 So, okay, little complicated, but I think, okay. So what I'm, I thank Daniela for her very enlightening introduction, because she already explained many of the things that I will use, so I can go faster. I will talk about multi-fluid, a multi-fluid description of astrophysical on space plasmas. So I will start from the very beginning, I mean single MHD, but then I'll move to a multi-fluid and show mostly in the next lecture some applications. Oh, okay, I need to, but before I actually start, because as you know, we are in the Chambiashi lecture room, and many of you might wonder who Chambiashi was, and it happens as it happens, I know, because he's from Argentina, from my university. So, oh, I still don't know how to use this, okay. So Chambiashi, Juan José Chambiashi, his friends call him Bocha, Bocha means a big head, and you can see why. He graduated in Buenos Aires, Argentina, just like myself, but many years before. In 1948, and he got his PhD two years later, several years after that, he became the director of my physics department in 1966, and at the time, he had to leave the University of Buenos Aires because there was a military coup, as we have several of those in Argentina regretfully, and during the so-called Night of Long Sticks, the Long Sticks are, you can see one here, is the method that our military has decided to use to invite the students and many professors to leave the university at the time, and some of them also the country, like Chambiashi. So then he left Argentina and never came back, and he settled in Brazil, and from there he pioneered the development of physics in, not just in Brazil, but in the whole Latin America. He was a, some years later, he became the director of the Central Latin Americano de Física, which is CLAF for several years. It's like a more modest ICTP settled in Latin America. He also participated in the early stages of the ICTP itself in the early 60s, and he was part of the Scientific Council from 1987 to 1995, and he was a bright theoretical physicist. He made an important contribution to dimensional regularization in collaboration with Carlos Bolini, another Argentinian mathematician in this case. So I wanted to do that because I feel honored to be able to talk at this lecture room. Then as a brief introduction, I have this cartoon that shows that you have magnetic fields almost everywhere in astrophysics, and if you have magnetic fields, someone has to generate this magnetic field. So chances are that you have plasmas all over the universe. You have magnetic fields in the earth, and most of, not all of the planets, but most of them, you have heard from Lina yesterday, some of magnetic effects on planets, the ionosphere and magnetosphere. Yes. What do you, what? Control what? Oh, it's already full screen here. What should, what else should I do? I don't know what to do because it's full screen here. Yeah, just, yeah, the full screen, that's, that's what we learned this morning. I'll leave it like that. I mean, no, no big deal. You can get it. Okay. Thank you. So you have magnetic fields in, on earth, of course, and most of the planets, as I was saying, also on the sun and on almost every star, not just like the sun, but in younger stars as well. And in the interstellar medium, we can measure that indirectly because of several, in some cases because of what I'll go faster. You have neutron stars with very terse magnetic fields that we call pulsars. Oh, great. You made it. Thank you. And we should pay him for the job he's doing, I mean. And we also have magnetic fields around accretion disks and turn out to be very important in the focusing of these jets coming out of many accretion disks. And also in galaxies, we know we have magnetic fields, which are tend to be aligned with these arms of spiral, spiral galaxies. So what one theoretical method that one can use is, as we have seen in most lectures, probably last week also. We use MHT, single fluid MHT, which is okay if you use it properly and in the right regime, it doesn't solve any old problems, of course. There's no theory that solves all your problems. So it's a fluid like theoretical description for the motion of matter. Most the baryonic matter, I won't talk about that matter because we don't even know what it is, but the baryonic matter, which is the one we observe, is mostly hydrogen. And when you heat hydrogen beyond several thousand degrees, then it becomes ionized. And then you have your electron proton plasma, which is the most usual to find in all of these places in the universe. The large scale, the very large scale behavior of the dynamic of this plasma is described by a fluid like MHT, fluid like description like MHT. This fluid is made of electrically charged particles, in that regard it's different from water in a neutral, electrically neutral fluid. So these charged particles, of course, suffer external electric magnetic fields, but not only that, since you have charges and since they are moving, then you generate charge density and a current that are sources for electric and magnetic fields. So your Navier-Stokes equation for the dynamics of matter, couples with the Maxwell equations. We have seen much of that during these days. So you have a more system of nonlinear equations, which is most of the time too complicated because you have the nonlinearities and all the complexities on Navier-Stokes, but they are