 Okay, so it's this one that we need to do. Okay, so last, where we ended up last time, let's see, okay, so, so I finished last lecture by concluding that for this, this type of turbulence, the dissipation can actually eat at large scales in all in all a basic similar fields to where the energy is injected. So, this next section I'll talk about another way to view that phenomenon. So the turbulence here is driven by some unstable eigen mode. This is called for this is going to be high on temperature gradient driven mode. I believe that you've heard about this last week from MJ. And some of you here in the room I believe are also. This example is studying instabilities like this. Now, for a long time. It was just assumed that this unstable eigen mode drives the turbulence, and then the cascade gift energy to smaller sales and case base. But since we're dealing with a high dimensional system. So there is this other dimension in base space that that can be represented by different items. In the last lecture, we represented this other dimension base base in terms of per meter polynomials. But you can also think of it in a different basis. One is per meter polynomials. One is different eigen modes when your operator. So that's what I'm going to talk about for this part of the lecture is conceptualizing the turbulence as an unstable driven by unstable eigen mode. And then driving a cascade in case space but simultaneously exciting sub dominant eigen modes at the same weight number as the ability. All right, so that's kind of abstract, I think, but maybe it'll become more clear with some concrete examples. One way to look at this is using a technique called proper photography composition. That's this is a term that's been used in fluid turbulence quite a bit. Other names for basically the same thing singular value decomposition is the major C composition. So a linear algebra technique. That's universal. I mean, universally useful in almost every area where one uses linear algebra, I'm sure many of you are familiar with it from various applications. But anyway, this sort of technique is widely used in various fields. It's useful because it's an optimal representative that produces an optimal representation. So what I mean by that is, if you decompose a matrix you can't be composed of matrix any matrix. You can take a matrix A, I, B, you can decompose it in terms of two sets of basis factors. So the other product, the other product between sets of basic factors. So, and then these are weighted by what's called a singular value. So this decomposition you can show. You can prove it's optimal in the sense that if you truncate the decomposition so instead of taking all n terms of this series, if you take some are less than so the, the, so this one term in the series, it captures more of the more of the information than any other possible decomposition as long as you can think of it as simply extracting an optimal basis set for whatever, whatever your interest in it. Okay, so if we look at this even more, it's a more concrete examples. This one I'm showing here is some of these basic factors from orthogonal decomposition. The one on the upper left, this is the unstable Eigen mode. And what this is actually showing is the Eigen structure of the electrostatic potential. So if you think of a Pokemon, big paroidal structure, then the what I'm showing here is the electrostatic potential from this unstable Eigen mode. It peaks here on the outside of the Tokamak and then it follows the field lines around like this. And as it gets to the inside it gets small. And there are reasons why I expect MJ come to this last big reason why it's, it's peaked here at the outboard side of Tokamak. So that's what I'm showing here is this unstable ITG mode. Now, from the turbulence, from a nonlinear simulation of the turbulence, we can extract, construct one of these proper orthogonal decomposition. I'm taking a single wave vector and performing this matrix decomposition. And these are the other structures that come out of this proper orthogonal decomposition, either just a few of them. This is, it looks like they've got, yeah, these are the first two, number one, number two, and then these are successively higher order ones to the competition. So that's this part of the decomposition. This is one, this is the basic vectors that cover the spatial, the spatial information. The other basis vectors cover the time information. So this tells you the spatial, the base space structure, and this tells you the time behavior of each one of these basic vectors. So I just have two examples here. The lower order vectors have kind of slower behavior and time and you can go to higher number becomes more, you know, faster and faster time. So the third part of this decomposition is the singular value and this tells you how large, how much energy is in each one of these fluctuations. And so here you can see them, the singular values. The virtue of this decomposition is that it's optimal, which means that you'd expect it to compete very strongly in this lower rate. That's what makes it an efficient decomposition. So this unstable, I demo right here is much larger than the others, but the others aren't negligible. This is number two right here. And you go down and they get smaller and smaller. So this is, this is the energy in each one of these eigenvectors. Okay, so just as an