 OK, got it. So as I mentioned, there are many global modes. These are DABS terry modes, m equal 1 and m equal 0 modes, affecting the dynamics of the RFP. And we have seen that it changes its topology. It changes its transport properties in nonlinear ways. But there are other modes which were already identified very early, but basically put aside saying that we will think about these modes later on in the research. And these are non-resonant modes. And we conventionally call them as non-resonantly internally or non-resonantly externally, in which these are called resistive mode modes. And I will show you in a minute however they are derived, basically. And today I'm going to show how with these active coils, we will control both of them. So now let me start with the resistive mode, which basically are described in the realm of linear MHD. This is the realm of engineers, actually. So they use all their machinery that allows to deal with a lot of intricate details and about the power supplies. But basically they are linear. And so this simplifies a lot of things. And then we start with a recall of the RFP ideal stability and the first evidences. And now, basically, ideal MHD stability of RFPs is hidden in very old papers, very difficult to read, actually, so full of analytics. But Friedberg, in his latest version of his ideal MHD book, actually summarized it with some simplifications but to make it understandable. And so in summary, the stability of ideal MHD modes within a given RFP configuration can be summarized in a theta-beta diagram. So clearly, the reversal is necessary, as we have seen in the first day. So if the Q has a minimum, then there are some resistive interchanges. So the F parameters need to be less than 0, which is the ratio between the edge-to-radle field and the flux produced by the plasma. Then internal pressure-driven modes are stable below very high beta, which is here, basically. And then next are the localized interchanged pressure-driven M-equal-1 modes, which are stable below this kind of threshold here, which is still very high in beta. It depends on the equilibrium. So for given theta, this is the F-equal 0 and F-equal-1 lines determine the region in which the RFP typically operate. And another characteristic is that the core pressure, in order to be stable, must be flat. And experimentally, actually, either in the shock states or in the multibody states in the center, the pressure and the temperature are always flat, actually. And the internal M-equal-1 modes have the same stability boundary as Sweden. So both Sweden and Yukon Craterium basically give the same threshold. And so at the beginning, it was thought that RFP was promising very high beta operation, because there is no stability, ideal stability limit for the RFP differently from tokamak. And all of this stability is for with so-called internal mode, meaning that they do not displace the last closed surface. So the plasma remains exactly cylindrical in this case. And if a conducting shell is located just in front of the plasma, that's it. So this is all the ideal stability that it is required for an RFP. But as practically, you cannot place a conductor just in front of a plasma. And if there is some vacuum between the plasma and this conductor, there is another kind of mode, which are so-called the external modes, which can be unstable. And here now it is less optimistic that's put in this way. So without a wall, there always exist some m equal 1 external modes, even at zero pressure. So they are current driven, no pressure driven, but are always unstable. And here is an example of how it is determined. So by doing some marginal stability analysis, so computing the variation of energy for a given displacement in the linear eigenfunctions, it turns out that this is a spectrum here shown. But for a certain equilibrium, either with beta equals 0 or beta equals 0 1, there is a number of modes here around n equals 0, which are delta W negative means that they are unstable. So they can grow unstable no matter how low is beta. And the only way you can stabilize this mode is to make a conducting wall sufficiently near by sufficient means that even if it is three times the plasma, so kind of far away, some of the modes can get positive 2 times 1.7. So in principle, if it is sufficiently near, but not that much, you can obtain delta W positive for all of them, so for a given equilibrium. Now, you have to check all possible equilibrium. And it turns out that clearly you get the one which grows fastest and you change the various data possibilities and you'll see that you will get at least 1.4 is the aspect ratio, which is still good actually. You can do a reactor with a conducting wall 40% far away from the plasma. So you can place inside all the machinery you need actually. So this in principle is something that can be dealt with in dealing with a reactor or a device. And so this is the main difference. So the pinch of the RFP requires for the ideal stability of the presence of a conducting wall. But there is another problem is that any conducting wall is not infinitely conducting. So in time, the eddy currents will decay. The time it takes for this eddy current to decay is related to the thickness of the wall and its resistivity. So the thicker the wall, the longer the time it takes for the eddy currents to decay and consistently this eddy currents keep the plasma from basically deforming itself. And in fact, given this thickness, the fixed shell experiments were the ones in which the discharge duration was shorter than the shell time. So a way to circumvent this problem was just to look at the plasma from a time window below the time it requires for this instability to grow. So why in the thin shell RFPs, as now it is in RFX mode, the discharge length is comparable longer than the wall time. Now, this ideal mode, whenever you have a resistive shell, get the begin to be resistive wall modes. They are called this way. So they do not grow on the ideal time scale of the microseconds of the half van square, in thin ways, but they grow much more slowly. But at that time, in which they were identified, they were considered as a serious problem for the RFP as a fusion concept. This is an example of an spectrum. This is a high aspect ratio machine. So