also coupled to the Maxwell equations, which are somewhat simpler, but not trivial. At small spatial scales and fast time scales, all plasmas will show their kinetic nature. You will get to see the lens scales, typical for instance, the electron ion, electron ion inertial lengths. You will get to see times like the ion cyclotron or electron cyclotron frequency. So when you get to those scales, you have gone too far and your fluid description will not be valid. It will be valid as long as your lens scales and time scales are longer than those. So that as a general remark. So these are the MHD equations. I can do this really fast because you have seen them many times already. You have your continuity equation for mass conservation. If you want to make it simple, you can assume a polytropic. You need these equations will not be enough. You need at least one more to close the system. The easy thing with Daniel also mentioned that. I mean, you can assume to be an adiabatic plasma or a compressible or maybe isothermal or maybe something else with a different gamma. And if you have a you want to have a more realistic description, then you need an equation for instance for temperature, putting the heating and the cooling times in the equation for the thermal evolution of temperature. You have your equation of motion, which includes well, nonlinearities, the pressure gradient. If you want to make it easier, if it's necessary, you need to remember that the pressure is in general a tensor quantity. And then you have a for a one fluid MHD, you have this magnetic force that gives you, I mean, if you know the magnetic field, then you know from this equation, you will get the dynamics of the flow. But also the magnetic field oftentimes is not external. The magnetic field is part of the unknowns of your problem. So you need an equation for the magnetic field as well. OK, I was saying you can have an external force like, for instance, a gravitational force or a Coriolis force if you are in a rotating reference frame. You have viscosity in this case, the non-isotropic part of the pressure tensor is here, usually associated to viscosity. And then for the magnetic field, you need an equation which is the so-called induction equation, which is this. So to solve for the magnetic field, you need the velocity field, which is an unknown in the other equation. So here you have equations to get, if you have a good code or if it's a geometrically sufficiently simple problem, then you can get your density, velocity, and magnetic field as a function of position and time. That's this equation is not, OK, sort of enter here from the window. What does this come from? Actually, this equation is nothing but the curl of the Ohm's law. Ohm's law, you can think of it as a phenomenological equation. For instance, the left-hand side here is the electric field measure in the reference frame of the fluid. If you are moving with the fluid at the speed, at the velocity u, then the left-hand side altogether is the electric field that you measured there. And so this is just saying that the electric field is proportional to the current density divided by the conductivity, electrical conductivity. Or else if you, instead of conductivity, you want to work with the resistivity, then this is the relationship. So you can, as I was saying, think of it as a phenomenological equation. There's no time here. So this is valid at any given time. But this is something tricky. I want to show you, in a way Daniela also showed you, where this actually comes from. It's not a new equation from nature. You can derive it, but I will show. So you take the curl of this and the curl of e. You use one of Maxwell's equation. You know that the curl of u is proportional to the time derivative of the magnetic field. So the curl of this will take you here. So this is the curl of the electric field. The curl of this cross product is here. And the curl of j, because j, in turn, is the curl of b. And the curl of the curl is the Laplacian. Plus another term, which is 0, because the divergence of b is, of course, 0. So that's sort of a summary of what MHT are. I'll skip this, because you already know that the magnetic force can be split into two terms, one of which you can regard as a magnetic tension. I mean, the magnetic fieldings don't like to be curved. If you curve it, there's a reaction force that tend to restore it to a straight line. And if you have a bunch of magnetic fieldings and try to squeeze it, then it reacts with the magnetic pressure. Which is this other term. And one more concept that you can get out of these MHT equations in the limit of zero resistivity. Plasmas are all very good conductors, because of course you have precharges, so it can conduct electricity very easily. That means that the conductivity is very large, or else the resistivity is very small. Sometimes you can make it zero. And if you do, then the induction equation reduces to this. And that there's a theorem that tells you what does this mean. This means the frozen, the so-called frozen in condition meaning that if you trace any close curve in a fluid, just pick up your preferred close curve. And of course this close curve, there are a number of bunch of magnetic fieldings crossing this close curve, these ones. Then let this close curve move along with the fluid, just let it move with the vector velocity field. And then