example, this is showing how these eigenvectors, how these singular vectors. So we're going to reconstruct this complex fluctuation data. So we have just one term in the series right here, you can see that it doesn't match very well, the, the actual fluctuation nonlinear fluctuation are the wiggly black line, and the, the most sense, or the first singular vector of this red one. So then if we add two of these, you can see it starts to become a little better and kind of evokes the same sort of general shape. So if you go up to 15, it's almost everything to distance me if you would look here and then on top of 100, it's almost a perfect reconstruction. Okay, so another example, lots of lots of the singular vectors. Now if you look at just the singular values, they take quite quickly. But if you look at the dissipation that's these. That's the dissipation from each one of these. It actually grows quite quickly with these with the, the vote number reason why because you're getting your as as the vote number increases these have finer and finer sales finer sales are what great participation. So, to summarize, the energy is contained in just a few of these large scale structures, but since they get smaller and smaller sales. The dissipation actually is distributed among a large number of these structures. And if you look at this in case space. So this is, if you think back to our first figure for Kolmogorov turbulence. This is basically the same picture here. The energy injection is here and large scales. And the Kolmogorov turbulence, the dissipation can't, can't happen until you get very small scales, but for this system. Once again, because you have this extra dimension in the state that can give you this patient. The dissipation that is actually mostly in the same scale ranges of the energy drive. It does expand somewhat out with the smaller scales, but predominantly happen the same scale range of the energy injection. And as I said, in the last lecture, this is really convenient for us doing numerical simulations because we don't have to. We don't have to deal with all the decades of sales in order to resolve resolve the dissipation, and it makes it a much more trackable problem that might be otherwise. Okay, so we have this kind of strange situation with energy injection and dissipation occurring in similar scale ranges. I think that these that the scaling arguments for the energy sector might not be valid and the energy sector might change. And, okay, we review these are the three assumptions from the, that we use to derive the Kolmogorov five thirds block separation of driving the station. So I saw three and local energy transfer. So which one of these have we violated. Exactly. We violated this one. There's no separation of the scale of the inertial range in between where you have this conservative conservative transfer of energy. So I touched on these two. But I saw for feeding. It's not perfect. So there's extreme anisopathy in this case between the sales perils magnetic field and perfectly done, but in the sales perpendicular magnetic field, it's roughly isotropic. So we can keep this guy. The transfer is going to be transferred local or nominal. What I mean by that is the cascade shift energy abruptly to some other scale range or is it cascading from one locally from one ceiling. I haven't touched on that yet. But we did look at that and found that indeed for this kind of turbulence, the transfer is local. So what this is showing is for a given scale range this is is looking at the energy tax within shells and the perpendicular wave number space. And red means that the energy is leading from nonlinear from the nonlinearly from the sale and blue means that it's going to this scale so for this wave number here or this shell of wave numbers. The energy is predominantly coming from the shell of wave numbers, just larger, just smaller and go into the scale of wave numbers just smaller so indeed the, the, the transfer is largely local. And this is don't close with with Lema studies. And I'll show a picture of that a video that I wasn't able to get back ready for this lecture, but maybe I can just show quickly at the beginning of the next lecture. But the cascade and pay the cascade small scales occurs basically local. Okay, so there's two of these assumptions that are valid and the survey limits. But the separation of drive administration is not not the case for this type of terms. So you might suspect to get much more exotic power laws and we've seen in a lot of fluid scenarios and and the prediction from the Kolmogorov scaling. And indeed you do. This is just a few examples. I mean, really, all that drop in a real you know, token mag turbulence, say, not only you get this patient in in all this this range of scales across here, but you also have different types of disabilities that materialize a different scale. So I've talked about the ITG instability that that comes in right here. But there are other instabilities, you know, with other acronyms, traffic electron modes that come in here often, and then even smaller traffic electron will be kind of this sale range and even smaller. There's the cousin of the ITE mode the exact same physical for electrons instead of ions. So, and you can see that these these spectra basically follow the power laws but it's very unpredictable. What exactly power law they will follow. Now, we have