 Hello everybody and welcome to the free online version of the Fusion Research Lecture. This is video number eight and we are in chapter two, Magnetic Field Configuration. And you might remember that in the last video we introduced the concept about magnetic flux surfaces and why they are important and what the consequences of them are. This video we will talk about the proof of existence for flux surfaces. The topic of today's video is the proof of existence, the proof of existence of flux surfaces. And the proof of existence is only possible if the magnetic vector potential A, which we introduced in the last video, features some kind of symmetry, some kind of symmetry. And just as a reminder, the magnetic vector potential A was defined as such that the curl of A defines the magnetic field B, meaning that the vector that it serves basically as a vector field, or it is a vector field serving as a potential for the magnetic field. That is a proper way to put it. Now the general method for the proof of existence of flux surfaces is as follows. So first we start with some coils. Coils define us the spatial distribution of the current. From that we can insert the definition of B via the vector potential so that the magnetic field B is defined as the curl of A into MPS law. And then we get an expression for A from that, which reads as follows, mu naught over four pi, then the integral, then J r prime over r minus r prime. So the distance to the point we are looking at D3r prime. And if we make the curl of that, as I said, it gives us a magnetic field, then we would end up with the Biosavars law here in that case. So we have magnetic field coils defining the spatial distribution of the current. That gives us the magnetic vector potential. Okay, having the magnetic vector potential, what we need to do next is find or rather guess flux surface. So some kind of function for the magnetic flux. So we need to guess or find flux surfaces, flux surfaces. And oops, sorry, that should recalculate and calculate psi being the line integral around the boundary of the surface which we had. We now have the area A, sorry, the vector potential A, of course, the magnetic vector potential A, and then the tangential unit vector along that curve surrounding the surface which we are looking at. And then three, we need to check the fulfillment of the gradient of psi times B is equal to zero inserting for B the curl of A. Those are the three steps we have to do if we want to prove for the existence of flux surfaces. Okay, now let's look at an example. So let's look at an example. And as an example, we take the case for a rotational symmetry in phi direction corresponding for example to a Tokamak. So we take the case of a rotational symmetry into phi direction, phi is the toroidal direction as you know. And then first we define or let's say we have made a good guess of a flux surface function of a function for the magnetic flux. So define or guess flux surfaces with a function psi in equal to 2 pi and then r a phi where a phi is a function of r and z that this is constant. And with this definition or guess, we now have to check the fulfillment of the gradient of psi times B equal to zero. So we have to check now if the gradient of psi times B and now inserting what we have for psi here meaning it's the gradient of and then 2 pi r a phi times then for B we insert the curl of a so nappler times a the magnetic vector potential. And now we are using the definition of nappler and cylinder coordinates sort of the gradient and the curl in cylinder coordinates. So we are using cylinder coordinates and cylinder coordinates. Then this equation reads so the first term is the derivative with respect to r. So when I'm writing something like I'm writing something like this, this is supposed to mean the derivative with respect to x to the partial derivative. So this is dr of r a phi and then 1 over r d phi of r a phi and then dz of r a phi corresponding to the first term and now the second one the curl of a then the corresponding definition in cylinder coordinate reads 1 minus r d phi a z minus dz a phi and then dz so the partial derivative as a partial derivative to z then a r minus dr a z and then 1 over r d r r a phi minus 1 over r d phi a r let's not forget the brackets here. And now if we have a close look at that, we know that we have rotational symmetry into phi direction meaning that the derivatives with respect to phi vanishes. So these are this is this term then we have here another derivative into phi direction and then we have here another derivative into phi direction. If we write that down now what remains so we have the first expression dr supposed to be an r and then a r a phi times there's a minus dz a phi the second row vanishes. So then that's a plus dz r a phi then we have 1 over r dr r a phi what is yes so here in this expression so in this expression the r does not depend on z so we can cancel it out with the 1 over r which we have here and then having a close look at it first of all we might see that we have missed the term 2 pi which is here at the top in the beginning but this does not matter because the expression vanishes anyway so that's no problem. Now let's have again a look first of all here we have dr r a phi this is what we have here dr r a phi and then this is dz a phi dz a phi so these terms are actually these two expressions are identical so what we get in the end is 0 and this is what we wanted to show right so that this equation here gives you zero and since this equation results in zero we have a proper definition of a flux surface now okay so having a look at the equation especially in the second row here we basically see since the first term is zero that a r and a z can be of arbitrary shape so we see here that a r and a z can basically be arbitrarily can be arbitrarily as long as the rotational symmetry is conserved meaning as long as the derivative with respect to phi vanishes so if the rotational symmetry is conserved and it basically the same applies for you could say the same for b phi in general which is defined by a r and a z okay that is as an example the case of rotational symmetry into phi direction which already results in proper flux