 Hello everybody and welcome to video number 25 of the online version of the fusion research lecture in the last video We chimmel finished sorry with chapter four particle trajectories in a in tokamak and stellarator and Today in this video. We will start this chapter five collisional transport So let's write that down as a starting point chapter five will be about Collisional transport in fusion plasmas Okay, we will distinguish or there are to distinguish three different main categories or mechanism of transport first of all We have classical transport classical transport In classical transport, we just have diffusion through gradients and density temperature and potential So we have a so classical transfer is basically diffusive or diffusion through gradients in density temperature and potential and What this means basically is that we have fixed law which says that the diffusion flux is Proportional to a concentration gradient. So you might remember from other lectures at fixed law. So this is basically the flux that this is proportional to a diffusion coefficient and then the Gradient in some concentration. So here the density for example. Yeah, so this is just as an example For the classical transport we assume B to be homogeneous the magnetic field is homogeneous Which means we have a magnetic field, but it is straight. So straight magnetic field If classical transport would be the only transport mechanism in a fusion plasma Then a reactor would have a small minor radius of a small radius on the order of 14 centimeters For a fusion reactor We know that this is not the only mechanism, but this is just to tell you How a reactor would look like and this is what people thought in the very beginning of future research So in addition to classical transport, we then have neoclassical transport Neoclassical transport and this is transport where we include effects due to the Inhomogeneity of the magnetic field in a toroidal geometry meaning we will use all the knowledge We hopefully gained in the last chapter where we talked about particle trajectories and tocomax and cell writers So the neoclassical transport neoclassical just means that it includes The inhomogeneity the inhomogeneity Of the magnetic field in toroidal geometry in toroidal geometry and It also means a toroidal geometry. This is important here for neoclassical for the expression neoclassical This also means we are taking into account for example mirror effects and we will see how these Mirror effects we discussed in the last chapter will lead to additional transport losses And then finally, there's the so-called anomalous anomalous or turbulent transport The expression anomalous was basically used because initially it was not clear what this is due to turbulent transport is maybe a more physical Meaningful expression because it tells us that this transport is due to turbulence going on in the plasma and Turbulence is driven by gradient in the pressure. So this is pressure the plasma pressure profile and The plasma pressure profile drives instabilities and the instabilities instabilities can lead to turbulence which then can lead to losses and this is the dominant transport Usually definitely in toka max The early stellarators had neoclassical losses as being the dominant losses are ever modern stellarators as for example W7x now also are turbulence more more turbulence dominated and one can maybe draw an error here something like this Which indicates the direction of Increasing strength. So if we were to write here, let me try if I can do this so in Where's my mouse here? So this is the direction of increasing strength Okay, how will we tackle that so what we are doing here in this chapter is an intuitive derivation and intuitive derivation of The transport coefficients The correct way would be to use the distributions functions perturbed them via transport And then see how this perturbation is balanced by the thermalization due to collisions. This will tell us the Mathematically Correctly derived transport coefficients. However, as we will see the intuitive derivation quite often gives the exact same result so Formally more formally more correct way would be to perturb the distribution functions perturbation of distribution function distribution function via transport caused by Gradients and see how this is balanced. Okay, clear how this is balanced by The thermalization again thermalization due to collisions Just that you know what would be the correct way Okay, let's start with talking about classical transport Classical transport to tackle or to to describe classical transport. We use a random walk approach Random walk approach now what is random walk? so for that I have a video Let me see where this is a random walk. Yes to this one I'm sorry. I am sorry about this. Okay, so What you can see now here is a 1d Simulatory a simulation of a one-dimensional random walk So what I have done here assume there you are standing at one position Somewhere along a one-dimensional line Then you cross a tome and if you get head you have to walk one step to the left and if you get a tail one step to the right and You do this a thousand times and then you do this multiple times and this is a one-dimensional random walk and Here I have made three simulations of that and you can see how for three examples the trajectories all ready look quite different So all these all of these three Walkers, let's say have the exact same probability the same starting conditions So 50 percent to get a step into the left direction 50 percent probability to get a step into the right direction This is by the way also sometimes called walk of a drunken man because you either get to the left or a step to the right or For the drunken man is often the example used a step