 Hello and welcome to video number 20 of the online version of the future research lecture. As you might remember in the last video we finished with chapter 3, parameter limits for fusion plasmas and thus today we will start with chapter 4, particle trajectories. Particle trajectories is important for a number of reasons. So particle trajectories are important for the understanding of, for example, perpendicular transport losses, perpendicular transport by perpendicular, of course, refers to the direction of the magnetic field. And understanding implies, of course, that we will also, would also like them having the understanding, trying to control it and reduce maybe the losses. Then it is important as we will see for toroidal electric current, toroidal electric current, which is, as you very well know, important in the tokamak as it generates part of the confining magnetic field. And then also for radial electric fields, as we will see, for radial electric fields generated inside of the plasma by the plasma itself. Now how do we treat the particle trajectories by the guiding center approximation? And the guiding center, oops, sorry, the guiding center, we take that actually as a superposition. So the guiding center motion is a superposition of the drift motion and of the parallel motion, which is the thermal motion and of the parallel motion. An important thing to note is that if we have magnetic configurations with a symmetry coordinate, then the trajectories stay on average on the magnetic surfaces, on the flux surfaces. So if we have configurations with a symmetry coordinate, like, for example, the tokamak, which is an axi-symmetric device, then the trajectories stay on average on the magnetic surfaces, on the flux surfaces. The magnetic surfaces. Remember, this is basically what the nöta, what can be reduced from the nöta theory, which I talked about a few lectures ago. If the trajectories stay on average on the magnetic surfaces, they don't get lost by drifts, which is important to note. So the particles don't get lost by drifts. Now on the other side, if we have no perfect symmetry coordinate, like in a stellarator, the particles can get lost just due to the drift motion. So if we have no perfect symmetry coordinate, like in a stellarator, then the particles can get lost without additional collisions. And so just due to the drift motion, as we will see, and thus particle confinement, or that's the reason why particle confinement is an important topic in stellarator optimization. That's the reason why particle confinement is an important topic in stellarator research and optimization and stellarator optimization. Okay, but before we talk about stellarators, let's first talk about particle trajectories in the tokamak field, because that's the simpler case. So we start with particle trajectories in the tokamak field. And we have basically, sorry, the motion of a particle consists of two elements. One inside is the drift motion for the perpendicular movement. And then parallel, we have the thermal motion and there we can have particles trapped in a mirror, because the magnetic field is non-homogeneous in the tokamak as we will discuss. So now let's, before we start to talk about particle trajectories further, let's draw a coordinate system and introduce the most important. Quantities here. So typically a coordinate system like this, which we had already a few times here. So this is the direction x. And here we have the tau-royal direction phi. This is capital R or sometimes also small y, depending if you're in cylindrical or Cartesian coordinate system. Then let's try to draw a circle here, maybe something like this. Then here we have r naught. Oops, sorry. So here we have r naught. Then here we have small r. This is the angle theta. So this is the angle theta. The magnetic field points into the board. Then we have the magnetic field gradient pointing into this direction. So this is the direction of the magnetic field gradient. And then if we are this position, for example, then the particle has a poloidal velocity component, that is v theta. And capital R now can in this coordinate system be expressed as the sum of r naught. Then plus small r times cosine of theta, which is then the same as r naught times one plus epsilon times the cosine of theta. Where epsilon is, as you might remember, the inverse aspect ratio r over r naught. So this is here the inverse aspect ratio, which we had on the board a while ago when we talked about magnetic configurations. Now with this definition of this coordinate system, we can now write down equations for v phi. So the total magnetic field component, which is v phi, you remember v phi naught. So the total magnetic field at the axis times r naught over r. And now this can be the Taylor and the Taylor expansion. Quitting it after the first element, we get v phi naught, one minus epsilon cosine of theta. So this is the Taylor expansion and this is valid, especially for a good approximation, let's say for a large aspect ratio device. For a large aspect ratio approximation, we usually say, which means that epsilon, the inverse aspect ratio is supposed to be small. And then the same can be done for the polar magnetic field component using the polar magnetic field on axis, which is a function of small r also. And then r naught over r, this can also be tailored, being roughly the same as the polar magnetic field on axis, times one minus epsilon cosine of theta. Just keep in mind that this is a typical talker mark where the polar magnetic field is much smaller than the toroid magnetic field. Okay, these are the main coordinates and we will need to add expressions for the magnetic field. Now let's have a look at the parallel only the beginning to begin with. So first of all, we will just look at the particles which have only a v parallel component. And we neglect all kind of inhomogeneities of B, we neglect inhomogeneity of the magnetic field. And then we can define a so-called poloidal transit frequency. A poloidal transit frequency, which reads omega theta be equal to two pi, so one poloidal transit, and then the