 the essential plasma, but there is a delay. Okay. There is a, I can try. It's muted. Okay. That's something different that you are asking. Okay. So I make stop share. And then I can do this. Okay. No, no. Yes, it is completely. You should. Guess this. Now. Okay. Okay. Yes. So now I do this. Okay. Nope. Okay, now it's working and we are recording. Okay. Thank you very much. It's not shared. I'm sorry. Okay. Okay. So what we were saying, well, but a fluid description can be obtained from the kinetic one averaging the Boltzmann equation on the velocity space and this introducing the moments of the distribution function. The fluid approach introduce a great simplification in the in the way you can deal and you can treat your problem. Although losing the details that might depend on the case of the distribution function. So it's some ways a democratic system. Okay. But despite this simplification, the system of equation that you have in the fluid constant context is in any case really difficult to solve analytically. And the numerical approach is mandatory in this research field. I will try to convince you showing many numerical results numerical simulation in the second part of this lecture. So let's start with the moments of the distribution function. The distribution function, you know, obeys the Boltzmann equation, the electric and magnetic fields are coupled with the Maxwell equations, as you know, which determine the charge and the current density distribution that must be expressed in terms of particles that generate the plasma and all the distribution function itself. So it's a loop. The analytic and numerical solution of this system of the question is really cumbersome. And so one may ask if and when really needs all the information which are contained in the distribution function or Satan and other in other words, can we rely on the information obtained from more easily measurable quantities? That's the question. And for instance, we can think, okay, let's suppose that all the information we need is related to the physical geometrical space. And we can average the distribution function over the velocity space. Okay. Here, maybe. Let's see. Okay, yes. Here we have the, okay, we integrate and we average the distribution function on the velocity space. Since the distribution function represents the average number of particles within an infinitesimal cell in the phase space. Once we average on the velocity space, this quantity will represent the number of particles you have with a position between R and R plus delta R, okay, regardless of the velocity. In other words, this is the density in ordinary space, okay, something which you can think you can measure. Okay. So how we define the average value of any function of the velocity, well, an average value on any function of the velocity is defined in this way, okay. So you integrate over the velocity space, the function which is indicated in general as a psi of velocity. And then you divide it by the density. The division by the density comes from the definition of the distribution function itself, which is not normalized to unity, but is normalized to the number of the particle, okay. So the most interesting function are represented about what are called the cave moments of the distribution function, which means integral over the velocity space of the powers of the velocity. Okay, so it is clear that, no, the density is the zero order moment, okay, of the distribution function. In the same way, the first order moment represents the average velocity, okay. Then consider you have a plasma composed by electrons and ions, each of these pieces have its own distribution function, and so you can define another quantity, which is measurable, the charge, okay, of the audio system, the charge density, and the current distribution density in this way. So for the charge you multiply by the common charge and you make the difference between the distribution function to take into account the different sign of the charge, and you get what is called the charge density, and then you can do by multiplying by the velocity, you get the current density, okay. Second and third order moments give the stress tensor and the heat flux, okay. So the moments of the distribution function have a clear physical meaning, okay, this simplifies, which simplifies things because they are related to measurable quantities in the experiment. So averaging on the velocity space, of course we are losing information related to those particles which have a, whose velocity differ significantly from the average value. So all this information is lost. So the quantities, the moments that we have seen obis some general equation for the moments itself, and how do you get this general equation for the moment? Well, you start from the, what is, you start again from the Boltzmann equation and multiply, okay, the Boltzmann equation times the function, the moments you want to obtain for which you want to obtain the equation and integrate on the velocity space. So first consideration, since the function psi is just a function of the velocity does not depend explicitly, explicitly on time. And so you can take this term and bring it under the derivative, the time derivative, okay. So then you remember the definition we gave of general, of the average value of any function and you end up with this derivative of the density times the average value of your function psi. Likewise, since psi does not depend on air either, the second terms gives in an analogous way because you can bring your function under the special derivative term and you can get the divergence in the end of the distribution function times the velocity for the moments you are considering. Let's see what to do with the third term and here we separate the part which is the velocity free force, the one which depends on the electric field and so it's quite easy in integration, this one because you take the velocity, the electric field outside, then you integrate by part and then you can send this term to zero thanks to the Gauss theorem because if you assume that these limits hold, okay, that the limit for psi which tends to infinity of psi times the distribution function is equal to zero. This is a reasonable assumption because you have to think that a supra integral will grow with the square velocity. When the real distribution like can be a Maxwellian distribution will likely go to zero much faster than that. So because no particle can have infinite velocity and so you can get rid of that term and you are left just with this part. What to do with the part will depend on the velocity. So here is a bit more tricky but actually you have just to think in terms of the component and you know that the height component of the Lorentz force is independent from the height velocity and so again you can bring this term under the derivative. Okay, and if you bring it under the derivative then you integrate by part and again using the Gauss theorem you can send one of the two integrals to zero and you are left with this, okay. For the collisional term we assume that in analogy with the temporal evolution of the distribution function we can bring these terms under the derivative, okay, and so we are left with this collisional term. Okay, so finally this is the form of the general equation for the moments, okay. Well F is the electric and the Lorentz force. So this averaging process we have introduced, although destroying a piece of information because we are losing the information on the tail of the distribution function allows us to find some general laws for the plasma and to show this we will evaluate the general equation for the zero, first and second order moments and we start with a simple example and we consider a neutral gas, okay. So I have reported here the general momentum equation and we consider a neutral gas where all the particle has the same mass, okay. So from the last equation if we take our psi function to be equal to V to the zero which means one for the mass we obtain this equation, okay. And well if we assume only binary collisions that conserve the number of particles and we define the density mass we get the continuity equation, okay. This one, very simple and I think you have seen it many, many times in your life. So now consider that the psi function is equal