forward and a step backward Yeah, but this was just an example for the 1d random walk now if in the 1d random walk Approach You would plot the probability for the position At the end of the random walk approach it would have a Gaussian shape So the probability So the probability for the final position The probability for the final position of a simple 1d random walk has a Gaussian shape centered around the starting position has a Gaussian shape centered centered around The starting position So if we would make a lot of these simulations, we would finally get up Gaussian position of the distribution Sorry a Gaussian distribution of the positions at the final end plane now If we vary the step size or the step time or the probability for each step then we get different shapes so varying the step size the step time stepping time and the probability For each step Then we get different shapes the Simple random walk approach as I said can be modified by varying the step size the stepping time or the probability And then this very simple model can be and is Successfully applied to a lot of areas in physics and just to name a few Brownian motion for example Very famous for that so brownie in motion then heat conduction And in brain research so how the neurons move in the brain Then when the walk plays a role in finances When you follow for example Stockmark price index and of course in plasma physics it plays a role to describe transport processes Sorry and plasma Physics and there are numerous more fields which I will not Write down here in general you can say that one can say that the diffusion Can be considered as the sum of many random walks Diffusion can be the sum of many or can be considered as described as the sum of many random walks by many particles by many Particles and that allows you then to write down a diffusion coefficient D diffusion coefficient D Equal to L squared the step size then over the stepping time and Again, this is the diffusion coefficient which you would insert in fixed law of diffusion So mine is the diffusion coefficient times the gradient of the concentration here the simply the density and In this diffusion coefficient as I already said L is the step size And since here we have a magnetized plasma so classical transport means transport in a plasma So there's a step size perpendicular to the flux surface perpendicular to the flux surface and One over tau is Is the step frequency? This is a step frequency Yeah, so it is important Depending well the main question let's say is Which yeah, let's write it like that. So the question the main question is what values to use for the step size and the step frequency what values to use here Okay, let's have a look at the classical transport where we are at the moment so to describe that One thing which you can easily imagine is that the particle Displacement the particle displacement of the guiding center of a particle is due to drifts and Collisions due to drifts and collisions Now here however in this example we have classical transport Which means we have a straight magnetic field and no electric field. So here We are considering the example of classical transport. That means we have a straight Magnetic field we have no electric field Which means there are no drifts There are only collisions to be considered Talk about collisions. You might remember from the plasma physics one lecture that in general small-angle collisions dominate in a plasma small-angle collisions dominate and You might also remember that this was expressed by the Coulomb logarithm and You might also remember the way how to describe the collision frequency we use the moment relaxation time and you can say that it takes the Sorry the momentum relaxation time of course wanted to say it takes the momentum relaxation time towel to rotate oops to rotate the momentum vector the momentum vector by 90 degrees So this was the definition of the momentum relaxation time Here we make the assumption to make it a bit simpler We make an assumption And this is a very often applied assumption that the particles experience one 90 degree Collision so instead of many many many small-angle collisions. We assume it's 190 degree collision within the time tau Without being perturbed otherwise without Being Disturbed or perturbed otherwise. Okay, so this is the simplified assumption which we Make here. This is an often applied and reasonable assumption now, let's have a look at collisions for same particles For same particles So first of all collisions for same particles, so let's try to draw this or sketch this again Circle Now we delete a bit of the circle like this And we copy it because we need it multiple times so We start with some Particle here which moves around in this direction and Then as you Might have guessed already Let's get rid of this here There is another particle. Let's use a different color for that That's another particle Let's draw another circle That's another particle example like this oops Let's copy this again move this to here and Get rid of it Oops, I'm sorry. I wanted to use the rubber like this The eraser I wanted to say of course and then we have the other trajectory here. Okay, so this is Another particle starting basically here, but it is the same particles. So we should probably use the same color So the same party is species. I'm wanted to say I Said same particles where same particles means Same particles species, right? So two electrons for example moving into this direction then This here, this is the point of collision. This is where the particles collide So after the collision they move on further for example to here or to here Now if we were to draw look at the center of mass for example, this one would be here This one would probably be here so there