surfaces now the the case is similar for translational symmetry into z direction so it is similar if we have the case of translational or if you prefer that linear translational symmetry into z direction then the flux surfaces or the flux many flux function is given by a z being a function of r and phi equal to constant so this also results in flux surfaces and then the case is also similar for having a helical symmetry having a helical symmetry around the z axis so this is something which you would have in a spring on a drill bit or in a screw in general it is more complicated or different if you have no symmetry so if we have no symmetry like for example in the complex geometry like in a stellarator then we can no longer make such a simple proof for the existence of flux surfaces what we have to do instead is either numerically or and experimentally a so-called Poincare plot the Poincare plot which basically means visualizing visualize visualize the intersection points or piercing points of a magnetic field line in one poloidal plane so visualize the piercing points or intersection points of the magnetic field line in a poloidal plane that is a Poincare plot this is something this is basically the first thing you have to do when you have built a stellarator check if you have closed flux surfaces by making such a Poincare plot okay so let's have a look at an example for this first of all before we look at the Poincare plot here you see a photography from the inside of w7x and here you see one magnetic field line how was that visualized so there is an electron gun basically a hot filament emitting electrons placed at one particular position then the magnetic field it was turned on ramped up and the electrons can move freely along the magnetic field line and along the magnetic field line if there are some energy so the filament is biased negatively such that the electrons are accelerated they move freely along the field line and then they can ionize a residual gas which is inside of w7x in this case it was hydrogen and in case you wonder about the color this is a false color image and this is a nice way to visualize magnetic field lines so what you can see here is one magnetic field line only one magnetic field line in hydrogen gas hydrogen gas and note these are false colors now this is a very nice way to visualize magnetic field lines however what we would like to have is a more quantitative way and this is the Poincare plot so let's have a look up at the Poincare plot now if we first have a look from the top at a toroidal experiment at our stellarator it might look like this so this is basically a toroidal sorry well a toroidal section a section of the toroidal experiment the top view onto our stellarator and then let's say we have an electron gun being located here emitting electrons which travel along the magnetic field line then somewhere here they intersect with a rod which is moving with a fluorescent rod and what then happens let's try to visualize that let's draw now a poloil cross section so here we have a poloil cross section now a poloil cross section corresponding to the position to this position where the rod is located so there's some kind of swiveling or sweeping rod to move around like this and every time in the electrons hit the rod for example it might be here the field line might intersect with the rod at this position and then again at this position and if the rod is at this position might be here and here here light is limited because the rod is coated with a fluorescent powder and now if the rod is really moving around and around oops for example in another position it is like this and then like this and like this and so on i guess you get it then we might have further intersection points here and here and here it might be here here and if it would move around further we would get maybe a lot of intersection points ending up with the visualization of one flux surface in a poloil cross section and this would be a Poincare plot so a Poincare plot basically is produced by having an electron beam being emitted by some hot and negatively biased filament and remember the electrons follow the magnetic field lines and stay on that basically so we have an electron beam hitting a move in the sweeping or let's say a swiveling rod and the rod has some fluorescent some fluorescent sorry some powder on it usually this is some kind of zinc oxide and this technique in general is also referred to as fluorescent rod technique technique and then what you do is making a long time exposure of the moving rod and thus of the fluorescence light fluorescence light and this then results in an image in an illustration of a flux surface as for example shown here on the left hand side you see a photography of w7x meaning i'm not precisely sure how the rod was moving here i think it comes from not from the top but that's just as you become from the top so there was some kind of rod moving around then you get all these intersection points here along the flux surface resulting in this nice illustration of the flux surface yeah on the right hand side you see a case for this is basically a processed image of w7as the predecessor of w7x and the electron beam so the electron emitting gun let's say has been positioned at different radial positions each position resulting in an image for one flux surface for example like this and another flux surface for example like this one and so on and if you put all these images on top of each other you get this very nice illustration of nested flux surface uh flux surfaces in a stellarator okay that's it for this video where we have talked about the proof of existence for flux surfaces which is only possible if your experiment of the vector potential the magnetic vector potential features some kind of symmetry that is usually only the case in tokamaks in stellarators you have to do something else to prove for the existence and or flux surfaces and this is for example a Poincare plot as illustrated here this example and this is usually the first thing which you do when you start experiments in a stellarator to check if you really have closed flux surfaces okay that's it for this video hope to see you in the next video