 Hello everybody and welcome to video number 6 of the free online version of the fusion research lecture. If you have followed the last videos, you might remember that we finished with chapter 1 fusion research, where we talked about the nuclear fusion process. You have hopefully understood why deuterium trisium fusion is the most attractive candidate for a fusion reactor on Earth. And we also talked about a few key parameters, namely the Lawson criterion, the plasma beta and a few others. In this video, we will start with chapter number 2 magnetic field configuration. And first of all, a short reminder why magnetic fields are important. So chapter 2, this is magnetic field configuration, magnetic field configuration. And as a reminder, the idea of the magnetic field is basically to isolate the plasma from the wall to keep the energy in the plasma. So this is the idea basically to isolate the plasma from the wall because we want to have as much energy as possible inside the plasma to achieve the fusion relevant parameters. In addition, magnetic field also allows us to control the particle and energy flux to certain regions because the plasma consists mostly of charged particles and those can move freely along a magnetic field line. So magnetic field also allows us to control the particle and energy flux to certain regions. And as we will see, usually the diverter is used as some kind of an exhaust region if you want for a fusion reactor. Some people call the diverter also the vacuum cleaner because it removes all the things you do not want to have in a fusion reactor. And we achieve all this basically by using nested flux surfaces, using nested magnetic flux surfaces. And we will talk about magnetic flux surfaces in detail mostly in the next video. For now, it is important to know that those flux surfaces are defined by the magnetic field lines. They are defined by closed magnetic field lines. So magnetic field lines which are traveling around a torus and after a certain number of circumferences, which can be a very large number, they close itself and thus defining a magnetic flux surface. Okay, so let's start with a few general properties trying to introduce the coordinate system, for example. So first we dare to draw some kind of a cross section, maybe like this, and then it spins around into this direction. Let's see, yeah, it doesn't look too bad. Then here we have our coordinate z and then this should look like this. So two polo cross sections, something like this roughly, right? Well, you get it. I guess so usually there are two type of coordinate systems used coordinates. One is the torus coordinate system. The other one is a cylindrical system cylinder. And in the torus coordinate system, there is usually a small r is used for the radial coordinate. So this will be, if we start from here, from the center, then going outwards, going outwards like this, then this coordinate will be small r. Then we have the theta, the polo angle, polo angle, meaning if we have our center approximately here. And then going around like this, this corresponds to the poloidal direction. The poloidal angle and then we have phi, which is the toroidal angle. This is the angle when we go around like this. This corresponds to phi. Now on the cylinder, we have capital R corresponding to the radial coordinate. This is this one here, which we have already included in the drawing. Then we have phi as for the toroidal coordinate already included in the drawing. And then z, the vertical coordinate, the vertical direction also already included. Good. A few more things. So first of all, this is what I have drawn or tried to draw there. This is supposed to be a flux surface. It's supposed to be a flux surface traced out by a magnetic field line. This is something we will talk about in the next video. For now, maybe just, you need to know, so along one of these flux surfaces, which look like a tube, we have a magnetic field line going around like this, for example. And it goes around and around and around and around and thus basically defining the flux surfaces. And as I said, these are, we have nested flux surfaces. Meaning there's another one inside, which goes around, oops, sorry. Also goes around toroidally like this, which is spanned up by another magnetic field line. So goes also, sorry, goes around like this, more or less a circle. So this is also a flux surface defined by different magnetic field line. Now a few more important parameters. So first of all, we have our knot, which is the major radius. This is the major radius. And let's rather call it the major plasma radius, because this is also often used. So it's a major plasma radius. And if you take a look in the drawing at the top, maybe I use a different color yellow. So this goes up to here. And this was correspond to our knot, the major plasma radius. Then we have a the minor plasma radius, a the minor plasma radius. And if we again use yellow for that, maybe it's easier to see. This is until this is along the small radial coordinate R and then until here, for example. So this distance would correspond to a the minor plasma radius. And the major and the minor plasma radius, oops, are often used to define the aspect ratio. A, which is our knot over a, or which you also often read to the inverse aspect ratio, which