 Hello everybody and welcome to video number 15 of the free online version of the future research lecture My name is Alph and this is a YouTube channel called Der Plasma and today is Sunday We are in chapter 3 parameter limits for fusion plasmas and in the last video as you might remember We had a look into the different types of instabilities occurring in the fusion plasma limiting its performance and how we actually analyze Which of some initial perturbations reside in unstable situations and which initial perturbations are simply stabilized The technique we will use is the energy sorry the energy principle as explained last time And in this video we will start to look at the first type of instabilities and these are the current driven instabilities so in this video We will have a first look at current driven instabilities Just as a reminder we will not derive all the Instability criterion in greater detail. This would require a lecture on its own We rather have a brief look on them and trying to understand the underlying physical principles So current driven instabilities are driven by a gradient in the plasma current Which should be no surprise So these instabilities are driven by a gradient in the plasma current and We will hear for now neglect the plasma pressure and Since the plasma current plays an important role These type of instabilities are mostly important in Tokamaks As Usually in cellarators and classical cellarators. We don't have large currents occurring there The change in potential energy is Due to a redistribution of the current profile. So the change of potential energy data you is due to a Distribution of the plasma current profile and this then leads to a change In the magnetic energy Sorry and We will first only look at ideal MHD modes Meaning we will have no break up of the magnetic surfaces So there will be no break up of The magnetic surfaces Which is the same as saying that we will have no magnetic islands occurring We only allow for deformations of those surfaces Only deformations and this These type this type of modes is referred to kink modes This is what we usually refer to as kink modes Okay, let's start with an example of an M equals one mode M equals one mode Here you see three drawings and these are poloidal cross sections on the left hand side This is an unperturbed situation. So this is an unperturbed Situation you can see various various flux surfaces on the middle. This is an ideal M equals one mode It's an ideal M equals one mode and you can see how Here's one flux surface which is perturbed and you can see it's only Deformed that's only shifted to the right nothing else is going on and on the right hand side We have a resistive M equals one mode and resistive means that here we have reconnection going on leading here in this case to a massive Island magnetic island And by the way, if you're interested in learning some German Insel is the German word for island Okay You should all know or might all know the kink instability from your plasma physics one lecture But to be sure that we're all on the same page here Let's make a brief reminder on the kink instability kink instability And this is easiest illustrated with some drawing. So let's draw a plasma column See like this Plasma column some kind of cylinder and then in this cylinder we have a Current flowing. I think for the current Used the color green current flowing along the z direction so into this direction the plasma current then we have Magnetic field lines going around it with according to the right hand rule. I think we use blue for that So going around like this These are magnetic field. This is a magnetic field line then another one Here's another one and so on And the shape of this is given by the equilibrium condition where we have the plasma pressure gradient being stabilized by j cross B forces Yeah This is the equilibrium condition. Now, let's assume we have an initial perturbation somewhere So let's say maybe something like this In our cylinder some initial perturbation Then magnetic field lines at the bottom still look like this going around but at the perturbation They are somewhat tilted Maybe looking like this so these are our magnetic field lines Here again like this And now if we have a look at this box this side the left-hand side compare it with the right-hand side Then we can see that the magnetic field Strength on the left-hand side is higher as the magnetic field lines are closer there. So this here corresponds to a higher Magnetic field strength Comparing with this side where we have a lower magnetic field strength and The j cross B forces now push the initial perturbation further outwards Meaning that the next time this step might look like this It exaggerated maybe maybe Magnetic field lines now be like this going around here like this Like this Like this like this and so on so the initial perturbation is growing it is enhanced and probably Here at the bottom This would maybe go a bit more But not be such a strong King baby something like this Anyway, so having an initial perturbation is growing if we have such a Geometry and this is the King instability leading to an immediate Disruption or break up of the Z pinch which we discussed at the very beginning of the lecture and how to stabilize this and this is stabilized by adding Strong Axial field adding a strong magnetic field along the Z axis corresponding to the Toroid direction and the Toroid experiment to the Toroid direction to stabilize it against small wave lengths perturbations to stabilize it against small wavelength perturbations corresponding to large mode numbers and This King instability Although we learned we can stabilize