identified if you kind of really idealize things and and construct the scenario a little bit artificial the scenario with really strong drive, then you can kind of find a scenario where you do get the Kolmogorov power law, but it's a little bit hard to do to do that. Okay, so we wanted to understand this behavior in more detail work on a more fundamental level. Um, we constructed a toy problem to look at what happens, how laws, the gas station hour laws, when you have this dissipation that is at some level, but kind of stands the whole range of your journal and cascade. So this, this paper. He was a graduate student when he did this work but then he came and worked with me and professors in Austin afterwards. So he. So that's a modified version of this equation that hermoto suggestions equation, kind of a simplified one D. Has anybody heard of this equation by the way. It's very well known to it, but it's kind of I am on citations, hundreds or thousands of sites really use as the, the model to understand laying, you know, fluctuations and playing so for example, the video actually at the beginning of the last lecture. I'm sure people have analyzed that sort of thing with the criminal though. So, okay, so this is just one dimensional. It has a nonlinear term, roughly similar, you know, I bet she's roughly similar to what you would see in the notes. And then to dissipation sort of terms. It's, although it's no one in the general community as the curve of citations equally than we at you feel I thought a lot from the heart of the equation. So. Yeah, so this was. Yeah, you actually, I mean, you can just tell us the history for three minutes slide. So, this this was the work of my unit. My first post office. And we were analyzing what was called that that that I know. And the idea is that I'm very fond of very appreciative of the world. But I believe that we try to post everything in the world or unless. Right, because the point is for the world, the situation from the distance, because of the period mission, and that is that if you know that you're giving me some time to take a few more minutes. And if you remember, so you have, let me just use some that. So you have the conductive very good data. All right, and some dissipation. And then post it. Right. And by definition, this whole scene is the only thing that was in the system. There is no instability in this situation. Right. So if you try to do some kind of a linear analysis, getting this one, all that you get is that this one. So you force it, but the important thing is that the dissipation takes place that is forced to take this a short wavelength and we see that. But the point is also, you know, some really low cap system. And as David has shown, let's suppose that we have been using the set of alpha, the one measure of this program, and this data. These are all physical terms, and that's further. And you knew we could do it. If you try to do a linear as a system, then I mean of course the clip. There is the conductive, which is really the most important part. So if you could write, you will find that in that is I is squared. So the spectrum is unstable at some range of is cut out at large game is cut out at short. Okay. And so what will happen is that then the energy is being so generated. It's not that you're forcing the oxide, there's an instability and that instability comes in the region. So if you look at this one. Okay. And the energy is in this way that this is greater than that being generated. And so there's a dissipation rain decide and dissipation rain on the outside. And in fact, the total amount of super initial rain, they want you to expect the power sector is not small. So, what we're going to do is a great guide, but in most of the physical problem, you will really have to figure out what exactly I look like as David is very nice. And this is another example of something very interesting. This paper was written. I think it was, I think it was promised to finally resuscitate. Yeah. It should be a really good example. You're a second paper. All right. So yes, I, I, we did this before swatting light and be before I even knew swatted. So we called it the curve of the citizenship equation unfortunately at this time. So we, for this purpose we modified it slightly. We added two terms. We added this. These two terms right here are the standard curve of citizenship. We have this term in the denominator. Can anybody tell what what happens now to this term as you go to large K, this whole term on the right, what's, what's the limit in large K. I keep you awake on right after a long two weeks. I'm just going to be. Yeah, I mean, if you were just this term, it would, it would blow up. But with this term, these two in large K limit these two terms, this one's negative or negative. And so then it just for it just goes some flat level, which is roughly what we're trying to do is if roughly mimic this situation that I showed in our kind of complex, you know, being a massive dark magnetic simulations all we're trying to do is roughly mimic this situation where the dissipation kind of flattens out and it's just a steady, but maybe small green from the energy throughout the whole ceiling. So this would be the standard. Locomodron equation, and we've modified it to go like this. And then we can modify this, this be term here to make this go up and down in comparison with the