to the mass times the velocity, okay. I am reasoning here on the if i component of the velocity and again I apply to this equation taking this definition, okay. And so we have the first time is the derivative respect to time and then we have the derivative respect to the special coordinate then we have the force term and the collision but the momentum and be the variant under collision because the momentum lost one from one particle will be gained by another particle so we can send to zero the second the right hand side of this equation, okay. And we are left with the first two terms which express the total variation of the momentum density, okay. And the last where keep note that in this there is the second order moment. So the system is not closed so far because in this equation which is the first order moment equation it appears the second order moment this is a problem we have to deal with and we'll try to see how to solve this problem. Okay, well I don't do all the calculation but for the if we if we set the size equal to the kinetic energy we get the second order momentum equation, okay. And if you keep in mind that kinetic agent energy is unaltered by collisions of course we can send to zero the second the right hand side of this equation and we get an equation which is an energy conservation equation. So this set of equation allows general consideration on the gas behavior but does not represent the closed system because the number of a non function always exceed the number of the equations. So so far we cannot solve the system unless we know the distribution function. So no help can come from higher order moments because we will face again this problem at the higher order. But for a neutral gas we can make some assumption and maybe we can close the system because for a neutral gas which is dominated by collisions we can assume that the system will evolve through an equilibrium represented by a Maxwellian distribution. In both cases of global and local equilibrium we can get rid of the off diagonal term of the stress tensor. And so we are left with a scalar with a pressure scalar. In this case so if we count the unknown and the equation and we can check it again together. We have an equation here. If we go up we have a second equation here and a five equation and a three equations here. So we have five equation for five unknown because we have the density, the pressure scalar and the three components of the velocity. So five to five and we are safe in a safe situation. So let's hi welcome. And I saw all the people from Galileo I guess has arrived now. Okay so just to to make a bit of a summary for you I started with the some basics so maybe you can just start from now on I derive the moments of the distribution function and the general equation for the moments. And we apply that this procedure to the case of a neutral gas. Now I'm going to apply this to the two fluid equations. Okay. We consider the case of a plasma composed of two species of protons and electrons. In this case we have to consider two distribution function one for the protons for the ions and the other for the electrons. Okay. And the order moment equation give two equations where the average are defined as respect to their own specific distribution function, of course. So we have these two continuity equation. Okay, electron one for the electron one for the ions. The same to me question. Again, we get the same equation as before, but now we cannot read off of the collision alter. Because each species will collide also with the other one. So we have to keep in. We cannot get rid of it, and we have to consider. We have to take into account for the collisions with the other species. We will see this term can be then specified better. Then if we introduce the definition of the stress tensor, we can write these terms. Okay, the terms which depends from the Lawrence force in in this way. And then the terms which depends from the stress tensor from pi of P, as you want to call it, is it gives this equation. Of course we can go on. Okay, so on with higher order moments, but keep in mind that we have not solved this this problem now for the natural gas we could, but now we have to close the system. We can see which are the typical closure later on this presentation during this presentation. Then we can simplify more things because we can think, for instance, that the two species have approximately the same temperature and can be treated as a single fluid. So we define the in this square here we define the total density. Okay, the total number of particles the total density and the total charge density and the total current density. Okay. The first equation, if we sum the two equation one for ions and one for electrons, we get this equation. And if we define a mean velocity. Okay, in this way, a mean velocity which is made on the average velocity velocities of the two species. We can write this equation as a density continuity equation. Okay, this one. If we multiply the charge by for the charge and we sum again the two first zero order momentum equation, we get the charge density continuity equation. Okay, where we have the divergence of the do I am able to go there. I don't know. Yes. The divergence of the current density. I'm sorry. Okay, the divergence of the current density is here. Okay. So adding the momentum equation, keeping in mind that the momentum lost in this case the momentum lost from one species is gained by the other, we can send to zero. We have the right side equation, the right hand side part, and we get the momentum equation where the stress tensor is now the sum of the stress tensor for the two species. Okay, in this box over there. Okay. Still we have not solved the closet the closet problem keep in mind that put it on your mental pinboard will come to that later. Another question, which is a really important one which can be obtained in this framework. And if we multiply each momentum equation by the charge divided by the day by the mass, and we subtract them. I can assure that with a bit of algebra and making this to assumption. And you are keeping in mind that the electron mass is much less than the iron mass, and that the plasma is. There is the quasi neutrality condition, you end up with this equation which is called the generalized almost low, where we have specified the resistivity. Okay, which depends on the electron mass and the collision frequency between electron and ions. Okay. Neglect all the right hand side. Okay, if you send to zero all this part, which can be done under particular yes particular assumption, you send to zero the electron mass. And then you consider for instance, and you consider the smaller larmor radius case. You send to zero this, this a second right hand side, you end up with what is called the home slow for a fluid conductor in the reference frame, which moves with the fluid. Which is something you have seen. I'm sure many times. It's important to note that in a plasma differently to what what happens for ordinary conductors where when you hit the conductors you make the lattice, the ion lattice to oscillate. And so you increase the collision frequency of your system. And so you increase the resistivity in a plasma it goes the other way around. So if you do to this. There is a dependence of the collision frequency, which goes like the temperature to the minus three over two. So the more you hit the plasma, the less collision it becomes. I make you to note this because in the following I will talk a lot about collision less plasmas because because they are really common actually is the most common situation. And, and this is the reason for hot plasmas they are no collision. So as I was saying the inertial term can be neglected when the current density very, very slowly on the collision at timescale. Okay. The j cross beta if you balance between the V cross B and the j cross B, you discover that this term can be neglected. If you in the approximation of small normal radius, which is an approximation valid for very strong guide field. And then, if you assume also a isotropic pressure. Okay, in such a way that the pressure reduce to a scholar. Okay, these are usually the approximation that can be done, or that are done. Okay, so how do I get the m hd equations. The m hd equation is the sum of these three