 Hello everybody and welcome to video number 27 of the online version of the future research lecture We are in chapter 5 collisional transport and in the last video We started looking to neoclassical transport that is transport in the toroidal or due to a toroidal magnetic feed geometry And we looked only at transport residing from passing particles Now we will look also at trap particles So we are still dealing with neoclassical transport Neoclassical transport and this time we will look at trapped Particles trapped particles now You might not be surprised for that, but particles trapped in banana orbits reside in increased transport so banana orbits reside in Somewhat increased Transport increased due to as in comparison to the passing particles now It is now important to note that there can be a transition from banana particles to passing particles by small angle collisions So particles so we have to look into this transition region as well so particles can change from Banana orbit Into the passing particle now how might that look let's make a simple drawing for that a sketch rather to access Maybe something like this where here we have the parallel velocity and then we have a magnetic field Strength maybe looking like this Down up again, so this supposed to be be the magnetic field strength Then here we are at a zero on the output side. So this is the poleoidal angle Theta then here where the maximum is We have minus pi and Accordingly here this would correspond to plus Pi and Now having a particle sitting here with that parallel velocity It would mean that this particle is trapped so it travels in this direction Then it bounces off the magnetic field and so on and so on so this would correspond to a trapped particle Now a passing particle would have a velocity a parallel velocity somewhat larger for example sitting here Then it could just pass around being a passing particle and By a collision which changes the parallel velocity just a little bit increases is trapped particle can transit into a passing particle and Just a reminder when we look at our typical pole oil cross section Then the banana orbit my oops a banana orbit Might look like this for example the pole oil projection. You remember this Then the width here, so this is supposed to be more or less the same this With this distance corresponds to the width of the banana orbit, which we will use as step size okay, so Before looking at one particle it transits from one Being trapped to being a passing particle First the obvious thing is that we will again and describe the transport with a random walk process where the random radial Motion corresponds to a had the step size of Data B a which is the banana width And that's sort of as a reminder. This was the whole Lord Lamoradius times the square root of epsilon or the Width of a passing particle of The displacement of the drift plane with respect to the flux of faces divided by the square root of epsilon Now there is an important thing to note An important thing to note as I said times already the step size is different from the collision time as we have usually Defined it so now the step time for this case is different Is different from The collision time Which was defined Which is defined for a 90 degree collision because Here we are looking at Smaller angles Here we are looking at small smaller angle which transfers Kicks if you want to transfer the trapped particle into a passing particle and that angle is Much smaller than 90 degrees This is much smaller than 90 degrees Well smaller angle. Let's write it make a More less proper sentence a small angle transferring the trapped particle into a passing particle is actually much smaller than 90 degrees now Instead of fully deriving it and just again outline the steps for using the conservation of Kinetic energy of kinetic energy that is one half MV squared and in addition the conservation of the magnetic momentum Something which you know from the discussion of the magnetic mirror That was used to derive the trapping condition U m v per squared over two times the magnetic field and from These two conditions we can or using these two Conservation laws we can derive estimated trapping condition The parallel and in the outboard side, this is why you say zero there Over the perpendicular at the outboard side Has to be smaller than the square root of two times the inverse aspect ratio and That allows us then to Write down a correction for the momentum relaxation time So I'm using this type of error here to indicate that this involves a bit of a Algebra as well so the momentum relaxation time Needs to be corrected needs to be corrected and It needs to be corrected by the ratio of the small Partial of the small momentum change compared to the 90 degree change Yeah, so the ratio of just a small angle change to a 90 degree change and then the Effective effective collision time You define that we call this an effective collision time can be given an approximated which is How effective is equal to tau ei times epsilon? Okay, so this is the effective collision time what we will use here Now let's first look Or we have to distinguish Two more cases and the first case is again like in the previous example where we looked at passing particles for a long collision time So for little collisions happen. So if we have long collision times This means that tau effective Which is given by the electron ion collision time times epsilon as I've just said is larger than the banana time Which was given by the transit time Over the square root of epsilon and we can rearrange that such that we can write that tau transit over tau Electron ion is actually smaller than Epsilon to the power of three half. So this is the condition of This is fulfilled. We speak of long collision times and We call this regime which we are in then the Banana regime Yeah, so this is the banana regime and There the effective collision time is larger than the reflection time of the bananas as it's written at the top Yeah, so basically means that the effective Doesn't hurt to write it down again that the effective collision time is Larger than the reflection time of the banana orbits and the reflection time of The banana orbits Just right bananas here now since we now have also Collision time so a step time we can define the diffusion coefficient of as an example of electrons in the banana regime So we can define a diffusion coefficient or derive the diffusion coefficient of electrons in the Banana regime and that is D banana approximately the Width of the banana orbits as stepping Size as step size and we use two times tau effective for the stepping time and And then We also account we multiply with a square root of epsilon This is to account for the fraction of trapped particles as you remember the square root of epsilon is Approximately