all what this equation tells you is that all these bunch of magnetic fieldings will have to go along with the cross curve. Meaning that the magnetic fields get frozen into the fluid. But also the other way around, because if you somehow move the magnetic fieldings, then the fluid has to go along as well. So the matter and the magnetic field go together if this approximation is true. I mean, for no resistivity. If you allow for a small resistivity, then you have, for instance, reconnection. Whoa, I skipped. Why do I do this? So there are, of course, a huge number of applications. We've worked in just a few of them. For instance, instabilities, as we have just heard from Danila, shocks, wave propagation. In MHD, incompressible MHD, you have the open waves. And also you have fast and slow magnetosonic modes. They propagate and transfer energy. And there are a large number of instabilities that transfer energy. And also you can generate shocks. And then these shocks produce acceleration of particles, heating, as we have heard from Servidio yesterday, for instance. Then another important issue in astrophysical classmas is to answer, OK, who put these magnetic fields in all these places in the stars, planets, and what's the origin? I mean, nobody believes that you have magnets in the center of stars or planets. So there has to be some physical process that generates magnetic field. And where does the energy to drive this magnetic field increase come from? So one possible, one of the few possible mechanisms is dynamo mechanisms. That is mechanical energy of matter transferred into magnetic fields. So the magnetic field can grow out of the kinetic energy of movement of matter. Another important issue is MHD turbulence, which is a particularly complex. How we understand when we hear the word turbulence. You can have hydrodynamic turbulence. And of course, you can have magneto hydrodynamic turbulence. So both the velocity field plus the magnetic field become turbulent. We have heard some of that in the previous days. And of course, magnetic reconnection. I won't talk too much because Daniela here has took that. Particular subject in more detail. So these are some of the applications I will show you. I guess what I will show you is dynamo. How do you generate it? Actually in MHD, I mean, you don't generate magnetic fields. I should say they have magnetic fields because I'll go back a bit. Here in the induction equation, you see that the right hand side is linear in B. So let's assume that initially at time equals zero, B is exactly zero. Then the right hand side is zero and exactly. And then this tells you that the time derivative of B is zero. So if you don't have a magnetic field, you won't have in the future either. That's true because we are working under this approximation. So if you don't have a magnetic field, MHD won't generate it for you. But what you can do is to enhance it. Let's assume that you have a small magnetic field. That is what is called usually a seat. I mean, a fluctuation of magnetic field that someone put it there. MHD can enhance this magnetic field for you. So that's what I will show as an example. So what we do, we have done in the past. I'm an old guy, so I have done this during, I don't know, 30 years. To integrate the MHD equations using a spectral scheme in all three spatial directions, spectral means in the MHD equations, as you have seen, you have time derivatives and you also have a large number of spatial derivatives. These spatial derivatives, you can solve them with finite differences. That is, you assume a derivative is just a ratio of increments, very small. That's one way. This turns out to be dissipative. I mean, you don't want the dissipation, but the numerical code puts it there. And or else you can do in a spectrum of method. I mean, you make a Fourier transformation of all three spatial coordinates. And then derivatives, curls, divergence is an old kind of derivatives in Fourier space are algebraic operations. Right. I mean, if you divergence is i times k dot the quantity or the curl product, if you are dealing with with the curl, the Laplacian is minus k square. So it's easier in Fourier space to derivate in space. So we transform and do all our derivatives in Fourier space. And then we transfer back to make the time evolution step. That one might think takes a lot of time, a lot of computing effort, and it does. But since we have the so-called fast Fourier transform, then we can do that Fourier transform back and forth very efficiently. So it turns out to be a good way of integrating things like MHD numerically. The time integration is performed using a very simple technique, which is second order Runge-Kutta, very classical. Of course, you need to make sure that your time step is small enough. Small enough means that you satisfy the CFL condition. And with that, you are done. And also what one does, as most people do, is to run this on a cluster on several machines linked together. Yes. Yeah, it's kind of technical. When you transform in Fourier space and then you want to do the nonlinear. If the equations were linear, then in Fourier space, you can do it pretty easily. But of course, the MHD equations are nonlinear because you have both the induction plus the Navier-Stokes equation are on linear. So to do the nonlinearities, you transform back when you have to do, if you, for instance, have