ingredients, single fluid, plus some approximation, plus Maxwell equation with a closure, and you get to with the end up with the m hd magneto hydrodynamics equations. Okay. So, which are the, which are the approximation the single fluid equation we have seen. We have seen the single fluid equation before so now which are the approximation well, usually the scale lines of many instabilities and ways are able to grow and provide a propagate in a system. So, sorry. Go. Yes, and I will talk about the constraint because he's. He's the frozen in condition, and I have a few slides dedicated to that the problem. So if you don't mind, I will come to this later, but very good question. Please. Collisionless. Yes. Sorry, I didn't know what. Collisional. Yes. Well, in in Tokamak it depends from the value of the resistivity but there are regimes where the value of the resistivity is not so high and so so low. In that case, you are in the regimes which can be defined as resistive regime. For instance, in the neoclassical terry mods are usually treated as a resistive with resistive equations. While, if you want to explain the faster connection process resistivity cannot give you the right answer. Unless you go to plasmoid mediated regime and a collisionless approach can be more, more useful. And I'm, I hope I will convince you with the second half of the other my presentation where I will show you example of all these things I'm talking about. Okay, so I was saying that if the phenomenon propagate on a scale. On a scale which is comparable to the scale of the to the plasma size. So we assume that the scale L is much greater than the by length, we can make the assumption of the quasi neutrality condition. Okay, and so we can neglect the displacement current in their home slow. Then the, also if we consider the low frequency phenomena that allow the displacement current to be neglected in the maximum equation. Okay, so omega 10 to 10 to zero and L 10 to infinity. Okay. Then if we consider only collisions dominated the phenomena that keep the system isotropic at all the times, we can have the pressure as a scholar. Yeah, okay. And so we are first order, zero order moment first order moment, the on slow. Okay, and then the maximum equation. This is not a closures closer system but that we have to adopt the closure and the in MHD there are three typical closure that are adopted the incompressible fluid, which means that in this equation. Okay, if you solve for the divergence that will be an equation which goes to zero the error over the t plus the total derivative of the top of the density goes to zero and you are left. You are left just with the divergence of the velocity. So the same things as incompressible fluid fluid or divergence of velocity equal to zero is the same thing, or you can have an adiabatic closure, or an isothermal fluid. Okay, it will depend on the situation that you are studying the problem you have you can adopt the one of these different closure. And then the other class of model which I, I put under under the general know name of extended the MHD, okay extended MH magneto hydrodynamics, which comes from the two fluid approach to fluid equation, plus approximation you have to decide which approximation you want to make, and how many terms that you want to keep in the home slow. Okay, which is here. This is a very much equation. Okay, so here I was retaining the mass, the electron mass of terms in the home slow. Okay, this part here. I retain it also the, the pressure and these pie of problem with this are the off diagonal part, the term of the stress tensor. And any momentum equation for the so the momentum equation for electrons, which comes from, which is the generalized on so and the momentum equation for the ions. And the Maxwell equation. Still, the problem of the closure is also here, of course, and some of the closure I explained before can be adopted also here. It's like easy life. And so we want to simplify more things, which is something that can be done because one often encounter situation in which the magnetic field is strong and most unidirectional magnetic field in loops in the solar corona or in the Tocamax, you know, you know, the Tocam is a big, big donuts with a strong magnetic fields along is a toroidal direction. So these circumstances allow one to consider approximation of the full mhd or extended the mhd equation, which are based on physical and geometrical considerations. So since we have a strong magnetic field, which is called the usually guide field. Okay. There is a, this is the guide component that the guide the field is much larger than the other component. Okay. And usually this guide field is denoted by be not and so the magnetic field can be written as be not the plus by be perp where be a perp is the magnetic field in the direction perpendicular to to the guide field. Okay. So we adopt a force simplicity a Cartesian geometry reference frame and assume that the strong component is directed along said the perpendicular will be of course in the x y plane. So the magnetic field that can be intensity can be denoted by by be which will approximately coincide with BZ and so would be not. The characteristic the ordering that we define a parameter which is the epsilon parameter which is be perp divided by be not in this parameter must be must be much less than one. Okay. Consequently, the other ordering of the problem are that the derivative in the x and y plane will be of order one, while the derivative along the Z direction will be of order epsilon. The component of the magnetic field that we thought we said here are order epsilon while since the guide field must be at all the times much greater than one eventual possible perturbation of the guide field must be of order epsilon where. Okay. The derivative. Yes, the meaning is that the, when you have a strong guide field that basically your dynamic is happening in the perpendicular plane. So it's fair that things happen, while the dynamic along Z that can be neglected. So that's why the derivative respect to said that are not important of order epsilon while the derivative in the plane of order one. Okay. Sorry. The derivative respect to Z. Yes. Yes. I'm saying it's of order epsilon, which means that you can treat some three dimensional problem, taking care that you are posteriori verify that what you have done is in the correct ordering for these equations. Absolutely, but you can also neglect and study two dimensional problem, which is something which is usually done and I will show you many studies into the approximation also. Okay. And then the assumption of the strong guide field. It does also to assume that the thermal and kinetic energy are much smaller than the magnetic energy. So the here is the, the beta parameter which is famous, at least for the tokamak physics, the beta parameter is most, most much important. We assume this ordering that the, the, the pressure, the thermal pressure divided by the magnetic pressure is much less than one. Okay, so it is of order epsilon. The dynamics parallel to the magnetic field of course on a much shorter time scale than the dynamics in the perpendicular plane, which is a reasonable issue if you think for instance to have the electrons so they will move very rapidly along the magnetic field line. You can assume that a dynamic in that direction is much is a is a faster. We saw we can we can expect that we can approximate the force balance to be maintained in the parallel direction on the time scale of the perpendicular dynamics. So we assume that the VZ is constants and without losing of any generality we can assume now that the VZ is equal to zero for our purpose. Okay. Then in the reduced model the field are assumed to be composed by an equilibrium part plus a perturbation which is of order epsilon again. Finally, a closure and so we in the reduced model, usually the closure is the incompressible fluid closure approximation. And then we for the magnetic field, since of course there is a divergence precondition together with the fact that the VZ is constant, we can write our magnetic field in this way, where we introduce here what is called the magnetic flux function. Okay, so BX and BY are related to the gradient of this function. Yes. Yeah. Is the plasma is the ions. The ions. The plasma means the