 Hello everybody and welcome to video number 14 of the free online version of the future research lecture You might remember that we started in the last video with chapter 3 parameter limits for fusion plasmas And this is where we will continue this time So we will first outline the general principle how to analyze the stability and What we do is we analyze stability limits by applying the so-called energy principle so we will Analyze We're analyzing the the stability Limits by Applying the energy principle The energy Principle I will explain that in a minute, so we have yes the ability limits and energy principle What does that mean so at first we will take an equilibrium and Perturbed that or deform it. So we will deform an equilibrium equilibrium by small perturbation and We use the Letter psi for that probably the one which is the hardest to write and this is definitely wrong This is better and Second we will then calculate the change in potential energy due to that deformation So the second step is to calculate the change in potential Energy Due to that perturbation now, let's try to illustrate that a little bit. So Let's assume we have a situation like this Where we have here our potential energy with the capital U of X then he is basically coordinate X and then we have potential like this and then inside of the potential we have our starting position here and Now when we move the That into negative X direction oops, sorry Into negative X direction or positive X direction. We are moving upwards of this potential wall meaning this the potential The potential energy changes and it's getting larger and this means we have a stable Situation a stable equilibrium now if on the other hand side It looks As follows again spatial coordinate X X or potential and now the spatial distribution of the potential might look like oops like this and If we have our initial situation sitting here on top and then perturbing this one so we would move into a negative X direction would going downwards into this direction positive perturbation into this direction then the change in potential energy is negative and This is an unstable situation and a change in potential energy Which is negative means the kinetic energy increases so it is a conversion of potential energy to kinetic energy Corresponding to an unstable situation Now what do we take as equilibrium since we have here a magnetic confinement situation? We take the flux surfaces as equilibrium so we take the flux surfaces as equilibrium as equilibrium surfaces Which we perturb Then what we do is we expand the perturbation of the deformation into modes to make life a bit easier so we expand the deformation or perturbation Into modes and what to do I mean by that let's quickly draw our equilibrium situation So it might look for example like this supposed to be a poor loyal cross section now let's Copy that a few times since we need a few examples like this and if we now We can perturb it multiple ways. I said we want to expand the deformation of perturbation into modes and modes are characterized by motor numbers and an M equals one perturbation Would mean just shifting the whole flux surface So this would be an M oops, sorry This would correspond to an M equals one perturbation So this M is here the poor loyal mode number since we're having a poor loyal cross section This is a poor loyal mode Corresponding to a poor loyal mode number in M equals to perturbation would look for example Like this so we have a two-fold Symmetry basically of the perturbation So roughly right? I guess you get the idea. So this would be an M equals two perturbation M equals three perturbation Would correspond to something like This roughly This would be an M equals three perturbation So you get the idea of that Now let's assume we have a linear Tokamak, so a screw pinch meaning the geometry would be something like What we have if this would be our Tokamak Expanded into this direction Oops into this direction then we would here have the length of Two pi capital R not linear Tokamak. Sorry for this kind of Very tight space here This direction here so going from here Into this direction This is the z direction then we would have Let's try to draw this this small R So this one is a small R and then we need an angle. This is the one. Sorry, which goes down here And this is the angle theta And then we can free a decompose Almost as such that the perturbation psi being a function of R theta and Z can be written as The sum Over the mode numbers M and N And we have an amplitude term Mn being a function of R and then the exponential Which has in the argument I times M times theta plus N times Z over R not so a Fourier decomposition of The modes something very handy and very useful because it allows to make the calculation of the actual Equilibrium and the perturbation of that and the resulting change in the magnetic energy much easier so What you need to do then is you need to analyze you need to analyze the change in potential energy Delta u for each mode separately for each mode and As I said, it's you have an unstable situation if the change in potential energy is negative and When you do that you find that the most unstable modes occur if the mode numbers are resonant with the flux surfaces, so the most unstable modes occur if The mode Numbers Mn are resonant With the flux surfaces so Where the modes appear? Yeah, so the most unstable modes occur if the mode numbers where these modes appear are resonant with the flux surfaces and This means that M or being resonant with the flux surface Surface means that M over N. So the mode numbers are equal to the safety factor or That the inverse of that N over M is equal to Yota bar and As a as an expression how to properly say that so if you have a deformation of Um If you have a deformation which is unstable, sorry if you have a deformation which is unstable against Against mode numbers M and N you call that an unstable an unstable and then you use brackets M comma N mode So this is meant by saying you have an unstable Mn mode And What I have written Here basically is a equivalent to say that Qs so the Ratio of M over N if you have an unstable mode there that this has to be rational for the most unstable modes So this is a very general property very general finding that if The mode numbers the ratio of the mode numbers is equal to the safety factor being a rational number Then you get you find the most unstable modes and remember the safety factor is not constant It is a function of the radius It means along its profile along its function of the radius. You certainly have positions Where you have a rational number for Qs and thus these are prone to instabilities of a certain mode number combination Okay, a few words on how to characterize unstable modes, so the correct characterization of Unstable modes characterization of unstable modes So first of all we can distinguish between different driving mechanisms, so first of all we have the drive and This is where it is always some kind of gradient But there are two types of gradient in the plasma one is to the pressure So these are then pressure driven modes or Of course, this is mostly a talker muck topic current driven modes Because only in a talker muck we have the large currents And then current driven modes can be further distinguished or categorized so current Driven modes Can be further distinguished