to do U dot grad U. And this is in Fourier space and this is in Fourier space. You will need to make a convolution product, which takes a lot of time and memory. So one doesn't do that. You transform this back into physical space and this one too. Okay, but what you transform back is this. You compute the gradient of this and this and make this product in physical space. But when you do that, there are some, what is called aliasing. Some of the modes, Fourier modes from this and this will adapt to produce a spurious effect. So you have to correct for that. So the two first, the aliasing is that if you, I'll put it, I'll draw just one direction, for instance, KX. So you have from minus K max over two, and because of course you are dealing with computers, so you have to use a finite number of grid points. So in Fourier space will give you also a finite number of career modes. So go from here to here and if you're integrating in a cubic box, then, well, you're going to be not only a finite number of modes, I mean it will be finding because it will be a discrete number of modes. So these modes with here will add up to some of the modes here and produce because this is periodic will give you a spurious effect. So what you do is to cancel out two thirds means that you, you, you keep two thirds of the modes, and the remaining one third you make it zero. It's just a technique. Otherwise, what you get is, it's spurious and people know that and so it's, yeah, the recipe is this, the justification will take longer but Okay, so we now you have to do to recover the of course we look. Ever since you discretize a problem you you already losing something because I mean of course the your fluid is continuous. But you have to do this otherwise what you get is wrong. So it's a spurious effect. Exactly. Yes, yes. Thank you. Yes, you. You have to do it. So this is, we all heard about turbulence, what enough, especially the ones that were here yesterday afternoon. Yesterday afternoon when I counted there were like 15 people, 10 of them were you women. So I wondered where were the man yesterday afternoon, most of them anyway some of them were here. So I can see more men now but some of them are still missing. What is the shame because I'm a man as well and I don't like it. So please, the ICTP on the lectures have done a big effort, Daniela took three trains yesterday to come here to give a lecture to you. So the ones, I'm not trying to patronize here to you because you are here but the ones that are not here should be here. Yeah, especially men, because all the women, I don't know how many women are in this, in this college. One, two, three, four, seven, eight, nine, 10, 11, 11 are here. So probably all of them. Okay. They are very courageous because yesterday they were here during Sergio was explaining this in greater detail and they stayed there. And I guess all they, all of them now know what turbulence is. So I'll keep it. The idea is that you, this is the paradigm of Conmogorov for hydro turbulence incompressible hydro turbulence that you put inject energy into this fluid by steering at large scales that is you inject this is the energy power spectrum energy per unit weight number as a function of weight number. You are injecting energy at large scales, for instance, if you have a fluid you steering at large scales that is small k. That's how the energy comes into the system. Then the non linear terms won't dissipate any energy in. So what non linearities do is to transfer energy from this scale, which is where it enters the system and cascades down to smaller scales, where the energy dissipated for instance by viscosity. This picture on the inertial range is a you to have strong turbulence you need a scale separation this injection scale the macroscopic scale. And this microscopic scales with energy dissipate had to be separated by orders of magnitude if possible. So in the in the middle, then you don't have any injection or dissipation the energy just cascades from large scale to smaller and smaller scales and so on. You can visualize this process as if at any K you identify this K with a given structure of a given diameter, it's diameter being one over K because length is inverse of weapon number. Then the non linearity of the problem splits eventually in a given time that one can estimate into two smaller vortices, then instead of one big vortex. Now you have two smaller vortices that your lens scales is one half that it used to be, but if the lens scale is reduced and the weight number is increased. So that's how you visualize in physical space this energy transfer in case space you're you're going moving energy from small K to large scale. That means large vortices to smaller vortices. And when you do your elementary mathematics and and say okay this in this initial range the the flow of energy is going to remain constant and it's going to be called epsilon, there's an epsilon somewhere. Then you get this famous power spectrum K to the minus five thirds the epsilon is the injection rate, how much energy per unit time you're injecting, which in a stationary regime is the same as how much energy per unit time you're dissipating. And also, it has to be how much energy per unit time you're transferring from K to say to K or larger K. So that's my cartoon for a turbulence hydro turbulence is for MHD is not too different only that you now have