fluid plasma. Okay, so the, the, like, so the ions are have a busy zero. Okay. Okay. No, this is the fluid as derived in the fluid approximately in the in this context of the velocity of the plasma, which means of the ions in this case along Z. So the plasma moves in the plane and the electrons because of the current density is basically related to the electrons. Okay, and so the current density is in the Z if you are in it to the approximation the current density is all along directed along Z. The velocity of the electrons is not zero along Z. Okay, so this is the plasma as if you are in the single fluid approximation. Okay. Okay. Okay, so BX and BY are related to the, to the side function. Okay, so reduce it. So if we apply our reduction scheme to the two fluid equation derived before, and we use the unfair slow and the Faraday's equation, we can end up with adopting the reducing scheme I've shown you before and we're relating with some algebra which I can give you all the details of the algebra but I, I think I already made many calculations, you end up with the system of equation so the first equation for the magnetic flux function comes from the home slow. Or, in other words, we can call it the electron momentum equation. Okay. While the second equation comes from the motion equation for the ions. Okay. And so the vorticity is related to the string function. Oh, I'm sorry, I use a small fire and the big fire here. I'm sorry, but there's the same pie. Okay. And so you can approximate the velocity as a perpendicular velocity in the plane, and where, of course, since you have adopted the divergence of the velocity is equal to zero the incompressible fluid approximation, you can write these terms in the plane, the velocity field, where you are these terms are basically the across be the drift term. Okay, the across be drift part of the velocity. Then you have so the question for the vorticity and then here there are a few parameters so there is the electron skin depth, which is related to the electronic skin depth is equal to the speed velocity divide the plasma frequency. It depends on the electron mass. Okay. And then you have the resistivity, which depends from the collisions between the two species. Okay. And then I kept also the last term, which is the rest, which is the ion sound Larmore ways. Larmore scale length. And these terms come from the gradient of the pressure. Okay. So we have three, three parameters in this equation and two of them, the electron skin depth and the resistivity can cause the magnetic reconnection process I'm talking in the next slide. And while the last one does not cause the electron magnetic reconnection because it can be reabsorbed in a definition of the stream function. But these two terms are really important. Okay. So we have introduced that the Poisson brackets. And so this tells you that this equation comes from the manipulation through taking the divergence and the curl of the electron momentum equation and all the iron momentum equation. Okay, I think I've said everything here. So let's come to your question. So if we can neglect any term here. Okay, so let's say that we can put to zero all the second the right hand side. Okay. Yes, five minutes. Okay, so maybe this is a good moment to stop. Yes, I think it's a perfect time for stopping. So if there are some other question and I'm sorry you have to wait. I made the questions during the talk so I can give you the slide. You don't have to take the picture I will give you the slides if you like. And then after I collect the misprints. Okay. And make it a bit shorter maybe 20 minutes. Okay. No, no, he stopped the post. Yes, he posed and then he resumes now right. Okay. Okay, okay. Okay, so thank you so much. And here we come to your question. What about the magnetic connection so let's take the idea of home slow, which means that we are neglecting the right hand side of the generalized home slow. That means that we are not neglecting the electron mass and then neglecting also the resistivity which is actually also common approximation that can be made. Okay, so if we take the curl of these equation here. Okay, and we get an equation evolution equation for the magnetic field. So if we integrate this equation on a surface, which is moving with the fluid. And we get this equation, which expressed the total variation of the magnetic flux flux sorry through a surface, which is in motion with the fluid. And this total variation, it happens to be zero. So this is a very big constraint on the possible motion of the plasma, because means that the two fluid elements, which are connected at the given times, they will be connected, they stay connected at later times. Okay. So this is an example so let's consider we are by by dimensional and we have a magnetic field. Okay, which has a shear. So the magnetic field that you see change changes the direction. Okay, across these the X equal zero surface, which is called rational surface. You see there is this change of direction. So if you we calculate the current density of these. And so we can balance this field and say that it is of the order of B zero B. A constant time sex divided L where L is the variation scale of your magnetic field. Okay. So if we evaluate the current density of this magnetic field that it will be directed along Z. So in the out of plane direction. And it will be of order be not divided by L. Okay, because we are deriving respect to the coordinate. Okay. So that will be also a J cross B force. Okay. And this J cross B force will be of order be not times X divided by L square. So now let's imagine that you are pushing in some way together these magnetic field lines. Okay. And so you make your variation of scale to become smaller and smaller. So you have to be of order delta. And let's send, let's send this delta scale to zero. What happens. It happens that the current density will diverge. And of course also your J cross B terms which act in the direction opposite to the motion you are with with which you are pushing the field line towards each other. And that will go to infinity. So, in the in an ideal plasma, if you try to make the lines to come closer and closer, you will have a infinite strength, which tend to opposite to this motion. And that will go to the frozen in condition, I told you before, because you cannot make since the magnetic field lines are connected to the particle. And so, since a particle cannot element fluid element cannot come penetrate. Of course, you cannot change the topology of the magnetic field. So, but if we relax this, let's see if I'm able to start. Yes, if you relax this constraint. So let's suppose that there is some of the terms on the right hand side of the generalized home slow, which is different from zero. Okay, so now your evolution equation changes. Okay, and, and so the magnetic flux is not conserved any longer. This means that the magnetic field line can decouple from the plasma motion and can reconnect. Okay, as in this cartoon is shown in this movie. There is a change of the topology. So let's think to the same topology we had before a magnetic field that we change is the sign across X equal to zero. And if we allow this process to happen, what happened well there is the formation of the magnetic what is called the magnetic island. Okay. The line which separates the open field line from the closed the thin line that this is a periodic in the y direction is called separatrix. You have an X point here, and you have a no point here. Okay. Let's suppose that in the second in the right hand side of the generalized home we are keeping the resistivity. Okay, so this parameter is proportional to the inverse of the resistivity. Okay. This is an equation of this equation of evolution for the plasma. So it's interesting to note that when you are closer to a surface where the magnetic field is no. Or when you are in a station point where the velocity field is no longer these second term disappear. Okay. And if the second term disappear these terms become even if it's really small become very, very important for my mathematical point of view, you can say you say that this is a singular perturbation, because it enters into the equation. With a higher order derivative respect to the other derivatives in the question. Okay. So, which what does it