the higher the aspect ratio, the wider the spectrum of all possible modes that can be unstable. So as I said, most of the RFPs avoided the RWM with fixed shells while several others investigated RWM physics just to verify whether linear theory were applicable to real plasmas. And the first experiments that performed a convincing experiment showing these RWM instabilities was the HBTX1C, so one of the first generation in the last modification of the experiment, in which, in a few millisecond time scale, clear exponential growth of many modes were observed. And it corresponded to three times the scales. So three times the wall time, which was one millisecond. So the time traces are shown here. And in that experiment, they also tried to feedback stabilize. Because in principle, the idea is that if you have these eddy currents into the shell, which are decaying, you can think to use some coils outside the shell that provide the current in the time scale it decays so that you continuously feed these eddy currents inside the shell. And why it is possible actually depends on the fact that the theory is linear. And so you can imagine that your coils that you are winding around the plasma basically generate a single Fourier harmonic. So you have a sheet of current of m equal 1 or generic n, which is located in a particular radius outside the plasma. And this helical eddy coin sustained the eddy currents. Now, the derivation is algebraic, it's kind of lengthy, but it can be seen that the helical coils and the conducting shell act as a boundary condition for the Newcombe's equation stability method to determine linear stability. So you can determine how the eigenfunction is valid by the presence of these two boundary conditions. The shell actually has some current which is induced due to the fact that the mode changes, so it induces some current. While the coils are changing the boundary condition depending on the current which is flowing, so the helical current which is flowing, the evolution is simple one degree ordinary derivative equation or the ease, which we have a component in which you have the growth rate of the mode and the contribution of the coil here. And so it is what is this called from the engineers are one pole model, so a very simple model, so an exponential growth. The same framework without plasma give you simply the penetration of an harmonic. Now, what the control variable here is the radial field at the radius of the shells on the coils. So pretty simple. So once you have your system which is described by linear growth as in the simplest case that I have shown, in control theory terms, you have given the open loop behavior of your system which is the process plant. So you have your output preference, you want your actuator which are very close to produce some current and then your RWM evolution equation gives you how the plant will process this current and it will tell you which is the current which is the amplitude of the radial field due to the mode. In closed loop, this reference actually is determined by a controller which is based on a sensor which measure the output. So the radial field in this case and it compares to the wishes of the experimenter. So basically having a zero value or a finite value so some kind of reference and then the controller to decide which action so which reference go to the coils. So they're very simple actually. So in the system, in the simplest system in which you have a one pole and stable system and the simplest idea that you can implement is that the far away you are from the reference, the more you want to push just to get near to your desired reference. So it is a proportional control and you can simply work it out in equations here. You can also try to give a finite reference as we did in the past, but now we are not discarding for simplicity. And the evolution of a plant in this case is very simply dictated by an exponential. So if your grow, oh, sorry. If the gain is zero, then you have gamma which is an exponential. So the solution is exponentially decreasing. If you increase your gain, the exponential will decrease at a certain time you get to the marginal stability and then you will stabilize the system and your steady state behavior will be either with a reference value or with zero if you have not set any different value. Clearly this is the simplest model that you can use and it always converge. If you have delays or if you have some non-ideal behavior or some components, it may be impossible to obtain a negative exponential but we are skipping this part now. Now, what they did in HBDX it was actually really to implement this kind of control. So they have to have a sensor. We have to have an actuator. So the sensors were analog constructed series and anti-series of pickup coils taken on a quarter of a machine. And in fact, they were targeting M equal one and equal two. So which has some kind of symmetry. And so we were able to obtain either the cosine and the sine component and feeding two sets of coils allowing to produce both the sine and the cosine component of a mode. And they were using analog feedback. So basically this was really the realm of engineers and not physicists there because every tool was analog. And they were successful in this aspect. So they were able to publish a paper in which they showed that the two components of the cosine component of the two one mode actually were kept at a low level in the milliseconds range. It was six milliseconds. But as they were targeting only one mode and there were 10 of them, the overall discharge performances were not improved. And the proof was that it can be controlled in feedback. So that was a good news. The bad news was that it was practically very difficult to implement different coils with different elicities for different modes, which may change. And so having all of the and so on and so forth. So as Fred says in his book, it was known that in theory it could be done but in practice, a lot of people were skeptical about the possibility of doing this kind of stabilization. So there was a smarter idea around but still with required some technological development. And this is the Intelligent Shell Compset. So LOSO in 77 but later on also Bishop proposed a different scheme. So instead of having helical coils