is, as the name suggests, just one over a. Then the confinement region is defined for every radial position being smaller than the minor radius. So this is how the confinement, confinement region is defined, confinement region is defined where every radial coordinate is smaller than the plasma minor radius. And it corresponds to a region of closed magnetic field lines. So this is a region of closed magnetic field lines, a principle closed and then the separatrix. This is given by the radial position corresponding to the minor plasma radius. And this is basically the last closed flux surface. This is the last closed flux surface. And then we have the scrape of layer, the scrape of layer often just abbreviated S, O, L. This corresponds to regions outside of the confinement region. And this is where the magnetic field lines intersect somewhere with a vessel wall. This is why there are sometimes loosely called open magnetic field lines because they intersect with the vessel wall. So this is where the field lines intersect with wall, or let's say rather general with wall, so with some wall components. So in our drawing in the top, this outermost flux surface here would actually correspond to the separatrix, because I have written there with indicated by the yellow label that this has lies at the radius A. Now the intersection with a vessel, we would rather like to have some means of control for that. So terminating the confinement region is something where we want to have some control and there are two methods for that. So terminating the confinement region, there are two ways to do that. And the first one is basically using limiters. So this is so-called a limiter configuration, limiter configuration. And in a limiter configuration you basically have, if we again draw some kind of a poloid cross section, it might look like this, then if you nested flux surfaces, maybe like this, roughly right. And then you have some kind of material which intersects there. So this would be some kind of material intersecting with a plasma corresponding to the limiter or to a limiter. Meaning a limiter is basically just some kind of protruding wall element. Some kind of protruding wall element which sticks into the plasma. It is, of course, or has to be designed to withstand very high particle and energy fluxes. So this is designed to withstand very high particle fluxes, particle fluxes. And it also, in addition, we also often have a very shallow inclination angle between the wall and the field lines to spread the region where the intersection actually happens over a larger area. So there's a very shallow inclination angle between the wall and the field lines. Nevertheless, however, using limiter configurations, the incoming power density is often above the material limits. So nevertheless, the incoming power density is often above the material limits. We have some well-infusion plasmas in small-scale devices which do not operate at fusion parameters. Limiters might be a very useful configuration but not for fusion plasmas. And if we have power densities being too high, we have erosion. Erosion basically means that there might be some maybe high Z material which enters the plasma. And if you remember our video about the power balance, having high Z material entering the plasma can lead to significant bremsstrahlung losses. And thereby, basically, you can completely radiate all of the plasma's energy away and thus lose your energy confinement completely. So this can lead to significant bremsstrahlung losses and we thus want to avoid this if possible. This is also why limiter configurations are not really used nowadays in fusion experiments. They have been used in the past and definitely had their use. But as I said, the major problem here is the significant bremsstrahlung losses and materials eroded from the limiter. And the alternative for that is basically number B, the diverter configuration. So the diverter configuration. So now what is this? So the general idea is to separate the wall contact area and the confinement region. And we do this by having very long open field lines which guide the particles away from the confinement region into the diverter. And we do this by having very long open field lines which guide the particles away from the confinement region into the diverter. We do this by having very long open field lines which guide the particles away from the confinement region into the diverter. So the ideas then are realized by having open field lines which guide the particles away from the confinement region away from the confinement region into the diverter. Now how can this look like? For example, if we just again draw some kind of polar cross section maybe looking like this. Then we have here usually some kind of box. This refers to as diverter. This is the intersection region where the field lines intersect with the vessel component. And this whole area here is called diverter. And there is a very long connection length along the field line until we reach the diverter. So a long connection length on the order of 100 meter in large scale fusion devices. And then we have a very high neutral density in the diverter usually by inserting gas there, neutral gas. And this allows the plasma to cool down before actually reaching the wall components there. So we