means these that the process is local. So it occurs in very narrow and in very tiny regions. But the changes that this process causes are global, because if a magnetic islands form, even if at the beginning that it will be very small, a very small magnetic island then it can grow. And so it can change all the topology of the of the plasma. So where magnetic reconnection occurs. Well, it occurs in the on the sun. So this is an image taken from the NASA a movie taken from the NASA where the NASA site and the subtle trace satellite. And, and you see there is a big explosion. So there is a big emission of plasma because when the magnetic field lines get close together and they form the magnetic island. There is an emission of plasma and the, so the magnetic field energy is transferred to particle energy and to hit the pending from the system you have another word does it occur magnetic field. And so there is a connection when when the solar wind. Yes, pushing the lines. Exactly. Yeah, now we get a way by which this line can be pushed and the particles actually that energy is pushing out is making the particles. Exactly. There is a relaxation of the constraints. So you're in the in some way you may, you must think that the infinite current you have is not infinite any longer because you are dissipating because there is a terms which goes like in the generalized which goes for instance for the resistive example you're making. We, it goes like a G J. And so there is this relaxation of the infinite current. Okay, yes. Well, what I was saying, well, another example where magnetic reconnection occurs is when the solar wind, which is a window for charges particle which is bringing its own magnetic field and counters the magnetic field of the magnetosphere. Okay. And so you can have again here it's in this region, you can have a reconnection events and what happens to do the particles that are accelerated along the magnetic field lines. They enter the atmosphere closer to the pulse in the causes of the aurora, as you know that's beautiful because they enter the atmosphere, they hit the, the, the particle the atoms of the, of the atmosphere, and there is an emission in the visible light. So we get the aurora fusion. The magnetic reconnection is really important also infusion because that's a, you want to avoid the magnetic reconnection infusion, because here is an example of a sort of crash. Okay. So to crash it means that when you are hitting your plasma, you have an increase of the temperature, and then something get wrong. And so you have a sudden decrease of the template, then you have another ramp and then another, another clash. Here is a tomographic reconstruction of the temperature now, and you see that you have the, at the beginning, there is a, the hot plasma is inside, then something get, get wrong, which means that the reconnection events occurs. And so you have a decrease of temperature. Here is a reconstruction of the magnetic surface, and you see so you are when you hit your plasma and the magnetic surfaces are still in the circular shape. But then the reconnection events of course, and you have the formation of the magnetic island. So when you have the formation of the magnetic island in the core of the plasma you have an expulsion of the particle of the heat. And so you have a decrease in the temperature. And so that's an example. So just for to give you an idea of the time so the time the typical time of the crash, or the sort of crash in in Tokamaka is of order or 10, which is the sweet Parker model which is a stationary model. So let's go into the detail but just to tell you that if you take the resistive term to dominate in the right hand side of the home slow, you get a time for the reconnection events of order or three millisecond. Today, if you take the electron inertia term so the terms related to the electron mass to dominate in the right hand side of your on slow, you get a, a reconnection time of order of 300 microsecond which is closer to 100 microsecond which is experimental time for a sort of crash. And this is why 20 years ago so maybe more than 20 years ago, almost 30 years ago, collisionless reconnection started to become really considered also in Tokamak physics, because it can give explanations of the, of the times of experimental time for the sort of crash. Yes. Yes, because in the, if I go. Oh, why is not working. I'm sorry. Oh, is not going up. Okay. Let's see. Sorry. I should be able to go to this. Okay, so the terms which depends from the electron mass. Okay, or these terms I'm neglecting the other in this because they do not cause a reconnection in any case the other two terms. Okay, so let's see if I can go. And now I have to say, I'm sorry. I am stuck with the movie. Okay. Yes, I know. Okay, so here we are. Now the equation I have derived the before in the reduced the mhd and the reduced the reduced extended mhd can be at least solved from the linear point of view. Okay, you can make a linear theory and you can get scaling for the growth or your reconnection in stability is okay. But you have to do well you have to linearize the system of equation and of the mhd model you assume an equilibrium with no equilibrium flow. You assume an equilibrium for instance the magnetic field which is make in the, this is the perpendicular plane and the magnetic field in the perpendicular plane plane because we are assuming sorry, a strong B, BZ the field. When you look for solution of this type. Okay, so solution where K is the wave vector, which is, we are by dimensional here so it's just the model number times two pi divided by your box in the y direction, and gamma is the growth rate of your stability. So when you have this kind of problem when I, I, as I said before you have a small region where your process is occurring, you are dealing with what is called the boundary layer problem. So far away from this region, you can adopt the ideal mhd, but you have to consider the real physics in the small region across your boundary layer. And so there are standard matching asymptotic techniques for these boundary layer problems. And in this adopting this procedure so looking for solution of this kind, you cannot. You can derive an expression for the free energy of your system, which depends on the magnetic field, you have from the initial equilibrium magnetic field. Okay, so how much energy magnetic energy can be converted into a particle acceleration or heat or during the reconnection event. And so the free energy is expressed by this parameter, which is called the standard delta prime parameter. And his name comes from a full killing and Rosenblut 1963, and it is related to the logarithmic derivative of the solution of the external solution, the one you determine in the ideal mhd framework. Okay. And so there is, of course, since you are in a boundary layer theory, you have a gap in there at the cross across the rational surface and these delta prime parameter express the free energy you are you have in these in the problem. And then you can derive a dispersion relation. So here I just give you the, the standard, the definition for the two regime, there is a small delta prime regime in which you do not, you do not have so much energy to convert in the large delta prime regime, where where you're in basically in the small delta prime regime, your island will not grow too much. While in the large delta prime regime, your magnetic island can grow to two dimension size, which are of the order of the equilibrium magnetic field scale lens. And these are just the scaling you get the painting from the resistivity or the electronic inertia. So, in the small delta prime regime, electronic inertia, which goes like the growth rate that you see go like D to the third power is very relevant. While it becomes relevant in the large delta prime regimes. Okay, compared to the resistive scale lens. So here I'm now I go, I'm going to present some of the results we obtained with my colleagues in the in the last 20 years. Yes. Yes. Well, I have normalized everything. I'm sorry, I forgot to say so that's a number. It is normalized to the scale to the equilibrium scale lens. Okay, that's just a number. Okay, so here are the summarize the results of the last 20 years in collision less magnetic connection which was which was the focus of my