I mean, I think I thought it was pretty simple there, right? Yeah, can throw, yeah, you can just cancel some things out there. Yeah. Is this working? Cool. Um, I was wondering about how you choose appropriate boundary conditions in simulations. Like if you're building an MHD simulation of like the magnetosphere or the solar wind, what are the factors that you need to consider to make sure you choose the right boundary conditions? The right boundary conditions, that's actually a loaded question. Uh, what we usually do, um, we choose boundary conditions that we think might be realistic because they come from some observations. Usually for magnetosphere, let me back up. Um, for magnetosphere simulation, we use, um, solar wind data from, uh, as were discovered that are L1. And we make the assumption that, right, that's one point. And we make the assumption that the boundary, the, um, outer boundary on the sun side, right, it's, um, a plane wave, right? Kind of the same thing over the entire, um, um, boundary. There's no variation. It's just, uh, whatever information, however solar wind propagates from L1 to or 32 re on the day side, it propagates the plane wave. And people say that works. Uh, well, it works because we get the solutions. And sometimes, uh, is the solution correct? Well, it is the solution to that set of equations for that boundary condition. But that's kind of how we chose it. That's the only way to infer. Um, we don't have options other than guessing. Use the data from observation. So that's the boundary solutions that use the appropriate, uh, appropriate. I don't know what that means. If it's feasible, we can say we know it's feasible. Uh, if it's appropriate from the perspective of, um, is the equation set has been very for like simple problems like we had, is the solution that it's yielded for something that we already know for like, let's say something that has a problem that has an analytical solution, is that approaching the numerical one? Yeah, we verified a lot of that. Validated with data other than the data that drives the initial and boundary conditions. We also validate against, uh, spacecraft data that, uh, it's basically just, uh, extracting damage parameters of location, space drift is, uh, um, measuring in the actual space and comparing against the observation. So that would be the validation part. Um, appropriate, I don't know what that meant. Can I just add? I think this question is a fundamental question in in all of our field. Um, the philosophical question too. That's why I think saying that there's no such thing as appropriate. It's philosophical in the sense that you're taking a subset of a volume and you're trying to find out that subset of a volume. And of course, the subset of the volume is influenced by what happens outside. And so you have to have information that's propagating into your volume. And you have to have some knowledge of what information propagates in. Unfortunately, there's not only information that's propagating into the volume, but your volume, but the dynamics inside your volume is also propagating outside. And there's a coupling between the information that's coming in and the information that's going out, which makes it a particularly difficult task with a set of nonlinear equations to make sure that the boundary conditions that you're choosing, not only are they good, as you said, but they have to be consistent. And there are methods to make the boundary conditions self-consistent. And the cases that she's talking about, I think are, are which you have an incoming flow which is super everything, which makes things easy because there's no information going out. If you have an inflow boundary where things are moving supersonically, super-Alfven, anything super everything, there's no information that's going the other way. And so you can def, you can get away almost with anything. But if you have a more general problem where you have some subsonic propagation of information, then you have to try to make sure that you capture correctly the information that's relevant to the dynamics that you want want, which is coming from outside. And there are methods for that. I don't know how good they are. Things like projected characteristics that people usually don't use because they're not very simple to impose in complicated geometries, but are the only real guarantee of consistency, I would argue. And not enough of it is done, I think, in our field. If I can make another comment, and we'll see an example of, uh, the solar wind, uh, at the, in the activity this afternoon, what, and what we use for, what we, what we cobble together for boundary conditions from that. One time, I think time for one more question or not. Uh, let's thank our speakers again, and we'll do lunch.