 Hello everybody and welcome to video number 22 of the online version of the fusion research lecture. We are still in chapter 4 and in the last video we learned about banana orbits and banana particles. And in this video we will talk about another important result partly which comes from these banana particles. And this is the bootstrap current. So finally we talk about the bootstrap current. So the bootstrap current is based on the gradient drift or the result of the gradient drift. And just as a reminder this is the drift and then graph B. This is minus the perpendicular energy over the charge. The perpendicular gradient of the magnetic field times the magnetic field over the magnetic field to the power of 3. And now we are looking at the minus grad. Let's use a different color for that. We are looking at the minus grad B times B theta so times the polar magnetic field direction component let's say. And we will see that this results in the toroidal precession of the particles. So let's try to explain again our typical coordinate system x in this direction the toroidal angle phi. This is the radial direction. We have the magnetic field pointing into the board. That's maybe a bit hard to see. So we have the magnetic field pointing into the board to the phi direction. Then we have the gradient of the magnetic field pointing to the left. Now let's try to draw a typical flux surface. Look for example like this. And then we also have where you are not have the plasma current pointing into the board as well like the magnetic field. This is the direction of the plasma current theta going into this direction. And what I have drawn here corresponds are supposed to be to illustrate a flux surface. And then we have a typical banana orbit which can look for example like this right so roughly like this. And then we have and now I use blue for this since we are looking at the result of the gradient drift looking at the polar magnetic field component. So the polar magnetic field points in this direction downwards. So this is the polar magnetic field to be theta points here into this direction here into this direction. So it goes around into this direction into this direction here like this oops sorry like this. And finally here like this. Ok, I forgot to label this. This red line should correspond to a banana orbit. And now let's look at the grad B times the polar magnetic field component. If you now make the cross product the grad B direction pointing to the left as indicated by the arrow on the bottom left. Then times the polar magnetic field component we end up with the direction of this drift pointing here into the board. Look at the minus sign in front of the drift. On the other side it points out of the board where the polar magnetic field pointing is pointing upwards. Since here it is still pointing upwards but it is a bit weaker pointing upwards. Here it is zero at the top and at the bottom. It is pointing into the board again pointing into the board. Thus the grad B times B theta drift changes sign towards the high field side direction. So this is important to realize the grad B times B theta drift changes sign towards the high field side. Changes sign at r zero. So here at r zero it changes sign. Now this has of course consequences. It means that if we look first at let's look just at ions so at positively charged particles we will get a drift into the plus phi direction. This is a unit vector to phi direction. If we are at radial positions larger than the major radius of the plasmas. So everything on this side here for this we will get a positive into positive phi direction at drift. We will get a drift in the other direction into negative phi direction for radial positions which are smaller than r naught. And of course well not of course but it's the opposite for electrons as we have a charge dependence in the drift. Now if you look up here you see there's a charge dependence. And this means that there is a well if you well not this but if you look at the banana orbit in the poloidal projection. You see that there is a net drift into the plus phi direction because the particles spend much more time on that part of the poloidal cross section. So there's a plus drift into the plus positive phi direction. This is the dominant part. And from this the net current parallel to the plasma current can rise. And this is the bootstrap current. Okay so the bootstrap current is a safe generated current and I will explain this picture in a minute. The bootstrap current is a safe generated current which amplifies the plasma current. So it flows in the same direction and since it is safe generated by the plasma this is called a bootstrap current. And the bootstrap in that sense that refers to somebody who pulls himself up by his own or her own bootstraps. And the German there's a German legend about not somebody who is using his own bootstraps but somebody and his name is Baron Münchhausen. So this refers to Münchhausen who is able to drag himself out of the swamp. So this is a swamp where his horse is stuck in. So this is a swamp where his horse is stuck in and then he just drags at his hair. He drags himself out of the swamp. So this is basically the same effect which is of course as you know not possible in contrast to the bootstrap current which is a physical effect observed in plasma dynamics in the Chokambak. So let's write down a quantitative estimation for the bootstrap current. So a quantitative estimation for the bootstrap current. This is J and then Bs for bootstrap minus then we have the square root of epsilon the inverse aspect ratio. One over the pole loyal magnetic field B theta and then the pressure gradient the radial pressure gradient dp dr. And what actually happens is the bootstrap current that is not the trap particles alone which drive the current but it's the trapped particles which transfer momentum via collisions to passing particles, to passing particles which are then carrying the current, the toroidal net current. And since as I said what happens here is that the plasma creates its own pole loyal magnetic field confining itself its own pole loyal magnetic field. And this is why it's called a bootstrap current. Now maximizing the bootstrap current as you can easily imagine maximize the bootstrap current in Tokamax allows for longer discharges and is therefore something which is of big importance. So maximizing the bootstrap current in Tokamax allows for longer discharges for longer discharge durations let's say for