two fields which are turbulent, not just the velocity field but also the magnetic fields. And so what we did many years ago is to. Okay, before that, and there is a theory called mean field theory for Dynamos that was developed by Krause and Bradley are many years ago in the 80s, 1980. And that's you from theory that if you want to enhance magnetic a small amount of magnetic field in a turbulent or not to burn regime then your fluid has to be helical, not any fluid motion but the fluid motion has to be somewhat helical If you have both is vortex moving like this, then you have, you need to, and of course the vorticity goes then like this, then there has to be a component of the velocity field, aligned with the with the board, vorticity. It doesn't need to be completely aligned but you have, you need to have a preference on this fluid has to have helical motions preferably like this. If you have as many as of this as the other way around and the total helicity will be zero, and then you won't have a magnetic field enhancement. So that's, that's a theoretical prediction. And so what we did is to do this type of simulations. In green you have their references I can give you the details reference, if you're interested, and many years ago like 20 years ago or something. I don't know if you can read it, is it too light. Oh, I'm sorry. In the PowerPoint you will see it. Sorry, I don't know why is it to like, never mind, you'll get the information with Pablo mini who was at the time and a student doing his PhD with me and also my husband. So I have to show this because I'm here and if my husband was here then I have to say show these results. They are pretty all but I think nonetheless they are interested. Interesting. So what we did is this we have, we made simulations of a spectral code in this cluster that I just showed these simulations are 256. If you're doing a spectral it better be the number of grid points be a power of two that will make it more. It doesn't need to be but it's going to be more efficient. So it's 256 cube. Nowadays, like we can do easily 1024 cube with no problem to see whether we obtain this. If you want to go beyond that and do 2048 or even further, then you need a really big computer or a big cluster. It gets very steepie to to go with larger and larger number of grid points. But then this shows the physics so I'll just show it. So what we do is to run a hydro part of the of the code. That means, as I showed him, I don't need to run a different code to do hydro. Because if my initial magnetic field is zero, then that's what I do. I initiate the magnetic field with zero and run the only code. So what that will do is to show a hydrodynamic evolution. So what were so these in blue is the kinetic part of the energy so it's, it's you square over two. It's also an incompressible flow. What we do is to put an external force in the, in the, in the question of motion that external fall. What we want to do is to not just inject the energy but also electricity. How do we do that. Okay, our vector field F the force is such that it's curl. It's proportional to F that there are many ways to do that that there are the so called ABC fields that will will tell you how to help to make curl of a given vector field. From F or to minus F, whatever if you want to inject positive or negative electricity. So we do that. And in all wave numbers within a narrow sphere is fear in in Fourier space in Fourier space we have KX KY cassette. We can pick up a narrow hollow sphere, I'd say, K equal for for instance, modulus of K equal for so you have a sphere of unit with. So all three MOs that have modulus around for, then you put a forcing in that. You put it in, in all three directions and you get sort of an isotropic force, which also introduces not just energy but also electricity. So, say, this K equal for is this the forcing. So you get energy into the system like this. If the questions were linear. You guys are going to be raising the velocity, the kinetic energy in this only web number but because they are nonlinear, then no linearities will put will scatter energy from this K equal for into larger and smaller web numbers, and eventually you develop a cascade. It's really still zero. And when you reach your viscous dissipation scale you viscosity is one of the parameters that you put, and we put viscosity sufficiently large, so that the dissipation takes on before you run out of modes. Okay, I said 256. So this is already K equal for five and so on. So 256, it has to be somewhere here. And you're because it because it has to be sufficiently large so that the energy quenches leaves the system before you run out of modes. Otherwise you get against previous results. But one can do that. So you get your call model of power law. It's fine minus five thirds we can check that the call model of result is robust robust enough that if this were not five minus five thirds. The call model of is wrong, but our code is wrong so but it does believe me. And then we stopped the running of the code. We put some seed. I mean, a small amount of magnetic fluctuations that doesn't matter that you put some flak and small magnetic amounts of magnetic field at much larger scales at the micro scale. It's not so large that you hit the viscous scale but so you put a small negligible amount of magnetic energy say this much smaller than this. The energy is the