drive to the, to the spectrum as we do this. And what we find is that you can, this is really the complexity in this problem comes in here this is reworking the nonlinear transfer coming up with a simple model for the nonlinear transfer between scales. And once we identified this form of a reasonable form of this term you can solve this equation, and find that I didn't actually put it up here but get to derive the spectrum. And show that the spectrum goes like a case negative delta, where delta is this term right here lambda is some roughly speaking some nonlinear energy transfer time, and new or be either these terms from here. This is just basically the dissipation level that you hope that you select. And so what happens is this. There's a natural. If you didn't have this term here you would have some natural power law, but with this term here. And the spectrum stephens in proportion to this dissipation level, but at least like. And so we did a whole range, large range of simulation large range of scanning this parameter be this patient level. And this is the prediction in blue, and the nonlinear simulations are in red. And so you can see that the effective quite accurate model for the, for the spectrum. So in short, the, it's maybe somewhat surprising, I mean, going into this we kind of intuitively expected that maybe you add this station you get some exponential today, instead of being a power law but no you actually do retain a power law. So basically you have this kind of, you know, steady constipation, maintain a power law but see things as you have this patient. So roughly speaking we think this is a good, a good paradigm for gas seeds and power loss sector in this fusion terms. You can get any number of any kind of spectrum, depending on how you know what this patient level or what even energy injection level, you have standing to see a range. So that was the end, that was supposed to be the end of the. But I've got lots of other things for. For lecture number two, which I don't think I'll be able to get. But we'll do it till the end of time. Okay. Okay. Okay, so this is another one. Yes. Okay, so this is another. So we're going to look at multiple the role of multiple ID modes in the turbulence. So the big question about 10 to 15 years ago was what do magnetic fluctuations do and talk about for a long time electromagnetic effects are fairly challenging in these simulations. For a long time, we mostly did electrostatic simulations. So we just give all the electric potential, and neglected the magnetic fluctuations. This is a reasonable assumption for some, some parameter for genes and dogma max but not for all parameter when we when we started doing like a magnetic simulations, we saw a kind of surprising thing. We saw that for this type of turbulence driven by this ITG instability, you actually get a reasonably large amount of electromagnetic heat loss. What I mean by electromagnetic heat loss, these magnetic fluctuations produce magnetic stoke activity. So the, I think this one's got earlier, and then the heat can escape the confinement simply by flowing along these particular magnetic field lines, except going across field lines. And the reason why this was surprising is that the reason why I'm showing here is for the ITG mode, the electrostatic potential and the magnetic potential. The reason that this is surprising is because the structure of this I can mode is such that you perturb the magnetic field. When you're still working on the blackboard, you perturb the magnetic field when you're on the bottom side of the document. So you're getting some, some, you have some magnetic fluctuation, but then due to the symmetry of the mode that's canceled out when you go to the other side of the blackboard. So the assumption was that this sort of turbulence would not create any sort of magnetic stoke activity, magnetic chaos, and you wouldn't have any, you'd also not have any electromagnetic heat loss. But in fact, we do the simulations we did find a substantial amount of electromagnetic heat fluxes is one example of kind of low beta. And, and the, the total flux is very strange looking this is a spectrum in case space and it's like I missed a delay where this is, this is wave number here, and this is the spectrum of the heat loss. And so what was found is that there is positive heat flux at most of the spectrum, but then this strange dip right here where the energy keeps. And this was completely unexplained for, for years was observed, but they can explain. So we looked at this from the lens of this proper. And we wanted to see what other nonlinear structures were excited and one of them, I'm going to work side of the turbulence. And we found that indeed, the second largest structure is this, it has exactly the opposite parody of the ITG mode. The electrostatic potential has odd parity and the magnetic potential has even parity. And this sort of structuring exactly what you need to produce magnetic chaos and produce electromagnetic heat flux. So what we found is when we do this proper orthogonal decomposition, the total heat flux syndrome this weird term here in black is the superposition of the contribution from the ITG mode which actually has a negative contribution. And this second mode here, this, we call it terrarium parity, which has a positive contribution. And we can track the energy in both of these modes throughout