would be Displacement from the center of mass from this particle to here Now looking at the other particle the center of Mars of this one might probably be here of that one here So that would be a displacement into this direction If we now let's quantify that So going down here Let's say this is L1 the displacement Then here we have a 2 and If I whoops would have made the drawing a Bit better than L1 and a 2 would be the same Meaning that the center of Mars of the whole system is not changing so there is no net That is important. No net displacement of The center of Mars of the center of Mars also referred to as Barry Mars and That means there is no net transport in this scenario. No net transport Okay, so how does it look like if we have unlike particles unlike particles? So from different particles species for example in electron colliding with an ion so once again Let's try to draw this start to draw the circle maybe like this Then let's copy this circle like this And let's use the eraser to get rid of This part here This part and Now we start with one particle again sitting here Circulating so gyrating around the magnetic field into this direction and then here Let's assume these would be electrons the electron would hit here with an ion Of course, these are not to scale. This is just an illustration Then the ion would barely move. I mean it would slightly move about barely the electron will just bounce off roughly continue its gyration into this direction Then if we now look at the center of Mars from the electron system This clearly has changed, right? So this is a displacement of Oops The center of Mars from the electron system. So now there is a net displacement of The electron by of the electron population in general by Lama radius Meaning there is net transport Okay Good, this is an illustration for why there's no Transport if the same particle types or if the like particles collide if there is Transport in contrast when unlike particles collide now. Let's try to quantify this a bit further so Let's assume we have collisions of Elections with Iance I Then The step size to get the diffusion coefficient is given as I've said on the previous slide by the electron Lama radius Which you can write as 2 m e T e the electron temperature here. I have inserted for the velocity the thermal velocity over eb and the Step time tau corresponds to the electron ion to the electron ion collision time which You have discussed in your plasma physics lecture and you might remember that it can be written as three times three square root of three times m e Times the electron temperature to the power of three half Over the Coulomb factor e squared over 4 pi epsilon not the Coulomb factor squared times 8 pi and e and the Coulomb logarithm and That this is abbreviated with GI the electron ion collision frequency and from that the classical Or that allows us to write down the classical diffusion coefficient The classical diffusion coefficient the classical diffusion coefficient D C l for classical Then row so the electron Lama radius squared Over two times the electron ion collision frequency In equal to so now we insert the standard expression for the electron Lama radius squared times one over two TI and this is proportional to the collision frequency which means that the classical diffusion coefficient Scales linearly with a collision frequency something important to be aware of Something we will discuss later on now we can insert the thermal velocity again and then write down for the Classical diffusion coefficient. We might need a bit more space for that. So inserting the thermal velocity So be thermally equal to the square root of two times the temperature divided by the mass then the classical diffusion coefficient reads e squared and e square root of M e times Coulomb logarithm divided by six times the square root of three times pi Epsilon not squared B squared and the electron temperature Square root it and this corresponds to if you want to Sorry insert some numbers 2.4 times 10 to the minus 23 or should correspond to this times N e times in the Coulomb logarithm divided by B squared and the electron temperature and In the end as usual as it should be the unit of the diffusion coefficient is meter squared over seconds Now if we would do the exact calculation, we will get the same results. So the exact Calculations They give the same results or they give the same result for Electrons and ions meaning that if we look at the iron Lama radii divided by two times tau i e that this Has an I missing that this is the same as the electron Lama radius which is shorter as the iron Lama radius But it is then balanced by the collision frequency the electron ion collision frequency and This means that the transport is MB polar that the transport is Is MB polar So the same for electrons and ions here to different It's also ballots. I should say probably So the transport is MB polar okay That's it for this video where we Introduced or start to talk about collisional transport in fusion plasmas You definitely should remember the order of the strength of the transport Processes so classical transport neoclassical transport and anomalous or turbulent transport We in this chapter further or sorry in this video talk about classical transport Which can be described using a random walk approach a widely used approach Where the diffusion coefficient is given by the Squared step size perpendicular to the flux surface Divided by two times the stepping time and in the classical transport We inserted for the step size the electron Lama radius and for the step time the collision frequency of electrons and ions or These versa for ions and electrons and that the classical transport is MB polar Okay, that's it hope to see you in the next video