integral under the core so the total kinetic energy is the integral under the blue curve and a magnetic energy, which is what we are adding now is the integral and this the under this record. So, and we run it before that I forgot to show if you want to measure the, the Helicity Helicity is you times that is kinetic Helicity, not magnetic Helicity, let's say kinetic Helicity is the integral of you times the magnetic Helicity, the magnetic Helicity is for a dot B. So per unit volume in any given read point you compute this. This doesn't have a definite sign it can be positive or negative but once you are at the end of your kinetic and hydro simulation. You check how much, which are the points with when you get an intense value of you dot Omega and it's great enough. Then you have these red spots and same with negative. So you have these green spots and you see an unbalanced in Helicity so they total Helicity I mean, you don't need to do the integral you will see that it's going to be positive. So, then we put the magnetic energy, and let's see what the magnetic energy does when you evolve the, the, the code in the second stage. So this is. So this is a magnetic energy in color is magnetic energy at the beginning. And this is the magnetic energy in the end. And what you will show is that the magnetic energy was at small scales and it was miles. And as you move in time the magnetic energy grows up in intensity but also it has larger scale the structures are larger. So, and what you will, wow. What you're looking at here this is the, the full line is the power spectrum of kinetic energy. And there is the five first somewhere here for reference. It's minus five thirds in this intermediate scales not when you're injecting the energy or and it's not when when you're dissipating so only in the intermediate scales and the magnetic energy, which are the curves that almost you don't see. And a very little time is like this it's the seed magnetic field. And as the energy, I don't know, can you, can we turn the lights off. I don't know how to do this. Any better or not. Oh, yeah. If I knew how to do that. Oh, no. Okay. Is it. All right. Okay. This is the seat. Power spectrum from magnetic energy and then it grows in times, and it grows in time, and so on until it reaches what it's called a repetition. Sometimes then you end up with a power spectrum of kinetic kinetic energy that are more very similar at all scales. So you have K to the minus five thirds, both for kinetic energy and the magnetic energy. That's what I intended to show here. Let's see if the next one. You can see the next one better. And this is the same. This is the same thing. The in green, this is the kinetic energy power spectrum in red is the magnetic energy. This is the seed for reference the minus five thirds in blue. This is the total a kinetic energy and magnetic energy in time. This is what's going to happen. And this is the total. When you integrate in time, but I can animate this and see how the magnetic, especially the red curve grows from almost nothing to a competition. That's what the simulation shows. So this is again, total energy as a function of time a kinetic energy is here almost it's not constant but because part of the energy that the magnetic field obtains comes from the, all of the energy of the magnetic field comes from here. The external driver puts energy into kinetic energy and then kinetic energy puts energy into the magnetic energy until the reach of the partition. And here you have a first stage of fast growing magnetic energy. This is what is called the kinematic stage of the dynamo. You can do a linear calculation because the magnetic field is very small. So if you have a very small magnetic field, you can make a linear calculation as I was saying and find that the solution is an exponentially growing magnetic energy as a function of time. We did that for different values of the Reynolds number. The Reynolds number here was like 100 300 very small. So if you want to have real turbulence you have to have at least a few thousand, but this is what we did at the time and, and this is. So it's a numerical confirmation of what the people doing mean field theory was predicting at the time. If I do the same numerical exercise but let's say I don't drive at K equal four, which is very large scale is one fourth of the size of the cube. So if I drive at smaller scales K equal 10. So my driving is at intermediate scales. What changes. So the, the peak of the kinetic energy power spectrum is here not not here now. They see this the same. So let's run this. What what is it different from before. Okay, power spectrum grows, but notice that at various at the largest scales of the size of the system, you now have larger magnetic energy than kinetic energy. And this is a log plot so that means that this is like two artists, two orders of magnitude in larger than this. So if you if you have a bad resolution instrument, and you look this at the very larger scales, you see magnetic energy and not too much motion of the fluid. So you might expect this magnetic field to be in a sort of an equilibrium. And if you look for equilibrium of magnetic fields you have the so called force free solutions. So if you look at the equations have it. If you look for equilibrium solutions that you have these ones, for instance, J parallel to be if J is parallel to be the magnetic