the, the simulation. What we see is that the ITG mode here in blue grows up at the rate of the growth rate of the instability, and the second mode right here starts out it's not unstable so it's start linearly unstable so it doesn't do it for a while, and then non linearly, it couples to the ITG mode, and this gives it twice the growth rate of the ITG mode and eventually it grows up and saturates at some level. Okay, yeah, I have a movie of this. Okay, so this is the magnetic fluctuations. You can see that switches from one parent to the other, kind of short movie. You can see that switches from, from being odd parity to the ease parity and it's traveling back and forth all the time. And what I'm showing here is the superposition of these two modes. So just two of these structures combined, and the red curve is the superposition of these two structures the black curve is the total fluctuation. And you can see that these two contributions, combined to reproduce the non linear fluctuation throughout the simulation in both layers of some of these characteristics. So really, we should not think of the turbulence as being completely characterized by one stability but the addition of multiple fluctuations that are non linearly driven. That explains some, some kind of puzzling features of the simulations. Okay, so this is a visualization I think somebody was asking about magnetic chaos yesterday. This is one way to visualize it. So, so what these blocks are showing, if you take a. So if you pick a point anywhere in here. And then you follow the magnetic field from that point and go around the Torah, you know, dozens or hundreds of times, then each time. It's like a loyal cross section of the top of that. So you pick some point, and then you follow the magnetic field around the top of that. And when it goes through this process you make them adopt. And if you keep on doing this hundreds and hundreds of times a billion below a plot like this it's called a long array surface of section plot. And if the, if there were no magnetic fluctuations at all, if it were just the pristine magnetic flux surfaces that we try to produce in coconut, what you would see is a single color so we're labeling. This is a radial location like this with a single color. So what you would see is that there would be individual bands very well separated vertical bands like this with the same colors. And what you would see in the simulations is that the. So you can see, for example, blue, blue dots scattered throughout the whole clock, same with red, red dots scattered throughout the whole clock. What this means is that the steer lines are wandering wherever they want to eventually as we go around the courts. So what happens with the full magnetic fluctuation from the simulation, if we just take out the part from the unstable eigen mode, there's a little bit of wandering, but not too much. So, the blue band is really limited to this region here, red band is limited to this region here there's even regions without any, the field lines just don't get to. This is a demonstration that if you just consider the single unstable eigen mode, the, the field does not become chaotic, but when you consider multiple eigen modes, even then stable like the linearly stable I'm excited. The field does become stochastic and you get this electromagnetic flux. Alright, so this is another look at the. picture for turbulence. This is going back to this simple reduced genetic model that I described yesterday with the basis in velocity space. We're doing away with a lot of the complexities of token max so we're not looking at any of the curvature very simple flat system. We're not even looking at a sheer kind of field so kind of the simplest, simplest system that has some connection with the token plan. And we can look at the, what we want to do here is look in detail at the eigen modes, the leader of the system. And see how they are managed at all in the turtles. So, the, I won't focus on a lot of the details but just looking right here. This is the. This is the unstable IDG mode so this is a, the complex plane, the real frequency right here, the growth rate on this side. This is the unstable IDG mode right here it's the only one above zero which means the only unstable mode. This one right here is some stable drift wave. And here are the land out groups so these are the groups that you're buying from. Now, play with the law transfer on the top way of the land out that embrace. Now, this is the, this is a full linear operator, the, the land out damping term, the collisions and the gradient dryer of bringing here this is a whole linear operator. I'm not going to talk about these guys but this guy is interesting because this is the linear operator without the collisions and the sector here, you have this continuum of modes with zero growth rate. So here's what you'd call case van canton modes, and then there's this. This mode right here I call this the mirror IDG mode because it has the same frequency and exactly the opposite growth rate as the IDG mode. So, I was kind of puzzled by why this mode is there in the collision system but this is the collision system. And, and it turns out that it reappears in the nonlinear simulation. So, I'll show you how that happens. We decided to, to use a technique called pseudo spectra. And basically what this is doing is, is realizing that in a system like