 Hello everybody and welcome to video number nine of the free online version of the fusion research lecture We are in chapter two magnetic field configuration and you might remember that in the last video we talked about the existence of flux surfaces and In this video, we will talk about something which you might have seen already towards the end of the last video And these are magnetic islands so to start with here are two photographies of magnetic of the flux surface structure and W7x on the left hand side. So this is W7x and on the right hand side we have W7as the predecessor of W7x and To start with on the right hand side, maybe this is a classical Poincare plot on the left hand side This is a Poincare plot Overlaid with a with an image of the field line made visible Having some residual gas inside of the plasma and then inserting an electron beam or applying an electron beam to it And you can see now again Back to W7x on the left hand side one flux surface and on this flux surface You see at the very edge a few structures appearing there and these are magnetic islands. So these are structures and encapsulated in the background configuration here and here and here as well and on the right hand side You can even see that maybe in a few cases here For example in this case you can even see that within these structures. They are Encapsulated flux surfaces as well So these magnetic islands are basically nested flux surfaces and encapsulated in the background configuration Okay, so let's write down a few facts about magnetic islands magnetic Islands magnetic islands are as I said nested Flux surfaces Encapsulated Encapsulated in the unperturbed background configuration in the unperturbed background configuration now Magnetic islands can be bad for the confinement Magnetic islands can be bad for the confinement Why is that the case? Now to understand that let's have a look back at The picture on the left hand side. So if we just look on this island here So at this island Here this one then we see that this via this island There are regions connected which would usually be radially Separated because these islands are also spent by a magnetic field line meaning that this position here is connected via this Position due to the magnetic field line going around the torus. So it's There's a radially shortcut if you want from here to here and this of course can lead to enhanced transport losses So it's bad for the confinement because we have the radio We have radio transport Via the parallel mobility remember that the parallel mobility so parallel We firstly to the direction along the magnetic field of course Is very high whereas perpendicular to it. It would be very slow and now having the radio transport via the parallel mobility Can be something very bad Now flux surfaces with a Rational qs are sensitive to perturbations Due to resonant defects and these perturbations are magnetic islands now in general flux surfaces flux surfaces with a rational with a rational safety factor qs for example one or two or three over two or whatever those flux surfaces are sensitive to perturbations due to perturbations Basically, this is a due to resonance effects So an initial perturbation can easily grow on on these flux surfaces and And then can reside in a magnetic island so magnetic islands Magnetic islands are more likely to appear the magnetic islands can appear on these flux surfaces Where the safety factor qs has a rational value Magnetic islands are thus also characterized by the mode number So magnetic islands are characterized characterized by The motor number m which is the pole oil motor number and the pole oil mode number and and n Being the tall royal mode number The tall royal mode number so the periodicity basically into each of these directions and Since I said the islands are more likely to appear at rational values of the safety factor at flux surfaces Where we have a rational value of the safety factor then m over n is equal to Qs at these flux surfaces. So this is an important thing we need to keep in mind Now, let's have a look at an example. So here you can see Oops, sorry You can see a magnetic field line So here let's first look at the magnetic field line which goes around the torus like this and it continues here Like this continues along the oops, sorry roughly right along the toroid direction now We have one toroid circumference So it continues here, of course, then goes along like this like this Like this and then it finishes its pole oil circumference as well Meaning that here this field line has a safety factor of of oops, sorry a safety factor of Qs equals to 2 So we needed two tall royal circumferences one pole oil circumference And then you see these four pole oil cross sections. So 1 2 3 4 right and on each of these cross sections you can see there are two islands appearing here So we have an m equals to Magnetic island structure here. So we have m equals to Magnetic Island structure appearing here as an example Now as another example here, we have unrolled the flux surface So here we have kind of unrolled the flux surface What do I mean by that? So if you have a close look you can see that Here the more or less horizontal coordinate r times phi. This is the toe royal direction so this is the toe royal direction unrolled the toe royal direction and The vertical