theta over two pi r. And now looking on the, so this is basically the frequency for one poloidal circumference. This is why it's called the poloidal transit frequency, although you might hardly be able to read it, but it should read poloidal transit. Sorry, let me just write that again. Poloidal transit frequency. Now, if you look on the coordinate system on the right, you see here this is the poloidal angle, the poloidal direction. Here we have the toroidal direction, then we have the magnetic field and the velocity. And from that we can, or we can write and we will need these expressions that the ratio of the parallel over B is the same as v theta over B theta. And that is the same as v phi over B phi. And using that expression, we can now insert for v parallel here, sorry, for v theta, we can insert v parallel. And then we need to insert v theta as well. And here we have r and B. So this is an expression for the poloidal transit frequency. Then we also have a poloidal transit time. So a poloidal transit time, which reads tau transit is equal to, so one circumference to pi over omega theta. And then inserting the value of omega theta of the poloidal transit frequency, which we have just defined, and making use of the definition of yota. Then we can write 2 pi times r naught over yota bar v parallel, where we have used the definition of yota bar being equal to r naught over r times B theta over B. So this was used for getting this expression for the poloidal transit time. So the time it takes for one particle to make a poloidal circumference, the transit. Also looking at the equation, we can use this to define a length of the field line. The length of the field line is basically 2 pi r naught over yota bar. This corresponds to the length of a field line to close. Or if it doesn't close to roughly make a full poloidal turn or circumference or transit. And of course v parallel, I haven't written that down here. But from a previous slide v parallel, we use here the thermal velocity for that. So we make the approximation that everything is in thermal equilibrium. And v parallel corresponds to the thermal velocity. Now for a fusion plasma, what are some typical numbers which we get from that? For those quantities. So in a fusion plasma, you might have temperatures of 1 keV, so not too hot. 1 keV might have a radius of 1.5 meters. So a medium-sized tokamaga, I would say, you have a typical value of yota bar of 1 over 3. And this results in a transit time for the electrons of roughly 15 microseconds. And for the ions, the time to make one full poloidal transit or circumference, this is roughly 50 microseconds. Okay, this is if we only have a parallel motion. So we need to take into account the effects of a movement perpendicular to the background magnetic field. So we need to take into account movement parallel, sorry perpendicular to b. And that then means that the particle motion is basically a superposition, as I said in the beginning, of the thermal parallel velocity plus the vertical drift velocity, which is governed or dominated, which is governed by two drifts. And that is the gradient drift and the curvature drift. Gradient drift and the curvature drift. So just as a brief reminder, let's write that down. So we would then have the drift velocity being the sum of the gradient drift of the grad b drift plus the curvature drift. Well, it's not right at such a little superscript. And now inserting the gradient drift, we have m over 2 q r v. Just as a reminder, q here is the charge and not the safety factor. So this one is the charge, which is important, of course. Then we have the perpendicular squared plus two times the parallel squared, then times the unit vector into x direction. And now for the perpendicular using the drawing we had on the previous slide, just rewrite it as v squared times sine squared. And for the parallel, we make a similar replacement, replace it by v squared times one minus sine squared alpha. And then we can write this, the drift velocity, the overdrift velocity as m v squared over 2 q r b times two minus sine squared alpha times the unit vector into x direction. So unit vector sometimes that's the headline thing. So let's stick to that notation. Okay, now if we have particles which experience a perpendicular, which have a perpendicular and a parallel component in the movement with respect to the magnetic field, we need to distinguish between two types of particles. The first type of particles are so-called passing particles. So let's first talk about passing particles. Now for passing particles to start with, we assume that the angle alpha, which is basically the pitch angle is either zero or roughly pi. And we start at the outboard side, at outboard side. Okay, so let's make a drawing to explain what happens then. So this is x, this is again, as you might have guessed it, the two-royal direction. Then let's draw our coordinate system, something like this. So this is r, oops, capital R. Then let's try to draw a flux surface. Okay, then here we would have r naught and we would have a magnetic field pointing into the board. So magnetic field pointing into the board. This would correspond to a flux surface and we let a particle start on the outboard side. So we would let it start here. And this particle is supposed to have a positive charge. So we start here and just so you can read it, this is supposed to have a positive charge. Okay, now then if it has a positive charge, let's have a look at the drift velocity. The drift velocity, as it's written on the top, has a Q here. So it means there's a charge dependence having a positive particle and ion, meaning that the drift velocity points upwards for the ions. And now we have to distinguish further if the particle moves along or against the magnetic field. So if the particle moves along the magnetic field, this representation, since we have a twisted magnetic field, it would mean that the velocity component, that the polar velocity component here would point downwards. For a particle moving along the magnetic field and these particles are called co-moving particles. Now if it's