this turbulent system, the linear spectrum is kind of an artificial thing, because there's a big nominary there as well. Right. So students to the sector is a linear approach, but kind of halfway in between the pure linear picture and the nonlinear picture, because it asks, what happens if you take the linear operator here this is a, and you perturb it. So just some arbitrary perturbation with some amplitude. So, and what we get this is the, the x's are the actual linear, the linear frequencies, linear eigenvalues. And the contours are all locations in the complex plane, where the linear operator, plus some perturbation is an ideal. So if you talk policy, a certain contour, everywhere on this contour there's some perturbation of the same amplitude that gives you an value on this contour. That's a little bit of a little bit of a abstract concept. So basically what what you get from this is, when you have a perturbed, a system that perturbed away from linear eigenvote to a certain degree anywhere on a single contour is equally viable as a linear resonance or eigenvalue. So you might expect a nonlinearity that if the nonlinearity perturbed something to a certain degree that you, you know, might have actual fluctuations or nonlinear eigenvalue, nonlinear eigenvalues that could be anywhere. You know, it very placed in the complex play much different from the actual eigenvalues. We took the turbulence data, and we looked at these pseudo spectra, along with the linear eigenvalues, and then we constructed what's called what we call a nonlinear eigenvalue and as far as I know, we invented this procedure. So what we did was, we took the singular value decomposition, and we took the singular vector from that. And we stand the complex plane. So this is kind of an eigenvalue problem right operator on vector but we prescribe the vector we get the vector from the nonlinear simulation data. And then we scan the complex plans for choosing different eigenvalues so to speak, and then we minimize this so basically given the nonlinear structure, we're asking where the complex plane is as close as to being an eigenvalue. And we found some interesting things first of all we found a couple things that are not, well, kind of reassuring I guess, namely that there are nonlinear eigenvalues that generally why quite close to the unstable IDP mode which would accept that really the dominant thing in the system. There's also one that is close to this drift wave. So basically an interesting thing is this guy right here. So this guy right here is basically in the position of this mirror. So we have this ITG mode if you remember, I showed the collisionless spectrum where there's this load of the opposite trophy, but it disappeared in the political. So the nonlinear turbulence, it apparently appears despite the fact that the linear operator doesn't find that as an eigenvalue in the linear spectrum. Once you go nonlinear, it appears and it's quite a prominent nonlinear structure in the system. We're not doing this here but what we found is that the. So for each place in the complex plane using this pseudo spectra spectrum approach, you can expect to correspond the eigenvector attack. The most single eigenvector at that point. And what we found is that, indeed, in this region right around this mode right here, the eigenvector eigenvectors from the pseudo spectra have properties very similar to this mirror IDG mode. So the, again, I guess we're talking about the current themes that be here is that the extra dimension the face space dimension in plasma turbulence has a lot of interesting phenomena. Okay. Another interesting thing here is, we found that, as you go to higher parallel wave numbers for land of that you'd expect land of that be quite strong. In nonlinear turbulence in Joe in various places that land of happening is not as effective not as strong as you would expect in a turtle. So in a turtle setting land of happening is not as strong as you would expect from a linear picture, and this this reinforces that what we found is that this is the linear IDG mode. It's very strongly that from land of happening, but you look at these concours and you see that they're very strongly distorted, right around this in the vertical direction. So a little probation from the nominee area. Anywhere, you know, quite, quite far away and complex playing it's still a viable resonance of the current system. And the only found is that the actual nominee structures tend to lie very close to zero so they're not there. They're really usually to have, despite the fact that the linear, the linear prediction would tell you that they're strongly that. Okay, so, getting away from fusion for a minute, just a few more examples of interesting applications of some of these techniques. This was, so at the University of Wisconsin they had an experiment trying to mimic or produce the dynamo of the dynamo in the earth that creates the Earth's magnetic field. And so they had, they had basically propellers. They had a spherical experiment. It's actually liquid sodium, which is actually very dangerous and put it miles and miles away from campus for safety reasons. So they have propellers that drive a certain flow pattern. And this in this region here. And the goal was to create a flow pattern that that that creates a large magnetic field in order to understand the production of the Earth's magnetic