 Hello everybody and welcome to video number 26 of the online version of the Fusion Research Lecture. In the last video, we started with chapter 5, Collisional Transport and I tried to explain you how the transport or how we will describe the transport based on random walk processes and we just started to talk about the classical transport last time. And in this video, we will start to talk about neoclassical transport because you might remember I told you that the Fusion Reactor just being based on classical transport would have a size which would fit on your table and you know that this is not the case and one of the reasons why this is not the case we will see in this video. So it is about neoclassical transport. So as I said last time, neoclassical transport, the expression neoclassical just means that we have a toroidal magnetic field geometry, toroidal B field geometry. The expression neoclassical just came from the fact that this is more than just classical transport and it is maybe a bit of an unfortunate expression because it sounds somewhat fancy but neoclassical transport does means having a toroidal field geometry. As you know from the chapter particle trajectories, having a toroidal field geometry means there are different type of particle trajectories possible. We have passing particles. We have particles on banana orbits and in Celerator for example, we also have helically trapped particles, helically trapped particles. All of these different orbits and different particles contribute to the neoclassical transport in a different way as we will see. And in this video, we will start to talk about passing particles. We start to talk about passing particles. And passing particles, the transport residing from passing particles is also referred to as Fersch-Schlüter transport, Fersch-Schlüter transport. So again these two important names, Fersch-Schlüter transport, so it is very likely that these names will be asked, that you will have to say these names in the exam. Now just as a reminder, having passing particles means that the particles do not stay locally on the flux surface. On average, yes, but locally they can deviate from it. And this displacement is what we will use as step size because also for the neoclassical transport, we will use the random walk approach, meaning we need a step size and a stepping time. And as a step size L, so this is a step size L in the random walk approach, we will use data P, which was the displacement of the guiding center, displacement of guiding center from the flux surface, from flux surface. And we call it these orbits, you might remember the drift orbits, the drift orbits. So it is basically the displacement of the center of the drift orbit with respect to the flux surface. And just to be precise and sure here, L, this is the step size in the random walk description that we are using. Okay, and we get the maximum displacement if, well, let's say the next quantity, we need the two quantities for getting a diffusion coefficient based on the random walk approach, the step size and the stepping time. So and we get the next step for the stepping time, we distinguish two cases. First of all, we get the maximum displacement if the collision time is much longer than the transit time. So we get the maximum displacement, maximum displacement which is data P if the electron ion collision time is longer than the transit time, transit time. This means that we allow the particle to fulfill or to make the orbit, the full orbit such that it can experience the maximum displacement from the flux surface. If the particle would collide a lot, then it would not experience a full displacement from the flux surface. This is why at a first as a first this distinguish, we have, we get the maximum displacement if the collision time is longer than the trapping, sorry, than the transit time. And this then results in a diffusion coefficient P s for first Schlüter, which simply reads data P squared, so the stepping time squared, a squared, sorry, the stepping size squared over two times tau electron ion where we have used, where we can now insert for, oops, sorry, for data P, the displacement, this was the Poloid-Lama radius times epsilon the inverse aspect ratio and the Poloid-Lama, sorry, the Poloid-Lama radius, this was given by the energy of the particle basically M times V over the charge and the Poloid magnetic field. Now if we insert these quantities, we get the following expression, we have one over two times tau EI, the collision time, then we have M V over Q B and then times yota bar as you will see, and this squared and this is basically, so this expression here you might remember, this expression corresponds to the classical diffusion coefficient as we had it in the video last time, just this expression. So this is then the classical diffusion coefficient times one over yota bar squared and we have used here the expression which we had, I think one or two times in the video that yota bar can be written as R naught over R and then B theta over B, meaning we now have an expression for the, or we can now get an expression for the particle diffusion coefficient in the Schluter regime, in the first Schluter regime and this is usually often abbreviated with D for NIO and then NIO in the index and the subscript sorry, so for neoclassical being the sum of the classical diffusion coefficient plus the Schluter diffusion coefficient and since we could express the Schluter diffusion coefficient in terms of the classical diffusion coefficient, we can now write the classical times and then parenthesis brackets one plus and now there's a two over yota bar squared, just if you're wondering where this two comes from, so this one, this comes just from the fact to make this simple description, this equation agree with the literature value, okay? So this then is proportional to the collision frequency since the classical diffusion coefficient is proportional to the collision frequency, meaning that we have again a linear scaling, linear dependency and maybe we just write that down here as a reminder this factor of two, this is to make our simple description, to make simple description agree with literature, so with a properly derived rigorously, I should say with a rigorously derived diffusion coefficient. Now we have an equation for the diffusion coefficient in the neoclassical regime, what does that mean? Let's have a look, so if we assume a typical value of yota bar of 1 over 3, then we get neoclassical diffusion coefficient on the order of if you insert this here on the order of 20 times the classical diffusion coefficient, okay? So the neoclassical diffusion coefficient is larger by a factor of 20 than the classical diffusion coefficient and this is just due to the toroidal magnetic field geometry, just due to the toroidal magnetic field geometry, so quite significant. Now in the beginning of this slide in the top, we made the case or the requirement that the collision time should be much larger than the transit time. Now what is if this is not the case? So what happens in the case of a collision time being smaller than the transit time? Collision time being smaller than the transit time, then the displacement is reduced, oops, then the displacement of the particle is reduced, which means that the diffusion coefficient is also reduced and the displacement is reduced because the particles drift now only between two collisions, so they do not fulfill a full orbit. So it is reduced basically due to the fact that the particles now drift between two collisions only such that the displacement is given by the drift velocity times the stepping, sorry times the collision time. Instead of what we had before, we had the drift velocity over the poloidal transit time but since the particles now no longer make a full poloidal orbit due to the short collision time, we have to replace that by the drift time and as you know by now probably the drift velocity simply mv squared over qrv. Inserting this results in a diffusion coefficient for Schluter equal to we have the drift velocity times the collision time and then this needs to be squared. So this is the step size, now the stepping time is the same 2 times tau ei, so it's the same expression. Now inserting this for the drift, inserting for the drift velocity we can write mv squared over qbr squared times tau ei over 2, since one of the ti cancels out, of course if you have a look here so this ti cancels out and one of the ti's in the nominator cancels out and this means now interestingly that the diffusion coefficient scales inversely with the collision frequency. So this means that the diffusion coefficient in the Schluter regime now decreases with increasing sorry collision frequency between electrons and ions. If we have a plasma which has short collision time and these plasmas often referred to as collisional plasmas, so the diffusion coefficient in the Schluter regime decreases with increasing collision frequency in a collisional plasma. Okay so that's it for the neoclassical transport being based just on passing particles and we call the resulting transport first Schluter transport and we have to distinguish between two cases. One case if the collision times are long meaning we have very little collisions because then the particles can fulfill the full poloidal transits then the other cases if the collision time is shorter than the transit time the particles can no longer fulfill their the full poloidal orbits and these two distinguished cases resulted in different scalings the first one where the collision times are long so where we have very little collisions very little collisions going on. The diffusion coefficient increased linearly with the collision frequency and for the other case when we have a lot of collisions the diffusion coefficient decreased so it increased with one over the collision frequency and the important result is that the particle diffusion coefficient in the first Schluter regime when we have little collisions going on when the collision time is long is approximately 20 times the classical diffusion coefficient so much larger. Okay that's it for this video see you in the next video.