 Hello and welcome to video number 28 of the online version of the Fusion Research Lecture. We are in chapter 5, Collisional Transport, and in the last video we talked about the Neoclassical Diffusion Coefficient in the Tokamak Geometry. We talked about passing and trapping particles, so banana particles and the influence of them. And in this video we will introduce the additional effects occurring in a Stellarator. So we are still talking about Neoclassical Transport, still Neoclassical Transport. And now it's the third category which we had, which is Helically Trapped Particles or Stellarator Specific Elements. So these are Helically Trapped Particles, meaning we are going to talk about Stellarator Specific Elements. Okay, to start with just a reminder on the Tokamak. In the Tokamak we had a toroidal ripple, and only a toroidal ripple. That was basically the shape of the magnetic field strength if you follow a field line, which has one minimum basically and one maximum if you follow it around the Tokamak. And this is the ripple, the waveiness if you want, the toroidal ripple. This can be described as a difference, understood as a difference of the maximum magnetic field strength and the minimum magnetic field strength along the field line. And for the ripple, this can be described as epsilon t as a function of r being equal to one half, then r0 over r0 minus r corresponding to the maximum magnetic field strength and then r0 over r0 plus r corresponding to the minimum magnetic field strength, roughly corresponding to small r over r0, and which then is the aspect ratio epsilon. This is how the toroidal ripple can be defined. In a Stellarator, you know that we have in addition the Helical Ripple, so we also, so in addition to the toroidal ripple we have the Helical Ripple, the Helical Ripple epsilon h and unfortunately there is no simple relation like the one in the Tokamak for the toroidal ripple. There's no simple relation for the Helical Ripple, however for a classical Stellarator we can estimate and implicitly define it as follows. So for a classical Stellarator, one can implicitly define it as the magnetic field b being equal to b0 and then we have 1 minus the toroidal ripple times cosine of the poleoidal angle minus epsilon h, so the Helical Ripple times cosine of the periodicity of the Stellarator, so l times theta minus m times phi and usually the toroidal ripple is smaller than the Helical Ripple and this is smaller than one just as a general expression. Okay, you might remember that in this Stellarator we have the direct loss of helically trapped particles, helically trapped particles and you might remember that this is due to the vertical drift motion, due to vertical drift and if I'm not wrong that was discussed in video 23 and this of course only happens if the collision time is long, such that the particles can fulfill their well helically trapped orbits where they don't have orbits but their movement. This loss process is not a diffusion process where collisions are involved so this is not a diffusion process but if we assume or if the effective collision time would be smaller than the lost time which was given by the minor radius over the drift velocity then the particles can be scattered back into passing particles or passing trajectories, then the particles can be scattered back into passing trajectories and this also allows us then to define a radial step size for a random walk approach so it allows us to define a radial step size data p for a random walk approach given by the drift velocity and the effective collision time for a random walk approach where for the effective collision time we assume in analogy to the banana particles in the toca mark we trapped the electron ion collision time times epsilon h, the helical ripple basically and for the vertical drift velocity we take the angle alpha being pi half which is sufficient to account for de trapping. Now these assumptions then allows us to define a diffusion coefficient so this allows us to define a diffusion coefficient then let's first look at the electrons in the so-called one over new regime and you will see in a while just a minute why this is called the one over new regime and for now we are first neglecting the radial or radial electric field then the diffusion coefficient d one over new in the index can be written as first of all the square root of epsilon to account for the trap particles then the drift where the step size as just explained squared since it is a random walk approach over the stepping time and then we can get from the tau effective the one epsilon out such that we can write epsilon h to the power of three half over two times the drift velocity squared tau electron ion and then this can be written as again epsilon to the power of three half times one half and then inserting the drift velocity we have the vertical drift velocity m v squared over two q with the charge for the electron r not b squared times the electron ion collision time meaning this scales with one over new and this is why it's called the one over new regime more interestingly though however we can rewrite that by inserting for the electron ion collision time the appropriate expression and replacing the velocity by the temperature then the one over new collision the diffusion coefficient in the one over new regime scales as the temperature of the electrons to the power of seven half over the electron density and this is in strong contrast to what we find in tokamaks so in contrast to transport coefficients in tokamaks this would be a severe problem for this would be a severe problem for the confinement of hot fusion plasmas and stellarators due to the scaling with the temperature to the power of seven half of hot fusion plasmas in stellarators let's underline hot here now there is however a simple solution to that and the solution is actually quite simple because the plasma creates a radial electric field to counteract the transport losses and this is precisely the radial electric field we already talked about earlier so the solution is that the plasma creates a radial e field to counteract the transport losses to counteract the transport losses now having said all that let's have a look at the diffusion coefficient in the stellarator so let's try to draw this so here we have d the diffusion coefficient here we have again the transit time over the electron ion collision time which is proportionate to the collision frequency of electron ions and then we can draw something like this where we start without a radial electric field then a radial electric field was set in diffusion coefficient would decrease then we have again the plateau regime and then the first Schluter regime so this would correspond to the plateau regime same as in the tokamak case this would be the first Schluter regime and here you would have remember the case where the transit time over the electron ion collision time