 Hello everybody and welcome to video number 23 of the online version of the fusion research lecture We are in chapter four particle trajectories and as you might remember in the last video We basically finished with discussing the Trajectories in a tockermak field. We finished with talking about the bootstrap current or by talking about the bootstrap current and the wear pinch and Today in this video, we will have a look at this delirator. So we will talk about particle trajectories in a stellerator field particle trajectories in a Stellerator field Now as I already said a few lectures ago Important is the structure of the magnetic field on the flux surface. Yeah, so this is important to be aware of the structure of The magnetic field on the flux surface is important in a tockermak The magnetic field was toroidally homogeneous. It was an axi-symmetric device There was only a poloidal variation of the magnetic field in a tockermak the magnetic field B Was toroidally or is toroidally? homogeneous it is an axi-symmetric device and there was only a poloidal variation Of the magnetic field Where the strength scales with 1 over r basically in a stellerator now, however Everything is more complex as you know, there is no general symmetry coordinate no general Symmetry coordinate and as we will see this means that particles can get lost due to drifts alone Something different as compared to a tockermak. It means particles Can get lost due to drifts alone due to drifts alone Now let's first have a look at the structure of the oops, sorry of the magnetic field and The tockermak in comparison. So let's say we have a coordinate system Which might look for example like this Where the horizontal coordinate is a Direction along a magnetic field line. So here we following a magnetic field line and the vertical coordinate system is the absolute value of the magnetic field strength now How does the magnetic field strength on a field line looks if we follow a tockermak? It has basically a sinusoidal shape. So let me now try to draw a sinusoidal shape Might look for example Well Roughly like this. That's so I guess you get it. So this is a tockermak having a Sinus Sinusoidal shape roughly and now as we discussed Particles can get trapped in This magnetic well, this isn't a magnetic mirror basically where we can have trapped particles These were the banana particles we discussed Now in a stellar radar, there's an additional harmonic of the magnetic field springs. So if we So we also have the toroidal variation, but in addition as I said, there's an additional Harmonic of Overlay to it super positioned Which might look for example like this. So this corresponds to the stellar radar magnetic field strength along a magnetic field line the stellar radar and We can get additional trapping in between these Magnetic welds basically so we have an additional Classification an additional class of particles which can get trapped here in these small We call this ripple in the small ripple here as we will see Okay, as an example here is shown the Magnetic field strength of a stellar radar. So shown this here. This is an example An example. This is the magnetic field strength magnetic Field or rather these are magnetic field contours of a torsachon of a torsachon and To be more precise of the TJK Stelerator the TJK torsachon which is located in at the University of Stuttgart in the lab and The coordinated systems are such that the vertical coordinated system is the poloidal angle and the toe sorry the I guess I said wrong the horizontal Coordinated here is the poloidal angle and the vertical coordinated corresponds to the toroidal angle. So it's again unfolding a talker Toroidal experiment if you want and as you might see there is a certain symmetry and in the Toroidal direction, you have a symmetry of one two three four five six here and in this direction I'm sorry a symmetry of one so the The the the mode numbers of this Delirad are L equals one and M equals six and The color code in this plot works such that brighter colors are stronger magnetic field strength So these areas have a higher magnetic field strength than this area see a for example and The magnetic field strength is higher when the flux surfaces are closer to the coils so in general magnetic field strength is larger near the coils and the you know that the coils turn around helically and the helical turn of the field Maximum which you can see here in this plot Basically reflects in the coil structure. It reflects the coil structure and The coil and this experiments winds around six times the torus helically and Thus here you can see a six-fold symmetry a six-fold symmetry now I Said that or introduced this magnetic field a contour plot because we will use it as an example To look at how the orbits in this Delirad are look like so we will classify the orbits into different Particles into different categories. So the classification of Orbits So first of all we of course also have passing particles. So we also have passing Particles assuming or if the parallel velocity is large enough and passing particles in the plot on the right-hand side Might look for example if we follow one magnetic field line then this Passing particles it might look for example something like well, maybe something like this And like this so This one corresponds to The magnetic field line well, let's Maybe So this corresponds to a Magnetic Field line for the case of a value of Yota bar equal to 1 over 3 and Thus it roughly also corresponds to passing particle assuming that the particle stays on average on the magnetic field line On on the on the flux surface as you know, this is not the case. It locally deviates from it but still on average it stays on on the flux surface and This is so this blue line here Corresponds to a magnetic field line or to a passing particle. Yeah, so Okay, what else now we can have trapped particles oops What was that? Can have also of course trapped particles and we can have these particles being trapped between Proximate neighboring helical maxima trapped particles between Proximate which is just the same as neighboring Helical maxima Oops, this should be an a sorry maxima and In the plot on the right-hand side this would correspond since again brighter or lighter colors Meets higher magnetic field strength. So trapping between helical maxima would correspond for example to a particle Which travels around like this and is Trapped in this area there And these particles are called Helically trapped particles these particles are called Helically trapped Particles which are very important as we will see These are helically trapped particles corresponding to This one here now a few words on helically trapped particles So Helically trapped Particles Just to indicate that we are talking about helically trapped particles now those Particles are restricted in the movement or are rather confined to only one side of the torros being at the top or the bottom Where they are bouncing between the helical maxima. So they helically trapped particles are Restricted to to being trapped there or confined to one side of the torros for example top or bottom top or bottom and This has some important consequences because in contrast to passing particles which are moving around the whole torros in contrast to passing particles the trapped particles their Vertical drift due to the