have a high neutral density in the diverter which requires of course to have very high or very efficient pumps there. So there are specialized vacuum pumps installed in the diverter region. So the high neutral density in the diverter allows the plasma to cool down before actually reaching the wall. Before reaching or impinging or hitting the wall before actually reaching the wall. The wall components are called Bethel plates. So the wall components where the plasma is actually making contact with them. So here this for example and this would be the Bethel plates. And in that region the plasma emits a lot of visible light there. So the plasma cools down, emits visible light, or lights in the visible region emits, let's call it visible light or this might sound a bit weird. And when you look at a typical picture of a fusion plasma you might realize that only the bottom area is emitting light in the visible range. That is only because there in the bottom area in the diverter region the plasma is so cool that it emits light in the visible range. The rest is just emitting light in the x-ray range which we cannot see. Okay and the diverter is definitely a key component for fusion devices. The diverter is a key component for fusion devices because it takes care that the impurity concentration is really low in the plasma. Thus, Bremstrahlung losses are reduced which increases our energy confinement, our energy content in the plasma. So this is something which you find in every large fusion experiment. Okay, another important thing is the shape of the plasma because often and I guess I could also write always nowadays instead of often we have non-circular cross sections or non-circular shapes. This is something which you find also in every new major or every large fusion experiments. And there are two main parameters to characterize that. One is kappa which is supposed to be a small kappa which is supposed to be a measure for the ellipticity or sometimes also called elongation. And kappa is defined as the height of the plasma. So it's the height of the plasma compared with its minor radius, compared with the plasma minor radius. Meaning that if we have a cross section which looks like this, for example, then we compare this one, the height with this value. And if we stick to our coordinate system which we have to find, this would correspond to z maximum minus z minimum. And then over 2 times a, where a here is defined as the maximum radial position minus the minimum radial position times one half. This is the ellipticity. As we will see later on, this is an important parameter because it allows us to get more stable plasmas to get higher beta values. Remember the beta is the plasma pressure per magnetic field, so it is something like the efficiency. And we will see in the next chapter that elongated plasma allows to achieve higher ellipticity, higher plasma betas. Therefore, your plasma is more efficient if you want. The same is true for the triangularity. And delta is a measure for the triangularity. Triangularity, this should be triangularity. So meaning having a plasma which looks, I don't know, something like this, somehow triangular. The measure for that, there are two quantities, the upper triangularity. This corresponds to r not minus r at z equals z max over the minor plasma radius. And then the bottom or lower, let's write lower because we have an upper there. The lower triangularity is defined as r not minus r z equals z minimum divided over the minor plasma radius. And of course r not here, the major plasma radius as I have defined it already. And you can define it here in the same manner as with the minor plasma radius as r max plus r minimum times one half. And using these two definitions of the triangularity, then the average triangularity simply is the mean of the upper, sorry, of the upper and the lower triangularity. So this is just the upper, some of the upper and the lower triangularity. Since the plasma cross section is now lower circular, we often talk about, we often use the expression effective plasma radius, effective plasma radius, effective. And this is defined via the equivalent area of a circle. So p, a squared, effective, so basically the average area if you want. Giving the same area as these kind of shaped cross sections. So shaping is also a very important optimization aspect in plasma to achieve higher performance in general. Okay, another important thing which we have to talk about is the rotational transform, the rotational transform, which is basically a measure for the twist of the magnetic field. You remember the magnetic fields need to be twisted. So this is a measure of the twist without a twisted magnetic field, there would be no confinement. Now let's have a look at the drawing on the right hand side. Again the same cross section as before. So let's follow a magnetic field line which starts here and let's follow this one around the torus. It goes around the torus somehow, ends up for example here. So this is B, the magnetic field line, or magnetic field line. And if we follow that, say we have started here, this is one position one, and we follow it to here. This is again one because it's positioned by field line one. Then we measure the angle we see from here basically to here. Then here we have a certain angle, maybe