 Hello everybody and welcome to video number 29 of the online version of the fusion research lecture We are in chapter 5 collisional transport at the last video We talked about stellarator specific elements for neoclassical transport and about the diffusion coefficient in a stellarator And in this video, we will just briefly talk a little bit about the mb polar electric field so The mb oops, sorry that should be a bit thicker Maybe like that The mb polar electric field topic of this video So to give a mathematically rigorous and correct derivation of the transport coefficients We need as I initially said a kinetic description so to be Mathematically Correct the kinetic description is required to derive the full particle flux and thus the transport and If one would do such a full treatment One would get for the particle flux an expression like the following So are for radial and then electron and iron and the density Then the diffusion coefficient As we had derived it. So this is the diffusion coefficient for the transfer processes we discussed Then the density gradient over the density So this is a radial derivative then the charge number of the The Where particle species Over the charge for the electron where this e looks a bit weird like that then the radial electric field over the temperature of electrons or ions Then The term including the thermo diffusion So here's sorry a diffusion coefficient for the thermo diffusion for electron or ion Meaning we have an expression here where we have the electron or ion temperature gradient the radial derivative of it Electron ion and finally expression where we have a diffusion due to the wear pinch meaning the Toroid electric field Is included here So as I said This basically corresponds to the diffusion coefficient we discussed over the last videos This is due to the diffusion coefficient this expression here is due to the this is called the thermo diffusion due to the Temperature gradient, which is the course of that This one here is as we discussed the wear pinch This is the wear pinch Okay, and the last videos, uh, you hopefully have realized that radial fluxes So radial transport processes radial fluxes are not necessarily ambipolar are not necessarily ambipolar especially when it comes to The transport of helically trapped particles if you see we've seen that these are not Not ambipolar to start with The ambipolarity which we finally lead is achieved by a radial electric field. So the ambipolarity is achieved by the ambipolar electric field And we get that um from the equations which we just had by setting The electron flux to be the same as the ion flux and The resulting electric field to get that is called the ambipolar electric field the ambipolar radial electric field now the calculation of The diffusion coefficient for the electron and ion flux in a fully kinetic description Is actually quite complex So the calculation of the diffusion coefficient in the expressions for the particle flux for electrons and ions in a fully kinetic description Is quite complex And this is why there are numerical codes for that and the famous one one example is the decast code Which stands for drift kinetic Equation solver Now let's finally have a look at an example. So here you see decast calculations for w 7 as So this is um w 7 as where we have assumed density and Ion temperature profiles to be constant. So not changing This is one radial position. So this is a fixed radial position But the electron temperature has varied. So the electron temperature has been varied from figure to figure So on the left hand side, it's an electron temperature of 500 e v in the center 2 k e v and on the right most figure 4 k e v Now what do we see here? Let's have a look. Let's first have a look at the ions so The ions correspond here to the dashed lines. So here you see the ions The radial transport profile of the ions Which of course is the same in every figure because the ion temperature does not change And you see there is a central peak and the central peak Is due to the direct losses Of helically trapped particles and Of helically trapped particles And we always have some collisionless particles in the maxwellian tail, which are directly lost if they are helically trapped Then there are secondary peaks so left and right secondary peaks and The secondary peaks are due to the two royal resonance Now you might wonder why there are Two secondary peaks because the electric field can point in either inwards or outwards So it's the sign of the electric field which leads to two peaks here and then there is The the solution the radial transport as a function of the wave of the radial electric field for the electrons shown which is the Solid line Which is the solid line the solid black line. So this one here in each of the pictures and as the Electron temperature has been varied this profile changes from graph to graph However, the general structure for the electrons is similar to the ions Similar to ions, but it has a different scale And what often can be neglected is the two royal resonance this Rarely this plays in principle no role for the electrons to a royal resonance Can be Neglected Okay, if we look again at this drawing we see there are these graphs We see there are intersection points here for example or here or here Here's an intersection point and somewhere on the right. Maybe somewhere here If we extend this would be an intersection point as well and the intersection points of both curves so the intersection points of both curves These are the stable solutions we have these are the stable solutions For the MB polar radial electric field So only if we are have a radial electric field of this order then we have a stable solution. Okay so there are A few comments for the stable solutions so we get a more Decreasing radial electric field if we have dominant ion flux As indicated on the left hand side, you see the stable solution is in more on to the negative Radial electric fields. So we get further decreasing radial electric fields for dominant Ion fluxes Sorry, sorry For dominant ion fluxes And this is the region which we were already introduced as ion route or regime which we introduce as iron route and we get an increasing Radial electric field if we have a dominant electron flux and this regime Was called was introduced in the last video as being called the electron route Okay, now looking at the examples here in the first case. We have the electron temperature If we have an electron temperature of 500 e v corresponding to the iron temperature And in this scenario, we have an iron route regime And this is basically the normal case for fusion plasma. So this is a normal case for a fusion plasma if we have decreased electron density and strong Electron heating Then we can get higher electron fluxes initially than iron fluxes And the increased losses would be mostly due to collision less transport So helically losses of helical triparticles And the two examples shown above from the middle and on the right on the very right hand side first We have an electron temperature of 4 k e v So 4,000 electron volt And there we are in the electron route Regime and we have a very large positive positive Radial electric field needed for getting finally ambipolar transport And in the the interesting case is maybe more in the center where we have two k e v 2000 electron volt Because there are multiple routes possible As you can see in the picture multiple routes are possible So there are more than one stable There's more than one stable solution and what actually can happen the sudden jumps Between or sudden jumps of the radioelectric field is possible So jumping from one stable solution to the other stable solution This is actually what also happens in the experiment. What is observed in the experiment these sudden? Sorry these sudden jumps up deserved not Too much of a difficult word Okay, that was just a final example of electron and ion transport in Shown at the w7as Delirator and that's basically it for the video where I just wanted to show you or make a few comments about the ambipolar electric field and show an example of w7as And that there are multiple solutions sometimes possible for getting a stable plasma a stable situation And that the radioelectric field can jump between these kind of solutions That's it for the video. Hope to see you in the next