field for example. The experiment was not as they did a lot of great work but it was not as successful as they expected. It wasn't successful in producing it wasn't successful in producing a dynamo. And so, this graduate student Angelo Limone did MHC simulations to try to understand what was happening and maybe understand what was preventing the dynamo from forming. So we took this simulation data and again we did a proper orthogonal decomposition, and the first mode of this decomposition turned out to be exactly what was expected from predicted from this propeller. And this was the flow that was predicted to produce a dynamo. And then it got interested interesting as you look at the secondary structures that are excited and turbulence. The color here shows the amplitude of the flow so just the magnitude of the flow. This first structure has the symmetry that you would expect you have propellers going in different directions so that the blue and the red mean different directions. But the second structure here is interesting because it violates that symmetry. So you would expect, you know, some odd symmetry going from here to here. It is interesting that in time the symmetry is kind of maintained because this shifts direction in time so it's just from being, you know, red blue at one point in time to being blue red at another point in time. And it turns out that the second, the second structure inhibits the dynamo mechanism. And so you can, so that they can, you know, you can take some frozen flow field and then calculate what dynamo you would expect from that. So we did that for this structure and it was much more proficient for dynamo, for the dynamo than both of these structures combined. So we went back to the experimentalists of Wisconsin, and they looked at this and said okay well we can let's build a little baffle right here to disrupt or to kind of get in the way of this flow pattern. And they did that, and it was actually good it actually, it actually produced the best results that they, that they had gotten on that experiment. So they turned up to get a really strong dynamo but they didn't get the best results that they had seen on that experiment. Okay, so we're almost out of time. I will just note that these ideas of mode decomposition proper to be done. They're being applied more and more in various station after physics problem. This is an example. We have an application of a application of a Helmholt, Helmholt, and so almost done, I'll leave there and we'll. Any questions, any questions. Yes. Okay, I didn't get what you're, what you know, you were 19 and when you went with, why is serious and you're right to deal with new with the collision upgrade. Yeah, that's actually a really good question and I never, I never figured that out. Hopefully frankly. I think we talked about this one that you said you have some ideas for why that happened. Okay. Anyway, we never did figure out figure that out it's probably needs, you know, linear algebra some sophisticated linear algebra. So I'm really more sophisticated math than I'm capable of. Operator theory knows what to really sort that out so it's an interesting question but I never have the time or inclination to the matter. The surprising interesting thing to us with that decided being gone from the linear sector that reappeared in the nonlinear hurdles. Yes. So I'm really into experience with that in a way, because that's my direction many and so I'm with my hands and they're just on you to the home to what we're hearing about the global climate but I need to go in fact to have the in more of a tricky way, so I can really touch knowledge through time. Would you mind letting us know if you think it's good for dark things, not from people at the same time? Yeah. Now are you looking for information about fluid turbulence or plasma turbulence or? Okay, yeah, sure. So I will put these slides on, you know, I'll post these slides for the meeting. And I'll add kind of bibliography with some material along those lines also. Yes. I'm going to read the details, but at the same time, I would like to find myself and I'm starting to build up my science. So I don't want to go into a difficult point. So it makes me crash with the information and just find myself. I understand and go back again. I don't want to get. So for you with that point, so I can read more at this point and try to go. Just one point. Okay, I didn't quite catch the question. I mean, you asked first of all, you asked for a run on. Yeah. Yeah. Okay, so that sounds very much like your first question though. Is that how's it different from your first question? Yes, yes, that you can talk to maybe say I'm starting something. You have a specific thing and that's not going to be like that. But by what you mean, stop. Yeah, please. Thank you for your questions. So I, I hope that you know what, in the case of any broad overview of various approaches to evidence. And so the subject of this is for me, but it's exactly there. And sometimes it seems to work. But the fact of the matter is that almost nothing remains in the so-called linear state. Although what was the question. What exactly does that mean. And so it's kind of interesting that you guys go over the simple lenses law. Right. So what it does is that if there is a pause, which is trying to drive the system somewhere, then the system generates reaction. All right. So