we use a different color for that. So here we have a certain angle. That angle measures the twist of the magnetic field. Since here we have only half a torus, this angle here is yota half and yota would be the angle if we go around the full toroidal circumference. Then we have the expression yota bar of news which is yota over 2 pi. This basically gives us the number of pulloidal transits per one toroidal transit. In tokamaks often the inverse of yota is used, which is the safety factor. So in tokamaks we often have the safety factor, the safety factor qs, which is defined simply as 1 over yota bar. And let's give that one of our famous yellow boxes, since this is such an important expression, the safety factor. And the safety factor basically is the number now of toroidal transits. The number of toroidal transits required for one full pulloidal transit. So the inverse of the rotational transform. Now in the linear tokamak approximation, something which we will often use in the linear tokamak corresponding of course to a cylinder, to a cylinder, the safety factor qs can be written as r b phi then over r not b theta, where b phi is the toroidal field, the toroidal magnetic field and b theta is the pulloidal magnetic field. And b theta, we can derive this from MPS law. Remember that there is a plasma current, a strong current flowing generating our pulloidal field. And from MPS law we can then get that b theta at the plasma boundary at A. So at the last law of flux surfaces is mu not Ip over 2 pi A, where Ip is the total plasma current. Now having defined the rotational transform here, this is not the full story. Actually, yota depends on the radius. As you can already see in the equation of the safety factor here, because you see there's a radial dependence included there. And that means if we look at the drawing above which we have there and we follow a different field line for this one, the field line on the inner flux surface which I have indicated here, it would also go around toroidally like this and then end up somewhere here as well and maybe end up somewhere like here in this position. So if we start following it again in the same plane, so 2, now field line 2 ends up here, sorry. Then you can see the angle here is quite different, right? So the angle here is quite different. And the radial change of the rotational transform is called magnetic shear. So we just said that yota, we just said that yota bar or the safety factor often or will always depends, well often on the radius. Sometimes the dependence is so small that it's hard to see. The experiment is designed in such way. So yota bar often depends on the radius and this is what we call a magnetic shear. This is what we call magnetic shear and it is basically defined as we use the letter s for that where we have r over qs and then dqs over dr. So it is basically the degree of change of the twist of the magnetic field. Now magnetic shear is good for confinement for a few reasons. So magnetic shear is basically is good for confinement. There are a few reasons for that. We will discuss a few more in more detail just to give you now one reason is that because if we have some radially extended structures appearing in the plasma, those are decorrelated due to magnetic shear. Those are decorrelated due to magnetic shear which reduces basically transport losses which improves thus our confinement. So these transport losses are reduced by that. What do I mean by that? So if I have some kind of density structure, this is a very big poloidal cross-section, some position in the plasma and now having starting here and let's say we have one magnetic field line starting here and another one starting here and if we follow now this density structure toroidly around, so let's just assume we follow it toroidly around then the magnetic field lines as we have magnetic shear they will be for example here and here and if we follow it further around then the density structure might look like this because the magnetic field lines now are further separated so this is further following it toroidly around and since the plasma can move freely along the magnetic field lines but not perpendicular these density structures are kind of strained out they are decorrelated and this can lead to a transport loss reduction so better confinement. Okay, so in this video we introduced the magnetic field geometry we talked about the major radius, the minor radius we defined the confinement region I tried to explain the importance of diverters as opposed to limiters so diverters are really a key component for fusion devices in order to reduce the losses due to bremstrahlung by impurities in the plasma then we really talked about plasma shaping, ellipticity and triangularity I just introduced the concepts we will talk later on why they are good but keep in mind you rarely find circular shaped plasma they are often somehow shaped and the shaping happens to increase the confinement to increase the energy in the plasma then I introduced the rotational transform a measure for the twist of the magnetic field line the safety factor on tokamaks often used and then that the rotational transform or the twist changes as a function of radius and this is called magnetic shear Okay, that's it for this video hope to see you in the next video