is roughly one here oops we can make this a proper line we have the case where this ratio is epsilon to the power of three half and this decrease here this is called the one over new regime and this would be the case without a radial electric field and if you might wonder why this is anyway decreasing so if we don't have a radial electric field so without a radial electric field then the diffusion coefficient d does not decrease until the resulting gap in the maxwellian distribution due to the losses can no longer be compensated fast enough by appropriate collisions so without a radial electric field d the diffusion coefficient does not decrease until the gap in the maxwell-bolzmann distribution which i probably just read maxwell here and the gap is of course due to the transport losses until this gap is no longer compensated fast enough by electron-electron or ion-iron depending on that which species you're looking at um collisions if this is no longer compensated then there's just lack of electrons or a lack of ions and then the diffusion coefficient starts to decrease now how does the example look like the diffusion coefficient if we have a radial electric field if we have like let's say a moderate radial electric field then we have some increase in the beginning of the diffusion coefficient but then there's a decrease so this is a moderate radial electric field and this decrease approximately starts to occur where the helical ripple is epsilon to the power of 3 half then there can be also cases where we have a very strong radial electric field for example something like this this would be a strong or maybe it looks more like that it goes up a bit faster than something like this this would be a strong strong radial electric field and as i said this radial electric field is nothing fancier magically it is generated by the plasma itself and why is that the case because the plasma charges up if there's for example only transfer of electrons so as we have seen in this drawing the radial electric field or a radial electric field changes the te dependence and as we will see now changes the te dependence drastically so an important fact to be aware of is that the transport due to helically trapped particles is not ambipolar so it's not the same for electrons and ions it's not ambipolar and this means a radial electric field is created by the plasma just due to the fact that this transport mechanism is not ambipolar and an example a possible case let's assume the electrons are in the one over new regime the ions are however in the plateau regime since the particle species can have different collisionalities due to having different temperatures and dense temperatures basically then we have strong electron losses and this these strong electron losses leads to the plasma charging up positively so the plasma charges up positively positively and this scenario such a scenario is called electron root so if we have strong electron losses in the beginning and the plasma charges up positively to contact these electron losses by a radial electric field this example is called the electron root and however in a typical fusion device usually the opposite is to the case meaning usually it's an it's the ion root case which we have so two important expressions electron root and ion root okay let's have a look at the deflection or at how this works with a radial electric field so on the right hand side you see a drawing which sketches um the linear um stellarator here so we have a helically trapped particle which bounces around like this and usually it would get lost quickly like what I have drawn here with the yellow color now however due to the radial electric field which is created by the initial fast losses we then get an e cross b drift and that e cross b drift leads to turning this around so this one leads to this is a drift in this direction so here's an r missing so this is the e radial cross b drift and this means that the radial electric field becomes important if it is strong enough to turn the helically trapped particles around politely so or to say it in a different way you remember that trapped collision less that is important to be aware of that trapped collision less particles so helically particles are lost directly oops directly so helically trapped collision less particles are lost directly however radial electric field can help to deflect them so they are deflected by a radial electric field okay so let's try to finally get a diffusion coefficient for that so we get the displacement so the step size the displacement from the drift velocity and the poloidal transit time the poloidal transit time which we estimate or assume or approximate with the er cross b drift the poloidal transit time due to the er cross b drift and the transit time omega e cross b or the frequency in this case of course is 2 pi for a full circumference then over tau e cross b the 2 pi cancels out because we also have a 2 pi in the tau and then we get er over rb and then for the step size l just use the drift velocity over omega e cross b and then these two expressions allow us to get the diffusion coefficient for a stellarator in the new regime and you will understand and I guess already guess why it's called new regime by now and you will understand this in a minute the diffusion coefficient d new then can be approximated with as I said we use here stepping size the drift velocity over omega e cross b and we need to square this then times the collision time sorry and then as a step size we use the collision times electron ion then this can be approximated with by the by epsilon squared and v squared so the drift velocity the charge of the electron the electric field the position squared times 1 over 8 times tau electron ions meaning this is proportional to nu now not to 1 over nu and this is why it's called the new regime and if we now express or insert for the collision frequency again sorry for the collision time the appropriate expression and replace the velocity by the energy then well inserted properly of course then we find that d new now scales with the square root of the electron temperature so it's a much weaker scaling for electrons of course okay in this video we talked about neoclassical transport in stellarators so we looked at helically trapped particles and we found that helically trapped particles can lead to very strong losses initially and we would get a diffusion coefficient which scales to the power of which scales with the electron temperature to the power of seven half which would be a very strong and unfortunate scaling for a fusion device however since the transport due to helically trapped particles is not ambipolar the radial electric field is created by the plasma such that the antitransport gets ambipolar and we introduced the two corresponding expressions electron root and ion root and that's it for this video and hopefully I'll see you in the next video