dedicated to doing helical harmonics, the targeting each mode. Okay, forget about the mode. Imagine that you want to, you see the shell, you see the shell, but it is actually frozen the flux. So what the shell does is no matter what mode you have, whenever the mode is penetrating an eddy current is trying to cancel it. And so imagine that you have so a subtle loop which is delivering the right amount of radial field that you require. Bishop's idea is do, okay, let's do it in feedback. So connect the subtle loop which is producing because the subtle coil which is producing the field with a flux loop which measure it. And so basically the idea is no matter what mode the system is generating, take the signal of a loop and feed it in feedback to the coil loop and make it to make it zero basically. So the idea no matter what mode will be there you will be have a zero flux. So an intelligent shell in itself. Clearly there are some limitations because the shell is can be thought as being produced the real shell, the continuum shell by an infinite number of coils which is not something that you can do with a real coil that you are winding. And so you have to limit to a discrete number of coils but how many of them do you require? Well, at the very least you need to have twice of the minimum at the maximum because you needed to deliver the radial field with a topology which is similar to the sinusoid that you are willing to cancel basically. So at least for an n equal one you need at least three coils in the polygonal direction. So that you can be sure that you can control all of the phases that the mode can grow but from practical reason it is basically taking from four. But the point is whenever you go instead to many modes the criteria get more complex. And here comes into play the concept of sideband harmonics because the fact that these subtle coils do not produce a continuum field but it is kind of square-ish. So in Fourier terms it is full of sideband harmonics instead of together with the dominant one. And so let's visualize. So there are people who prefer to see formulas why prefer to see pictures just to give an impression of what a sideband is. And so this is a cartoon showing what does a system a control system, how did it behave whenever he want to fight again a radial field which is located in the last closed surface. So this is a one-seven mode say one particular eliciting which the color encodes the radial field. So it is nicely sinusoidal but then it is picked up by our control system and then the control coils which are that discrete let's try to count to zero the flux which is below each and every control coil. And the field broadly resembles the n equal one n equals seven but it is a kind of sum of square-ish as you see in the picture because it is rich of what I call the sideband. So many multiples of the dominant frequency which is being generated here, so seven. So in both in the m direction and n direction and the multiples is the number of coils. So basically, so the first for the seven in the phase of 48 it is with 55 and so on and so forth. So the point is whenever you have a spectrum of harmonics if your system is trying to cancel the harmonics say minus 11, you have to have enough toroidal coils so that the first harmonic, which is after NC should not be in the unstable part of the spectrum. Otherwise it will couple these two modes through the sideband. And this actually was actually done whenever we were in extra T2, there's more machine in which the controller was developed before RFX model was actually operating. So during the installation of the commissioning and as we did not have all of the audio amps that were required actually, this is a small machine. So the amplifiers that were feeding the power supply units that were feeding the power, the coils were actually audio amps are used for constant say so they were supposed to be modified in order to have low frequencies but it was the standard technology. And having half of them available we already tried the experiment and it turned out that it visualizes what means that you are coupling two modes basically. So below the control coils, the zero, the mode, the measurement was zero but the two modes combined together so that they were known zero only in the region where without coils basically. And so this is the coupling visualized say once we had all of the amplifiers in place and the system worked actually both in extra P2R and in RFX model it was possible basically to routinely remove and stabilize the RWM for in the case of extra P2R which is a small machine high aspect ratio. The increase was also more dramatic because it also canceled a lot of error fields which were present in the machine where in RFX still it was good and actually it allowed also going to two mega amps. So this is the first part on the resistive world modes. Now I'm skipping a lot of details and now I'm giving for granted that they behave linearly but the people was skeptical about that because actually it is true that whenever you pick one mode it behaves linearly but whenever it is inserted in an RFP which is full of non-linear interacting resistive carrying modes there was some skepticism about the fact that they would continue to behave linearly also when in your phases but it turned out that we have plenty of experimental evidences that they still are in the real domain so they can be dealt with the control theory and the control technology. Different story is the control of carrying modes because they are non-linear. I just gave you some example at the very beginning in the first lesson of the fact they are wall locked all of these kinds of behavior or also the dynamic relaxation events they are all non-linear. They cannot be dealt with the linear tool theory but they still respond to what you do from the outside with your control course. The point is that the modeling requires some kind of non-linear doc modeling which cannot rely on the standard control tool technologies. Yes? I think it's possible that many of the people in the fields don't quite know what the carrying mode is. They don't know how it works. Unfortunately, because actually these are what I call the global modes in the first lessons. I only showed that these global modes basically I have to resort