force is equally zero. And if the velocity is negligible as it is, then you can check that the right hand side of your MSD equations are almost zero. It is an equilibrium and the force free solutions are observed for instance in the corona of the sun. There are a lot of structures that if you compute the force free solution out of what what are called magnetograms which is the information of the distribution of magnetic field on the surface of the sun. Then you can compute the force free extrapolation in the upper part. Yeah, so here you can then show that a turbulent system that, like the corona might show a very large case. Things that don't look as turbulence at all. I mean this, these structures are sitting there for days. There's no one we said this is turbulent, but the tuberous is the one that brought the magnetic field there. And if you look at smaller scales. Now if you have a better telescope and you look at much smaller scales, both for magnetic field and velocity field then you eventually you will get to see this I mean here the magnetic energy is comparable to the kinetic energy in the fluctuations that are sufficiently small scales. And then you realize that this is turbulent. I mean, if you manage to compute the kinetic and magnetic spectrum, it will be minus five thirds. So this is the micro physics of this. It's hard to tell but it might well be turbulent, only that that large scales, you only see the, I mean this tail of magnetic energy. Now I will move to. I don't know. I should finish in like 10 minutes or anybody's counting time. 10 minutes. Okay. Thank you. To multi fluid because Daniela already explained that I will go again but I don't have too much time anyway. If you want the description of multi fluid, then there's the book of Goldstone and rather for that's one of them. Let's call s to any species. Most of the time is going to be electrons and protons but you can have alpha particles you can have. I don't know, oxygen or carbon ions and depends on where we're in in the universe you are sitting, you might have another kind of charge particles, and you can also of course have neutral particles as well. So, the idea of a multi fluid description is to add more physics to just one fluid mxt which is the one we have seen more physics means that you are going into the regime of kinetic physics. You can get there, but then you, you can obtain some of the effects of kinetic physics and then of course you won't have a distribution function per species so you don't get all that information but I'll show you what what the physical If you assume that for any species you have the, you know, your set of fluid equations a continuity for the species s that is the number of s particles. It's, it's, it's constant. If you are, you are assuming that you don't have for instance electron positron collisions or you are not ionizing or recombining species because in that case the number of any given particle species might not be conserved. You don't have any of that so you have a mixture of particles and there are no processes that transmute one into another. The question of s particles is conserved. The equation of motion for s particles ms is the mass of this particular kind of particles a q is, is the, is there charge electrical charge. You have a velocity field for this particular species and pressure viscous effect. This is a Daniela show this is the exchange of linear momentum from a space from species s it to to the species s prime. It's a collision of exchange of linear momentum. And of course if you are studying your species s, you have to consider collisions with all other species so there's some summation of over the targets that you are considering. For example, of course this is a simplification of how you can take into account the collisional momentum exchange in collisions that okay this is the relative velocity of between the two species that are colliding. This is the mass of the particles that you're considering their number per unit volume and this is the, the new s s prime is the collision frequency between species s and s prime, which should be given somehow because, or, or else you can compute it for a more kinetic, a more basic description. So if you have all these species and you somehow know what their dynamics is going to be, then, of course this charge particles will generate a charge density at any given point and time, which is just the sum of all the, this is the individual charge that per unit volume number of particles per unit volume this is your charge density, which is almost zero because of the quasi neutrality condition which is very strong for non relativistic plasmacy if you're all your species move at velocities much smaller than the speed of light. Then the net charge density is almost zero. Basically the electrons which are much lighter move almost intense instantaneously to quench any charge and balance to satisfy this. Same for the electric current density. This is a current. This is the charge story for an individual particle times the number of them per unit volume times the speed. This is the charge flow for each species use some overall species and you get your J. This is the actual equation. We are neglecting here the displacement current, again, because if you're dealing with no relativistic problems then the displacement current can be also neglected. And so if you if