 Hello everybody and welcome to video number 10 of the free online version of the future research lecture. My name is Alf and this is the YouTube channel called Der Plasma. We are in chapter 2 of the lecture called magnetic feed configuration and you might remember that in the last video we talked about magnetic islands and in the videos before about flux surfaces. And in this video we will start to talk about configurations for the confinement looking at the tokamak. So today's topic is to start with, oops, let's make it a bit thicker, to start with confee durations for the confinement and we will first look at the tokamak. And we will in particular look in this video how to build a tokamak, what we need to build a tokamak. So we will build a tokamak and for building a tokamak we need basically three ingredients. One is we need a conducting ring or a current loop if you prefer that expression. Then we need some vertical magnetic field, a vertical magnetic field and in addition we need a toroidal magnetic field, a toroidal magnetic field. And we start right away with the conducting ring. So the first ingredient is the conducting ring or as I said some you might be more familiar with the expression current loop or flat coils also often used expression for that. Let's underline that the conducting ring. Now the conducting ring is a model for the tokamak current. That is a model for the strong current flowing in a tokamak. You remember that we need this current to create part of the confining magnetic field and we model that tokamak current by a current filament, by a current filament. And to draw that to indicate that to illustrate that, sorry, let's draw a coordinate system. Here we have z and this is the z direction. This is supposed to be the y direction then into this direction. This is the x direction and then the current filament might look something like, let's try to draw it going through this point, this point, this point, maybe this point and then might be something like this. So lying in the, this is supposed to lie in the xy plane and is supposed to be a circle flowing into this direction. And it's supposed to have the magnitude, the current density, the corresponding current density is IP, the plasma current. And as for the coordinate system, this is the Cartesian system, however often cylindrical coordinates are used as well or spherical coordinates and that would correspond to an additional vector r or coordinate r, sorry, capital R, which is supposed to lie in the xy plane. This angle here is the angle phi corresponding to the toroidal direction and then small r would be going into this direction. This would be small r and lie in the same plane as r is located. And this angle corresponds then to the poloidal angle theta. And now the current filament can be modeled by j being equal to j phi times e phi, the unit vector into phi direction, where j phi is defined as being IP. So the plasma current, the current density of the plasma current times the data function of r minus r naught, where r naught is the position, so it would correspond to this value where the current filament is located and the data function of z. And in case you don't know, the data function, the data function is generally defined as it is everywhere 0 for every x not equal to 0 and the integral, if you take the integral from minus infinity, sorry for, it's probably hard to read, so I try to pronounce it properly from minus infinity to plus infinity, then the integral over the data function dx, that this is 1. So this just ensures that it is infinity is really small and located at the position we want to have it. Now using this definition of the current, we can insert it into the magnetic vector potential to calculate the resulting vector potential. So we will insert it into the magnetic vector potential, which we talked about in I think it's two videos ago, and this was given by a being a function of r equal to mu naught over 4 pi, then times, oops sorry, times the integral over j, sorry again, over j, then r prime over the absolute value of r minus r prime to have it located at a certain position, d3r prime, then we use spherical coordinates here, we use spherical coordinates here to make use of some symmetry properties, meaning this would be r theta and phi, we, as I said, we insert j as we have defined it in the previous slide, and that will then result doing so in a vector potential, which only has a phi component being a function of r and theta, and it will not go through this calculation, just say the result here that it is, it just gives us an a phi component, and if we have a vector potential, which is just an a phi component, we know from video number eight that this vector potential, such a vector potential exhibits rotational symmetry into phi direction. And also this means we have rotational symmetry into phi direction, and we showed that in video number eight, and we also talked about how corresponding magnetic flux function would look like, that was psi equal to 2 pi capital r a phi being a function of r and z equal constant, that this flux function results in flux surfaces, this corresponds to flux surfaces, we showed this by proving, but by making the proof for that, the mathematical proof, and of course we need now to translate it into r and phi, but the major point, the main point is that this all results in already closed flux surfaces, and to royally closed flux surfaces. So we get already to royally closed, well, let's write that word again, that looks particularly ugly, closed flux surfaces, which is interesting, right, because just having the conducting ring, we already get closed flux surfaces. And let's now have a look at these flux surfaces. To do that, we will now make a drawing. So let's try to make a drawing where we here have the z coordinate, and then into this direction, we have the r coordinate capital R, and we will look at the flux surfaces of the conducting ring now, and let's suppose the conducting ring is located somewhere here, and this is in the RZ plane, then it would go around like this, and then going back here and down here roughly, but we will now just look at this plane. Now having this conducting ring here results in magnetic field lines, and the magnetic field lines look like, but you might remember that from a high school experiment where you put iron filings around a current carrying wire, indicating