it with such a strong extra magnetic field to something which still plays a role in a talk about Guess we will see this is why we will first look at the external King instability So we will first look at the external King instability because this is the one which is the way more important stability the external King instability and for the external King instability we start with an perturbation at the plasma boundary and This these perturbations tend to be unstable if they have a long Waif length so a large wavelength or a low mode number so long Waif length Deformation at the plasma boundary so psi at the plasma boundary a different from zero having a long wavelength corresponding to a low mode number and this tends to be Unstable Compared to short wavelength perturbation which would require locally more bending of the feed lines That's not forces. So these are easier to stabilize. This is why the long wavelength deformation of perturbation tends to be unstable and The strong currents the strong plasma current can lead to such unstable situations flowing in a plasma in a tocamac so the strong currents flowing in a plasma can lead to such unstable situations Because with a strong current We also get a pool oil magnetic field this gets larger than And having a larger pool oil magnetic field corresponds to having a small safety factor small qs corresponds to the small qs This is why we call this is a safety factor because the safety factor should be Large and then we have a more stable plasma So a larger number here for the safety factor for qs generally resides in the more stable plasma and remember qs The safety factor was the number of two royal transits for a field line required for a full whole loyal transit Okay, this is why This is small if we have a larger pool oil magnetic field component Now for the description of the external King instability. We neglect the stabilizing effect of the walls so we neglect the stabilizing effect of The vessel wall Or we can say in other words that the wall is Far away from the plasma which is usually not too wrong and then the stability analysis yields that the Resonant Qs values the values Which are resonant with the perturbation m over n Located at the plasma boundary at the plasma boundary Well for the instability remember, this is how we described and the mode which is unstable The safety factor is resonant with the perturbation m over n with the perturbation mode numbers stability and Keep in mind that the safety factor in the tokamak increases with the radius Qs increases with Radius, so this is not a constant number And from this we can then derive a condition necessary for stability so necessary for stability then in the linear tokamak is basically that the Safety factor At the edge Qs over a is larger then the Mode numbers m over n and the safety factor at the edge is or can be calculated or described as 2 pi a squared over mu not are not and then times the Toro magnetic field to be phi Divided by the plasma current so this is the Criterion which has to be fulfilled to get a stable situation That the safety factor at the edge is larger than the ratio of the mode numbers and m over n. Sorry now an Unstable kink mode an unstable external kink mode a few words about it. Oops. Sorry an unstable external kink mode those are likely to occur when There are strong gradients in the plasma pressure profile Sorry, the plasma current profile element, which is not too surprising because strong gradients Trigger instabilities right if you have a strong rating you can easily imagine that it's more likely for instability to occur and These are those strong gradients Up here appear for example in the ramp up face of a discharge when we ramp up the plasma current And let's have a look at such an example Here at the top you can see time traces. First of all, we have a time trace of the plasma current being ramped up and Since The plasma current is ramped up We get a stronger pole loyal magnetic field. That's the safety factor Decreases this is a safety factor at the edge QA it decreases over time And during this decrease it passes several m over n values during the ramp up face And here on the bottom are depicted various Scenarios assuming n equals 1 then m equals 9 in the beginning so here we have 9 a mode number of 9 the very beginning of the discharge and Then we have m equals 8. So you have an 8-fold symmetry. So 1 2 3 4 5 6 7 Yeah 8 1 2 3 4 8 yes, and then afterwards is m equals 7 then m equals 6 And with a further decrease of the face safety factor, it's an m equals 4 and finally m equals 2 here And one has to pay attention that the discharge does not stay long in this Situations otherwise the whole system will get unstable and of special importance is the external King mode With m equals 1 so the external kink With m equals 1 this one is special one is special and this is Unstable or because this is unstable if now inserting what we had two three slides ago the equation if Now m is 1 so if 1 over n If the safety factor at the edges smaller than 1 over n the safety factor at the edges often abbreviated with Q a and Remember this is proportional to the inverse of the plasma current as I had on the board on the slides on the board three slides ago Thus here we get an upper limit for the plasma current so this results in an or corresponds to an upper limit For the total Plasma current and now if we have a look at the n equals 1 situation This results in a special limit and this limit is called the Kruskal-Shafranoff Limit, this is the Kruskal-Shafranoff limit. This is for n equals 1 thus Qs Safety factor sorry at the edge has to be larger than one for stability and this