to the first session. First, let me... Yes, I switched to jargon actually. So in the first lessons I was careful it's not using jargon. So I was calling them just like global modes and then... That's exactly what exactly is being told that the magnetic surface and we will start with the question of use. Because carrying modes are characterized by the fact that they do have a resin surface. They do have islands but they also deform the last closed surface. And so, as I said, they are to be on precise resistive king carrying modes because they do islands and they displace also the edge and they produce this kind of behavior basically. And with the control course, so in a sense they behave like resistive world modes in the sense that they are deforming the last closed surface but they do have differently from resistive world mode a resin surface with some current flowing into it. So it is a more degree of freedom and they interact with one with each other and it is doing a different, they can rotate. And so this is why they are tougher to control, let's say. Okay. So without control in, without control, these three modes are observed to rotate spontaneously. So at frequency in the kilohertz range, but they tend to wall lock in conditions which were actually widely different from experiment to experiment at the beginning. And an example here is from the MSD in which actually the second trace is the velocity, the magnetic phase velocity and also the velocity of the plasma as measured with spectroscopy, so Doppler shift basically and so they go together. But whenever this mode grows compared to the others and it goes to a single elicity as I showed in the first day. So it's amplitude increases at a certain point it eventually with the magnetic phase it eventually with the rotation frequency locks. So it goes to zero and also the plasma is arrested. And this is occurring more or less around 200, 300 kilo amps or even 400 in different regimes. While in RFX, we never saw a rotation in RFX. I was a student, I was working on the soft X-ray tomography. I prepared a lot of correlation analysis technique and trying to look at the correlation in frequencies between the various signals and during my PhD thesis and nothing was there, it was a desperation. And we only were able to find the rotations in RFX mode whenever we did the experiment at very low current to compare to what was actually the design. So we were able to push our partial pi on the low side. So in the 100 kilo amp range actually some rotation has seen here both in the flow of plasma and in wavelets of internal measurements that were occurring. So the threshold for wall locking in RFX was incredibly low compared to MST. And it comes that this wall locking phenomenon is a non-linear phenomenon which is due to the eddy currents in the, it was thought it was in the shell, MST and RFX had aluminum shell both. But it turned out that it is the first conducting structure around the plasma that matters. And depending on the resistivity of this structure there is much or less breaking torque. And such a breaking torque is a balance between the drug by the plasma and the electromagnetic torque. And so it depends on the mode amplitude on the resistivity of the shell and also on the gas. So in deuterium and in hydrogen the torque behaves differently due to the fact that viscosity is different due to the plasma coefficient. And as I said the first day this is a way by which we can estimate viscosity basically by measuring the breaking whenever we apply some error fields. And so Fitzpatrick did the phenomenological breaking torque model, which is written here but it is full of awkward amount of analytical formulas with a non-linear, quasi-linear approach. But it turned out that he was able to explain the fact that RFX and MST were very different due to the fact that RFX had an inconel vessel. This inconel vessel was actually designed in order to be nuclear. And it's even more robust than a stainless steel. It was designed to be light and so it had a nice, a lot of nice engineering properties but this resistivity actually was underestimated. Actually, they were thinking that the more resistive the more similar to vacuum and so that's it. So forget it about it. But it turned out that actually it determined the very low locking threshold. Actually in RFX module we are getting rid of all this shell. So how did we try to active control to mitigate to mitigate this wall locking in RFX mod? Basically to remind in RFX from the beginning of discharge the deformation due to the wall locking was always in the same place changing from shot to shot here as a function of the territorial position in time the deformation of the VR is always the same. And then ways which don the coils, the 48 times four control coils, we call it virtual shell but in the intelligent shell approach actually this measurement the fluxes were basically zero and so they were happy. So we removed the wall locking but still the cameras and the influxes were showing some clearly nice signature of the interaction so something was not going well. So it turned out that these measurements were polluted by still again the side bends as I showed you in the RFX session but the side bends were polluting the measurements so they were affecting the control in a more subtle way. So at first you have to remove the aliasing and we show you why it is and it turned out that the intelligent shell was doing good but not that good. So there was a residual radial field deformation which was causing a localizable interaction and the idea was the fact that you have the same number of actuators as the same number of sensors. So Bishop said yeah it was very nice so you have your coil, you want to zero the flux by applying exactly the same amount of flux so you measure all zero and you are happy but whenever you are harmonics this zero is actually done by the harmonic itself when all of the other side bends there so they all pile up so your measurement is this one so part of it is the other harmonic part of it is actually different harmonic so your zero is not a good zero so in this way the real solution would be just to increase the number of cells so get rid of this intrinsic limitation of the intelligent shell approach and use much more sensors or as