longer discharge durations. And for an economic device for economic steady state operation this is of vital importance. So this is of vital importance for an economic steady state operation of a Tokamax as it reduces the need for an externally driven current. Installerators, there is a counter contribution, there's a counter contribution from helically trapped particles and we will discuss that in the next or definitely in the next lectures in one of the next videos. There's a counter distribution from helically trapped particles which goes in the opposite direction and W7X for example is actually optimized for a minimum bootstrap current such that these contributions cancel out such that there's no additional bootstrap current which modifies a magnetic field further. Here you can see a time trace of an Aztec upgrade discharge so this is Aztec upgrade from a series of experiments performed by Andreas Bock and in different colors you can see the different contributions to the overall plasma current. So in blue we have ECCD which is heating due to electron cyclotron resonance heating then we have neutral beam injection in red then we have in gray the ohmic current and then in yellow the bootstrap current. And you can see that there's an area here for example where the majority of the plasma is not provided by ohmic heating so not by an externally induced current but actually by the sum of the bootstrap current and the neutral beam injection and then partly of the microwave driven current. So this is a very promising result because this was an experiment at reactor relevant conditions as an upgrade experiment at say reactor relevant parameters or condition scenarios I should say at reactor relevant scenarios. And this shows that we are ready to run these type of experiments in ETA with a clever setup where we do not can operate long discharges if we would only rely on the ohmic current so we need additional current in the bootstrap current the combination of the bootstrap current the neutral beam current and in addition the ECCD the microwave current is the way to go here. So these were very promising experimental results from two years ago. Okay there's another important interesting effect. This is the so-called wear pinch. The wear pinch is a result of the fact that in a tokamak the plasma current IP is driven by a toroidal electric field E5 which is of course due to the transformer effect. Which is induced. Let's again try to explain that we are drawing. So this is X. This is the phi direction. And we have here our typical tokamak coordinate system. This is R. Oops sorry. This is the R direction. This is the electric field direction which drives the plasma current. So let's now draw the typical cross section looking for example like this. Plasma current pointing into this direction. So let's now again draw the unperturbed banana. Unperturbed banana orbit for example might look like this. So this would correspond to an unperturbed banana orbit. And now the toroidal electric field results in an acceleration if the particle moves along the magnetic field. So parallel to B the particle will be accelerated by that. And this means that the reflection of a trapped particle will happen at smaller radias so deeper in the plasma. So let's try to illustrate that. If we start somewhere maybe now here for example and the particle moves and instead of being reflected here it moves a bit further like here. Then it moves upwards again against the magnetic field and now it moves upwards against the magnetic field. And since it's decelerated moving against the magnetic field since it moves against the electrical field the decelerate as the reflection happens at larger radias. Then it moves down again maybe something like this up again something like this down again and so on. So let's continue that. We said parallel to B the particle is accelerated the reflection happens deeper where deeper here means at smaller radias. Whereas on the other hand side if a particle moves antiparallel to B so against B so against the electric field it will be decelerated. That's a deceleration and that means that the reflection happens earlier or to say the same thing is at larger radii. And the result is a tilt as I have tried to indicate it here of the banana orbits. A tilt of the banana orbits and a net inward drift and this drift is called the wear pinch named after the physicist wear. Wear pinch. Now let's try to estimate this with a few numbers here. So the change of the banana with between two reflections this quantity can be estimated. It's right now data because it changes the banana with changes can be estimated by minus e phi over V theta times the transit time times the square root of epsilon. So this corresponds to the displacement of particles per transit time. And from that there is a negative sign so these patterns are displaced into values of smaller radii. An inward velocity resides. An inward directed velocity resides where this can be approximated then by data B. Over the transit time corresponding to minus e phi over B theta times the square root of epsilon. And note that this is the same for ions and electrons. So the wear pinch is an inward directed movement of electrons and ions as a result from the toroidal electric field which drives the plasma current induced by the transformer coils. Ok so we introduced in this video the concept of the bootstrap current which is a very important current in tokamaks as this will be the current which drives most of the plasma current in large scale tokamak devices together with the NBI the new tube beam injection. And it allows to extend the operation of a discharge and it is of vital importance for an economical point of view as this allows to make long discharge. It's only this bootstrap current which is safe generated by the plasma this is why it's called bootstrap current. Only this current allows to make the steady state operation which you need for a reactor. Then finally we talked about the wear pinch and I've also shown you by the way experimental results indicating or illustrating that the bootstrap current is something which we can very well control already in a tokamak. I showed in the last two slides briefly introduced the wear pinch and inwards directed drift of electrons and ions in a tokamak as a result of the toroidal electric field which is induced by ramping up or down a voltage in the primary circuit and the plasma current. That's it for video number 22 hope to see you in the next video.