moving downwards, so along the magnetic field downwards is this projection, it means it gets a constant drift upwards because this drift, as I've written here, points upwards. So it has an upwards directed force and that means that it deviates from the flux surface and actually will move to smaller radii, maybe something like this. Then when it's here, since the drift is pointing upwards, it will again deviate and move to larger radii. And in the end, we will get something like this. So this is the example for a co-moving particle, for a particle moving along the flux surface, sorry, along the magnetic field lines. If we have a particle moving against the magnetic field lines, so a counter particle, a counter-moving particle, then the polar velocity component points upwards here for a counter-moving particle. Since it's pointing upwards, it moves to larger radii because it's still the drift velocity which it experiences which points upwards. And this means it will again deviate from the flux surface but to larger radii. If it's here, it will again deviate, going to smaller radii again. Then we get something like this. This is for a counter-moving particle. So we have particles which deviate from the flux surface just when they are moving along the magnetic field lines. And these are called, oops, sorry, let's try to make that a bit nicer. So the green one and the red one are both called so-called drift planes. So these are drift planes. This is an important result that there is a displacement of the drift orbits from the flux surface. Although of course this is a local displacement if you want because on average it kind of stays on the flux surface because it ends up where it started but locally it can strongly deviate from that. And now having a look at that, usually the velocity is treated with respect to the plasma current IP and the plasma current IP points into the same direction as the magnetic field here. And now this has consequences for NBI, for neutral particle injection. At NBI we inject neutral particles at very high energy and usually those particles are ionized at the outboard side due to the relatively high density of the plasma. They cannot really go deep into the plasma. Instead they are getting ionized at the outboard side of the plasma. And if we inject now positively charged ions and if we inject them along the magnetic field we talk about co-NBI. So we have co-moving particles and as you have seen in the drawing on the left hand side this results in a more central heating because the drift plane deviates from the flux surface such that it moves to smaller radii. So this is for more central heating whereas if we inject against the magnetic field so if you have counter NBI this is less central. Instead it can happen that the deposition now happens at the vacuum vessel which is something which we do not want. And of course it's the opposite for electrons because if you look again on the equation for the drift here we have the charge included there thus the drift, sorry this is supposed to be pointing downwards, thus the drift velocity for the electrons is pointing downwards here. Thus it's the opposite as what I have drawn here. Okay this is important now let's look if we can somehow quantify that. So now let's try to quantify the displacement which is usually abbreviated with delta p and that is the drift velocity over the poor loyal transit frequency because that is how it is defined and it is the absolute value of that and the poor loyal transit frequency and the drift velocity and insert the equations for that using our initial assumption that alpha is equal to zero then we end up with m v squared over two and then q again this is a charge and not the safety factor times two rb over v parallel b theta and now there is an often used abbreviation which is rho L theta the poor loyal Lama radius times epsilon the inverse aspect ratio so this is the equation we get for the displacement this is the displacement of the center of the drift plane of the center and here we have introduced the so-called poor loyal Lama radius rho L theta being equal to m v parallel over q b theta which can be approximated with two m times the temperature so just inserting the thermal velocity for that over q b theta so assuming we have a thermal plasma and this is an expression for the poor loyal Lama radius and to give you an idea let's again have a look at an example which would be having iron temperature of one kilo electron volt assuming we have hydrogen so we just have protons here make it very simple have a typical for a medium sized tokamak poor loyal magnetic field of 0.2 Tesla which results in a poor loyal Lama radius on the order of two centimeters and then having an inverse aspect ratio of 0.1 this would result in a displacement of the drift plane with respect to the flux surface and a displacement of the center of the drift plane with respect to the flux surface of two millimeters so not too much okay so in general as a very last statement then we can write down that the drift planes for co-moving ions and counter-moving electrons are smaller smaller than the flux surface where they started on and it's the opposite for counter-moving ions and co-moving electrons there the drift planes are larger okay that's it for video number 20 and we started with chapter 4 particle trajectories and introduced the very important concept of passing particles passing particles which are not trapped in the magnetic mirror but being able to move around so fulfill the poor loyal or make full poor loyal transits full poor loyal circumstances and then we have learned that if they are just moving along the magnetic field line or again the magnetic field line they deviate from the flux surface they are moving on so-called to drift planes and here we have written down an expression to estimate the displacement of the center of such a drift plane with respect to the flux surface and this has consequences for NBI for neutral beam ejection which we use for heating up the plasma to future relevant conditions okay that's it for video number 20 see you in the next video