and What we can say about them is that they depend on the resistivity resistivity and First of all we have so-called ideal modes We have ideal modes to start with an ideal modes means that we have no resistivity no resistivity or Well, perfect conductivity Perfect conductivity Which means basically we have an ideal mhd situation And that means that the flux surfaces are only deformed so flux surfaces are only Deformed They are not broken apart remember from maybe plasma physics one lecture that in such situation the magnetic field is frozen in the plasma So an ideal mhd now We can have ideal modes, but we can also have resistive modes resistive resistive modes and You might be surprised by that but resistive modes mean that the resistivity is finite now You might have guessed that probably and this means that That reconnection is possible that reconnection of the magnetic field lines Reconnection of field lines Is possible That means magnetic islands can be formed That allows for magnetic Island formation Now what happens during? Magnetic field line reconnection maybe just a few words on that So if we have the following situation so to start with we have Two field lines for example one field line going like that Then another field line going like that So they are close but Separated then there is a strong plasma flow from the top. Let's say pushing downwards Then there is another plasma flow here pushing upwards. So one two You one you two So we have a situation where the field lines do not interact with each other due to the strong plasma flow They are pushed Together such that the situation might be for example now like this. So they are very close to each other so the We have an area here where they are very close and if this continues if you have finite resistivity then Reconnection can finally happen and Then the field lines will look in this example this one will go like this and Here we will have one going like this so the field lines are basically reorganized here and Magnetic energy here is converted into kinetic energy. So this is a process thought to be responsible for acceleration of cosmic particles for example, so if we think about MHD phenomena in space So this is a very interesting topic and still a subject of very active research. So this is about reconnection here Sorry about resistivity and reconnection Then there is a third Current driven mode which we will not talk about in the lecture, but we definitely should mention it here and these are wall boats and Here the resistivity of the vacuum vessel wall plays a role. So here the resistivity of the vacuum Vessel wall is taken into account and What happens here is that currents can be induced in the metal Vessel wall and this can affect plasma dynamics. So currents induced in the metal can affect the plasma dynamics and You can for example, if you think of a metal wall of an ideal wall with a with a negligible resistivity being very close to the plasma Then the currents induced in that ball can actually help to stabilize growing modes inside the plasma Due to mirror currents being induced in the wall This is something which is researched in the reverse field pitch in powder for example also medicine and a very interesting concept And for burning fusion plasma However, you might have the problem that the walls cannot be so close to the plasma But it's definitely these are definitely very interesting experiments to study The effect of such modes and how to stabilize modes and something and we can learn definitely something from these experiments for ETA Okay So that's it basically for this lecture But I think I should maybe make a very brief reminder on ideal mhd since I said that a few times Just that we are all on the same page here So feel free to skip that last slide, but I guess I mean feel free to skip anywhere the whole lecture, but I Would be more happy if you don't do so, but of course Feel free to do so Just a brief reminder about ideal mhd so in ideal mhd we have a few initial assumptions You have a few initial assumptions so a We assume that the characteristic timescales we are looking at those are Larger than the iron cyclotron frequency cyclotron frequency or by the inverse of that The timescale given by the iron cyclotron frequency meaning that tau Typical timescale is larger than the inverse of the iron cyclotron frequency then B that typical length scales are Much larger than the iron gyro radii and the iron gyro Radius meaning that L for a typical length scale Much larger than the iron Lama radius row li then C We assume to have a collisional plasma Which means we have a maxwellian distribution Maxwellian This basically means that the length scales are much larger than the mean free path so the average distance between two collisions and Then D that we have no dissipation with no DC patient Meaning that the viscosity Is basically zero now Applying these assumptions allows us to make a few simplifications during calculations and A few remarks on ideal mhd because ideal mhd is something very useful So for example the ideal instabilities which we find within this Within these assumptions within these limitations are generally the most fastest and most violent modes So it's definitely already very important to know just these modes so the so-called ideal Instabilities Or modes so modes or instabilities described with an ideal mhd they are Generally the most fastest and Violent Modes and if we have an ideally unstable plasma Such a plasma won't last long and ideally unstable plasma won't last long so we definitely have to take care that the plasma Where to start with is ideally stable now in addition The non ideal instabilities are often just some variations of the ideal instabilities So often so there are of course a few exceptions, but often the non ideal Instabilities insta bill it I think there are a few letters missing insta bill it Tease those are Often just some variations of Ideal instabilities and then Finally ideal mhd works generally in hot and collisionless plasmas. So fusion relevant plasmas ideal mhd Works and this means holes and can be used can be applied in hot And hot often means that basically it's a collision less plasma Hot plasmas if the Typical perpendicular length scale because this is what is important in a magnetic confinement experiment is larger than the iron Lama radius Yeah, so the perpendicular length scale Okay Finally, that's it for this video where we talked about the general stability or the general way how we analyze stability limits namely by applying the energy principle talked about that we Expand the deformation or perturbation into modes that we fully decompose them and I told you that the most unstable modes occur if the mode numbers mn being the polar lentor oil mode number Equal to the safety factor being a rational number. Yeah, so these are then the most unstable modes then we can Correctorize or distinguish the unstable modes usually into How they are driven so either pressure driven or current driven and then the current driven modes can be further distinguished or separated into Ideal modes where we do not have resistivity. So this is ideal mhd then resistive modes where we have resistivity Which means reconnection is possible and then wall modes and finally I gave here know of you about or remind about ideal mhd Okay, that's it for this video. Hope to see you in the next video