activity for a while. And here I have considered the equation I, the extended the mhd equation in which I'm considering the contribution of the electron inertia, and I am neglecting the resistivity, and also the contribution of the pressure gradient to the thermal, which gives these sound the larmor radius k length. Okay, and this is the question of motion so this is the question for the magnetic flux function which comes from the generalized almost low this equation current density and that the main results. Well, first of all, we, we, we have, we have seen that non linearly, there is a quasi explosive behavior of the reconnection events. So, that was important to to for instance to to explain the sort of crash. I mean electric pressure can give an explanation of these a faster growing of the instability. And you see here we have the straight line is the linear growth rate, the linear part of the growth rate and then you see that you depart, depending from the parameter but you depart from the from a linear growth. So you have a more than exponential behavior. Okay. And then a second important process is related to the related to the phase mixing process because this equation can be written. So maybe can I use the chalk because I forgot to write on on the user the chalk just so these are two equations for the magnetic flux and for the current density can be written in a very compact way in this way. Okay, equal to zero. And let me write here that when you have a question for the magnetic flux function which is equal to this. Okay. So these expressed the conservation of the frozen in condition for a magnetic field, which is made like this. Okay. Okay. And, and so you see that in the collisionless case, you can write for these two fields, which are g plus g minus is equal to psi. Plus the current density plus or minus. Over the EU. No, I square you I don't, I don't remember exactly the definition but that is like something like this and three plus three minus are equal to plus minus. Over the five. Okay, so what I want to say is that the in a collisionless limit that the frozen in condition is not removed is just transferred to other fields. Okay, so this g plus and g minus fields are are still invariance. So they are topology cannot change. And this process causes a phase mixing, because the topology of this field which is shown here g plus and g minus. So they can cannot change. They, they are rotating. Okay, so they are rotating generating a smaller smaller case until they reach a course graining process like a face in like in face mixing. And this, when you go back from the g fields to psi you average on this so the magnetic flux can reconnect but still your ideal constraints in some sense are retained by these field g plus and g minus. Okay. Another important things that happens on I want to do to notice also that there is a deformation here is the current density and here is the vorticity of intense current density and vorticity sheets. Okay, very intense. And these current density and vorticity sheets are related to secondary instabilities because they are so narrow that they are unstable to magnetic or fluid instabilities. Okay, so here are an example of the plasmoid instability. The magnetic when you have an intense current sheet that form during the nonlinear development of a reconnection events. And these current sheets are characterized by a small thickness. And so they can become unstable to the formation of plasmoid if they are also really elongated so an elongated current sheet that very, very narrow can become unstable to the plasmoid instability. Here is an example of the formation of a plasmoid for a case of resistive reconnection. Okay. Recently, and actually this is a paper of a PhD students who is working with us with we just published, we analyzed these instability also in the context of the collision less regime. Again, we have small current sheets, but whose thickness can be so narrow that they become unstable to these plasmoid instability. And when the plasmoid are generating in a cascade, they can make their connection process to become really, really fast. Okay, so the plasmoid instability is some ways is a fashionable our topic in these days, but because it can explain the faster connection process, because the more you are connected and so you are forming a small and smaller magnetic islands and these increase your reconnection rate. Here is an example of the secondary instability which is not. Yes. Yes. Well, the, I don't know if this will start anyway. Just let me see. Okay. Yes, I can what elaborate more on plasmoid. Yes. What, what. Yes. Okay, so the plasmoid is a magnetic instability. Okay. Now, suppose in this case, for instance, you start with a single magnetic islands, which can be seen here you see the steal the primary magnetic Island. Okay. There was a current sheet that very, very elongated and very, very narrow. So if the ratio of this current sheet is of a certain order, which can be 200 according to some theory. I don't want to give numbers because they can depend on many parameters, but it must be a very narrow and very elongated. When this becomes unstable to secondary order, order magnetic instability, which means that if you have only one magnetic Island, and then you have the KY, you in the linear theory, just to. Okay, you, you express these is. Okay, when you look for solution linear solution, this is your magnetic, your KY vector. Okay. So when you have this elongated current sheet, this elongated current sheet that can become unstable to higher order. So when you have one magnetic Island, we have just an equal one mode. Okay, but when these become unstable, you can have M equal to on and so on. Okay, and it is still a magnetic instability. Okay, so you have the formation of more islands like in this picture. Okay, in the sorry to use. So you have a first the formation of a small island, and then the formation of smaller smaller island. Okay. So in the plasmoid theory, usually what do you do you treat. It must be specified that this in the plasmoid theory, the plasmoid theory. The plasmoid course in the non linear phase of the reconnection events, but from the analytical and theoretical point of view, they can be treated linearly, because they are treated like a forced the problem, because you have these elongated current sheet, and the plasma flow around it. So there is a plasma flow which is pushing and there is narrowing this current sheet more and more. And so that's why I'm talking about higher order magnetic islands like I can treat them a linearly, because the theory can be can be made linearly to explain these plasmoid formation. Okay, and the the talk of tomorrow will be on plasmoid and Kelvin MOTC stability as well so we will come again to this topic if you if you like it and we can discuss also later. I have to escape the next slide because otherwise I cannot go on with the presentation so let me. Let me just make this and the. I escape that slide and I go to this that maybe works. Okay. Maybe, or maybe not. I'm sorry. It was supposed to be a movie, which is not working. Okay, it was an interesting movie but I can show you this way. Let me. Okay. Let's see if I can show you. Okay. Okay, this way it works. Right. Okay, so this is an example of the different instability. This is an instability of the fluid type. So we have a magnetic island. Okay, very big, so you don't see all the magnetic island in the in the in the picture, and you have an instability which is growing so this is the reconnecting instability this black blue dot just expresses the. The flux, the magnetic flux variation starting from the beginning to the end of the simulation. And it's interesting because you see there are the formation of vortex on the border of the magnetic island. So when it flew the instability. This is the vorticity. Okay, this is the vorticity, and it, what happens is that you can imagine that the, the, there are transport properties related to these phenomena. So what is in this is an asymmetric magnetic Islander. And usually when you have a symmetric magnetic Islander that was I wanted, I wanted to show before. How do I close this one. I don't know. Okay, this way, and the other movie which is. Okay, which works like this. So here you see the you have a symmetric magnetic