it was impossible in RFX compute it in real time basically this is a visualization of what the spectrum was so the control system without any correction was thinking that basically the red curves were zero so and all of the diagrams that I have shown were basically with these measurements but whenever you remove with the alias the measurement the real spectrum was here so there was a residual significant one and so we had to correct it and ok long story short when we corrected this kind of systematic error basically switching from the intelligent shell control to the mode control so changing the gains in a different space doesn't matter the details it turned out that the deformation was reduced but to a level because it began to jump and so we experienced for the first time since I was waiting since I was PhD I had some rotation it was in the Hertz range range instead of kilohertz but some rotation was there and given that we implemented in 2007 this new control it was possible to reach the two mega amps which was impossible basically due to the locket mode that we had and reach the highest temperature in RFX and now we also realized that still there are some limitations in this kind of control and we had to model this the control by expanding the old 5th patrick theory of the passive the effect of the passive boundary on the wall locking and so by enhancing this approach using the quasi linear this is the modeling of the Eigen function of a given mode for example it is a 1.7 so it is not a full code this means that the radial field at the resonance is an input to the system but this code tells you how it rotates the subject to the fact that you have a vacuum vessel, you have a boundary you have a coils so given a complex structure of boundary conditions it can give you which is the minimum value of the radial field at the edge that you can obtain taking into account that if you implement some kind of algorithm it will begin rotating more or less and it will also interact with other modes now our figure, the parameter of merit is this value at the edge the lower this value the lower the interaction it turned out that the model explained what we were seeing experimentally so by increasing the gain of a various mode we were able to reduce the radial field at the edge, we reduce it so we do a gain scan but at a certain point actually the reduction doesn't proceed but what happens is that the plasma begins to rotate so you try to fight against the plasma and then it switches it to a position and the faster you push the faster it goes it escapes from you and this depends on the passive structure that you have in the system and also on the real system so in the feedback delays the discretization and so on and so forth and so there is a limitation so we are stuck with a certain limit we cannot do better than an ideal shell with the control coils even if that was expected basically it turned out that the vacuum vessel still was playing another bad role say and so this minimum as a function the minimum as a function of the gain was limited but by replacing the vessel and making the plasma nearer to the copper shell it allowed us in principle to in theory in the model to decrease by a factor of 2 the minimum value and so this was a further motivation to motivate to change from RFX model to RFX model 2 and to get rid of the incremental vessel basically and so on the last part my presentation RFX model was designing in this way so it had the mechanical structure the passive stabilizing shell and then the vacuum vessel and the graphite so the distance between the shell and the plasma was this way 1.11 something like that in RFX model which is now being built and assembled the plasma actually is touching ties and the ties are attached directly to the copper shell which is here so the plasma is bigger by a few centimeter and so the ratio between the distance of the shell the plasma is 1.04 and so we expect an improvement and actually the first improvement by using the code the RFX locking code is that the amplitude of the radial field at the edge should be in principle decreased by a factor of 3 clearly the code takes the input the value of the resonant modes at the resonant surface as given so we pessimistically assume that they are not increasing also the simulation can be run in order to see which is the value at which wall locking will occur while in Inconel this is actually a very synthetic run in which modes are supposed to grow linearly and the plasma rotation is measured here and with Inconel actually whenever the modes get to a level which is consistent with the low amps they lock while in copper it is expected that such a threshold would be much higher clearly, this is optimistic we will see in the real experiment so we should have a possibility to have some experimental regimes with rotating modes just like in MST and finally not relying only on the semi-empirical mode in which the terry modes amplitude were taken as given some runs with this resistive special code were done at lundquist number 10 to the 5 with some resistive shell con boundary conditions so with some vacuum and so they simulated an ideal feedback by changing the location of the ideal shell and by performing runs at different wall proximities in particular going from RFX mode to RFX mode do both the edge value which is the black line clearly is decreased now in the code whenever the wall touches the plasma it is zero but also the average energy of the modes decreases by say a 25% and so assuming very simple stochastic transport confinement which we know which is not correct and knowing that this is for MH regime which we know we hope to do better but would correspond to a 40% increase of confinement and so we should see something different in RFX mode too and so in summary I quickly showed you but in RFX mode with the active control we were able to either solve the problem of resistive wall mode but was fought as technologically impossible by many people in the 70s and 80s and this is taking us for granted while for tearing modes we succeeded in getting to the 2 mega amps we succeeded in mitigating the issue of wall mode we identified some limits and with RFX mode too we are going to hopefully restart operations and investigate RFP farther that's it thank you