you, you're multi fluid, let's stop at two fluids, electrons and protons, then you have this is dimensionless the same same equation but dimensionless I took away dimensions, I consider a problem of a given scale as zero, whatever scale you decide a density particle density and zero, and an alpha and speed you you have a given which magnetic field be zero so you can compute a constant of the velocity so you take away units velocities by dividing by this and the length by this and times, of course from L zero and VA you get something with time units so you take away all units and you end up with this equation for the dynamics of ions, your ion species, this is the electric magnetic force, the pressure and the, the, this exchange of momentum turned into because it's proportional to the difference of velocities it turned into the current density, because the current density for two species is the charge of electrons, number of electrons per unit volume velocity of electrons, plus the charge of protons, number of protons, velocity of protons, but because the charge is the same and the neutrality tells you that any it's equal to the charge, the net net charge density is this plus this right. So, the density of protons and electrons are the same. So you assume this here, and you end up with your current density will be e times n n is both protons and electrons, and your exchange of momentum was already proportional to this relative speed between the only two And so that exchange of momentum can be written in terms of J, which is already here. And of course, the now the momentum that the protons lose because of this interaction is exactly the one that electrons gain and around here you have a few dimensionless numbers. One of them is the ion inertial length dimensionless because I take away dimensions by using L zero, the beta of the plasma is a ratio between the thermal thermal pressure divided by the magnetic pressure and the resistivity, which is a dimensionless version of it. Actually, this is the inverse of the Reynolds number. If you add these two equations for the two species then you end up with this one fluid m h t notice that which is you can recognize the terms only for this. There's a curl of shape, which is weird. But if you do this rigorously that that's what you get epsilon E is the electron inertial length. It is the epsilon that I show before but multiply by the square root of the mass ratio. It's very small, because the electron masses very small. So if you are ready to neglect the mass of electrons then this goes to zero, and this time you can forget about it, and you get one fluid m h t for massless electrons. If you neglect the mass, this is the fluid flow, the bulk flow, which is a mixture of the two velocities of the two species but again see if the the electron masses negligible. Then you this goes away and your bulk flow will be the proton flow. So this is the answer to that. And so this is will be the last slide to show you the that the arms law is actually nothing but the equation of motion for electrons. So this is the equation of motion for electrons that I show before only only that now. The mass ratio. I'm just calling it new. It's a very small number. Now I don't neglect the mass I just assume that it's small but I retain it just in case. So this is the bulk, the fluid velocity in terms of the velocity of each species, this is the current density, again, in terms of the relative velocities. And then if I want to forget okay I have these two fluid plasma but I don't carry how many electrons how many protons I just to want to use a fluid description. What I do from these two equation is to obtain you E and UI in terms of these fluid quantities that the bulk flow and J. You can derive UI and you E as a function of you and J. And then you replace and forget about the velocity of protons or or electrons you only have a one one vector fluid velocity flow. And one current density running through the plasma. So you replace this in this equation of motion. And now you can make assumptions, let's assume that the electron mass is negligible so new goes to zero, then this acceleration of electrons goes to zero. There is no electronic energy anymore electrons instantaneously adapt to a force equilibrium, because if this is zero, but this equation tells you that all the forces of electrons instantaneously balance one another. And so you end up with this. This is the so called generalized on slow. You have the electron pressure as one of the terms for this extended on slow. So you can take cross be term here in the, in arms law is the whole, whole term. And then, but you can neglect this also, as long as the your dealing with scales, much larger than the ion inertial links. So your epsilon also goes to zero. So the length is very small so you don't want to take care of that. And so it's not if epsilon goes to zero then you end up with it with arms law. So it's not a phenomenology. It can be meant phenomenologically. Introduce if we want to do it that way by this it's, it's an equilibrium of forces of electrons. So I can I'll stop here. And talk about this and the next lecture tomorrow. Any questions. Sure. Thank you. Thank you. Otherwise, that's trying to speak well. Now, I have a question to ask all of you, do you want to delay your lunch by 15 minutes and I tell you the solution of that problem. Sure. So we'll do that.