Islander if I can show it again. No. Okay, no. Okay, and you see, you have a deformation of the tool. The vortex sheet, which are coming together. They collide and then they spread and they generate turbulence all around in inside the magnetic Island. So this is a fluid instability which is occurring on the top of the reconnection in stability is. In the third case I showed you before the turbulence is developing along the magnetic Island. And so, in this case it was developing on the on the border of the magnetic Island. So this opens transport properties different because you can go from inside the island to outside the island and this is something really interesting for magnetosphere problem. I will show you later. No electron inertia here. Okay, electronic inertia. Okay, so now I can go to the next slide. I hope. No. Sorry. How do I do this. I am sorry for this, because the movie are. How do I do. It's stuck in there. I don't know what to do. Oh, maybe I can. No. Yes, I can, but it's not going to the next slide. Why may I ask maybe. Okay, okay, we are. Okay, sorry. So now I can again go to this presentation. Okay, so the movie are are done with the movies and everything should be easier. So, collision with the connection that we we are the. So we have a talk so far about two dimensional model. What's happen in three dimensional model. Well, there are two important. Well, there is important issue that you know, if this is the format for your magnetic field. And this is equivalent to an amytonian system. Okay. And so you can write at the fixed dynamical time, you can write the field line equation where the third dimension is playing the role of the field line time. Okay, so it's clear that if you are bidimensional. Okay, this is an, this is an integral in integrable system, and you can solve and you have a magnetic Island, very clean structures. But when you, you cannot neglect the dependent from said, you have a non integrable system. And so you develop chaos. This is an example. And these were really a pioneer work of my colleague, Dario Borgono in 2005. And here we studied the, the problem with two magnetic islands and we were looking at what the goal was happening. So, as far as the two islands can be considered are so small that they be considered independent the one from the other. You, you see, they are growing. Okay, here, they are growing independently one from the other. So then, when they start to to grow to match and they start to overlap according to the Chiric of a criteria. The first magnetic surface which is going to be destroyed in the separatrix and you have the formation of chaos around the separatrix. Then if you go on, go on with time. These are one kind of plot. Okay. Exactly. Exactly. And so you have chaos is spreading all over your domain. It's interesting that the one kind of plots are important because they can provide you a very detailed information on the structure at a fixed dynamical time, of course, providing you integrate a very long time Z and providing you have many initial conditions, but they do not give any information on the transport properties of your system. And so, what we have done in the past is that we borrowed techniques from fluid dynamics. And so these techniques are very used in the oceanography when they studied the distribution of the plankton. And they, they observe that the plankton does follow particular streams in the in the oceans. Because in the chaos, there are hidden structures. And so we, we look at the for these hidden hidden structures. So here is the system where we are taking into account the dependence of Z. Okay, our system of the question. They are always the same equation where we are adding one set of time ingredients. So we are here we added the, the, the, the dependence from the four dimensions. And here is an example of a current density distribution inside the chaotic C, and you see that there are very intense current sheet, either in the chaotic C, you have. And here we studied how particles with a part with a certain distribution are distributed along in the chaotic C. And we noticed that that there are the green particle which stays exactly on the current density. Okay, while the blue particle are not involved, they stay in the chaotic C. So, this is suggested also that we have to to understand the what's what's going on in this magnetic in this chaotic being chaotic C. So, from the start of the same simulation of before we extracted a certain dynamical time in which there was the transition between the local chaos and the global chaos. And then we went to study the coherent structure with inside the chaos. So these are Lagrangian coherent structure that you see they enclosed particular area. So then we put initial conditions. Okay, in this, along in these particular regions, and we demonstrated that this condition remain bounded in this region exactly like the plankton is related to particularly streams in the stream flow in the oceans. So, this was a very interesting work and if you are interested we can discuss about this in the of today, whenever you like. Well, exactly in some way exactly for in the passage cover. Yes. Okay, then. Now I talk about theory I showed you how many interesting thing can be studied. And then now I want to show you where we can apply it to this stuff. Okay, all we have done about the reconnection. But, well, in, you know, as I told you before when a reconnection events occur in Tokamak and you have also small magnetic islands. They can be very detrimental for the Tokama performance, because they can grow too much they can bring you to a disruption into the loss of confinement. So you want to avoid this. And so there are many studies of how to control them, the small magnetic islands that form in the during the Tokama cooperation. And here we you see the same set of the question is just written in the different format that this is the Poisson brackets of the fee with you. Okay, where we have added a control current. Okay, this is called the electron cyclotron current drive. So what means that we inject some current in the whole point of the magnetic island in such a way to restore the ideal situation in which no highlands form. Okay, so here we have the evolution of the process. This is the magnetic island the area of the magnetic island, which it grows indefinitely. If we we do not have this term, but when we have the other the control current, we are able to bring the islands to shrink. Okay, we can avoid that we can avoid the formation of the growing of the island, and we can shrink it to zero. Then we have a phenomenon which is a flipping stability which is called flipping stability, because the island, the accent that the point are exchanged. And so the island start to grow again. But since you are injection, but it is smaller and you are injecting the tuning the parameter which define the current drive, you can shrink it again. It's interesting because in this phenomenon, you have again the formation of along the region which enclose the magnetic island of the Kelvin animals instability. Okay, so this was in some way the revolutionary work in in this field that because usually people when when control the islands relies on the rather for the question, which is a zero dimensional, it has just the amplitude of the magnetic island. So we made these two dimensional studies and we believe that we draw the attention on some issue that cannot be neglected that when you want to design the control strategy for the reconnection for the magnetic island control in Tokamak. Another job we are dealing with another problem is the problem of magnetic reconnection and the effect of the runaway current in Tokamak. So during the disruption in Tokamak, you may form many high speed electrons, so a runaway electrons. And so we investigated the problem of the magnetic reconnection instability of a post disruption weekly collisional plasma, where the current is completely carried by the runaway electrons. So the plasma current has disappeared. And now you have a runaway electron current. And these runaway electron current, is it able to drive the reconnection process? Yes it is. And we again here is our system of equation. This is a two dimensional solution of the problem. So we are neglecting this term. Sorry. We are neglecting this term. And