coordinate you see our theta though. This is the pole oil direction To which we have unrolled at it Then you can see we have three islands appearing here. So one two Three a periodicity of three a mode number of three meaning we have here an m over n equal to Three and then the periodicity into the toe royal direction is only one So it's a three one magnetic island, which we have here the three one magnetic island And here you can again nicely see the structure the structure of the Flux surfaces inside of the island. So you have closed flux surfaces inside of the magnetic island as well Okay, let's have a closer look at this structure by making sorry by making a drawing So let's try to draw on magnetic island with an appropriate coordinate system Something like this and then Here into this direction So here we have the radial coordinate and then here the vertical coordinate This is size which corresponds roughly to the pole oil direction, but is more generally referred to a magnetic coordinate To a magnetic coordinate Then in the center of the island We have a magnetic axis as well and around it the first flux surface and then another one and then another one and Then another one where the next island kind of connects For example like this, then we have a few distinguished points here. So this one and And this one These are the x points. So this is the x point of the island Here's another x point Then in the center, this is called the o point of the island. So the magnetic axis of the island is called the o point and then the last Flux surface of the island if you want this is called the separatrix So here we have the separatrix. He has its own separatrix the island Which is basically this flux surface here Yeah, so this one Okay the island now how is the island generated so to understand how the Magnetic island is generated. Let's draw next to it the Oops, sorry, let's draw next to it Required magnetic perturbation which we need to generate the island because the island is generated by perturbation in the background magnetic field So now we have Again the coordinate Xi and now we have a radial magnetic field perturbation here. So and because in general Magnetic islands Generated by A periodic periodic radial perturbation Of the magnetic field a radial magnetic field perturbation How can this look like? So let's first look where we have the characteristic points. So this is the x point Here we have the o point This is the the next x point And then the perturbation in the magnetic field Might look for example like this. So some kind of a sinusoidal which is periodic perturbation where we have here a Negative one we are being smaller than zero here a positive one we are being larger than zero and This as you can probably easy imagine easily imagine can be caused by a Current filament along the field line so this can be caused by a Current filament by a current filament along Well, let's write the proper sentence here. So this Can be caused by Current filament along The rational field line along the corresponding rational we assume you have along the rational field line Basically in the o point That's how we generate or how magnetic islands are generated now an important parameter To correct us islands is the width of the island. So the width is this Size here. So this one is defined defined as Sorry the width of the island w and A few words to say about the island width the island width now By calculating the magnetic fluxes it can be shown which we are not doing here that the width of the magnetic islands can be Approximated by the following expression four times and then our nut squared times br where are not is the coordinate of the resonant surface the coordinate of the Resonant surface so back to the formula for the denominator then have M the motor number b theta the pole oil magnetic field and the shear At the coordinate and then the square root of that expression now looking at that equation we can deduce a few things from that first of all the island size increases with The arms of the perturbation of the radio magnetic field not surprising Maybe more interesting is that it decreases With a pole oil mode number m decreases with a pole oil mode number n and in addition it decreases with Magnetic shear s and that is interesting, right? So Just to be explicit here shear Limits the island width shear limits the island width and remember when we talked about Potential benefit of magnetic shear by introducing it in the beginning of chapter 2 here you can see the potential benefit of Magnetic shear which can limit the width of magnetic islands magnetic islands typically rotate in a plasma so magnetic islands can rotate and Can rotate and the frequency is on the order of 10 kilohertz Okay, now few words about courses and consequences of magnetic islands. So Let's talk about courses and Consequences Consequences of magnetic islands magnetic islands so Let's focus on tokamaks for now Because you know that tokamaks are generated by large current flowing in the plasma So it's maybe more likely that islands play a more important role there as we will see they play a part role in Stellarators as well, but let's first look at tokamaks So in tokamaks You can easily imagine as well that there might be some kind of initial perturbation Some initial perturbation in the current profile