Corresponds to a Limit on the plasma current then so we can just rearrange the equation and getting a limit for the plasma current IP which has to be smaller than 2 pi a squared V phi over Mu naught R naught then times an additional factor of one half So this is a limitation For the plasma current and the factor one half oops, this was basically experimentally found so it was Experimentally found that the plasma is also unstable against the To one mode so against the m equals to n equals one mode Which means that the safety factor qs At the edge has to be larger than two For stability and this has to be larger than two oops for Stability Now what does this mean for a typical experiment? So let's have a look at Aztec's upgrade Medium-sized Tocca mark located at the Max Planck Institute in Garching at the IPP Garching So Aztec's upgrade Gets or it results for Aztec's upgrade having a maximum again, well for Aztec's upgrade Sorry, it resides in a maximum current of one Mac amp one mega amp here and The actual limit is however a bit higher. So it's actually not one mega amp, but one point four mega amp. This is due to Having non Circular cross Sections as we will just discuss in future lecture Because everything the Krusgeshafran of limit as it is written up above there seems to have a circular cross section of a plasma Now a question that might arises now What happens if we pass that limit if we don't care about the limit then the plasma will very likely disrupt And what is a disruption? So what is a Disruption now in a disruption the plasma Typically hits the top or bottom vessel wall. So the plasma typically hits the top or bottom Vessel wall and then the plasma energy is lost in a very short amount of time actually a few milliseconds the plasma energy is Lost in a few Millie Seconds Resulting in a power loss on the order of gigawatts and Since the plasma energy is lost in a few milliseconds It also means that the plasma current decays in a very short amount of time. So the plasma current Decays a short amount of time and that is usually in on the order of 10 milliseconds and This means that we induce so-called halo currents surrounding the original plasma current. So we induce so-called Halo Currents because Changing current the time it uses a magnetic field which then again induces the current. These are the halo currents surrounding The last close flux surface Which is usually abbreviated as ACF which is short for last close flux surface And these halo currents then flow Along open field lines that it is outside of the last close flux surface along field lines along sorry open field lines and That means they will eventually intersect with the vessel and deposit energy and Particles on the vessel and then we also induce Well, they also get any currents Induced in the vacuum vessel Induced in vacuum vessel Due to the change of current in the plasma and the associated forces are really large They can be on the order of hundreds of tons so the associated Associated forces And you have the order of hundreds of Tons which is enough to lift up your vacuum vessel by a very short amount a short distance So this is something you really want to avoid a large scale experiment And then due to the Release of magnetic energy part of that energy gets transferred to a relativistic relativistic electron beam so part of the Magnetic energy so initially part of the plasma current actually part of the plasma current gets transferred to a Relativistic Electron beam meaning we have an electron beam consisting of electrons which have relativistic energies And this is really a threat to the surrounding vessel of those electron those electrons hit it this is a threat to the surrounding Vacuum vessel and and or components attached to it Now how to handle this disruption one way which is Usually done nowadays is to do disruption Prediction with the neural networks during disruption disruption prediction with neural networks Which of course needs some kind of training at the training then is Are a lot of disruptions which actually happened at small scale to come up with night might not be that harmful for the machine Then you extrapolate that knowledge to larger scale to come up Or you do a lot of maybe trying to make small disruption events Using that for a disruption prediction Network based on neural networks, and then if this network for example would detect during the operation a disruption will occur then to Mitigate the disruption so for mitigation What is done is? We have a massive Really massive gas injection of Some kind of noble gas trying to terminate the discharge Before and radiate all the energy away before the disruption occurs Okay, so on this in today's lecture we started to look at current driven Instabilities, and we only started to look at kink modes. We had a brief reminder on the kink instability Which you know from plasma physics one lecture We used to that knowledge and talked about the external kink instability and in particular the Kruskal-Schafranoff limit is an important Limitation resulting from the external kink instability because it imposes limitation on the plasma current and it is important to keep in mind that that's the safety factor at the edge Qs has to be larger than two for stability And on this last slide which is here still I can still be seen we talked about the disruptions What are disruptions and how to handle disruptions and? Just keep in mind a disruption is something you want to avoid at all cost at large scale experiments like eta You really don't want to have a major disruption not a single one at eta going on Okay, that's it for today's video hope to see you again in the next video