in this and this. And we come to this system with a fluid equation for the runaway current. Okay. And we studied, first we studied the linear growth rate of this instability, which is pretty well, we are in pretty good agreement with the theory. We are analytical theory that can be carried out. And then we studied the saturation of the problem. And again, our saturation of the problem, the saturation magnetic islands, we can recover the linear growth, the linear saturated island that we derived from the from the theory with for the case, I guess that this is without runaway electrons. No, the blue line is with the runaway electron and the red line is without runaway electrons. Okay, so this is the theory. And again here, the blue is with the runaway current and the red is without the runaway. No. The blue is without and the red is with the runaway. And here I'm sorry we exchanged the color. I'm sorry. Okay, so this is an example where this problem can be important application of this problem. Another application we carried out. And this is related to the chaotic problem. I saw also that problem has an application. Increasing the plasma current in the RF mode experiment in Padua show two things. The transition from a chaotic configuration to order one, which is named quasi single electricity state and the formation the consequent formation of strong transport barrier and closing the high temperature. Okay, here we have the temperature barrier. Okay. And what we showed is that the existing correlation between the transport and the magnetic barrier. I will talk in you about before. In the chaotic see there appear to be some these Lagrangian coherent structure that they limit the motion of the particles, because in the first approximation you may think that the particle are moving along the magnetic field lines very fast. Okay. And here we have determined the Lagrangian coherent structure in the chaotic see derived from a simulation made from a special code in Padua, and then we put the bunch of initial condition in different location of respect to these coherent magnetic structures. And you see what happens is that the, the bunch of this bunch of initial condition, they stayed confined. And they stay confined and the colors are related to this transport magnetic transport temperature barrier. Okay. So this equation is related to the problem I showed you before when the solar wind impact the magneto. The magnetosphere. And this is the reconnection, the asymmetry reconnection occurring at the health day side. There is a lot of literature, which analyzed the data from the multi scale magnetosphere mission. Okay, which explained the particle emission in the out of page direction by the passage of the space craft inside the asymmetric magnetic island. There was, there is a peak simulation that have been carried out. I'm forgot to write the reference I'm sorry it's a paper of you, I can give the right reference of 2015 or 18 I do not remember exactly. Sorry, the date. But anyway, here is a peak simulation. And you see that you have an asymmetric magnetic Island, which I hope to resemble you, the one I was showing you before this is our snapshot to take in a different times of the movie I have shown you before. And, okay, this peak simulation, we're invoking the lower dive read the drifting stability to explain this, the stability, the stabilization of the shear flows. Here we have tried within our fluid model to explain the this mechanism, and we were able to explain the mechanism with the asymmetry of the magnetic Island, and with the presence of the ROS parameter, the ion sound Larmore radius. So this is just to tell you that fluid model can be really powerful, because they allows you to reduce the problem you are studying to a few essential ingredients. Of course, you must be careful in treating this and in interpreting this, but it is, they are really powerful instruments, because you can switch the on and off the parameter one at a time. And so you can really ascribe to a phenomenon what you are to a to a physical effect, what the phenomenon you are trying to understand a to interpret. Okay, so this is the last part I have still five minutes I will go really fast and it's necessary I will come back tomorrow. The Kelvin elements instability because I'm talking a lot about Kelvin animals shear flow instability so I hope I convinced you that the magnetic recognition processes always lead to the formation formation of regions with high velocity shear, confined to narrow sheet. And so I just want to give you a few ingredients for the starting with the natural gas, if you have a natural gas, and you have a vortex sheet. So you have you have just the equation for the vorticity, which I brought, because I take this from the base camp. This is the theory you can find on the base camp book. Here is the linearized equation for the vorticity you have a vortex sheet so a value of the velocity on one side and then other on the other side. And if you look for the linear solution on this problem, you find the dispersion relation. So, you see that since the view is different from the door from the one is different from the two sorry for the Italian. You always ended up with a vortex sheet is always unstable to perturbations because the one is different from the two. Okay. Now consider and this is what is called the Kelvin animals instability. Okay. Now consider the, the case of the weekly jet. Okay, the weekly jet is a jet that which is made in this way. So epic something which is made like this, if you look. Okay. Okay, the velocity pick and the in this case there are two class of solution and even more, which correspond to the kinky got the jet. Okay, so you are moving this way, you're, you're instability and then all the model we correspond to a pinching of the jet so you are pinching. And so it's a, it reminds of the reconnection, a reconnection instability. Why I, these are the growth rate of the, of the mode, this is the event mode, which is more stable and this is the odd mode. Okay. What happens when you put because I'm talking about the reconnection and Kelvin animals instability which are occurring at the same time. So I want to just to give you what to do to tell you what happens when you have a magnetized Kelvin animal so you have also now the magnetic field. If you have the magnetic field, now you have in green is the magnetic field and what happens that if this is the dispersion relation we ended up with. And so you see that if your alpine velocity is greater than this parameter, the Kelvin animals instability is completely suppressed. Okay, so this is important for what I'm, I will tell you tomorrow and now consider you have a big legit and a magnetic field, which is made in this way. Okay, the magnetic fields is the, okay, like this. Okay, the tangent. Okay. What happened in this case, so you have a magnetic field that we change the sign. Well, both the king can the pinch model so the even and the odd model are stabilized by the presence of the magnetic field because here is the curve without the magnetic field and you see that increasing the magnetic field that you stabilize the the model number are the wave vector decrease. So your unstable modes are less but the pinch model are more unstable so are less stabilized by the presence of the magnetic field. So here is the summary of what we have done all during these two lectures. So we see that we have seen that the flu fluid equation allows, allow to simplify the problem and that to switch a physical parameter one at a time. In plasma physics, the problem of magnetic connection can be firstly addressed in the fluid framework. Okay. I hope that I give you an overview of many problem that are can can be explained by magnetic connection and also the application that can be treated in these fluid content context, the application of to the study of magnetic connection. And I hope I convince you that the process of magnetic connection is intimately linked to fluid instability. They cannot be the couple. And as I said before, tomorrow, we will see how these two classes of instability coexisting compete when we are in a turbulent context. Okay, so I hope that's all for today. I hope I don't know what happened. Thank you very much. I want to be a thank you that I don't know what I did.