the shape of the magnetic field lines, and the field lines then look like this, going downwards if the current flows into the board, then like this maybe, whoops, that's supposed to close itself, makes it a bit better, let's at least try, so this is supposed to close, putting downwards and one more, something like this, and if we would dare to draw more, then this might look for example like this, and like this for example. These would be magnetic field lines, and as I said, you are probably familiar with that from a high school experiment where you put iron filings around a current carrying wire, and these iron filings range itself such that they indicate the shape of the magnetic field lines. Okay, these are the field lines, remember when we talked about the toe-royal and the po-royal flux, now these field lines from the current carrying wire, which is just occurring into the toe-royal direction, already define a po-royal flux, so let's try to draw or indicate the po-royal flux to a particular surface, so if I were to draw a surface like this going around, and coming back here maybe, something like this, and the magnetic, sorry the po-royal flux through that surface would be psi one, so this would correspond to psi one, then let's draw another surface on the same magnetic field line, maybe starting somewhere around here, this would also go around something like this, something like this, so this now is supposed to be psi two, these two surfaces are supposed to lie in an r-z plane, so sorry for the bad drawing, but they are supposed to lie in the r-z plane, meaning that the magnetic field lines, which I have drawn here, this one for example, at a certain position, it pierces through the field line, it intersects with it, sorry with the flux, with the surface which I'm drawing here, for example here, then it is behind psi one, well this is psi two, sorry I said psi one, this is supposed to be psi two, and then this field line intersects here with the surface, this one with this surface, and remember when we talked about the flux being a measure for the number of magnetic field lines sticking through a surface, the magnetic field line sticking through surface psi one and psi two are the same, which means that psi one and psi two have the same magnetic flux, so psi one and psi two are the same, the constant number because basically the same number of magnetic field lines in it, let's say, so sticking through it, right, and you might also remember that we defined the poloid flux psi theta as the surface integral, S, over the surface S, we have the magnetic field, this was the poloidal magnetic field component times the surface normal dS theta, the surface normal is the normal to that surface here, so this would be dS theta point, being perpendicular to the surface, and if you look at the definition here of the poloidal flux, if we increase the surface of the poloidal flux, so going further outwards, so psi two for example has a larger surface than psi one, but the angle changes between B, between the magnetic field line, between Bp, so the poloid magnetic field component times the surface normal, since this is a scalar product, the final result of that product changes and thus psi two is then still the same as psi one, okay? Good, so we have closed flux surfaces, what is the problem with that? So problems, there are two problems with that, first of all, the shape is inappropriate for a vessel, the shape of the flux surfaces is inappropriate, inappropriate, what a difficult word, for a vessel, because it is basically, these are basically too large, if you have again a look at the drawing, let's go, let's follow that one field line and that one flux surface, which here corresponds to the shape of the field line, if we go around like this, so this is where we have a constant poloidal flux at each position, then you see that we need a relatively large vacuum vessel to confine this, and if we would go further outward, that vacuum vessel would be much larger, and this is inappropriate, something which is very, which is not useful. In addition, there is the Laurentian force, which pushes the plasma outwards, so the J cross B force basically, so if you use your right hand, then you can calculate the J cross B force, and this pushes the plasma outwards, it's the Laurentian force, which pushes the plasma outwards, and that force is also called hoop force sometimes. Okay, what to do if this is inappropriate? Well, let's continue building our tokamak and add the next ingredient, and that is a vertical magnetic field, so we now need to add a vertical magnetic field to get a more appropriate plasma shape so far, plasma shaping, whoops, add a vertical magnetic field. Now let's have a look how to do this, so the corresponding vector potential in cylinder coordinates to get this vertical magnetic field reads A, and then a V for the vector, sorry, not for the vector potential, but for the vertical magnetic field, which we want, one-half BZR, and then the unit vector into phi direction, meaning it only has a component into phi direction. In calculating the resulting magnetic field, BV, then is of course done, as you know, by due to the definition of the magnetic vector potential, by calculating the curl of the vector potential. Now we need to calculate the curl of the magnetic vector potential in cylinder coordinates, so this is very similar to what we have done in Vdu8, and don't worry, I will not do such a thing again here, so I will just show you the final result, this is 1 over r, then the derivative into r direction, r times a phi. Now we need to insert for the phi component of the vector potential, which is exactly what we have written above, and then we just get in the end BZ times the unit vector into Z direction, which is what we wanted to have. Now with this result, the symmetry of the current loop is maintained, meaning the symmetry into phi direction is not broken, so the symmetry of the current loop is maintained, so the symmetry into phi direction is maintained because here we have no further component into that direction, the symmetry is not broken. We get however a contribution to the toroidal flux, we get a contribution to the toroidal flux, and that reads psi and then a v for vertical to pi r, then a vertical, what we have to find above, and inserting that we get pi r squared BZ, which is then now the second ingredient which