gradient drift or the diamagnetic drift, however, you want to Derive it. This is not Kansas. This is not cancel out. This is not compensated which is usually Achieved by the twist of the magnetic field. So They are vertical The vertical drift Just for example. Oops due to the grubbed be drift Cannot Be Compensated or does not cancel out and this means That the particles can escape the confinement region. They can get lost particles can escape The confinement region and the lost time can be estimated from the Drifts, so the lost time can be estimated from first. Let's indicate. This is an estimation From the minor radius and then the vertical drift velocity Which reads then q a Rb over mv squared there Oops where the D is the vertical Drift velocity as you might have guessed and For a stellarator with an electron temperature of five kilo electron volt This means We have a lost time of a few hundred micro seconds. So a very short Lost time. So this means that a stellarator the conventional stellarator has poor performance. Yeah, so the stellarator performed Very bad in the 1960s because it was not initially clear that these helically trapped parties lead to additional losses So that they are there. So it's a poor stellarator performance in the 1960s And This is why The interest shifted to tokamak with that. I'm also as you remember. I told you about the breakthrough experiments of the Soviet Union physicists of the T3 tokamak where they achieved the record temperature of electrons and ions and As I said, this is why the stellarator Why there was only marginal interest in stellarators Only very few countries kept working at stellarators Germany and Japan basically being those two countries Now there is an additional Orbit type of orbits and these are particles Particle orbits particles oops Particle orbits With a full Toroidal Turn between the reflections of the full Toroidal turn Between the reflections so this would correspond to particles for example Let's say if we start here and then it goes around like this and then so be reflected here and Then going further down to here and then being reflected here So these particles have some characteristics of features of banana particles so these particles have some characteristics of Or these orbits have characteristics of banana orbits from the tokamak, but their reflection Will happen at different positions Due to the lack of axis symmetry, which we do not have here in a stellarator Happened at different positions because as you might remember also from the banana particles as a toroidal recession I'm sorry a pre-session and in a stellarator without the lack of where we do not have an axis symmetric Device this leads to reflections at slightly different positions Okay, and then there's another class of orbits so D because Helically trapped Particles can be trapped also in the toroidal ripple Not only in the poloidal one. So helically trapped particles Can be trapped also in the Toroidal ripple via the poloidal drift component in the poloidal drift component and these particles are called Well, they are these orbits are called super banana orbits Because these are basically very fat bananas. So with a very Large banana width and this is why they're called super banana orbits and if we Were to draw these particles Then this might for example be a particle which bounces around Like this So following here the magnetic field strength like this and then it goes Continues here. So it also travels around toroidally as you can see Much more than just a helically trapped particle would do now as I said these particles are called super banana particles also they have super banana orbits due to their shape and What happens is basically they have a very long duration duration Of stay in regions of the at the reflection points and You know from banana particles that the longer they stay there the Thicker the banana so the larger the banana width becomes Because this results in a large radial displacement of the particle as compared to the flux surface Where the particle originally started so a large radial displacement displacement and This leads to Enlarged Banana radii and arch banana Radiuses and this is why they are sometimes In the very few publications you also find the expression fat bananas for that fat banana Or with fat bananas Okay, let's now have a look at an example So I have a video for that Oops, you can probably not really see that So I have a video for that. So this Video shows first of all color coded the magnetic field strength in the Vega Stellarator and you can see a particle which has started somewhere You can also see the gyro motion in this video of the party you can see the particle Let's start and let's start at the very end or the particle is first Reflected and going around to roid Lee. So this is a particle which is Following is long to royal part then it gets trapped between a helical maxima and then it's lost immediately Yeah, so this is a particle which is first trapped in a large toroid ripple and then transits into a helical ripple and being immediately lost there This is the same Graphic again graphic again. This is from the Vega Stellarator which was Operated for quite a while at the IPP grives wide and what you have seen in this video was the transition from Toroily trapped particle from a Toroily trapped particle to a Helically trapped particle and the resulting immediate loss of that and You could see that the particle Gets lost so the simulation basically stopped where the particle get lost and being lost was defined here as hitting the vessel wall Okay, I have another example another video No, not this one What is it here? Okay, so Again color coded magnetic field strength Then we have three particles all starting at the exact same position, but with a different pitch angle meaning the Ratio between the parallel and the perpendicular velocity varies and Here you can see three these three particles The red one immediately transits or goes into a helically trapped particle and then it's lost to the blue one is a passing particle and The yellow one is moving in the other direction against the magnetic field and you can see how this is trapped so this has a similar shape if you would look at the poor Lloyde Projection like a banana particle in a talk on mark. Okay So what you have seen there was a video a simulation from an L equals two and equals five Stelerator and These were three particles Which started Three particles which started at The same position but with different Pitch angles, but with different Pitch angles Okay That's it for this video. We have talked about particle trajectories in the stelerator field We have hopefully learned that in addition to a talk on mark where we only have the Toroil ripple Resulting in trapped particles In the stelerator, we also have a helically ripple which results in a number of few possible orbits possible trapped particles We have looked at these different type of particles classified them and have seen that Hedically trapped particles can lead to very fast losses in a stelerator Which leads to very poor performance in normal and classical Stelerators or torsatrons as it was the case in the 1960s in the next videos We will see how to compensate for that and why modern stelerators reach similar confinement times as modern talk on marks Okay, that's it hope to see you in the next video