 Hello everybody and welcome to video number 24 of the online version of the fusion research lecture We are in chapter 4 particle trajectories and in the last video we talked about Particle trajectories in a stellarator field and in this video We will talk about the influence of a rage electric field on the trajectories both in talker mark and stellarator so we will talk about the influence of Radial Electric field now a Radial electric field and you might not be surprised by that Induces drifts on a flux surface You know you have the E cross B drift so every time when you have an electric field and a magnetic field Which we always have a magnetic field in a toroidal confinement experiment this electric field induces drift So a radial electric field now Induces Drifts on the flux surface and Radial electric field can for example be created as we will see by fast losses of electrons as compared to ions so if you have initially more electron losses than ions and a Radial electric field builds up and we will discuss that in one of the next videos But for now we just discuss the influence of such a radio electric field so if We have to write a symmetry like in a talker mark so add toroidal symmetry Then the toroidal drift component is not of interest then the toroidal Drift component is not of interest however There is also a poor loyal drift component There's also a poor loyal drift Component as we have Tilted magnetic field as you know, so there's a poor little drift component and this can induce or rather influence The particle trajectories Particle trajectories Strongly that's what it's the Poor loyal drift component. We are looking at D for drift and then theta the poor loyal component That is given by the radial electric field Over the B phi so the toroidal magnetic field Okay Let's try To explain that with the drawing so again, we try to draw a poor loyal cross section Example like this and then elongated into this direction Longated into this direction Then here we have the center somewhere Then going down here is the That's the radio coordinate and then we have Particles sitting somewhere here Which moves along a magnetic? Oh well, if the corresponding magnetic field line then looks like this Right going around the torus here So this is B the magnetic field line and then the particle here has the Parallel component Oops a parallel velocity component The parallel Which is as you can see hopefully along the magnetic field line and It has also assuming now we have a radio electric field So assuming now we have a radio electric field. So something which points somewhere to the center So this continues here So this is a radio electric field And now e cross B leads to a Drift component Which is in this Coordinated system pointing downwards Has a drift component pointing downwards and of course it also has a Poor loyal velocity component pointing upwards. So if we were to Project this this area here As sorry to this yeah to this area here to this poor loyal cross-section then we have Velocity component here pointing upwards of the particle itself So this is v theta the particles velocity component and then We also have a velocity Opponent has sorry a vector pointing downwards and this is the drift of the induced Poloidal drift component of the e cross B Drift from the radio electric field Yes, so as a reminder u d theta is given by the radial electric field over the Poloidal sorry the toroid magnetic field and as you might have already guessed a critical critical situation arises if the drift the Poloid drift component Cancels out the thermal Motion component the thermal the particles thermal motion component v theta That would correspond to v theta being v parallel over Be theta over be By the parallel so this is just rearranged the equation where you can write v theta over v parallel is equal to be theta over v Parallel and be parallel is v phi so here we have then be Getting them this equation and again as a reminder be theta is much smaller than v phi Just as a reminder. Yeah, so this is our typical talk amok situation and now the Drift induced velocity is given above oops. Sorry is given above here and if the thermal particle motion is now Given by that velocity component by er Over be phi which is roughly be this situation is then called Toroidal resonance. This is what we call Toroidal resonance and for that case the resonant radiated electric field It's just by rearranging the equation v theta times Sorry v parallel times v theta Which then is approximately? epsilon b phi Yota bar v parallel so this is the Toroidal resonance of the electric field is strong enough such that the Resciting Poloidal component of the drift velocity cancels out the thermal Poloidal Component where the thermal sorry the polar component of the thermal motion of the particle, okay now um The resonant This Toroidal resonance the electrical Resulting or corresponding field eras is critical for The particle confinement is critical for the particle confinement and that is both in Stellarator and Tokamak both in Stellarator And Tokamak Because now the particles Do not Make or no longer make the full Poloidal turn They're basically staying at one polo angle because the their poloidal Velocity component with they initially had is cancelled out by this rate electric field induced drift velocity and this means that the grad B induced drift or We are the diameter to drift as we discussed it is no longer Canceled out. It's not compensated which is usually achieved by the twist of magnetic field, right and This means we get kind of open orbits These are referred often to as open orbits because they end up at the vessel wall So we get open orbits similar Where they are not close to the orbits open orbits similar to The helically trapped particles to helically trapped particles This means Direct particle Losses can occur. So this is something to be aware of but This is not the end of the story. So but the Poloidal drift component can have beneficial influence it can have beneficial influence on Helically trapped particles so in a stellar a town helically trapped Particles Occurring in the stellar as you know Can be in that sense beneficial for them helping the particles that they escape sort of escape Their poloidal trapping condition their Poloidal Trapping condition and we will discuss that in more detail in one of the next videos So for now, let's look at the basic requirement for that. So the requirement is that the lost time that the lost time due to the grad B drift is larger than the Poloidal transit time and The Poloidal transit time due to due to the Poloidal drift component Yeah, I'll cross be drift Which we previously have written as UDC and In if we want to write that in a bit more formal way the lost time due to the grad B drift Can be estimated by the minor radius a Over the residing or the corresponding drift velocity and this should be larger than 2 pi a so one poloidal circumference and Then B over er so over the resulting drift velocity in the inverse of it And now inserting for the drift velocity the approximation And we have here the temperature Rather the energy as you know q r0 B we insert that and We find the expression that the radial electric field has to be larger than 2 pi The energy of the particle the temperature Over q r0 and if this is fulfilled If this is fulfilled then Helically trapped Particles are Not lost Are not lost. Yeah, so this is a very important Fact for a stellarator Having said that let's now have a look at a stellarator So this is W7x color coded again the magnetic field strength So let's have a look at this video so the magnetic field is kind of unfolded here We only look at one period And we will see now one particle which is helically trapped But instead of leaving the confinement region this helically trapped particle just bounces around so the Resulting radially electric. Sorry radio electric field in the Stellarator and W7x leads to a Turning around of the particle which is not lost So this is a strong enhancement in the performance or lead to a strong enhancement in the performance So here you can see it so this is was a simulation for W7x Illustrating how to reduce or illustrating that we can reduce the loss of Helically trapped Particles by a Radial electric field and this is then also called an optimized stellarator because these Losses are strongly reduced which we had in the original stellarator design Okay, that's it for this video where we talked about the influence of a radial electric field We will talk about more how this feed is created in one of the next videos For now, you should have learned that there are electric field of course can lead to drifts and this drift Can be important if the drift velocity The polo drift velocity component is the same as the polo thermal motion component Then we speak of a toroidal resonance partless can get lost then in a stellarator However, the radio electric field can also be beneficial because it can help to reduce the losses of helically trapped particles Illustrated in this movie here by for example turning the motion of the particles basically poloedly around so that they are no longer lost That's it for this video. Hope to see you in the next video