we need for a tokamak, for building a tokamak in that is to get the vertical magnetic field. Here is an example on how the vertical field changes or modifies the shape of the surfaces, on the, you see three drawings here on the left hand side, so this drawing, this is just the current loop or the flat coil or the conducting ring, so this is the current loop with the shape of the flux surfaces as I have indicated it in the drawing as well, then in the second drawing, so in this drawing here, you see what happens if we add a vertical magnetic field pointing downwards, so at a vertical field pointing downwards and you see that the result is that we get more useful shape of flux surfaces because now the flux surfaces are much more compact, they are much more smaller as you can see, so compare this surface here for example with this one which should be the same on the right hand side, the ones are much smaller, so we can build much more compact vacuum devices, so we get now a more useful shape of psi equal constant surfaces, which are flux surfaces as you know, and then just for the sake of completeness on the right hand side, you see drawing what happens when we add a vertical magnetic field pointing upwards, something which has also been done in Japanese tokamak experiments for example, then we get such an x-point at the right hand side here, which allowed the people back then, it's been a few years since this was done, to install limiters there, but as you know from one of the early videos the preferred configurations nowadays is to not use limiters but use a diverter instructor instead for a configuration, okay there's only one ingredient missing and that is a toroid magnetic field, so number three is now add a toroid magnetic field add a toroidal magnetic field add a toroidal magnetic field and then we will have our tokamak, so this then results in a tokamak, now the toroidal magnetic field is generated by a current into z direction or modeled by being created of a current flowing into z direction into the vertical direction and this current reads j r prime being equal to i z and then to locate it at a certain position data x, so these are data functions again, data y and then the unit vector pointing into z direction, what we need to do is now calculate the vector potential, calculate the vector potential az, then making use of the symmetry that our system features, meaning there's no change of the flux surfaces because we do not have an a phi, so there's no additional vector potential with a phi component and then the resulting magnetic field which we get from that b reads finally minus data sorry dr so derivative into r direction az times e phi unit vector into phi direction and that can be then calculated as mu not i z over 2 pi r times the unit vector into phi direction e phi or which is a very often used way to define that or using the toroidal field strength at the magnetic axis at the magnetic axis this is often called b phi and then a not for indicating that this is at the magnetic axis then b phi can be written as b phi not and then you might have guessed this times r not over r sort of referring to the field strength at the magnetic axis and as you can see this is proportional then to 1 over r and that is basically the typical the typical 1 over r decay of the toroidal field which we see in tokamaks the toroidal field and now we have our three ingredients together to build a tokamak we need a conducting ring a current loop then we need to add a vertical magnetic field on top of that to get appropriate shape of the flux surfaces and then a toroidal magnetic field for the confinement and you can see those three ingredients here so first of all this corresponds to the plasma and in the plasma we have the current flowing so this is basically the current loop this is the current loop then the vertical magnetic field is created by these green coils here oops let's make a proper line so this corresponds to the vertical vertical magnetic field of this these coils create the vertical magnetic field needed to get appropriate shapes of the plasma of the flux surfaces and the red magnetic field coils those create the toroidal magnetic field the toroidal magnetic field and then we have our tokamak sounds easy right however the real challenge often consists in putting a physicist and an engineer together physicist you on the left hand side engineer the right hand side and trying to make them agree on the shape of the tokamak which they want to build despite all these potential disagreements a lot of tokamaks have been built and this is a diagram showing basically the tokamak genealogy where you have on the horizontal axis the time and then you see a lot of polar cross sections of different tokamaks which have been operated throughout the last 50 years basically and when you have a close look you can see that in the beginning most of the tokamaks had a circular cross section a circular polar cross section nowadays however this has been changed and more and more tokamaks and basically all modern tokamaks nowadays have elongated shapes like this one here for example so these are no longer a circular but rather elongated and you will see that the tokamak evolution has sort of converged to the shape of eta which you see on the very right hand side so eta is basically the result if you want of the evolution of the tokamak line okay that's it for video number 10 where we talked about one particular configuration for magnetic confinement that was the tokamak we talked about how to build the tokamak you need three ingredients the conducting ring a current loop which already creates two royally close flux surfaces with an inappropriate shape however this is why you have to add a second ingredient that is the vertical magnetic field which shapes the plasma results in more appropriate flux surfaces which can fit into a vacuum vessel and then for stability reasons you need to add to the third ingredient that is a strong toroid magnetic field and we will discuss about why you need such a strong field strong such a strong toroid field and how that helps you to get a more stable plasma in future videos and finally then i have shown you here this tokamak genealogy plot where you can see how the tokamak evolution the tokamak line has converged basically into the shape of eta okay that's it for this video and i hope to see you in the next video