what it does is to kill that box. So you have, you have a large power, that's supposed to simplify the large number of days. So somehow the plasma generates forces processes by which the millions. So the entire business that we are trying to do there. A state, which is a unlaxative, so to say, but the moment you created it, it resists. It basically wants to go back to its uniform distribution at the end. So most of the phenomena that you're finding are the reaction to the conditions that are important. Last one wants to be free, like a bird, like a bee. And every time they are correct. Okay. So, so, so what you have to often understand is that the instability that we're talking about. So instability mechanism by which the system wants to get back to its linear stage. I mean, but it will be different. It will be different. Right, because I mean, it doesn't put the. So eventually everything wants to be in some large space to force me, nobody likes to be under pressure. And so you can philosophically begin to think that, you know, it wants to free itself from those constraints. Okay, and therefore it will generate motions, you know, in order to conduct the things which have been doing it. And turbulence is exactly the instability is exactly a mechanism by which the system is trying to react to things which is being stopped from doing it's match tennis. And David introduced several things which are, don't give up on it. And I think that with the methodologies that he understood is developed. So, they have done a very, very nice penetration into various modes of canonizing techniques. You know, as he said, this is so random, so difficult, so making a proper analysis to extract all the information from that is very, right. So just as much technology of extracting information is just as important as understanding physics. Otherwise, how can you understand the physics if you don't even quite know what is it. There is a, David also introduced you to the fact that you have a linear system, and you understand quite well. And then, without doing anything, you kind of try to make it somewhat non-linear. And that, that changes the fundamental nature of the system that you are thinking. Now, there are many problems which you could build with your head in order to really understand that. I'm going to give you one problem you may follow up on. So, let's suppose that a perturbation is part of it. You don't have to take notes, it's so simple, so you know. And then our linear theory gets to the point that this is in the project. Right, and then you find the roots of this here. And then you say that these are linear matrices. But really speaking, the system is not this, because this is true only when file coding at A goes to 2. So, those of you who might have read every article on what's from the response theory, know that you take a perturbation and you let it go to 0. And then the response that you get can be part of it. Okay. But these three things we have, some may get M and some, let's suppose there was a quadratic non-linearity. So then you will have, so the right hand always has to be second thing that one, omega prime and K prime are summed up. What's left is a function we have in the directory. Okay. So you say, look, what's the simplest non-linear correction I generate the system. And again, following what David said. And then we change it's linear dispersion. Because linear dispersion tells us the dynamic property of the system. Okay. And so we want to change the dynamic. So anything in this for us to introduce is an imaginary part is very important, or which will take away the imaginary part of this thing and cancel it from that is very important. Right. So there are, there are well known procedures for this to play with it. So you take your simple system. This way, I'll train there or whatever had. So what you do is out of all this junk of non-linearity. Okay. If you could pick up one single term. Okay, which is of the phones that there is some spectrum and then pi omega. Start here. Right. So what this is all the term which is faced with the linear. This has more meaning than everything else, because this changes effectively the linear dispersion. Okay. And so what you do is if you combine this one, your, you can say that I have an effective dispersion relation. Okay. Minus alpha. This will be a sum. And both. This is now a non-linear modification to the linear dispersion. This is simple. And from this, you can understand very great amount of it. This particular vote is immersed in the boss of all of the votes. Okay. And then contributed and imagine a real part of that. And this change is more fundamental than the scattering, which might take from the other votes. So this is always good idea, whether you can estimate that. I don't know if anyone of you then was posted. There's an example of this study that she's trying to do. And, and it's, it's a simple, very simple. Nothing is, nothing is simple, not yet. But then you want to add this thing. I don't know if you should try. I haven't done it yet. Yeah. And that is a linear. You say that it is a linear, even high. Yes. I mean, I mean, it's not a long period solution. In fact, this person is a distance. It's not even satisfied. Only then you can have normalization, but I must say. All right. I think you guys should have tea. And then we'll come back. Thank you. Thank you. Thank you.