corresponds approximately to the fraction of trapped particles the fraction of trapped particles trapped particles now inserting for the Banana width we can then write First of all, let's bring the square root of epsilon to the front and then insert then we have MV over E times B theta Times the square root of epsilon and this squared Times one over and then now for tau effective we insert two times tau electron iron times epsilon and If we have a close look at this now and remind ourselves of the last video we see that the Diffusion coefficient for the first Schluter regime appears here. So we have one over First of all the epsilons. This is epsilon to the power of three half Times the diffusion coefficient for the first Schluter regime, which means that in the banana regime We also have a scaling a linear scaling with the electron iron collision frequency like we had in the first Schluter regime But it is larger than the first Schluter regime due to this additional factor one over epsilon to the power of three half Okay, so an increase As compared to the first Schluter regime Now when the collision time increases it transition occurs Okay, so a transition occurs and this transition occurs when Well when the collision time Decreases When it is epsilon when epsilon three half is smaller to the power of three half is smaller than the ratio of the transit time To the electron iron collision time being smaller than one Then we have no longer full complete Or well closed. I should probably say closed banana orbits Which means that we have a reduced step size So we get a reduced step size but The number of collisions increases so to counteracting processes so a reduced step size, but also the number of Collisions increasing and this means that the diffusion coefficient D is approximately constant here and this is why the corresponding transport regime is called the Plateau regime so this corresponds to the plateau regime and in this regime we can make an estimation that the Effective collision time Corresponds to the banana time which means basically that we have just one banana of it just one banana of it and then the Diffusion coefficient in the plateau regime Can be estimated by again the square root of epsilon to account for the trap particles and then it's the banana width Over the To be a the banana stepping time But now the transit time for the banana particles sorry And then inserting then we get that the square of this Data P the displacement of passing particles over two times the Transit time and this then can be written further as one over Utah bar squared times the Lama radius squared over two times The transit time and the main point however is as I said that it does not scale with the Sorry that it does not scale with the electron ion collision frequency Meaning scale to the ad electron ion collision frequency to the power of zero Yeah, now what happens when the collision time further decreases So if we have the case that the electron ion collision time is approximately the same as the transit time then we have practically No banana orbits anymore, we have practically no Banana orbits or let's say banana particles anymore and What then happens is we have a transition into the first Schluter regime so we have a transition into the first Schluter regime Okay, so now we have everything together for the neoclassical diffusion coefficient in the talker mark We have passing particles and we have trapped particles and we have distinguished Depending on their Visionality of the particles on the collision frequency. So let's have a look how it looks if we combine everything together. So let's look at the neoclassical Diffusion coefficient in Talker mark now, let's first draw the coordinate system. Let's make it a bit larger Like this Then into this direction maybe like this so The vertical axis is D the diffusion coefficient the horizontal axis we have the transit time over the electron ion collision time being proportionate to the electron ion collision Frequency Okay, um, so let's start first with a banana regime So the banana regime it starts somewhere here Maybe up to here then what we have here is a position where Epsilon where the transit time over the electron ion collision time corresponds to epsilon to the power of three half This corresponds here to the banana regime so this is the Banana regime and in this regime The diffusion coefficient can be approximated by Data be a squared so the banana with squared over two times tau effective times in the square root of epsilon Next we have the plateau regime as I said the plateau regime the diffusion coefficient is constant as a function of the frequency of the collision frequency And it stops roughly where we have a ratio of the transit time to the electron ion collision time of one Which means that at this position We have practically no bananas Anymore This regime is called as I said plateau regime This is the plateau regime and the diffusion coefficient The plateau regime is given by data be a squared over two times tau and then the banana time times the square root of epsilon and then finally We have the first shooter regime Those approximately like this at first and then in the end it continues to decrease again So this is the fish schlüter regime the fish Schlüter regime Where the diffusion coefficient is first in the linear increase Region given by the polar llama radius squared times epsilon squared Over two times the electron ion collision time. Of course, this can be this increases linearly if we would extend this something like this And then if we were to include the classical diffusion coefficient here This would look something like this so a linear increase But being much much smaller, so this is actually Bad scaling here. This is not to scale. So it would be much smaller actually as You remember from the lecture where I said it is a factor of 20 at least smaller So this is a classical diffusion coefficient where the Where it is given by the llama radius squared over two times the electron ion collision time Okay, that's it for this video where we looked into Trapped particles and how they contribute to the neoclassical transport We distinguish between particles Which experience almost no collision so they stay in their trapped banana orbits particles Which experience some small angle collisions thus can transit from being trapped to passing particles and particles which Transit which experience a lot of collisions. So they are the first Schlüter regime and in the end we get a full Neoclassical diffusion coefficient in a tokamak as shown on this plot where we have three different regimes the banana regime the plateau regime and the first Schlüter regime In the next video, we will look into What changes in a stellarator, but for this video, that's it. See you the next video