Just due to for example a fluctuation of the electron temperature variation of the electron temperature changing resistivity Which changes the current profile? Now a perturbation in the current profile as we have just discussed can generate magnetic islands and generate magnetic islands Now magnetic islands as I have also tried to explain at the beginning of this video can lead to increased Radial transport can lead to increased radial transport and This can lead to a local flattening of the profiles can lead to a local flattening of profiles and If we just look at the temperature profile The flattening of a temperature profile flattening of the temperature profile This changes resistivity This changes resistivity and This enhances so further enhances the initial perturbation enhances perturbation and Then we have something which is a t-ring mode something we will discuss in the next chapter But it is important enough to mention it already here a t-ring mode which is a type of an instability a T-ring mode and This instability as we will see Can be very dangerous Because it can lead potentially potentially to disruptions and Disruption is something which we want to avoid at all costs in a large scale experiment because the disruption Basically means the sudden loss of confinement It's the sudden loss of confinement Within a very short amount of time and if you imagine the sudden loss of confinement of Mega amp of current circulating in the plasma Then it immediately induces mirror currents in the vessel wall in the metallic structure the metal structure surrounding the plasma That of course can induce a lot of mechanical stress And this is what we want to avoid in large scale experiments And don't worry. We will talk about disruptions in more detail in one of the next videos Now let's look at an example So here is illustrated is so-called Ntm So-called Ntm which is a neoclassical tiering mode in Aztec's upgrade And Aztec's upgrade. This is a a neoclassical tiering mode is a certain type of instability Just called neoclassical because there is the so-called bootstrap current involved and these are all things which we will Discuss in the future videos, but I just wanted to show it here because I think it's a nice example So first of all, we have the polar cross-section on the left-hand side where we have an M equals 3 island structure. So 1 2 3 and then you can see that over the On the right-hand side is a plot which shows two electron temperature profiles The black filled circles are for the unperturbed case without magnetic islands the open Circles are for the case with the island and you can see that across the island So this is the width. This is across the width of the island the temperature profile Flattens and then the overall temperature profile is reduced due to the island being present there now a few last words so in Tokamaks and Stellarators islands are caused by external can be caused by external error fields for example due to Some metal structure surrounding the experiment. So in Tokamaks and Stellarators and Tokamaks and Stellarators external error fields External error fields From some kind of metal structures surrounding the experiment can cause Magnetic islands oops can cause magnetic islands So before you install something at your experiment you have to be aware of how this Installation interacts with a magnetic field of the Tokamak or Stellarator and be sure that you do not create unwanted islands Now in Stellarators in addition in Stellarators in addition Magnetic islands can naturally occur They can naturally occur from the vacuum configuration from the vacuum configuration and They can be actually beneficial for in Stellarators for example. There is the concept of a magnetic Island diverter Remember when I talked about diverters being important basically as an exhaust For the plasma somebody once said it's like a vacuum cleaner for the plasma getting out the impurities And if you have now in a Stellarator a magnetic island Around the structure so at the very edge sorry around the plasma So at the very edge of the plasma and then if the Outermost side of the island ends at the diverter plate This is a very can be a very efficient way to direct your impurities to a diverter structure and this was first explored in W7AS and is also explored in W7X Just as an example That islands can also be very beneficial in Stellarators for example Okay, that's it for this video where we talked about magnetic islands which are nested flux surfaces encapsulated in the unperturbed background configuration these are generated by radial periodic Distribution of the sorry perturbation of the background magnetic field which can be caused for example by a current filament Along the field line. We talked about the island width how this can be decreased with increasing magnetic shear and I also Told you that magnetic island can lead to a disruption Which is something which we want to avoid or have to avoid at all coast and large-scale experiments like ETA I also told you that islands can however be beneficial in Stellarators for example This is the concept of a magnetic island a diverter being explored in W7X which is actually helpful to Direct the unwanted impurities in the plasma towards the buffer plates into the diverter structures That's it hope to see you in the next video